repositorium.sdum.uminho.pt · summary \if you can’t explain it simply you don’t understand it...

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Models and Algorithms for Hard Optimization Problems

Universidade do Minho

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Programa Doutoral em Engenharia Industrial e de Sistemas

Trabalho efectuado sob a orientação doProfessor Doutor Cláudio Manuel Martins Alvese doProfessor Doutor José Manuel Vasconcelos Valério de Carvalho

Rita Alexandra Santos Gonçalves de Macedo

Janeiro de 2011

Models and Algorithms for Hard Optimization Problems

Universidade do Minho

Escola de Engenharia

É AUTORIZADA A REPRODUÇÃO PARCIAL DESTA TESE APENAS PARA EFEITOSDE INVESTIGAÇÃO, MEDIANTE DECLARAÇÃO ESCRITA DO INTERESSADO, QUE A TAL SECOMPROMETE;

Universidade do Minho, ___/___/______

Assinatura: ________________________________________________

Summary

“If you can’t explain it simply

you don’t understand it well enough.”

Albert Einstein

This thesis is devoted to exact solution methods for NP -hard integer program-

ming models. We consider two of these problems, the cutting stock problem and

the vehicle routing problem. Both problems have been studied for several decades

by researchers and practitioners of the Operations Research field. Their interest

and contribution to real-world applications in business, industry and several kinds

of organizations are irrefutable.

Our solution approaches are always exact. We contribute with new lower

bounds, families of valid inequalities, integer programming models and exact al-

gorithms for the problems we explore. More precisely, we address two variants of

each of the referred problems.

In what concerns cutting stock problems, we analyze the one-dimensional pat-

tern minimization problem and the two-dimensional cutting stock problem with

the guillotine constraint. The one-dimensional pattern minimization problem is a

cutting and packing problem that becomes relevant in situations where changing

from one pattern to another involves, for example, a cost for setting up the cut-

ting machine. It is the problem of minimizing the number of different patterns of

a given cutting stock solution. For this problem, we contribute with new lower

bounds. The two-dimensional cutting stock problem with the guillotine constraint

and two stages is also addressed. We propose a pseudo-polynomial network flow

model, along with some reduction criteria to reduce its symmetry. We strengthen

the model with a new family of cutting planes and propose a new lower bound.

For this variant, we also consider some variations of the problem.

i

ii

Regarding vehicle routing problems, we address the vehicle routing problem

with time windows and multiple use of vehicles and the location routing problem,

with capacitated vehicles and depots and multiple use of vehicles. The first of

these problems considers the well know case of vehicle routing with time windows

with the additional consideration that vehicles can be assigned to several routes

within the same planning period. The second variant considers the combination

of the first problem, without time windows, with a location problem. This means

that the depots to be used must be selected from a set of available ones. For both

of these variants, we propose a network flow model whose nodes of the underlying

graph correspond to time instants of the planning period and whose arcs correspond

to vehicle routes. We reduce their symmetry by deriving several reduction criteria.

For the vehicle routing problem with time windows and multiple use of vehicles,

we propose an iterative algorithm to solve the problem exactly.

Our proposed procedures are tested and compared with other methods from

the literature. All the computational results produced by the series of experiments

are presented and discussed.

Resumo

“Se nao es capaz de o explicar de forma simples

e porque nao o compreendes suficientemente bem.”

Albert Einstein

Esta tese e dedicada a metodos de resolucao exacta para problemas de pro-

gramacao inteira NP -difıceis. Sao considerados dois desses problemas, nomeada-

mente o problema de corte e empacotamento e o problema de encaminhamento

de veıculos. Ambos os problemas tem vindo a ser abordados por investigadores

e profissionais da area da Investigacao Operacional ha ja varias decadas. O seu

interesse e contribuicao para aplicacoes reais do mundo dos negocios e industria,

assim como para inumeros outros tipos de organizacoes sao, hoje em dia, inegaveis.

A nossa abordagem para a resolucao dos problemas descritos e exacta. Con-

tribuimos com novos limites inferiores, novas famılias de desigualdades validas,

novos modelos de programacao inteira e algoritmos de resolucao exacta para os

problemas que nos propomos explorar. Em particular, abordamos duas variantes

de cada um dos referidos problemas.

