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EU/MEeting 2009 1

Table of Contents

Welcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Information for Conference Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Program Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Scientific Program Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Invited Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Electronic design: a new field of investigation for large scale optimization

Marc Sevaux, Andre Rossi, Kenneth Sorensen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Applications of metaheuristics to optimization problems in sports

Celso C. Ribeiro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Annual planning of harvesting resources in the forest industry

Mikael Ronnqvist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27SLS Algorithms Engineering

Thomas Stutzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Applications of meta-heuristics to traffic engineering in IP networks

Bernard Fortz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Extended Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.1 – Genetic Algorithm and Memetic Algorithm with Vocabulary Building for the

SONET Ring Assignment ProblemAna Silva, Eberton Marinho, Wagner Oliveira, Dario Aloise . . . . . . . . . . . . . . . . . . . . . . . . 69

1.2 – A memetic algorithm for multi-ob jective integrated logistics network designMir Saman Pishvaee, Reza Zanjirani Farahani, Wout Dullaert . . . . . . . . . . . . . . . . . . . . . . 75

1.3 – Creativity, Soft Methods and MetaheuristicsJose Soeiro Ferreira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.1 – Computational Experience with GRASP for a Maximum Dispersion TerritoryDesign ProblemRoger Z. Rıos-Mercado, Elena Fernandez, Jorg Kalcsics, Stefan Nickel . . . . . . . . . . . . . . 89

2.2 – A Binary Particle Swarm Optimization Algorithm for the Maximum CoveringProblemBruno Prata, Jorge Pinho de Sousa, Teresa Galvao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

3.1 – On Portfolio Selection using MetaheuristicsAbubakar Yahaya, Mike Wright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.2 – Heuristic Search for the Stacking ProblemRui Jorge Rei, Joao Pedro Pedroso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

4.1 – Application of Pareto Local Search and Multi-Objective Ant Colony Algorithmsto the Optimization of Co-Rotating Twin Screw ExtrudersC. Teixeira, J.A. Covas, T. Stutzle, A. Gaspar-Cunha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2 – Multi-Objective Memetic Algorithm using Pattern Search Filter MethodsF. Mendes, V. Sousa, M.F.P. Costa, A. Gaspar-Cunha . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.3 – Metaheuristics for the Bi-Objective Orienteering ProblemMichael Schilde, Karl F. Doerner, Richard F. Hartl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Porto, Portugal, April 29-30, 2009

2 EU/MEeting 2009

5.1 – Iterated Density Estimation Evolutionary Algorithm with 2-opt local searchfor the vehicle routing problem with private fleet and common carrierJalel Euchi, Habib Chabchoub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.2 – Solving a bus driver scheduling problem with randomized multistart heuristicsRenato De Leone, Paola Festa, Emilia Marchitto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.1 – An hybrid approach to the Rectangle Packing Area Minimization ProblemMarisa J. Oliveira, Eduarda Pinto Ferreira, A. Miguel Gomes . . . . . . . . . . . . . . . . . . . . . 147

6.2 – A Combined Local Search Approach for the Two-dimensional Bin PackingProblemT. M. Chan, Filipe Alvelos, Elsa Silva, J. M. Valerio de Carvalho . . . . . . . . . . . . . . . . . 153

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Porto Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164


EU/MEeting 2009 3

Welcome

Dear Friends,

Welcome to the Polytechnic School of Engineering of Porto (ISEP – Instituto Superior deEngenharia do Porto) and to the EU/MEeting 2009, the annual meeting of EU/ME – TheEUropean chapter on MEtaheuristics. On behalf of the Organizing Committee of EUMEeting2009 I wish you a pleasant stay in Porto.

This is the 10th edition of a set of successful workshops on Metaheuristics. The meetingkeeps the same high quality standards of previous editions, and shows the diversity and sci-entific potential of the area of Metaheuristics. We wish that it will allow all participants toexchange interesting scientific ideas and experiences.

The scientific program that has been put together guarantees that these will be rather fruit-ful days: fourteen standard communications were accepted for presentation and five prominentresearchers were invited. The meeting will close with a prospective discussion session on thetheme of the workshop: “Debating the future: new areas of application and innovative ap-proaches”.

With more than 70 participants, from over 10 different countries, once again the goalsof disseminating this field of research, of bringing new people to this scientific community,of being a forum of discussion and intellectual stimulation for all of us interested in theseproblems, have been fully achieved.

Our wish is that you may leave Porto thinking already on your participation in theEU/MEeting 2010.

All the best.

Ana VianaISEP / INESC Porto

Organizing Committee Chair


4 EU/MEeting 2009

Organizing Committee:

Ana Viana, (chair) Polytechnic School of Engineering of Porto / INESC-Porto

A. Miguel Gomes, Faculty of Engineering of the University of Porto / INESC-Porto

Celso C. Ribeiro, Fluminense Federal University, Brazil

Joao Pedro Pedroso, Faculty of Science of the University of Porto / INESC-Porto

Jorge Pinho de Sousa, Faculty of Engineering of the University of Porto / INESC-Porto

Maria Teresa Costa, Polytechnic School of Engineering of Porto

Organized by:

EU/ME — EUropean chapter on MEtaheuristics

ISEP — Polytechnic School of Engineering of Porto

INESC Porto — Institute for Systems and Computer Engineering of Porto

FCUP — Faculty of Sciences of the University of Porto

Sponsored by:

EURO — The Association of European Operational Research Societies

IFORS — International Federation of Operational Research Societies

UP — University of Porto

ISEP — Polytechnic School of Engineering of Porto

FCT — Fundacao para a Ciencia e a Tecnologia


EU/MEeting 2009 5

Information for Conference Participants

The City of PortoPorto, also Oporto in English, is Portugal’s second city and capital of the North region. Thecity is located in the Atlantic coast in the estuary of the Douro river in northern Portugal.

Historic references to the city go back to the 4th century and to Roman times, althoughCeltic and Proto-Celtic remnants of ancient Citadels were found in the heart of where Portonow lies. The major touristic attraction is the historic city centre which is classified by UN-ESCO as a World Heritage site. The major architectural highlights include the Oporto Cathe-dral (the oldest surviving structure), the gothic Igreja de Sao Francisco (Church of SaintFrancis) with its elaborate gilt work interior decoration in the baroque style, the Igreja e Torredos Clerigos (Church and Tower of Clerics). The neoclassicism and romanticism of the 19thand 20th centuries also added interesting monuments to the landscape of the city, like the StockExchange Palace (Palacio da Bolsa), with its magnificent Arab Room, and the tile-adornedSao Bento Train Station. Another touristic highlight is Serralves, a contemporary art museum,with its surrounding gardens and Art Deco Villa. No visit to Porto would be complete withouta stroll to Gaia (on the south bank of Douro), a visit to the Port Wine Cellars and a taste ofthe world famous Port Wine.

If you have the opportunity to extend your stay in Porto you can explore the Minho region,just north of Porto. There you can visit Guimaraes, with its city centre also classified as WorldHeritage Site, and Braga. Alternatively, you can take a river cruise upstream the Douro riverand visit the home of the Port Wine: the Alto Douro wine region, with its typical terracedvineyards along the Douro valley.

During your stay in Porto you will also be able to try the ample traditional Portuguesecuisine based on the Mediterranean diet, rich on vegetables, fresh fish and shellfish. Theking of portuguese dishes’ is the codfish (bacalhau), cooked in more than a thousand differentmanners. Traditional Portuguese cooking is a five star pleasure! Typical dishes from Porto arethe Tripas a Moda do Porto (Tripes Porto style) and Francesinha (literally Frenchy), a kindof sandwich with several meats covered with cheese and a special sauce made with beer andother ingredients.

Useful Information

• ClimateAverage temperatures in April: High 17oC (63oF) and Low 8oC (46oF).

• Shopping HoursShops are generally open from 10.00 to 13.00 and from 14.30 to 19.00 hours, Mondaysthrough Saturdays. Department stores and malls usually open from 10.00 to 23.00 anddo not close for lunch.

• BanksBanks open from Monday to Friday from 8.30 to 15.00. Almost every bank has a 24-hourscash machine.


6 EU/MEeting 2009

• Public TransportationAn extensive public transportation network covers the whole city of Porto, including fivemetro lines and public buses which travel to any part of the city. Tickets, called Andantecard, are valid for metro and buses and can be bought at metro stations and major busstops. You can buy a card with a single ticket (Z2: 0.90 e) or with 10 + 1 tickets (Z2: 9e). You need to buy the first Andante card (0.5 e), afterwards you can recharge it withadditional tickets. The lowest fare is Z2, which is valid to travel within the city limits.There also 1 day (5 e) and 5 day (11 e) travel passes, valid in the entire network. Themetro runs from 6:00 am to 01:00 am.Please note: each Andante card can only be used by one person per trip and must alwaysbe validated – whenever a journey is commenced, whenever the means of transportchanges and independently of ticket type.

• RestaurantsPortuguese eating habits are characterized by heavy/full and long meals. Lunch, oftenlasting over an hour is served between noon and 2 o’clock and dinner is generally servedlate, around or after 8 o’clock. Most restaurants open from 12:00 to 14:00 for lunch andfrom 21:00 to 23:00 for dinner. Some tourist and informal places have longer openinghours.

• Electricity SupplyElectricity is supplied at 220 V - 50 Hz AC with European norm plugs.

How to reach Porto

• by planeGetting to Porto is easy and convenient. There are direct flights from all major Euro-pean cities, and from all over the world via Lisbon. You will have no problem checkingwith your travel agent.The airport is about 15 km from the city centre. You can reach the city centre by taxi,by metro or by bus.A ride to the conference hotel and city centre is around 25.00 e. Volumes exceeding55x36x20cm needing to be carried in the luggage compartment are charged 1.50 e.There is also a metro line from the airport to the city centre: line E (violet). To reachthe conference location you need to transfer at the Trindade Station and catch the Dline (yellow), direction Hospital Sao Joao. The metro runs from 6:00 am to 01:00 am.You need to buy a Z4 ticket (1.45 e), see details above.

• by trainThere are comfortable trains from Lisbon, Alpha and Intercidades. The train station isEstacao de Campanha, where you can take the metro to the city centre. For informationabout timetables and prices visit the site of CP (http://www.cp.pt).

Conference LocationThe venue for the EU/MEeting 2009 is the Polytechnic School of Engineering of Porto (ISEP).It is located in the north of Porto and very close to the conference hotel and the metro line D(yellow) station IPO.


EU/MEeting 2009 7

Address:Instituto Superior de Engenharia do PortoRua Dr. Antonio Bernardino de Almeida, 4314200-072 PortoPortugalhttp://www.isep.ipp.pt

RegistrationThe registration desk will be held at room Secretariado (on the 1st Floor) on Wednesday 29thApril, from 9:00 a.m.

PresentationsPresentations should not take more than 30 minutes, including discussion. The conferencewill be on room Auditorio E (also on 1st floor), which is equipped with video projector andcomputers.

Internet AccessISEP is member of the EDUROAM initiative, the roaming infrastructure used by the inter-national research and education community that allows EDUROAM users to access a wirelessnetwork at a visited institution. If you are member of a EDUROAM institution just open yourlaptop and be online. There will be also a room (Sala de Reunioes) with ethernet sockets toconnect laptops (we suggest participants to bring their own ethernet cable).


8 EU/MEeting 2009

Lunches and Coffee-Breaks

• LunchesParticipants can have lunch at the conference location for e5.50 per day. Lunch includessoup, main dish, dessert and water. Other beverages have to be paid separately. Allparticipants willing to have lunch at the conference venue, should send an e-mail to

[email protected] that. Payment will have to be made on the 29th of April, during registration.Lunches will be on room Sala de Refeicoes.

• Coffee BreaksCoffee and biscuits will be served during scheduled coffee breaks.

Get-TogetherAn informal get-together will take place on Wednesday, at 20:00, April 29 (on a pay yourselfbasis) at a restaurant on the left bank of Douro river, facing the oldest area of Porto. Doplease let us know if you intend to join, so that we can make the necessary arrangements.

Restaurant:Republica da CervejaCais de Gaia4400-161 Vila Nova de GaiaTelefone: 22 374 74 00Fax: 22 374 74 09

Take the metro (line G, yellow) toEstacao de Sao Bento, walk down toRibeira on the river bank, cross theriver on bridge D. Luıs I (lower level)and walk down to the buildings onthe river bend. You will need around45 minutes to reach the Republica daCerveja restaurant.


EU/MEeting 2009 9

Accommodation

The Organizing Committee of EU/MEeting 2009 has pre-booked some rooms in Hotel IbisS. Joao, close to the conference location. Please remember that the number of rooms is limitedand that rooms are only guaranteed until March 30, 2009. When booking at this hotel doplease refer to ”ISEP - EU/MEeting2009”.

Hotel Ibis Porto Sao JoaoRua Dr Placido Costa4200-450 - PORTO - PORTUGALTel : (+351)22/5513100Fax : (+351)22/5513101http://www.ibishotel.com/gb/hotel-3227-ibis-porto-sao-joao/index.shtml

This hotel is at 2 minutes walking from the conference location. There is a direct metroline to the center of Porto, with a station 1 minute away from the hotel.

Useful Links

• EU/MEeting 2009 – http://www.dcc.fc.up.pt/eume2009/

• EU/ME — http://webhost.ua.ac.be/eume/

• ISEP – http://www.isep.ipp.pt/

• Hotel Ibis Sao Joao – http://www.ibishotel.com/gb/hotel-3227-ibis-porto-sao-joao/index.shtml

• Porto Turismo – http://www.portoturismo.pt/

• Metro do Porto – http://www.metrodoporto.pt/

• STCP– http://www.stcp.pt/

• Aeroporto F. Sa carneiro – http://www.ana.pt/portal/page/portal/ANA/AEROPORTO_PORTO/

• CP – http://www.cp.pt/

• Google Map — http://maps.google.com/maps/ms?hl=en&ie=UTF8&oe=UTF8&msa=0&msid=114152777408396939022.000467ea9c2431d413dca&ll=41.166379,-8.607359&spn=0.069524,0.114155&z=13


http://www.dcc.fc.up.pt/eume2009/

http://webhost.ua.ac.be/eume/

http://www.isep.ipp.pt/

http://www.ibishotel.com/gb/hotel-3227-ibis-porto-sao-joao/index.shtml

http://www.ibishotel.com/gb/hotel-3227-ibis-porto-sao-joao/index.shtml

http://www.portoturismo.pt/

http://www.metrodoporto.pt/

http://www.stcp.pt/

http://www.ana.pt/portal/page/portal/ANA/AEROPORTO_PORTO/

http://www.ana.pt/portal/page/portal/ANA/AEROPORTO_PORTO/

http://www.cp.pt/

http://maps.google.com/maps/ms?hl=en&ie=UTF8&oe=UTF8&msa=0&msid=114152777408396939022.000467ea9c2431d413dca&ll=41.166379,-8.607359&spn=0.069524,0.114155&z=13



EU/MEeting 2009 11

Program Overview


EU/MEeting 2009 13

Scientific Program Schedule

Wednesday 29th April9h30 – 9h45

Opening SessionChairperson: Ana Viana

Welcome

9h45 – 10h30

Invited Talk 1Chairperson: A. Miguel Gomes

Electronic design: a new field of investigation for large scale optimizationMarc Sevaux, Andre Rossi, Kenneth Sorensen

11h00 – 12h30

Session 1Chairperson: J. Soeiro Ferreira

1.1 – Genetic Algorithm and Memetic Algorithm with Vocabulary Building for the SONET RingAssignment ProblemAna Silva, Eberton Marinho, Wagner Oliveira, Dario Aloise

1.2 – A memetic algorithm for multi-ob jective integrated logistics network design

Mir Saman Pishvaee, Reza Zanjirani Farahani, Wout Dullaert

1.3 – Creativity, Soft Methods and Metaheuristics

Jose Soeiro Ferreira

14h00 – 15h00

Invited Talk 2Chairperson: M. Teresa Costa

Applications of metaheuristics to optimization problems in sportsCelso C. Ribeiro

15h00 – 16h00

Session 2Chairperson: Teresa Galvao

2.1 – Computational Experience with GRASP for a Maximum Dispersion Territory Design ProblemRoger Z. Rıos-Mercado, Elena Fernandez, Jorg Kalcsics, Stefan Nickel

2.2 – A Binary Particle Swarm Optimization Algorithm for the Maximum Covering Problem

Bruno Prata, Jorge Pinho de Sousa, Teresa Galvao


14 EU/MEeting 2009

16h30 – 17h30

Session 3Chairperson: Mike Wright

3.1 – On Portfolio Selection using Metaheuristics

Abubakar Yahaya, Mike Wright

3.2 – Heuristic Search for the Stacking Problem

Rui Jorge Rei, Joao Pedro Pedroso

17h30 – 18h30

Invited Talk 3Chairperson: Jorge Pinho de Sousa

Annual planning of harvesting resources in the forest industry

Mikael Ronnqvist

Thursday 30th April9h30 – 10h30

Invited Talk 4Chairperson: Joao Pedro Pedroso

SLS Algorithms Engineering

Thomas Stutzle

11h00 – 12h30

Session 4Chairperson: Richard Hartl

4.1 – Application of Pareto Local Search and Multi-Objective Ant Colony Algorithms to the Opti-mization of Co-Rotating Twin Screw Extruders

C. Teixeira, J.A. Covas, T. Stutzle, A. Gaspar-Cunha

4.2 – Multi-Objective Memetic Algorithm using Pattern Search Filter Methods

F. Mendes, V. Sousa, M.F.P. Costa, A. Gaspar-Cunha

4.3 – Metaheuristics for the Bi-Objective Orienteering Problem

Michael Schilde, Karl F. Doerner, Richard F. Hartl

14h00 – 15h00

Invited Talk 5Chairperson: Celso C. Ribeiro

Applications of meta-heuristics to traffic engineering in IP networks

Bernard Fortz


EU/MEeting 2009 15

15h00 – 16h00

Session 5Chairperson: Paola Festa

5.1 – Iterated Density Estimation Evolutionary Algorithm with 2-opt local search for the vehiclerouting problem with private fleet and common carrier

Jalel Euchi, Habib Chabchoub

5.2 – Solving a bus driver scheduling problem with randomized multistart heuristics

Renato De Leone, Paola Festa, Emilia Marchitto

16h30 – 17h30

Session 6Chairperson: Eduarda Pinto Ferreira

6.1 – An hybrid approach to the Rectangle Packing Area Minimization Problem

Marisa J. Oliveira, Eduarda Pinto Ferreira, A. Miguel Gomes

6.2 – A Combined Local Search Approach for the Two-dimensional Bin Packing Problem

T. M. Chan, Filipe Alvelos, Elsa Silva, J. M. Valerio de Carvalho

17h30 – 18h15

RoundtableChairperson: Jose F. Oliveira

Debating the future: new areas of application and innovative approaches

Bernard Frotz, Mikael Ronnqvist, Kenneth Sorensen, Thomas Stutzle

18h15 – 18h30

Closing SessionChairperson: Ana Viana

Closing notes


EU/MEeting 2009 17

Invited Talks


EUMEeting 2009 19

Electronic design:

a new field of investigation for large scale optimization

Marc Sevaux ∗ Andre Rossi ∗ Kenneth Sorensen †

∗ University of South Brittany – UEBLab-STICC, Centre de Recherche, F-56321 Lorient France

Email: {marc.sevaux,andre.rossi}@univ-ubs.fr

† University of AntwerpFaculty of Applied Economics, Antwerp, Belgium

Email: [email protected]

1 Introduction

Mobile phones, music players, personal computers, set-top boxes and countless other digital electronicsitems are part of our daily life. These nomad devices offer more and more functionality. For example,recent mobile phones allow to communicate, to take pictures, to play video, to listen to music, to watchTV, to browse the WEB and so on. Thus, integrated circuits (IC) achieving all these functions becomemore and more complex.

Figure 1 shows the evolution of architectures (blue), standards (yellow) and throughput (red) overthe last decade. Architectures, first based on System on Chip (SoC), went through Multi-ProcessorsSoC, Network on Chip (NoC), Crypto-Processors and now Adaptive MP-Soc. For example, TV Stan-dards at the end of the previous century were MPEG2 and MPEG4. Now the HD-TV is based on H264standard. Furthermore, these new standards require increasing throughput, up to 1 GBit/s and moretoday.

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Figure 1: Evolution of architectures, standards and throughput

Designers aim at developing all these products taking into account several axes:


EUMEeting 2009 20

• Time to market reduction

• Integrated circuits miniaturization

• Low power consumption

• Throughput increase

This leads to a large increase in the global complexity of the problems and causes IC design method-ologies to change. Of course, one can see that these objectives are conflicting and appropriate strategiesshould be used to deal with them smartly.

Concerning architectures complexity, FPGAs were a major evolution twenty years ago. A FPGAis a chip that can be configured as desired using libraries. The designer can decide which parts of thechip will be registers (memory banks), which part will be processors (or co-processors), and how theywill be connected together. This is a big change compared to an ASIC which is a dedicated circuitthat cannot be self-configured. Figure 2(a) shows a classical FPGA framework using a bus-hierarchyand IP-based designed. IP here refers to Intellectual Properties or bricks to be integrated to a globaldesign.

(a) FPGA (2003) (b) MPSoc (2005) (c) TeraScale (2007)

Figure 2: Chip complexity

A Multi-Processor System on Chip, as in Figure 2(b), is a more complex structure where someprocessors are dedicated to some tasks and connected together. Those MP-SoC are often designedfor a class of tasks like Digital Signal Processing. They include processors for audio and/or imageprocessing, 2D and 3D images, cryptography, antenna transmissions, wireless protocols, etc. Figure2(b) is the MP211 chipset from 2005.

Adaptative Multi-Processors System on Chip are available since in 2007. The main difference isthat some parts are dedicated to certain tasks (as in a MP-SoC) but the rest of the architecture isprogrammable (like in a FPGA) but can be reconfigured on-line. This means that if a part of theprocessor fails or is unable to reach the requirements (typically real-time constraints satisfaction), thetasks normally processed by this part of the MP-SoC can be transfered to another part of the processorby a partial or complete reconfiguration. Figure 2(c) depicts the last processor TeraScale from IntelCorporation.

2 Electronics community and optimization

For a long time, designers have been using a wide variety of optimization techniques such as IntegerLinear Programming, heuristics and metaheuristics. An observation of the use of these techniquesover the last two years has led to the following conclusions: even if the electronics community know


EUMEeting 2009 21

these methods, there is a great need for more formal techniques, more appropriate models and efficientsolution approaches tailored for specific problems in the electronics industry.

In the sequel, we will introduce several problems where the help from the metaheuristic communityis needed and for which collaborations are necessary. This talk will be the starting point for the creationof a network of long term collaborations between metaheuristic and electronics communities.

Several applications in electronic design require optimization, to name a few:

• Solving data allocation to memory banks (graph coloring)

• Reducing latency in data transfer in a NoC - Network on Chip (Shortest Path Problems withconstraints)

• Multiobjective optimization in High level Synthesis (multiobjective scheduling)

• Optimizing parameters in DSP filters (non-linear optimization)

• Reducing treatment under satellite norm data transfer (Hamiltonian path problems)

• Robustness and flexibility analysis for above problems

In the rest of the paper, we present some of these applications with their main characteristics andwhat has been done up to now, either by people from the electronics community or by us.

3 Potential applications

3.1 Network-based problems

Network-on-Chip (NoC) are electronic chips where several parts are considered as processors and anetwork between them feeds the processors with data and computations to run.

There are many common links with computer network based problems. For each NoC, synchro-nization, routing, security and reliability are also required. But at the same time, information that issent through the NoC consists most of the time of repetitive tasks and the quantity of information isseveral gigabytes over a very short timespan (less than a few milliseconds). Communications have tobe handled either in a centralized or decentralized manner as well as memory management.

One of the problem that interests electronic designers is latency reduction in data transfer in a NoC.This can be seen as a Shortest Path Problems with constraints in terms of operations research

Problem statement A Network on Chip (NoC) [2] is an internal structure of an electronic componentthat serves to transfer data and executes operations on distant processors. A graph represents the NoCwith specific input and output ports. Data has to be exchanged between these ports. Of course, theshortest path between source and sink nodes is always the best path unless some arcs are occupiedwith other data. In that case, classical shortest path algorithms cannot be used anymore and shouldbe replaced with different approaches for handling these constraints. The goal is to make an efficientimplementation of such an algorithm taking into account the specific structure of the graph and of theconstraints [1].

Main characteristics of the problem The number of data units to be transfered is very large(about 3/5 gigabytes of data). Fortunately the network itself is rather small (less than 100 nodes and≈ 200 edges).


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First steps of a general approach A first MILP model of this problem has been proposed, mainlyfor the sake of modeling it. The NoC has n nodes, it is modeled as a non-directed graph G = (V,E).There are o packets to be transported by this NoC, each packet k having an origin and a destination,modeled as follows: for all node i in V , oi,k = 1 if packet k has i as its origin, di,k = 1 if packet k has ias its destination. Edges are modeled with a matrix c defined by ci,j = 1 is and only if node i and nodej are connected (i.e. if (i, j) ∈ E). Time is discretized, and the problem is solved in a finite horizonH where H is assumed to be given. There are n× n× o×H + 1 decision variables defined as follows:xi,j,k,t = 1 if packet k moves from node i to node j at time t, and time T , that is the maximum arrivaltime over all the packets. The problem can be formulated as follows:

(MILP ) :

Minimize T∑

k∈[1,o]

xi,j,k,t ≤ 1 ∀i ∈ [1, n],∀j ∈ [1, n],∀t ∈ [1, H] (1)

xi,j,k,t ≤ ci,j ∀i ∈ [1, n],∀j ∈ [1, n],∀k ∈ [1, o],∀t ∈ [1, H] (2)∑

t∈[1,H]j∈[1,n]

xi,j,k,t = oi,k ∀i ∈ [1, n],∀k ∈ [1, o] (3)

∑

t∈[1,H]i∈[1,n]

xi,j,k,t = dj,k ∀j ∈ [1, n],∀k ∈ [1, o] (4)

xi,j,k,τ ≤ 1−∑

t<τh∈[1,n]

xi,h,k,t ∀i ∈ [1, n],∀j ∈ [1, n],∀k ∈ [1, o] (5)

∑

t∈[1,τ−1]l∈[1,n]

t× xl,i,k,t ≤∑

j∈[1,n]

t× xi,j,k,t ∀i ∈ [1, n],∀k ∈ [1, o],∀τ ∈ [2, H] (6)

∑

t∈[1,H]i∈[1,n]

t× xi,j,k,t ≤ T ∀k ∈ [1, o], j|dj,k = 1 (7)

T ≥ 0, xi,j,k,t ∈ {0, 1} ∀i ∈ [1, n],∀j ∈ [1, n],∀k ∈ [1, o],∀t ∈ [1, H]

Constraint (1) specifies that at any time, a packet can only move across one edge, a packet canonly move across existing edges (2), each packet starts at its origin, i.e. packet k must leaves node i(its origin) to another node (say j) at some time instant t (3), each packet ends at its destination, i.e.packet k must reach node j (its destination) from another node (say i) at some time instant t (4), apacket k cannot move across edge (i, j) at time τ if it left node i earlier (at any time t such that t < τ)(5), a packet k cannot move across edge (i, j) at time τ if it did not reach node i at some time t in[1, τ − 1] (6) and each packet k reaches its destination before time T (7). The goal is to minimize thetotal time T .

Of course, this approach cannot be practically used since it involves too many variables as it growswith o, the number of packets. So it allows space for applications of several heuristics and metaheuristics.

3.2 Sizing and assignment problem

How many memory banks are needed, and once defined, in which of these should data be placed tominimize the number of remaining conflicts while reading the data? That is the type of question adesigner can ask when designing an architecture. This problem can be solved using variants of thegraph coloring problem.

Problem statement In electronic design, several tasks have to be performed in order to obtain anarchitecture that meets the needs of the final consumer. That goal is reached after a series of synthesisand modelling steps. One of these steps concerns the data memory allocation. The designer has to


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decide in which memory banks data will be stored in order to minimize access conflicts (two pieces ofdata cannot be accessed at the same time if they are in the same memory bank and this implies a delayin data transmission). In an early work, we have modelled the problem as a variant of the graph coloringproblem where the nodes of the conflict graph have to be colored in such a way that two adjacent nodeshave different colors (colors represent the memory banks). The goal is to extend this model to takeinto account specific constraints of the electronic application and find the best solution method. Twokinds of problems can be stated. Either, the number of memory banks has to be determined (withoutadding a delay in data processing) and the problem to be solved is the classical graph coloring problem,or the number of memory banks is fixed and we have to find a coloring that minimize the weights ofthe conflicting edges [4].

Main characteristics of the problem Instances for this kind of problems come mainly from elec-tronic applications (e.g. MPEG or HD-TV encoder/decoder). This results in huge amount of sourcecode written in C or C++. The size of the conflict graph can count up to several thousands of nodesand have a high density. This kind of instances is know to be easy for the classical problem, but forthe k-weighted graph coloring problem, it is not the case.

First steps of a general approach Up to now, we have again modeled the problem as a MILP andused it for resolution:

Minimizeo∑

k=1

dk.yk

m∑

j=1

x(i, j) = 1 (∀i ∈ [1, n])

x(k1, j) + x(k2, j) ≤ 1 + yk (∀j ∈ [1,m])(∀k ∈ [1, o])x(i, j) ∈ {0, 1} (∀i ∈ [1, n])(∀j ∈ [1,m])yk ∈ {0, 1} (∀k ∈ [1, o])

The objective is to minimize the weights of the conflicting edges, the first constraint forces eachnode to have a single color and the second constraint fixes the value of yk for a conflicting edge.

For this problem, a PhD program has been started recently and the student has already producedinteresting theoretical results (some complexity results, a new upper bound on the number of colors andsome new properties on conflicting edges). Now the work is extended in the metaheuristics direction byderiving classical approaches of the graph coloring problem for the k-weighted graph coloring problem.

A new extension is also studied: a set of n data structures {a1, a2, ..., an} has to be allocated to aRAM page bj (j is in [1,m] as there are m pages available). The data structure ai has a size si, andthe page bj has a capacity cj . Moreover, there are o pairs of data structures that should be placed inthe same page. A pair pk is defined by pk = {ak1, ak1, dk}, where ak1 and ak2 are two data structures,and dk is the pair cost. Each pair has also a status: it is OFF if the two data structures share the samepage, it is ON otherwise. If a pair is ON, the cost dk should be paid. If it is OFF, then no cost shouldbe paid. The purpose of this problem is to allocate one page to each structure, while minimizing thetotal cost.

3.3 Multiobjective optimization

In High level Synthesis, we aim at solving at the same time, resource allocation, data assignment andscheduling sub problems for different criteria [5]. This leads to a multiobjective resource schedulingproblem.


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Problem statement High Level Synthesis (HLS) is a modelling phase of electronic design whichconsists in allocating resources, scheduling operations and binding data into memory. During thescheduling phase, it is possible to find starting time of operations in order to minimize latency (totalduration of the execution). It appears that a very good schedule for this objective can be rather poorin power consumption, in silicium (silicon) surface or in other criteria. Using a method based on anestimator for the solution cost, we are able to find different solutions and measure their value forthe different objectives. We aim to implement a multiobjective optimization method to find the besttrade-off between these solutions and let the designer choosing the best solution [3].

Main characteristics of the problem The precedence graph of the operations, which is the maininput data of our problem counts for some instances more than 45 000 nodes (to be compared withthe ≈200 tasks scheduling problems generally dealt in production). This is challenging, even just forhandling data.

Steps of a general approach We are currently concluding a thesis on that general problem where aVNS is returning very competitive results and will be coupled with a GRASP approach. But up to now,a linear combination of the objectives is taken into account. Moreover the power consumption criteria isstill not addressed in this work. The VNS algorithm improves initial solutions up to 67% and is alwayssuperior to practitioners ad-hoc heuristics as “left edge”. However, “modified left edge” algorithmssometimes performs better than VNS but this happens for instances that have been specifically designedfor MLE.

4 Conclusion and future work

This paper has aimed at presenting optimization problems raised by electronic practitioners and de-signers and aimed at attracting new partners for long term collaboration. It has been decided, as astrategic plan for our lab, to start common PhD programs with interested people and make effortswith the metaheuristic community to model and solve these problems more efficiently. In a mid-termhorizon, an European project should be started on those topics as well.

References

[1] R. Dafali, J-Ph. Diguet, and M. Sevaux. Key research issues for reconfigurable network-on-chip. InProceedings of the International Conference on ReConFigurable Computing and FPGAs, ReCon-Fig’08, pages 181–186, Cancun, Mexico, 3-5 December 2008.

[2] S. Evain, R. Dafali, J-Ph. Diguet, Y. Eustache, and E. Juin. µSpider cad tool: case study of NoCIP generation for FPGA. In Proceedings of conference DASIP’07, France, 2007.

[3] D.D. Gajski, N.D. Dutt, A.C-H. Wu, and S.Y-L. Lin. High-level synthesis: introduction to chip andsystem design. Kluwer Academic Publishers, Norwell, MA, USA, 1992.

[4] T.R. Jensen and B. Toft. Graph coloring problems. Discrete Mathematics Optimization. Wiley-Interscience, New York, USA, 1994.

[5] B.M. Pangrle. On the complexity of connectivity binding. IEEE Transactions on Computer-AidedDesign of Integrated Circuits and Systems, 10, 1991.


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Applications of metaheuristics to optimization problems in

sports

Celso C. Ribeiro ∗

first institution ∗ Universidade Federal Fluminense / Instituto de Ciencia da ComputacaoRua Passo da Patria 156, Niteroi, RJ 24210-240, Brasil


Abstract

Professional sport leagues involve millions of fans and significant investments in players, broadcastrights, merchandising, and advertising. Multiple agents, such as the organizers, media, players, fans,security forces, and airlines, play important roles in the leagues and tournaments. Professional sportsleagues are therefore part of a major economic activity and face challenging optimization problems.On the other side, amateur leagues usually do not involve impressive amounts of money, but insteadthe number of tournaments and competitors can be very large, also requiring coordination and lo-gistic efforts. The field of sports scheduling and management has been attracting the attention ofan increasing number of researchers in multidisciplinary areas such as operations research, schedulingtheory, constraint programming, graph theory, combinatorial optimization, and applied mathematics.Different optimization techniques have been applied to solve problems arising from sports schedulingand management. The hardness of the problems in the field lead to the use of a number of exact andapproximate approaches, including integer programming, constraint programming, metaheuristics, andhybrid methods. Problems associated with the scheduling of round robin tournaments are of partic-ular importance, due to their relevance in practice and to their interesting mathematical structure.We review some applications of metaheuristics to different scheduling problems in sports, such as thetraveling tournament problem, referee assignment, and the minimization of the carry-over effect. Wealso report real-life applications to tournament scheduling. Recent advances in metaheuristics are alsoillustrated in the context of these applications.


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Annual planning of harvesting resources in the forest industry

David Bredstrom ∗ Petrus Jonsson ∗ Mikael Ronnqvist ∗ †

∗ The Forestry Research Institute of SwedenUppsala, Sweden

Email: [email protected], [email protected]

† The Norwegian School of Economics and Business AdministrationBergen, Norway


Abstract

A cost efficient use of harvesting resources is important in the forest industry. The main planningis made in an annual resource plan which is continuously revised. The harvesting operations aredivided into harvesting and forwarding. The harvesting operation fells trees and put them inpiles in the harvest areas. The forwarding operation collects piles and moves them to storagelocations adjacent to forest roads. These operations are done by machines (harvesters, forwardersand harwarders) and these are operated by crews living in cities/villages which are within somemaximum distance from the harvest areas. Machines, harvest teams and harvest areas have differentcharacteristic and properties and it is difficult to come up with the best possible match throughoutthe year. The aim with the planning is to come up with a cost efficient plan The total cost isbased on three parts; production cost, traveling cost and moving cost. The production cost is thecost for the harvesting and the forwarding. The traveling cost is the cost for driving back andforward (daily) to the harvest area from the home base. Moving cost is associated with movingthe machines and equipment between harvest areas. The Forest Research Institute of Sweden hastogether with a number of Swedish forest companies developed a decision support platform for theplanning. An important aspect is to come up with high quality plans within short computationaltime. A central part is an optimization model which integrates assignment of machines to harvestareas and scheduling of the harvest areas during the year for each machine. The problem is complexand we propose a two phase solution method where we first solve the assignment problem and ina second stage the scheduling. In order be able to control the scheduling also in phase 1, we haveintroduced an extra cost component which balances the geographical spread of the assignments inphase 1. We have tested the solution approach on a case study from one of the larger Swedishforest companies. This case study involves 46 machines and 968 harvest areas representing a logvolume of 1,33 million cubic meters. We describe some numerical results and experiences from thedevelopment and tests.


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1 Introduction

An important planning problem for many forest companies is how to best use their own and/or con-tracted harvesting resources during the year. There are two main operations; harvesting and forwarding.Harvesting operations are in turn classified into final felling and thinning. Final harvesting is whenall (or most) of the trees are cut down. This happens when the trees have reached a maturity age,say 80-120 years. Thinning operations on the other hand removes only a proportion of the trees inorder for the remaining trees to grow better. Thinning operations may occur several times before thematurity age is reached, for example at the age 15 and 30 years. In Sweden, industrial harvesting andforwarding operations are done 100% with mechanized machines. The machines used for the operationsin a harvest area are called harvesters and forwarders. A harvester fells the trees and cuts (buck) itinto logs of various dimensions with respect to length and diameter. As a result, there will be manypiles of logs distributed in the harvest areas. These piles are then picked up by forwarders and movedto storage locations adjacent to forest roads. Here they are stored until logging truck transport themto mills (saw-, pulp-, paper-mills and heating plants).

The planning of harvesting and thinning operations must consider several goals and restrictions.The planner has information about which harvest areas that should be harvested during the next year.These areas have different properties such as geographical location, ownership size, volume, averagetree size, forwarding distance and terrain accessibility. The machines have different sizes, e.g. small,medium, large and very large. To compute the actual production by a machine in a harvest area, weneed to use so-called performance functions. These functions takes the characteristic of both machinesand harvest areas and compute the production volume per hour. From this, we can establish how longtime it takes for each machine to do the operations at each harvest area. The machines typically worksin teams of two; a harvester and a forwarder. Other combinations are also possible but less frequent.Over the last years, a new type of machine has been developed. It is called a harwarder and it canperform both harvesting and forwarding. A harwarder has its own performance functions and willperform better and worse than a team of harvester and forwarder for some type of harvest areas.

The crews of the machines are either long term employed by the forest company or work as inde-pendent entrepreneurs on a contract basis. Each crew has a home base from where they travel by carto the harvest area where the machines are located. Once its shift is finished, they travel back home.A machine generally has more than one team in order to use it more efficiently during day and week.

It is important to keep the cost of the harvesting and thinning operations as low as possible. Thereare three main cost components with the operations. The first component is the machine operation cost.For each machine, there is an hourly rate. From the performance functions, it is possible to computethe time to harvest and forward each harvest area. Given an hourly cost rate for each machine, we cancompute the overall cost for the operations. The second component is the traveling cost. There is agiven contract on how much to pay each crew for their travel back and forward between harvest areaand home base. The third component is a machine moving cost. Once a harvest area is harvested andforwarded, the machines are moved to the next area. If this distance is short, the machines are movedon their own wheels to the next site. However, if the distance exceeds a given limit, a trailer is used tomove the machines. Moreover, it is important to note that the moving cost depends on the schedulingof each machine during the year.

In this paper, we describe a planning system developed close together with SCA, which is one of thelarger Swedish forest industry companies. The aim was to develop a tool that could be used to makea number of different scenario analysis within a short computational time. First, it should minimizethe total cost for all harvesting operations during one year given a fleet of machines. Second, it shouldbe possible to find out the best configuration of the fleet. For example, how many harwarders shouldbe contracted. Third, where is the best location of the home bases of the teams. Fourth, what are thebest size of the machines given the existing teams. This is the case when some older machines need tobe replaced by new ones.


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The overall planning problem is an integrated allocation and routing problem. This is known to bedifficult to solve even for small instances. In order to meet the limited solution time, we have developeda two phase solution approach. In the first phase we make an assignment between machines and harvestareas. In this part, we include production and traveling costs but not the moving cost. In the secondphase we make a scheduling of the harvest areas assigned to each harvest team. In order to havea connection between the two phases we include an approximate moving cost. This cost componentmeasures the geographical spread of the harvest areas in the first phase.

Another requirement from the company was to provide detailed result reports. This include sum-mary reports, reports for each machine and aggregated teams, reports for each harvest area and aggre-gated areas, summary of costs, and specific production indexes used in the company. The implemen-tation uses an interface in Excel, where input data are provided through one Excel sheet and outputreports is written into another Excel sheet. The optimization models and methods are developed inthe modeling language Ampl using Cplex as a solver. Ampl is connected directly to Excel for inputand output. In addition we have connected a map system which provides result maps of the solutionfor each harvest team.

We have tested the software on a case from one of the larger Swedish forest companies which involves46 machines and 968 harvest areas representing a log volume of 1,33 million cubic meters. From thiscase, we have tested a number of scenarios about potential savings and improvements. Typical solutiontimes are within 20 minutes.

The outline of this report is a s follows. In Section 2 we describe some planning problems found inthe literature and how our problems fits within these. In Section 3 we describe our problem and itscharacteristic in detail. We also discuss our assumptions made. In Section 4 we give the integratedmodel and in 5 we describe our two-phase solution approach. In Section 6 we describe our implementa-tion, case study and some numerical results. In Section 7 we discuss some planning aspects and issuesfrom using the system in practice. We finish by making some concluding remarks in Section 8.

2 Literature review

Supply Chain (SC) planning in the forest products industry encompasses a wide range of operations anddecisions, from strategic to operational. SC in the forest industry has been discussed in several papersand a recent survey is found in D’Amours et al. [3]. There are many Decision Support Systems (DSS)developed for different planning problems in the forest SC. These are often integrated into application-specific databases holding all the information needed for the models and the Geographical InformationSystem (GIS) used to visualize the input data and results. In addition, many DSS include OperationsResearch (OR) models to support the planning. Ronnqvist [14] presented a series of typical planningproblems found in the forest products industry, with comments about the time available for solvingeach of these problems. In [14], it can be observed that, while operational planning problems usuallyneed to be solved rapidly, within seconds or minutes, strategic planning problems can be solved over alonger period of time, sometimes taking many hours. For this reason, heuristics, meta-heuristics andnetwork methods are generally used for operational problems, whereas Mixed Integer Programming(MIP) and stochastic programming methods are better for tactical and strategic planning problems.

Strategic forest management puts the emphasis on the relationship between decisions connected toforest use (e.g., harvesting areas, allocations, silvicultural treatments) and their different socio-economicconsequences (e.g., environmental problems, non-declining yield, continued employment, forest accessand industrial competitiveness). Numerous models have been developed to aid forest managers andpublic forestry organizations in their decision-making. These models typically cover several rotations,which in Sweden corresponds to 100-200 years. Once the forest management strategy has been estab-lished, the tactical and operational planning decisions are made, integrating the needs of the differentsupply chains.


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Tactical models in forest management are used to decide which harvest areas to cut in each yearand is a prerequisite for annual planning. Here, spatial data becomes important and GIS is used tocollect necessary data. Increasing importance is put on richness and diversity of wildlife, creatingfavorable habitats for flora and fauna, ensuring the quality of soil and water, preserving scenic beauty,and guaranteeing sustainability. One of the primary ways that spatial relationships and environmentalconditions have been modeled at the tactical level is by using adjacency restrictions with green-uprequirements. Specifically, a maximum local impact limit is established to restrict local activity for agiven period of time. In the case of clear cutting, for example, this corresponds to a maximum open area,which is imposed on any management plan. Another important example for wildlife is the requirementthat patches of mature habitats (i.e., contiguous areas of a certain age) must be maintained to allowanimals to live and breed. To ensure this, potential areas must be grouped to form patches, see e.g.Ohman and Eriksson [12]. A number of models incorporate the maximum open area and adjacencyconstraints, see e.g. McDill et al. [11] and Goycoolea et al. [8].

