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Organiza(on*

9th*ESICUP*Mee(ng*La*Laguna,*Spain,*March*21=23,*2012*

9th ESICUP Meeting 3

Table of Contents

Welcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Information for Conference Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Program Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Scientific Program Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Social Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

La Laguna, Spain, March 21-23, 2012

4 9th ESICUP Meeting

Local Organizing Committee:

Coromoto Leon, (chair) Universidad de La Laguna

Jésica de Armas Adrián, Universidad de La Laguna

Yanira González González, Universidad de La Laguna

Gara Miranda Valladares, Universidad de La Laguna

Eduardo Segredo González, Universidad de La Laguna

Carlos Segura González, Universidad de La Laguna

Program Committee:

José Fernando Oliveira, (chair) University of Porto

A. Miguel Gomes, University of Porto

Gerhard Wäscher, Otto-von-Guericke-Universität Magdeburg

Ramón Alvarez-Valdes, University of Valencia

Julia Bennell, University of Southampton

Coromoto Leon, Universidad de La Laguna

Organised by:

ESICUP— EURO Special Interest Group on Cutting and Packing

University of La Laguna



Welcome

José F. Oliveira Coromoto Leon

Dear Friends,

Welcome to the 9th Meeting of ESICUP – EURO Special Interest Group on Cutting and Packing. Since itsformal recognition as an EURO Working Group in 2003, ESICUP has run a series of annual meetings thathave joined researchers and practitioners in the field of cutting and packing. Wittenberg, Southampton,Porto, Tokyo, L’Aquila, Valencia, Buenos Aires and Copenhagen have hosted our past meetings andthis 9th meeting is now held in San Cristóbal de La Laguna. The scientific program that has been puttogether guarantees that these will be rather fruitful days.

With 24 presentations organized in an unique stream, we will be able to hear and enjoy all presentations,in a friendly and relaxed environment propitious to fruitful scientific discussions. Once more this meetingwill be an instrument for the dissemination and the development of our field of research.

San Cristóbal de La Laguna has an exceptional, universal value due to the concept of its town plan.This historical site is the archetype of city-territory. It is the first example of a non-fortified town,conceived and built according to a plan inspired by navigation, the science of the time. Its layout isbased on peaceful new social order, inspired by the religious doctrine of the millennium of the year 1500.The World Heritage Committee of the United Nations Educational, Scientific and Cultural Organization(UNESCO) approve San Cristóbal de La Laguna in Tenerife as a World Heritage City in 1999.

The University of La Laguna, that is hosting this meeting, has its roots in 1701 when Augustine monksestablished a centre for higher learning in the city of La Laguna, but the university opened its doors asan academic institution on January 12, 1817, although under a different name. After a turbulent periodthat led to the university abolishment in 1845, in 1913 it was reestablished and sice then it did not stopgrowing. Today the University of La Laguna has 25000 students, 1700 professors and 800 administrativeand service employees and offers 56 undergraduate degrees, 41 masters and 33 doctorate programmes inall different fields.

We would like to spend a word of gratitude to our colleagues, members of the Scientific and OrganizingCommittees, for the their important contribution for the existence and success of this meeting.

Our wish is that you may leave La Laguna and Tenerife island looking forward to come back on holidaysand for the 10th ESICUP Meeting, in 2013.

All the best.

José F. Oliveira Coromoto Leon,University of Porto, FEUP / INESC Porto Universidad de La Laguna

Program Chair Local Organizer



Information for Conference Participants

MEETING VENUEThe 9th ESICUP Meeting will be held at ULL, Universidad de La Laguna (Campus Anchieta), building“E.T.S. Ingeniería Informática”, in San Cristóbal de La Laguna, Tenerife, Canary Islands, Spain.

Address:E.T.S. Ingeniería InformáticaCamino San Francisco de Paula, s / n.Campus de Anchieta38271 La Laguna

CampusAnchieta (C - E.T.S. de Ingeniería Informática)

REGISTRATIONTakes place on Wednesday 21th March, 18:30 to 20:00, and Thursday 22th March, 9:30 to 9:50 at ULL.

NOTES ON PRESENTATION• Equipment

The conference room is equipped with an overhead projector and with a video projector and laptopcomputer.We suggest that you bring your own computer and/or transparencies as a backup.

• Length of Presentation 22.5 minutes for each talk, including discussion. Please note that weare running on a very tight schedule. Therefore, it is essential that you limit your presentation tothe time which has been assigned to you. Session chairpersons are asked to ensure that speakersobserve the time limits.

INTERNETFurther details on how to access wireless network will be given at registration.

GET-TOGETHERTakes place at the conference venue, “E.T.S. Ingeniería Informática” - ETSII, Camino San Francisco dePaula, s/n. There will be several cold snacks and soft drinks.The get-together takes place on Wednesday 21th March, 18:30 to 20:00.

CONFERENCE DINNERTakes place at “Nivaria Hotel”, Plaza del Adelantado, 11, San Cristóbal de La Laguna, Tenerife. To get


http://www.ull.es/view/institucional/ull/Campus_de_Anchieta/es


there from ETSII, walk 10-15 minutes. Take Avenida Francisco Sanchez Astrophysical toward the city,crossing the bridge, and you will be in Avenida of La Trinidad. Turn right onto Calle Barcelona. Crossthe Plaza de la Milagrosa and take Calle Santo Domingo to the Plaza del Adelantado. There is the HotelNivaria. The conference dinner starts on Thursday 22th March, 20:30. http://www.hotelnivaria.com/.

Takes place at “Nivaria Hotel”, Plaza del Adelantado, 11, San Cristóbal de La Laguna, Tenerife. To getthere from DIKU, take any bus to Nørreport station, and then walk 10 minutes to Kongens Nytorv. Theconference dinner starts on Thursday 22th March, 20:30.http://www.hotelnivaria.com/

TOUR AROUND LA LAGUNAThe guided tour to La Laguna starts at 16:00 on March 23th,. The meeting point is “Plaza del Adelan-tado”, 11, San Cristóbal de La Laguna, Tenerife.

LA LAGUNA, A WORLD HERITAGE CITYSan Cristóbal de La Laguna has an exceptional, universal value due to the concept of its town plan.This historical site is the archetype of city-territory. It is the first example of a non-fortified town,conceived and built according to a plan inspired by navigation, the science of the time. Its layout isbased on peaceful new social order, inspired by the religious doctrine of the millennium of the year 1500.The World Heritage Committee of the United Nations Educational, Scientific and Cultural Organisation(UNESCO) approve San Cristóbal de La Laguna in Tenerife as a World Heritage City in 1999.

MOVING AROUND• Busses (Guagua - autobús)

The name of the bus company is TITSA (http://www.titsa.com/). The Bus Stop is in PadreAnchieta, 5 minutes walk to the ETSII Building.

• Tram (Tranvía)The name of the tram company is METROTENERIFE. You can see the map of lines on the web:http://www.metrotenerife.com/. For further informations: http://www.visitlalaguna.es/node_468.jsp.