Em relacao ao problema de corte e empacotamento, analisamos o problema de

minimizacao de padroes a uma dimensao e o problema de corte e empacotamento

a duas dimensoes, com restricao de guilhotina. O problema de minimizacao de

padroes a uma dimensao e pertinente em situacoes em que a mudanca de padrao

envolve, por exemplo, custos de reconfiguracao nas maquinas de corte. E o pro-

blema de minimizacao do numero de padroes diferentes de uma dada solucao de

um problema de corte. Para este problema contribuimos com novos limites in-

feriores. O problema de corte e empacotamento a duas dimensoes com restricao

de guilhotina e dois estagios e tambem abordado. Propomos um modelo pseudo-

polinomial de rede de fluxos, assim como criterios de reducao que eliminam parte

iii

iv

da sua simetria. Reforcamos o modelo com uma nova famılia de planos de corte e

propomos novos limites inferiores. Para esta variante, consideramos tambem outras

variacoes do problema original.

No que se refere ao problema de encaminhamento de veıculos, abordamos um

problema de encaminhamento de veıculos com janelas temporais e multiplas via-

gens, e tambem um problema de localizacao e encaminhamento de veıculos com

capacidades nos veıculos e depositos e multiplo uso dos veıculos. O primeiro destes

problemas considera o conhecido caso de encaminhamento de veıculos com janelas

temporais, com a consideracao adicional de que os veıculos podem ser alocados a

varias rotas no decurso do mesmo perıodo de planeamento. A segunda variante

considera a combinacao do primeiro problema, embora sem janelas temporais, com

um problema de localizacao. Isto significa que os depositos a usar sao selecciona-

dos de um conjunto de localizacoes disponıveis. Para ambas as variantes, propomos

um modelo pseudo-polinomial de rede de fluxos cujos nodos do grafo correspon-

dente representam instantes de tempo do perıodo de planeamento, e cujos arcos

representam rotas. Derivamos criterios de reducao com o intuito de reduzir a sime-

tria. Para o problema com janelas temporais e multiplas viagens, propomos um

algoritmo iterativo que o resolve de forma exacta.

Os procedimentos propostos sao testados e comparados com outros metodos

da literatura. Todos os resultados obtidos pelas experiencias computacionais sao

apresentados e discutidos.

Acknowledgments

“If I have seen further than others,

it is by standing upon the shoulders of giants.”

Isaac Newton

This thesis is the culmination of a three years journey of hard work but also

of great professional and personal growth. It represents a milestone in my life and

there are many people to whom I need to thank and acknowledge for their positive

influence and for contributing in many ways to the accomplishment of this project.

First of all, I feel truly fortunate for the opportunity I had to work with my

supervisor, Professor Claudio Alves, and my co-supervisor, Professor Valerio de

Carvalho. What I have learned from them, first as my teachers and then as my

supervisors, provided me strong foundations for my work. I thank them for all the

valuable opportunities they offered me, for all the people they introduced me and

for all the important experiences they provided me. This thesis would not have

been possible without their guidance and support, and for that I owe them my

deepest gratitude.

Professor Claudio has been for me an example of competence, dynamism and

hard, serious work. He is an outstanding professional, whose profound knowledge,

experience and know-how guided me through my research, enhancing my work sig-

nificantly. He was demanding but very supporting, always gave me the confidence

to pursuit any project, and I just could not have asked for a better supervisor. His

commitment to this project went far beyond his obligation, and his kindness and

friendship provided me a fruitful and enjoyable work environment. I just hope I

can reward him for his generosity by making him proud of my work in the future.

Professor Valerio introduced me to the field of combinatorial and integer pro-

gramming. It was his exceptional expertise in this subject that inspired me to

v

vi

love and appreciate it. He has been a great role model and a source of inspiration

to my work. I am profoundly grateful for his precious help, advice and constant

availability.

The friends I made in the lab also played a major role in the pursuit of this

project. Each one of them, in their peculiar and special way, enriched my work

days and my whole experience as a PhD student. One of them deserves special at-

tention: Elsa enrolled in this adventure with me from the beginning, and together

we faced all kinds of new challenges. It was side by side that we found our way in

this journey, and the precious friendship and companionship we built will certainly

remain forever. Without forgetting any of the others, I must also mention Pedro,

Telmo, Raid, Tiago, To and Carina. They were there for the good and bad days,

always willing to cheer me up in my whining moments and celebrate my accom-

plishments. We share countless memorable stories and these years would not have

been the same without them. Thank you all guys!