When we study annual planning, there are several problems that cover a similar time horizon. Inthis review, we focus on systems developed in Sweden as they relates closely to the problem we focuson in this paper. In road maintenance planning we want to decide which roads to upgrade in orderto be able to make sure we can harvest the areas in the correct time period, e.g. season or month.In Sweden, Olsson [13] and Henningsson et al. [9] have presented MIP models that include decisionsabout restoring existing forest roads and transportation in order to provide access to available harvestareas during the spring thaw when only certain roads are practicable. The model used by Henningssonet al. [9] is the basis for the decision support system, RoadOpt (Frisk et al. [7]), developed by theForestry Research Institute of Sweden.

Forest operations, refers to the actions that affect harvest operations directly, see Epstein et al.[4]. Transportation is an important part of forest operations, constituting up to 40% of the operationalcosts. In some cases, harvest planning is combined with transportation and road maintenance planning,with an annual planning horizon. A MIP model to solve this multi-element planning problem has beenproposed by Karlsson et al. [10]. Other important issues in transportation include the possibility ofintegrating truck transport with other modes of transportation, specifically ship and train (Forsberg etal. [5]). Transportation operations provide the operational link between the forest supply chain andother supply chains.

Operational routing is also an important planning problem. There are several papers describingdifferent solution approaches and DSS systems, see Weintraub et al. [15], a DSS for logging trucks,received the Franz Edelman Award in 1998. This DSS is currently used by several forest companiesin Chile and other South American countries. It exploits a simulation-based heuristic to produce aone-day schedule. The Swedish system, RuttOpt (Andersson et al. [1]), establishes detailed routes forseveral days and integrates a GIS with a road database, using a combination of tabu search and an LPmodel. Testing of this system has shown cost reductions between 5% and 20% compared to manualsolutions. Forwarding operations are another type of routing problem. Flisberg and Ronnqvist [6] haverecently proposed a system designed to support forwarding operations at harvest sectors. Using a DSS,this system improves forwarding operations about 10% by establishing better routing. In addition, itproduces better information on supply locations and volumes that can be used in subsequent trucktransportation planning.

We want to position our annual resource planning to the above planning problems. The roadmaintenance planning cover 1-5 years and our resource planning require that this plan is available.This is required as we need to know the accessibility of the harvest areas during the year. The annualharvesting planning problem in Karlsson et al. [10] is closely related to our problem. However, thereare some differences. The main differences are that we consider the moving between harvest areas,special constraints on teams and machines, and that we have a detailed description of machines andharvest areas. The tactical transportation problem is related but it follows once we have a plan for theharvesting teams. As such it follows after our resource planning problem. The routing problems coverdaily and weekly planning and is not related.


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3 Problem description

The Forest Research Institute of Sweden has done earlier development work and case studies for resourceplanning at Swedish forest companies. However, in these studies most of the planning and evaluationwas done manually or semi-manually. Different companies view their annual resource planning some-what different. The aim with the project described in this paper is to generate a planning platformthat can be used at several companies. Therefore, we have made an attempt to develop a general modeleven though the case study reported in this paper is based on properties from one specific company.

3.1 Harvest machines

There are two main machine types; harvester and forwarder. These are illustrated in Figures 1 and 2,respectively. A harvester fells the trees and cuts (buck) it into logs of various dimensions (length anddiameter). This cutting operation is supported with on-board optimization routines to find the bestpossible cutting pattern given a list of log values. The system on-board also creates information abouttrees harvested and a detailed description of each log. This information also include GPS coordinatesfor the log piles. These piles are picked up by forwarders and moved to the side of forest road. Theseforwarders must follow in the tracks of the harvesters to avoid additional damage to the ground. Theycan also use the information provided by the harvesters, e.g. the GPS coordinates, to find efficientroutes for the forwarding operations.

Figure 1: An example of a harvester


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Figure 2: An example of a forwarder

Over the last years, a new type of machine, called a harwarder, has been introduced. It is a machinethat can perform both harvesting and forwarding. This machine can be more cost efficient in sometype of harvesting areas, in particular, in thinning operations. An illustration of a harwarder is givenin Figure 3.

There are several manufacturers of machines (harvester, forwarder and harwarder) and they comein different size and characteristic. The machines are typically sorted into size classes and we use thecommonly used classes small, medium, large and very large. Each machine has an hourly price (SEKper hour) and an available annual capacity expressed in hours. The capacity is based on the numberof drivers available, typically two drivers per machine that works in shifts.

3.2 Harvest teams

A harvest team is generally made up by one harvester and one forwarder. This is also the generalgrouping when a contract is negotiated between a forest company and a contractor. However, othergroupings are available but in our case study, we only have harvest teams consisting of one harvesterand one forwarder. A harvest team has a home base, typically the town or village where the driverslive. To limit the traveling, each team has a maximum operational radius. This is measured as the


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Figure 3: An example of a harwarder


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distance from the home bas to harvest areas. A team may have different goals with their operationsand it may be included directly in a contract. A team may be primarily focused on either final felling orthinning operations depending on their skills. Therefore, there is a limit set on the minimum amountof final felling and/or thinning operations. Some teams are full time contracted and hence there is aminimum activity level expressed as a percentage of its available annual time.

3.3 Harvest areas

As a basis for the annual planning, there is a set of harvest areas available for final felling or thinningoperations. These areas are selected from an earlier more long term planning and the set is called (inSwedish) traktdatabank (bank of areas) and is illustrated in Figure 4. The areas may be owned by theforest company or by an external organization or private owner. In the latter case, the forest companyis only responsible for the harvesting and there may an requirement that the harvest areas must beharvested during the year. This is not the case for their own areas in the case the capacity of the teamsis not enough.

Figure 4: Illustration of the set of harvest areas ready for harvesting during one year

Each harvest area has a set of properties and characteristic. These are listed below.

GPS coordinates The GPS coordinates are used to snap the areas to the closest road segment. Giventhis, we can compute distances between then and distances to home bases of the harvest teams.

Operation The operation to be done can be either final felling or thinning.

Accessibility The terrain accessability describes when it is possible to do the harvesting operations.For example, some areas can only be harvested in the winter when the soil is frozen. In otherseasons, the damage done by the machines would bee to great (and costly). Therefore, each areahas a coupling to a season. In our case we use winter, spring, summer and autumn.

Area The areas are different in size and each area has an area expressed in hectares.

Volume Each area has its own silviculture treatment and ground characteristic. Therefore, the treesgrow differently and each area is measured to have a particular volume expressed in cubic meters.

Average tree The trees at different areas have different sizes die to the particular ground conditionsat the area. The machines work different with different sized trees. For example, a small harvestermay not be capable to hold a large tree. The average size is expressed in cubic meters.


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Forwarding distance The area may not be located adjacent to a road and this affect the performanceof the forwarder that must travel a forwarding distance each time when it unloads the logs a theforest road.

3.4 Performance functions

To compute the time it takes for a particular machine to harvest or forward a particular harvest,we need to have match properties between machines and harvest areas. This is done by so-calledperformance functions. There are many possibilities form such functions but most companies usequadratic polynomials as they are easy to use. The performance function for the machines can beexpressed as

pharvester(x) = aso ∗ x2 + bso ∗ x+ cso

pforwarder(y) = dso ∗ y2 + eso ∗ y + fso

pharwarder(x, y) = gso/(hso + iso/x+ jso ∗ y)

where

x : average tree size (m3sub, solid under bark)y : forwarding distance (m)s : size: small, medium, large or very largeo : operation: final felling or thinning

All coefficients aso, . . . , jso are company specific coefficients which are measured through many testson machines in different settings. Each function provides how many cubic meters harvested/forwardedper standard hour. In Figures 5 and 6, we illustrate typical performance functions for harvesters andforwarders, respectively. A harwarder is more difficult to illustrate graphically as it depends on bothaverage tree size and forwarding distance. In Figure 7, we illustrate typical performance functions aharwarder where the forwarding distance is fixed at certain values.

Figure 5: An illustration of the performance function of a harvester


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Figure 6: An illustration of the performance function of a harvester

Figure 7: An illustration of the performance function of a harwarder with fixed forwarding distance

3.5 Cost components

The aim with the planning is to minimize the overall cost. The three main cost components aredescribed below.

Production cost This is the combined cost of harvesting and forwarding each harvest area. Giventhe volume in each area, the performance function, the hourly cost for each machine, we cancompute the overall production cost. Included in the production time (and cost) is also the timeit takes to clean the harvest area before moving to the next. Note that our production time isbased on the slowest of the two machines (harvester and forwarder) in the team for each harvestarea.


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Travel cost This is the traveling cost for crews between home bases and harvesting areas. For eachharvest area, we can compute the number of times they need to travel back and forward. Thistogether with given traveling allowances (SEK/km) we can compute the traveling cost.

Moving cost This is the cost to move machines between harvest areas. If the distance is long enough,there is a need to use a trailer to move the machines. Otherwise, the machines can be drivendirectly to the next area. There are specific parameters to describe the maximum distance,average speed (to get the moving time) and costs for the moving. There are two parts; a fixedcost and one depending on the distance between the harvest areas.

4 Model formulation

In this section we will develop our optimization model. Before we provide the model we discuss someissues and assumptions made in the modeling.

4.1 Modeling issues and assumptions

In general we construct harvest teams with any groupings of machines. However, in most cases (includ-ing our case) they work in a pair of a harvester and a forwarder. In the proposed model, we assumethis characteristic. We will discuss how to proceed otherwise in Section 7.

We use four seasons (winter, spring, summer and autumn) to describe when the harvest areas canbe harvested and forwarded. Each machine has an available capacity (hours) in each season. This iscomputed as an proportion of the annual capacity based on the length of each season. In order to toallow some overlap between seasons, we use a so-called overlap time, which describes how much of thecapacity in e.g. the winter that can be used in the spring. This is used in practice and it ensures thatthe machines are not stopped at the end of e.g. winter because the capacity was not enough to finisha particular area. In our case, we use an overlap of one week between the seasons.

If the harvesting capacity is not enough to harvest all harvest areas we need put them in a so-calledpool at the end of the year. If this happens, additional teams can be contracted part time. Otherwise,in case there are too many harvest areas available, the pool of harvest areas are kept to the next year.However, externally owned harvest areas must follow special agreements. Hence, we need to ensurethat the volume of externally owned volumes are limited in the pool. There are different limits for finalfelling and thinning volumes. The pool creates some problem in the modeling as we need to keep themnon-harvested. We can not set a cost of 0 for these as the optimal solution would be to put all harvestareas in the pool. Instead, we need to set a penalty cost so that as many harvest areas as possible areharvested. As the areas have different size, we aim to keep the volume in the pool as low as possible.It is important that this penalty parameter is large enough to not balance against other solutions.

4.2 Model

The model consists of input data, decision variables, objective function and constraints. Each of theseare described below.

Sets and input data

The sets and input data are described below.


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Notation DescriptionI Set of harvest areasT Set of time periods (seasons)M Set of machines (Mh ∪Mf ∪Md)Mh Set of harvestersMf Set of forwardersMd Set of harwarders

Supporting setsIf Set of harvest areas for final fellingIt Set of harvest areas for thinningIe Set of harvest areas with external ownershipIm Set of potential harvest areas for machine m ∈MNi Set of machines that can harvest and/or forward harvest area i

Notation Descriptiondij Distance between harvest area i and harvest area jthmi Harvesting time for machine m in harvest area itfmi Forwarding time for machine m in harvest area ichmi Harvesting cost for machine m in harvest area icfmi Forwarding cost for machine m in harvest area itmt Available time for machine m in season tgmij Cost to move machine m between harvest area i and harvest area jhmi Cost for machine m to travel from home base to harvest area i and backaf

m Minimum level (proportion) of final felling for machine mat

m Minimum level (proportion) of thinning for machine mbf Maximum volume of externally owned harvest areas for final felling in the poolbt Maximum volume of externally owned harvest areas for thinning in the poolvi Volume in harvest area i)tmax Maximum overlap time between seasonsγ Penalty cost for harvest areas in the pool (per volume unit)

Comments:

• A harwarder has both a harvesting and a forwarding time. A harvester has value 0 for forwardingand a forwarder value 0 for harvesting.

• The moving cost include both fixed cost and the cost related to the distance between areas.

4.3 Decision variables

The main decisions are to assign machines to harvest areas and scheduling of machines between harvestareas. We also need to decide any overlap between the seasons and whether a harvest area is in thepool or not. The definition of the variables are as follows.

ymit ={

1, if machine m is used in harvest area i during season t0, otherwise

si ={

1, if harvest area i is in the pool0, otherwise

xmjk ={

1, if machine m moves between harvest area i and harvest area j0, otherwise

omtt′ = overlap time used for machine m between time period t and t′


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Objective function

We have four cost components in the objective. The first three are associated with real costs whereasthe fourth is a penalty cost for the harvest areas put in the pool.

zproduction =∑

m∈M

∑

i∈Im

∑

t∈T

(chmi + cfmi)ymit

ztraveling =∑

m∈M

∑

i∈Im

∑

t∈T

hmiymit

zmoving =∑

m∈M

∑

i∈Im

∑

j∈A

gmijxmij

zpool =∑

i∈I

γvisi

Model

The model can be stated as follows.

min z = zproduction + ztraveling + zmoving + zpool

s.t.∑

m∈Mh

∑

t∈T

ymit +∑

m∈Md

∑

t∈T

ymit + si = 1, i ∈ I (1)∑

m∈Mf

∑

t∈T

ymit +∑

m∈Md

∑

t∈T

ymit + si = 1, i ∈ I (2)

∑

i∈Im

(thmi + tfmi)ymit ≤ tmt, m ∈M, t ∈ T (3)∑

t∈T

(∑

i∈If

(thmi + tfmi)ymit−

awm

∑i∈I(thmi + tfmi)ymit) ≥ 0, m ∈M,w = t, f (4)∑

m∈M

∑

t∈T

∑

i∈Ie

vmsm ≤ bw, w = t, f (5)∑

m∈M

∑

i∈Im

∑

j∈Im

xmij =∑

t∈T

ymjt, j ∈ I (6)

∑

m∈M

∑

i∈Im

∑

j∈Im

xmij =∑

t∈T

ymit, i ∈ I (7)

∑

i∈S

∑

j∈S

xmij ≤ | S | −1, 2 ≤| S |≤| I |,m ∈M (8)

ymit ∈ {0, 1}, ∀m ∈M, i ∈ Im, t ∈ T (9)xmij ∈ {0, 1}, ∀m ∈M, i, j ∈ Im (10)si ∈ {0, 1}, ∀i ∈ I (11)

ymit = 0, ∀m ∈M, i /∈ Im, t ∈ T (12)

The description of the constraints are summarized below.


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Constraint set Description(1) each harvest area should either be harvested (by a harvester or a harwarder)

or put in the pool.(2) each harvest area should either be forwarded (by a forwarder or a harwarder)

or put in the pool.(3) limit on available time for each machine in each season.(4) minimum proportion on final felling and thinning for each machine.(5) maximum volume of external owned harvest areas for final felling

and thinning operations.(6) a machine can only move to a harvest area where it does any operation.(7) a machine can only move from a harvest area where it does any operation.(8) states all subtour elimination constraints for each machine.(9) binary requirements on the harvesting area assignment decisions.(10) binary requirements on the routing decisions.(11) binary requirements on the pooling assignment decisions.(12) remove all infeasible combinations of assignments.

For clarity we did not include the overlap possibility in the model. In the implemented model, thisis added in the following way. Suppose we study constraint (3) for machine m in the spring. Thenwe simply add a variable om,spring,summer in the right-hand-side of the constraint for the time periodspring. At the same time we need to compensate for earlier overlap decisions in the winter periodby removing such decisions. Otherwise, we would not model the total annual capacity correctly. Theright-hand-side becomes

. . . ≤ tm,spring + om,spring,summer − om,winter,spring

The model is an integrated assignment and routing problem. One part is to make an assignmentand the second part is to make the routing. The problem is more complicated than a standard vehiclerouting problem, which is known to be very hard to solve even for small instances. In our case, we havea large problem as the number of harvest areas is 968.

5 Solution approach

Given the requirement of limited solution time, it is not possible to solve the problem directly. Insteadwe need to develop a heuristic approach. We propose a two-phase approach where we make use of theunderlying structure in the decomposition. If we ignore the moving of machines between harvest areas,we have a model structure which close to a generalized assignment problem (GAP). If we have donethe assignment between machines and harvest areas, we have a traveling salesman problem for eachmachine. With this as a basis, we suggest the following solution approach.

Algorithm:

Phase 1 Solve the problem to assign machines to harvest areas. We use the variables ymit, si andomtt′ . In the objective we use all parts except zmoving. As constraints we use (1)-(5), (9)-(10)and (12).

Phase 2 Given a solution from Phase 1, each machine is assigned to a set of harvest areas. We thenneed to solve a problem with variables xmij , objective zmoving, and constraints (6)-(8) and (10).This problem will separate into one traveling salesman problem (TSP) for each machine.

The main disadvantage with this approach is that we can not take into the moving cost in Phase 1as we do not control the geographical distribution of areas.. The traveling cost tries to keep the harvestareas close to the home base but they may located around the home base instead in a particular region.


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In order to include some relation between the two phases we have included a fifth cost part whichmeasures the spread of harvest areas for each machine. Suppose that we identify a ”central” harvestarea a for machine m, then we can introduce a cost depending on the distance from this central harvestarea. The new cost for machine m can be expressed as

∑

i∈Im

α1 ∗ αd[i,a]2 ymit

where α1 ≥ 0 and 1 ≤ α2 ≤ 1.1 are chosen parameters. The notation d[i, a] is the distance between aand i.

A critical question is how we can identify a central harvest area. We choose to solve Phase 1 problemwithout zspread. Given the solution, we determine the most central harvest area for each machine. Thisharvest area is then used to compute zspread and added into the model. We then resolve Phase 1including the spread term. In Figure 8 we illustrate a possible solution without the spread component,and in Figure 8 we illustrate one with the spread component. From the figures it is clear that themoving cost will be lower using the spread component.

Figure 8: An illustration of a team’s set of harvest areas when the spread component is not used

Figure 9: An illustration of a team’s set of harvest areas when the spread component is used


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6 Results

6.1 Implementation

The implementation of the platform is done in a combination of Excel and the modeling language Ampl.All input data is defined through tables in Excel. The solution approach is implemented in a script usingAmpl where we make use of the solver Cplex. All result reports are written into a separate Excel sheetwhere company specific reports are designed. A graphical report is also produced where we generateone map for each harvesting team. The distance calculations are done using the forest National RoadDatabase for Sweden, which is a GIS system for all forest companies and includes detailed descriptionof all roads in Sweden. In order to better control the solution time, we have introduced parameters tolimit the time used for the first and second problem in Phase 1 and for each of the TSP problems inPhase 2. To make sure we quickly find a feasible TSP tour, we have also implemented a TSP heuristic.

6.2 Case study

The case study comes from one of the larger Swedish forest companies. The case study consists of 46machines in 23 teams. There are 22 small, 6 medium and 18 large harvesters. There are 33 small,10 medium and 8 large forwarders. There are 14 different home bases. The average capacity for eachmachine is about 2,400 hours per year. The hourly machine price is between 600-1100 SEK per hour.The travel cost is based on a tariff of 2,50 SEK per km. The fixed cost to move a machine is 1,800 SEK.The trailer cost to move machines longer than 5 km is about 900 SEK per hour. There are 968 harvestareas. The total volume of the harvest areas is 1.33 million cubic meters covering 8,971 hectares. Wehave four seasons with specified length; winter - 18 weeks, spring - 9 weeks, summer - 16 weeks andautumn - 8 weeks.

6.3 Results

From the original case study, we have created two sets of instances. The first set is based on theoriginal data. These instances have a limited harvesting capacity and there will be some harvest areasin the pool. This makes it difficult to compare the instances with respect to the objective functionvalues. In the second set, we have introduced five additional harvest teams evenly distributed over thedistrict. Here, it is easier to make comparisons between the objective function values. We have usedthe following limits on the solution times. For the first problem in Phase 1, we have used 2 minutesand for the second problem 10 minutes. The time used to compute TSP routes is set to 20 seconds perteam.

6.4 Original case

We have created four instances (C1-C4) and the results are presented in Table 1. Column ”spread”indicate whether we use the spread component or not. Column ”Production limits’ indicate if we useproduction limits in equation (4) of the model or not. Column ”Travel distance” states the maximumoperational radius from the home base. A radius of 200 km means that you can reach any harvest areafrom any home base. The column ”Total cost” gives the overall function value, whereas columns ”Pro-duction cost”, ”Traveling cost” and ”Moving cost” gives the objective functions for zproduction, ztraveling

and zmoving, respectively. All objective function values are given in millions SEK. Column ”No. inpool” gives the number of harvest areas in the pool, i.e. not harvested. Instance C1 corresponds tothe original case and C2 when we remove the production limit. The objective function values are quiteclose but it is hard to evaluate any difference as there are different number of harvest areas in the pool


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and the cost to harvest the difference is hard to evaluate. Instances C3 and C4 give the solution whenwe have two different settings of the spread. We have used the following setting for low, medium andlarge: low - α1 = 100, α2 = 1.05, medium - α1 = 500, α2 = 1.1 and high - α1 = 1000, α2 = 1.1. If we usehigh in the spread component, the costs goes up and it results in fewer harvest areas being harvested.

Production Travel Total Production Traveling Moving No.Case Spread limits distance cost cost cost cost in poolC1 No Yes 200 102,707 96,287 6,420 2,981 67C2 No No 200 102,500 96,920 5,580 2,852 65C3 Yes (low) Yes 200 102,111 96,390 5,722 2,557 76C4 Yes (high) Yes 200 101,043 95,949 5,093 2,190 113

Table 1: Results from the four instances based on the original set of machines.

6.5 Modified case

The difference in this modified case is that we have added five extra harvest teams. With these extracrews, we will ensure that the capacity is enough to harvest all harvest areas and we have no areas inthe pool. We have created seven instances (E1-E7) to describe what type of analysis that is possible.The results are given in Table 2. Case E1 has no spread and the same restrictions as in the originalcase. The reason for this to have a lower objective than C1 is that a better combination betweenmachines and harvest area is possible. In C1, the model tries to push the volume in the pool down andhence we try to harvest large areas that may be located far away from the home bases. If we removethe production limits (E2), the decrease in objective is only about 0,18 million SEK. If we lower theoperational radius for the teams, we start to get an effect when it is down to 120 km. An decrease ofthe operational radius down to 100 km increase the cost with about 0,41 million SEK.

When we add the spread component, we can clearly see that the moving cost decreases in E5-E7.However, the cost decreases slowly as compared to the increase in production cost. This is an interestingobservation as the planners believe that the moving cost has a large impact. From these tests, it is clearthat the production cost dominates and is more important. From this we may conclude that it is moreimportant to match machines with harvest areas than grouping the harvest areas together for loweringthe moving cost. The reason for this is that the production cost represent such a large proportion ofthe total cost and hence is more important. We have also done some tests with other setting of themaximum solution times but it turns out to be very stable in generating high quality solutions. Evenif we decrease the overall solution time to about half, we get the same solution.

Production Travel Total Production Traveling MovingCase Spread limits distance cost cost cost costE1 No Yes 200 99,811 94,926 4,888 2,620E2 No No 200 99,633 94,826 4,807 2,580E3 No Yes 120 99,833 94,950 4,882 2,615E4 No Yes 100 100,228 95,368 4,860 2,603E5 Yes (low) Yes 200 99,945 95,022 4,923 2,466E6 Yes (medium) Yes 200 103,252 98,309 4,943 2,331E7 Yes (high) Yes 200 107,008 102,206 4,801 2,314

Table 2: Results for seven instances where we added five harvesting crews.

In Figures 10 and 11, we provide a solution for one team with and without the spread component.The red squares denote home bases and the circles the allocated harvest areas. It is clear that using aspread component makes set of harvest areas to be much closer to each other.


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Figure 10: An example of a team’s set of harvest areas when the spread component is not used

Figure 11: An example of a team’s set of harvest areas when the spread component is used

7 Discussion

The results are dependent on the input data and in particular of the performance functions. Theparameter settings for the performance functions varies among companies. The data is also collectedand the parameter set within each company. In our tests, we found that the variation in productionefficiency (m3 per hour) between different machines sizes were low. In some cases, we made optimalcombinations between machines and harvest areas that would not be the case in practice. This eithershows that the functions are to similar or that historical assignments were not efficient.

At most companies, the machines are classified into different sizes. However, although they do havesimilar size, they may perform quite different in different type of harvest areas. In our model, it isstraightforward to use unique performance functions for each machine type. The advantage is that webetter describe each machine. The disadvantage is the need for more input data. However, in manycases, the machine manufacturer can support this as they do many tests on their own machines.

When we have harvest areas in the pool, it is difficult to make a comparison between differentsolutions. By introducing extra harvest teams (which also is the case in practice) we can ensure thatthe pool is empty. Moreover, the interpretation of the pool and its effect is quite straightforward forpersons with an OR modeling background. However, for people with no OR background it is difficult


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to understand this concept. It is therefore very important to have a dialogue to discuss these issuesbetween the planners and people responsible for the models..

When we are using the proposed two phase approach it is important to have some connectionbetween phase 1 and phase 2. In our case, we make use of the suggested spread component. However,this needs to be used with some caution as it may provide solutions that look good because the areasare close together but are in fact worse. The reason for this is that the moving cost is small as comparedto the production cost. If we use a high spread cost, we can lower the moving cost. However, the overallproduction cost increases more quickly than the moving cost decreases. This is an aspect that is noteasy to visualize but it is important to understand for the planners. A good match between machinesand harvest areas is more important than lower the moving cost. We should however note that this isthe case for the input data. If the planners come up with another model of the moving cost, than thesituation may change.

In our case study, we have harvest teams consisting of one harvester and one forwarder. In othercompanies, the construction of harvest teams can be more flexible. For example, we may have aentrepreneur using, for example, two large harvesters supported by three forwarders. In this case, weneed to add some constraints in the model stating that harvesting and forwarding must also be doneby the same entrepreneur. Moreover, we also need to include additional restrictions in the routing sothat the forwarding is done just after the harvesting. In this case, we do get a synchronization of theoperations, see e.g. Bredstrom and Ronnqvist [2].

In the implementation, we have also included an efficiency value for each machine. The reason forthis is that there are large differences in skills and performance between machine crews. The efficiencyis expressed as 100% if they work as computed in the performance function. Typical values are in therange 80-120% where 120% means that they are more efficient.

In the annual resource planning we do not consider the demand of different assortments during theseasons. This is typically dealt with in more short term harvesting planning. However, it would be ofgreat interest to include also this aspect in our resource planning model.

8 Concluding remarks

We have proposed a two phase method for an integrated assignment and routing problem appearingin annual resource planning in the forest industry. The first phase is a GAP structured problem toassign machines to harvest areas. The second phase is to schedule the assigned areas for each machinethroughout the year. This is a TSP problem for each machine. To make a connection between thephases, we have introduced a cost component that controls the distance spread between harvest areasin the first phase. By grouping harvest areas together, we can keep the moving cost in phase 2 lower.

The implemented platform is based on input and output using Excel. This makes it very easy forplanners to use the system. The optimization part uses Ampl and Cplex but is integrated with Excel.The solution times can be set and from the case study, we have noted that we get very good solutionsin a short time, in our case less than 20 minutes. We have a flexible way to generate customer specificresult reports, including maps, which makes it more interesting for planners to use it.

We have tested the system on a large case from a larger Swedish forest company.. With the system,it is easy for the planner to test different scenarios when making an entire new annual plan or revisingan old. We have found it important to have close discussion on how the OR models work and how tointerpret different aspects of the proposed solution method and the results.

For further work, we plan to include a number of additional features including better flexibility indefining harvest teams, synchronization of harvesting and forwarding operations and controlling theproduction levels of different assortment.


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Acknowledgement

We thank Goran Tjernberg and Tomas Johansson for providing the case study and many valuablediscussions and comments around annual resource planning.

References

[1] Andersson, G., P. Flisberg, B. Liden and M. Ronnqvist RuttOpt - A decision support system forrouting of logging trucks. Canadian Journal of Forest Research, Vol. 38, 1784–1796, 2008.

[2] Bredstrom, D., M. Ronnqvist. Routing and scheduling with synchronization constraint. EuropeanJournal of Operational Research, Vol. 191, pp. 19-29, 2008.

[3] D’Amours, S., M. Ronnqvist, A. Weintraub. Using Operational Research for supply chain planningin the forest product industry. INFOR, Vol. 46, No. 4, 47–64, 2008.

[4] Epstein, R., J. Karlsson, M. Ronnqvist, A. Weintraub. Harvest Operational Models in Forestry. InA. Weintraub, C. Romero, T. Bjørndal and R. Epstein, editor, Handbook on Operations Researchin Natural Resources, Kluwer Academic Publishers, New York, Chapter 18, 2007.

[5] Forsberg, M., M. Frisk, M. Ronnqvist. FlowOpt - a decision support tool for strategic and tacticaltransportation planning in forestry. International Journal of Forest Engineering, Vol. 16(2), 101–114, 2005.

[6] Flisberg, P., M. Ronnqvist. Optimization based planning tools for routing of forwarders at harvestareas. Canadian Journal of Forest Research, Vol. 37, 2153–2163, 2007.

[7] Frisk, M., J. Karlsson, M. Ronnqvist. RoadOpt - A decision support system for road upgrading inforestry. Scandinavian Journal of Forest Research, Vol. 21, Suppl. 7, 5–15, 2006.

[8] Goycoolea, M., A.T. Murray, F. Barahona, R. Epstein, A. Weintraub. Harvest scheduling subject tomaximum area restrictions: exploring exact approaches. Operations Research, Vol 53(3), 490–500,2005.

[9] Henningsson, M., J. Karlsson, M. Ronnqvist. Optimization models for forest road upgrade plannin.Journal of Mathematical Models and Algorithms, Vol. 6(1), 3–23, 2007.

[10] Karlsson, J., M. Ronnqvist, J. Bergstrom. An optimization model for annual harvest planning.Canadian Journal of Forest Research, Vol (8), 1747-1754, 2004.

[11] McDill, M.E., S.A. Rabin ,J. Braze, J. Harvest scheduling with area-based adjacency constraints.Forest Science, Vol. 48, 631–642, 2002.

[12] Ohman, K, L.O. Eriksson. The core area concept in forming contiguous areas for long-term forestplanning. Canadian Journal of Forest Research, Vol. 28, 1032–1039, 1998.

[13] Olsson L. Optimization of forest road investments and the roundwood supply chain. Ph. D. Thesis,The Swedish University of Agricultural Sciences, Sweden, 2004.

[14] Ronnqvist, M. Optimization in forestry. Mathematical Programming, Ser. B, Vol. 97, 267-284,2003.

[15] Weintraub, A., R. Epstein, R. Morales., J. Seron, P. Traverso. A truck scheduling system improvesefficiency in the forest industries. Interfaces, Vol. 26(4), 1–12, 1996.


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Some Thoughts on Engineering Stochastic Local Search

Algorithms

T. Stutzle ∗

∗ IRIDIA, CoDE, Universite Libre de Bruxelles (ULB)Brussels, Belgium


1 Stochastic Local Search Methods

SLS algorithms [7] are among the most prominent and successful techniques for solving computationallyhard problems in computer science, operations research, engineering, and business administration [1, 7].SLS techniques range from rather simple constructive search and iterative improvement algorithms togeneral-purpose SLS methods, which are also widely known as metaheuristics [5]. General-purposeSLS methods can be seen as general algorithmic frameworks that can be applied to specific problemswith relatively few adaptations. They include simulated annealing, tabu search, iterated local search,variable neighborhood search, evolutionary algorithms, and ant colony optimization (ACO). General-purpose SLS methods have received enormous attention, as witnessed by thousands of publications anddedicated conference series, such as the Metaheuristics International Conference (MIC). In fact, someSLS methods, including evolutionary algorithms and ACO, are the focus of rather independent areaswith their own series of conferences, such as GECCO, CEC, PPSN, and ANTS.

There are a number of reasons for these successes. These include advancements in the algorithmicmethods, such as new techniques and hybridizations of existing search techniques. Additionally, SLSalgorithms are typically easier to implement than systematic search algorithms and, thus, they requiretypically lower development times. Finally, SLS techniques are flexible and applicable to slight varia-tions of a problem. Developments that supported these advances in SLS are in the stronger usage ofeffective data structures that significantly contribute to the efficient implementation of the algorithmictechniques, especially when tackling large problem instances.

Despite these significant successes, there are several important issues that are neglected in most ofthe ongoing research efforts on SLS algorithms (and, in particular, metaheuristics).

• There do not exist general guidelines on how to design and implement efficient SLS algorithms.Current practice is to implement, based on some underlying, basic heuristic one specific SLSmethod. However, an SLS method is not a fully-defined recipe: it leaves many implementationchoices open, and it is restricted to a limited number of concepts for the implementation ofeffective algorithms. Even worse, the underlying basic heuristics have a tremendous influence onthe final performance, and this influence is frequently neglected. In fact, SLS algorithms are oftendeveloped in an ad-hoc fashion instead of following a principled engineering process that is basedon established guidelines and practice.

• The current understanding of the relationship between algorithm components, the performanceof SLS algorithms and problem or search space characteristics is not good enough as to allow fora systematic development of SLS algorithms. This is not surprising, since state-of-the-art SLS


EUMEeting 2009 48

algorithms are complex, stochastic algorithms that are difficult to analyze theoretically as well asexperimentally.

• There are significant shortcomings in the experimental methodology used in many research stud-ies; this makes it difficult, for example, to compare the results reported by different researchersand to reach final conclusions based on these results. Therefore, there is an urgent need for theuse of a scientifically well founded experimental methodology and its systematic application, asalso argued in [6].

• If high performance is required, SLS algorithms often involve significant development times andneed a significant amount of expert work for developing appropriate search strategies and for fine-tuning their parameters. Few specific tools exist for improving aspects of this problem; examplesare tools for the (automatic) tuning of parameters [2, 3]. However, further advancements in theperformance of these techniques are necessary to make them more practical.

Another aspect of the high development times is that many aspects of an SLS algorithm arere-designed and re-implemented from scratch when considering different application problems.Hence, there exists significant overhead when tackling modifications of a problems.

• Despite some initial steps, as reviewed in [8], much work remains to be done to combine system-atic search techniques and SLS techniques, since both are developed in quite strongly separatedcommunities. Rather than seeing these approaches as competing techniques, there is a largepotential for synergies when combining the algorithmic techniques from these two worlds.

• Last, there are several application domains that involve multiple objectives, dynamic modifi-cations of the data or objective functions and stochastic information on instance data. Whentackling these problems, the current deficiencies in the state-of-the-art in SLS algorithms arefurther accentuated, because the design and the evaluation of the algorithms becomes even morecomplex.

These deficiencies were also clearly visible in the research conducted in the context of the Meta-heuristics Network, an EU FP5 Research and Training network. One conclusion of the research effortswas that a significant part of the success that is obtained when implementing SLS algorithms for agiven problem is due to (i) the creative use of general ideas and insights into the algorithm behaviorand its interplay with problem characteristics, (ii) the level of expertise of the person who develops andimplements the algorithm on the problem, (iii) on the time that is invested in the implementation andthe tuning of the algorithms. While fundamental issues such as the choice of the underlying heuristicor local search neighborhoods were found to be essential, the strict adherence to the rules of a specificSLS method were less important.

2 Towards a Discipline of Engineering SLS Algorithms

The mentioned deficiencies of the current state of the art in SLS algorithms and the insights of theMetaheuristics Network suggest that a principled approach to the process of designing, analyzing,implementing, tuning, and experimentally evaluating SLS algorithms is a key component to success.In fact, the goal of SLS Algorithms Engineering ultimately is to develop an engineering methodologyfor the design and implementation of stochastic local search algorithms that guides researchers andpractitioners in the development of such algorithms for solving challenging optimization problems.The establishment of such methodological developments will help to shift the impression that thedevelopment of SLS algorithms is rather an art towards the perception that their development processcan be guided by a principled methodology. We call the area defined by the research towards thisoverall goal SLS Algorithms Engineering, reminiscent of related areas such as Algorithm Engineering[4] and Software Engineering [9], where similar methodological issues are tackled in a different context.


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Probably the closest area to SLS Algorithms Engineering is the area of algorithm engineering, whichhas, in fact, a close relation to our goals. Algorithm engineering has been conceived as an extension tothe more theoretically oriented classical research in algorithms. According to [4], algorithm engineeringdeals with the iterative process of designing, analyzing, implementing, tuning, and experimentally eval-uating algorithms. Essentially, SLS Engineering deals with a very similar high-level process. However,the algorithms that are dealt with in SLS typically have a much more complex and unpredictable be-havior than those typically handled in algorithm engineering. This is due to the fact that (i) NP-hardproblems pose a significant challenge to algorithmic techniques, (ii) SLS algorithms often use heuristicsfor guiding the search which makes their behavior difficult to predict, (iii) there are typically many moredegrees of freedom to consider in the development of SLS algorithms than in the case of algorithms forpolynomially solvable problems, and (iv) the stochasticity of the algorithms increases even further thedifficulty of their analysis. Hence, we firmly believe that SLS Algorithms Engineering is a particularlychallenging field that deserves special attention.

3 Aspects of SLS algorithm engineering

Doubtlessly, in several existing work principled methods for the development of SLS algorithms havebeen applied. However, these have mainly been limited to specific applications or SLS methods. Hence,considering the establishment of an engineering methodology for SLS algorithms puts these worksin a significantly wider context. In fact, it can be seen as a logical next step of such initial, moresimplistic attempts. In what follows, we outline a few important aspects of a discipline of SLS algorithmsengineering.

3.1 SLS Algorithms Engineering Procedures

Clearly, a main goal of the SLS algorithms engineering is to develop a sound and systematic methodologyfor the implementation of effective SLS algorithms that can guide researchers and practitioners intheir development of such algorithms. Hence, an important step in SLS algorithms engineering isthe definition of systematic procedures that can lead to high performing SLS algorithms. (Note that,obviously, the procedures are a function of the application context.)

As a simple example of a very high-level procedure, let us give an example that follows a bottom-upapproach, where SLS algorithms are built by iteratively adding complexity to simple, basic algorithms.More concretely, a tentative first guess of an engineering procedure for designing and implementingSLS algorithms could be as follows (how tools assist this process or what procedures or experimentaldesigns are to be followed at each step is not defined here but should nevertheless be an integral of acomplete engineering procedure): (i) study the knowledge on the problem being tackled and problemscharacteristics; (ii) implement basic and advanced constructive and iterative improvement algorithms;(iii) improve step-by-step the performance of SLS algorithms by adding features from simple SLSmethods (examples could be simple tabu search or simulated annealing type moves); (iv) add morecomplex concepts from hybrid SLS techniques (such as perturbations, prohibitions, populations); (v)further fine-tune the current SLS algorithm. Clearly, such a procedure will very likely not be executedas a linear procedure, but one may need to iterative through these steps.

Note that an important aspect of such a procedure is that it adopts a component-wise view of SLSmethods. For example, the SLS method iterated local search (ILS) uses perturbations to diversify thesearch and acceptance tests (components: perturbations, acceptance tests); or evolutionary algorithmsoffer (among others) the usage of a population of solutions (component: population of solutions). Theabove procedure should suggest experimental designs or other types of criteria to guide users whichpotential components to consider. Hence, such an SLS algorithm engineering procedure can be seenas a methodology that guides a modular design for the generation of SLS algorithms from individ-ual algorithm components: add iteratively appropriate algorithmic components taken from different


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metaheuristics (or even from different techniques such as mathematical programming, as exemplifiedin [8]) if they contribute significantly to some aspect of algorithm performance. Hence, the fact thatthe resulting SLS algorithms are in some sense “hybrid” algorithms will be rather a normal result ofthe engineering procedure rather than a noteworthy special characteristic of the algorithm.

3.2 Tools

A definition of possible procedures and experimental designs to follow is just one aspect of SLS al-gorithms engineering. These procedures need to be supported by a number of tools that assist thedevelopment process. Useful existing tools for SLS engineering range from software frameworks (exam-ples are Paradiseo [http://paradiseo.gforge.inria.fr/], EasyLocal++ [http://tabu.diegm.uniud.it/EasyLocal/], HotFrame, over libraries of data types like LEDA [http://www.algorithmic-solutions.com/enleda.htm], to statistical tools such as the comprehensive, non-commercial package R (http://www.r-project.org/). While several of such tools are used, what is currently missing in the liter-ature is a systematic integration of such tools into a possible SLS algorithm engineering environment.

A particularly important class of tools are automatic tuning procedures [2, 3]. In fact, parametertuning tasks arise in many steps of the development of SLS algorithms and, ultimately, the design ofan SLS algorithm can itself be seen as a specific tuning task. This is the case since choices about,for example, whether to use a population of solutions or not can simply be seen as an instantiationof a specific categorical parameter. From SLS algorithms engineering’s perspective, tools for the auto-mated tuning of SLS algorithms will support the algorithm designer in many of the decisions that aretaken in the development of SLS algorithms (and in some occasions also take crucial design decisions).Therefore, one particularly important line of research in SLS algorithms engineering will be on thefurther development of such tuning tools and in their systematic integration into the SLS algorithmsengineering process.

3.3 Knowledge aspects

SLS

Appli-cations

ComputerScience

OperationsResearch

Statis-tics

SLS

Figure 1: SLS is an interdisciplinary researcharea, which is in the intersection of operationsresearch, computer science, statistics, and ap-plications.

Research in SLS algorithms is a strongly inter-disciplinary research, although actually the disciplinesafafected are usually not too far apart from each other.For the field of SLS algorithms many aspects of opera-tions research, computer science, artificial intelligence,and statistics play a significant role (see also Figure 1for a graphical illustration). Important aspects fromoperations research include knowledge about partic-ular optimization problems, the use of methods andtechniques developed in mathematical programming,and the knowledge of the importance of modeling prob-lems in an appropriate way. Relevant aspects of com-puter science stem from the areas algorithmics (and, inparticular, algorithm engineering), software engineer-ing, and parallelization. Relevant subareas of artifi-cial intelligence are search, machine learning (to en-hance the solution process), constraint programming,and knowledge-based techniques (such as decision sup-port systems). From statistics, of main importanceare techniques from descriptive statistics, from infer-ential statistics such as hypothesis testing and experi-mental design and analysis, as well as more advanceddata analysis techniques. In addition to these, detailed


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knowledge about the problem being tackled is crucial for the development of effective SLS algorithms.

In summary, besides the knowledge on all aspects of SLS algorithms, knowledge in a wide varietyof related areas can be of crucial importance for SLS algorithms developers. This fact can also betranslated into another goal for SLS algorithms engineering, namely, to raise the awareness of theinterdisciplinarity of the area and to identify important knowledge of which researchers in the fieldshould be aware. Ultimately, this may lead to the definition of a curriculum for SLS algorithmsengineering.

3.4 Interplay between theory and practice

Despite the fact that we talk about an engineering methodology, it should be stressed that in the areaof SLS also pertinent scientific questions arise on theoretical guarantees, the role of randomness in SLSalgorithms, and the examination of the relationship between problem characteristics, algorithm com-ponents, and performance. It should be clear that there are synergies between more theoretical aspectsand SLS engineering : Insights gained from scientific studies on SLS algorithms may inform an SLSengineering process and lead to more principled decisions in SLS design. (Note that an SLS algorithmsengineering process can be defined and useful without actually having a deep, insightful understandingof, for example, the interplay between problem characteristics and algorithm components; however,it is clear that such an understanding would help and open new possibilities for an SLS engineeringprocess.)

3.5 Cross-sectorial aspect

Finally, it is to be stressed that SLS algorithms are cross-sectorial techniques that are applied fortackling computationally hard problems in many different fields ranging from all natural sciences toimportant applications in engineering and business. Hence, any advancements in the methodologicalaspects of the design, development and experimental testing of SLS algorithms will have significantrepercussions in these application areas.

Another aspect to be mentioned here is on the general applicability of SLS algorithms. The situationin SLS algorithms is very different from, for example, integer programming, where very powerful general-purpose (commercial) packages exist. The area of SLS algorithms is largely lacking such general purposepackages, an exception to some extent being Comet [http://www.comet-online.org/]). Probably a middleway will be the most effective to increase the applicability of SLS technology. One may envision systemsthat provide some general modeling language for specific application domains (such as vehicle routingor scheduling) and develop general SLS solvers able to tackle efficiently problems that can be expressedwithin the defined modeling language. In fact, various software packages for vehicle routing are actuallyalready following such a direction.