• AirportTenerife island is served by two international airports: Tenerife North Airport (Los Rodeos) andTenerife South Airport (Reina Sofia).La Laguna lies near the Tenerife North Airport (Los Rodeos’), it is just 5 minutes to the city center.The South Airport (Reina Sofia) is 80Km from La Laguna. It takes 2 buses and more or less 1 hourto reach the north.


http://www.hotelnivaria.com/

http://www.hotelnivaria.com/

http://www.titsa.com/

http://www.metrotenerife.com/

http://www.visitlalaguna.es/node_468.jsp

http://www.visitlalaguna.es/node_468.jsp


Program Overview



Scientific Program Schedule

Thursday March 22th9:50 – 10:00

Opening Session

Welcome Adress

10:00 – 11:00

Session 1 Chair: A. Miguel Gomes

1.1 – A Reduction Approach for Solving the Rectangle Packing Area Minimization ProblemAndreas Bortfeldt

1.2 – New Results on Conservative ScalesGleb Belov, Guntram Scheithauer

1.3 – Tracing the Origin of Cutting and Packing Problem’s NP-HardnessYiping Lu, Jinmin Wang

11:30 – 13:00

Session 2 Chair: Gerhard Wäscher

2.1 – Models and Algorithms for Selecting the Best Shipper Sizes for Sending Products to CustomersF. Parreño, M. T. Alonso, R. Alvarez-Valdes, J.M. Tamarit

2.2 – Mixed Integer Programs for Cutting and Scheduling OptimizationFabrizio Marinelli, Claudio Arbib

2.3 – An Hierarchical Approach to Shelf Space AllocationTeresa Bianchi-Aguiar, Maria Antónia Carravilla, José F. Oliveira

2.4 – The cutting stock/leftover problemMarcos Arenales, Andrea Vianna, Adriana Cherri, Carla Lucke

15:00 – 16:30

Session 3 Chair: François Clautiaux

3.1 – Automated Storage DesignFrançois Clautiaux, Ines Bahri, Luce Brotcorne, El-Ghazali Talbi

3.2 – One Dimensional Cutting Stock and Suborder OptimizationMihael Cesar, Mirko Gradišar

3.3 – Constraint Programming Approaches for Higher-Dimensional Orthogonal PackingMarat Mesyagutov, Gleb Belov, Guntram Scheithauer

3.4 – Automatic Generation of Heuristics for the On-line 1D Bin Packing Problem with Biased Random KeyGenetic ProgrammingJosé Fernando Gonçalves

17:00 – 18:30

Session 4 Chair: Coromoto Leon

4.1 – A Flexible Framework for Pallet and Container LoadingJoaquim Gromicho, Gerhard Post, Jeroen van Wolffelaar

4.2 – Parallel Models for the Multi-Objective Container Loading ProblemYanira González, Jesica de Armas, Gara Miranda, Coromoto León



4.3 – Parallel Hyperheuristics for a MultiObjectivised 2D Packing ProblemCarlos Segura, Eduardo Segredo, Coromoto Leoón

4.4 – Two Phase Loading in Presence of Multiple ConstraintsM.T. Alonso, R. Álvarez-Valdés, J.Gromicho, F.Parreño, G.Post, J.M. Tamarit

Friday March 23th9:30 – 11:00

Session 5 Chair: Ramón Álvarez-Valdés

5.1 – A New No-Fit Polygon Generator for Rectilinear PolygonsMarisa Oliveira, Pedro Rocha, Eduarda Pinto Ferreira, A. Miguel Gomes

5.2 – Using Circle Covering to Tackle Nesting Representations LimitationsPedro Rocha, Rui Rodrigues, A. Miguel Gomes

5.3 – Packing of Arbitrary Shaped Objects into a Convex n-Polygonal Container of Minimal AreaT. Romanova, Yu.Stoyan, A. Pankratov

5.4 – Matheuristics for the Two-Dimensional Nesting ProblemA. Martinez, R. Alvarez-Valdes, M.A. Carravilla, A.M. Gomes, J.F. Oliveira, J.M. Tamarit

11:30 – 13:00

Session 6 Chair: José Fernado Oliveira

6.1 – Comparing Encoding Schemes for Multi-objective Two-dimensional Guillotine Cutting ProblemsJesica de Armas, Gara Miranda, Coromoto León

6.2 – A Dynamic Ordering Rule Based on Size Matches for Solving Packing ProblemsJinmin Wang, Qi Yang, Yiping Lu

6.3 – A 5/3-Approximation for Strip PackingLars Prädel, Rolf Harren, Klaus Jansen, Rob van Stee

6.4 – A Decomposition Algorithm for the Exact Solution of the Strip Packing ProblemJean-François Coté, Mauro Dell?Amico, Manuel Iori

13:00 – 13:10

Closing Session

Closing Notes



Social Program

• Registration and Get-togetherMarch 21, from 18.30 to 20.00“E.T.S. Ingeniería Informática”ETSII, Camino San Francisco de Paula, s/n.

• Conference dinnerMarch 22nd, from 20:30 to 22:30“Nivaria Hotel”,Plaza del Adelantado, 11, San Cristóbal de La Laguna, Tenerife

• Guided tour to La LagunaMarch 23th, from 16:00Plaza del Adelantado, 11,San Cristóbal de La Laguna, Tenerife



Abstracts

1.1A Reduction Approach for Solving the Rectangle Packing Area

Minimization ProblemAndreas Bortfeldt∗

∗ University in Hagen, Germany

In the rectangle packing area minimization problem (RPAMP) we are given a set of rectangles with known di-mensions. We have to determine an arrangement of all rectangles, without overlapping, inside an envelopingrectangle of minimum area. The paper presents a generic approach for solving the RPAMP that is based on twoalgorithms, one for the 2D container loading problem (CLP), and the other for the 2D strip packing problem(SPP). In this way, solving an instance of the RPAMP is reduced to solving multiple SPP and CLP instances.A fast constructive heuristic is used as SPP algorithm while the CLP algorithm is instantiated by a tree searchmethod and a genetic algorithm alternatively. All these SPP and CLP methods have been published previously.Finally, the best variants of the resulting heuristics are combined within one procedure. The guillotine cuttingcondition is always observed as an additional constraint. The approach was tested on 15 well-known RPAMPinstances (above all MCNC and GSRC instances) and new best solutions were obtained for eleven instances. Thecomputational effort remains acceptable.Keywords: Rectangle packing area minimization problem, Floor planning, Open dimension problem,MCNC, GSRC, Metaheuristic, Tree search

1.2New Results on Conservative Scales

Gleb Belov∗, Guntram Scheithauer†∗ Universität Duisburg-Essen, Germany, † Technische Universität Dresden, Germany

Conservative scales (CS) are modified sizes of items in packing problems which allow for construction of alter-native volume bounds. CS can be obtained, e.g., by dual-feasible functions and other methods. We present aconvergence result for a linearization heuristic for a multi-linear programming problem over CS. Furthermore,some greedy methods to obtain strong CS can be accelerated by a modification of dynamic programming. Testresults are presented on specially constructed instances and compared with other methods from the literature.Keywords: Conservative scales, Dual-feasible functions, Linearization, Dynamic programming