I cannot forget to sincerely thank all the precious technical support, but also

the good mood and kind presence of Eng. Acacio, and the administrative work

and patience of Conceicao. They were always there and helped me in numerous

ways through all these years.

I am greatly indebted to my coauthors, Francois Clautiaux and Saıd Hanafi, for

our joint work in this thesis. Working with them was a precious experience that

I truly cherish. I must also thank them dearly for their hospitality and friendship

during my two months stay in France. I do not forget the friends I made in the lab

during that time, who welcomed me with so much warmth and kindness. I would

like to thank especially to Mathieu, for his generous assistance running instances

for me, and to Jean Claude, for all his priceless help, encouragement and wonderful

friendship.

In addition, I would like to acknowledgement FCT (Fundacao para a Ciencia e

Tecnologia) for financially supporting my research with a grant (SFRH/BD/39607/

vii

2007), and to University of Minho, in particular to the Production and Systems

Department of the Engineering School, which provided me all the technical support,

equipment and work conditions for the development of my research.

A very special thanks goes to my close friends, for their caring and affection,

and to Gabriel, for his patience and understanding.

Finally, and perhaps most important, this thesis would not have been possible

without the unquestioning encouragement and love of my family, especially of my

parents and brothers.

I dedicate this thesis to my parents, for all the opportunities they have provided

me and for their constant support in everything I choose to do.

Braga, January 2011

Rita Macedo

Para ser grande, se inteiro: nada

Teu exagera ou exclui.

Se todo em cada coisa. Poe quanto es

No mınimo que fazes.

Assim em cada lago a lua toda

Brilha, porque alta vive.

Fernando Pessoa

(Odes de Ricardo Reis)

ix

Contents

1 Introduction 1

1.1 Motivation and Contributions . . . . . . . . . . . . . . . . . . . . . 2

1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Combinatorial and Integer Programming 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Algorithms and Computational Complexity . . . . . . . . . . . . . . 11

2.3.1 Complexity Classes . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Big-O Notation . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Methods for Linear Programming . . . . . . . . . . . . . . . . . . . 15

2.4.1 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Interior Point Methods . . . . . . . . . . . . . . . . . . . . . 15

2.5 Branch-and-Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Network Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Dantzig-Wolfe Decomposition . . . . . . . . . . . . . . . . . . . . . 19

2.8 Constraint Programming . . . . . . . . . . . . . . . . . . . . . . . . 24

2.9 Heuristic Solution Methods . . . . . . . . . . . . . . . . . . . . . . 26

3 Cutting & Packing Problems 29

3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 One-dimensional Cutting Stock Problems . . . . . . . . . . . . . . . 34

3.3 Two-dimensional Cutting Stock Problems . . . . . . . . . . . . . . 39

4 New Lower Bounds based on CG and CP for the PMP 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Mathematical Programming Models: State-of-the-art . . . . . . . . 52

xi

xii CONTENTS

4.2.1 A Nonlinear Compact Formulation . . . . . . . . . . . . . . 52

4.2.2 The Gilmore and Gomory Model with Setup Variables . . . 53

4.2.3 The IP Model of Vanderbeck [193] . . . . . . . . . . . . . . 54

4.2.4 An Arc-flow Formulation . . . . . . . . . . . . . . . . . . . . 55

4.3 Exploring a New IP Formulation . . . . . . . . . . . . . . . . . . . 56

4.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2 Strengthening the Model . . . . . . . . . . . . . . . . . . . . 61

4.4 A CP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.2 Deriving Lower Bounds from the CP Model . . . . . . . . . 69

4.4.3 Using the CP to Derive Valid Inequalities for SP1 . . . . . . 69

4.5 Comparative Computational Results . . . . . . . . . . . . . . . . . 71

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Models and Bounds for the 2D-CSP and its Variants 79

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Network Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Arc Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 A New Family of Cutting Planes . . . . . . . . . . . . . . . . . . . 89

5.6 A New Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7 Variants of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7.1 The Non-Oriented Case . . . . . . . . . . . . . . . . . . . . 90

5.7.2 Orientation of the First Stage’s Cuts . . . . . . . . . . . . . 91

5.8 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 The Vehicle Routing Problem 115

6.1 Problem Definition and Some of its Variants . . . . . . . . . . . . . 116

CONTENTS xiii

6.1.1 Depot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.2 Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1.3 Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1.4 Planning Period . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.1.5 Type of Input Data . . . . . . . . . . . . . . . . . . . . . . . 120