3.6 Structuring aspects

The establishment of an SLS algorithm engineering discipline has also the positive side-aspect thatit would give an orientation to the research in the SLS field, which at the moment appears to berather scattered into different directions. The main areas of interest for SLS engineering could be (i)methodological developments on the engineering aspects, (ii) systematic, in depth experimental studiesand case studies to inform an SLS algorithm engineering methodology, (iii) the development of newtools for SLS engineering and their integration into an engineering process, (iv) the development ofnew algorithmic techniques to widen the possible design choices available, (v) theoretical insights andadvances to inform the engineering process, etc.


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3.7 SLS Algorithm Engineering Workshops

In 2007, the first workshop on Engineering Stochastic Local Search Algorithms — Designing, Imple-menting and Analyzing Effective Heuristics was held, organized by Thomas Stutzle, Mauro Birattari,and Holger H. Hoos [10]; the second edition of this workshop will be held in September 2009, againin Brussels, Belgium; for details see http://iridia.ulb.ac.be/sls2009/. The aim of this workshopseries is foremost to serve as a forum for researchers who are interested in the establishment of ap-proaches towards the engineering of SLS algorithms. We hope that this workshop series can help tomake progress in the integration of relevant aspects of SLS research into a more coherent engineeringmethodology and to foster the exchange of ideas of researchers who work in various fields relevant foran SLS algorithms engineering methodology.

Acknowledgments

The thoughts expressed in this extended abstract have profited from discussions with Holger H. Hoos,Mauro Birattari, and Marco Dorigo on this topic. This work was supported by the META-X project, anAction de Recherche Concertee funded by the Scientific Research Directorate of the French Communityof Belgium. Thomas Stutzle acknowledges support from the F.R.S-FNRS of the French Community ofBelgium of which he is a Research Associate.

References

[1] E. H. L. Aarts and J. K. Lenstra, editors. Local Search in Combinatorial Optimization. John Wiley& Sons, Chichester, UK, 1997.

[2] T. Bartz-Beielstein. Experimental Research in Evolutionary Computation. Springer Verlag, 2006.

[3] M. Birattari, T. Stutzle, L. Paquete, and K. Varrentrapp. A racing algorithm for configuringmetaheuristics. In W. B. Langdon et al., editors, Proceedings of the Genetic and EvolutionaryComputation Conference (GECCO-2002), pages 11–18. Morgan Kaufmann Publishers, San Fran-cisco, CA, USA, 2002.

[4] C. Demetrescu, I. Finocchi, and G. F. Italiano. Algorithm engineering. Bulletin of the EATCS,79:48–63, 2003.

[5] F. Glover and G. Kochenberger, editors. Handbook of Metaheuristics. Kluwer Academic Publishers,Norwell, MA, USA, 2002.

[6] J. N. Hooker. Needed: An empirical science of algorithms. Operations Research, 42(2):201–212,1994.

[7] H. H. Hoos and T. Stutzle. Stochastic Local Search—Foundations and Applications. MorganKaufmann Publishers, San Francisco, CA, USA, 2005.

[8] V. Maniezzo, T. Stutzle, and S. Voß, editors. Matheuristics: Hybridizing Metaheuristics andMathematical Programming. Operations Research / Computer Science Interfaces. Springer Verlag,New York, 2009.

[9] I. Sommerville, editor. Software Engineering. Addison Wesley, 7 edition, 2004.

[10] T. Stutzle, M. Birattari, and H. H. Hoos, editors. Engineering Stochastic Local Search Algorithms,International Workshop, SLS 2007, volume 4638 of Lecture Notes in Computer Science. SpringerVerlag, Berlin, Germany, 2007.


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Applications of meta-heuristics

to traffic engineering in IP networks

Bernard Fortz ∗

Universite Libre de Bruxelles, Faculte des Sciences, Departement d’InformatiqueCP 210/01, Bld du Triomphe, 1050 Brussels, Belgium


Abstract

Intra-domain routing protocols are based on Shortest Path First (SPF) routing, where short-est paths are calculated between each pair of nodes (routers) using pre-assigned link weights, alsoreferred to as link metric. These link weights can be modified by network administrators in ac-cordance with the routing policies of the network operator. The operator’s objective is usually tominimize traffic congestion or minimize total routing cost subject to the traffic demands and theprotocol constraints. However, determining a link weights combination that best suits the networkoperator’s requirements is a difficult task.

This paper provides a survey of meta-heuristic approaches to traffic engineering, focusing onlocal search approaches and extensions to the basic problem taking into account changing demandsand robustness issues with respect to network failures.

1 Introduction

Provisioning an Internet Service Provider (ISP) backbone network for intra-domain IP traffic is a bigchallenge, particularly due to rapid growth of the network and user demands. At times, the networktopology and capacity may seem insufficient to meet the current demands. At the same time, there ismounting pressure for ISPs to provide Quality of Service (QoS) in terms of Service Level Agreements(SLAs) with customers, with loose guarantees on delay, loss, and throughput. All of these issues pointto the importance of traffic engineering, making more efficient use of existing network resources bytailoring routes to the prevailing traffic.

1.1 The general routing problem

Optimizing the use of existing network resources can be seen as a general routing problem defined asfollows. We are given a directed network G = (N,A) with a capacity ca for each a ∈ A, and a demandmatrix D that, for each pair (s, t) ∈ N × N , tells the demand D(s, t) in traffic flow between s andt. We sometimes refer to the non-zero entries of D as the demands. The set of arcs leaving a nodeu is denoted by δ+(u) := {(u, v) : (u, v) ∈ A} while the set of arcs entering a node u is denoted byδ−(u) := {(v, u) : (v, u) ∈ A}

∗The author has been supported by the Communaute francaise de Belgique - Actions de Recherche Concertees (ARC).


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With each arc a ∈ A, we associate a cost function Φa(la) of the load la, depending on how closethe load is to the capacity ca. We assume in the following that Φa is an strictly increasing and convexfunction. Our formal objective is to distribute the demanded flow so as to minimize the sum

Φ =∑

a∈AΦa(la)

of the resulting costs over all arcs. Usually, Φa increases rapidly as loads exceeds capacities, and ourobjective typically implies that we keep the max-utilization maxa∈A la/ca below 1, or at least below1.1, if at all possible.

In this general routing problem, there are no limitations to how we can distribute the flow betweenthe paths. With each pair (s, t) ∈ N ×N and each arc a ∈ A, we associate a variable f (s,t)

a telling howmuch of the traffic flow from s to t goes over a. Moreover, for each arc a ∈ A, variable la representsthe total load on arc a, i.e. the sum of the flows going over a. With these notation, the problem canbe formulated as the following multi-commodity flow problem.

min Φ =∑

a∈AΦa(la)

subject to

∑

a∈δ+(u)

f (s,t)a −

∑

a∈δ−(u)

f (s,t)a =

D(s,t) if u=s,

−D(s,t) if u=t,

0 otherwise,

u, s, t ∈ N, (1)

la =∑

(s,t)∈N×Nf (s,t)a a ∈ A, (2)

f (s,t)a ≥ 0 a ∈ A; s, t ∈ N. (3)

Constraints (1) are flow conservation constraints that ensure the desired traffic flow is routed froms to t, and constraints (2) define the load on each arc.

As Φ is a convex objective function and all constraints are linear, this problem can be solvedoptimally in polynomial time. We denote by ΦOPT the optimal solution of this general routing problem.

In our experiments, Φa are piecewise linear functions, with Φa(0) = 0 and derivative

Φ′a(l) =

1 for 0 ≤ l/ca < 1/3,3 for 1/3 ≤ l/ca < 2/3,

10 for 2/3 ≤ l/ca < 9/10,70 for 9/10 ≤ l/ca < 1,

500 for 1 ≤ l/ca < 11/10,5000 for 11/10 ≤ l/ca < ∞.

(4)

The function Φa is illustrated in Figure 1, and can be viewed as modeling retransmission delays causedby packet losses. Generally it is cheap to send flow over an arc with a small utilization la/ca. Thecost increases progressively as the utilization approaches 100%, and explodes when we go above 110%.With this cost function, the general routing problem becomes a linear program.

The objective function was chosen on the basis of discussions on costs with people close to the AT&TIP backbone. Motivations on our choice for the objective function and the different model assumptionsare discussed in detail in [17], and, for a closely related application, in [15]. A description of the generalinfrastructure behind this kind of traffic engineering is given in [13].


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Figure 1: Arc cost Φa(la) as a function of load la for arc capacity ca = 1.

1.2 The OSPF weight setting problem

The most commonly used intra-domain internet routing protocols today are shortest path protocolssuch as Open Shortest Path First (OSPF) [20]. OSPF does not support a free distribution of flowbetween source and destination as defined above in the general routing problem. In OSPF, the networkoperator assigns a weight wa to each link a ∈ A, and shortest paths from each router to each destinationare computed using these weights as lengths of the links. In practice, link weights are integer encodedon 16 bits, therefore they can take any value between 1 and 65,535. In each router, represented by anode of the graph, the next link on all shortest paths to all possible destinations is stored in a table.A flow arriving at the router is sent to its destination by splitting the flow between the links that areon the shortest paths to the destination. The splitting is done using pseudo-random methods leadingto an approximately even splitting. For simplicity, we assume that the splitting is exactly even (forAT&T’s WorldNet this simplification leads to reasonable estimates).

More precisely, given a set of weights (wa)a∈A, the length of a path is then the sum of its arc weights,and we have the extra condition that all flow leaving a node aimed at a given destination is evenlyspread over the first arcs on shortest paths to that destination. Therefore, for each source-destinationpair (s, t) ∈ N ×N and for each arc a ∈ δ+(u) for some node u ∈ N , we have that f (s,t)

a = 0 if a is noton a shortest path from s to t, and that f (s,t)

a = f(s,t)a′ if both a ∈ δ+(u) and a′ ∈ δ+(u) are on shortest

paths from s to t. Note that the routing of the demands is completely determined by the shortest pathswhich in turn are determined by the weights we assign to the arcs.

The quality of OSPF routing depends highly on the choice of weights. Nevertheless, as recommendedby Cisco (a major router vendor) [11], these are often just set inversely proportional to the capacitiesof the links, without taking any knowledge of the demand into account.

The OSPF weight setting problem is to set the weights so as to minimize the cost of the resultingrouting.

In Section 2, we survey meta-heuristics that have been proposed to solve the basic version of theproblem. Section 3 presents extensions taking into account changing demands and robustness issueswith respect to network failures.

2 Survey of meta-heuristics

Shortest path routing problems are NP-hard. Direct formulations are extremely hard to solve, andinteger programming approaches can typically solve only small to medium size problems. Moreover,


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in an operational setting, additional constraints can appear that are difficult to integrate in a mixed-integer programming formulations. Therefore, for large networks instances, heuristics can be necessaryto find good feasible solutions in a limited computing time. In Section 2.1, we present a local searchapproach to the problem. Section 2.2 examines further some issues that need to be considered in orderto obtain an effective heuristic, and Section 2.3 presents a method for finding good starting solutions.In Section 2.4, other approaches are briefly discussed.

2.1 A local search heuristic

One of the first heuristic approaches to the OSPF weight setting problem is a local search approachdeveloped by Fortz and Thorup [14, 17]. Recently, a similar implementation has been made availablein the opensource TOTEM toolbox [19].

In OSPF routing, for each arc a ∈ A, we have to choose a weight wa. These weights uniquelydetermine the shortest paths, the routing of traffic flow, the loads on the arcs, and finally, the value ofthe cost function Φ.

Suppose that we want to minimize a function f over a set X of feasible solutions. Local searchtechniques are iterative procedures that for each iteration define a neighborhood N (x) ⊆ X for thecurrent solution x ∈ X, and then choose the next solution x′ from this neighborhood. Often we wantthe neighbor x′ ∈ N (x) to improve on f in the sense that f(x′) < f(x).

In the remainder of this section, we first describe the neighborhood structure we apply to solvethe weight setting problem. Second, using hashing tables, we address the problem of avoiding cycling.These hashing tables are also used to avoid repetitions in the neighborhood exploration. While theneighborhood search aims at intensifying the search in a promising region, it is often of great practicalimportance to search a new region when the neighborhood search fails to improve the best solution fora while. These techniques are called search diversification. We refer the reader to [17] for a descriptionof the diversification techniques we use.

A solution of the weight setting problem is completely characterized by its vector w = (wa)a∈A ofweights, where wa ∈ W , the set of possible weights. We define a neighbor w′ ∈ N (w) of w by one ofthe two following operations applied to w.

Single weight change. This simple modification consists in changing a single weight in w. We definea neighbor w′ of w for each arc a ∈ A and for each possible weight t ∈ W\{wa} by settingw′(a) = t and w′(b) = wb for all b 6= a.

Evenly balancing flows. Assuming that the cost function Φa for an arc a ∈ A is increasing andconvex, meaning that we want to avoid highly congested arcs, we want to split the flow as evenlyas possible between different arcs.

More precisely, consider a demand node t such that∑s∈N D(s, t) > 0 and some part of the

demand going to t goes through a given node u. Intuitively, we would like OSPF routing to splitthe flow to t going through u evenly along arcs leaving u. This is the case if every arc in δ+(u)belongs to a shortest path from u to t. More precisely, if δ+(u) = {ai : 1 ≤ i ≤ p}, and if Pi isone of the shortest paths from the tail of ai to t, for i = 1, . . . , p, then we want to set w′ such that

w′ai+ w′(Pi) = w′aj

+ w′(Pj) 1 ≤ i, j ≤ p,

where w′(Pi) denotes the sum of the weights of the arcs belonging to Pi. A simple way of achievingthis goal is to set

w′(a) ={w∗ − w(Pi) if a = ai, for i = 1, . . . , p,wa otherwise.

where w∗ = 1 + maxi=1,...,p{w(Pi)}.


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A drawback of this approach is that an arc that does not belong to one of the shortest pathsfrom u to t may already be congested, and the modifications of weights we propose will sendmore flow on this congested arc, an obviously undesirable feature. We therefore decided to chooseat random a threshold ratio θ between 0.25 and 1, and we only modify weights for arcs in themaximal subset B of δ+(u) such that

wai+ w(Pi) ≤ waj

+ w(Pj) ∀i : ai ∈ B, j : aj /∈ B,lwa ≤ θ ca ∀a ∈ B,

where lwa denotes the load on a resulting from weight vector w. The last relation implies that theutilization of an arc a ∈ B resulting from the weight vector w is less than or equal to θ, so thatwe can avoid sending flow on already congested arcs. In this way, flow leaving u towards t canonly change for arcs in B, and choosing θ at random allows to diversify the search.

This choice of B does not ensure that weights remain below wmax. This can be done by addingthe condition maxi:ai∈B w(Pi)−mini:ai∈B w(Pi) ≤ wmax when choosing B.

The simplest local search heuristic is the descent method that, at each iteration, selects the bestelement in the neighborhood and stops when this element does not improve the objective function.This approach leads to a local minimum that is often far from the optimal solution of the problem, andheuristics allowing non-improving moves have been considered. Unfortunately, non-improving movescan lead to cycling, and one must provide mechanisms to avoid it.

Our choice was to use hashing: hash functions compress solutions into single integer values, sendingdifferent solutions into the same integer with small probability. We use a boolean table T to record if avalue produced by the hash function h() has been encountered. At the beginning of the algorithm, allentries in T are set to false. If w is the solution produced at a given iteration, we set T (h(w)) to true,and, while searching the neighborhood, we reject any solution w′ such that T (h(w′)) is true. Checkingthat a solution has been encountered is therefore performed in constant time. The hash function weused is described in [17].

2.2 Effectiveness issues

To evaluate the cost of a solution represented as a set of weights, we have to compute the shortestpaths for all origin-destination pairs, then to send the flows along the shortest paths according to theECMP splitting rule. This could be a bottleneck in the search for good solutions as computing thiscost function from scratch is computationally expensive.

To overcome that difficulty, we can apply a fast algorithmic approach to compute the flows. Weuse a two-step algorithm based on the shortest path computation. In the first step we compute all theshortest paths with respect to the current weights for all node pairs, and then, in the second step, werecursively assign flows to the paths computed in the first phase (see [17] or Algorithm 7.1 in [23]).

In most heuristic approaches, the number of changes in the shortest paths graph and in the flowsis very small between neighboring solutions. Hence, using fast updates of shortest paths and flows iscrucial to make heuristics effective. We now briefly review these approaches.

With respect to shortest paths, this idea is already well studied [24], and we can apply their algorithmdirectly. Their basic result is that, for the re-computation, we only spend time proportional to thenumber of arcs incident to nodes s whose distance to t changes. In typical experiments there wereonly very few changes, so the gain is substantial - in the order of factor 15 for a 100 node graph. Animproved algorithm was recently proposed by Buriol et al [10].

To update the flows, a similar approach, described in [17], can be used. Experiments reported inthat paper show that using dynamic updates of shortest paths and flows make the algorithm from 5up to 25 times faster, with an average of 15 times faster.


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2.3 Improvements using column generation

The general routing problem introduced in Section 1.1 provides a good lower bound for the IGP weightsetting problem in practice, as shown in [17]. This general routing problem can also be formulated withpath flow variables, and solved with column generation.

The main advantage of this approach is that dual variables used in the pricing problem are naturalcandidates as link weights. Umit and Fortz [25] managed to significantly improve the results obtainedby the heuristic by warm-starting the local search with the best solution obtained with dual variablesof the multi-commodity flow relaxation of the problem.

2.4 Other heuristic approaches

Ericsson et al [12] have proposed a genetic algorithm for the same problem. Solutions are naturallyrepresented as vectors of weights, and the crossover procedure used is random keys, that was firstproposed by Bean [4]. To cross and combine two parent solutions p1 (elite) and p2 (non-elite), firstgenerate a random vector r of real numbers between 0 and 1. Let K be a cutoff real number between0.5 and 1, which will determine if a gene is inherited from p1 or p2. A child c is generated as follows:for all genes i, if r[i] < K, set c[i] = p1[i] otherwise set c[i] = p2[i]. They also implemented a mutationoperator that randomly mutates a single weight.

This approach was improved by Buriol et al [9]. They added a local search procedure after thecrossover to improve the population. This hybrid approach, combined with dynamic updates of shortestpaths and flows, lead to results competitive in quality to the local search of Fortz and Thorup, with aslightly faster convergence. Another application of genetic algorithms to SPR design can be found in[22].

A simulated annealing approach was proposed by Ben-Ameur [6] for the single path routing case.Another line of heuristic approaches comes from using Lagrangean relaxations of the MIP models (seee.g. [8] for single path routing and [18] for ECMP routing).

3 Extensions

The basic traffic engineering problem is very important, but usually does not meet the requirements ofnetwork operators as changing demands or network failures are not taken into account. We now discusssome extensions to the basic problem. In Section 3.1, we consider fast re-optimization with a fewweight changes. Then, in Section 3.2 we consider the simultaneous optimization over multiple demandmatrices. This comes as a building block for heuristics dealing with failure scenarios (Section 3.3) aswell as for polyhedral demand uncertainty (Section 3.4).

3.1 Re-optimization with few weight changes

The problem of optimizing OSPF weights was mostly studied with a given fixed set of demands and afixed network topology.

However, demand matrices and networks change. In case of changes that degrade performance toomuch, most network operators do not like to make many weight changes, for two basic reasons:

(i) Weights are typically not changed centrally, so changing a lot of weights may require substantialnetwork management overhead, and creates the risk of a big protocol overhead.


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(ii) Network operators are often not comfortable with many weight changes. There may be manyaspects to good routing beyond the simple objectives presented in this paper. Therefore majorweight changes are not welcomed when a first weight setting giving a satisfactory routing hasbeen established.

Therefore, we want to make as few weight changes as possible. The local search from [17] workswith a single solution that is iteratively improved by small changes in the current solution. It typicallyperforms a lot of iterations (5000 in our practical experiments), and therefore produces a solutioncompletely different from the starting one. This can be seen as a depth-first search in the solutionspace, but since we want to make as few changes as possible, our approach should rather be a breadthfirst search.

Another heuristic was proposed in [15] for improving an input weight setting w0 with as few weightchanges as possible. It works as follows: first we consider about 1000 single weight changes to w0,corresponding to about 5 weight changes for each arc in our largest networks. The number of weightchanges considered is limited by applying random sampling to the neighborhood structure, as exploringthe full neighborhood is too time-consuming. Instead of selecting only the best weight change as inclassical local search heuristics, we keep the 100 best weight changes in a family F of “best weightsettings”. The process is iterated with F instead of w0: we consider 1000 single weight changes for eachweight setting in F and a new F is selected containing the 100 best of the old weight settings in F .After i iterations, including the start from w0, the family consists of weight settings with up to i weightchanges from w0. The size of F corresponds to the breadth of our search. All the above numbers arejust parameters that experimentally were found to give a good compromise between quality of solutionand time.

This technique for few changes has, to our knowledge, not been used before. Its main interest isthat it provides a general framework for optimizing with few changes that can be easily adapted forother applications. It has the advantage that if a local search heuristic is available, the main ingredientssuch as the changes applied to one solution to get a neighbor of it, or the procedures to evaluate a newsolution, can be reused in this new framework, saving a lot of implementation work.

3.2 Multiple demand matrices

Our motivation for working with multiple demand matrices is the general experience from AT&T thattraffic follows quite regular periods with a peak in the day and in the evening. The network operatorsdo not want to change weights on a regular basis so we want just one weight setting which is good forthe whole period. We then collect a peak demand matrix for the day and one for the evening. A weightsetting performing good on both performs good on all convex combinations, and hence it has a goodchance of performing well for the whole period. This approach was also introduced in [15].

Given a network G = (N,A, c) with several demand matrices D1, ..., Dk, we want to find a singleweight setting w := (wa)a∈A which works well for all of them. In general, we will use à(G,D,w) todenote the load on link a with network G, demand matrix D, and weight setting w. Similarly for ourcost function, we have Φ(G,D,w) =

∑a Φa(à(G,D,w)) with Φa as defined in (4).

Now consider a demand matrix D dominated by a convex combination of D1, ..., Dk, that is D ≤α1D1 + · · ·+ αkDk where α1 + · · ·+ αk = 1. Here everything is understood to be entry-wise, so for allx, y, D[x, y] ≤ α1D1[x, y] + · · ·+ αkDk[x, y].

Since the routing for each source-destination pair is fixed by the weight setting w, for each arca ∈ A, à(G,D,w) ≤ α1à(G,D1, w) + · · · + αkà(G,Dk, w). In particular, it follows that the max-utilization for D is no worse that the worst max-utilization for the Di. Furthermore, since each arccost function Φa is convex, Φa(à(G,D,w)) ≤ α1Φa(à(G,D1, w)) + · · · + αkΦa(à(G,Dk, w)), andhence Φ(G,D,w) ≤ α1Φ(G,D1, w) + · · · + αkΦ(G,Dk, w). Thus, our weight setting w does no worse


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for D than for the worst of the Di, neither with respect to our cost function Φ, nor with respect tomax-utilization. Note that the same observation holds true with MPLS, as long as the routing for eachsource-destination pair is fixed.

To optimize simultaneously for several demand matrices D1, ..., Dk, we simply modify the localsearch heuristic to minimize

Φ(G,D1, ..., Dk, w) =∑

i≤kΦ(G,Di, w) (5)

As in our original motivation for defining Φ, this has the the effect of penalizing highly loaded links,this time, for all the demand matrices instead of just one.

3.3 Robust optimization for single link failures

The heuristic of Section 2.1 has been designed to provide a weight setting for a single demand matrix anda fixed network. We now consider the possibility of link failures. The approach below was proposed in[16]. We define a state of the network as a subset S ⊆ A of arcs containing the arcs that are operational.The states considered here are the normal state A and the single link failure states A\{a} for each linka ∈ A. In SPF protocols, the network operator assigns a weight to each link, and shortest paths fromeach router to each destination are computed using these weights as lengths of the links. These shortestpaths are updated each time the network state changes. Given a network state S, a weight setting wand a vector of arc capacities c, we denote by Φ(S,w, c) the cost of the routing obtained with weightsetting w in state S of the network, using the piecewise linear cost function described in Section 1.1.

We suppose that once the operator has fixed the set of OSPF weights, he does not want to changeit whatever the state of the network is. The possibility of getting a better routing in case of failures byallowing a few weight changes has been studied in [15].

The cost function we use has been designed in such a way that it tries to keep the flow on each linkbelow the capacity of that link. This is an objective we want to maintain for each link failure. In thenormal state, however, the operator usually wants the flows to remain much more below the capacity,in order to be more robust in cases of increasing demand and to ensure capacity will be available toperform the rerouting in case of failure. Let α be the maximal ratio of the capacity the network operatorwants to use in the normal state. In our experiments, we assumed α = 0.6. Therefore, in the normalstate, the operator wants w to minimize Φ(A,w, αc). We suppose here that the operator gives an equalimportance to the quality of the routing in the normal state and to its robustness (i.e. the quality ofthe routing in all the single link failure states). Moreover, we assume all the link failures have the sameimportance, but it is trivial to extend our results by giving a weight to each link failure.

Putting it all together, we want to find a weight setting w∗ that solves

minw

Ψ(w) :=12

(Φ(A,w, αc) +

1m

∑

a∈AΦ(A \ {a}, w, c)

)(6)

where c is the capacity vector and m := |A| the number of links in the network. A brute force approachwould be to use the heuristic presented in Section 2.1 to optimize Ψ(w). However, this would requirethe evaluation of all the m + 1 scenarios for each weight setting encountered, therefore increasing thecomputing time by a factor m.

Our approach to reduce the computing time is the following. We hope that only a few link failureswill be representative of “bad cases” and will contribute for a large part of the total cost in (6). Ateach iteration, we maintain a list of critical links C and only evaluate the cost function restricted to thecorresponding states, i.e. we evaluate each weight setting w in the neighborhood of the current iterate


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with the cost function

Ψ(w, C) :=12

(Φ(A,w, αc) +

1|C|∑

a∈CΦ(A \ {a}, w, c)

).

The heuristic starts with C = ∅. It is essentially the same as before, adapted to optimize Ψ(w, C),with the addition that C is updated every T iterations. We update C as follows. Let u(a,w) be themaximum utilization with weight setting w when arc a has failed, i.e.

u(a,w) = maxb∈A\{a}

l(b, w, a)cb

,

where l(b, w, a) denotes the load on arc b with weight setting w when a has failed, and let u be theaverage maximum utilization over all scenarios in the critical set, i.e. u := 1

|C|∑a∈C u(a,w). We first

choose the arc a not in the critical set that maximizes u(a,w). If u(a,w) > u, we add a to C. Moreover,we want to keep C of small size. To this end, we fix a maximal size K and we remove the arc thatminimizes u(a,w) over C from the critical set each time its size exceeds K.

3.4 Polyhedral demand uncertainty

For a given network, the traditional routing problem deals with selecting paths to transfer a ‘given’ setof demands from their origins to destinations. In this general definition, there is no restriction on thestructure of the paths to be used, and it is assumed that the amounts of traffic between all origin anddestination pairs are already known. However, several restrictions are imposed on the path structurein telecommunication networks, and designing a reliable network using a single demand matrix strainscredibility as the network size and the service variety increase in the contemporary business world. Itis not likely to anticipate fluctuations in demand expectations without overestimations, which wouldlead to the waste of network resources or a high service cost. A well-known online approach to handlesuch shifts is to update routes adaptively as some changes are observed. However, the additionalbenefits of these methods are not for free since excessive modifications might ruin the consistency anddependability of network operations. At this point, off-line methods based on optimizing over a set oftraffic matrices have started to win adherents [1, 3, 5, 7, 21].

The general method is to use either a discrete set and hence a scenario-based optimization or apolyhedral set defined by network characteristics. Then the motivation is to determine the routingwhose worst case performance for any feasible realization in this set is the best. Such a routing is calledoblivious since it is determined irrespective of a specific demand matrix. Applegate and Cohen [3]discuss the general routing problem with almost no information on traffic demands. Later, Belotti andPinar [5] incorporate box model of uncertainty as well as statistical uncertainty into the same problem.Ben-Ameur and Kerivin [7] study the minimum cost general oblivious routing problem under polyhedraldemand uncertainty and use an algorithm based on iterative path and constraint generation as a solutiontool. Mulyana and Killat [21] deal with the OSPF routing problem, where traffic uncertainty is describedby a set of outbound constraints. Finally, Altın et al. [1] study polyhedral demand uncertainty withOSPF routing under weight management and provide a compact MIP formulation and a Branch-and-Price algorithm.

In this section, we discuss oblivious OSPF routing with weight management and polyhedral de-mands, and describe the solution strategie proposed in Altın et al. [2]. Optimizing weights wouldenable traffic engineering with OSPF since link metric is the only tool we can employ to manipulateroutes so as to make OSPF more comparable to other flexible protocols like MPLS. Moreover, polyhe-dral demands make the problem more practically defensible by ensuring a design robust under a rangeof applicable shifts in traffic demand.

Let us relax the assumption of a fixed TM d and consider a polyhedron D of feasible demandrealizations. Now, the concern is to determine paths such that any d ∈ D can be accommodated


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efficiently. This means that our ‘optimal’ routing will have the best worst case performance for Dindependent of a specific TM. The main impact of such a shift will be a change in the definition of loadon each link a ∈ A. It is now defined as a function of d, which can be any vector in D. Consequently,for the polyhedral case, the load definition becomes

la ≥∑

(s,t)∈Qdstf

sta d ∈ D, a ∈ A (7)

where D is an arbitrary polyhedron. However, this change leads to a semi-infinite optimization problem.We can eliminate this difficulty by using a duality transformation. Details can be found in [2].

We now discuss our algorithmic approach to tackle polyhedral demands. It has two main steps,namely the TM enumeration and weight optimization. We use IGP-WO, the TOTEM weight optimizer,for the latter step, whereas we use mathematical programming for the first part. As the representationtheorem for polytopes suggests, any TM d ∈ D can be represented as a convex combination of theextreme points of D. Hence, we could equally write the link load constraint (7) for each extreme pointof D, which are in finite but exponential number.

For a given routing f , the motivation in the TM generation step is to enumerate the extreme pointsof D which correspond to the ‘most challenging’ traffic demands in terms of the arc utilization or therouting cost. Since D is a polyhedral set, the algorithm will terminate after a finite number of iterations.Besides, the greedy choice of extreme points would lead to much fewer iterations before termination.We provide the pseudo codes of two different strategies in Algorithm 1 and Algorithm 2, respectively.

The first step INITIALIZE and the final step CHALLENGE are common for the two strategies.To start, we need an initial d0 ∈ D.

Let D = {d ∈ R|Q| : Ad ≤ α, d ≥ 0} be the polytope of feasible TM s with A ∈ RK|Q| and α ∈ RK .Then (7) implies that

la ≥ maxd∈D

∑

(s,t)∈Qdstf

sta a ∈ A. (8)

To create d0, in INITIALIZE, we solve maximization problem (8) for an arbitrary arc a ∈ A bysetting fsta = α for all commodities (s, t) ∈ Q, where α can be any positive constant. Then we createD to hold all the TM s that we generate throughout the algorithm. On the other hand, the aim of thestep CHALLENGE is to determine a ‘challenge’ case to compare the routing cost and the maximumlink utilization of the two routings. For this purpose, we take dmax = argmaxd∈D

∑(s,t)∈Q dst as our

challenge TM. Notice that we choose dmax independent of any performance measure or any topologicalinformation. At this stage, we are only interested in the TM that requires the utmost use of networkresources. Although we could use some other criteria at this stage, we believe the current choice is fairenough since we will use dmax for comparison. Between INITIALIZE and CHALLENGE, there is theMAIN step, where the two strategies differentiate.

The first strategy is based on the greedy search of a new feasible TM based on total routing cost.Algorithm 1 outlines this strategy, which we call CM in the rest of the paper. At iteration cnt ofCM, we have an OSPF routing g∗ with the minimum average routing cost ΦD for the TM s in D. Thequestion we want to answer is: Does there exist another demand d ∈ D \ D that costs more than ΦD,


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if we route it using g∗? To tackle this question, we first solve the following MIP model (PMaxCost):

max∑

a∈Aφa

s.t. φa − uzla ≤M(1− yza)− vzca a ∈ A, z ∈ Z∑

z∈Zyza = 1 a ∈ A

la −∑

(s,t)∈Qdstg

st∗a = 0 a ∈ A

∑

(s,t)∈Qastk dst ≤ αk k = 1, ..,K (9)

φa, la ≥ 0 a ∈ Ayza ∈ {0, 1} a ∈ A, z ∈ Zdst ≥ 0 (s, t) ∈ Q

where ya variables show the segment of the objective function that each φa lies in and (9) ensures thatwe obtain a feasible TM dnew ∈ D. Notice that g∗ is not a variable anymore in PMaxCost. Thus, thelink load la is defined as a linear function of the demand variables d ∈ D with coefficients gst

∗a obtained

in the most recent TABU iteration. In consequence, the solution of PMaxCost will be the worst caseTM dnew leading to the highest routing cost

∑a∈A φ

∗a for g∗.

Since D is nonempty, PMaxCost will always yield a feasible TM dnew. However, there is no guaranteethat we will get a new dnew /∈ D at each iteration since (9) ensures dnew ∈ D but not dnew ∈ D \ D.To shun fake updates, we keep track of all matrices in D using a hashing table. This is similar to whatwe use to avoid cycling in the tabu search algorithm for optimizing link weights. In brief, we use ahashing function to map each dnew to an integer hdnew and we mark its generation in the hdnew entryof a boolean table. Each time we solve PMaxCost, we decide whether or not we should update D usingthe boolean table and continue with the next iteration only if we have a new TM dnew /∈ D for whichthe routing cost

∑a∈A φ

∗a is higher than the current average cost ΦD.

On the other hand, the second strategy is greedy in the sense of traffic load on arcs. It uses linkutilization as the determining factor for new TM generation. Basically, given an OSPF routing g∗

optimal for D, it looks for a TM dnew, which makes some arc a ∈ A overloaded or increases the currentcongestion rate of the network, that is Umax = maxa∈A,d∈D

laca

. We use the hashing function that wehave described above to keep track of the TM s in D and avoid cycling. A framework of this strategyis provided in Algorithm 2. We will refer this strategy as LM from now on.

The main difference between CM and LM is the domain of the challenge. CM generates a demandmatrix d∗ that puts the network in a worse situation as a whole for a given routing configuration on thebasis of the total routing cost. On the contrary, in LM, the new TM is at least ‘locally’ challenging,since we consider the worst case for each arc individually. In both strategies, we enumerate at mostone TM at each iteration. However, we can modify Algorithm 2 easily to generate multiple TM s,namely at most one for each arc. Finally, each time the algorithm performs a tabu search, it startswith the optimal weight metric of the most recent iteration. This is useful to reduce the time spent forre-optimizing the weight metric in the TABU stage.

4 Conclusion

We have surveyed meta-heuristics for traffic engineering in the framework of intra-domain routing. Theavailable techniques cover the basic problem and some important extensions that seem sufficient forthe needs of most network operators. Moreover, experiments have shown that there would be little togain from switching to other routing protocols, if traffic engineering is the only concern.


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Algorithm 1 Strategy 1 with Cost Maximization - CMRequire: directed graph G = (V,A), traffic polytope D, link capacity vector c;Ensure: minimum cost OSPF routing f∗ and metric ω∗ for (G,D, c);

INITIALIZE:Find an initial feasible TM d0 ∈ D;drec ← d0; // drec : the most recently enumerated TM;D ← d0; // D : current set of TMs enumerated so far;NewTM ← TRUE;cnt = 0;

MAIN:while (cnt ≤ cnt−limit) and (NewTM = TRUE) do

TABU: Find an optimized oblivious OSPF routing g∗ for D and the associated metric ω∗T ;Get ΦD : the average routing cost for D;NewTM = FALSE;Solve PMaxCost to get

∑a∈A φ

∗a and dnew;

if∑a∈A φ

∗a > ΦD and dnew /∈ D then

D ← dnew;NewTM = TRUE;cnt← cnt+ 1;

f∗ ← g∗;ω∗ ← ω∗T ;CHALLENGE:

Find the challenge TM dmax = argmaxd∈D

∑(s,t)∈Q dst;

Get Φ∗dmax // the cost of routing dmax with f∗;Get U∗dmax // the congestion rate for dmax with f∗;

Algorithm 2 Strategy 2 with Arc Load Maximization - LMRequire: directed graph G = (V,A), traffic polytope D, link capacity vector c;Ensure: minimum cost OSPF routing f∗ and metric ω∗ for (G,D, c);

INITIALIZE // As in Algorithm 1MAIN:while (cnt ≤ cnt−limit) and (NewTM = TRUE) do

TABU: Find an optimized oblivious OSPF routing g∗ for D and the associated metric ω∗T ;Umax = maximum link utilization for drec;NewTM = FALSE;a = 0 // start with the first arc of G;while (a < |A|) and (NewTM = FALSE) dodnew = argmaxd∈D(g∗ad); // dnew : worst case TM for a with routing g∗;if (g∗ad

new > ca) or ( g∗ad

new

ca> Umax) then

if dnew /∈ D thendrec = dnew;D ← drec;NewTM = TRUE;cnt← cnt+ 1;

if NewTM = FALSE thena← a+ 1;

if NewTM = TRUE thencnt← cnt+ 1;

f∗ ← g∗;ω∗ ← ω∗T ;CHALLENGE //As in Algorithm 1


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Nevertheless, there is still room for improvement. In particular, understanding the interactionsbetween inter-domain and intra-domain routing and developing tools that allow for inter-domain orjoined inter/intra-domain optimisation remains a big challenge.

References

[1] A. Altın, P. Belotti, and M.C. Pinar. Ospf routing with optimal oblivious performance ratio underpolyhedral demand uncertainty. Technical report, Bilkent University, 2006.

[2] A. Altin, B. Fortz, and H. Umit. Oblivious OSPF routing with weight optimization under poly-hedral demand uncertainty. Technical Report 588, ULB Computer Science Department, 2008.Submitted.

[3] D. Applegate and E. Cohen. Making intra-domain routing robust to changing and uncertain trafficdemands: understanding fundamental tradeoffs. In SIGCOMM ’03: Proceedings of the 2003 con-ference on Applications, technologies, architectures, and protocols for computer communications,pages 313–324, New York, NY, USA, 2003. ACM.

[4] J.C. Bean. Genetic algorithms and random keys for sequencing and optimization. ORSA J. onComputing, 6:154–160, 1994.

[5] P. Belotti and M.C. Pinar. Optimal oblivious routing under statistical uncertainty. Optimizationand Engineering, 9(3):257–271, 2008.

[6] W. Ben-Ameur, E. Gourdin, B. Liau, and N. Michel. Optimizing administrative weights for efficientsingle-path routing. In Proceedings of Networks 2000, 2000.

[7] W. Ben-Ameur and H. Kerivin. Routing of uncertain demands. Optimization and Engineering,3:283–313, 2005.

[8] A. Bley. A Lagrangian approach for integrated network design and routing in IP networks. InProceedings of the 1st International Network Optimization Conference (INOC 2003), Paris, France,pages 107–113, 2003.

[9] L. S. Buriol, M. G. C. Resende, C. C. Ribeiro, and M. Thorup. A hybrid genetic algorithm for theweight setting problem in ospf/is-is routing. Networks, 46(1):36–56, 2005.

[10] L. S. Buriol, M. G. C. Resende, and M. Thorup. Speeding Up Dynamic Shortest-Path Algorithms.Informs Journal On Computing, 2007. To appear.

[11] Cisco. Configuring OSPF, 1997. Documentation at http://www.cisco.com/univercd/cc/td/doc/product/software/ios113ed/113ed cr/np1 c/1cospf.htm.

[12] M. Ericsson, M. G. C. Resende, and Pardalos P.M. A genetic algorithm for the weight settingproblem in ospf routing. J. of Combinatorial Optimization, 6:299–333, 2002.

[13] B. Fortz, J. Rexford, and M. Thorup. Traffic engineering with traditional IP routing protocols.IEEE Communications Magazine, 40(10):118–124, 2002.

[14] B. Fortz and M. Thorup. Internet traffic engineering by optimizing ospf weights. In Proceedingsof IEEE INFOCOM’00, pages 519–528, 2000.

[15] B. Fortz and M. Thorup. Optimizing OSPF/IS-IS weights in a changing world. IEEE Journal onSelected Areas in Communications, 20(4):756–767, 2002.

[16] B. Fortz and M. Thorup. Robust optimization of OSPF/IS-IS weights. In W. Ben-Ameur andA. Petrowski, editors, Proc. INOC 2003, pages 225–230, October 2003.


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[17] B. Fortz and M. Thorup. Increasing Internet capacity using local search. Computational Opti-mization and Applications, 29(1):13–48, 2004.

[18] K. Holmberg and D. Yuan. A Lagrangian heuristic based branch-and-bound approach for thecapacitated network design problem. Operations Research, 48:461–481, 2000.

[19] G. Leduc, H. Abrahamsson, S. Balon, S. Bessler, M. D’Arienzo, O. Delcourt, J. Domingo-Pascual,S. Cerav-Erbas, I. Gojmerac, X. Masip, A. Pescaph, B. Quoitin, S.F. Romano, E. Salvatori,F. Skivee, H.T. Tran, S. Uhlig, and H. Umit. An open source traffic engineering toolbox. ComputerCommunications,, 29(5):593–610, 2006.

[20] J. Moy. OSPF: Anatomy of an Internet Routing Protocol. Addison-Wesley, 1998.

[21] E. Mulyana and U Killat. Optimizing IP networks for uncertain demands using outbound trafficconstraints, to appear. In Proc. INOC 2005, pages 695–701, 2005.

[22] E. Mulyana and U. Killat. An alternative genetic algorithm to optimize ospf weights. In InternetTraffic Engineering and Traffic Management, 15th ITC Specialist Seminar, pages 186–192, July2002. Wurzburg, Germany.

[23] M. Pioro and D. Medhi. Routing, Flow, and Capacity Design in Communication and ComputerNetworks. Morgan Kaufman, 2004.

[24] G. Ramalingam and T. Reps. An incremental algorithm for a generalization of the shortest-pathproblem. Jounal of Algorithms, 21(2):267–305, 1996.

[25] H. Umit and B. Fortz. Fast heuristic techniques for intra-domain routing metric optimization. InProc. INOC 2007, April 2007.


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Abstracts


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Genetic Algorithm and Memetic Algorithm with Vocabulary

Building for the SONET Ring Assignment Problem

Ana Silva ∗ Eberton Marinho † Wagner Oliveira ∗ Dario Aloise ∗

∗ PEP, UFRNCampus Universitario Lagoa Nova, Natal/RN

Email: [email protected], [email protected], [email protected]

† DIMAp, UFRNCampus Universitario Lagoa Nova, Natal/RN


1 Introduction

Communication services have always had an important role in society, and they have become essentialwith the advent of globalization. If we examine the evolution of telecommunications network, we canobserve: the technology evolves, the network expands, the supply and the demand for new servicesand products increase; consequently, the problem of planning telecommunication network becomesincreasingly larger and complex, despite the technological development ([1] and [2]).

This article analyses the SONET Ring Assignment Problem or SRAP, which is a NP-hard [3]combinatorial optimization that arises during the planning stage of the physical network. The SRAPconsists in selecting a set of connections between locations (customers) satisfying a series of restrictionsat the lowest possible cost. Between these restrictions, the survivability of the network is essential.

The main motivations for the development of this work were: to propose new algorithms from evo-lutionary methods which work out a solution to the SRAP, and to test the technique named vocabularybuilding on a problem of telecommunications. Thus, two metaheuristics were developed in this work -genetic algorithm (GA), and memetic algorithm (MA) - and another hybridization - memetic algorithmwith vocabulary building (MA+VB).

The second part of this article is organized as follows: in Section 2, we show the SRAP Problem.In Section 3, we detail the genetic algorithm. The memetic algoritm and memetic algorithm withvocabulary building are detailed in Section 4. Section 5 presents experimental results. Finally, insection 6, we write the conclusions.