1.3Tracing the Origin of Cutting and Packing Problem’s NP-Hardness

Yiping Lu∗, Jinmin Wang†∗ School of Mechanical Engineering, Beijing Jiaotong University, Beijing, 100044, China, † Tianjin Key

Laboratory of High Speed Cutting & Precision Machining. TUTE, Tianjin, 300222, China

The landscape of NP-completeness/NP-hardness of cutting and packing problem is still unclear for researchers.In this communication, the authors scan cutting and packing literature papers as well as their references, andtry to trace to the original NP-hardness proofs. The aim is to find the essence and origin of the declarationsof NP-hardness of cutting and packing problems. It is found that the proofs are far from simply unique, andare of many types. We reviewed these proofs and tried to summarize them. The object is to make the view ofNP-complete/NP-hardness aspect of C&P problems clearer to related application researchers. The original proofswe reviewed are of following types: Type 1 is traced to Karp (1972), Garey & Johnson (1979) and Martello &Toth (1990), where mainly 1-dimensional C&P problems are proved to be NP-hard; Type 2 is traced to manyresearchers (for example, Holthaus (2003), Serairi & Haouari (2010), and Allen, Burke & Kendall (2011)) thatuse the result of Type 1 proof, and argue that higher-dimensional problems (or problems with other factors) arethe generalization of (or being more complex than) 1-dimensional problems, and thus to be NP-hard; Type 3 istraced to Fowler, Paterson & Tatimoto (1981), where packing 2-dimensional squares in orthogonal polygon onplane is proved to be NP-hard; Type 5 is traced to Li & Cheng (1989) and Leung & Tam et al (1990), wherepacking squares in a rectangle or in a square is proved to be NP-hard; Type 6 is traced to many researchers (forexample, Jarray, Costa and Picouleau (2008), Rinaldi and Franz (2007), and Avriel, Penn and Shpirer (2000)),



where special cases (or “C&P problem variants” according to the typology defined by Wäscher, Haußner, andSchumann (2007)) are proved to be NP-hard; Type 7 is traced to Nelißn (1993), where we see that the palletloading problem (PLP) is possibly not in NP. During the process, we find a wider group of C&P problems thatare possibly not in NP, and a simpler and understand-easier proof of the NP-completeness of the square packingproblem.Keywords: Cutting and Packing, NP-complete, NP-hard, Square Packing

2.1Models and Algorithms for Selecting the Best Shipper Sizes for Sending

Products to CustomersF. Parreño∗, M. T. Alonso∗, R. Alvarez-Valdes†, J.M. Tamarit†

∗ University of Castilla-La Mancha. Department of Mathematics, Albacete, Spain, † University of Valencia,Department of Statistics and Operations Research, Burjassot, Valencia, Spain

A distribution company in Spain has to send products, packed into shipper boxes, from the store to the retailshops. The problem is to decide the sizes of the shipper boxes to be kept at the store so as to minimize the costof packing all the demands along a given planning horizon. The number of different box sizes is fixed beforehand,looking for a balance between transportation costs and stock and procurement costs.In this work we describe two integer linear programming formulations for the problem. The first one uses asvariables the possible packing patterns and the other is based on the p-median problem, adapted to the charac-teristics of this case. As the number of possible boxes is very large and consequently the models may have a hugenumber of variables, we have designed several heuristics to reduce the set of possible boxes. We only use feasibleand efficient partitions. In order to avoid symmetrical solutions, due to the rotation of the products, we onlyconsider shipper boxes whose length is lower or equal to their width. If we have orders with the same quantitiesof products, the order is repeated and we can remove one of these orders of the formulation, doubling its cost inthe objective function.Even with these reductions, the integer problems are usually too large to be solved in reasonable times, but solvingthe first formulation over some subsets of packing patterns gives us good feasible solutions while some relaxationsof the second one can be used in order to obtain lower bounds.With the aim of improving the solutions obtained with the previous methods, we have also developed a meta-heuristic algorithm based on reducing and increasing the sizes of shipper boxes. A computational study conductedon real instances provided by the company is presented and discussed.Keywords: Assortment; Column generation; Integer formulations; Metaheuristics

2.2Mixed Integer Programs for Cutting and Scheduling Optimization

Fabrizio Marinelli∗, Claudio Arbib†∗ DII - Università Politecnica delle Marche, † Dipartimento di Informatica, Università degli Studi dell’Aquila

Classical stock cutting calls for fulfilling a given demand of part types minimizing trim loss. With the productionof each part type assimilated to a job due by a specific date, a problem arises of scheduling cutting operations.The problem of minimizing a combination of trim-loss and weighted lateness of jobs recently received a syntheticinteger linear programming formulation by Reinertsen and Vossen. As can be shown by examples, this formulationis not exact and thus provides in general approximate solutions.We here propose an exact pattern-based formulation of the problem that models the time horizon as a set ofconsecutive production periods and part types production as a network flow. We solve the linear relaxation bycolumn generation and discuss the special case of scheduling of single parts. A computational study completesthe talk.Keywords: Scheduling, Integer Linear Programming, Column Generation

2.3An Hierarchical Approach to Shelf Space AllocationTeresa Bianchi-Aguiar∗, Maria Antónia Carravilla∗, José F. Oliveira∗

∗ INESC-TEC, Faculdade de Engenharia, Universidade do Porto

Shelf Space Allocation (SSA) is the problem of distributing the limited and scarce shelf space of a store among agiven set of assorted products. In the Waescher et al improved typology of Cutting and Packing (C&P) problems,SSA falls into the category of Placement Problems. Shelves represent the multiple large objects, and each assorted



product represents a subset of similar small rectangle items, from which a certain quantity must be assigned to theshelves, limited by an upper and lower bound. More than just displaying the merchandise available in the store,the clever arrangement of the products in the shelves is seen as a tool to improve the store’s financial performance.Therefore, besides the C&P aspects of this problem, additional marketing variables are associated, in order tointegrate the impact of different allocation decisions in products demands. The objective of the problem is thento allocate the products items, which can be regarded either as a 2D or a 3D problem, maximizing the financialperformance of the available space.To solve the Shelf Space Allocation problem, we propose an hierarchical structure which decomposes the probleminto three main levels. (1) The first level returns which products to display in each shelf and their sequence. Ittakes into consideration the best locations to boost the sales and the financial performance of the products. (2)The second level allocates space to each one of the products, filling all the space available in the shelves. Thepast sales in the store determines the space requirements. (3) Finally, the third level places products items intothe allocated spaces. Note that product items can be placed in multiple orientations, with a preferential ordergiven beforehand.In practice, when considering companies with multiple stores, the effort for space management increases consid-erably. However, most of the times, little differences are observed in product assortment decisions and spaceavailable among the stores. In this talk, we will present the last two levels of the hierarchical structure, adaptedto replicate a shelf space allocation solution for a store throughout similar stores. The objective is to fit thesame products into different existing spaces, following the allocation process received as input. In this approach,a mixed integer goal programming model tackles the space assignment for each product while a lexicographicconstructive heuristic determines the best allocation for the items of each product. All the work developed wasmotivated by a case study of a supermarket chain with over 150 stores.Keywords: Shelf Space Allocation, Retail, Hierarchical DecompositionSupported by Fundação para a Ciência e Tecnologia (FCT) grant SFRH/BD/74387/2010