6.1.6 Objective Function . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Mathematical Formulations . . . . . . . . . . . . . . . . . . . . . . 121

6.2.1 Two-index Network Flow Model . . . . . . . . . . . . . . . . 121

6.2.2 Multicommodity Flow Model . . . . . . . . . . . . . . . . . 122

6.2.3 Set Partitioning Model . . . . . . . . . . . . . . . . . . . . . 123

6.3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3.1 Exact Approaches . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3.2 Heuristic and Meta-Heuristic Approaches . . . . . . . . . . . 125

7 An Exact Algorithm for the MVRPTW 127

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.2 Integer Programming Models: State-of-the-art . . . . . . . . . . . . 129

7.2.1 Problem Definition and Notation . . . . . . . . . . . . . . . 129

7.2.2 A Compact Formulation . . . . . . . . . . . . . . . . . . . . 130

7.2.3 A Column Generation Model . . . . . . . . . . . . . . . . . 132

7.3 A New Pseudo-Polynomial Network Flow Model . . . . . . . . . . . 133

7.3.1 Vehicle Routes . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.3.3 Arc Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.4 Iteratively Refining the Discretization . . . . . . . . . . . . . . . . . 141

7.4.1 Initial Rounding Strategy . . . . . . . . . . . . . . . . . . . 142

7.4.2 Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . . 144


7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xiv CONTENTS

8 A Network Flow Model for the MLRP 155

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.3 Mathematical Programming Models . . . . . . . . . . . . . . . . . . 161

8.3.1 Three-index Commodity Flow Model . . . . . . . . . . . . . 161

8.3.2 Set-partitioning Model . . . . . . . . . . . . . . . . . . . . . 162

8.4 Network Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.4.1 Arc Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 165


8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

9 Conclusions 173

9.1 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . 174

9.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Bibliography 177

List of Figures

3.1 Cutting and packing typology [200] . . . . . . . . . . . . . . . . . . 32

3.2 1D-CSP representation . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 1D-CSP representation . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Solution representation (example 3.1) . . . . . . . . . . . . . . . . . 43

4.1 One-dimensional pattern minimization problem . . . . . . . . . . . 47

5.1 Non-exact two-stage guillotine cutting patterns . . . . . . . . . . . 85

5.2 Graph G0 of the network flow model (example 5.1). . . . . . . . . . 88

5.3 Graph G2 of the network flow model (example 5.1). . . . . . . . . . 88

5.4 Rotating items in a wood stock sheet . . . . . . . . . . . . . . . . . 91

7.1 First and last beginning and ending instants of route r . . . . . . . 134

7.2 Example of one path on the network flow model . . . . . . . . . . . 135

7.3 Complete network flow graph (example 7.1) . . . . . . . . . . . . . 138

7.4 Optimal solution (example 7.1) . . . . . . . . . . . . . . . . . . . . 139

7.5 Arc reduction (example 7.2) . . . . . . . . . . . . . . . . . . . . . . 141

7.6 Node disaggregation (example 7.3) . . . . . . . . . . . . . . . . . . 145

7.7 Node disaggregation (example 7.4) . . . . . . . . . . . . . . . . . . 146

xv

List of Tables

3.1 LP formulation (example 3.1) . . . . . . . . . . . . . . . . . . . . . 42

4.1 Restricted master problem (example 4.1). . . . . . . . . . . . . . . . 59

4.2 Benchmark instances [193]. . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Comparing the new IP model with the model of [193]. . . . . . . . . 74

4.5 Comparing the new IP model with cuts (4.40) with the model of [193]. 75

4.6 Deriving lower bounds using the CP model. . . . . . . . . . . . . . 76

4.7 Benchmark instances randomly generated using CUTGEN1. . . . . 77

4.8 Comparing the lower bounds for the randomly generated instances. 78

5.1 Description of the set A instances. . . . . . . . . . . . . . . . . . . 95

5.2 Description of the set B instances. . . . . . . . . . . . . . . . . . . 97

5.3 Results for the set A instances. . . . . . . . . . . . . . . . . . . . . 101

5.4 Results for the set B instances. . . . . . . . . . . . . . . . . . . . . 102

5.5 Variants for the set A instances. . . . . . . . . . . . . . . . . . . . 107

5.6 Variants for the set B instances. . . . . . . . . . . . . . . . . . . . 109

7.1 Instances description (example 7.1) . . . . . . . . . . . . . . . . . . 137

7.2 Feasible routes (example 7.1) . . . . . . . . . . . . . . . . . . . . . 138

7.3 Arc reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.4 Computational results for 25 customers and tmax 75 and 220 . . . . 149