2 SONET Ring Assignment Problem

The SONET Ring Assignment Problem or SRAP rises during the designing of a physical network oftelecommunications and deals with the determination of subsets of localities which will form SONETrings (Synchronous Optical NETwork). A SONET ring topology is based on communication betweenmany local rings which are formed by localities from the network, and connected through a specialring named Federal Ring (FR). The localities are connected to local rings through a dispositive called


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Add-Drop Multiplexer (ADM) and another called Cross Connect System (DCS). DCS is responsible toprovide communication between rings of FR. The cost of a SONET ring topology can be very expensive,since depending on how many local rings the network has, the cost with DCSs becomes very high (anexpensive equipament). However, this topology has a great benefit that is to make sure the survivalof the network if any fail happens or rupture of an optical fiber cable. In this case, the multiplexersautomatically send the services affected through an alternative way.

The SRAP deals with n-localities of network distribution between the local rings in order to solvesome constraints: each locality has to belong to a unique local ring and the maximum capacity of eachring cannot be greater than a constraint B, which represents the band width of ADMs and DCSs. Theobjective is to obtain a configuration of localities that minimize the quantity of local rings forming theFederal Ring. The Figure 1 shows an example of SONET network.

Figure 1: Ring Topology of a SONET network.

The SRAP can be described as a graph partitioning problem. In this case, the vertices representclients of the network and the weights of edge show required traffic between them. Thus, the problemcan be formulated as follows:

Let G = (V,E) be a non directed complete graph and a B positive integer. Associated to each edge(u, v) ∈ E there is a non negative integer duv. A feasible solution for the problem is a partition of thevertexes in G into disjoint sets V1, V2,. . . , Vk, such that:

∑

u,v∈Vi,u<v

duv +∑

u∈Vi,v /∈Vi

duv ≤ B onde i = 1, 2, . . . , k (1)

k−1∑

i=1

k∑

j=i+1

∑

u∈Vi

∑

v∈Vj

duv ≤ B (2)

The constraint determined by equation 1 establishes the limit of B capacity to local rings of thenetwork, and the equation 2 imposes B as a maximum traffic of Federal Ring. The integer duv is thetraffic demand between localities u and v in both directions. If there is no demand between u and v,the value of duv is null.


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3 Genetic Algorithm

The proposed GA chromosomes are represented through a list of vectors. Each vector has a localityu ∈ V which was assigned to local ring Vi of solution S. An initial population was created based onRandom Edge-Based and Random Cut-Based heuristics developed by [4], and each one is responsibleby generation of half of individual. These are extensions of constructive Edge-Based and Cut-Basedheuristics introduced by [3]. The newest versions include a randomly factor in some steps to increase thediversification, making the processes more appropriated to the construction stage of initial populationof GA.

A solution is feasible if each rings, including the federal ring, were feasible, that is if the totaldemand of all rings is less or equal to B. A good fitness function needs to discern feasible solutionsfrom infeasible solutions. As a feasible solution is always better than an infeasible one, we added theFederal Ring demand to its fitness function as a punishment factor. In this way, let S be any solutionto SRAP, the fitness function of S is given by equation 3:

F (S) ={

B ×K(S) feasible solutionB ×K(S) + DemAF (S) infeasible solution (3)

Where K(S) and DemAF (S) denote, respectively, the number of local rings and demand of FR ofsolution S.

We used a Bin packing crossover or BPX [7]. Let two solutions Father1 and Father2, selectedthrough roulette method, and a local ring from Father1 randomly inserted in Father2. A probabilisticcomponent was added to the selection step of Father1. The larger number of local rings localities, thegreater the chance of selection. Since every local rings are feasible (characteristic of Random Edge-Based and Cut-Based heuristics), it is plausible to conclude that rings with many localities are goodrings to form a new solution. After the ring is inserted in Father2, the repeated localities are excluded.This makes possible to appear lost localities which were isolated in a ring. Then, we develop a localsearch to improve the solution, relocating each locality to another ring, if these rings are infeasible.

The operator of mutation was not used because it did not cause significant changes during testphase. The operator of elitism was used because it improved the results.

The Algorithm 1 shows the pseudo-code of GA proposed to SRAP. Note that the Algorithm 1was bult up based on Genetic Algorithm and adding the Local Search that characterizes the MemeticAlgorithm (lines 5 and 16 in Algorithm 1) and adding the Local Search that charaterizes the VocabularyBuilding approach (lines 4 and 15 in Algorithm 1).

4 Memetic Algorithm

This implementation differs from the former one just considering the addition of local search in itsprocess of optimization. Let S be a solution to the problem, a neighborhood search for N is a subsetof solutions N(S). When there is no solution in N(S) better than S, then it is said this is an localoptimum. As there is no certainty that the solutions of the initial population of GA are local optima, theapplication of a local search procedure is required for the process of improving the generated solutions.MA was applied to the proposed neighborhoods N1 and N3 according to Bastos [4].

The neighborhood N1 search for a vertex u belonging to a ring r and try to relocate it to anotherring t, so that the demand in the federal ring is reduced without making infeasible any local ring. Inother words, r and t are two rings of the solution S, and a location u ∈ r is moved to ring t if demTt∪{u}keeps less than or equal to B and demAF is reduced. The neighborhood N3 aims to empty a ring,


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specifically the smallest total demand S. The procedure tries to redistribute the localities in a ringr, among others. Therefore, it searches the better ring t to receive a vertex u ∈ r, so that t remainsviable.

4.1 Memetic Algorithm with Vocabulary Building

The technique of optimization so-called vocabulary building was developed by Fred Glover and may beconsidered as a variation of path relinking [5], once it also uses the concept of constructive neighborhoodsto generate new solutions from existing ones. However, the difference is the possibility of using partialsolutions together with complete solutions. The technique has this name by analogy with the processof formation of words.

The basic idea of this approach is firstly to identify good partial solutions (meaningful fragments ofsolutions which will be called fragments from now on) for being used in the search for global optimumlater. A good partial solution may be, for example, a configuration which is present in several elitesolutions. From these fragments, it is possible to reach more complex and useful combinations.

The implementation of the vocabulary building worked with the notion of contraction of vertices(in this case, locations) introduced by Guedes [8]. The author employs the idea proposed by Glover [6]that from a pattern identified in various solutions you can reach more complex and useful combinations.These are meaningful fragments of the best solutions which will be identified, and then condensed. Fromthese new nodes, an auxiliary graph is created to be used at some point in the process of solving theproblem. Figure 2 (A) shows an illustration of the identification process and condensation of locationsin a set of elite solutions for the formation of an auxiliary population formed by fragments. Theprocedure for local search using neighborhoods N1 and N3 on the solutions containing the fragmentsis illustrated in Figure 2 (B).

Figure 2: (A) Identification and generation of the fragments and (B) Neighborhoods on solutions withfragments.

The procedure 1 shows the behavior of the searching phase with VB. In step 2 the best solutionsare separated to form an elite population. In step 3, a procedure for identifying the locations thatappear together is run, that is in the same ring for all elite solutions. After that, groups of identifiedlocations are condensed; in other words, each group will have its locations embodied into a single node.These single nodes are the fragments. In step 4, one runs the local search on the solutions belonging toauxiliary population; it must be observed that a node of this population may be a fragment (numberof customers) or simply a location. In step 5 the process of condensation of the localities is reversedand the population is updated, and finally in step 6 the population is updated.

The Algorithm 1 shows the pseudo-code of Memetic Algorithm with Vocabulary Building to SRAP.


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Procedure 1: VB Phase1: procedure optimize populationVB2: pop elite ← find population elite(pop actual);3: pop aux ← contraction of vertices(pop elite); //Generation fragments4: optimize population(pop aux); //Local Search5: pop aux ← revert contraction of vertices(pop aux); //Unmake fragments6: pop actual ← (pop actual − pop elite) ∪ pop aux;7:end-procedure;

Algorithm 1: Memetic Algorithm with Vocabulary Building to SRAP1: procedure MemeticVBSRAP2: pop actual ← generation population initial();3: avaliar population(pop actual);4: optimize populationVB(pop actual); // LS with fragments VB Algorithm5: optimize population(pop actual); // LS Memetic Algorithm6: while (conditions of unmet stops) do7: new pop ← ∅;8: elitism(pop actual, new pop);9: while (new pop not full) do10: s1, s2 ← select parents(pop actual);11: s1’, s2’ ← crossover(s1, s2);12: new pop ← new pop ∪ {s1’} ∪ {s2’};13: end-while;14: evaluate population(pop actual);15: optimize populationVB(pop actual); // LS with fragments VB Algorithm16: optimize population(pop actual); // LS Memetic Algorithm17: end-while;18: return pop actual.better;19: end-procedure;

5 Results

The algorithms described were implemented in C++ language. The tests were performed on two classesof instances - C1 proposed by [3] and C2, by [9] - in a computer-type PC with processor Core2Duo of2.2 GHz, 4MB of cache, and 2 GB of RAM in the Linux Operating System Ubuntu 8.04.

To investigate the performance of the proposed algorithms, ten independent runs on one of eachinstance of classes C1 and C2 were carried out. The parameters used were the same for the three testedprocedures. The used criterion for stopping was that to find a viable solution to the number of ringsequal to the optimal solution for each instance. If not found, the algorithm ends after 150 iterations.Table 1 shows the percentage of optimal solutions (*) found in the three proposed algorithms and theirmeans of computational time (minutes:seconds:miliseconds) spent in executions.

Table 1: Overview

Instance GA MA MA+VBClass #Inst. *(%) Mtime *(%) Mtime *(%) MtimeC1 118 98,30 0:04:794 99,15 0:14:317 100 0:15:995C2 230 88,26 0:11:720 94,34 0:37:567 99,56 0:34:064


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6 Conclusions

In this article, two metaheuristics were presented - Genetic Algorithms (GA) and Memetics Algorithm(MA) - and a third hybrid metaheuristic - Memetic Algorithm with Vocabulary Building (MA+VB)to the SONET Ring Assignment Problem (SRAP).

Extensive computational experiments were performed on instances of classes C1 and C2. We foundout that the algorithm MA+VB had the largest number of instances solved optimally for both testedclasses followed by MA and GA, respectively.

References

[1] C. G. Omidyar and A. Aldridge. Introduction to SDH/SONET. Communications Magazine, 31:30–33, September 1993.

[2] O. J. Wansem and T. H. Wu and R. H. Cardwell. Survivable SONET networks - design metho-dologies. IEEE Journal on Selected Areas in Communications, 12:205-212, January 1994.

[3] O. Goldschmidt and A. Laugier and E. V. Olinick. SONET/SDH ring assignment with capacityconstraints. Discrete Applied Mathematics, 129:99-128, June 2003.

[4] L. O. Bastos. Solucoes Heurısticas para o Problema de Atribuicao de Localidades a Aneis em RedesSONET . Dissertacao (Mestrado em Computacao) – Programa de Pos-Graduacao em Computacao,UFF, 2005.

[5] F. Glover. New Ejection Chain and Alternating Path Methods for Traveling Salesman Problems.Computer Science and Operations Research, 449-509, 1992.

[6] F. Glover. Parametric Combinations of Local Job Shop Rules. Chapter IV, ONR Research Memo-randum no. 117, GSIA, Carnegie Mellon University, Pittsburgh, PA.

[7] E. Falknauer. Genetic Algorithms and Grouping Problems. John Wiley & Sons, 1998.

[8] A. B. C. Guedes and D. J. Aloise. Um Algoritmo Memetico Para O Problema Do Caixeiro ViajanteAssimetrico: Uma Abordagem Baseada em Vocabulary Building. In: XXXVIII Simposio Brasileirode Pesquisa Operacional, 2006, Goiania. Anais, 2006.

[9] R. Aringhieri and M. Dell’Amico. Comparing Metaheuristic Algorithms for Sonet Network DesignProblems. Journal of Heuristics, 11:35-57, 2005.


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A memetic algorithm for multi-objective integrated logistics

network design

Mir Saman Pishvaee ∗ Reza Zanjirani Farahani † ‡ Wout Dullaert § :.

∗ Department of Industrial Engineering, Faculty of Engineering, University of Tehran16th of Azar Street, Enghelab Ave., Tehran, Iran

Email: ms [email protected]

† Department of Industrial Engineering, Amirkabir University of Technology424 Hafez Ave., Tehran, IranEmail: [email protected]

‡ Department of Civil Engineering, National University of SingaporeNo.1 Engineering Drive 2, Singapore, China


§ Institute of Transport and Maritime Management AntwerpUniversity of Antwerp, Keizerstraat 64, 2000 Antwerp, Belgium


:. Antwerp Maritime AcademyNoordkasteel Oost 6, 2030 Antwerp, Belgium


1 Introduction

Reverse logistics network design includes determining the number of collection, recovery and disposalcenters needed, their location and capacities, buffer inventories and the product flow between the facil-ities. In many cases, logistics networks are designed for forward logistics activities without consideringthe reverse flow of return products and most of them are not equipped to handle return products inreverse channels [5]. The configuration of the reverse logistics network has, however, a strong influenceon the forward logistics network, such as transportation capacity. Due to the fact that designing theforward and reverse logistics separately leads to sub-optimal designs with respect to costs, service levelsand responsiveness, the design of the forward and reverse logistics networks should be integrated [6].

Apart from the necessity of integrating forward and reverse processes, real world problems ofteninvolve multiple objectives. Because of the increasing importance of network responsiveness in supplychain management, this has recently been considered as a significant additional objective in logisticsnetwork design [1, 7, 3]. Objectives such as network costs and network responsiveness are, however,typically conflicting and as a result, considering the multiple objectives concurrently is the most fa-vorable option for most decision makers. Previous research often limited itself to only considering asingle capacity level for each facility and often does not address how capacity levels can be determined[2]. Nevertheless, capacity levels are important decision variables in real-life applications due to theirstrong influence on logistics network efficiency and responsiveness.

To the best of our knowledge, this paper is the first to offer an integrated design of the forward and


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reverse supply chain, including capacity decisions to minimize network costs and maximize networkresponsiveness. To find the set of non-dominated solutions, an efficient multi-objective memetic algo-rithm is developed. The proposed solution algorithm uses a new dynamic search strategy by employingthree different local searches. To assess the quality of the novel solution approach, the quality of itsPareto-optimal solutions is compared to those generated by an existing powerful multi-objective geneticalgorithm from the recent literature.

2 Problem formulation and solution structure

The integrated forward/reverse logistics network (IFRLN) problem discussed in this paper is a multi-stage logistics network including production, distribution, customer zones, collection/inspection, re-covery and disposal centers with multi-level capacities. Because the IFRLN design problem includes acapacitated p-median facility location problem which is known to be NP-hard, metaheuristic approachesare appropriate for tackling real-life problems. Because the population-based nature of genetic algo-rithms is useful when exploring Pareto solutions [4], we develop a multi-objective memetic algorithm(MOMA) to enrich the approach with dynamic local search.

To encode the solution structure of the IFRLN problem, we adjust the priority-based encodingmethod of Gen et al. (2006) [4] to account for the more complicated structure of the supply chain andthe presence of different capacity levels for each facility.

As illustrated in Figure 1, a chromosome for the IFRLN model is presented as a 2 · (I + J + J +K + L + K + L + M + L + I + L) matrix, in which each segment is related to one echelon of theIFRLN problem. As an example, consider the first segment I-J , in which the first row contains theunique priority levels (from 1 to 5) assigned to each of the facilities in the segment (2 factories and 3distribution centers). The second row contains the capacity levels, implying that for the first plant,the first capacity level is chosen and for the second distribution center the third capacity level for thatparticular distribution center is selected.

Fifth segment Fourth segment Third segment Second segment First segment I-L M-L L-K J-K I-J

2 1 2 1 2 1 2 1 3 2 1 2 1 3 2 1 3 2 1 3 2 1 2 1 node

4 2 1 3 3 1 4 2 5 2 3 4 1 5 4 2 1 3 6 3 5 1 4 2priority

3212312312221 12321133121capacity

levels `

Fifth segment

Second segment First

segment

Third segment

Fourth segment

Figure 1: An illustration of the IFRLN model chromosome


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To generate a solution for the IFRLN problem, both the forward and the reverse loop need to bedetermined. To decode an IFRLN chromosome in the forward network, the second segment shouldbe decoded before the first segment. In other words, decoding the first segment is impossible beforethe second segment is decoded. Also in the reverse network, decoding the fourth and fifth segments isimpossible before the third segment is decoded. Each segment is decoded as follows: (1) For a givensegment, choose the cell the highest priority; (2) If the cell is a source, then choose depot with lowestcost; (3) If the cell is a depot, then choose the source with lowest cost; (4) the amount shipped ismin(available, requested); (5) Update remaining demand and capacities. The way in which the priorityand capacity levels are set for all facilities in the network is described in the next section.

3 MOMA algorithm

3.1 Initializing the population - selecting parents

In the proposed algorithm, the well-known roulette wheel selection method is used for selecting parentsfrom the old population, making the selection probability of chromosome proportional to its fitnessvalue. If pop size denotes population size and pareto size denotes the number of tentative Paretosolutions to preserve, then the inner loop in each iteration must repeat (pop size− pareto size) timesto generate sufficient individuals to complete the new population. The selection process is based onspinning the wheel 2 × (pop size − pareto size) times, each time selecting a single chromosome fromthe old population.

3.2 Evaluation and fitness

In the proposed multi-objective memetic algorithm (MOMA), the fitness evaluation of chromosomesfor survival is calculated by the random-weight approach of Murata et al. (1996)[8]. The advantageof this approach compared to the traditional weighted-sum approach is that it gives the algorithm atendency to demonstrate a variable search direction, enabling it to sample the solution space uniformlyover the entire frontier [4].

At the beginning of each evolution loop (generation), each objective function value is normalizedand a new set of random weights is specified, then the fitness value of each solution (chromosome) iscalculated. Based on the fitness values, a tentative set of Pareto solutions (selected among currentpopulation) is stored and passed on to new population for elite protection. The other individuals ofthe new population are generated through an evolutionary loop employing the crossover operator andlocal search operators.

3.3 Crossover

MOMA uses a segment-based crossover, randomly selecting the corresponding segments of parents withequal probability. Because production and recovery processes are performed in the same facilities, thevalue of the fifth position of the binary array must be equal to first one. Each time a crossover operatoris employed, two offspring solutions are generated but only one of them is selected for the local searchphase based on their fitness values.


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3.4 Dynamic local search

In the crossover phase, two children are created. The child with the best fitness value is selected forfurther improvement during the subsequent local search phase in which three local search methods areused: 3-opt, 2-opt and CA (capacity adjustment). The first two local search operators are employed onthe priorities (first row of the chromosome) and they include two stages. In the first stage, each segmentof the solution chromosome is randomly selected with equal chance by a binary array. In the secondstage, the values of 3 and 2 cells respectively from each selected priority segment will be exchanged bythe first (3-opt) or the second (2-opt) local search operator. The third local search method (CA) isapplied to the capacity levels (second row of the chromosome). In the first stage, cells are selected bya random binary array and subsequently new capacity levels are assigned to them randomly, based ona uniform distribution over possible capacity levels.

During the search, the probability of applying each of the three local searches is adjusted to controlthe diversification and intensification of the search. As the algorithm approaches the last iterations,the share of 3-opt local search is decreased in favor of the 2-opt and CA local searches being increased.This strategy is used to obtain a larger variety of solutions and a faster improvement path during thefirst iterations and to explore more carefully and in greater detail the obtained solutions at furtheriterations.

After applying a local search operator, a new neighborhood solution is obtained. If the fitness valueof this solution is better than the old solution, the old solution will be replaced by the new one. To makethe search more diversified and to escape from local optima during the local search, the probabilisticacceptance strategy is used if the new solution has a worse fitness value than the old solution. To usethis strategy in a more guided way, the probability of accepting a worse solution is gently decreasedduring the search as specified below. The individual that comes out of the local search stage will bedirectly added to new population.

The number of iterations is set to 12 + (I + J + J + K + L + K + M + L + I + L)/60, increasingthe number of iterations with problem size as expressed by the number of potential production andrecovery center locations I, potential distribution center locations J , the customer zones K, potentialcollection/inspection center locations L, potential disposal center locations M and the possible capacitylevels for the facilities N . Other programme parameters such as the number of local searches periteration, the probability of accepting a worse neighborhood solution cannot be discussed in detailwithin the scope of this abstract.

4 Computational results

To evaluate the performance of MOMA for integrated forward/reverse logistics network design problem,MOMA is compared to the multi-objective genetic algorithm (MOGA) of Altiparmak et al. (2006)[1].An additional reason for selecting MOGA as a basis of comparison is its similarity to MOMA in usinga priority-based encoding method and random-weight approach and the fact that it uses a differentevolutionary search strategy. MOGA uses a (µ + λ) selection strategy and a 0.5 and 0.7 probabilityfor its crossover and mutation operators respectively. To compare the performance of MOMA andMOGA, both algorithms are coded in MATLAB 7.0 and the parameters of MOGA are tuned forbest possible performance on the IFRLN design problem. Both algorithms are also tested on 7 testproblems of different sizes (see Table 1), in such a way that the sizes are selected in the range of recenttest problems in the recent literature (e.g. [6, 3]).

Table 2 shows that MOMA obtains higher quality Pareto-optimal solutions than MOGA with respectto the average ratio of Pareto-optimal solutions. The average ratio of Pareto-optimal solutions rangesin MOMA between 60.58% and 78.81% and for MOGA it varies between 54.67% and 73.15%. In allproblems except for problem No.2, MOMA dominates MOGA. Moreover the total average number


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Problem No. of potential No. of potential No. of No. of potential No. of potentialNo. production/recovery distribution customer collection/inspection disposal

centers centers zones centers centers1 2 3 5 3 12 5 10 10 5 43 5 10 15 10 44 10 15 20 15 55 20 40 50 40 106 20 40 70 40 107 30 50 100 50 15

Table 1: Test problems

CPU time (secs) Avg No. of Pareto solutions Avg ratio of Pareto-optimal solutionsMOGA MOMA MOGA MOMA MOGA MOMA

1 24.3 28.24 6.2 5.6 73.15 78.812 57.31 60.58 9.93 8.33 61.12 60.583 63.57 74.45 12 10.4 60.73 71.424 106.6 125.2 9.46 8.93 59.64 63.845 197.49 247.71 11.93 13.06 58.08 70.666 327.96 422.84 8.93 8.46 63.33 67.027 497.73 650.3 9.53 9.46 54.67 65.92

average 9.71 9.17 61.53 68.32

Table 2: Single capacity instances

of Pareto-optimal solutions is 68.32% for MOMA and 61.53% for MOGA, suggesting that MOMA issuperior to MOGA, especially where larger problems are concerned. Based on the average numberof Pareto-optimal solutions, MOGA finds slightly more Pareto-optimal solutions than MOMA. Theaverage number of Pareto-optimal solutions ranges in MOGA lie between 6.20 and 12 and for MOMAthis changes to between 5.60 and 13.06. The total average number of Pareto-optimal solution is 9.17 forMOMA and 9.71 for MOGA. These results could be explained by MOGA’s limited search intensification,causing it to find more Pareto-optimal solutions of a lower quality. On the other hand, being equippedwith dynamic local searches, MOMA appears to have sufficient capability to intensify the search andexplore the neighborhood of high quality solutions more carefully, leading to higher quality solutions.Computation times for MOMA are higher than for MOGA. For small problem sizes, the gap betweencomputation times is not very significant, but the gap tends to widen as problem size increases.

The results for the multi-level capacity instances in Table 3 show that MOMA computation timeis only 1.5 to 2.5 times higher than the computation times for single-capacity level instances. Theincreased computational requirements are probably due to the additional local search on capacitylevels (CA) during the local search stage. Moreover a sensitivity analysis on the capacity levels (N),confirms the strong influence of considering multi-level capacity for facilities on the overall performanceof the logistics network.

5 Conclusions

Most of the existing models from the literature only consider a single capacity level for each facility.However, in real-life problems, capacity levels are important decision variables due to their stronginfluence on logistics network efficiency and responsiveness. To the best of our knowledge, this paperis the first to offer an integrated design of the forward and reverse supply chain, including capacity


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Problem No. of capacity CPU time Avg. No.No. levels (N) (seconds) Pareto solutions1 2 59.32 7.662 2 102.18 9.333 3 175.76 9.664 2 301.34 9.335 2 581.43 9.666 3 1126.20 11.507 3 1890.23 11.25

average 9.77

Table 3: Multilevel capacity instances

decisions to minimize network costs and maximize network responsiveness.

As the MOGA algorithm of [1] is one of the best for multi-objective integrated supply chain design,we use it as a basis of comparison. The numerical results show that our MOMA algorithm outperformedthe existing MOGA in terms of average ratio of Pareto-optimal solutions obtained. The main contri-bution of the new MOMA heuristic, however, consists of its ability to easily handle multiple capacitylevels for the facilities within the forward/reverse logistics network and applying dynamic local searchstrategy. Future research could be aimed at developing robust models to accommodate the changingparameters of the business environment during the life-time of the logistics network. In addition toaddressing the demand uncertainty, the supply of return products in a multi-product integrated logis-tics network is a promising research avenue with significant practical relevance. Although our memeticdynamic search strategy proved to be competitive for the IFRLN network problem under consideration,other multi-objective metaheuristics algorithms such as multi-objective tabu search or multi-objectivescatter search could offer promising avenues for developing richer integrated logistics networks as well.

References

[1] Altiparmak F., Gen M., Lin L., Paksoy T., A genetic algorithm approach for multi-objectiveoptimization of supply chain networks. Computers and Industrial Engineering, 51: 197-216, 2006.

[2] Amiri A., Designing a distribution network in a supply chain system: Formulation and efficientsolution procedure, European Journal of Operational Research, 171: 567–576, 2006.

[3] Du F., Evans G. W., A bi-objective reverse logistics network analysis for post-sale service. Com-puters & Operations Research, 35: 2617 - 2634, 2008.

[4] Gen M., Cheng R., Genetic algorithms and engineering optimization. New York, Wiley, 2000.

[5] Jayaraman V., Guige Jr V. D. R., Srivastava R., A closed-loop logistics model for manufacturing.Journal of the Operational Research Society, 50: 497–508, 1999.

[6] Lee D., Dong M., A heuristic approach to logistics network design for end-of-lease computer productsrecovery. Transportation Research Part E ,44: 455–474, 2007.

[7] Melachrinoudis E., Messac A., Min H., Consolidating a warehouse network: a physical programmingapproach. International Journal of Production Economics, 97: 1-17, 2005.

[8] Murata T., Ishibuchi H., Tanaka H., Multi-objective genetic algorithm and its applications toflowshop scheduling. Computers and Industrial Engineering, 30(4), 957–968, 1996.


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Creativity, Soft Methods and Metaheuristics∗

Jose Soeiro Ferreira ‡ †

‡ INESC Porto – Instituto de Engenharia de Sistemas e Computadores do PortoEmail: [email protected]

† FEUP – Faculdade de Engenharia da Universidade do PortoRua Dr Roberto Frias, Porto, Portugal


1 Introduction

Some ideas, methods and tools of Creativity and Soft Methods, in connection with Metaheuristics, willbe presented. The correspondent work, still going on, investigates the advantages and the utilization ofcreative thinking and soft Operational Research (soft OR) to resolve difficult Optimization problemsand to evaluate and compare dissimilar approaches based on Metaheuristics (Mh). We believe itconstitutes an innovative challenge by proposing to combine, articulate and merge diverse proceduresand techniques, from different areas. Relevance, power and success of Mh are well-known for decades.But open questions are still around: the choice of a Mh? In the presence of a concrete optimisationproblem – which ‘effective, efficient’ Mh (able to produce an ‘optimal’/acceptable solution), at thecost of a ‘reasonable’ computing time, should be selected? And after making a selection (how to doit?), there is no universal way to improve it, to elect adequate strategies or to tune its parameters.The choice of a ‘good’ Mh and the adjustment of the correspondent parameters suggest, or call uponinnovative ideas and tools, eventually out of the specific area. Obviously, expertise and experience ofthe users are of great value.

‘Quality’ of solutions and computational times are not the only and necessarily most importantcriteria for analysing or selecting a Mh. Very often, effectiveness of a solution approach has to beevaluated in the context of practical problem solving. Solutions and methods cannot be isolated fromthe problematic of understanding the right problem and from the agreement on a convenient (butapproximate) model. Flexibility, easiness, robustness, appeal, experience may represent other criteriato take into account.This paper also sketches a framework for a coherent and comprehensive comparativeevaluation of Mh.

Mh are themselves the outcome of fantastic creative processes. How many metaphors from natureand/or social behavior inspired these general heuristic methods! Who could imagine, some decades ago,that procedures based on natural evolution were competing to solve hard combinatorial optimizationproblems?

Section 2 refers to creativity, mentions well-known talents/abilities of creative persons/groups andintroduces the principles of divergent thinking and convergent thinking in connection with the Cre-ative Problem Solving method. Section 3 proposes and somehow integrates methods and tools fromcreativity and soft OR in the environment of optimization with Mh. The objective is to contribute for∗Supported by Fundacao para a Ciencia e Tecnologia (FCT) Project PTDC/GES/73801/2006 (CROME)


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better working procedures. An outline of a framework for a coherent and comprehensive comparativeevaluation of Mh is proposed. Finally, a list of references is included.

2 Creativity

The book ‘The art of problem solving’ [1] is probably the first book on creativity in Operational Research[1]. Another interesting paper of the same author is ‘Creativity in problem solving and planning: areview’ [2]. Paul Torrance has been a pioneer in creativity research and education for more than 50years. When referring to creativity he says:

“Creativity defies precise definition. This conclusion does not bother me at all. In fact, Iam quite happy with it. Creativity is almost infinite. It involves every sense - sight, smell,hearing, feeling, taste, and - even perhaps the extrasensory. Much of it is unseen, nonverbal,and unconscious. Therefore, even if we had a precise conception of creativity, I am certainwe would have difficulty putting it into words.” E.P. Torrance (1988), in [10].

[9] gives a short definition of creativity that encapsulates many other definitions presented in theliterature:

“Among other things, it is the ability to challenge assumptions, recognize patterns, see innew ways, make connections, take risks, and seize upon chance.”

Talents and tools

There is a variety of talents/abilities that characterises creative individuals or groups [16]:

• Fluency: the production of multiple problems, ideas, alternatives or solutions. It has been shownthat the more ideas we produce, the more likely we are to find a useful idea or solution.

• Flexibility: the ability to process ideas or objects in many different ways given the same stimulus.It is the ability to delete old ways of thinking and begin in different directions. It is adaptivewhen aimed at a solution to a specific problem, challenge or dilemma.

• Originality means getting away from the obvious and commonplace or breaking away from rou-tine bound thinking. Original ideas are statistically infrequent. Originality is a creative strength,which is a mental jump from the obvious. Original ideas are usually described as unique, surpris-ing, wild, unusual, unconventional, novel, weird, remarkable or revolutionary.

• Elaboration is, in particular, the capacity of ‘doing’, of structuring, composing and preparingcomplex situations.

Special techniques/tools may be used in creative processes: Brainstorming, Verbal checklists, Provoca-tive questions, Visual stimulation (Pictures, Objects), Analogies and Metaphors, Mind Maps, CognitiveMaps [5], Rich Pictures [3]. Talents and tools may be associated.

Creative Problem Solving

During a creative process it is convenient to start with divergent thinking to produce as many ideas orsolutions as possible and thereafter to switch to convergent thinking to select the few most promisingideas (Fig. 1).


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Figure 1: Divergent thinking and Convergent thinking

There are well established rules for these procedures and they will be mentioned, after an introduc-tion, in the context of their relevance to working with Mh. Divergent thinking and Convergent thinkinghave regularly been exploited in workshops and future conferences to promote creativity, innovationand strategy consensus. In a specific context, the report [15] summarises a workshop on ‘Facilitatingto Deal with Combinatorial Optimization Problems’.

Divergent thinking and Convergent thinking phases are an expected part of creative processes. Theygo along with the six-step model for a systematic approach to CPS – Creative Problem Solving [11],[4], [12]:

1. Mess finding, 2. Fact finding, 3. Problem finding, 4. Idea finding, 5. Solution finding and 6.Acceptance finding.

A description of each of the steps and the list of the more relevant competences (during each step)or what to do or think about are condensed in [17].

3 Creativity, Soft Methods and Metaheuristics

“Assuming that the way experts or others formulate problems will lead to a solution is oftenwrong. Experts are only experts within the box that defines their expertise. But solutionsto most problems that arise within the box are found outside of it. What is needed isout-of-the ordinary thinking, “crazy” ideas, without fear of the ridicule. Encouragement of“crazy” ideas ought to be the norm at organizational meetings.”

Ackoff, R.L. and Rovin, S. (2005) Beating the System, Using creativityto outsmart bureaucracies, Berret-Koehler Publishers, Inc., SF, USA.

We think that there are some advantages and a seminal value in the utilization of creative thinkingand soft methods to deal with difficult optimization problems and to evaluate and compare differentapproaches based on Mh. Understanding what is the problem/model, discussing the approaches andimplementing the solution methods should not be isolated. Especially if one is facing a real complexproblem. Emphases of soft OR methods are: structuring messy, complex problem situations ratherthan solving well-defined problems; exploring the differing views of the participants; and facilitatingparticipation and engagement, rather than analyzing quantitative aspects and models.

Mh are the outcome of unlikely creative processes, merging ideas of different areas. Their relevance,power and success are well-known but the open question is still here: the choice of a Mh? In the presence


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Figure 2: Resolution process

of a concrete optimisation problem – which ‘efficient’ Mh (able to produce an ‘optimal’/acceptablesolution), at the cost of a ‘reasonable’ computing time, should be selected? Moreover, even aftermaking a selection, there is no universal way to improve it, to elect adequate strategies or to tune itsparameters. On the other hand, the weak theoretical results about Mh are of almost no practical use!In conclusion there is no complete ‘rational’ guide to select or to implement a Mh, what is not a goodsign.

Quality of solutions, number of iterations/computational time are not the only and necessarily mostimportant criteria for analysing or selecting Mh. And even if they were, the legitimate question ‘WhichMh for a specific problem?’ could not get a clear recommendation. Certainly they are well-suitedfor questioning the efficiency and efficacy, but the effectiveness of a solution approach should also beevaluated in the context of practical problem solving. Other criteria should be taken into account and,quite often, they are intuitively used for relevant practical purposes. In this context a new tool will bepresented later.

The process depicted in Fig. 2, will be a basis for a more detailed discussion. It is about theproblematic of dealing and structuring the resolution of problems with Mh. Comprises phases ofdivergent and convergent thinking, as part of a CPS process – the researcher should perceive that thereis a dimension of creativity in a project with Mh. Brainstorming sessions involving at least the elementsof the team, are a convenient way for divergent phases.

Comparing, choosing and evaluating Mh

Mh are very powerful and flexible means applicable, potentially, to any optimisation problem. Butcomparing and choosing them is a hard job. Evaluating a particular Mh is also complicate. This iswell known but there is a risk of credibility if no ‘solution’ is found. We do not think a ‘solution’ willemerge from the ‘Mh context’ alone.

The ideas involved in the previous sections, together with various schemes and procedures takenfrom soft OR, may be of some help to structure and capture the environment of comparing and choosingMh and to facilitate the handling of quantitative and qualitative/soft information, simultaneously.

We briefly outline a few methods and tools to simultaneously organise properties and characteristicsof a set of Mh – those that are under study. Information may arise from personal experience, frompublications and other sources.

SWOT analysis may also be adapted to study and analyse advantages and drawbacks of eachmethod, as a complement, to the overview of the Mh eventually considered in Table 1. As stated in[14]: ‘the primary goal of the SWOT analysis is to assist in selecting the Mh with the best possibilitiesof implementing a divergent/convergent search strategy’. Obviously, identifying opportunities, threats,strengths and weaknesses (concepts which should be adapted) are of importance to create a good


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implementation.

Table 1: SWOT Analysis for Memetic AlgorithmsOpportunities Threats

Memetic Algorithms easily applied to ### long time inintensification by local search; local search— —

Strengths Strategies S-O Strategies S-Tconvergence by survival of the fittest; possibilities for various LS ???hybridisation (LS); proceduresfinds local optima; ——Weaknesses Strategies W-O Strategies W-Tconvergence by survival of the fittest; ??? ???—

These tables may be completed for a Mh, in general, or for a very specific application. Table 1,which is quite incomplete, points up the idea of an application of Memetic Algorithms [13]. Observethat besides assisting to organise and clarify thoughts these tables may also be useful for team discussionand decision making.

Strategic Choice Approach (SCA) [8] is another problem structuring/soft method ‘which deals withthe interconnectedness of decisions problems in an explicit yet selective way. Its most distinctive featureis that it helps people working together to make more confident progress towards decisions by focusingtheir attention on possible ways of managing uncertainty as to what they should do next’ [6]. Theframework distinguishes four complementary modes: shaping mode, designing mode, comparing modeand choosing mode. We suggest that some of the ideas involved in SCA, in particular in the modes ofcomparing and choosing, could be of some help. Without going into the details, tools such as the onesillustrated inTable 2 and Fig. 4 could be employed by individuals or groups to study, improve, classifyand evaluate Mh.

Table 2: Comparing different schemesQuality Comp. time Simplicity Adaptability Robustness Theory ? ?

Mh1 * * * t t s s XXXX r r r r + + -Mh2 * * t t t s s s X r r r + + + -Mh3 - - - - - -Mh4 * * t t s XXX r r r + -

Table 2 may include qualitative and quantitative data – this is a very significant point. The relativeimportance of each criterion may be defined/agreed. Just looking at this table, it may be concludedthat Mh4 is dominated by Mh2.

Associated with SCA, there is the STRAD2 software. In particular, windows equivalent to anadvantage comparison grid, such as the one of Fig. 3 (a balance window), allow for adjustments ofpositions and changes in ranges. A ‘combined’ row presents the mean position of all the advantageassessments in the rows above. Of course a formula will aggregate the ranges.

Note that a grid such as this one may be used for personal/group work carried on but also forstructuring and comparing published works. Other tools such as the progress package could be takeninto account.

Independently of the relevance of other issues (not necessarily quantitative), we understand thatthe development of statistical tests, to get a more ‘scientific comparison’, due to the lack of relevanttheoretical analysis of Mh, should be encouraged. A reference about ongoing work in this line is in


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Figure 3: Advantage comparison grid for a selected pair, Mh1 and Mh2

[7]. Instruments as those introduced before (see for example Table 2 and Fig. 4) are a contribution tointegrate soft and hard data.

Implementing Metaheuristics

Implementing a Mh is a pervasively imprecise decision-making process – quantitative aspects may bea good support, rational choices would be interesting if they are not overtaken, experience and intu-ition are frequently used. As proposed earlier, divergent/convergent thinking, as part of a creativeattitude, may be helpful to plan, design, and parameterize a given Mh. The procedures of intensifica-tion/diversification, associated to search strategies, may also benefit. The designer should be consciousthat there are many aspects involved in the implementation of Mh that are not susceptible of accu-rate/deterministic planning – a support from other areas of knowledge may be convenient. For instance,from the use of Memetic Algorithms in [13], the following elements could be identified and grouped forfurther study:

Population structure – influences the number of agents, Crossover, Mutation, Local search:Choice of edges, facultative or not . . . , Number of local search iterations

andRandom start, No improvements of best agents, Minimum improvements, Number of iter-ations without improvements, Change between diversification/intensification according toparameters.

Mutation is influenced by two parameters: k (the frequency of this operation); d (thedeepness of the Mutation).

In [14] the reader may find a complete report of divergent and convergent thinking as part of asearch strategy. Fig. 4 and Fig. 5 are taken from this work.

In a project of using a Mh, the correspondent activities may undergo numerous revisions, betweenthe interesting concentration on the set of principles of the Mh, such as: Neighbourhoods – simple,


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Figure 4: The shifting between convergent and divergent selection strategy

Figure 5: The overall convergence by decreasing threshold ε - the minimal difference in the objectivevalue of individuals in a population.

extended, . . . , Memories – populations of solutions, tabu list, pheromone trails, . . . and the tuning ofparameters.

Aspects to take into account in each of the phases have already been pointed out. Phases shouldbe visited in turn (Fig. 2 and Fig. 4); for instance, if one is following a convergent thinking phasein tuning the parameters, it may be convenient to undergo a new divergent phase on tuning or evencoming back to a divergent thinking phase in ‘guiding tools’.

References

[1] R. L. Ackoff. The art of problem solving. John Wiley, New York [etc., 1978.

[2] R. L. Ackoff and E. Vergara. Creativity in problem solving and planning: a review. EuropeanJournal of Operational Research, 7(1):1–13, May 1981.

[3] P. Checkland. Rational Analysis for a Problematic World Revisited, chapter Soft Systems Method-ology. J. Wiley & Sons, 2001.

[4] J. Courger. Creative Problem Solving and Opportunity Finding. Boyd & Fraser Pub., 1995.

[5] C. Eden and F. Ackerman. Rational Analysis for a Problematic World Revisited, chapter SODA– The Principles. J. Wiley & Sons, 2001.

[6] C. Eden and F. Ackerman. Rational Analysis for a Problematic World Revisited, chapter TheStrategic Choice Approach. J. Wiley & Sons, 2001.

[7] J. S. Ferreira. Creativity in optimisation – some ideas. In Proceedings IV SELASI - EuropeanLatin American Workshop on Engineering Systems, Havana, Cuba, December 2008.

[8] J. Friend and A. Hickling. Planning under Pressure: the Strategic Choice Approach. Pergamon,Oxford, 1987.

[9] N. Herrmann. The Whole Brain Business Book. McGraw-Hill, New York, 1996.

[10] G. Miller. E. Paul Torrance - ‘The creativity man’. NJ: Ablex Publishing, 1997.

[11] A. Osborn. Applied Imagination. Charles Scribner’s &Sons, New York., 1953.

[12] S. Parnes. Optimize: The magic of your mind. Bearly Limited, Buffalo, NY.


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[13] A. Rodrigues and J. S. Ferreira. Solving cutting path problems by memetic algorithms. In IISELASI - European Latin American Workshop on Engineering Systems, Porto, Portugal.

[14] P. Schulz. Creative design in optimization - metaheuristics design to multi-modal continuousfunctions. Master’s thesis, Technical University of Denmark, 2006.

[15] G Soares, A Rodrigues, and J. Ferreira. Facilitating to deal with combinatorial optimizationproblems. Report of a workshop, INESC Porto, 2008.

[16] R. V. Vidal. Creativity for operational researchers. Investigacao Operacional, 25, 2004.

[17] R. V. Vidal. Creative and participative problem solving – the art and the science.http://www2.imm.dtu.dk/ vvv/CPPS/index.htm, 2006.


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Computational Experience with GRASP for a Maximum

Dispersion Territory Design Problem

Roger Z. Rıos-Mercado ∗ Elena Fernandez † Jorg Kalcsics ‡

Stefan Nickel ‡

∗ Graduate Program in Systems EngineeringUniversidad Autonoma de Nuevo Leon, Mexico


† Department of Statistic and Operations ResearchUniversitat Politecnica de Catalunya, Spain


‡ Chair of Operations Research and LogisticsUniversitat des Saarlandes, Germany

Email: {j.kalcsics, s.nickel}@orl.uni-saarland.de

1 Introduction

We present a heuristic approach to a territory design problem motivated by a real-world applicationarising in the recollection of waste electrical and electronic equipment (WEEE) in European countries.The problem may be stated as follows. Given a set of recollection points for waste electrical andelectronic goods, a central agency must assign each recollection point to a corporation that would takecharge of the recollection. In countries like Germany and Spain, for instance, the coordination andsupervision of the collection is done by a central registry which also determines the market share of allcompanies that sell equipment in the respective country. According to the law, this assignment mustmeet several requirements. For white goods (i.e., dish-washers, fridges, etc.), the assignment of stationsto corporations should be made such that the average amount of returned WEEE items is proportionalto the market share of the corporation. Moreover, the points assigned to a corporation should be evenlydispersed all over the country to avoid a monopolistic concentration. Note that this criterion is exactlythe opposite of the usual compactness criteria of classical territory design problems [2]. For a generaloverview on territory design models and an introduction into the topic the reader is referred to [2].