2.4The cutting stock/leftover problem

Marcos Arenales∗, Andrea Vianna†, Adriana Cherri†, Carla Lucke‡∗ ICMC/USP, Brazil, † UNESP/FC, Brazil, ‡ UNICAMP/IMECC, Brazi

Minimizing the waste is usually the main concern in cutting problems, and the waste is highly dependent on thequantity and diversity of the item sizes to be cut. Small and medium size industries face the problem of cutting,within a period, non-standard items with a low demand. If they could postpone the cutting phase, orders ofother items could arrive, increasing the quantities and diversities of the items which may lead to a decrease in thewaste. However, because of the due dates, items have to be cut on time. On the other hand, if the large objectsto be cut are also available in large quantities and diversity of sizes, the waste decreases as well. Simple strategiesare used in practice to artificially create diversities of large object sizes, for instance, partially cutting some largeobjects and leaving usable leftovers to be cut to meet future demands. In this paper, when cutting an object instock (bars, reels for one-dimensional problems, or rectangular plates for two-dimensional problems) we considera smaller large object, a usable leftover, obtained by cutting a large object. In order to retain leftovers in stockfor a short time, leftovers are weighted to give them priority in use. A cutting pattern based mathematical modelis proposed and applied to real world instances from a small metallic framework industry and a small furnitureindustry.Keywords: Leftovers, Minimizing waste, Cutting stock

3.1Automated Storage Design

François Clautiaux∗, Ines Bahri†, Luce Brotcorne‡, El-Ghazali Talbi∗∗ Université de Lille, † Institut Supérieur de Gestion de Tunis, ‡ INRIA Lille Nord Europe

The problem to solve is the following: given a large object, and a list of demands for items, cut the large objectinto compartments in such a way that the total profit of the items that can be packed in the compartments in agiven time interval is optimized.Therefore, there are two decision levels: 1) cutting the compartments, 2) assigning the items to the compartmentsat time unit. We suppose that the item demand is known in advance for each time unit.We show that the problem can be reformulated as a knapsack problem, where the items are the compartments,and the large object is the knapsack. The profit of selecting k compartment has to be computed a-priori for eachcompartment size w and each possible value of k. We study several possible scenarios (anytime delivery, fixed timedelivery and anytime delivery with storage costs). For each possible scenario, we show that the profit of creating



k compartments of size w can be computed in polynomial time. We compare our approach experimentally witha mathematical programming model.Keywords: Knapsack problem, Mathematical models, Automated storage devices

3.2One Dimensional Cutting Stock and Suborder Optimization

Mihael Cesar∗, Mirko Gradišar∗∗ Faculty of economics, University of Ljubljana

We focus on the one-dimensional cutting stock problem where orders are large and stock-to-order average sizeratio is low (bellow 5:1). Such orders occur in various industries. Our research is based on the example from theconstruction industry - a construction of high voltage transmission towers. Literature usually defines large ordersas those where exact methods fail to deliver solutions without any production delay / obstruction or create otherinsurmountable difficulties for the business process of the company. The exact solution can always be calculatedfor all orders that consist of approximately up to hundred items of ten different lengths. Heuristic methods aregenerally used for larger orders than described. Nevertheless, the main objective of any one-dimensional cuttingstock optimization is to process the whole order and to minimize the material waste (in the best case scenario“scrap”), i.e., the trim loss. If the trim loss is longer than some arbitrarily set length T, it is no longer calledthe trim loss but a leftover. Leftovers are returned back to stock and reused in subsequent orders. In the casewe present, there is a combination of several several technological and logistical reasons that hinder the filling ofincoming orders at once. Some of the most important are the following: it is not possible to transport all necessarystock objects to the cutting machine to process the entire order; handling and sorting uncut and cut material closeto the machine has its boundaries; items are not cut in the order that matches technical and logistical proceduresafter the cutting activity. Therefore, all incoming orders need to be broken down to suborders that can be filledat once. Suborder generation is not a simple division of the order but an additional optimization by itself soas to enable the lowest possible trim loss for the whole order. A heuristic method for such suborder generationand a subsequent minimization of trim loss is proposed. We include the necessary mathematical framework, apseudo code of the method, a detailed solution development of the case provided from practice (suborder gen-eration and subsequent minimization of trim loss for the whole order) and the results of ninety automaticallygenerated problems (with the one-dimensional cutting stock problem generator called CUTGEN). The method isalso implemented in the form of the G-CUT computer program. Final thoughts and future research possibilitieson cutting stock optimization are discussed.Keywords: One-dimensional cutting stock problem, Discrete optimization, Suborders

3.3Constraint Programming Approaches for Higher-Dimensional Orthogonal

PackingMarat Mesyagutov∗, Gleb Belov†, Guntram Scheithauer∗

∗ Technische Universität Dresden, Germany, † Universität Duisburg-Essen, Germany

We consider the feasibility problem OPP in higher-dimensional orthogonal packing: Given a set of d-dimensional(d = 2, 3) rectangular items, decide whether all of them can be orthogonally packed in the given rectangularcontainer without rotation. OPP-d is a decision or an auxiliary problem which is successfully used in order tosolve some interesting orthogonal problems, e.g., d-dimensional strip packing problem, SPP-d. In order to solveOPP-d we investigate some known constraint programming approaches for OPP-2, improve and adapt them forOPP-3.The well-known 1D bar linear programming relaxation of OPP-2 reduces the latter to a 1D cutting-stock problemwhere the packing of each stock bar represents a possible 1D stitch though an OPP-2 layout. The similar relaxationis applied for OPP-3. Moreover for the case d = 3 we apply so called 1D slice relaxation where we determineslices within which we apply 1D bar linear programming relaxation. Based on the relaxations we determinepruning rules which application reduces in most cases the number of branching decisions. The pruning rules areapplied within the constraint propagation procedure of the constraint programming and can be considered as anamplification of the propagation. Further amplification techniques are also under consideration.The numerical results on the instances from the literature as well as on the self-generated instances showthe efficiency of the proposed algorithms and of the combination of the constraint propagation and the linearprogramming-based pruning rules.Keywords: Constraint programming, Linear programming, Column generation



3.4Automatic Generation of Heuristics for the On-line 1D Bin Packing Problem

with Biased Random Key Genetic ProgrammingJosé Fernando Gonçalves∗

∗ LIAAD, Faculdade de Economia, Universidade do Porto,Rua Dr. Roberto Frias, 4200-464 Porto, Portugal

This paper addresses the on-line one-dimensional bin packing problem. In on-line bin packing the items arepresented to the packing algorithm sequentially, and each item must be packed in a bin before the next item isknown.The heuristics are generated automatically by a Genetic Programming algorithm which combines intelligentlybase components and uses a biased random key evolutionary process.The analysis of the heuristics generated is based on an average-case performance where the inputs are assumedto follow known distributions.The computational experiments demonstrate not only that the approach performs very well on several typesof input distributions but also that the evolved heuristics are highly competitive when compared with humandesigned heuristics.Keywords: On-line Bin Packing, One-dimensional, Genetic programming, Biased Random KeysSupported by Fundação para a Ciência e Tecnologia (FCT) project PTDC/EGE-GES/117692/2010.