7.5 Computational results for 40 customers and tmax 75 and 220 . . . . 150

7.6 Computational results for 25 customers and tmax 100 and 250 . . . 151

7.7 Computational results for 40 customers and tmax 100 and 250 . . . 152

8.1 Instances description. . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.2 Computational results for 20 customers. . . . . . . . . . . . . . . . 167

8.3 Computational results for 25 customers. . . . . . . . . . . . . . . . 169

xvii

Chapter 1

Introduction

“However high we climb in the pursuit

of knowledge we shall still see heights

above us, and the more we extend

our view, the more conscious we shall

be of the immensity which lies beyond.”

William Armstrong

Contents

1.1 Motivation and Contributions . . . . . . . . . . . . . . . 2

1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . 3

1

2 1. Introduction

1.1 Motivation and Contributions

Integer programming problems are a special case of the broader area of optimization

or mathematical programming. From the computational complexity theory, we

known that many of these problems are NP -hard, which means that they are easy

to state but very hard to solve, even becoming intractable for instances that are

not very small.

The applicability of these problems in both theoretical issues and practical

real-world situations is widespread, but despite all the recent advances in math-

ematical programming theory and computer technology, these problems remain

very difficult to solve exactly. Approaches like the Dantzig-Wolfe decomposition

and column generation represent techniques designed to overcome the complexity

of these problems, and have been explored by researchers, who have so far applied

them to several integer programming problems.

In this thesis, we intend to contribute with new models and exact solution tech-

niques for integer programming problems. We explore pseudo-polynomial models

for some of these difficult problems. We also propose a column generation model

and a constraint programming model for one of them, as well as new lower bounds.

Two challenging NP -hard integer programming problems are addressed. They

have been widely approached in the literature, probably because of their difficulty

to solve and practical application in the real-world, where more and more complex

decisions must be taken in order to efficiently manage scarce resources. This rich

literature may also be the consequence of the fact that their standard versions can

be adapted and extended to numerous variants, addressing several practical issues

that are overly simplified in the standard versions of the problems.

The focus of our work goes to the cutting stock and vehicle routing problems.

We propose a column generation model and a constraint programming model to

derive lower bounds for the pattern minimization problem. We also propose three

1.2. Outline of the Thesis 3

network flow models, one for the two-dimensional cutting stock problem, one for

the vehicle routing problem with time windows and multiple routes and one for

the location routing problem with multiple routes. The three network flow models

are pseudo-polynomial and are solved exactly. Our approach consists of taking

advantage of that characteristic and using existing solvers to directly solve the

integer problem. To improve the algorithms’ efficiency, we derive several reduction

criteria, lower bounds and families of valid inequalities.

In order to investigate the effectiveness of our proposed methods, they are tested

on instances from the literature and, in the particular case of chapter 5, on real

instances from the furniture industry. All the computational results produced by

the series of experiments are presented and discussed.

1.2 Outline of the Thesis

In this thesis, we explore four variants of two well known integer programming

problems, the cutting and packing problem and the vehicle routing problem. It is

is organized as follows.

Chapter 2 presents an overview of some aspects of integer and combinato-

rial problems, providing a theoretical basis for the forthcoming chapters. In this

chapter, we present some definitions and notations regarding these problems. We

present the concept of lower bounds, refer to important computational complexity

issues, outline some linear programming solution methods, refer to some models

and introduce important solution techniques, namely branch-and-bound, Dantzig-

Wolfe decomposition, column generation and constraint programming. Although

this thesis concerns exact solution methods, we also briefly refer to some heuristic

solution approaches.

Chapter 3 introduces the cutting and packing problem, providing a description

of its main variants. Generally speaking, in cutting and packing problems there is

4 1. Introduction

a set of smaller items that must be placed, without overlapping, in a set of larger

objects. Given its great connection to a variety of commercial and industrial real-

world situations, cutting and packing problems have been widely approached in

the literature, which originated a variety of different definitions and notations for

problems with similar logical structure. Therefore, many surveys and categorized

bibliographies on this subject have emerged, and we describe a recent typology for

these problems. Finally, we present different formulations for the one and two-

dimensional versions of the cutting stock problem and a general literature review.