In this paper, we present an empirical evaluation of three GRASP heuristics for this NP-hardcombinatorial optimization problem. Three different construction mechanisms and three different localsearch schemes are derived and evaluated over a range of randomly generated instances. The resultsare very good. The local search strategies improve considerably the solutions found in the constructionphase particularly in terms of their infeasibility status. All three procedures reported feasible solutions.

2 Problem Description

Description: White goods are further subdivided into devices that have freezing capabilities andthose that do not (product type 1 and 2, respectively). The average amount of WEEE of a basic


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area (BA) is assumed to be proportional to the number of households. Moreover, all basic areas areclassified into three groups: good, mediocre, and bad. A good basic area is for example one, which has arelatively small geographic extent, contains few collection stations, and possesses a good infrastructure.The motivation for this classification is that the actual costs for the recollection and recycling shouldalso be (more or less) proportional to the market shares of the corporations. As the market sharesmay differ for the two product types, it is allowed to split basic areas, i.e., for some basic areas thecorporation that collects type 1 products may not be the same as the one that is responsible for thetype 2 products. The task is then to assign basic areas fully or partially to corporations such that (i)for both product types all basic areas are assigned to a corporation; (ii) for each corporation, the totalnumber of households of all basic areas assigned to the corporation is proportional to its market sharefor each of the two product types; (iii) the good, mediocre, and bad basic areas are evenly distributedamong the corporations relative to their market shares; (iv) the number of split basic areas is not toolarge; and (v) for each corporation, all basic areas that are fully or partially assigned to the corporationare as dispersed as possible. The set of all basic areas assigned to a corporation for at least one of thetwo product types is called a territory.

3 Model

A combinatorial optimization model is given due to the fact that it is more appealing for the heuristicprocedures to be described later.

Indices and sets: Let V be the set of basic areas (BAs), with |V | = n, and C the set of corporations,with |C| = m. Let P be set of product types. In this work |P | = 2. Let Q be the set of quality indices,where q ∈ Q = {1, 2, 3} stands for good (1), mediocre (2), and bad (3). Let V q be the set of BAs ofquality q, q ∈ Q.

Parameters: Let wi represent the number of households of i-th BA, i ∈ V . Let qi be the qualityof the i-th BA, i ∈ V . Let MSp

k denote the market share of corporation k for product p, k ∈ C,p ∈ P . Let dij represent the Euclidean distance between i and j, i, j ∈ V . Let τ and β denote therelative tolerance associated with the balance of number of households and fair quality distribution ofhousehold, respectively. Let S be an upper limit on the number of allowed split BAs.

Computed parameters: Let w(B) (=∑

i∈B wi) be the size of set B with respect to node activity,B ⊂ V , with w(V ) = W . Let cq(B) (= |B∩V q|) be the cardinality of set B with respect to BA qualityq, q ∈ Q, B ⊂ V .

Decision sets: Xpk is the set of BAs assigned to corporation k for product p, k ∈ C, p ∈ P ;

Xk is the set of BAs assigned to corporation k for at least one product p ∈ P , k ∈ C, that is,i ∈ Xk ⇔ i ∈ Xp

k for some p ∈ P ; Xs is the set of split BAs, that is, i ∈ Xs ⇔ ∃k1, k2 ∈ C, k1 6=k2 such that i ∈ X1

k1∧ i ∈ X2

k2. Note that for a given value of p, Xp = (Xp

1 , . . . , Xpm) is a m-partition of

V . Let Xk be the set of all BAs assigned to territory k for at least one product p, i.e., Xk = ∪p∈PXpk .

Let Π be the collection of all |P | m-partitions of V , that is, Π = {X1, X2}, where Xp is a validm-partition of V , p ∈ P .

Model: The problem consists of finding X = (X1, X2) ∈ Π, so as to

maxX∈Π

f(X) = mink∈C

mini,j∈Xk

{dij} (1)

subject to1Ww(Xp

k) ∈ [(1− τ)MSpk , (1 + τ)MSp

k ] k ∈ C, p ∈ P (2)

1|V q|c

q(Xpk) ∈ [(1− β)MSp

k , (1 + β)MSpk ] q ∈ Q, k ∈ C, p ∈ P (3)

|Xs| ≤ S (4)


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Objective (1) measures territory dispersion. Constraints (2) represent the territory balance of thenumber of households with respect to the market share proportion for each product (measured bytolerance parameter τ). Constraints (3) assure a fair distribution of the basic areas regarding itsproduct quality (measured by tolerance parameter β). Constraint (4) sets a limit on the number ofmaximum split BAs allowed. We call this problem the Maximum-Dispersion Territory Design Problem(MaxD-TDP). MaxD-TDP is NP-hard since we can reduce the well-known NP-hard Partition Problemto it.

4 Description of Heuristics

GRASP [1] is a well-known multi-start metaheuristic that captures good features of both pure greedyalgorithms and random construction procedures. Each GRASP iteration consists of two phases: con-struction and local search. The construction phase builds a solution, and the local search phase attemptsto improve it. The best overall solution is kept as a result. GRASP has been successfully used for solvingmany combinatorial optimization problems, including territory design [3].

Construction Phase: Two of the procedures (1 and 1R) try to assign basic areas that are relativelyclose to each other to different corporations. The third one (Procedure 2) is based on the complementaryidea of trying to assign to the same corporation basic areas that are relatively far away from eachother. Given that feasibility with respect to the balancing constraints is very hard to achieve by theconstruction schemes, a greedy function that incorporates a penalty term for these violations is used.

Procedure 1: After sorting the pairwise distances in non-decreasing order, we go through this list stepby step, starting with the smallest non-zero value. At a given iteration, the next largest distance dij

is considered and if i and/or j are yet unassigned, we try to allocate them to different corporations.The allocation decision is hereby based on a greedy function that takes a distance-based measure andthe sum of the relative violations of the upper balancing constraints into account. More precisely, ifX = (X1, . . . , Xm) is the partial solution obtained so far, for a given basic area i, a corporation k, anda product p, the relative violation of the upper bound (UB) of balancing constraints (2) and (3) is givenby

Gpi (k) = max

{w(Xk) + wi

W ·MSpk

− (1 + τ), 0}

+ max{cqi(Xk) + 1|Vqi| ·MSp

k

− (1 + β), 0}

where qi is the quality index of BA i. Note that it makes no sense to include the lower bound violationsat this point, as they will always be violated until the very end of the construction procedure. Thegreedy function is then defined as φi(k) = λFi(k)−(1−λ)Gi(k), where λ ∈ [0, 1], Gi(k) = G1

i (k)+G2i (k)

and Fi(k) = di(Xk) := minj∈Xkdij (in case Xk = ∅, then di(Xk) :=∞). The restricted candidate list is

defined as RCL = {k : φi(k) ≥ φmaxi −α(φmax

i −φmini )}, where φmin

i = mink φi(k) and φmaxi = maxk φi(k).

Procedure 1R: As the UB of constraints (2) and (3) have been taken into account in Procedure 1through the greedy function only, it is likely to obtain infeasible solutions. Therefore, in a variant ofConstruction Procedure 1, we try to obtain solutions with no or at least fewer upper bound violations.Construction Procedure 1R has two phases. In the first one, basic areas are assigned to corporationsas in Construction Procedure 1, provided that they do not violate the UB of (2) and (3). This canbe achieved easily through a modified restricted candidate list: RCL = {k : Gi(k) = 0 ∧ φi(k) ≥φmax

i − α(φmaxi − φmin

i )}. As some basic areas may remain unassigned at the termination of the firstphase, these areas are assigned to corporations in a second phase in a different fashion. Throughout thisphase, to reduce the violation of the UB of (2) and (3), we use as greedy function just Gi(k). Splittingof basic areas is allowed in the second phase when no more than |S| basic areas remain unassigned.

Procedure 2: Using a different rationale, we now try to assign to the same corporation basic areasthat are relatively distant from each other. The procedure has two phases. In the first, the iterativeprocedure builds m territories, one at a time, using a farthest insertion greedy function that assigns tothe territory Xk currently being built BAs relatively “far away” from Xk. Since, again, no violation of


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the UB balancing constraints is allowed, the RCL only contains unassigned BAs for which Gi(k) = 0.Note that φmax = maxi φ(i) and φmin = mini φ(i). If some BAs remain unassigned at the terminationof this phase, we proceed to a second phase that is exactly as in Construction Procedure 1R.

Local Search: In this phase we attempt to recover feasibility as well as to improve the objectivefunction value. Solutions are now evaluated by means of a function that weighs both infeasibility withrespect to the balancing constraints as well as the smallest pairwise distance among the territories.This function is similar to the greedy function φi(k) used in the construction phase, however, with thedifference that now the sum of relative infeasibilities Gi(k) takes into account the violation of the lowerbounds of balancing constraints (2) and (3) as well. The types of exchange moves that we considerare the following: Type A1: For all products, reassign a basic area i that is currently assigned to someterritory Xk of corporation k to the territory Xk′ of a different corporation k′ 6= k. The size of theneighborhood is nm. Type A2: Reassign a basic area i just for product p from its current territory toa different corporation’s territory. Splitting is allowed. The size of the neighborhood is 2nm. Type B:Exchange the assignment of basic areas i and j currently allocated to different corporations for one orboth products. The size of the neighborhood is 2n2.

5 Implementation Issues and Experimental Results

We generated problem instances using real-world data. Basic areas correspond to German zip-codeareas with their respective number of households. The instances range from 100 up to 300 basic areasin steps of 50, and with four up to seven corporations. We generated five instances for each number ofbasic areas, except for the last, where we have just four instances. In combination with the four differentvalues of m, we have 96 instances. The tolerance values for the balancing constraints are set to τ = 0.05and β = 0.2, respectively. The pairwise distances were computed as the Euclidean distance betweenthe polygonal representations of the zip-code areas. In that way, neighboring basic areas have distancezero. The logistics indices were chosen randomly such that we have approximately the same number ofgood, mediocre, and bad basic areas. We set S to 20% of the value of n for each instance. Finally, themarket shares of the companies are computed independently for the two products. A market share isdrawn uniformly from the interval [0.75

m , 1.25m ]. At the end, the market shares are normalized to obtain a

total sum of 1. The heuristic procedures were coded in C++ and compiled with the Sun C++ compilerworkshop 8.0 under the Solaris 9 operating system. They were run on a SunFire V440 with 8 GB ofRAM and 4 UltraSparc III procesors at 1062 MHz.

Parameter Tuning: For fine-tuning the RCL parameter α and the weight parameter λ of the greedyfunction, we run the different versions of the GRASP with an iteration limit of 500 and no local search,and measure both the degree of violation of the balancing constraints and the value of the objectivefunction. The set of values for λ is {0.5, 0.6, . . . , 1.0}, and for α we take {0, 0.2, . . . , 1.0}. Figure 1,displays results for Procedure 1 with λ and α fixed, respectively. In these figures, the left verticalaxis measures the average (over all instances) relative infeasibility with respect to the upper and lowerbalancing constraints, whereas the right vertical axis measures the average (over all instances) percentgap between the objective function value of the obtained solution and the best solution found withall the values of the tested parameters. Similar figures for the other two construction procedures areomitted here for space reasons, but will be presented at the conference. For Procedure 1, a value ofα = 0.2 consistently finds the best compromise between infeasibility and deviation from maximumdispersion for each tested value of λ. Comparing the values of λ we observe that λ = 0.5 yields a goodcompromise. Due to their mechanism for building solutions, λ plays a minor role in Procedures 1R and2. Thus, only α was evaluated. We observe that α = 0.2 and α = 0.8 yielded the best compromisebetween deviation from maximum dispersion and infeasibility for each of these procedures.

Local Search Strategies: Next, we investigate the behavior and effect of the local search. In theprevious section we introduced three different neighborhoods. For our tests we consider three differentcombinations of these neighborhoods: LS1 (apply type A1 and then type A2 neighborhood), LS2


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0.4

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Figure 1: Construction Procedure 1 with λ = 0.5 as a function of GRASP quality parameter (left-handside) and with α = 0.2 as a function of the merit function weight parameter.

(apply type A2 and then type A1 neighborhood), and finally LS3 (apply type A2, then type A1, andthen type B neighborhood). Since the neighborhoods are polynomially bounded and not too large,we use a best improvement strategy. For Constructive Procedure 1, 1R, and 2, we use (α, λ) valuesof (0.2, 0.5), (0.2, 1.0), and (0.8, 1.0), respectively. The GRASP iteration limit was set to 2000. Theresults (omitted for space reasons) indicate that, independently of the construction procedure, bothLS1 and LS2 were able to obtain always feasible solutions, which represents a significative improvementwith respect to the construction phase where no feasible solutions were found (although solutions withrelatively small deviations from feasibility were obtained). This shows the effectiveness of the localsearch strategies for repairing the infeasibility issue. The results also indicate that LS2 outperformsLS1 in terms of relative deviations from the best known value for the dispersion objective. In addition,LS2 uses considerably less CPU time than LS1 (about 40% less in average). Therefore, we concludethat it is more advantageous to explore A2 first, and then A1.

n Statistic Proc 1 Proc 1R Proc 2

100 Avg. rel. dispersion gap (% from best known) 0.6 1.2 21.7Worst relative dispersion gap (%) 6.4 16.6 34.4Number of best solutions 14 16 0Average CPU time 128 120 173





Total Avg. rel. dispersion gap (% from best known) 0.7 0.9 24.6Worst relative dispersion gap (%) 13.3 16.6 53.2Number of infeasible solutions 0 0 0Number of best solutions 76 76 0

Table 1: Comparison of construction procedures under LS2.


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Heuristic Comparison: Table 1 shows a valid comparison among the heuristics under local searchstrategy LS2. This time the relative dispersion gap is computed with respect to best known value ofthe dispersion objective found by any procedure under any local search strategy. We observe that, interms of solution quality, Construction Procedure 2 exhibits a very poor performance compared to theother two. Procedures 1 and 1R perform very similarly with respect to both dispersion objective andCPU time, with Procedure 1 being slightly better than 1R in terms of the dispersion objective. Inparticular, for the largest problems (300 basic areas), both heuristics find the same solution for 75% ofthe instances. For problem instances below 300 basic areas, Construction Procedure 1 finds the bestsolution. Overall, Procedures 1 and 1R each find 76 best solutions. It is important to highlight thetremendous benefit reported by the local search phase in each case. Most of the solutions found in theconstruction phase were infeasible (feasibility success of 5% and large deviations from feasibility in somecases). After the local search, feasibility was always recovered in all cases. Neighborhoods A1 and A2helped to bring this figure up to 100.0%. This suggests further work on the local search schemes couldbe worthwhile. We also tested strategy LS3 for the 300-node instances with Procedure 1. For 11 outof 16 instances, no improvement was found. Moreover, while the overall average relative improvementis less than 1.5%, the average CPU time for LS3 is 19778.9 seconds, which represents, compared to the1056.5 seconds for LS2, a very large increase that barely pays off in terms of solution quality.

6 Conclusions

We have presented a computational study of a MaxD-TDP motivated by a real-world case in therecollection of WEEE. To the best of our knowledge, this is the first model based on maximum dispersionin the territory design literature. We have evaluated three construction procedures and three localsearch schemes within a GRASP framework over a range of instances randomly generated accordingto real-world scenarios. This included heuristic fine tunning and a comparison of several local searchstrategies among heuristics with very good results. There are several challenging avenues for futureresearch, for instance, the design of more sophisticated metaheuristics, and the study of an extensionthat would combine the design of the territories with the planning of the collection routes.

Acknowledgments: This research has been partially supported through grants MTM2006-14961-C05-01,of the Spanish Plan Nacional de Investigacion Cientıfica, Desarrollo e Innovacion Tecnologica (I+D+I),SEP-CONACYT 48499-Y, of the Mexican National Council for Science and Technology, and PAICYTCA-1478-07, of the Universidad Autonoma de Nuevo Leon.

References

[1] T. A. Feo and M. G. C. Resende. Greedy randomized adaptive search procedures. Journal of GlobalOptimization, 6(2):109–133, 1995.

[2] J. Kalcsics, S. Nickel, and M. Schroder. Towards a unified territorial design approach - Applications,algorithms and GIS integration. TOP, 13(1):1–56, 2005.

[3] R. Z. Rıos-Mercado and E. A. Fernandez. A reactive GRASP for a commercial territory designproblem with multiple balancing requirements. Computers & Operations Research, 36(3):755–776,2009.


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A Binary Particle Swarm Optimization Algorithm for a Variant

of the Maximum Covering Problem

Bruno Prata ∗ † Jorge Pinho de Sousa † Teresa Galvao †

∗ Center of Technological Sciences, University of Fortaleza - UNIFORAvenida Washington Soares, 1321, CEP 60811-905. Fortaleza, CE - Brazil


† FEUP, Faculdade de Engenharia da Universidade do PortoRua Dr. Roberto Frias, 4200-465, Porto, Portugal

Email: {bruno.prata, tgalvao, jsousa}@fe.up.pt

1 Introduction

The Maximum Covering Problem (MCP) is a widely studied Combinatorial Optimization problem,with several applications, such as facility location (including health centers, emergency vehicles andcommercial bank branches) and scheduling (flexible manufacturing systems, mass transit services,telecommunications)[1], [3], [4]. Practical instances of the problem are in general quite difficult tosolve. Thus, approximate heuristic methods are used to achieve satisfactory solutions in acceptablecomputational times.

A GRASP approach for the MCP is presented in [1], while in [3] and [4] Genetic Algorithms are ap-plied for the problem. The GRASP metaheuristic proposed by Feo and Resende [2] has been successfullytested in randomly generated instances of MCP. Arakaki and Lorena [3] propose a constructive GeneticAlgorithm for the Maximum Covering Location Problem, using both real and theoretical instances intheir computational experiments. Park and Ryu [4] present a Genetic Algorithm with unexpressedgenes for the MCP, using real instances of a subway system. In the unexpressed genes, the chromo-somes consist of an expressed part and an unexpressed part. The expressed part is used in evaluationof an individual and the unexpressed part preserve the information of an individual, maintaining thediversification.

Park and Ryu [4] describe a MCP approach for the crew pairing problem. The MCP can be agood model for the Vehicle and Crew Scheduling Problem (VCSP), if the changeovers (the change of avehicle of a driver) are forbidden.

Given a matrix A, in which aij ∈ {0, 1}, the MCP consists in covering the largest number of rowsof matrix A with a number of columns of matrix A less or equal to d. The variables yi are associatedwith the rows of matrix A, so that yi = 1 if the ith row is not covered in a solution, being yi = 0otherwise. The variables xj represent the columns of matrix A, so that xj = 1 if the jth column is partof the solution, being xj = 0 otherwise. Obviously the xj are decision variables, the yi being auxiliaryvariables required for the definition of the objective function.

It should be noted that our version is a simplified version of the MCP, as the weights of the rows(considered by other authors) are here ignored. In this formulation, a maximum cover occurs when aminimum number of rows are left uncovered.The abovementioned formulation was adopted in schedulingproblems of transport planning [4].


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The variant of the MCP can be formulated mathematically as follows:

min z =n∑

i=1

yi (1)

subject to,m∑

j=1

xj ≤ d (2)

n∑

j=1

aijxj + yi ≥ 1, i = 1, . . . ,m (3)

xj ∈ {0, 1} (4)yi ∈ {0, 1} (5)

(6)

The objective function represented by equation (1) aims at minimizing the number of uncoveredrows of matrix A. The set of constraints (2) impose that a maximum number of d columns of matrix Abe selected in the solution. The constraints of type (3) are used to define the auxiliary variables yi. Itis easy to show that constraints (5) are redundant and can therefore be replaced by simply bounds onthe variables yi. Therefore in our computational implementation, the number of binary variables is m.

The MCP is similar to the Set Covering Problem (SCP). While in the SCP the goal is to cover therows of matrix A with minimum cost, in the MCP the goal is to minimize the number of uncoveredrows of A, with no type of cost being measured. It seems that constraint (2) makes the problem reallydifficult to solve.

The objective of the work reported on this paper was to create a Binary Particle OptimizationAlgorithm for the Maximum Covering Problem. This work should obviously be viewed as a preliminarystage of a larger research project.

2 Particle Swarm Optimization

Evolutionary Algorithms are inspired by the theory of evolution of species, by recombining solutions,and thus aiming at performing an intelligent search of the solution space. As examples we have GeneticAlgorithms (GA) and optimization based on colonies of socials insects, such as ants, bees, and termites.

Kennedy and Ebehart [5] proposed a new meta-heuristic for solving unconstrained problems of non-linear optimization they have called Particle Swarm Optimization (PSO). The philosophy of PSO isthe following: an initial population of particles is generated, each one in an initial position of the spaceof solutions and with a certain initial velocity. Each particle i is characterized by its position and avector of change in position called velocity [9].

A particle is a candidate solution of the swarm in a step of the search. The swarm is the whole set ofparticles in a given iteration. The velocity is a number that leads the movement of the particles. Eachparticle will go through the n-dimensional search space and will have its velocity updated according tothe velocity of the other particles.

For each particle, its best fitness in the search process is stored, and this value is named pbest. Theoverall pbest value, obtained evaluating all the particles in the population, is the best solution of thesearch, which is called gbest. The values of pbest and gbest are used for the updating of velocities ofthe particles in the search space. The particles that are far from the promising regions of the search


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space will have their velocity increased, while the other particles will have their velocity decreased.

PSO has a search strategy extremely efficient [5], [6]: it has an easy computational implementation,requiring few lines of code; it uses little memory and requires few processing speed; and the searchprocess is enhanced by the continuous learnship of the particles. After the promising results achievedby PSO in non-linear programming problems, its application to combinatorial optimization problemsshowed also to be very promising.

Kennedy and Ebehart [7] developed a binary version for PSO and concluded that the meta-heuristicis also flexible and robust for this class of problems. Tasgetiren and Liang [8] applied a binary PSO forthe Lot Sizing Problem, obtaining results of better quality than a Genetic Algorithm.

3 Proposed heuristic

For the resolution of the MCP with PSO, the following coding was used: each particle consists ofa binary vector of m elements, representing the values of the xj . Therefore, the solution space isx = {x1, x2, . . . , xm}, xj ∈ {0, 1}. It should be noted that many solutions may be unfeasible for theoriginal problem, due the constraints (4) and (5). The neighborhoods are the solutions obtained fromx by a moving. A movement is a flip in a bit of the binary vector. Therefore, the total number ofsolutions is 2m.

Since it is an evolutionary algorithm, PSO operates on a population of solutions (particles), withsize maxpop. The search is performed in maxgen iterations (generations). The initial population isgenerated in a random way. The values P [xj = 1] = 0.05 and P [xj = 0] = 0.95 were adopted, becausethe vector of decision variables is very sparse. Therefore, it is likely that constraint (2) is satisfied.

This policy for the generation of the initial population does not prevent the generation of unfeasiblesolutions. However, it hopefully guarantees a good diversification of the search. An infeasible solutioncan be a neighbor of a high quality solution.

The evaluation function (fitness of a solution) is the number of rows of matrix A uncovered in acertain solution, to be minimized. In order not to violate constraint (2), a penalty factor equal to 20was adopted, per extra column, to penalize particles that use more than d columns in a solution. Thispenalty factor was adjusted empirically, after some initial computational experiments.

The velocities are stored in a matrix of maxpop ×m order. The values are generated through thefollowing expression: velocity[i, j] = vmin+ (vmax− vmin) ∗ random[0, 1]. The parameters vmax andvmin are the maximum and minimum values allowed for the velocity of each particle (the followingvalues have been adopted: vmax = ln(m) and vmin = −ln(m)).

Be pbest and gbest the best solutions obtained, respectively, by the ith particle and by the swarm. Bexpbest and xgbest the positions of such solutions, the velocity variation is given through the expression:

∆v[i, j] = c1 ∗ random[0, 1] ∗ (xpbest − x) + c2 ∗ random[0, 1] ∗ (xgbest − x)

c1 is the cognitive constant and c2 is the social constant. The first is related to the learnship of theparticle, based on its own move in the search space, while the social constant is related to the learnshipof the particle concerning the behavior of the swarm as a whole. A position of a particle is a binaryvector in the solution space is x.

As in a binary optimization problem the decision variables only take the values 0 and 1, a functionthat transforms a real velocity, defined in the interval [vmin, vmax], into the interval [0, 1] is required.The sigmoid function is usually used for this purpose [7], [8]:

sigmoid(v[i, j]) =1

1 + e−v[i,j]


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Thus, the value of the sigmoid for the velocity of the ith particle in the jth dimension (bit) iscalculated. If this value is smaller than a random number uniformly distributed, the jth bit takes thevalue 1; otherwise, it takes the value 0. Algorithm 1 presents the pseudo code for updating the positionof the particles.

for i = 1 to maxpop dofor j = 0 to m do

r ← [0, 1];if r < sigmoid(v[i, j]) then

x[i, j]← 1;else

x[i, j]← 0;end

endend

Algorithm 1: Movement of particles

After the accomplished moves, the fitness of each particle is calculated and the unfeasible solutionsare penalized. If the value of gbest does not improve during k iterations of the algorithm, a restart ofthe velocity of the particles, except from xgbest. After that, at Algorithm 2, the pseudo code of theheuristic proposed is presented.

According to [1], both constructive algorithms and local search can play an important role in the per-formance of algorithms for the MCP. However, the authors employed a PSO without the hybridizationof such techniques aiming to demonstrate that the mechanisms of social learnship of PSO, by them-selves, can incur in the obtaining of good solutions in combinatorial binary optimization problems, asit is the case of MCP.

Generate initial population ;Generate initial velocities ;Generate initial values for pbest and gbest;while gen ≤ maxgen do

Find pbest and gbest ;Calculate velocity of the ith particle in the jth dimension;Update velocity of the ith particle in the jth dimension;Update each bit in string using a sigmoid function (update position) ;Evaluate fitness of each particle ;Assign penalization to unfeasible solutions ;Restart velocities for each k iterations without improvement of gbest;gen← gen+ 1 ;

endAlgorithm 2: Pseudo code of proposed PSO heuristic

4 Computational Results

For the computational experiments, 10 instances of medium size were generated randomly. Theseinstances were solved in an exact way with LINGO 8.0. The values n=m=200 have been adopted toallow the resolution of the instances to optimality. These instances are presented in Table 1 as follows:number of rows of matrix A (n), number of columns of matrix A (m), maximum number of columnsto be taken into the solution (d) and density of the matrix A (ρ), i.e. number of ”1”s divided by thetotal number of elements, and t(s) is the time needed to solve the problem.


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Instance n m d ρ GlobalOptimum

t(s)

MCP01 200 200 16 4.02 48 5434MCP02 200 200 10 3.82 91 48MCP03 200 200 18 3.94 37 5906MCP04 200 200 18 3.83 40 2896MCP05 200 200 16 3, 93 44 232MCP06 200 200 6 9.22 65 396MCP07 200 200 17 3.96 41 443MCP08 200 200 32 3.83 0 1621MCP09 200 200 16 3.95 52 2148MCP10 200 200 23 4.12 15 19193

Table 1: Test instances

The algorithm was implemented in Pascal. The experiments were made in an AMD Semprom2400 + 1.67 GHz, 504MB RAM. The following parameters were utilized: maxpop = 15, maxgen = 2500and c1 = c2 = 1. For the restart routine of the velocities, k = 500 generations was adopted. It shouldbe emphasized that the performance of PSO is not so much dependent on parameters.

For each instance, 20 runs of the algorithm were made. The results are presented in Table 2. Thefirst column of the table identifies the instance. The second and third columns present, respectively,the best and the worst results obtained by the heuristic. The fourth and the fifth column present themean and the standard deviation of the results obtained by the algorithm, for the 20 runs. In the sixthcolumn, we have the number of times that the algorithm found the optimal solution of the instance.Finally, in the seventh column, the average computational times are presented.

Based on these results, we can say that, although the initial solutions generated are of low qualityand there is no repair heuristics, PSO could obtain good solutions in acceptable computational times.For 5 out of the 10 instances considered, the optimal solutions were achieved. Such fact confirms thepotential of PSO in solving this type of Combinatorial Optimization problems.

The strategy of generating initial solutions in a random way, without taking into consideration thespecific characteristics of the problem, turned out to be very inefficient. i.e., the quality of the initialsolutions has a considerable impact on the quality of the final solutions.

Constraint (2) leads to feasible vectors x with a few number of ”1”s. The several local optima aretherefore quite ”different”, being separated by a large Hamming distance. Thus, migration to globaloptima becomes, in general, quite difficult.

The quality of a solution for the MCP can be assessed through the calculation of the coveringpercentage p, determined by the following expression, in which n is the number of rows of matrix Aand z is the value of the objective function obtained, which represents the number of uncovered rows:

p =(n− z)n

Based on the mean of the values of the solutions obtained by the heuristic, it was possible tocalculate the deviation (gap) between the covering percentage p obtained by PSO and by the exactmethod, for the instances analyzed. For this class of instances the results seem to be very promising(Table 3).


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Instance GlobalOptimum

Best solu-tion

Worstsolution

Mean Stddev.

n.opt. sol.in 20 runs

tm(s)

MCP01 48 48 55 50.10 1.76 4 23.71MCP02 91 92 96 93.90 1.22 0 23.78MCP03 37 37 44 40.35 1.77 2 23.74MCP04 40 42 46 44.10 1.70 0 23.77MCP05 44 44 50 46.45 1.60 4 23.94MCP06 65 65 74 70.15 3.35 5 23.85MCP07 41 41 48 43.70 2.85 9 23.86MCP08 0 4 8 5.85 1.39 0 23.79MCP09 52 54 58 55.65 1.62 0 23.60MCP10 15 19 24 21.20 1.03 0 23.82

Table 2: Computational Results

Instance GlobalOptimum

poptimum pbest pmean pworst gapbest gapmean gapworst

MCP01 48 76.00 76.00 74.95 72.50 0.00 1.05 3.50MCP02 91 54.50 54.00 53.05 52.00 0.50 1.45 2.50MCP03 37 81.50 81.50 79.83 78.00 0.00 1.67 3.50MCP04 40 80.00 79.00 77.95 77.00 1.00 2.05 3.00MCP05 44 78.00 78.00 76.78 75.00 0.00 1.23 3.00MCP06 65 67.50 67.50 64.93 75.00 0.00 2.58 4.50MCP07 41 79.50 79.50 78.15 63.00 0.00 1.35 3.50MCP08 0 100.00 98.00 97.08 76.00 2.00 2.93 4.00MCP09 52 74.00 73.00 72.18 71.00 1.00 1.83 3.00MCP10 15 92.50 90.50 89.40 88.00 2.00 3.10 4.50

Table 3: Comparison between heuristic results and optimal solutions.


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5 Conclusions

The main goal of this work was to demonstrate the potential of PSO in solving the Maximum CoveringProblem. This new way to handle the MCP is interesting because the MCP can be quite usefulin modeling important practical problems such as those occurring in simultaneous vehicle and crewscheduling.

The heuristic proposed was tested in a set of 10 randomly generated instances of medium size thatwere solved to optimality with LINGO 8.0.

The accomplished computational experience showed that the algorithm could produce solutions ofgood quality with a low computational cost. The process of collective learnship of the particles seemsto be a good way to guarantee a comprehensive search of the solution space.

Unfortunately, there are not available instances for the variant of Maximum Covering Problemstudied in the paper. Therefore, the comparison to other meta-heuristic is difficult. This comparisonis a topic for future studies.

Future research should also cover the design of heuristics for repairing unfeasible solutions, the gen-eration of an initial population, and the hybridization of the PSO heuristic with local search procedures.

Comprehensive tests with larger and heterogeneous instances are required, in order to fully validateand tune the proposed general approach. In line with this research area, the authors are developingan application of PSO for the Vehicle and Crew Scheduling Problem (VCSP). The MCP model for theVCSP is likely to have a good performance, when compared with the more traditional Set Covering/SetPartitioning approaches.

References

[1] Mauricio G. C. Resende. Computing approximate solutions of the maximum covering problem withGRASP. Journal of Heuristics, 4: 161-177, 1998.

[2] T. A. Feo and Mauricio G. C. Resende. A probabilistic heuristic for a computationally difficult setcovering problem. Journal of Operations Research Letter , 8: 67-71, 1989.

[3] R. G. I. Arakaki and L. A. N. Lorena. A constructive Genetic Algorithm for the Maximal CoveringLocation Problem. In Proceedings of 4th Metaheuristics International Conference, 13-17, Porto,2001.

[4] T. Park and K. R. Ryu Crew pairing optimization by a genetic algorithm with unexpressed genes.Journal of Intelligent Manufacturing, 17: 375-383, 2006.

[5] J. Kennedy and R. C. Eberhart. Particle swarm optimization. In Proceedings of the IEEE Inter-national Conference on Neural Networks, Piscataway, 1995.

[6] J. Kennedy and R. C. Eberhart. A new optimizer using particle swarm theory. In Proceedings ofthe 6th International Symposium on Micromachine and Human Science, Nagoya, 1995.

[7] J. Kennedy and R. C. Eberhart. A discrete binary version of the particle swarm algorithm. InConference on Systems, Man and Cybernetics, Hyatt, 1997.

[8] M. F. Tasgetiren and Y. C. Liang. A binary Particle Swarm Optimization Algorithm for the lotsizing problem. Journal of Economic and Social Research, 5: 1-20, 2003.

[9] J. Dreo, A. Petrowksi, P. Siarry and E. Taillard. Metaheuristics for hard optimization. Springler,Heidelberg, 2006.


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[10] N. Nedjah and L. M. Mourelle (Eds.). Swarm Intelligent Systems. Springler, Heidelberg, 2006.


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On Portfolio Selection Using Metaheuristics

Abubakar Yahaya ∗ Mike Wright ∗

∗Department of Management ScienceManagement School, Lancaster University, UK{a.yahaya, m.wright}@lancaster.ac.uk

1 Introduction

One of the main objectives of asset management is the process of intelligently combining set of attractiveassets into a single asset often called portfolio of assets. These portfolios are strongly required to beoptimal in the trade-offs between the two conflicting objectives of maximizing returns at the mostminimum possible risk. Before the advent of modern way of portfolio selection, management andtracking procedures - now known as Modern Portfolio Theory (MPT); assets selection decisions arepurely based on qualitative attraction of the available assets. Consequently, it was practically impossiblethen to incorporate some real-life constraints into such processes.

The portfolio selection model pioneered by Harry Markowitz for more than half a century ago isunarguably considered as the corner stone of the MPT. The model basically showed how investors canmake rational decisions for constructing portfolio of assets in an uncertain condition. It adopts the firsttwo moments of a normal distribution to respectively characterize the gain and risk associated with aninvestment. Hence, this mean-variance (E-V) model can be solved as a two-way quadratic programming(QP) optimization problem whose objective is either to maximize the expected return of a portfoliogiven that the risk does not exceed a given tolerance level or alternatively, minimize the portfolio risksubject to achieving a specified level of expected return.

Despite its enormous contribution in providing the foundation of mathematical framework in port-folio selection; the then newly found model/theory was practically handicapped by some underlyingassumptions guiding its implementation coupled with other limitations that makes it incapable of cap-turing the actual realism of events in an investment settings; some of which are:

1. The normality of asset returns: Under this framework, the asset rate of returns are assumed to benormally distributed; hence the characterization of the two conflicting elements (return and risk) of anyinvestment by respectively the first moment (mean) and the second moment (variance) of the normaldistribution. This assumption, in particular, has been vehemently criticized by several academics andpractitioners in the financial/investment sector, as it has been widely researched and believed that assetreturns do not exhibit a normal, but rather asymmetric behaviour.

2. It is a single period model: This means once investors have made their decisions concerning theallocation of Capital to different securities at the start of time period, they cannot take any furtheraction until the next time period. This is risky, because in reality portfolios are constructed such thatthey can be traded and rebalanced at any time the investor wishes (Portfolio Rebalancing).

3. The estimation of the underlying parameter inputs (return, variance and covariance) is consideredto be another downside to this model. The calculation of mean, variance and covariance of returns areconsidered to be vital for accuracy reasons.


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4. Variance, as a measure of risk has been attacked over the last few years. Some investors view itas a false and unfit indicator of investment risk. Since investors dislike negative deviation and embracepositive deviation from the mean, they argue that downside risk measures fit better.

5. Failure to capture real life investment constraints: In constructing a portfolio of assets, there arebasically some important decisions that have to be taken into consideration; some of which are meantto reduce administrative costs, avoid over-dependence upon one of the portfolio’s constituent assetsand/or at times to comply with the rules of some investment regulatory agency.

It should be noted that, the Markowitz’ E-V model is composed of three main features, namely theobjective, constraints and variables. It should also be understood that, most of the researches conductedon portfolio selection problem (henceforth, abbreviated as PSP) centered on some modifications madeto either one or more of these features. Researchers who argue that, Variance (the objective) is not anappropriate measure of risk have substituted it for another alternative measure. Konno and Yamazaki[2] used Mean Absolute Deviation (MAD) and Ballestero [4] adopted semivariance below the mean.

But modifying the objective alone is not sufficient to fix the flaws inherent in the original model,as there are other issues to do with investment constraints. However, any attempt to incorporate theserealistic practical constraints into the original model has some accompanying consequences, such as thevulnerability of the problem to become an NP-Hard with exponential time complexity which cannotbe solved within a reasonable time period. Consequently, the problem can now be solved by usingApproximate (Heuristic) algorithms. There are however, two classifications of heuristic algorithms:the simple heuristics (whose search terminates at local optimum), such as Simple Descent, Multi-start,e.t.c., and Metaheuristics (those that develop extra mechanisms for escaping local optimum), suchas Simulated Annealing, Tabu Search, Variable Neighbourhood Search, Genetic Algorithms, ParticleSwarm Optimization, Ant Colony, e.t.c.

Speranza [5] linearized the objective by using Mean Deviation below Average and using techniquesinvolving Branch & Bounds to solve the problem after incorporating additional constraints dealingwith transaction cost, transaction units, cardinality and integer variables. Hamza and Janssen [6] usedSeparable Programming techniques to solve the problem by adopting Semivariance as the objectivewhile maintaining all the constraints used by Speranza [5] except for the Cardinality constraint. Changet al [3] applied Genetic Algorithms, Tabu Search and Simulated Annealing to solve the problem,maintaining the original objective (variance) and (real) variables used by Markowitz [1], but introducedadditional practical constraints: the Cardinality and Floor & Ceiling constraints. Crama and Schyns[8] used Simulated Annealing algorithm to tackle the problem by introducing the turnover (Purchase& Sales) and trading constraints to the Chang et al [3] model.

2 Implementation Issues

2.1 The Cost FunctionThe objective function was originally designed as a single-criterion, minimizing portfolio risk subject toachieving a desired target return. However, recent modifications visualized the objective from multi-criteria perspective, where the portfolio risk is minimized and simultaneously maximizing the portfolioreturn [9]. In order to force the algorithm achieve a given target, we decided to penalize violation of thereturn constraint by a return penalty factor, λ; we then incorporated the constraint into the objective,which results in our problem formulation to take the form:

Minimize (1− λ)(∑n

i=1

∑nj=1 wiσijwj) + λ(|∑n

i=1 wiri −RT |)Subject to∑n

j=1 wi = 1 and 0 ≤ wi ≤ 1.Where ri is the expected rate of return for asset i, wi is the weight allocated to asset i, σij is thecovariance associated with assets i and j, RT is the target return to be achieved, λ is the returnpenalty factor.


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We set our return penalty factor to take a value between [0.5, 1] inclusive, so that there is always atrade-off between achieving the desired target and minimizing the risk.

2.2 Parameter choice for the Search TechniquesWe designed and coded (in C++) 2 Metaheuristic Search Algorithms: Particle Swarm Optimization(PSO) and a hybrid of Particle SWarm and Simulated ANnealing (SWAN). The success achieved byour algorithms is due to (some extent)the right choice of parameters associated with the respectivesearch methods.

For PSO, the acceleration coefficients were set to C1 = 1.25 and C2 = 9.75. The inertia factor,w was set such that it dynamically takes values between 0.40 and 0.90 as in [10]. A total of 31particles (equal to the magnitude of the problem dimension) were used and the algorithm terminatesafter conducting a total of 2000 iterations or when there are 500 consecutive non-improving functionsevaluations.

The parameter settings for SWAN combined the parameters for PSO (except for the stoppingcriteria) and Simulated Annealing. The cooling schedules were chosen in such a way that: (i) Theinitial temperature is set to T0 = 1, (ii) The cooling rate was set to α = 0.7, (iii) The Markov Chainlength was set at 100 iterations per temperature change, and (iv) The final temperature was set to Tf

= 0.001. The search stops also when there are 500 consecutive non-improving objective evaluations.

2.3 Search Initialization & Neighbourhood StructureThere are basically 2 important factors (namely, Initialization and neighbourhood generation system)that play a vital role in determining the efficiency of any search method.

2.3.1 Search InitializationThe fact that, implementation of PSP deal with investment constraints and working with values incontinuous domain strongly indicate that the process of initializing a search in this structure shouldnot be taken lightly. In view of this, we decided to look at the process of initialization from 4 differentperspectives (henceforth, referred to as scenarios and abbreviated as: S1, S2, S3 and S4 respectively),with the intent of comparing their effects on algorithmic performance. The 4 scenarios are defined asfollows:S1: involves starting from a randomly generated infeasible solution (

∑nj=1 wi 6= 1).

S2: involves beginning the search from a feasible solution (∑n

j=1 wi = 1), ensuring that assets’ weightsare all equal (i.e: wi = 1

n , ∀i).S3: involves starting from a randomly generated feasible solution, ensuring that assets’ weights areNOT all equal (i.e: wi 6= 1

n , ∀i).S4: involves starting from a randomly generated feasible solution for the uppermost point on theefficient frontier, while subsequent points use their immediate predecessors’ near-optimal configurationas their starting solution.It should be noted that, all the assets’ weights (w′is) generated in the scenarios are uniformly distributedtaking values only in the interval [0, 1].

2.3.2 Neighbourhood structureTo generate a neighbouring solution, we adopt an enhanced update mechanism as proposed in [7] givenby:Repeat ∀i = 1...numParticlesRepeat ∀j = 1...variables dimensionR1 = randomNumber ∈ [0, 1], R2 = randomNumber ∈ [0, 1]V ∗i,j = w × Vi,j + C1R1 × (XLB

i,j −Xi,j) + C2R2 × (XGBj −Xi,j)

X∗i,j = Xi,j + V ∗i,jendNormalize X∗i,j

end;where, Vi,j and V ∗i,j are respectively particle i

′s velocity at the jth variable prior to, and after theupdate; Xi,j and X∗i,j are the values taken by particle i at the jth variable before and after the update


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respectively. XLBi,j is particle i

′s personal best value at the jth variable and XGBj is the global best

position ever attained by any particle; while C1 and C2 are the acceleration coefficients.

2.4 The Unconstrained Efficient Frontier (UEF) & Benchmark OutperformanceWe run an AMPL (A Modelling Language for Mathematical Programming) program (invoking CPLEXSOLVER) to solve the same PSP (with equally-spaced target returns) as tackled by our algorithms andas a result, frontier of ’efficient’ points were generated.In order to objectively report the performance of our algorithms; we decided (for each of the 2 algorithmsand across all the 4 scenarios) to run 5 experimental trials on each of the 300 points (computed) onthe UEF, which amounts to a total of 12,000 experimental trials (5 trials×300 points×4 scenarios×2algorithms). We then compute the average of these points (for both return and risk) before computingsome performance metrics including MAD, Benchmark Domination (points in which our algorithmsoutperform the benchmark) and Average Execution Time in seconds.One of the vital questions this study is aimed to answer, is whether our algorithms are capable ofproducing results comparably efficient to the benchmark’s; we decided to answer that by testing:

IF (Rai = RB

i ) & (σai < σB

i ) THENAlgorithm (a) outperforms the benchmark at point i [∀i, i = 1...300]

ELSEAlgorithm (a) is beaten by the benchmark at point i [∀i, i = 1...300]

where: Rai and RB

i are respectively the ith portfolio returns by algorithm (a) and the benchmark;while σa

i and σBi are the corresponding respective ith portfolio risks by algorithm (a) and the benchmark.