4.1A Flexible Framework for Pallet and Container Loading

Joaquim Gromicho∗, Gerhard Post∗, and Jeroen van Wolffelaar∗∗ ORTEC

We describe a flexible algorithmic framework that enables the user to configure the packing algorithm from theoutside, based on an algorithm script. This script can dinstinguish different loading equipments (containers,pallets, boxes, . . .) and different package items (products from orders, packed boxes, loaded pallets, . . .). Basedon the situation and algorithm the user can add loading rules, constraints, and objectives he thinks are important.The script enables multi-phase planning, like products on pallets, and pallets in containers.This work is under construction at this moment; we hope to be able to give a demo to show the effect of differentscripts on the solution of, for example, the BR instances.Keywords: Container Loading, Pallet Loading

4.2Parallel Models for the Multi-Objective Container Loading Problem

Yanira González∗, Jesica de Armas∗, Gara Miranda∗, Coromoto León∗∗ Dpto. Estadística, I. O. Computación, Universidad de La Laguna, Avda. Astrofísico Fco. Sánchez s/n, 38271

La Laguna, Santa Cruz de Tenerife, Spain

The Container Loading Problem (CLP) belongs to an area of active research and has numerous applicationsin the real world, particularly in container transportation and distribution industries. In general, this probleminvolves the distribution of a set of rectangular pieces (boxes) into one large rectangular object (container) so asto maximize the total volume of packed boxes, i.e., trying to obtain a pattern of packaging that uses the containerspace as much as possible. The single-objective formulation of the problem - which seeks to optimize the totalvolume of the packed pieces into the container - has been widely studied in the related literature. However, arather common aspect in the scope of this problem is the weight limit of the containers, since they normally cannot exceed a certain weight for their transportation, and they should make the most without exceeding that limit.For this reason, we have focused on a multi-objective formulation of the problem which seeks to maximize thevolume at the same time that weight, without exceeding the container weight limit.In order to afford such a formulation of the problem, we have applied multi-objective optimization evolutionaryalgorithms (MOEAs) given their great effectiveness with other types of real-world multi-objective problems. Fur-thermore, with the aim of improving performance and delivering quality results in less time, we have made astudy of some parallelizations using island-based models, such as homogeneous and heterogeneous models. Theobtained results show the great effectiveness of MOEAs when applied to such kind of problems. In first place, wehave achieved to improve the only known results of a referenced work which deals with the same multi-objectiveproblem. Such related work is based on an approach which uses different weights for each objective, thus in-volving several executions to obtain the whole set of solutions. However, using strategies which can address themulti-objective problem as such, only one execution is required, and then the decision maker can select the mostappropriate solution, depending on the demands at the time. Secondly, when we have used the homogeneousmodels, results demonstrate the good behaviour of these parallel schemes, improving the quality of the sequential



ones. Nevertheless, when we apply the heterogeneous models - which involve the execution of different MOEAs- with migration of individuals among islands, the results can’t improve the sequential ones. Only when themigrations are not used, the heterogeneous models behave as expected. We have demonstrated that this happensbecause the different applied MOEAs are searching on different areas of the search space, so results are in com-pletely different areas of the solution space. Therefore, when an island migrates individuals to another island, itis introducing individuals from another area of the solution space and thus, difficulting the right evolution of thepopulation.Keywords: Container Loading Problem, Multi-Objective Optimization, Evolutionary Algorithms, Island-Based Models, Parallelization

4.3Parallel Hyperheuristics for a MultiObjectivised 2D Packing Problem

Carlos Segura∗, Eduardo Segredo∗, Coromoto Leoón∗∗ Dpto. Estadística, I. O. Computación, Universidad de La Laguna, Avda. Astrofísico Fco. Sánchez s/n, 38271

La Laguna, Santa Cruz de Tenerife, Spain

Bin Packing Problems are combinatorial np-hard problems in which objects with different shapes must be packedinto a finite number of bins. In this work, the 2D Bin Packing Problem (2dpp) proposed in the Genetic andEvolutionary Computation Conference 2008 has been considered. The best results for the 2dpp contest instancewere obtained with a mono-objective Memetic Algorithm (ma). However, subsequent studies concluded thatstagnation may occur with this approach for other instances.Multiobjectivisation refers to the reformulation of originally mono-objective problems as multi-objective ones. Itcan be performed by aggregating an alternative objective function. Multiobjectivisation has been successfullyapplied to the 2dpp. However, the approach has two main drawbacks: the proper artificial objective depends onthe instance to solve, and the time required to obtain high-quality solutions is very high. In order to avoid suchdrawbacks parallel hyperheuristics might be applied. The main aim of the research has been to solve the 2dppwith a model that avoids the drawbacks of the aforementioned strategies.A parallel approach (dyn) that merges the island-based model with the hyperheuristics principles has been ap-plied. The hyperheuristic allows changing in an automatic way the algorithms, parameters and artificial objectivefunctions that are used in the islands. The selected hyperheuristic has been the one named HH_Imp in theliterature. The low-level configurations have been constituted by combining two different mas with eight differentways of defining the artificial objective for the multiobjectivisation. Computational experiments with two differentinstances have demonstrated the validity of the approach.In a first experiment the sensitivity of the model regarding the adaptation level of the hyperheuristic has beenanalysed. It has shown that the best behaviour of the approach is obtained when a global adaptation level hasbeen used. In the analysis, the approach has been compared with a model that assigns the resources followinga uniform distribution (uni) instead of applying the hyperheuristic principles. The dyn model has converged tohigh-quality solutions about three times faster than the uni approach.A scalability analysis has also been performed. The parallel model has been executed with up to 128 slaveislands. The analysis of the run-length distributions has shown that a speedup factor approximately equal to 30has been achieved with 128 slave islands when the dyn model has been compared with the best low-level sequentialconfiguration. Since the selection of the proper alternative objective and ma is performed in an automatic way,and in a single run, the time saved is higher than the one that can be calculated with the run-length distribution.Therefore, the benefits of the proposal are clear.The main achievement of the research has been to design a model with the following benefits. First, the timerequired to obtain high-quality solutions has been drastically reduced. Moreover, the parallel model has madepossible to integrate the usage of several optimisation strategies with different artificial objectives in a single run.Therefore, the usage of the hyperheuristic has highly improved the usability of the model.Keywords: Parallel Hyperheuristics, Adaptation Level, Multiobjectivisation, 2D Packing ProblemThis work has been supported by the ec (feder) and the Spanish Ministry of Science and Innovation as part of the ’PlanNacional de i+d+i’, with contract number tin2011-25448. The work of Carlos Segura has been funded by grant fpu-ap2008-03213. The work of Eduardo Segredo has been funded by grant fpu-ap2009-0457 The work has also been fundedby the hpc-europa2 project (project number: 228398) with the support of the European Commission - Capacities Area -Research Infrastructures. This work made use of the facilities of hector, the uk’s national high-performance computingservice, which is provided by uoe hpcx Ltd at the University of Edinburgh, Cray Inc and nag Ltd, and funded by theOffice of Science and Technology through epsrc’s High End Computing Programme.