Chapter 4 addresses the pattern minimization problem. It is a cutting and

packing problem that consists of finding a cutting plan with the minimum num-

ber of different patterns. This objective may be relevant when changing from one

pattern to another involves a cost for setting up the cutting machine. Most of the

approaches described in the literature are based on heuristics and only very few ex-

act solution methods have been reported so far in the literature. In this chapter, we

intend to contribute to the exact solution of the pattern minimization problem with

new lower bounds. We explore a different column generation model, and describe

different strategies to strengthen it. Among the different procedures discussed in

this chapter, one is based on constraint programming. An exact branch-and-price-

and-cut algorithm is also described. We test our approaches on a set of benchmark

instances from the literature and present the computational results.

In chapter 5, we describe an exact model for the two-dimensional cutting stock

problem with two stages and the guillotine constraint. It is an integer linear pro-

gramming network flow model, formulated as a minimum flow problem. In this

chapter, we explore the behavior of this model when it is solved with a commercial

software, explicitly considering all its variables and constraints. We also derive a

new family of cutting planes and a new lower bound, and consider some variants

of the original problem. The model is tested on a set of real instances from the

wood industry and the computational results are discussed.

1.2. Outline of the Thesis 5

With chapter 6, we intend to address the vehicle routing problem and its vari-

ants. This problem, which was formally stated for the first time more than fifty

years ago, has a very wide application area and a major practical relevance for

several different types of organizations. It is, generally speaking, the problem of

scheduling a fleet of vehicles to visit a set of customers, taking into account certain

operational constraints. The standard problem is defined, and we discuss some of

its variants. This chapter also provides an overview of several standard formula-

tions of the problem. A literature review of both exact and heuristic methods for

the vehicle routing problem and its variants is also presented.

In chapter 7, we explore a variant of the vehicle routing problem with time

windows and multiple use of vehicles. It considers that a given vehicle can be

assigned to more than one route per planning period. In this chapter, we propose

a new exact algorithm for this problem. Our algorithm is iterative and it relies

on a pseudo-polynomial network flow model, whose nodes represent time instants,

and whose arcs represent feasible vehicle routes. This algorithm is tested on a set

of benchmark instances from the literature and the corresponding computational

results are presented.

Chapter 8 addresses a variant of the location routing problem where vehicles

can perform several routes in the same planning period. This problem represents

a combination of two problems, a vehicle routing problem, which determines the

optimal set of routes to fulfill the demands of a set of customers, and a location

problem, as the depots from which the vehicles performing the routes can be asso-

ciated must be chosen from a set of possible locations. In this chapter, we explore

a new network flow model, whose nodes represent time instants, and whose arcs

represent feasible vehicle routes. It is tested on a set of adapted instances from the

literature.

Finally, in chapter 9 some conclusions are drawn, regarding the research results

of each of the addressed problems and the overall work in the thesis. The contribu-

6 1. Introduction

tions to the state-of-the-art of the tackled subjects are outlined, and some future

work directions are discussed.

Chapter 2

Combinatorial and Integer

Programming

“True optimization is the revolutionary contribution

of modern research to decision processes.”

George Dantzig

Contents2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Algorithms and Computational Complexity . . . . . . 11

2.3.1 Complexity Classes . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Big-O Notation . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Methods for Linear Programming . . . . . . . . . . . . 15

2.4.1 The Simplex Method . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Interior Point Methods . . . . . . . . . . . . . . . . . . . 15

2.5 Branch-and-Bound . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Network Flow Models . . . . . . . . . . . . . . . . . . . . 17

2.7 Dantzig-Wolfe Decomposition . . . . . . . . . . . . . . . 19

2.8 Constraint Programming . . . . . . . . . . . . . . . . . . 24

2.9 Heuristic Solution Methods . . . . . . . . . . . . . . . . 26

7

8 2. Combinatorial and Integer Programming

2.1 Introduction

Optimization or mathematical programming is a branch of applied Mathematics

that roughly consists of finding the best possible scenario for a given system. This

system is usually defined by an objective function f (to be maximized or mini-

mized), a set of variables whose values must be chosen in order to optimize the

objective function, and a domain, X, called the solution space, which represents

the intersection of all the problem’s constraints, defining the complete set of fea-

sible solutions. It is an important branch of the Operations Research field, and a

useful tool to choose the very best solution among a potentially exponentially large

number of feasible options. Optimization programming arises in a wide variety of

practical problems, from management to engineering, and can be applied to im-

prove all kinds of processes in the industrial and business fields, in several fields of

applied science and technology, and even in a variety of everyday life problems.