3 Data & Computational Results

3.1 Dataset UsedThe dataset used in this research is the 31 stocks (Hang Seng, Hong Kong) data used by Chang et al[3] and available online at the OR-Library - http://people.brunel.ac.uk/ mastjjb/jeb/orlib/files/.

3.2 Results & DiscussionsThe following figures provide the results and the various UEF’s obtained by our algorithms and thebenchmark.

Figure 1: Summary of Results obtained from the search methods across the 4 different scenarios


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Figure 2 Figure 3

Figure 4 Figure 5

From figure 1, it can be observed that the Combined Average of MAD for the 2 search algorithmsin the first 3 scenarios are noticeably bigger than their corresponding values in S4; indicating thesuperiority of S4 over the others. This is further supported by the overall MAD values across scenarios,in which 2.3344, 0.5274 and 0.4308 respectively for S1, S2 and S3 are almost 123×, 28× and 23× biggerthan S4’s value of 0.0190. Similarly, figures 2, 3 and 4 show that the 2 algorithms form an area ratherthan an ’efficient’ line, which indicates presence of many ’inefficient’ portfolios that are far from beingclose to optimality. A mere glance at figure 5 reveals a remarkable improvement on the part of S4, andhence, the most efficient of them all.Although, PSO was able to outperform the benchmark in almost 50% ( 148

300 ) of the points on the UEFin S4; its overall MAD value of 4.8402 is almost 3 times bigger than that of SWAN.From the results presented above, it can be inferred that: the SWAN is the best algorithm, becauseapart from maintaining lowest MAD values across all scenarios, it also outperforms the benchmark byalmost 33% ( 99

300 ), 58% ( 173300 ), 59% ( 177

300 ) and 97% ( 292300 ) in S1, S2, S3 and S4 respectively.


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4 Conclusions

In this study, 2 search algorithms implemented on 4 different scenarios of search initialization werestudied and their results presented.The study reaffirms the validity of the fact that, a wise choice of a starting solution can immenselycontribute positively to the performance of a search technique.Furthermore, it can be inferred that, metaheuristics (especially, Evolutionary Algorithms - such asPSO) will continue to play significant roles in tackling complicated optimization problem, such as PSP.The results returned by SWAN are very encouraging, and this attest to the fact that, hybridized algo-rithms - especially those combining the features of Local Search techniques and Evolutionary Algorithmsare viable candidates for solving PSP.

5 Future Works

As part of our future work, we plan to implement the modified portfolio problem by: (i) substitut-ing the conventional risk measure (variance) with the more investor-preferred downside risk measure(semivariance), (ii) imposing additional real life (Cardinality and Floor & Ceiling) constraints to thedesigned Mean-Semivariance model, (iii) Hybridizing more of the already designed algorithms, to seeif any better performance can emerged from such hybrids, and (iv) Using a freshly downloaded datasetin order to evaluate the robustness of our algorithms.

References

[1] Markowitz, H. M. (1952). Portfolio Selection. Journal of Finance, 7:77–91.

[2] Konno, H. and Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model andits Applications to Tokyo Stock Market. Management Science, 37:519–531.

[3] Chang, -T. J., Meade, N., Beasley, J. E. and Sharaiha, Y. M. (2000). Heuristics for cardinalityconstrained portfolio optimization. Computers & Operations Research, 27, 1271–1302.

[4] Ballestero, E. (2005). Mean-Semivariance Efficient Frontier: A Downside Risk Model for PortfolioSelection. Applied Mathematical Finance, 12, 1:1–15.

[5] Speranza, M. G. (1996). A heuristic algorithm for a portfolio optimization model applied to theMilan stock market. Computers & Operations Research, 23, 5:433–441.

[6] Hamza, F. and Janssen, J. (1998). The Mean - Semivariances Approach to Realistic PortfolioOptimization subject to Transaction Costs. Applied Stochastic Models and Data Analysis, 14:275–283.

[7] Shi, Y. and Eberhart, R. (1998) A modified particle swarm optimizer. Proceedings on IEEEInternational Conference on Evolutionary Computation, 69–73.

[8] Crama, Y. and Schyns, M. (2003). Simulated annealing for complex portfolio selection problems.European Journal of Operational Research, 150, 546–571.

[9] Schaerf, A. (2002). Local search techniques for constrained portfolio selection problems. Computa-tional Economics, 20, 3:177–190.

[10] Kendall, G. and Su, Y. (2005). A particle swarm optimization approach in the constructionof optimal risky portfolios. Proceedings of the 23rd IASTED International Multi-Conference onArtificial Intelligence And Applications, Austria.


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Heuristic Search for the Stacking Problem

Rui Jorge Rei ∗ Joao Pedro Pedroso ∗ †

∗ Faculdade de Ciencias da Universidade do Porto: Departamento de Ciencia de ComputadoresRua Campo Alegre, 1021/1055, 4169-007 Porto


† INESC Porto – Instituto de Engenharia de Sistemas e Computadores do PortoRua Dr. Roberto Frias, 378, 4200 - 465 Porto, Portugal

1 Introduction

The stacking problem is a combinatorial optimization problem where the placement of items in a fixedset of positions is sought, so as to minimize the number of unnecessary item movements. Arising in thesteel industry, the problem has wide areas of application, since similar problems often appear in thecontext of item storage. This problem is very difficult, as its structure does not easily allow traditionaloptimization methods to be used.

In its generic form, the problem consists of a series of placement decisions for a set of items, withknown dates for entrance and exit from a storage location, where the items are organized in stacks.Placement choices should attempt to minimize retrieval effort when items’ exit dates are reached. Spaceconstraints also play an important role in these problems, since storage locations may have limited areaand/or height1. Hence, stacking policies optimizing resource and space usage are desirable and havehigh practical interest.


Consider the problem of stacking n items in a warehouse with p positions, or stacks. Let r ∈ Nn bethe list of release dates, where ri denotes the release date of item i, i.e., the time at which i is availablefor storage. In the same way, let d ∈ Nn be the list of due dates, i.e., the times at which items shouldbe delivered to customers. We assume that due dates are greater than release dates for every item:ri < di, for i = 1, . . . , n.

The objective is to store all the items, and deliver them to customers, with the minimum numberof moves. A move consists in taking a single item, either from the top of a stack or from the releasepoint, and placing it on top of another stack, or delivering it. Hence, there are three types of moves:

• A release move occurs when an item i enters the warehouse (at t = ri) and is placed on top ofa stack s. The release move of item i into stack s is represented by i : s0 → s, where s0 denotesthe release point.

1However, the problem handled in this paper does not impose a height limit to stacks.


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• A delivery move occurs when an item i, located at the top of a stack s, is removed from thewarehouse (at t = di). The delivery move of item i from stack s is represented by i : s → sp+1,where sp+1 denotes the delivery point.

• A reshuffling move occurs when an item i is shifted from the top of a stack s to the top ofanother stack s′ (with ri ≤ t ≤ di). The reshuffling move of item i from stack s to stack s′ isrepresented by i : s→ s′.Note that each item must have exactly one release move and one delivery move, so in order tominimize the total number of moves, it is enough to consider only reshuffling moves, or reshuffles.

A solution is represented by an ordered list of moves, which is constructed by selecting a stack foreach item, in the arriving order, using a heuristic function h. When an item is due, any items above itin the same stack are reshuffled (using h), and the item is delivered. This construction process, usingsimulation, allows the creation of an unambiguous solution to the problem. If there is some randomnessin the placement heuristic h, multiple constructions will lead, in general, to multiple solutions. Hence,we can run several simulations and select the one that leads to the minimum number of reshuffles, i.e.,use a simulation-based optimization process.

3 Background

The classic Towers of Hanoi (ToH) problem, with three posts and n disks, is optimaly solvable with asimple recursive algorithm, taking 2n − 1 moves to convey the stack from the first post to the third.However, this is a very hard problem in its general formulation, e.g. the four-post ToH problem hasno known proven optimal algorithm. Stockmeyer [7] surveys some variations of the ToH problem,including a discussion of their hardness. Korf and Felner [3] apply heuristic search methods combinedwith multi-objective pattern databases to optimaly solve the four-post ToH problem for up to 30 disks.

It should be noted that the stacking problem is in fact a variation of the ToH. Hence, it inherits thedifficulty of the four-post (or p-post) ToH problem, while gaining further complexity due a wider searchspace. While the four-post ToH is optimaly solvable with state-of-the-art techniques for up to 30 disks,we will study stacking policies that should be applicable for hundreds of items and dozens of stacks.Hence, full-search algorithms like branch-and-bound are likely to be unusable due to the dimension ofthe solution spaceLocal search is also inapplicable because defining the neighborhood of a given list ofmoves (or schedule) is too costly. This is so because altering a move in the move list may influence anarbitrary number of other moves, from that point onward.

In another similar area, Dekker et al. [2] study container terminal operation policies, utilizing asimulation environment to compare category and residence time stacking policies for containers. Thepaper also mentions the lack of published work on stacking problems. Though the problem handledin this paper is a rather simple stacking variant, with very few restrictions, it is nonetheless NP-hard(NP-completeness has been shown in [1] for a similar problem, arising in ship stowage planning).

In prior work on this problem [5], several different stacking policies were compared in a discrete eventsimulation environment. Among the compared policies, one of them, namely flexibility optimization(see 5.2), proved to be much better for all instances analyzed.

4 Test instances

The policies compared in [5] were tested in a set of 1500 different instances, with varying sizes (numberof stacks and number of items), and different values for parameters controlling the sparsity of releaseand due dates, as well as the maximum residence time of items.


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The instances were divided into easy and hard, with the main difference of having many or fewstacks. A reduced number of stacks usually leads to a higher number of reshuffling moves, hencemaking the instance harder.

The parameters controlling release and due date generation also influence the hardness of instances.It was observed (empirically) that concentrated release dates generally lead to easier instances, whileconcentrated due dates have the opposite effect. As for residence time, as a natural consequence ofmaintaining items in the system for longer periods, more inversions are likely to occur, causing anincrease in difficulty. Detailed explanation of the relations between generator parameters and instancehardness is out of the scope of this paper, and is left for future work.

5 Heuristics

In [5], it was shown that among the tested stacking policies, a greedy method based on flexibilityoptimization (detailed in 5.2) performed better in both easy and hard instances. However, the policy’sstrictness leads to few variations on the move list over several simulations, greatly reducing variety ofthe solutions. On the other hand, another solution construction method, conflict minimization (detailedin 5.1), allows a wider range of solutions by imposing less restrictions to the stack choice.

Motivated by the lack of variety in one method, and the poor quality of solutions in the other, anew heuristic was developed and compared with both. The new heuristic attempts to achieve betterperformance relying both on the flexibility optimization principle and increased solution variety, byincluding second-best alternatives in the placement choice.

5.1 Conflict Minimization (CM)

We say that a conflict, or inversion, occurs whenever an item i is placed on a stack containing at leastone item j, such that di > dj . Since i occupies a higher position in the stack, when t = dj , i will haveto be reshuffled to allow the retrieval of j for delivery.

This very simple heuristic uses only this notion to take placement decisions. The placement of eachitem i is randomly chosen from all stacks where i does not cause an inversion. If no such stacks exist,i.e., an inversion is unavoidable, the destination stack is randomly chosen among all available stacks.

5.2 Flexibility Optimization (FO)

We begin by defining flexibility for a stack s as the minimum due date of the items in s:

F (s) = min{di : i ∈ s}

The larger F (s), the more items with different due dates can be placed in stack s without inversions,thus making it more “flexible”. When placing an item i, we are interested in the variation of flexibilitycaused by that item on a given stack s:

∆F (i, s) = di − F (s)

A non-positive variation, ∆F (i, s) ≤ 0, means that i can go to s without inversion. On the otherhand, if ∆F (i, s) > 0, then placing i in s will cause an inversion, since we know that s contains at leastone item j such that di > dj . Note that ∆F (i, s) > 0 means an inversion, but not an increase in stackflexibility, since the flexibility F ′(s) after placing i in s is still equal to F (s).


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In case there are two or more non-inversion stacks, i.e., ∆F (i, s) ≤ 0, the FO heuristic will choosethe stacks with smaller |∆F (i, s)|, therefore preventing loss of flexibility. For instance, in the exampleshown in figure 1a, item C has two possible non-inversion choices. However, stack s2 is the choice withlower loss of flexibility.

∆F (C, s1) = −3 < ∆F (C, s2) = −1

The choice of saving flexibility does not represent an immediate gain, and may in fact yield no gainat all, but it seems nevertheless a good idea for preparing future arrivals. Suppose that, in the sameexample (figure 1a), an item D arrives after C, with dD = 5. If we had selected s1 for C, we wouldnow be at hand with a forced inversion, or a preparation move would be required (C : s1 → s2). Thissituation is avoided by selecting C : s0 → s2 in the first place.

6 4

stack 1 stack 2

3A B

C

5 4

stack 1 stack 2

6A B

C

Figure 1: a) Flexibility based decision in a multiple no-inversion case (left). b) Conflict delaying choice,forcing the inversion to show up later (right). A, B, and C identify the items, and the numbers insidethe boxes represent their due dates (dA, dB , and dC).

It seems thus that it is desirable to minimize the flexibility reduction when placing an item, incases where non-inversion stacks exist, i.e., to select a stack minimizing |∆F (i, s)|. Similarly, when anew inversion is unavoidable (∆F (i, s) > 0), in many cases, choosing the minimum ∆F (i, s) is better.Figure 1b shows such a case. Item C will forcibly cause an inversion in both stacks, but one reshuffleis avoided if stack s1 is chosen. If item C is placed above B and no more items arrive, B must bedelivered first, forcing a reshuffle (C : s2 → s1) which could be avoided by choosing s1 previously.

The FO heuristic is then built based on these two cases:

1. if there is a stack s such that ∆F (i, s) ≤ 0, in order to prevent loss of flexibility, stacks whichminimize |∆F (i, s)| are preferred.

2. if there is no stack s such that ∆F (i, s) ≤ 0, then a new inversion is unavoidable. The reshufflemove caused by this new conflict is deferred to the latest possible date by choosing stacks whichminimize ∆F (i, s).

Note that case 2 only occurs if case 1 is not possible, meaning that preference is always given tostacks where no new inversions are created. These rules are used by the heuristic for the selection ofthe destination stack as follows: ∆F (i, s) is calculated for each stack, building a candidate list ofstacks with the best (equal) variation; finally, a stack is randomly chosen from the candidate list.

5.3 Parameterized Flexibility Optimization (PFO)

As presented above, for each item i, the FO heuristic leads to a candidate list of stacks which resultin similar ∆F (i, s). However, in many cases this list has a single element; thus, repeated constructionwith this method is likely to lead to very similar or identical solutions.

A way to tackle this problem is the following: when the candidate list has a single item, then extendit with another stack which leads to the second-best flexibility variation. Then, select the first stackwith probability 1 − P , and the second with probability P . P is thus a parameter that controls howlikely the second-best candidate is chosen; in particular, if P is zero, this heuristics is equivalent to FO.The value used for P in this paper’s experiment was 0.01.


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5.4 Solution variety

As mentioned before, the PFO heuristic was created to compensate for the lack of variety in thesolutions produced by the FO heuristic. Figure 2 shows the comparison of these two heuristics alongwith CM, in two instances over 100 independent simulations.

Figure 2: Comparison of heuristics performance in a 40-stack instance (left) and a 2-stack instance(right). The dashed lines represent the actual number of reshuffles made by a given heuristic in eachof the 100 simulations, while the solid lines correspond to best number of reshuffles found so far.

It can be easily noted that FO has little variety in the number of reshuffles in both instances. Inthe 40-stack instance (left) we observe much worse performance from CM, while PFO and FO revealsimilar performance. However, taking advantage of increased variety, PFO leads to a lower minimumat the end of the 100 runs. As for the 2-stack instance (right), both PFO and CM perform better thanFO after the 100 runs. This is justified by their ability to generate very different solutions (throughless restrictive placement choices), as opposed to FO which creates fewer different solutions. We exploitthis capability by running a high number of independent simulations and selecting the best solutionfound. The results above support these ideas, by showing that a slight variation of FO can obtain alower number of reshuffles both in easy and hard instances with a relatively small number of runs.

5.5 Computational results

Using the instance generator with new combinations of parameters, 1536 new test instances werecreated. All instances have n = 1000; however easy instances have a number of stacks p ∈ {10, 20, 40},and hard instances have p ∈ {2, 3, 4}. Figures 3a and 3b show the minimum number of reshuffles foreach heuristic on 100 simulations in each of the easy and hard instances, respectively.

Figure 3: a) Heuristic performance on easy instances (left). b) Heuristic performance on hard instances(right). Each dot represents the best performance (minimum number of reshuffles) over 100 simulationsin one instance using the same heuristic. To improve visualization, the dots were randomly shifted inthe x-axis.


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In the set of easy instances we observe CM as the worst-performing heuristic. This, however, changesin the hard instances, where it’s performance is comparable to the others. As for FO and PFO, theirplots are quite similar in both sets of instances. As it is difficult to graphically determine which one isbetter, we provide table 1 to allow an accurate comparison of the heuristics.

Heuristic p = 2 p = 3 p = 4 p = 10 p = 20 p = 40CM 25341 22650 19869 10414 5099 1930FO 26141 12292 7876 2035 564 14

PFO 25390 11640 7425 1890 523 10

Table 1: Average best performance per instance size (number of stacks) after 100 simulations.

Note that for 2-stack instances, FO is actually the worst heuristic, while PFO is only the secondbest. However, on the remaining instance sizes, PFO is consistently better than FO and CM, leadingto the conclusion that increased solution variety leads to consistent improved results, thus confirmingthe hypothesis set in 5.4.

6 Conclusions and future work

We presented a hard combinatorial optimization problem with direct application in real life, studiedproperties of the problem instances, and used discrete event simulation to compare the performance ofdifferent heuristics for placement decisions.

The outcome of this work is an improved version of the previous best heuristic (FO), using theidea of creating a larger range of different solutions by introducing second-best stacks in the placementchoice. This, combined with running several simulations, led to a consistent better performance of thePFO heuristic on all instance types.

Future research includes testing different parameterizations of PFO, and to modify PFO in orderto adapt the value of P during simulation, e.g. according to the number of reshuffles up to the currentpoint in the simulation, or the difference in flexibility between best and second-best candidates.

References

[1] Mordecai Avriel, Michael Penn, and Naomi Shpirer. Container ship stowage problem: complexityand connection to the coloring of circle graphs. Discrete Applied Mathematics, 2000.

[2] Rommert Dekker, Patrick Voogd, and Eelco van Asperen. A general framework for schedulingequipment and manpower at container terminals. OR Spectrum, 2006.

[3] Richard E. Korf and Ariel Felner. Recent progress in heuristic search: A case study of the four-pegtowers of hanoi problem. In International Joint Conference on Artificial Intelligence, 2007.

[4] Michael Pidd. Computer simulation in management science - fourth edition. Wiley, 1988.

[5] Rui Rei, Mikio Kubo, and Joao P. Pedroso. Simulation-based optimization for steel stacking.In Le Thi Hoai An, Pascal Bouvry, and Pham Dinh Tao, editors, Modelling, Computation andOptimization in Information Systems and Management Sciences, Communications in Computerand Information Science (CCIS). Springer, 2008.

[6] Stewart Robinson. Simulation - The practice of model development and use. Wiley, 2004.

[7] Paul K. Stockmeyer. Variations on the four-post tower of hanoi puzzle. In Congressus Numerantium,1994.


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Application of Pareto Local Search and Multi-Objective Ant

Colony Algorithms to the Optimization of Co-Rotating Twin

Screw Extruders

C. Teixeira ∗ J.A. Covas ∗ T. Stutzle † A. Gaspar-Cunha ∗

∗ IPC - Institute of Polymers and Composites, University of MinhoGuimaraes, Portugal

Email: {cteixeira,jcovas,agc}@dep.uminho.pt

† IRIDIA, Universite Libre de BruxellesBrussels, Belgium


1 Introduction

Co-rotating twin screw extruders are extensively used in the polymer compounding industry mainlydue to their good mixing capacity. Given its modular construction, this type of machines can easilybe adapted to work with different polymeric systems, e.g., polymer blends, nanocomposites or highlyfilled polymers. Nevertheless, the performance of these machines is strongly dependent on the screwconfiguration and geometry. As a result, the definition of a best screw geometry to use in a specificpolymer system is an optimization problem that involves selecting the location of a set of available screwelements along the screw axis. This optimization problem is a sequencing problem, where a permutationof a specific number of different screw elements must be determined. The sequence determines theposition of screw elements along the screw axis aiming at maximizing performance. In fact, there aretypically a number of different, often conflicting optimization goals for defining the performance of ascrew configuration. Hence, the screw configuration problem is actually a multi-objective combinatorialoptimization problem (MCOP).

In previous work, a Multi-Objective Evolutionary Algorithm (MOEA) has been used to determinethe best sequence of a pre-defined number of screw elements in co-rotating twin screw extruders [1].In this work, we develop alternative algorithms for tackling this problem. In particular, we developeffective Stochastic Local Search (SLS) algorithms following the Pareto local search and two-phase localsearch frameworks [2, 3] as well as a Multi-Objective Ant Colony Optimization (MO-ACO) algorithm.We have carried out a detailed investigation of the sensitivity of the algorithms performance to changesof their parameters and a comparison to the previously designed MOEA using different objectives.This paper is organized as follows. The twin-screw extrusion configuration problem is described inSection 2 and in Section 3 we give details of the algorithms we use. Some experimental results withthese algorithms are presented in Section 4 and we conclude in Section 5.


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Figure 1: Twin-screw extrusion configuration problem

2 Twin-screw Extrusion Configuration Problem

The choice of an optimization methodology depends mainly on the type of problem to solve. Weconsider the case, where the problem of determining the best screw configuration consists in findinga sequence of a fixed number of screw elements of a twin screw extruder. We denote this problemas Twin Screw Configuration Problem (TSCP). The TSCP is illustrated in Figure 1; it involves thedetermination of the position along the screw of 10 transport elements, 3 kneading blocks (with differentstaggering angles) and one reverse element.

The flow behavior is induced by the different screw elements in dependence of their geometricalcharacteristics. Right handed elements have conveying properties while left handed and kneadingblocks with a negative staggering angle induce restriction (generating pressure) to the flow. These arecalled restrictive elements. After the solid polymer is fed into a hopper, it will flow under starvedconditions through transport elements. When a restrictive element is reached, the channel starts to fillup and the melting process takes place. When all polymer is melted, the flow occurs with or withoutpressure in the rest of the screw elements, depending on whether it is totally or partially filled; overall,pressure is determined by the location of the restrictive elements. The evaluation of the performanceof each screw configuration is made by an elaborated computer simulation of the polymer flow throughthe screw elements that takes into account the relevant physical phenomena. Each evaluation of ascrew configuration takes about one to two minutes on current CPUs. Hence, the high computationaleffort allows only a rather limited number of function evaluations. In this example we consider theoptimization goals average strain, specific mechanical energy (SME) and viscous dissipation.

3 Multi-Objective Optimization

3.1 Iterative improvement algorithms

SLS algorithms [5] have been successfully applied to single objective problems and, less frequently toMCOPs. Successful single-objective based SLS algorithms can be straightforwardly extended in twoprincipled ways to multi-objective optimization problems [2, 3]. One possibility is to apply single-objective SLS algorithms to aggregations of the various objective functions into a single one. Analternative is to adopt a component-wise acceptance model, where a neighboring solution is accepted ifit is non-dominated. Both strategies were tested using as underlying SLS methods iterative improvementalgorithms. We use Pareto Local Search (PLS) as an example using a component-wise ordering searchmodel [2], and the Two-Phase Local Search (TPLS), based on the scalarized acceptance criterionsearch model [3]. In both cases, the 2-swap operator was used to define the neighborhood relation: twosolutions are neighbors if one can be obtained from the other by swapping the position of two screwelements [5].


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Pareto Local Search The main ideas of PLS are the use of an archive, where all non-dominatedsolutions found so far are kept, and the exploration of the neighborhood of each of these solutionsbased on non-dominance criteria [2]. The algorithm starts with a random initial solution. This solutionis added to the archive and its neighborhood is explored. All non-dominated solutions identified in theneighborhood exploration are added to the archive, if they are not dominated by any of the solutionsin the archive. If any solution in the archive would become dominated, it is eliminated. These steps ofsolution selection and archive update are iterated until the neighborhood of all solutions in the archivehas been explored. In order to avoid a too strong increase of the number of solutions in the archive,an archive bounding technique was used [6]. This bounding technique divides the objective space by agrid into hypercubes and allows only one non-dominated solution to occupy a given hypercube.

Two-Phase Local Search In the TPLS algorithm, the various objectives are aggregated using aweighted sum and the local search process generates a sequence of solutions through occasional changesof the weights [3]. TPLS starts from a random initial solution optimizing one specific objective (i.e.,all weights are equal to zero except one). The resulting solution of this first phase is added to thearchive and it is taken as the initial solution for the second phase. In the second phase, at each step aweighted sum aggregation is considered and the starting solution is the final solution from the previousstep. For each weight vector, local search is applied and the final solution returned is added to thearchive. The search process stops when all weight vectors are explored. As in [3], we follow a strategythat minimally changes the component-wise differences between the weight vectors. All non-dominatedsolutions found during this search process are added to the archive.

3.2 Multi-Objective Ant Colony Optimization

Ant Colony Optimization is a population-based algorithm that is inspired by real ants’ foraging behav-ior [4]. In recent years, several approaches have been proposed to apply this algorithm to multi-criteriaoptimization problems [7]. The ACO algorithm is based on a probabilistic solution construction, wherethe probabilities are a function of the pheromone strengths on the ants’ “trail”. The solution com-ponents of the better or the best solutions receive reinforcement through the deposit of an amount ofpheromone that typically depends on the solutions’ quality. Simultaneously, the pheromone trails ofall solution components are decreased by a pheromone evaporation mechanism. This process induces asearch around the best solutions found.

ACO algorithms can be applied to multi-objective problems, for example, by changing the way thepheromone information is considered. Two different approaches are usually followed. The first approachconsists in using a single pheromone matrix for all the objectives, while in the second approach onematrix for each objective is used [7]. When using a single pheromone matrix the solution constructionfollows the usual steps of the ACO algorithm. However, if for each objective one pheromone matrixis used, then the various pheromone informations are typically aggregated by using a weighted sumbetween the two pheromone matrices. In this case, for a specific ant m a screw element j will be chosenafter a screw element i with a probability equal to:1

pmij =τλ1ij · τ ′λ2

ij∑l∈N l

i

[τλ1il · τ ′λ2

il

] if j ∈ Nmi ,

where Nmi is the feasible neighborhood of the ant m (i.e., the screw elements which are still available)

and λ1, λ2 with λ1 + λ2 = 1, are the weights given to each pheromone matrix. The exploration ofdifferent regions of the search space is accomplished by attributing different sets of weights to thedifferent ants to be considered during the search process. All non-dominated solutions generated along

1A pheromone model, where τij refers to the desirability of assigning a screw element i to a position j in the sequencewas also tested. However, this pheromone model was found to be inferior to the successor based one described above.


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Screw 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Length 97.5 120 45 60 30 30 30 60 30 120 30 120 37.5 60 60 30Pitch 45 30 KB-45 30 -20 60 30 20 KB-60 30 30 60 KB-30 45 30 20

Table 1: Configuration of the individual screw elements

Case Study Objectives Aim Xmin Xmax

1Average Strain Maximization 1000 15000

Specific Mechanical Energy (SME) Minimization 0.5 2

2Average Strain Maximization 1000 15000

Viscous dissipation Minimization 0.9 1.5

3Specific Mechanical Energy (SME) Minimization 0.5 2

Viscous dissipation Minimization 0.9 1.5

Table 2: Optimization objectives, aim of optimization and prescribed range of variation used ia eachcase

the successive iterations are stored in an archive. When one pheromone matrix for all objectives is used,the pheromone update is done using at most five non-dominated solutions that each deposit an amountof pheromone of 1/mu, where mu is the number of ants that deposit pheromone. If one pheromonematrix for each objective is used, only the best ant for each objective is allowed to deposit pheromoneon the corresponding matrix.

Two different strategies were tested in what concerns the pheromone update. One is based on thenon-dominated solutions found in the current iteration (iteration-best strategy) and the other considersall non-dominated solutions found so far (best-so-far strategy). The best strategy is the latter one and,for this reason, it was adopted in the remainder of this work.

4 Results and Discussion

The performance of the algorithms presented above were tested using the individual screw elementspresented on Table 1, for a Leistritz LSM 30-34 extruder, and the objectives presented on Table 2.Each optimization run was performed 10 times using different seed values. The comparison betweenthe algorithms was made using the attainment functions methodology [8, 9].

4.1 MO-ACO results

To test the ability of the MO-ACO algorithm to deal with the TSCP problem, an initial study consider-ing various values for the algorithm parameters (such as, pheromone evaporation, use of one or variouspheromone matrices, different types of assignment of screw elements, use of the probability summationrule and use of various colonies) was performed. Figure 2 presents the results obtained for the EmpiricalAttainment Functions (EAFs) when comparing the use of one versus various (one for each objective)pheromone matrices for case study 1. The left plot represents the region of the Pareto frontier wherethe EAF obtained with one matrix is higher than the results obtained with various matrices, while onthe right are represented the regions of the Pareto frontier where the second method is better. As canbe seen, the performance when using various matrices is higher. Identical results were obtained for theremaining case studies.

Figure 3 represents three different screw configurations taken from the Pareto frontier shown. Asexpected, when the SME (objective to be minimized) increases, the restrictive elements are locateddownstream of the screw configuration (in case C the polymer will melt earlier because the restrictiveelements are located earlier, and consequently, the energy necessary to rotate the screws is higher).The opposite is true when considering the Average Strain objective that is to be maximized.


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5500 6500 7500 8500 9500Average Strain

0.8

0.9

11.

1

SME

One matrix

(0.8, 1.0](0.6, 0.8](0.4, 0.6](0.2, 0.4](0.0, 0.2]

5500 6500 7500 8500 9500Average Strain

0.8

0.9

11.

1

SME

Several matrices

(0.8, 1.0](0.6, 0.8](0.4, 0.6](0.2, 0.4](0.0, 0.2]

Figure 2: Location of differences in terms of EAFs between the use of one or several pheromone matricesin MO-ACO. The shades of grey indicate the strength of the positive differences.

Figure 3: Pareto front for case study 1 and the screw configuration for solutions A to C

4.2 Comparison with SLS algorithms

Figure 4 shows an example of the comparison between MO-ACO and TPLS considering identical numberof evaluations of the objective functions. Similar results are obtained for the remaining case studies.As can be seen, the TPLS algorithm appears to be significantly better than MO-ACO especially in thecenter of the fronts. In fact, further comparisons have shown that typically the SLS algorithms alsoimprove upon the performance of the previously applied MOEA.

5 Conclusions

Simple SLS algorithms and MO-ACO algorithms have been applied with success to the Twin ScrewConfiguration Problem. The solutions obtained are in agreement with the knowledge about the processand have physical meaning. The good performance obtained with the simple SLS algorithms indicatesthat the incorporation of iterative improvement methods into the MO-ACO algorithm may further


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6000 6500 7000 7500 8000 8500 9000 9500Average Strain

0.75

0.8

0.85

0.9

0.95

1

SME

MO!ACO

(0.8, 1.0](0.6, 0.8](0.4, 0.6](0.2, 0.4](0.0, 0.2]

6000 6500 7000 7500 8000 8500 9000 9500Average Strain

0.75

0.8

0.85

0.9

0.95

1

SME

TPLS

(0.8, 1.0](0.6, 0.8](0.4, 0.6](0.2, 0.4](0.0, 0.2]

Figure 4: Location of differences in terms of EFAs between MO-ACO and TPLS for case study 1

improve its performance.

References

[1] A. Gaspar-Cunha, J.A. Covas and B.Vergnes. Defining the Configuration of Co-Rotating Twin-Screw Extruders With Multiobjective Evolutionary Algorithms. Polymer Engineering and Science,45:1159–1173, 2005.

[2] L. Paquete and T. Stutzle. A study of stochastic local search algorithms for the biobjective QAPwith correlated flow matrices. European J. Operational Research, 169:943–959, 2006.

[3] L.Paquete and T. Stutzle A Two-Phase Local Search for the Biobjective Travelling SalesmanProblem. In C. M. Fonseca, P. J. Fleming, E. Zitzler, and K. Deb and L. Thiele, editors, LectureNotes in Computer Science, pages 479–493, 2003.

[4] M. Dorigo and G.Di Caro. The Ant Colony Optimization Metaheuristic. In D. Corne, M. Dorigoand F. Glover, editors, New Ideas in Optimization. McGraw-Hill, London, UK, 1999, 11–32.

[5] H. Hoos and T. Stutzle. Stochastic Local Search - Foundations and Applications. Morgan KaufmannPublishers, San Francisco, California, USA, 2005.

[6] E. Angel, E. Bampis and L. Gourves. A dynasearch neighborhood for the bicriteria travelingsalesman problem. In X. Gandibleux et al., editors, Metaheuristics for Multiobjective Optimisation,volume 535 in Lecture Notes in Economics and Mathematics Systems, pages 153–176, 2004.

[7] C. Garcıa-Martınez, O. Cordon and F. Herrera. A taxonomy and an empirical analysis of multipleobjective ant colony optimization algorithms for the bi-criteria TSP. European J. OperationalResearch, 180:116–148, 2007.

[8] M. Lopez-Ibanez, L. Paquete and T. Stutzle. Hybrid population based algorithms for the bi-objectivequadratic assignment problem. J. of Mathematical Modelling and Algorithms, 5: 111-137, 2006.

[9] V. G. da Fonseca, C. Fonseca and A. Hall. Inferential performance assessment of stochastic opti-misers and the attainment function. In E. Zitzler et al., editors, Proceedings of EMO’01, volume1993 in Lecture Notes in Computer Science, pages 213–225, 2001.


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Multi-Objective Memetic Algorithm using Pattern Search Filter

Methods

F. Mendes † V. Sousa ∗ M.F.P. Costa ∗ A. Gaspar-Cunha †

† IPC/I3N - Institute of Polymers and Composites, University of MinhoGuimaraes, Portugal

Email: {fmendes,agc}@dep.uminho.pt

∗ Mathematics for Science and Technology Department, University of MinhoGuimaraes, Portugal

Email: {vmariano,mfc}@mct.uminho.pt

1 Introduction

Solving real world Multi-Objective Optimization Problems (MOOP) often involves the use of complex“black-box ”modeling routines, where the resolution of the process governing equations requires the useof expensive numerical methods [1]. Multi-Objective Evolutionary Algorithms (MOEA) is an excellenttool to deal with the multi-objective nature of these problems. They do not need the calculation ofderivatives which are not available for this type of problems [2, 3]. Simultaneously, due to the fact thatthe MOEAs are based on the use of a population of solutions that evolves during successive generations,they are a good global search method and are particularly suitable to solve multi-objective problems[2]. The concept of non-dominance is used in MOEA in order to establish a trade-off between thesolutions, i.e., the Pareto frontier [2]. The major difficulty in applying MOEAs to real optimizationproblems lies on the large number of evaluations of the objective functions necessary to obtain anacceptable solution. Due to the usual high computation times required by the numerical methodsused, reducing the number of evaluations necessary to reach an acceptable solution is thus of majorimportance [4]. Different approaches have been pursued in the literature to solve this problem. One ofthese methods consists in using approximate objective functions, such as statistical methods or ArtificialNeural Networks (ANN), to evaluate the solutions [5]. An alternative consists in coupling MOEAs withlocal search methods, where in each generation of the MOEA some (good) solutions using an efficientlocal search algorithm are generated [6].

The objective of this work is to present the details concerning the development and application ofa hybrid optimization methodology where a local search algorithm was incorporated on a MOEA. Theaim will be to accelerate the global search of the optimization algorithm through the generation oftentative individuals by the use of a local search algorithm. For that purpose a Pattern Search FilterMethod (PSFM) [7] was used to perform the local search in the neighborhood of some non-dominatedsolutions selected from the Pareto frontier. The sensitivity and the performance of the methodologyproposed was tested using some benchmark problems from the literature. The results obtained indicatethat this methodology is able to reach to good results.


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2 Pattern Search Filter Method

Growing interest in efficient optimization methods has led to the development of news techniques forNonLinear Programming (NLP). Fletcher and Leyffer [8] proposed the filter methods, as an alternativeto merit functions, as a tool to guarantee global convergence in algorithms for NLP. In a NLP problemis necessary to minimize two objectives: the objective function f(x) and a measure of the constrainedviolation θ(x). A filter is defined as a set of non-dominance pairs, (θ(xl), f(xl)), that correspond to acollection of points xl. A point x′ on the search space is accepted if it is not dominated by any of othersthat are in filter. The filter methods has been extended successfully to different areas of optimization [9].Pattern search and direct search methods are one of most popular classes of derivative-free methods [10].They are appropriate when some functions are given as “black-box”. They are based on generatingsearch directions which positively span the search space. Direct search methods for unconstrainedminimization generate a sequence of iterates {xk} ∈ RN with non-increasing objective function values.At each iteration, the objective function is evaluated in a finite set of trial points (computed by simplemechanism). Pattern search methods can be seen as direct search methods for which the rules ofgenerating the trial points follow stricter calculations. A detail review of direct and pattern searchcan be found in [10]. The central notion in direct or pattern search is positive spanning [10]. One ofthe simplest positive spanning set is formed by the vectors of the canonical basis and its counterparts:D⊕ = {e1, . . . , eN ,−e1, . . . ,−eN}. The direct search method based on this positive spanning set isknown as coordinate or compass search and its structure is basically all we need from direct or patternsearch in this work.

3 Memetic Multi-Objective Algorithm

3.1 Multi-Objective Evolutionary Algorithms

The capacity of EAs in exploring and discovering Pareto-optimal fronts for multi-objective optimizationproblems is well recognized [2]. The ability of the MOEAs in finding the Pareto front in a single run is animportant characteristic of this method. The aim of a MOEA is to have an homogeneous distribution ofthe population along the Pareto frontier together with an improvement of the solutions along successivegenerations. Generally MOEAs replace the selection phase of a traditional EA by a routine able todeal with multiple objectives [2, 3]. In this work the Reduced Pareto Set Genetic Algorithm (RPSGA)is adopted [3]. The main steps of this algorithm are illustrated below (Algorithm 1).

Algorithm 11. Random initial population (internal).2. Empty external population.3. while not Stop-Condition do

a) Evaluate internal population.b) Calculate the Fitness of the individuals using clustering.c) Copy the best individuals to the external population.d) if the external population becomes full then

Apply the clustering to this population.Copy the best individuals to the internal population.

end ife) Select the individuals for reproduction.f) Crossover.g) Mutation.

end while

Initially, an internal population of size N is randomly defined (step 1) and an empty externalpopulation formed (step 2). At each generation, i.e., until a stop condition is not found (step 3) the


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following operations are performed: the internal population is evaluated using the modeling routine(step 3a); a clustering technique is applied to reduce the number of solutions on the efficient frontierand to calculate the fitness of the individuals of the internal population (step 3b); a fixed number ofbest individuals are copied to an external population (step 3c); if the external population is not totallyfull, the genetic operators of selection (step 3e), crossover (step 3f) and mutation (step 3g) are appliedto the internal population to generate a new better population; when the external population becomesfull (step d) the clustering technique is applied to sort the individuals of the external population, anda pre-defined number of the best individuals are incorporated in the internal population by replacinglowest fitness individuals. More information about this algorithm can be found elsewhere [3]. Theinfluence of some important algorithm parameters, such as size of internal and external populations,number of individuals copied to the external population in each generation and from the external tothe internal population and the limits of the indifference of the clustering technique, have been studied[3].

3.2 Hybrid Algorithm Proposed

The methodology proposed consists in coupling MOEAs with a local search method. The main ideais, in each generation of the MOEA, some (good) solutions are to be obtained using an efficient localsearch algorithm. For that purpose an additional step will be added to the RPSGA algorithm after theselection phase (step e of Algorithm 1). Some solutions belonging to the non-dominated front will beselected. Starting in each one of these solutions a local search procedure will be carried out and newbetter solutions will be generated and incorporated in the main population. The local search methodselected was the PSFM referred above. This methodology will provide the way to improve the speed ofthe search since some selection pressure was introduced. The implementation of the hybrid methodologyproposed implies the insertion of the following steps after the selection phase above (Algorithm 2). Afterrunning the MOEA alone during a certain number of generations (Ngen) a pre-defined number of non-dominated solutions (Nsol) are selected from the current population. For that purpose the Clusteringprocedure (similar to step b of Algorithm 1) is used. For each one of the solutions the PSFM is applied.In this case, the first filter method objective (objective 1) corresponds to objective 1 (of the originalproblem) and the second objective (objective 2) is obtained through the transforming the remainingoriginal objectives in restrictions. Then, the application of the PSFM will produce a new generation.At the end, all the new solutions are incorporated on the main population. This procedure is carriedout every Ngen generations.

Algorithm 2f) If generation = Ngen then

- Select the Nsol using the clustering technique- Set S = {}

- for i=1 to n. of objectives do. Objective 1 = i. Objective 2 = restriction (for a objective different of i). Apply the PSFM to get new set of Nsol solution Si

. Set S = S ∪ Si

- Incorporate S, the solutions generated, in the main population- Ngen = 0

4 Results and Discussion

4.1 Benchmark test Problems

The approaches proposed will be tested using several difficult benchmark problems, such as ZDT1,ZDT2, ZDT3, ZDT4 and ZDT6 two and three-objective functions. These functions cover various types


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of Pareto-optimal fronts, such as convex (ZDT1), non-convex (ZDT2), discrete (ZDT3), multimodal(ZDT4) and non-uniform (ZDT6) [2]. Table 1 presents the number of decision variables (N), the objec-tive functions and the interval of variation of the different decision variables. All objective functions areto be minimized. The performance of the methods was assessed using a hypervolume metric developedfor comparing different Pareto sets [11]. Since the most significant is to assess the performance in whatconcerns the number of evaluations of the objective functions, the hypervolume will be plotted againstthe number of evaluations.

Table 1: Bi-objective test problemsName N Objective x

ZDT1 30 f1(x) = x(1)

f2(x) = g(x)(

1−√

f1(x)g(x)

)

where g(x) = 1 + 9PN

i=2 x(i)

N−1

x ∈ [0, 1]

ZDT2 30 f1(x) = x(1)

f2(x) = g(x)(

1−(

f1(x)g(x)

)2)


i=2 x(i)

N−1

x ∈ [0, 1]

ZTD3 30 f1(x) = x(1)

f2(x) = g(x)(

1−√

f1(x)g(x) −

(f1(x)g(x)

)sin (10πf1(x))

)


i=2 x(i)

N−1

x ∈ [0, 1]

ZTD4 10 f1(x) = x(1)

f2(x) = g(x)(

1−√

f1(x)g(x)

)

where g(x) = −9 + 10N +∑N

i=2

(x(i) 2 − 10 cos(4πx(i))

)

x(1) ∈ [0, 1]x(i) ∈ [−5, 5](i = 2, . . . , N)

ZDT6 10 f1(x) = 1− exp(−4x(1)) sin6(6πx(1))

f2(x) = g(x)(

1−(

f1(x)g(x)

)2)

where g(x) = 1 + 9( PN

i=2 x(i)

N−1

)0.25

x ∈ [0, 1]

4.2 Study of the influence of Algorithm Parameters

Figures 1 and 2 shows the evolution of the hypervolume metric with the number of evaluations as afunction of the number of solutions (Nsol) and number of generations (Ngen) parameters for the ZDT1test problem with 2 objectives, respectively. As can be seen in Figure 1 the performance of the hybridmethod proposed is much higher than that of the RPSGA alone. The same level of the hypervolumemetric is attained after around 5000 evaluations in the case of the hybrid method while this level isattained after 12000 generations when the RPSGA is used alone. Also, the number of solution selectedin each generation where the PSFM is applied does not present almost any effect on the algorithmperformance. However, the results obtained when the number of solutions is equal to 10 are slightlybetter. A different conclusion is achieved when the number of generation’s parameter is considered(Figure 2). In this case the best result is obtained when the number of generations is 2% of the totalnumber of generations (in this case 6 generations).