4.4Two Phase Loading in Presence of Multiple Constraints

M.T. Alonso∗, R. Álvarez-Valdés†, J.Gromicho‡, F.Parreño∗, G.Post‡, J.M. Tamarit†∗ University of Castilla-La Mancha. Department of Mathematics, Albacete, Spain, † University of Valencia,Department of Statistics and Operations Research, Burjassot, ‡ ORTEC Logistics, Gouda, The Netherlands

We study the inter-depot transportation problem where trucks will drive from a depot to another depot. Truckswill contain at least all orders for the next day. The main objective is to drive with as few trucks as possible.All orders are placed on pallets of dimensions 40 x 48 inches with a height of at most half the truck height.Depending on their weight, two pallets can be stacked of not. Moreover, the problem has a set of constraints:• Priority restriction: The orders of each day have to be sent. We must send first the orders of the day y,

then the day y+1 and so on. If the items of day y do not fill the truck completely, items of the day y+1can be used to fill up the truck.

• Stackability restriction: The height of pallets is usually half the height of the truck. Hence pallets can bestacked of at most 2 high. However not any 2 pallets can be stacked. That depends on certain propertiesof the pallets, like the weights.

• Pallets in the truck: The trucks have dimensions such that 2 pallets can be placed width-wise, and 14 or15 pallets length-wise, leading to a maximum of 56-60 pallets per truck.

• Weight restriction. There is a maximum weight the truck can transport.• Stability constraint: The load into the truck must be well spread to avoid movements of the pallets during

the journey.• Axle weight: The truck has two axles, and each axle can bear a maximum weight. For stability reasons,

the weights on the axles have to be balanced.• Centre of gravity: The centre of gravity should not deviate too much from its ideal position, i.e., the centre

of gravity should preferably be inside a centre box.For solving the problem we have to decide first how to build the pallets, their height and number of layers,and then how to place the pallets into the trucks, according to the constraints. Both decisions are related, andtherefore we propose a constructive algorithm in which the two decisions are connected when the pallet is builtand placed into the truck.Our proposal is to solve the problem using a metaheuristic based on a GRASP algorithm combining two phases,one for finding a solution and the other for improving the solution while satisfying the constraints.Keywords: Container loading, Stackability, Axel weight, Priority restriction

5.1A New No-Fit Polygon Generator for Rectilinear PolygonsMarisa Oliveira∗, Pedro Rocha†, Eduarda Pinto Ferreira∗, A. Miguel Gomes†

∗ Instituto Superior de Engenharia do Porto, Instituto Politécnico do Porto, † INESC-TEC, Faculdade deEngenharia, Universidade do Porto

Cutting and Packing (C&P) problems are combinatorial optimization problems with a strong geometric compo-nent, that have been extensively studied mainly due to their numerous real world applications. The combinatorialcomponent arises in the selection of items, where to place the items, the selection of the large object, while thegeometric component arises when positioning the items without overlap and inside the large object.In real world applications pieces may have a large variety of sizes and shapes. Here we are interested in 2D C&Pproblems where the items are rectilinear polygons (i.e., polygons with 90o or 270o internal angles). This typeof items may occur in facility layout problems and in the placement of modules on Very Large Scale Integration(VLSI) circuits. For example, in VLSI designing, rectilinear polygons are introduced to improve both chip areausage and modules inter-connectivity.One common issue to all C&P problems is how to ensure feasible layouts. While it is relatively easy to ensure it forrectangular shapes, things become more complicated for irregular shapes (where irregular means both non-regularand non-convex). One option widely used to ensure the non-overlapping constraints between irregular shapes isthe No-Fit Polygon (NFP), which gives the set of admissible relative placements between two irregular shapes.Existing approaches to compute the NFP are very time consuming and not efficient for rectilinear polygons.We introduce a new procedure to derive NFP between pairs of rectilinear polygons. This procedure is fast andvery easy to use when pieces are rectilinear polygons with only convex concavities. The main idea behind it is tomake successive cuts in the enclosing rectangle that contains the two polygons. Additionally, this new procedureis more robust than existing ones since it is more imune to rounding errors. We will present computational teststo demonstrate the validity of the proposed procedure.Keywords: Nesting, No-fit polygon, Rectilinear polygonsSupported by Fundação para a Ciência e Tecnologia (FCT) project PTDC/EME-GIN/105163/2008.



5.2Using Circle Covering to Tackle Nesting Representations Limitations

Pedro Rocha∗, Rui Rodrigues∗, A. Miguel Gomes∗∗ INESC-TEC, Faculdade de Engenharia, Universidade do Porto

The development of effective and efficient Nesting algorithms have a strong dependence on the geometrical rep-resentation methods available for irregular outlines. Current geometrical representations are based on discretizedmatrixes, rectangular bounding boxes, polygons and mathematical expressions of basic objects. State of the artalgorithms use polygonal representations and rely on no-fit-polygons (NFP) to ensure feasible layouts. NFP com-putation is usually done in a pre-processing phase, for a set of discrete orientations, since it is very computationallyexpensive to do at runtime.Efficient overlapping detection algorithms based on these geometrical representations suffer from limitations onthe placement orientations of the pieces, since the representations are strongly tied to specific orientations, andchanging them is not trivial. A major challenge is to find geometrical representations that are independent ofpiece orientations. These representations will allow collision detection and overlapping detection algorithms totake full advantage of free rotations of the pieces, leading to the development of a new generation of Nestingalgorithms.An option to tackle this challenge is to represent geometrical shapes through a set of circles, since they are orien-tation independent and have simple and fast overlap detection operations. This idea leads to Circle Covering(CC)representations, where an irregular outline is approximated by a set of circles. This coverage is done dealing withtwo opposite objectives: minimizing the number of circles and approximating the outline with the least amountof error. Finding the optimum CC is still an open problem. The circles may have different sizes, with overlapamong them while nearly fully covering the geometrical shape.The approach that we propose is based on computing the Medial Axis (MA) for each irregular outline. TheMA is a topological skeleton that consists of the set of all points having more than one closest point on theobjects boundary. The greatest advantage of the MA is that it allows the placement of the biggest circles thatfit inside the irregular outline, by placing their center over the MA. The MA may be composed by straight linesand parabolic arcs. The circles are placed with their centers on the MA, while adjusting their position to controlthe approximation error, with the maximum radius possible without exiting the outline of the piece. The mainadvantage of the CC representation is the low overhead in overlapping detection computations. This allows forreal time adjustments by Nesting algorithms in the piece orientation and discarding the restriction of definingand imposing a finite set of discrete orientations.Keywords: Nesting, Circle-covering, Irregular shapes, Medial axisSupported by Fundação para a Ciência e Tecnologia (FCT) project PTDC/EME-GIN/105163/2008.

5.3Packing of Arbitrary Shaped Objects into a Convex n-Polygonal Container

of Minimal AreaT. Romanova∗, Yu.Stoyan∗, A. Pankratov∗

∗ Department of Mathematical Modeling, Institute for Mechanical Engineering Problems of the NationalAcademy of Sciences of Ukraine, Kharkov, Ukraine

We consider a finite set of arbitrary shaped objects bounded by line segments and circular arcs and a convexn-polygonal container. Each object may be free translated and rotated. In addition allowable distances betweenobjects may be given. The container is defined as a convex hull of its vertices with variable coordinates. Theproblem in question lies in arranging the given set of objects within the container in order to minimize areaof the container taking into account distance constraints. In particular, we consider a subsequent problem forconstructing a convex hull of a set of non-overlapping objects. In this case we make rigid demands of theobject shapes (polygons and concave arc objects). In order to model non-overlapping, containment and distanceconstraints we employ the main tool of our studies ? phi-function technique. We also provide a mathematicalmodel of the problem as a nonlinear constraint optimisation problem. Based on features of phi-functions weconstruct a solution tree. Each terminal node of the solution tree corresponds to a system of inequalities involvinginfinity-differentiable functions. Here we discuss our solution strategy. The algorithm involves the algorithm ofconstructing starting points and methods of local and global optimisation. We offer some of our computationalresults.Keywords: Irregular shapes; Arcs, Phi-functions, Containment, Rotations, Distances



5.4Matheuristics for the Two-Dimensional Nesting Problem

A. Martinez∗, R. Alvarez-Valdes∗, M.A. Carravilla†, A.M. Gomes†, J.F. Oliveira†, J.M. Tamarit∗∗ University of Valencia, Department of Statistics and Operations Research, Burjassot, Valencia, Spain,

† INESC-TEC, Faculty of Engineering, University of Porto, Porto, Portugal

The two dimensional nesting problem involves the placement of a given set of polygons, not necessarily convex,into a container so that polygons do not overlap and do not protrude from the container. In our problem, thecontainer is a strip of fixed width and the objective is minimizing the required length.In this work we develop a matheuristic algorithm based on a new mixed integer formulation. This model isa variant from the Fischetti and Luzzi formulation in which we define the slices in a horizontal way and thecontainment constraints have been lifted.In practice solving to optimality this model is not possible for instances with many pieces, because the numberof binary variables needed to describe the model is too large.We propose a constructive algorithm in order to obtain different initial solutions. The idea is to solve to optimalitythe model with fewer pieces and then, iteratively, insert the remaining pieces by solving similar models. We havealso developed a local search procedure in which we make different decisions among the subset of binary variablesof the model that should be fixed. We propose three different ways based on the reinsertion of one piece, theexchange of two pieces, and the compaction of a solution.The efficiency of the proposed procedures is assessed by solving a set of instances from literature and comparingthe results with the most recent and successful heuristic approaches.Keywords: Nesting problems; Matheuristics

6.1Comparing Encoding Schemes for Multi-objective Two-dimensional

Guillotine Cutting ProblemsJesica de Armas∗, Gara Mirandas∗, Coromoto Leóns∗

∗ University of La Laguna. Dpto. Estadística, I.O. y Computación. Av. Astrofísico Fco. Sánchez, s/n, 38271La Laguna, Spain

Most research on Two-Dimensional Strip Packing Problems (2DSPP) and Two-Dimensional Cutting Stock Prob-lems (2DCSP) are focused on single-objective formulations of the problems. However, in this work we deal withmore general and practical variants of the problems, which not only seeks to optimize the usage of the raw ma-terial, but also the overall production process. The problems target the cutting of a large rectangle in a set ofsmaller rectangles using orthogonal guillotine cuts. Common approaches are based in the minimization of thestrip length required to cut the whole set of demanded pieces (for 2DSPP) and in the maximization of the totalprofit obtained from the available surface (for 2DCSP). However, in this work we also deal with an extra objectivewhich seeks to minimize the number of cuts involved in the cutting process, thus maximizing the efficiency of theglobal production process.In order to obtain solutions to these problems, we have applied some of the most-known multi-objective evolution-ary algorithms, since they have shown a promising behaviour when tackling multi-objective real-world problems.We have designed and implemented two different encoding schemes: a direct encoding scheme based on postfixnotation, and a hyperheuristic encoding scheme as an alternative to combine heuristics in such a way that aheuristic’s strengths make up for the drawbacks of another. The goal of this work is to find out if differentencoding schemes can be successfully extended to different cutting and packing problems. So, firstly, we haveproved that the direct approach is able to improve the results of the single-objective approaches when dealingwith small and medium size instances for the 2DSPP. When using the hyperheuristic-based encoding scheme, wehave been able to reduce the search space initially handled by the direct encoding. This way, we have obtainedresults which improve the profit values given by the single-objective approaches and what is more important: wehave designed a quite general solution for the encoding of guillotine cutting problems. In order to better verifyour conclusions, we have also checked both multi-objective approaches with the 2DCSP. Results demonstratethat the 2DCSP hyperheuristic encoding scheme is able to obtain competitive solutions when compared to singleheuristics. But, even for large instances, the direct encoding scheme provides better results. This may be becausethe applied hyperheuristic is based on single-objective heuristics which don’t provide high quality results for thisproblem because originally, they have been created to deal with level-oriented cutting problems, which better fitsto strip packing problems. Moreover, it should be taken into account the complexity of the problems. Althoughthe instances used for both problems have approximately the same number of pieces, the search space for the2DCSP is smaller, since for this problem the rotation of the pieces is not allowed.Keywords: Strip Packing and Cutting Stock Problems, Guillotine Cutting Problem, Multi-ObjectiveOptimisation, Encoding Schemes, Evolutionary Algorithms



6.2A Dynamic Ordering Rule Based on Size Matches for Solving Packing

ProblemsJinmin Wang∗, Qi Yang∗, Yiping Lu†

∗ Tianjin Key Laboratory of High Speed Cutting & Precision Machining. TUTE, Tianjin, 300222, China,† School of Mechanical Engineering, Beijing Jiaotong University, Beijing, 100044, China

The packing problem is derived from practices that occur in industrial production and in real life, and the solutioncould rely on experience from these environments. In real life, people often place a pair of items or a set of pieceswith the same size together. This action makes the remaining packing region simple and regular and benefitslater layouts. Based on experience, we have designed a dynamic ordering rule that is based on size matching,this algorithm is presented in this paper. The ordering rule selects a rectangular piece by the match level of therectangular piece and the sizes of the remaining packing region. Throughout a series of experiments, we provethat the ordering rule can obtain better packing solutions compared to previous approaches.Keywords: Packing problem, Attractive factor approach, Size matching, Dynamic ordering rule

6.3A 5/3-Approximation for Strip Packing

Lars Prädel∗, Rolf Harren†, Klaus Jansen∗, Rob van Stee†∗ University Kiel, Germany, † Max-Planck-Institut für Informatik, Germany