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0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000 30000N. of evaluations

Hyp

ervo

lum

e

RPSGAn. sol.= 5n. sol.=10n. sol.=30

Figure 1: Influence of the number of solutions selected on the algorithms performance (for ZDT1problem)

0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000 30000

N. of evaluations

Hype

rvol

ume

RPSGAn. ger.=1%n. ger.=2%n. ger.=5%

Figure 2: Influence of the interval of generations where the local search is applied on the algorithmsperformance (for ZDT1 problem).

4.3 Results for Bi-Objective Test Problems

Figure 3 shows an example of the results obtained for the two objective ZDT2 problem. As can beseen, in this case the hybrid algorithm used is able to reach a solution much better than the RPSGAalone and, as in the previous case (ZDT1 problem), is able to reach the same level of the hypervolumemetric (0.8) after 5000 generations, while the RPSGA alone only is able to reach this level after 12000generations. Identical general results have been obtained for the remaining (two and three objective)problems tested.

5 Conclusions

A memetic multi-objective algorithm, based in the incorporation of a PSFM within a MOEA, wasproposed. The results produced using some difficult test problems indicate that the hybrid methodologyproposed is able to improve the final solution obtained but, simultaneously, to reach to this solution inless than half the number of evaluations of the objective functions necessary.


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0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000N. of evaluations

Hyp

ervo

lum

e

RPSGA

Memetic

Figure 3: Comparison between the performances of the RPSGA alone an of the Memetic algorithm(for ZDT2 problem).

References

[1] J.A. Covas, A. Gaspar-Cunha and P. Oliveira An Optimization Approach to Practical Problems inPlasticating Single Screw Extrusion, Polym. Eng. and Sci., 39:443–456, 1999.

[2] K. Deb Multi-Objective Optimization using Evolutionary Algorithms Chichester, Wiley, 2001.

[3] A. Gaspar-Cunha, J.A. Covas - RPSGAe - A Multiobjective Genetic Algorithm with Elitism: Ap-plication to Polymer Extrusion In X. Gandibleux, M. Sevaux, K. Sorensen, V. T’kindt, Editorss,Metaheuristics for Multiobjective Optimisation, Lecture Notes in Economics and Mathematical Sys-tems, Springer, 2004.

[4] Y. Jin, M. Olhofer, B. Sendhof A Framework for Evolutionary Optimization with ApproximateFitness Functions IEEE Trans. on Evolutionary Computations, 6:481–494, 2002.

[5] A. Gaspar-Cunha, A. Vieira A Multi-Objective Evolutionary Algorithm Using Neural NetworksTo Approximate Fitness Evaluations International Journal of Computers, Systems, and Signals,6:18–36, 2005.

[6] C. Grosan, A. Abraham, H. Ishibuchi Hybrid Evolutionary Algorithms Studies in ComputationalIntelligence, 75, 2007.

[7] C. Audet, J.E. Dennis A Pattern Search Filter Method for Nonlinear Programming without Deriva-tives SIAM Journal on Optimization, 14:980–1010, 2004.

[8] R. Fletcher and S.L. Leyffer Nonlinear Programming without a Penalty Function MathematicalProgramming, 91:239–269, 2002.

[9] Fletcher R., Leyffer S. and Toint P. A Brief History of Filter Methods Mathematics and ComputerScience Division, Preprint ANL/MCS-P1372, 2006.

[10] Kolda T.G., Lewis R.M., Torczon V. Optimization by Direct Search: New Perspectives on SomeClassical and Modern Methods SIAM Review, 45(3):385–482, 2003.

[11] E. Zitzler, K. Deb, L. Thiele Comparison of Multiobjective Evolutionary Algorithms: EmpiricalResults Evolutionary Computation, 8:173–195, 2000.


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Metaheuristics for the Bi-Objective Orienteering Problem

Michael Schilde Karl F. Doerner Richard F. Hartl

Department of Business Administration, University of ViennaBruenner Strasse 72, 1210 Vienna, Austria

Email: {michael.schilde, karl.doerner, richard.hartl}@univie.ac.at

1 Introduction

In this paper heuristic solution techniques for the multi-objective orienteering problem (MOOP) aredeveloped. The motivation stems from the problem of planning individual tourist routes in a city.Each point of interest (POI) in a city provides different benefits for different categories (e.g. culture,shopping). Each tourist has different preferences for the different categories when selecting and visitingthe points of interests (e.g. museums, churches). Hence a multi-objective decision situation arises.

To determine all the Pareto optimal solutions two metaheuristic search techniques are developedand applied. The Pareto ant colony optimization algorithm is adapted and the design of the variableneighborhood search method is modified and extended to the multiobjective case. Both methods arehybridized with path relinking procedures. The performances of the two algorithms are tested onseveral benchmark instances as well as on real world instances from different Austrian regions and thecity of Vienna and Padua. The computational results show that both implemented methods are wellperforming algorithms to solve the multi-objective orienteering problem.


The MOOP is the multi-objective extension of the Orienteering Problem (OP). The OP, introduced byTsilirigidis [14], is a generalization of the well known Traveling Salesperson Problem (TSP). The OP isalso known as selective traveling salesperson problem. Each vertex of the problem provides a certainbenefit. The aim of a traditional OP is to select a subset of vertices within a given complete graph suchthat the achieved sum of revenues collected by visiting these vertices is maximized subject to a giventour length restriction.

Like the standard OP, the MOOP can be defined as a directed graph problem on the graph G =(V,A). This graph consists of a set of vertices, V = {v0, v1, v2, . . . , vn+1}, and a set of arcs, A ={(vi, vj) : vi, vj ∈ V ∧ i 6= j ∧ i 6= n+ 1∧ j 6= 0}. A certain number of K benefit values sik (k = 1 . . .K)is assigned to each vertex i ∈ V \ {v0, vn+1}. The starting vertex v0 and the ending vertex vn+1 areassigned benefit values of zero for each objective. Additionally, every arc (vi, vj) ∈ A is assigned a costvalue cij which can be interpreted as time, money spent, or distance traveled when going from vertex ito vertex j. As usually the starting and ending vertex refer to the same point (v0 = vn+1), we denotea solution to the MOOP as a tour instead of using the more general term path.

A solution x dominates a solution x′ if x is at least equally good as x′ with respect to all objectivefunctions, and better than x′ with respect to at least one objective function. In formal terms: For


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(f1, . . . , fK) to be maximized, x dominates x′, if fk(x) ≥ fk(x′) for all k = 1, . . . ,K, and fk(x) > fk(x′)for at least one k. In this case, we write x � x′.

A solution x∗ is called Pareto-efficient (or: non-dominated) if there is no feasible solution thatdominates x∗. In addition, if x∗ is Pareto-efficient, then z∗ = f(x∗) = (f1(x∗), . . . , fK(x∗)) is calleda non-dominated vector. The set of all non-dominated vectors is referred to as the Pareto front (or:non-dominated frontier). We extend the relation � from the solution space to the objective space bydefining, for two vectors z = (z1, . . . , zK) and z′ = (z′1, . . . , z

′K), that z � z′ holds iff zk ≥ z′k for all

k = 1, . . . ,K and zk > z′k for at least one k.

The aim of the MOOP is to find all Pareto efficient tours that start at vertex v0, end at vertex vn+1

and do not violate the maximum tour length restriction Tmax.

3 Solution procedures

The following subsections provide a brief summary of the implemented solution procedures ParetoAnt Colony Optimization (P-ACO) and Pareto Variable Neighborhood Search (P-VNS) as well as forthe used Path Relinking (PR) procedure. For further implementation details, the interested reader isreferred to the technical report by Schilde et al. [12].

3.1 Pareto Ant Colony Optimization

The P-ACO was originally developed and applied to a multi-objective portfolio selection problem byDoerner et al. [3] as a Pareto extension to the general ACO framework [5]. The P-ACO algorithm wasthen applied to different problem classes, like the multi-objective activity crashing problem [4].

The initialization phase of the implemented P-ACO algorithm starts with creating a population ofartificial ants which initially are positioned at the starting vertex v0. Each ant is assigned an initial tourx = {v0} containing only the starting vertex and a deterministically generated set of objective weightsp = {p1, . . . , pK}, which is used to evaluate the vertices to be added during construction. In the currentpaper we present algorithms which are applied to problems with K = 2 objective values. Therefore,the weights are generated in a way such that for all ants within a population the weights are equallyspread between the two extreme weight values (p1 = 0, p2 = 1 and p1 = 1, p2 = 0). Additionally, eacharc (vi, vj) ∈ A is assigned an initial pheromone value τijk for each objective value k.

During the construction phase of the algorithm each ant constructs a feasible tour from the startingvertex to the ending vertex applying a pseudo-random proportional decision rule. For the calculationof the probability to visit the next customer two ingredients are used; the heuristic information ηijkon the one hand and the pheromone information τijk on the other hand. After each construction step(i.e. the inclusion of an additional POI in the current partial solution) for each of the objectives k, alocal pheromone update is performed on the edge (i, j) chosen. At the end of the construction phase,after each ant has reached the terminating vertex, an iterative improvement is applied. For all thegenerated solutions of a population non-dominance is checked and all the potentially efficient solutionsare stored in an external memory. Finally, a global pheromone update procedure is performed, takinginto account the best and the second-best solution found regarding each objective value k during thecurrent iteration. After each iteration, a new colony of ants is used.

3.2 Pareto Variable Neighborhood Search

The P-VNS is an extension to the well known multiple-neighborhood local search procedure VNSdeveloped by Mladenovic and Hansen (c.f. [10, 11]) to tackle multi-objective problems. The basic


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idea behind the traditional VNS solution method is to make use of a simple iterative improvementmethod to explore the local neighborhood of an initial solution in combination with a set of differentneighborhood operators to avoid getting stuck in local optima. The main important modifications ofthe standard VNS are the use of randomly generated weight vectors and a storage for the potentiallyefficient solutions. The random weight vectors are used for solution evaluation in both, shaking anditerative improvement.

The steps of the P-VNS are shown in Figure 1. Nκ(κ = 1, . . . , κmax) is the set of neighborhoods.The general stopping condition can be a limit on CPU time, a limit on the number of iterations, or alimit on the number of iterations between two improvements.

Initialization: Construct an initial solution x;Repeat the following steps until the stopping condition is met:

1. Set κ← 1;

2. Repeat the following steps until κ = κmax:

(a) Generate Random Weights pk.

(b) Shaking . Generate a point x′ at random from κth neighborhood of x (x′ ∈ Nκ(x));

(c) Iterative Improvement . Apply iterative improvement method with x′ as initial solution; theobtained local optimum is denoted with x′′; the evaluation is done by using the currentweight vector.

(d) Store x′′ in an external memory if it is potentially efficient.

(e) Move or not . If this local optimum x′′ is better than the incumbent by using the currentweight vector, or if some acceptance criterion is met, move (x← x′′), and continue the searchwith N1(x) (κ← 1); otherwise, set κ← κ+ 1;

Figure 1: Steps of the P-VNS

3.3 Path Relinking Procedure

PR has first been introduced by Glover and Laguna as an approach for integrating intensification anddiversification strategies (cf. [6]). The main idea behind PR is to search for additional solutions byexploring trajectories that connect solutions that are known to be of high quality. Therefore startingat one solution, the so called initial solution, a path through its neighborhood space is created, thatleads to the other solution, the so called guiding solution. This is a promising concept especially inmulti-objective optimization to find additional potentially efficient solutions by relinking two proposedpotentially efficient solutions. In our approach PR is applied as a post-processing stage, after thesolution procedures in order to improve the approximations of the Pareto front produced by P-ACO orP-VNS, respectively.

PR is applied to all pairs of solutions. The algorithm starts at the current solution being theinitial solution. Vertices not visited in the guiding solution are iteratively removed from the currentsolution and replaced by vertices included in the guiding solution. The algorithm stops when thecurrent solution equals the guiding solution. To increase performance, the procedure does not alwaysenumerate all possible combinations of removal and insertion steps between the initial and the guidingsolution. If there are more than 16 possible removal/insertion combinations, one of them is selectedand performed on a random basis and the procedure starts over. Otherwise, all possible combinationsare performed. All intermediate potentially efficient solutions are stored. The value of 16 was chosenbased on previous experiments as a tradeoff between speed and solution quality.


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Both solution approaches are implemented in C++ using the GNU compiler g++ in its version 4.1.0on a 64-bit Linux platform. All test runs are performed on an Intel Pentium 4 D CPU with 3.2 GHzand 4 GB of RAM.

4.1 Test instances

In a first analysis the algorithms are applied to the standard benchmark instances generated for singleobjective optimization. Tsiligirides generated 49 test instances (c.f. [14]) and Chao, Golden, and Wasil(CGW) published 40 problem instances (c.f. [1]).

For the performance evaluation of the developed bi-objective optimization algorithms these test in-stances are augmented with additional benefit values. Moreover 127 new bi-objective problem instancesbased on real world data of different Austrian regions, the city of Vienna and the town of Padua weregenerated. The real world problem sets consist of 97 POIs in Padua, 273 POIs in Vienna, and 559, 562and 2143 POIs in the Austrian regions Styria, Carinthia and Lower Austria, respectively. The benefitvalues have been randomly generated. Different Tmax values are used for the creation of different realworld based instances. The two city problem sets have been split into 20 test instances with maximumdistance restrictions between 1 and 20 kilometers. For each of rural problem sets (559, 562 and 2143POIs), 29 test instances with maximum distance restrictions between 10 and 150 kilometers have beencreated. The distance matrix is based on the real road network provided by Teleatlas, except for theinstances of Padua. Here Euclidean distances were used. In the real world problem sets service timeis added for each vertex. This service time can be interpreted as time spent for visiting a POI. Theservice time is 0.5 km for all problem sets.

4.2 Performance measures

A set of six state of the art performance measures published in literature was used to perform the anal-ysis of the results. All performance measures used in this paper have been calculated using normalizedobjective function values. For the unary performance measures a reference set is required. As the setof all Pareto optimal solutions can not be computed we have to generate an approximation set as areference set. All potentially Pareto efficient solutions found by any of the two solution methods in anyof the ten runs found have been stored in the reference set R. In the comparison of the solution qualitydifferent attainment surfaces were used. For comparison of the obtained approximations of the Paretofront, we use the performance measure values when considering the 20%, 50%, and 80% attainmentsurface of the ten runs (see [7, 9]).

The hypervolume indicator IH , as described by Zitzler and Thiele [15], measures the hypervolumeof the objective space of a set A that is weakly dominated by an approximation set or the set containingthe non-dominated frontier of an approximation method. The unary epsilon indicator Iε was describedby Zitzler et al. [16] and we use the multiplicative unary version which calculates the minimum factor ε,in that way when every point in the reference set R is multiplied by ε, then the resulting approximationset is weakly dominated by the approximation set A. The R3 indicator IR3 was described by Hansenand Jaszkiewicz [8] and is used to assess and compare the approximation sets found by our two solutionmethods, based on a set of utility functions. The pareto front approximation indicator IA, as proposedby Czyzak and Jaszkiewicz [2], serves as a measure of how well an approximation set A approximatesthe Pareto frontier. The spacing indicator IS proposed by Schott [13] measures how uniformly thepoints in an approximation set A are distributed in objective space. The range covering indicatorIR measures how well the whole possible range of the Pareto frontier is covered by the points in anapproximation set A.


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4.3 Results

The results presented are always average values over 10 independent runs of the solution method withone parameter setting. Both solution methods were given the same runtime. Computational resultsshow, that both developed solution techniques provide competitive results regarding the single objectiveproblem sets presented in literature. For the three test instances presented by Tsiligirides [14] (21, 32,and 33 vertices), with both algorithms results are found that are equal to the best results reported sofar. For the newer test instances published by Chao et al. [1] (64 and 66 vertices), the two methodseven find new best results for 12 out of 40 instances with an average improvement in the objective valueof 0.41% and 0.46%, respectively.

When looking at the bi-objective and real world results, it can be seen that for most of the per-formance measures, the P-ACO algorithm provides better results than the P-VNS on average over allinstances. This is consistent for all three attainment surfaces (20%, 50% and 80%).

For small size instances, the P-ACO procedure provides better results for the IH , Iε, IS , and IRmeasure on average. The values for measures IR3 and IA are comparable. For medium size instances,the P-ACO outperforms P-VNS in all the measures except for IA. The same picture can be observed forlarger instances. The P-ACO provides better results in almost all performance measures on the averagevalues except for the measure IS . The average values over all sizes for 80% attainment show, that theP-ACO algorithm always outperforms the P-VNS algorithm. Only for the spacing indicator IS , it canbe seen, that for 20% attainment and for 50% attainment the P-VNS provides a better distribution ofthe solutions on the pareto frontier. For all the other measures, the P-ACO algorithm is superior.

Summarizing the computational results, we can observe that while P-ACO seems to work slightlybetter, the average indicator values for both methods are rather similar. Hence we can conclude that:

• Both methods are appropriate approaches for approximating the Pareto frontier.

• Only if uniform dispersion of the solutions in the objective space is very important, the P-VNSapproach shows some advantages.

• For all other measures, like, for example, the attainment of extreme points or the size of theregion covered by the found solution set, P-ACO is the method of choice, since it provides slightlybetter indicators on average.

• There is no class of instances, where one method is always better on all instances. While e.g.P-ACO is in general better also in the class of medium sized instances, there exist examples thatshow clear superiority for P-VNS when plotted.

• There is also no clear picture that for small, medium or large instances the conclusion is different.The same holds true also for random (Tsiligirides), geometric (CGW) and real world instances.

• The comparison might be biased slightly in favour of P-ACO since the run times were chosenappropriate for P-ACO and then P-VNS was given the same run time. It might be possible thatfor smaller or larger run times, P-VNS can catch up with P-ACO.

References

[1] I-Ming Chao, Bruce L. Golden, and Edward A. Wasil. A fast and effective heuristic for theorienteering problem. European Journal of Operations Research, 88(3):475–489, 1996.

[2] Piotr Czyzak and Jaszkiewicz Andrzej. A multiobjective metaheuristic approach to the localizationof a chain of petrol stations by the capital budgeting model. Control and Cybernetics, 25(1):177–187, 1996.


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[3] Karl F. Doerner, Walter J. Gutjahr, Richard F. Hartl, Christine Strauss, and Christian Stummer.Pareto ant colony optimization: A metaheuristic approach to multiobjective portfolio selection.Annals of Operations Research, 131(1):79–99, 2004.

[4] Karl F. Doerner, Walter J. Gutjahr, Richard F. Hartl, Christine Strauss, and Christian Stummer.Nature-inspired metaheuristics for multiobjective activity crashing. Omega, 36(6):1019–1037, 2006.

[5] Marco Dorigo and Gianni Di Caro. The ant colony optimization meta-heuristic. New Ideas inOptimization, pages 11–32, 1999.

[6] Fred Glover, Manuel Laguna, and Rafael Martı. Scatter search and path relinking: Advancesand applications. In Fred Glover and Gary A. Kochenberger, editors, Handbook of Metaheuristics,volume 57 of International Series in Operations Research and Management Science, pages 1–35.Springer New York, 2003.

[7] Viviane Grunert da Fonseca, Carlos M. Fonseca, and O. Hall, Andreia. Inferential performanceassessment of stochastic optimisers and the attainment function. In Eckart Zitzler, KalyanmoyDeb, Lother Thiele, Carlos A. Coello, and David Corne, editors, First International Conference onEvolutionary Multi-Criterion Optimization, volume 1993 of Lecture Notes in Computer Science,pages 213–225. Springer New York, 2001.

[8] Michael P. Hansen and Andrzej Jaszkiewicz. Evaluating the quality of approximations to thenon-dominated set. Technical Report IMM-REP-1998-7, Technical University of Denmark, 1998.

[9] Joshua D. Knowles, Lothar Thiele, and Eckart Zitzler. A tutorial on the performance assessment ofstochastic multiobjective optimizers. Technical Report 214, Computer Engineering and NetworksLaboratory (TIK), ETH Zurich, Switzerland, February 2006. revised version.

[10] Nenad Mladenovic. A variable neighborhood algorithm: A new metaheuristic for combinatorialoptimization. In Abstract of papers presented at Optimization Days, page 112, Montreal, 1995.

[11] Nenad Mladenovic and Pierre Hansen. Variable neighborhood search. Computers and OperationsResearch, 24(11):1097–1100, 1997.

[12] M. Schilde, K. F. Doerner, R. F. Hartl, and G. Kiechle. Metaheuristics for the bi-objectiveorienteering problem. Technical report, University of Vienna, 2009.

[13] Jason R. Schott. Fault tolerant design using single and multicriteria genetic algorithm optimization.Master’s thesis, Department of Aeronautics and Astronautics, MIT, Cambridge, 1995.

[14] T. Tsiligirides. Heuristic methods applied to orienteering. The Journal of the Operational ResearchSociety, 35(9):797–809, 1984.

[15] Eckart Zitzler and Lothar Thiele. Multiobjective evolutionary algorithms: A comparative casestudy and the strength pareto approach. IEEE Transactions on Evolutionary Computation,3(4):257–271, 1999.

[16] Eckart Zitzler, Lothar Thiele, Marco Laumanns, Carlos M. Fonseca, and Viviane Grunert daFonseca. Performance assessment of multiobjective optimizers: An analysis and review. IEEETransactions on Evolutionary Computation, 7(2):117–132, 2003.


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Iterated Density Estimation Evolutionary Algorithm with 2-opt

local search for the vehicle routing problem with private fleet

and common carrier

Jalel Euchi ∗ Habib Chabchoub ∗

∗ GIAD Laboratory, Faculty of Economics and Managementroute de l’aerodrome km 4.5, B.P 3018, Sfax, Tunisia


1 Introduction

Numerous organizations are involved in the production and distribution of goods. Very often truckswith different physical, operational and cost characteristics are available to distribute such goods, andthe shipper has to decide which shipments to assign to each truck for delivery. Many manufacturersand distributors use private fleets, or common carrier, for the purpose of collecting and deliveringshipments for their facilities. In addition to offering greater control over goods movement, privatefleets may reduce costs over common carrier prices. Whereas common carriers typically require thatshipments be processed at consolidation terminals, private fleets can transport shipments directly fromorigin to destination via multiple stop routes.

There has been growing interest in truck service selection and the deregulation of truck commoncarrier. This has had particular impact on organization that use privately-owned vehicles, since theyare responsible for the utilization of the fleet and must make choices that determine the balance betweencommon carrier and private carrier usage.

If private vehicles are used, routing the vehicles for best utilization, sizing the vehicles, and de-termining the number needed are common choices that must be made. If a common carrier is used,rate negotiation, shipment consolidation, and routing are important considerations. Owing to its scaleeconomies, a common carrier may be able to offer a lower price, for small shipments in particular.

Despite its wealth and abundance of work that are devoted to him, the Vehicle Routing Problemwith Private fleet and common Carrier (VRPPC) represent only a subset of a larger family known as”Vehicle Routing problem- VRP”. The VRP is one of the optimization problems most studied. Thisproblem holds the attention of several researchers for many years, and everywhere in the world. Severalauthors have made a literature review that deal with vehicle routing these include those of Bodin et al.[2], Laporte [12, 13, 14], and Toth and Vigo [22]. Like other authors, Golden and Assad [10] presentedthe problems of vehicle routing as problem easy to explain but difficult to solve.

More specifically, two types of problems have been addressed in literature: the vehicle routingproblem with limited fleet and the vehicle routing problem with private fleet and common carrier.Several authors have studied the vehicle routing problem with limited fleet (e.g. Osman and Salhi [19],Gendreau et al. [9], Taillard [20], Tarantilis et al. [21], Li et al. [16] and Choi and Tcha [6]). Allthese authors considered a mixed fleet limited but sufficient capacity to serve all customers. At thelevel of routing problem with external carrier are Volgenant and Jonker [23], which demonstrated that


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the problem involving a fleet of one single vehicle and external carriers can be rewritten as TravelingSalesman Problem - TSP. This problem also have been studied by Diaby and Ramesh [8] whose ob-jective was to decide that customers visited with external carrier and optimize the tour of remainingcustomers. Several approaches have been used to solve the classical VRP, exact methods, heuristicsand metaheuristics solution principally are proposed. The VRPPC is more complex problem becauseit involves an internal fleet of several vehicles. To our knowledge, the VRPPC was introduced by Ballet al. [1], Klincewicz et al. [11], Chu [7] and Bolduc et al. [3, 4]. Precisely in the VRPPC, Chu [7]present the mathematical model of the problem and solve it with a heuristic economies improved byinter and intra routes. There after, Bolduc et al. [3, 4] have improved the results of Chu using moresophisticated exchanges.

Evolution computation (EC) motivated by evolution in the real world, has become one of the mostwidely used techniques, because of its effectiveness and versatility. It maintains a population of solution,which evolves subject to selection and genetic operators (such as recombination and mutation). Eachindividual in the population receives a measure of its fitness, which is used to guide selection.

Iterated Density Eestimation Evolutionary Algorithm (IDEA) family (see Larranaga [15]) is a newtype of metaheuristic which has attained interest during the last 5 years. IDEA is relatively recenttype of optimization and learning techniques based on the concept of using a population of tentativesolutions to iteratively approach the problem region where the optimum is located (e.g. Larranaga [15],Muhlenbein and Mahnig [18], Bosman and Thierens [5]).

The IDEA is good at identifying promising areas in the search space, but lacks the ability of refininga single solution. A very successful way to improve the performance of IDEA is to hybridize it withlocal search techniques (Lozano et al. [17]). We propose 2-opt local search for the vehicle routing withprivate fleet and common carrier.

The remainder of this paper is structured as follows: In §2, a problem statement and description ofthe vehicle routing problem with private fleet and common carrier is provided. In §3, we give the mainparadigm of the IDEA metaheuristic. In §4, we present our proposed approach to solve the VRPPC.In §5, we perform experiments results. We conclude this paper in §6.

2 Problem statement

The VRPPC can be described as follows: Let G = (V,A) be a graph where V = {0, 1, ..., n} is thevertex set and A = {(i, j) : i, j ∈ V, i 6= j} is the arc set. Vertex 0 is a depot, while the remainingvertices represent customers. A private fleet of m vehicles is available at the depot. The fixed cost ofvehicle k is denoted by fk, its capacity by Qk, and the demand of customer i is denoted by qi. A travelcost matrix (cij) is defined on A . If travel costs are vehicle dependent, then cij can be replaced withcijk, where (k ∈ 1, ...,m) . Each customer i can be served by a vehicle of the private fleet, in whichcase it is called an internal customer or by a common carrier at a cost equal to ei , in which case it iscalled an external customer. The VRPPC consists of serving all customers in such a way that:

1. Each customer is served exactly once either by a private fleet vehicle or by a external transportervehicle.

2. All routes associated with the private fleet start and end at the depot.

3. Each private fleet vehicle performs only one route r.

4. The total demand of any route r does not exceed the capacity of the vehicle assigned to it.

5. The total cost is minimized. In practice, several common carriers may be used to serve any ofthe customers unvisited by the private fleet.


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Typically, the one selected is the lowest cost carrier. It is not necessary to specify the routes followedby the external transporter because it charges a fixed amount ei for visiting customer i, irrespective ofvisit sequence.

3 Main paradigm of the IDEA metaheuristic

IDEA is listed as a type of evolutionary algorithms in which an initial population of individuals, eachone encoding a possible solution to the problem, is iteratively improved by the application of stochasticoperators. Every individual encodes a solution that is weighted with respect to the others by assigninga fitness value according to the objective function being optimized.

IDEA iterates the three steps listed below, until some termination criterion is satisfied:

1. Select good candidates (i.e., solutions) from a (initially randomly generated) population of solu-tions.

2. Estimate the probability distribution from the selected individuals.

3. Generate new candidates (i.e., offspring) from the estimated distribution.

4 Hybrid IDEA to solve the VRPPC (IDEA / 2-opt)

We propose an IDEA algorithm (e.g. Larranaga [15]) to solve the VRPPC. It is recommended tohybridize it with a local search (see Lozano et al. [17]). In this way, we propose to use a 2-optlocal search to improve the solution generated after the creation of the initial population and afterthe generation of new solution. With each generation t, IDEA algorithm maintains a populationpop(t) =

{π1, π2, ..., πN

}of N solutions and the probability matrix is where p(t) models the distribution

of promising solutions in the search space. More precisely, pkj (t) is the probability that vehicle k isassigned to customer in the assignment.

Below the implementation of each part of the IDEA to solve the VRPPC is described, vehiclerouting representation, initialization, selection operators, probabilistic model, replacement and stoppingcriterion.

Our proposed approach, for the VRPPC with theMaxiter and tmax criterion, start with the insertionheuristic for obtaining an initial solution. In order to obtain an initial feasible solution, the followingalgorithm is used.

sort all available vehicles in increasing order of capacityfor each available vehicle k := 1 to mmax loop

for each vi ∈ V (i = 1, ..., n) loopif (vi ∈ rki and qi < Qk) then

insert a customer vi into route rki using the insertion methodend ifExecute 2-opt local search

endend

Algorithm 1: Algorithm of initial solution

The N resultant solutions{π1, π2, ..., πN

}constitute the initial population Pop(0). The initial

probability matrix p(0) is set as pij = 1N . Then the probability matrix p(t) can be updated as follows:


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pij(t) = (1− β) 1N

∑Nk=1 Iij(πk) + βpij(t− 1), (1 ≤ i, j ≤ n)

where

Iij (π) ={

1 ifπ (i) = j0 otherwise

0 ≤ β ≤ 1 is a learning rate. The bigger β is, the greater the contribution of the solutions in pop(t)is to the probability matrix p(t).

Then we use the selection Pressure Towards Diversity of Bosman and Thierens [5]. One of thesuccessful way of IDEA is the use of a probabilistic model that captures the important correlations ofthe search distribution, assigning high probability values to the selected solution. The IDEA builds aprobabilistic model with the best individuals and then sample the model to generate new ones.

In order to create a new solution based on the probabilistic model we proceed as follows:

1. Divides the vehicles into two groups based on their capacity. The first group has customers withthe greater demands and the second one has the remaining customers.

2. Vehicle k is assigned to location π(k) , which is the location for this customer in solution π.

3. Arranges the vehicles in the second group sequentially, based on the probability matrix .

4. Customers not served are assigned to the external transporter.

To improve the solution and after the probabilistic model, we apply the 2-opt local search and weuse the tournament replacement. A subset of individuals is selected at random, and the worst one isselected for replacement. We use a maximum number of iteration Maxiter and maximum of executiontime tmax as a stopping criterion.

Figure 1 give the pseudo code of our proposed approach.


In this section we discuss the performance of our IDEA algorithm when applied on a wide set ofinstances taken from the literature.

We consider two sets of instances to evaluate the performance of IDEA algorithm. The first setis composed of a limited number of homogeneous vehicles, and the second set is composed of a lim-ited number of heterogeneous vehicles. In this extended abstract we are interested to some difficultheterogeneous instances.

The algorithm described here has been implemented in C++ using Visual Studio C++ 6.0. Exper-iments are performed on a PC Pentium 4, 3.2 GHz with 512MB of RAM.

To illustrate the effectiveness and performance of IDEA algorithm for the VRPPC, the algorithmwill be compared to the results of the SRI and RIP metaheuristic specified in Bolduc et al. [4], whereSRI algorithm is Selection - Routing - Improvement and RIP algorithm is Randomized construction-Improvement- Perturbation are both contains three main steps.

In order to verify the effectiveness and efficiency of proposed IDEA, we choose a few large instances.

Table1 gives the solution values. This table compare the results obtained with IDEA to thoseobtained with the SRI and RIP metaheuristic of Bolduc et al. [4] for each instances.


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The results showed that the quality of our algorithm solution is better than the metaheuristicpresented by Bolduc et al.[3,4]. The IDEA algorithm runs faster than the RIP metaheuristic.

6 Conclusion

We have introduced the vehicle Routing problem with Private fleet and common carrier. We propose theIDEA algorithm with 2-opt local search to solve this variant of the VRP. The proposed IDEA algorithmprovides the best results. And it outperforms those obtained by the SRI and RIP metaheuristicsproposed in Bolduc et al. [4] on the same problem sets.

The algorithm requires minimal computation time and it is very performing according to similarexperiment presented in the literature.

References

[1] Ball MO, Golden A, Assad A and Bodin LD. Planning for truck fleet size in the presence of acommon-carrier option. Decision Sciences, 14, 103–120, 1983.

[2] Bodin LD, Golden BL, Assad AA, and Ball MO. Routing and scheduling of Vehicles and crews:Thestate of the Art. Computers and operations research, 10, 69–211, 1983.

[3] Bolduc MC, Renaud J and Boctor FF. A heuristic for the routing and carrier selection prob-lem.Short communication. European Journal of Operational Research, 183, 926–932, 2007.

[4] Bolduc MC, Renaud J, Boctor FF and Laporte G. A perturbation metaheuristic for the vehiclerouting problem with private fleet and common carriers. Journal of the Operational ResearchSociety, 59, 776–787, 2008.

[5] Bosman PAN and Thierens D. Multi-objective optimization with diversity preserving mixture-based iterated density estimation evolutionary algorithms. International Journal of ApproximateReasoning, 31, 259–289, 2002.

[6] Choi E and Tcha DW. A column generation approach to the heterogeneous fleet vehicle routingproblem. Computers and operations research 34, 2080–2095, 2007.

[7] Chu CW. A heuristic algorithm for the truckload and less-than-truckload problem. EuropeanJournal of Operational Research, 165, 657–667, 2005.

[8] Diaby M and Ramesh R. The Distribution Problem with Carrier Service: A Dual Based PenaltyApproach. ORSA Journal on Computing, 7, 24–35, 1995.

[9] Gendreau M, Laporte G, Musaraganyi C and Taillard ED. A tabu search heuristic for the hetero-geneous fleet vehicle routing problem. Computers & Operations Research, 26, 1153–1173, 1999.

[10] Golden BL and Assad AA. Vehicle Routings: Methods and Studies North-Holland, Amsterdam,1988.

[11] Klincewicz JG, Luss H and Pilcher MG. Fleet size plannnig when outside carrier services areavailable. Transportation Science, 24, 169–182, 1990.

[12] Laporte G. The Vehicle Routing Problem: An overview of exact and approximate algorithms.European Journal of Operational Research, 59 (3), 345–358, 1992.

[13] Laporte G. The traveling salesman problem: An overview of exact and approximate algorithms.European Journal of Operational Research, 59 (2), 291–247, 1992.


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[14] Laporte G. Computer aided routing CWI tract.75:M.W.P. Savelsbergh Mathematisch Centrum,Amsterdam, 1992, 134 pages, DFL.40.00, ISBN 90 6196 412 1. European Journal of OperationalResearch, 71(1), 143, 1993.

[15] Larranaga P. A review on estimation of distribution algorithms. In P. Larranaga and J. A. Lozano,editors. Estimation of Distribution Algorithms. A New Tool for Evolutionary Computation, pp.80-90. Kluwer Academic Publishers, 2002.

[16] Li F, Golden BL and Wasil EA. A record-to-record travel algorithm for solving the heterogeneousfleet vehicle routing problem. Computers and Operations Research, 34, 2734–2742, 2007.

[17] Lozano JA, Larranaga P,Inza I, and Bengoetxea E. Towards a New Evolutionary Computation.Springer, Berlin Heidelberg New York, 2006.

[18] Muhlenbein H and Mahnig T. FDA - a scalable evolutionary algorithm for the optimization ofadditively decomposed functions. Evolutionary Computation, 7(4), 353–376, 1999.

[19] Osman IH and Salhi S. Local search strategies for the VFMP, in: Rayward-Smith VJ, OsmanIH, Reeves CR, Smith GD (Eds.)Modern Heuristic Search Methods. Wiley, New York, pp. 131-153(1996).

[20] Taillard ED. A heuristic column generation method for the heterogeneous fleet VRP. RAIRO 33(1), 1–14, 1999.

[21] Tarantilis C, Kiranoudis C and Vassiliadis V. A threshold accepting metaheuristic for the het-erogeneous fixed fleet vehicle routing problem. European Journal of Operational Research,152,148–158, 2004.

[22] Toth P and Vigo D. Models, relaxations and exact approaches for the capacitated vehicle routingproblem. Discrete Applied Mathematics, 123, 487–512, 2002.

[23] Volgenant T and Jonker R. On some generalizations of the traveling salesman problem. Journalof the Operational Research Society, 38, 1073–1079, 1987.


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0

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end

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Figure 1: Pseudo code of IDEA algorithm


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SRI RIP IDEAInstances Z CPU time (s) Z CPU time (s) Z CPU time (s)Chu-H-01 387.50 0.00 387.50 0.35 387.50 0.00Chu-H-02 586.00 0.02 586.00 1.90 586.00 0.08Chu-H-03 826.50 0.03 826.50 3.50 826.50 1.02Chu-H-04 1389.00 0.08 1389.00 5.85 1389.00 3.00Chu-H-05 1444.50 0.09 1444.50 10.40 1441.50 5.70B-H-01 423.50 0.02 423.50 1.85 423.50 1.30B-H-02 476.50 0.02 476.50 3.50 476.50 2.00B-H-03 804.00 0.03 778.50 4.75 777.00 2.80B-H-04 1564.50 0.09 1564.50 15.85 1564.50 10.23B-H-05 1609.50 0.13 1609.50 12.90 1609.50 10.29

CE-H-01 1220.72 0.00 1192.72 26.00 1168.23 15.90CE-H-02 1858.24 0.00 1798.26 72.00 1753.35 53.26CE-H-03 1999.91 1.00 1934.85 105.00 1861.93 90.70CE-H-04 2615.95 1.00 2493.93 251.00 2492.32 195.91CE-H-05 3248.26 3.00 3195.66 490.00 3126.99 407.33G-H-01 14599.16 4.00 14408.31 647.00 14206.00 426.72G-H-02 18945.77 13.00 18663.15 1254.00 1888.31 736.51G-H-03 26151.24 13.00 25561.55 2053.00 25324.12 1804.12G-H-04 36519.42 22.00 35495.66 2049.00 35252 1799.89G-H-05 17173.22 3.00 16138.50 512.00 16109 439.81

SRI: Selection Routing Improvement algorithm.RIP: Randomized construction Improvement Perturbation algorithm.IDEA: Iterated Density Estimation Evolutionary Algorithm with 2-opt local searchCPU time en seconds

Table 1: Results for heterogeneous limited fleet instances


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Solving a bus driver scheduling problem with randomized

multistart heuristics

Renato De Leone ∗ Paola Festa † Emilia Marchitto ∗

∗ DMI, University of CamerinoVia Madonna delle Carceri, 62032 Camerino (MC), Italy

Email: {renato.deleone, emilia.marchitto}@unicam.it

† DMA, University of Napoli FEDERICO IICompl. MSA, Via Cintia - 80126 Napoli, Italy


1 Introduction

The Bus Driver Scheduling Problem is an extremely complex part of the Transportation PlanningSystem (e.g., [12]) that generally can be divided in different subproblems due to its complexity:Timetabling, Vehicle Scheduling, Crew Scheduling (Bus and Driver Scheduling), and Crew Rostering(see Figure 1 for the relationship between these subproblems). The transportation service is composedof a set of lines that corresponds to a bus traveling between two locations of the same city or betweentwo cities. For each line, the frequency is determined by the demand. Then, a timetable is constructed,resulting in journeys characterized by a start and end point, and a start and end time. The VehicleScheduling Problem consists in finding a schedule for the buses, each schedule being defined as a busjourney starting at the depot and returning to the same depot. The objective is to minimize the totalcost given by the cost of buses used for the service and running costs. Running costs can be minimizedavoiding unnecessary deadheads, i.e., trips carrying no passengers. The daily schedule of each singlebus is known as a running board (or vehicle block).

Bus schedules are several running boards whose lengths are determined by the total time the bus is

timetabling

vehicle scheduling

trips

running boards

crew scheduling

tasks

crew shifts

crew rostering crew rosters

collective

labour rulesand

agreements

Figure 1: Transportation Planning System.


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operating away from the depot. The Bus Driver Scheduling (BDSP) is the second phase of the generalplanning of a public transportation company’s operations. This problem is important from an economicpoint of view and it is related to the costs of the drivers. It consists of determining the shifts (i.e., afull day of work) for the drivers at a certain depot to cover all the running boards assigned. Since adriver exchange can occur at various points along a running board, the entire running boards is dividedin units (Pieces-Of-Work) that start and finish at relief points, i.e., designed locations and times whereand when a change of driver may occur. Different types of shift can be taken into account, whichare composed of a different number of Pieces-Of-Work whose length is variable and whose beginninghappens at distinct possible starting times. For example, a shift can contain a single Piece-Of-Work,two Pieces-Of-Work or more Pieces-Of-Work. A set of Pieces-Of-Work that satisfies all the constraintsis a feasible shift. A break (a time slot of no work) can also be inserted between two different Pieces-Of-Work or in a single Piece-Of-Work (i.e. when the vehicle is stop to a location for a period of time thatthere isn’t a relief point). There are two different types of break: rest and idle time. Rest is unpaid,while idle time is paid (in Italy, for example, its rate is 12 percent of ordinary wage). The feasibility ofa shift not only depends on breaks and duration of the Pieces-Of-Work, but also on the total workingtime and on the spreadover. The total working time is the sum of the duration of the Pieces-Of-Work(excluding idle time) while the spreadover is the total time between the start and the end of the shift.A feasible solution for the BDSP is a set of feasible shifts for the drivers. To each shift a different costcan be associated. The aim is to minimize not only the total cost but also the total number of shifts.For a description of the terminology used in Bus and Driver Scheduling, the reader is referred to theglossary of [8].

The BDSP is a NP-hard problem even when there are only spreadover and working time constraints([4, 5]). By formulating and solving it as a Set Partitioning Problem or a Set Covering Problem, solutionsare often generated that contain very little or no over-covering at all and in general the over-coveringcan be eliminated a posteriori using, for example, a heuristic procedure ([11, 2]). Recently, Huismanet al. [9] proposed a combination of column generation and Lagrangian relaxation and a new generatorof random instances such that their properties are the same as for our real-world instances.

The goal of this paper is to propose and experimentally compare several new hybrid metaheuristics tosolve a Bus Driver Scheduling Problem under special constraints imposed by Italian transportation rules.To model and satisfy particular collective agreements and labour rules stated by the Italian governmentbut also to generalize the state-of-the-art problem including constraints for a variety of trade-unionrules and regulations for transportation companies, in [1] we have proposed a new mathematical modelwith a set of additional constraints originated by our collaboration with PluService Srl, leading Italiangroup in software for transportation companies. Moreover, depending on the choice of appropriate costcoefficients, the objective function of our model minimizes the total number of shifts and/or the totaloperational costs. To link our investigation to real world scenarios, in the experimental investigation,we have used several Italian transportation instances provided by PluService Srl.

The rest of the paper is organized as follows. In Section 2, we describe the implementation detailsof the proposed pure and hybrid heuristics. Computational experiments on real data sets provided byPluService Srl and concluding remarks are presented in Section 3.

2 Randomized heuristics

For finding approximate solutions of the special BDSP here studied, we propose various randomizedheuristics based on the instantiation of several metaheuristics and their hybrids.


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2.1 GRASP

Greedy Randomized Adaptive Search Procedure (GRASP) is a randomized multi-start metaheuristicfor combinatorial optimization problems [3]. Starting from the feasible solution obtained in the con-struction phase, a local search procedure is applied in which a local optimum in the neighborhood ofthe constructed solution is sought. The best solution produced from all GRASP iterations is returnedas output.

algorithm GRASP(MaxIter,α,c,W )1 T∗ := ∅; best := +∞;2 for k = 1 to MaxIter do3 T :=ConstGrRandomSol(W ,α);4 T :=LocalSearch(T );5 if (c(T ) < best)) then6 T∗ := T ; best := c(T );7 endfor8 return(T∗);end GRASP

algorithm GRASP+PathRelinking(MaxIter,α,c,W ,MaxElite)1 T∗ := ∅; best := +∞; E := ∅;2 for k = 1 to MaxIter do3 T :=ConstGrRandomSol(W, α);4 T :=LocalSearch(T );5 if (c(T ) < best) then6 T∗ := T ; best := c(T∗);7 if |E| = MaxElite then8 Randomly select a solution S ∈ E;9 T :=Path-Relinking(T, S);10 AddElite(E, T );11 if (c(T ) < best) then12 T∗ := T ; best := c(T∗);13 else E = E ∪ {T};14 endif15 endfor16 return(T∗);end GRASP+PathRelinking

Figure 2: Pseudo-codes of GRASP and Path-Relinking for our BDSP.

Figure 2 depicts the pseudo-code of GRASP for our BDSP. Given a set W of Pieces–Of–Work(POWs), T denotes the feasible schedule corresponding to a set of N feasible shifts Ti, i = 1, . . . , N ,and c(T ) denotes the sum of the operational costs of the shifts that are in T . The GRASP constructionphase iteratively builds a feasible schedule. Starting from an empty schedule, at each iteration a Piece–Of–Work is added to the schedule under construction in a greedy, randomized, and adaptive way. Ateach iteration, the candidate POWs are ranked according to their greedy function values and wellranked candidate elements are placed in a restricted candidate list (RCL), from which an element isselected at random and is added to the partial solution. Once a POW is selected, the set of remainingcandidates must be updated to take into account that the currently selected POW is now part of thepartial solution. To build the RCL a value based scheme has been applied, i.e., in the RCL are placed allcandidate elements with greedy function values within g+α(g−g), where g and g are the worst and thebest greedy value, respectively. The parameter α is a value in [0, 1]. Since a drawback of the constructionphase of GRASP is the characteristic lack of memory of the solutions computed in previous iterations, wehave applied the learning strategy used in the Reactive GRASP [10], in which the parameter α changesaccordingly to the quality of the solutions computed in previous iterations. In particular, at eachiteration α is randomly selected from a discrete set A containing m predetermined values. In our case,we use a set A = {0.0; 0.025; 0.05; 0.075; 0.1; 0.125; 0.15; 0.175; 0.2; 0.25; 0.3; 0.35; 0.4; 0.5; 0.6; 0.8; 1.0},since we have obtained better results in correspondence to smaller values of α. Once the current shiftis completed, a new shift is constructed. The procedure ends when all the POWs have been assigned.Starting from the feasible solution obtained in the construction phase, a local search procedure isapplied until a local optimality solution is computed. The local search phase applies swaps, i.e. triesto interchange compatible partial shifts (i.e., a partial shift is a sequence of compatible POWs).

2.2 Hybrid GRASP with Path-Relinking

Path-Relinking (PR) is an enhancement to the basic GRASP procedure, leading to significant im-provements in solution quality. It was originally proposed by Glover [6] as an intensification strategyexploring trajectories connecting elite solutions obtained by tabu search or scatter search. Starting fromone or more elite solutions, paths in the solution space leading towards other guiding elite solutionsare generated and explored in the search for better solutions. This is accomplished by selecting movesthat introduce attributes contained in the guiding solutions.

We now briefly describe the integration of PR into the Reactive GRASP algorithm above described.


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procedure VNS(MaxIter, c, kmax)1 for i = 1 to MaxIter do2 k := 1;3 Generate a starting solution T at random;4 while k ≤ kmax do

5 Generate T ′ ∈ Nk(T ) at random;6 T ′′ := LocalSearch(T ′);7 if c(T ′′) < c(T ) then

8 T := T ′′; k := 1;9 else k := k + 1;10 endif11 endwhile12 endfor13 return (T );end VNS

procedure GRASP+VNS+PR(MaxIter, α, c,W, kmax)1 for i = 1 to MaxIter do2 k := 1;3 T :=ConstGrRandomSol(W, α);4 while k ≤ kmax do

5 Generate T ′ ∈ Nk(T ) at random;6 T ′′ := LocalSearch(T ′);7 if c(T ′′) < c(T ) then

8 T := T ′′; k := 1;9 else k := k + 1;10 endif11 endwhile12 if i = 1 then E := {T}; T∗ := T ;13 else14 Randomly select a solution S ∈ E;15 T ← PR(T ,S);16 Update E with T ;17 if c(T ) < c(T∗) then T∗ := T ;18 endif19 endfor20 return (T∗);end GRASP+VNS+PR

Figure 3: Pseudo-code of a generic VNS heuristic and a hybrid GRASP with VNS and PR.

The pseudo-code is shown in Figure 2. In this context, PR is applied to pairs (T, S) of solutions, whereT is the locally optimal solution obtained by local search (initial solution) and S (guiding solution) israndomly chosen from a pool with a limited number MaxElite of high quality solutions found alongthe search. Let us suppose that T and S are composed of N and M shifts, respectively:

T =N⋃

i=1

Ti, Ti =ki⋃

l=1

POWl; S =M⋃

i=1

Si, Si =hi⋃

l=1

POWl,

where ki and hi are the Pieces-Of-Work that composed Ti and Si, respectively. Given the (T, S) pair,the corresponding difference set is given by

∆(T, S) = {l = 1, . . . , N such that Tl 6= Sl, l = 1, . . . ,M},

and define distance d(T, S) := |∆(T, S)|, with 0 ≤ d(T, S) ≤ N , which is a measure of closeness betweenthe two solutions. Different strategies have been considered to insert a schedule into the elite set E(AddElite(E , Tc)). Let Tc be the current solution, Tb the best solution, and Tw the worst elite solutioninto the elite set, respectively. The procedure replaces Tw in the elite set with the current schedule Tc

if c(Tc) < c(Tw) and at least one of the following conditions is verified:

• d(Tc, Ti) ≥ dmin, ∀i = 1, . . . , |E| such that Ti 6= Tw, where dmin is the minimum distance betweenschedules;

• there is only k = 1, . . . , |E| such that d(Tc, Tk) < dmin but c(Tc) < c(Tk) and moreover d(Tc, Tw) ≥dmin;

• c(Tc) < c(Tb).

2.3 VNS and hybrid GRASP with VNS and PR

Variable Neighborhood Search (VNS) [7] is a local search procedure based on a simple principle: system-atical changes in the size and type of neighborhood during the search procedure, rather than the usageof a single neighborhood. The idea of using different neighborhood structures is motivated by both thedesire of escaping from local (non global) optima by widening the dimension of the neighborhood andintensifying the search in more promising areas by reducing the dimension of the neighborhood. Todesign a variable neighborhood procedure, it is necessary to define the sets Nk(T ), the neighborhoodof T of dimension k = 1, . . . , kmax. The pseudo-code of VNS is depicted in Figure 3. An easy wayto construct different neighborhoods is to take into account a basic type of move and apply such a


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Table 1: Summary of the computational results obtained on real-world BDSP problem instances.

Problem rgrasp rgrasp+pr vns vns+pr rgrasp+vns rgrasp+vns+pr

55 POWs Cost 7564132 7398012 9029981 80030047 8030020 7332007Time in sec. 6 8 1227 2127 117 98






move k times starting from the solution T ; the set of all solutions obtained in this way correspondsto Nk(T ). In our case, the k-th order neighborhood is defined as the set of all solutions that can beobtained from the current solution by switching partial shifts (hereafter, this operation is called a cut).Hence, Nk(T ) is the set of possible cuts of order k (for instance, k = 1 is the cut that divides into twoparts the current shift, thus creating two partial shifts) according to the current solution T , for eachk = 1, . . . , kmax. In our case, to speed up the search we have set kmax = 2. In principle, of course onecan also consider the Nk(T ) neighborhood for k ≥ 3, but in our experience already N3(T ) is alreadytoo large and computationally not affordable.

A hybrid GRASP with VNS is simply obtained by replacing the local search phase of GRASP byVNS and PR intensification can be added to the GRASP with VNS, resulting in the hybrid GRASPwith VNS and PR, whose pseudo-code is shown in Figure 3.

3 Experimental results and conclusions

In this section, we describe computational experience with the heuristics proposed in this paper:

I. a Reactive GRASP (rgrasp);

II. a Reactive GRASP with PR after each local search for intensification (rgrasp+pr);

III. a VNS (vns);

IV. a VNS with PR after each cycle for intensification (vns+pr);

V. a Reactive GRASP and with VNS as local search (rgrasp+vns);

VI. a Reactive GRASP with VNS as local search, and PR for intensification (rgrasp+vns+pr).

The computational experiments have been performed on a Pentium 4 with CPU 3.20 Ghz and 1.00Gb RAM and are summarized in Tables 1- 2. All runs were done using a single processor. Our codesare written in C and compiled with the DEV-C++ compiler. For each instance, the table lists its name,followed by the solution values obtained by each heuristic.

It can easily be seen that the procedure that has given best results is Reactive GRASP combinedwith VNS and PR since for 12 cases out of 13 it finds better solutions. As future work, we are planningto adapt our heuristics to solve classical BDSP instances and to compare them with solutions obtainedby Huisman et al. [9].


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Table 2: Summary of the computational results obtained on real-world BDSP problem instances.

Problem rgrasp rgrasp+pr vns vns+pr rgrasp+vns rgrasp+vns+pr








References

[1] R. De Leone, P. Festa, and E. Marchitto. The Bus Driver Scheduling Problem: a new mathematicalmodel and a GRASP approximate solution. Tech. Rep.-22-DMA-UNINA, 2006.

[2] T. G. Dias, J. P. Sousa, and J. F. Cunha. A Genetic Algorithm for the Bus Driver SchedulingProblem. 4th Metaheuristics International Conference, 2001.

[3] T. A. Feo and M. G. C. Resende. A Probabilistic Heuristic for a Computationally Difficult SetCovering Problem. Operations Research Letters, 8:67–71, 1989.

[4] M. Fischetti, S. Martello, and P. Toth. The Fixed Job Schedule Problem with Spread-TimeConstraints. Operations Research, 35:849–858, 1987.

[5] M. Fischetti, S. Martello, and P. Toth. The Fixed Job Schedule Problem with Working-TimeConstraints. Operations Research, 37:395–403, 1989.

[6] F. Glover. Tabu search and adaptive memory programing – Advances, applications and challenges.In R.S. Barr, R.V. Helgason, and J.L. Kennington, editors, Interfaces in Computer Science andOperations Research, pages 1–75. Kluwer, 1996.

[7] P. Hansen and N. Mladenovic. A Introduction to Variable Neighborhood Search. In S. Voß,S. Martello, I. H. Osman, and C. Roucairol, editors, Meta-heuristics, Advances and trends in localsearch paradigms for optimization, pages 433–458. Kluwer Academic Publishers, 1998.

[8] T. Hartley. A Glossary of Terms in Bus and Crew Scheduling. In A. Wren, editor, ComputerScheduling of Public Transport, pages 353–359. North-Holland, Amsterdam, 1981.

[9] D. Huisman, R. Freling, and A. P. M. Wagelmans. Multiple-Depot Integrated Vehicle and CrewScheduling. Transportation Science, 39(4):491–502, 2005.

[10] M. Prais and C. C. Ribeiro. Reactive GRASP: An Application to a Matrix Decomposition Problemon TDMA Traffic Assignment. INFORMS Journal on Computing, 12 (3):164–176, 2000.

[11] J. M. Rousseau and J. Desrosiers. Results Obtained with Crew-Opt: A Column Generation Methodfor Transit Crew Scheduling. In I. Branco J. R. Daduna and J. R. Paixao, editors, Computer-AidedTransit Scheduling, pages 349–358. Lisbon, 1993.

[12] A. Wren. Scheduling Vehicles and their Drivers-Forty Years’ Experience. Technical report, Uni-versity of Leeds, School of Computing Research Report Series, Report 2004.03, 2004.


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A hybrid approach to the Rectangle Packing Area Minimization

Problem∗

Marisa J. Oliveira § † Eduarda Pinto Ferreira § A. Miguel Gomes ‡ †

§ ISEP – Instituto Superior de Engenharia do PortoRua Dr. Antonio Bernardino de Almeida, 431, Porto, Portugal


† INESC Porto – Instituto de Engenharia de Sistemas e Computadores do PortoRua Dr Roberto Frias, Porto, Portugal

‡ FEUP – Faculdade de Engenharia da Universidade do PortoRua Dr Roberto Frias, Porto, Portugal


1 Introduction

The Rectangle Packing Area Minimization Problem (RPAMP) belongs to the more general class ofcombinatorial optimization problems: the Cutting and Packing problems. This general class of problemsis characterized by the existence of a geometric problem where one wishes to pack a set of small objectswithout overlap and totally inside one or more resource objects. The objective function in Cuttingand Packing problems can be: minimizing the resources needed to pack all of the small objects ormaximizing the value of the small objects packed on a limited number of resource objects. In the caseof RPAMP, the objective is to pack a set of given rectangles without overlap so that the enclosingrectangle area is minimized.

The RPAMP is applicable to a wide range of activities, such as Very Large Scale Integration (VLSI)module placement, for which finding a compact placement for circuit modules on chips is an importantobjective, and facility layout (FL) which is concerned with finding the most efficient non-overlappingarrangement of n indivisible departments with unequal area requirements within a facility. In thesereal-world applications there are additional objectives that one needs to take into account (minimizewire length, transportation cost minimization, distance minimization, . . . ). Specific constraints on theaspect ratios1 of the rectangles must be imposed in order to ensure physically useful solutions. A moredetailed description of FL and VLSI problems can be found in [2, 15] and [1, 4, 5, 9, 11, 13, 14, 18],respectively.

To solve the RPAMP, we propose a hybrid approach between the metaheuristic Iterated Local Search(ILS) with Linear Programming (LP) models. The ILS is used to guide the search over the solutionspace, while LP models are used to ensure feasible solutions and to obtain local optima layouts.

The organization of this paper is as follows. In section 2 we briefly present the RPAMP. In section3 we propose a hybrid approach to solve the RPAMP. In section 4 we present the computational results∗Supported by Fundacao para a Ciencia e Tecnologia (FCT) Project PTDC/GES/73801/2006 (CROME)1by aspect ratio we mean the ratio of the length of the longer side to the length of the shorter side


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and in Section 5 some conclusions are presented.

2 The Rectangle Packing Area Minimization Problem

In the RPAMP one wishes to pack a set of non-overlapping rectangles while minimizing the enclosingrectangular area (the layout area). The dimensions of the enclosing rectangle are variable. It is atwo-dimensional problem and according to the recent typology of cutting and packing problems [17],it is classified as an open dimension problem. In this problem, rectangles must be placed orthogonally(i.e. parallel to the horizontal and vertical axes), the dimensions of rectangles are given, and rotationsare allowed. This problem is known to be NP-hard [8].

We can find the RPAMP considered in many papers including [6, 7, 12, 13, 18]. Layouts can bedivided into two types: slicing [18] and non-slicing structure [6, 7, 12, 13]. Wong et al. [18] presenteda normalized polish expression to represent a slicing structure. Nakatake et al. [13] present a methodbased on the bounded slice line grid structure for the RPAMP. In [12], Murata et al. used two sequencesof rectangles, called sequence pair (SP), to represent the geometric relations among modules. Imahori etal. proposed in [6] a decoding algorithm based on dynamic programming to obtain an optimal packingunder the constraints specified by the coded solution. In [7] the same authors proposed improvedlocal search algorithms using the SP representation to represent a solution, and speed-up techniques toevaluate solutions in various neighborhoods.

Since this problem appears in real world applications (VLSI and FL), additional constraints areconsidered on subsets of rectangles to facilitate the usage of the area. Besides rectangles with fixeddimensions (hard rectangles), various types of rectangles with additional characteristics are introduced.Soft rectangles have a fixed area with continuously variable aspect ratio within given bounds. Apreplaced rectangle is defined as a rectangle with fixed x and y coordinates. There are other typesof rectilinear shapes, T-shaped and L-shaped, which can be partitioned into two (or more) rectangles,where their boundaries are aligned horizontally or vertically.

3 Hybrid Approach

To solve the RPAMP we used a variation of the method proposed by Gomes and Oliveira [3]. Theydeveloped a hybrid approach between the metaheuristic Simulated Annealing (SA) and LP models tosolve the irregular strip packing problem. The SA algorithm is used to guide the search over the solutionspace, while the LP models ensure the solution feasibility and obtain local optima layouts. Two differentLP models are used coordinately, the compaction and the separation models. The compaction modelneeds to start from a feasible solution and is used to obtain local optima solutions (i.e., compacted andfeasible layouts). The separation model is used to obtain feasible solutions from unfeasible ones. BothLP models are built from relations derived from the relative positions between all pairs of pieces. Themain difference between these two LP models is the objective function: in the compaction model is thelayout length, while in the separation model is a measure of the overlap. The task of generating theinitial layout is performed by a greedy bottom-left placement heuristic and pieces are represented bynon-convex polygons.

In our work we used a hybrid approach of ILS with LP models. We used only hard rectangles thatare allowed to rotate and must be placed orthogonally (parallel to the horizontal and vertical axes).The dimensions of the enclosing rectangle are variable and the objective is to minimize the enclosingrectangular area without overlapped rectangles. This leads to a nonlinear objective function, whichis linearized by approximating the area of the enclosing rectangle with the perimeter of the rectangle.This approximation works quite well since the area of a rectangle increases (decreases) as the perimeterof the rectangle increases (decreases), except in extreme situations where one of the dimensions is very


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small. Details on how the compaction and separation models are derived are available in [3]. Theoption to build the compaction and separation models as in [3] will allow us to extend our approach todirectly deal with rectilinear shapes, since the original paper already deals with non-convex polygons.

ILS is an iterative algorithm composed by the following main components: initialization, localsearch, acceptance rule, and perturbation. The initial current solution is obtained by the initializationprocedure. In each of the ILS iterations a local search is applied to the current solution. Then theacceptance rule is used to update or not the current solution with the one obtained by the local search,and a perturbation movement is used to obtain a new current solution. The initialization starts bygenerating a completely random solution, i.e. a layout where the rectangles are randomly positionedon a big square (sufficiently large to have all the rectangles in one row). Afterwards we apply the LPmodels to obtain a feasible and compacted layout. In the local search, it is necessary to make smallchanges in the relative positions of rectangles to derive different compaction and separation from layouts(i.e., layouts with different relationships between pairs of rectangles). The small change introduced ina layout to generate a new neighbor is obtained by exchanging the positions of two rectangles in thelayout. Usually this leads to overlap which is afterwards removed by applying the separation model.Afterwards, the layout is compacted by the compaction model. Since the size of the neighborhood usedin the local search grows exponentially with the number of rectangles, only a partial neighborhood isinvestigated. Two different acceptance rules were used: one that always updates the current solutionand another that only updates it if the new solution is better than the current one. The reasoningbehind the first rule is to drive the search to another region of the solution space (i.e. to diversifythe search) and with the second rule the idea is to intensify the search around the best solution foundso far. The perturbation randomly moves a percentage of the rectangles of the current layout to newpositions.


To evaluate the hybrid approach we used the MCNC benchmark set introduced by the Microelectron-ics Center of North Carolina (http://vlsicad.eecs.umich.edu/BK/MCNCbench/HARD/), which is thestandard set of benchmark instances in this field. This benchmark set is composed by 5 instances withdifferent characteristics (total number of rectangles and total area of the rectangles). These character-istics and the best-known solution for each instance are presented in table 1.

Table 1: MCNC benchmark instancesRectangles Best-known solution

Instance Area Area UsageNumber (mm2) (mm2) (%) Ref.

apte 9 46.56 46.92 99.23 [1]xerox 10 19.35 19.80 97.73 [16]hp 11 8.83 8.95 98.69 [16]ami33 33 1.16 1.20 96.33 [10]ami49 49 35.45 36.43 97.29 [7]

Two different versions of the hybrid approach were implemented and evaluated: ILS CS andILS BS. ILS CS is based on the first acceptance rule described earlier (always update the currentsolution), while ILS BS is based on the second one (only update the current solution when the newsolution is better). To further emphasize the natural behavior of the two versions (ILS CS tries todiversify the search by moving to different solutions, while ILS BS tries to intensify the search alwaysaround the best solution), two different percentages values are used in the perturbation: 10% of therectangles in the ILS CS and 5% in the ILS BS. The remaining parameters are the same in bothversions and are the following: 20 ILS iterations and the neighborhood in the local search is restrictedto a maximum of 200 exchanges or 25% of the total possible exchanges (these values were fixed after


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some preliminary computational experiments).

The computational tests were performed on a Linux workstation equipped with an Intel XEONDual Core 5160 running at 3GHz and the ILOG Cplex 10.1 was used to solve the LP models. Thesecomputational tests consisted in running ILS CS and ILS BS on 20 independent runs for each instance.The results obtained are shown on table 2. The table presents for each instance the best solution, theworst solution and average of the 20 runs, as well the average running time for one run.

Table 2: Computational resultsILS CS ILS BS

Instance Avg. Best Worst Avg. time Avg. Best Worst Avg. time(%) (%) (%) (sec.) (%) (%) (%) (sec.)

apte 97.59 98.41 93.62 8.7 97.56 98.41 93.62 9.4xerox 96.42 97.57 94.52 13.6 96.57 97.58 95.25 14.2hp 95.41 97.77 92.25 12.5 95.71 97.77 92.25 12.1ami33 95.92 96.77 95.18 699.9 95.72 96.52 94.78 876.3ami49 95.95 96.98 95.18 2750.4 95.98 96.71 95.10 2579.1

The results obtained by the two versions of the proposed hybrid approach were very consistent interms of the obtained results. In both versions, the results showed differences between the best andaverage solutions around 1% for 4 instances. Only in instance hp this difference was above 2%. Thecomparison between the two versions showed similar results. Differences between the average, the bestand the worst solutions between the two versions are always smaller than 0.5%, with a small advantagefor ILS BS on the average solutions and for ILS CS on the best solution. As expected both versionspresented similar computational running times, since they both share the same structure.

In what concerns the comparison with the best-known solutions, both versions showed differencesinferior to 1% for four of five instances used. The exception was instance ami33 where the best-knownsolution was improved by both versions. For the average solutions, the difference is smaller than 0.5%for instance ami33, over 3% for instance hp and around 2% for the remaining ones.

Finally, the best layouts obtained for each instance are presented on figure 1.

5 Conclusions

We have described the RPAMP and proposed a hybrid approach to solve it. The main idea behind thisapproach is hybridizing the ILS metaheuristic with LP models, used to compact and separate layouts.The ILS algorithm is used to guide the search over the solution space, while the LP models are usedto ensure feasibility and to obtain local optima for a particular set of relative positions between therectangles.

A set of computational tests, based on the MCNC benchmark set, was used to validate the pro-posed hybrid approach. The computational results obtained clearly showed the potential of the hybridapproach, with the best solution within a 2% distance to the best-known solution. In the near future,we also want to use other metaheuristics to guide the hybrid approach.

Another important characteristic of the proposed hybrid approach is the flexibility to easily incor-porate additional constraints and objectives. This is particularly important since we want to extendthe hybrid approach to tackle VLSI module placement problems and the Facility Layout problems.


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(a) apte (98.41%) (b) xerox (97.58%) (c) hp (97.77%)

(d) ami33 (96.77%) (e) ami49 (96.98%)

Figure 1: Best layouts


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[10] J. M. Lin and Y. W. Chang Corner Sequence – A P-Admissible Floorplan Representation With aWorst Case Linear-Time Packing Scheme. IEEE Trans. on VLSI Syst., 4:679–686, 2003.

[11] H. Murata, K. Fujiyoshi, S. Nakatake and Y. Kajitani Rectangle -packing based module placement.Proc. ICCAD, 472–479, 1995.

[12] H. Murata, K. Fujiyoshi, S. Nakatake and Y. Kajitani VLSI module placement based on rectangle-packing by the sequence-pair. IEEE Trans. Comput. Aided Des., 15(12):1518–1524, 1996.

[13] S. Nakatake, K. Fujiyoshi, H. Murata and Y. Kajitani Module placement on BSG-structure andIC layout applications. Proc. Intl. Conf. Comput. Aided Des., 484–491, 1996.

[14] R. H. J. M. Otten Automatic floorplan design. Proc. DAC, 261–267, 1982.

[15] S. P. Singh and R. K. Sharma A review of different approaches to the facility layout problems. IntJ Manuf Technol, 30:425–433, 2006 .

[16] X. Tang and D. Wong FAST-SP: a fast algorithm for block placement based on sequence pair.Proc. on Asia P. Des. Autom, 521–526, 2001.

[17] G. Wascher, H. Haußner and H. Schumann An Improved Typology of Cutting and Packing Prob-lems. European Journal of Operational Research, 183:1109–1130, 2007.

[18] D. F. Wong and C. L. Liu A new algorithm for floorplan design. Proc. DAC, 101–107, 1986.


EUMEeting 2009 153

A Combined Local Search Approach for the Two-dimensional

Bin Packing Problem

T. M. Chan ∗ Filipe Alvelos ∗ † Elsa Silva ∗

J. M. Valerio de Carvalho ∗ †

∗ Centro de Investigacao Algoritmi, Universidade do Minho4710-057 Braga, Portugal


† Departamento de Producao e Sistemas, Universidade do Minho4710-057 Braga, Portugal

Email: {falvelos, vc}@dps.uminho.pt

1 Introduction

A bin packing problem is a common combinatorial optimization problem which is frequently tackled byindustries. The objective is to pack a given set of rectangular items to an unlimited number of identicalrectangular bins such that the total number of used bins is minimized and subject to three constraints:(i) all items must be packed to bins, (ii) all items cannot overlap, and (iii) the edges of items areparallel to those of bins. Items should be packed into bins based on the required cutting method whichcan be a free cutting or guillotine cutting. A free cutting means that the cutting does not have anyrestriction. However, a guillotine cutting means the one from an edge of the rectangle to the oppositeedge. A guillotine cutting which is applied n times is referred to an n-stage guillotine cutting. For n =2, horizontal cuts are applied in the first stage and then vertical cuts are applied in the second stage.If trimming is adopted (i.e. non-exact case), further horizontal cuts will be performed after the secondstage in order to separate items from wastes. In this study, two-stage guillotine cutting with non-exactcase and items with the fixed orientation are considered and a combined local search approach whichcomprises two local search heuristics, Variable Neighborhood Descent (VND) and Random NeighborSelection (RNS), is proposed to improve the solution given by a constructive heuristic (CH). Note thatRNS means the random selection of a solution’s neighbor among all of its possible neighbors.

Some related work about application of heuristics to solve bin packing problems and local searchheuristics to solve other problems will be presented as follows. Berkey and Wang dealt with the two-dimensional guillotine bin packing problem with prohibiting item rotations by utilizing the developedfour heuristic methods, Finite next-fit, Finite first-fit (FFF), Finite best-strip (FBS), and Finite bottom-left [1]. Monaci and Toth tackled the two-dimensional bin packing problem with free cutting and withfixing item orientations by capitalizing a set-covering-based heuristic [2]. Hayek et al. proposed a newheuristic based on a best-fit algorithm to deal with the two-dimensional bin packing problem with freecutting and item rotations [3]. Moreover, Parreno et al. attacked the two and three dimensional binpacking problem by using a newly proposed hybrid algorithm called Greedy Randomized AdaptiveSearch Procedure (GRASP) with VND [4]. Gao et al. coped with the flexible job shop schedulingproblem by using a hybrid genetic algorithm and VND which involved the movement of operations [5].Degila and Sanso dealt with the topological design problem of a yottabit-per-second multidimensionallattice network by means of tabu search with VND which involved the movement of facilities [6].


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Figure 1: An example of constructive heuristic.

Lodi et al. solved the two-dimensional bin packing problem by using a tabu search (TS) approachover a lower level heuristic [7]. In this paper, both deterministic neighborhood structures and RNSoperators are utilized to further improve the solution given by a constructive heuristic, CH.

2 Constructive heuristic

In this section, a CH for solving the two-dimensional guillotine bin packing problems is presented. Thedesign of this heuristic is simple but effective and can satisfy the guillotine-cutting constraint. Theidea of the heuristic is based on the sorting of item types for defining an initial packing sequence anditerative trials of packing items into existing stacks, shelves or bins according to the following criteria.For the former one, the three criteria considered to sort the item types in the descending order are: (i)by width, (ii) by height, and (iii) by area. For the latter one, it is possible that more than one existingstack, shelf or bin can accommodate each item. Therefore, criteria should be established to determinewhich existing stack, shelf and bin should be selected for packing the items. The three criteria usedto achieve this purpose are: (i) minimize the residual width after packing the item, (ii) minimize theresidual height after packing the item, and (iii) minimize the residual area after packing the item. Notethat different criteria can be adopted for stack, shelve and bin selection.

The first step of the heuristic is to define a packing sequence for items by sorting the item typesbased on a specified criterion. Then, iteratively, each item which can be rotated or non-rotated ispacked into an existing stack which minimizes a specified criterion. If this is not possible, we try topack it into an existing shelf which minimizes a specified criterion. If this is not possible, we try topack it into a used bin which minimizes a specified criterion. If this is not possible again, it is placedinto a new bin. Finally, a solution is obtained by running the heuristic with several sets of criteria andthen selecting the best one among the obtained solutions.

Figure 1 illustrates an example of the CH. Assume that the items are sorted by height and fouritems have already been loaded into the bin. The item 5 currently considered can be loaded into anexisting stack (free space A) (for three-stage problems only), an existing shelf (free space C or D), onthe top of shelves of the bin (free space E) or a new empty bin. However, the free space B cannotaccommodate it. The place where the item is packed depends on the pre-selected criterion and thetype of the problem tackled.

Suppose that a two-stage problem with trimming is now being solved. In this case, only threepossible choices of empty spaces, C, D and E, are available for packing item 5. If the criterion (i) is set,it will be packed into C to fulfil the criterion; if the criterion (ii) is selected, it will be packed into D; ifthe criterion (iii) is chosen, it will be placed into D. Note that the same applies to a three-stage problemwithout trimming. Nevertheless, if it is a three-stage problem with trimming, and the criterion (i) or(iii) is set, it will be placed into A instead of C or D. More details of this approach can be found in [8].


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3 A combined local search approach

A combined local search method is proposed to improve the solution given by the CH in Section 2. Thiscombined approach consists of two local search heuristics, VND and RNS. VND is a meta-heuristicproposed by Hansen and Mladenovic [9]. The concept of VND is to systematically utilize differentneighborhood structures. VND used in this study embraces three neighborhood structures named”swap adjacent item types”, ”swap adjacent item subsequences”, and ”reverse item subsequences”. Itis a sequential VND which is different from the common one that the three neighborhood structuresarranged in the fixed order are implemented one by one.

RNS is adopted in a way similar to VND. The differences between RNS and VND are that RNS(i) impose the restriction of using all random neighbor selection operators rather than deterministicor mixed ones, and (ii) use the fixed number of iterations to explore better neighbor solutions foreach of all neighbor selection operators. In each RNS operator, instead of finding the best neighborof the current solution by complete enumeration, only one neighbor is randomly generated in eachtime and then compared with the current solution. The advantage of this modification is that a largecomputational load is not required to search all neighbors of the current solution, especially whenthe problem size is huge. RNS adopted in this study comprises three neighbor selection operatorstitled ”Cut-and-Paste”, ”Split-and-Redistribute”, and ”Swap Block”. Like those in VND, these threeneighbor selection operators arranged in the fixed order are implemented one by one.

The three neighborhood structures of VND are firstly executed, followed by the three RNS operators.In the following, the details of the three neighborhood structures and three neighbor selection operatorswill be given.

”Swap adjacent item types” is aimed at swapping all items in two adjacent item types. Figure2(a) illustrates an instance of this neighborhood structure. Assume that two highlighted adjacent itemtypes, 2 and 4, are chosen. One item in the first type and three items in the second type are exchangedto produce the new solution.

The mission of ”swap adjacent item subsequences” is to swap two adjacent item subsequences, bothof which have the same size. A size parameter is used to define the size of the neighborhood. Figure 2(b)portrays an instance of this neighborhood structure for the size of the neighborhood, 2. Two shadedadjacent item subsequences (1, 3) and (3, 2) are selected in the current solution and then swapped toproduce the new solution.

The objective of ”reverse item subsequences” is to reverse the order of an item subsequence with agiven size. A size parameter is utilized to define the size of the item subsequence. Figure 2(c) shows anexample of this neighborhood structure for the size of the item subsequence, 3. The item subsequence(3, 2, 4) is selected and then their packing orders are reversed to generate a new solution.

”Cut-and-Paste” was initially proposed as a genetic operation which was applied in the Jumping-Gene paradigm to solve multi-objective optimization problems [10]. Its implementation is that the”jumping” element is cut from an original position and pasted into a new position of a chromosome.However, since this operation can also be used as a random neighbor selection operator, in this study,”Cut-and-Paste” is applied to the solution in a way that there is only one ”jumping” segment in thesolution, and the length, original position and new position of the ”jumping” segment are randomlychosen. Figure 2(d) depicts an example of this neighbor selection operator. Given that the randomlygenerated length is 4, the original position is 6 and the new position is 2. In other words, the highlightedsegment (4, 2, 1, 4) is randomly selected. The segment is cut from the original position and then pastedinto the new position to complete the operation.

The objective of ”Split-and-Redistribute” is to split various blocks with a given length from thesolution and redistribute them to the solution. The total number, length, original positions, and newpositions of the blocks are randomly selected and all blocks have the same length. Figure 2(e) illustratesan instance of this neighbor selection operator. Suppose that the randomly selected total number is 3,


EUMEeting 2009 156

Figure 2: Examples of different deterministic neighborhood structures and random neighbor selectionoperators used.

the length is 2, the original positions are 1, 5 and 10, and the new positions are 10, 1 and 5 for thethree blocks respectively. That is, the three highlighted blocks (1, 3), (3, 4), and (4, 4) are randomlychosen. These three blocks are split and then redistributed to their new positions to acquire the newsolution.

”Swap Block” is aimed at exchanging two different blocks with a given length in the solution. Thelength and the positions of the two blocks are randomly selected and the lengths of these two blocksare equivalent. Figure 2(f) portrays an example of this neighbor selection operator. Assume that therandomly selected length is 3 and the randomly chosen positions are 4 and 8. That means the twohighlighted blocks (2, 3, 4) and (1, 4, 4) are randomly selected. Then, these two blocks are swapped toproduce the new solution.


In order to show the effectiveness of the proposed method (i.e. CH + combined local search method),it is compared with other approaches, FFF, FBS, Knapsack Packing (KP), TS with FBS, TS with KP,CH + pure VND, and CH + pure RNS. The computer program for the suggested method was writtenin C++ and run on a PC with 2.00 GHz Intel R©CoreTM 2 CPU with 1 GB memory. Six classes ofproblems named B&W from the Berkey and Wang study [1] and four classes of problems named M&Vfrom the Martello and Vigo study [11] were adopted. Each class contains five subclasses, each of whichhas ten instances and therefore there are a total of 300 and 200 instances respectively. As mentionedin the Section 1, the scenario, two-stage guillotine cutting with non-exact case and items with the fixedorientation, was considered. In pure RNS and the proposed method, the order of neighbor selectionoperators used in the computational tests is the same as that described in the previous Section. Also,the total number of iterations used is 500 for each RNS operator. Since pure RNS and the proposedapproach adopt RNS operators, they were conducted 30 runs to obtain the best, average, and worstsolution in each problem of each subclass in order to allow a statistical analysis.

Figure 3 shows the computational results of various approaches in terms of the ratio of heuristicsolution to lower bound [7] for all 50 subclasses. z and LB denote a heuristic solution value and alower bound value respectively. Note that the results of FFF, FBS, KP, TS with FBS, and TS withKP were obtained from [7]. The lowest ratio is bold-faced in each subclass. By comparing the z/LBof the leftmost six methods with the best z/LB of pure RNS and the proposed approach, it can beseen that the proposed approach is the best performer which can obtain lower ratios in a total of 29subclasses. By comparing the best z/LB, average z/LB and worst z/LB of the proposed approach tothose of CH + pure RNS, the average z/LB and worst z/LB of the former one are generally lower thanthose of the latter one and closer to the best z/LB, and thus the former one is more stable and robust.


EUMEeting 2009 157

Figure 3: Computational results of various approaches.

Overall, the results reveal that all the deterministic neighborhood structures and RNS operators of theproposed approach are able to explore many different neighbor solutions that increase the chance offinding promising solutions. The deterministic neighborhood structures in the proposed method canlocate a better starting solution for the RNS operators to further enhance the quality of the solutionand thus it has a slight improvement over both CH + pure VND and CH + pure RNS.

Besides, note that the leftmost five methods from the past studies and the remaining methodsare run by means of different computers with different models, their computational time cannot becompared with each other and therefore only that of the rightmost three methods is given in the table.The computational time of CH + pure RNS and the proposed approach in each subclass is obtained bysumming the total time of 30 runs per instance for all 10 instances and then dividing the sum by 10. Itcan be observed that, as expected, the proposed method requires longer computational time than theremaining two because both deterministic neighborhood structures and RNS operators were used.


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5 Conclusions

In this paper, a combined local search method comprising VND and RNS has been studied. The ob-jective of employing the proposed approach is to further improve the solution given by the constructiveheuristic (CH). Three deterministic neighborhood structures and three RNS operators were designed inthe proposed approach. The quality of solutions found by the proposed approach are quite good as itobtains lower ratios in a total of 29 subclasses out of 50 subclasses and outperforms three heuristics, twotabu search meta-heuristics, and CH + pure VND. By comparing the best, average and worst resultsof the proposed approach to those of CH + pure RNS, the former one is more stable and robust.

References

[1] J. O. Berkey and P. Y. Wang. Two-dimensional finite bin-packing algorithms. J. Oper. Res. Soc.,38:423–429, May 1987.

[2] M. Monaci and P. Toth. A set-covering-based heuristic approach for bin-packing problems. IN-FORMS J. Computing, 18:71–85, Winter 2006.

[3] J. E. Hayek, A. Moukrim, and S. Negre. New resolution algorithm and pretreatments for thetwo-dimensional bin-packing problem. Computers Oper. Res., 35:3184–3201, Oct. 2008.

[4] F. Parreno, R. Alvarez-Valdes, J. F. Oliveira, and J. M. Tamarit. A hybrid GRASP/VND algorithmfor two- and three-dimensional bin packing. Ann. Oper. Res., to be published.

[5] J. Gao, L. Sun, and M. Gen. A hybrid genetic and variable neighborhood descent algorithm forflexible job shop scheduling problems. Computers Oper. Res., 35:2892–2907, Sep. 2008.

[6] J. R. Degila and B. Sanso. Topological design optimization of a yottabit-per-second lattice network.IEEE J. Sel. Areas Commun., 22:1613–1625, Nov. 2004.

[7] A. Lodi, S. Martello, and D. Vigo. Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems. INFORMS J. Computing, 11:345–357, 1999.

[8] F. Alvelos, T. M. Chan, P. Vilaca, T. Gomes, E. Silva, and J. M. Valerio de Carvalho. Sequencebased heuristics for two-dimensional bin packing problems. Eng. Optim., accepted, 2009.

[9] P. Hansen and N. Mladenovic. Variable neighborhood search: Principles and applications. EuropeanJ. Oper. Res., 130:449–467, May 2001.

[10] T. M. Chan, K. F. Man, S. Kwong, and K. S. Tang. A jumping gene paradigm for evolutionarymultiobjective optimization. IEEE Trans. Evol. Comput., 12:143–159, Apr. 2008.

[11] S. Martello and D. Vigo. Exact solution of the two-dimensional finite bin packing problem. Man-agement Sci., 44:388–399, Mar. 1998.


EU/MEeting 2009 159

List of Participants

Abreu, PedroLIAAD/INESC [email protected]

Afonso, Luıs M.Instituto Superior de Engenharia do [email protected]

Almada-Lobo, BernardoUniversity of [email protected]

Almeder, ChristianUniversity of [email protected]

Aloise, DarioUniversidade Federal do Rio Grande do [email protected]

Amorim, JoaoINESC [email protected]

Araujo, [email protected]

Araujo, SılvioUNESP/[email protected]

Bagalkote, JayasimhaUniversity of [email protected]

Bastos, JoaoPolytechnic School of Engineering of [email protected]

Borges Lopes, RuiUniversity of [email protected]

Braz Martins, Luıs RicardoInstituto Superior de Engenharia do [email protected]

Cardoso Costa, AntonioPolytechnic School of Engineering of [email protected]

Carravilla, Maria AntoniaFaculty of Engineering of University of [email protected]

Carvalho, PiedadePolytechnic School of Engineering of [email protected]

Chan, Tak MingUniversity of [email protected]

Chibante, RuiPolytechnic School of Engineering of [email protected]

Correia, AldinaESTGF-IPP, [email protected]

Costa, CarlosPolytechnic School of Engineering of [email protected]

Costa, Maria TeresaPolytechnic School of Engineering of [email protected]


160 EU/MEeting 2009

Coutinho, SerafimESMAE,Instituto Politecnico do [email protected]

Dullaert, WoutUniversity of Antwerp - [email protected]

Euchi, JalelGIAD,Faculty Of Economics and Management of [email protected]

Festa, PaolaUniversity of Napoli Federico [email protected]

Fonseca, CarlosPolytechnic School of Engineering of [email protected]

Fortz, BernardUniversite Libre de [email protected]

Galvao, TeresaFaculty of Engineering of University of [email protected]

Gaspar-Cunha, AntonioUniversity of [email protected]

Gomes, A. MiguelFaculty of Engineering of University of PortoINESC [email protected]

Gomes, BrunoPolytechnic School of Engineering of [email protected]

Gomes, RuiFaculty of Engineering of University of [email protected]

Gomes Alves da Silva, JosePolytechnic School of Engineering of [email protected]

Hartl, Richard F.University of [email protected]

James, RossUniversity of CanterburyNew [email protected]

Leite, TiagoPolytechnic School of Engineering of [email protected]

Lopes, Isabel CristinaESEIG-IPP, University of [email protected]

Macedo, MarkINESC [email protected]

Morais, HenriqueMorinfo-Consultores, [email protected]

Moreira, Maria do [email protected]

Moura, AnaUniversity of [email protected]

Neto, TeresaInstituto Politecnico de [email protected]


EU/MEeting 2009 161

Oliveira, Jose FernandoUniversity of Porto/INESC [email protected]

Oliveira, MarisaPolytechnic School of Engineering of [email protected]

Oliveira, WagnerUniversidade Federal do Rio Grande do [email protected]

Pedrosa, TiagoPolytechnic School of Engineering of [email protected]

Pedroso, Joao PedroFaculty of Science of University of PortoINESC [email protected]

Pereira Lopes, ManuelInstituto Superior de Engenharia do [email protected]

Pinheiro, AlexandrePolytechnic School of Engineering of [email protected]

Pinho de Sousa, JorgeFaculty of Engineering of University of PortoINESC [email protected]

Pinto Ferreira, Maria EduardaPolytechnic School of Engineering of [email protected]

Raa, BirgerGhent [email protected]

Rais, AbdurINESC [email protected]

Rei, Rui JorgeFaculty of Science of University of [email protected]

Ribeiro, Celso C.Universidade Federal [email protected]

Rios-Mercado, RogerUniversidad Autonoma de Nuevo [email protected]

Rocha, MartaFaculty of Engineering of University of [email protected]

Rodrigues, Ana MariaInstituto Politecnico do Porto/INESC [email protected]

Ronnqvist, MikaelNorwegian School of Economics and [email protected]

Santos, ValerioPolytechnic School of Engineering of [email protected]

Schilde, MichaelUniversity of [email protected]

Sevaux, MarcUniversite de [email protected]


162 EU/MEeting 2009

Silva, AnaUniversidade Federal do Rio Grande do [email protected]

Silva, ElsaUniversity of [email protected]

Silva, [email protected]

Soeiro Ferreira, JoseFaculty of Engineering of University of PortoINESC [email protected]

Sorensen, KenethUniversity of [email protected]

Stutzle, ThomasUniversite Libre de [email protected]

Teixeira, CristinaUniversity of [email protected]

Viana, AnaPolytechnic School of Engineering of PortoINESC [email protected]

Wright, MikeLancaster University Management [email protected]

Xambre, Ana RaquelUniversity of [email protected]

Yahaya, AbubakarLancaster UniversityManagement [email protected]


164 EU/MEeting 2009

Porto Map

• ISEP: Rua Dr. Antonio Bernardino de Almeida, 431, 4200-072 Porto (yellow mark)

• Get-together (Republica da Cerveja): Cais de Gaia, 4400-161 Vila Nova de Gaia (blue mark)

• Ibis Hotel: Rua Dr Placido Costa, 4200-450 Porto (red mark)


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