We study the strip packing problem which is one of the most classical two-dimensional packing problems: given aset of rectangles I = {r1, . . . , rn} of specified widths wi and heights hi, the objective is to find a feasible packingfor I (i.e. an orthogonal arrangement where rectangles do not overlap and are not rotated) into a strip of width1 and minimum height. This scenario has many practical applications in manufacturing, logistics, and computerscience. Since strip packing includes bin packing as a special case (when all heights are equal), the problem isstrongly NP-hard and there is no algorithm with approximation ratio better than 3/2, unless P = NP.As the problem is studied for decades, there are several known results. Among others there are the Next-Fit-Decreasing-Height and First-Fit-Decreasing Height algorithms by Coffman et al. with proven approximation ratiosof 3 and 2.7, respectively. Schiermeyer and Steinberg independently presented algorithms with approximationratio 2. Recently, Harren & van Stee presented an algorithm with a ratio of 1.9396. In the asymptotic settingthere is an AFPTAS by Kenyon & Rémila with additive constant O(hmax/ε

2) and an APTAS with an additiveconstant of hmax by Jansen & Solis-Oba (hmax is the maximal height of the rectangles in an instance).Our main result is the following significant improvement between the lower bound of 3/2 and the previous bestapproximation ratio of 1.9396.

Theorem 1 For any ε > 0, there is an approximation algorithm A which produces a packing of a list I of nrectangles in a strip of width 1 and height A(I) such that

A(I) ≤(53+ ε

)OPT (I).

Here OPT (I) denotes the minimum height of a packing of I.

Techniques. In the main phase of the algorithm we use a result by Bansal et al., a PTAS for the so-calledrectangle packing problem with area maximization as a subroutine. It computes a packing of a subset of therectangles with total area at least (1 − δ) times the total area of all rectangles in I. Hence, after this step a setof unpacked rectangles with small total area remains.The main idea of our algorithm is to create a hole in the created packing and use it to pack the remaining rectangles.Finding a suitable location for such a hole and repacking the rectangles which we have to move out account forthe largest technical challenges of this paper. To achieve a packing of the whole input we carefully analyse thestructure of the generated packing by the subroutine and use interesting and often intricate rearrangements ofparts of the packing.Keywords: Strip packing, Rectangle packing, Approximation algorithm, Absolute worst-case ratio



6.4A Decomposition Algorithm for the Exact Solution of the Strip Packing

ProblemJean-François Coté∗, Mauro Dell?Amico†, Manuel Iori†

∗ CIRRELT, University of Montreal, Canada, † DISMI, University of Modena and Reggio Emilia, Italy

Given a set of rectangular items, each with a given width and height, and a strip of given width and a very largeheight, the Strip Packing Problem (SPP) calls for the packing of the items into the strip by using the minimumheight possible. The items should not overlap and be packed with their sides parallel to those of the strips, norotation is allowed.The SPP is one of the most important problems in the literature on cutting and packing. It models the problemof cutting materials (e.g., fabric, steel, glass and wood) by minimizing waste, and packing items into an area bymaximizing the usage of such area (e.g., cells in warehouses, advertisements in journals). The SPP is also verydifficult in practice.Relevant exact algorithms for the SPP usually rely on a relaxation obtained by cutting each item into vertical“slices”, having unit width and height equal to the height of the item, and packing these slices by ensuring thatthose originating from the same item are contiguous each other. Such relaxation is known as the One-dimensionalContiguous Bin Packing Problem (1CBPP) and gives good lower bound values.In our work we exploit for the first time the full potentiality of the 1CBPP relaxation and embed it into adecomposition algorithm to solve exactly the SPP. The decomposition algorithm is composed by a master anda slave problem. In the master problem we solve the 1CBPP. Whenever an integer solution is found we invokethe slave problem, in which we try to reconstruct a feasible SPP solution starting from the 1CBPP one. If wefind a feasible solution then we decrease the SPP upper bound. If not, then we add a cut to the master problem,possibly increasing the SPP lower bound.Both slave and master problems are NP-complete, and may be solved more than once to reach the optimal SPPsolution. However they are easier to be solved in practice than the SPP and this fact makes our algorithm well-performing. We present several ad-hoc algorithms for the master and slave problems, taking advantage of theparticular combinatorial structures of both problems. In particular we propose branch-and-cut, branch-and-boundand Benders? decomposition algorithms, and make them more efficient in practice by the use of preprocessing,heuristics and cut-lifting techniques.We have preliminary yet very promising computational results. Our algorithm solves for the first time to optimal-ity the benchmark instances cgcut02, cgcut03, gcut04 and gcut11. It also produces several new proven optimalsolutions for the classes of SPP instances by Berkey and Wang and Martello and Vigo.Keywords: Strip Packing Problem, One-dimensional Contiguous Bin Packing Problem, DecompositionAlgorithms, Exact Algorithms



List of ParticipantsAlonso, Maria TeresaUniversidad de Castilla La [email protected]

Alvarez-Valdes, RamonUniversity of [email protected]

Arenales, MarcosUniversity of Sao [email protected]

Belov, GlebUniversität Duisburg-Essen, GermanyGermany–-

Bianchi-Aguiar, TeresaINESC-TEC, Faculdade de Engenharia,Universidade do [email protected]

Bortfeldt, AndreasUniversity in [email protected]

Cesar, MihaelFaculty of Economics, University of [email protected]

Clautiaux, FrançoisUniversité de Lille 1, INRIA Lille Nord [email protected]

de Armas, JesicaUniversidad de La [email protected]

Gomes, A. MiguelINESC-TEC, Faculdade de Engenharia,Universidade do [email protected]

Gonçalves, José FernandoLIAAD, Faculdade de Economia, Universidade [email protected]

Gonzalez, YaniraUniversidad de La [email protected]

Iori, ManuelUniversity of Modena and Reggio [email protected]

Kulak, OzgurNam?k Kemal UniversityÇorlu Engineering [email protected]

Leon, CoromotoUniversidad de La [email protected]

Lu, YipingSchool of Mechanical Engineering, Beijing [email protected]

Marinelli, FabrizioUniversità Politecnica delle [email protected]

Mesyagutov, MaratDresden University of [email protected]

Miranda, GaraUniversidad de La [email protected]

Oliveira, José F.INESC-TEC, Faculdade de Engenharia,Universidade do [email protected]

Oliveira, MarisaISEP ? School of Engineering/Polytechnic of [email protected]

Parreño, FranciscoUniversidad de Castilla La [email protected]

Post, GerhardORTEC bv and University of TwenteThe [email protected]

Prädel, LarsUniversity of Kiel, [email protected]



Romanova, TatianaDepartment of Mathematical Modeling, Institute forMechanical Engineering Problems of the NationalAcademy of Sciences of [email protected]

Segredo, EduardoUniversidad de La [email protected]

Segura, CarlosUniversidad de La [email protected]

Silva, ElsaINESC-TEC, Faculdade de Engenharia,Universidade do [email protected]

Wang, JinminTianjin Key Laboratory of High Speed Cutting &Precision Machining. [email protected]

Wäscher, GerhardOtto-von-Guericke-Universität [email protected]



Notes


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