When the objective function and the constraint functions that define the so-

lution space are linear, the problem is said to be a Linear Programming (LP)

problem, and X is a polyhedron (or a polytope, if X is bounded).

Optimization problems can also be divided into two categories: the ones in

which x ∈ Rn+,∀x ∈ X, called continuous optimization problems, and the ones in

which x ∈ X can only take a finite number of discrete values, called Combinatorial

Optimization (CO) problems. The latter are in general much more difficult to

solve.

This thesis concerns combinatorial integer programming problems. In [53], the

author provides a general overview of the main developments of this particular area

in the last five decades. A thorough theoretical overview of combinatorial and inte-

ger programming problems is provided in [175, 201]. In [106], Grotschel describes

several examples of applications of CO problems that arise in the manufacturing

industry.

2.1. Introduction 9

Let A ∈ Rm×n be a real matrix, c ∈ Rn and b ∈ Rm real vectors of constants,

and x ∈ Rn a real vector of variables. A general optimization problem can be

defined as

zOP = McTx : x ∈ X, X = x ∈ Rn : Ax ./ b,

with M representing minimization or maximization, and ./ representing one of

the three symbols, ≤, ≥ or =. Set X ⊆ Rn is a polyhedron that represents the

feasible region, i.e., the set of all points in Rn that are feasible for the optimization

problem.

Some problems require the use of variables that are indivisible. For example,

if a variable represents a person, a machine or a project, it is clear that its value

must be integer. An Integer Programming (IP) model can be generally stated as

zIP = mincTx : x ∈ XIP, XIP = X ∩ Zn+.

Set X is a formulation of set XIP . There can be several different formulations for

XIP . If X ⊆ Rn and X ′ ⊆ Rn are two formulations for XIP (XIP = X ∩ Zn+and XIP = X ′ ∩ Zn+), and X ⊂ X ′, formulation X is a stronger formulation than

formulation X ′. The convex hull of a set of points is the smallest convex set that

contains it. Therefore, the convex hull of XIP , conv(XIP ), is the strongest possible

formulation for XIP . This concept can be very important when we are solving IP

problems, as we will discuss later. In fact, the developments in polyhedral combi-

natorics theory [176] and its application to IP problems have greatly contributed

to the increasingly efficiency of their solution algorithms.

A special case of IP problems is the one that requires that all variables are

binary, the Binary Integer Programming (BIP) problems, which are defined as

follows,

zBIP = mincTx : x ∈ XBIP, XBIP = X ∩ 0, 1n.

When only some of the variables have the integrality constraint, a model is said


to be a Mixed Integer Program (MIP), and can be stated as

zMIP = mincTx+ dTy : (x, y) ∈ XMIP,

XMIP = (Zn+ ×Rp+) ∩ (x, y) ∈ Rn

+ ×Rp+ : Ax+By ≤ b,

being B ∈ Rm×p a real matrix, d ∈ Rp a real vector of constants and y ∈ Rp a real

vector of variables. When the integrality constraint does not apply to any of the

problem’s variables, it is said to be a LP problem.

zLP = mincTx : x ∈ X, X = x ∈ Rn+ : Ax ≤ b.

LP problems are much easier to solve than IP problems. In fact, some exact

solution techniques for IP problems, like branch-and-bound, include the solution of

LP-relaxations within their algorithms, which represent the integer problem with

the integrality constraints relaxed.

Generally speaking, given an optimization problem

(P ) z = mincTx : x ∈ X ⊆ Rn,

a problem RP is called a relaxation of P if

(RP ) z = minϕ(x) : x ∈ XRP ⊆ Rn,

with X ⊆ XRP and ϕ(x) ≤ cTx, ∀x ∈ X. The optimal solution of a relaxation

of a minimization problem is, therefore, a lower bound for the original problem. If

the optimal solution of RP, x∗, is such that x∗ ∈ X and ϕ(x∗) = cTx∗, x∗ is the

optimal solution of the original problem. The difference between the optimal value

of an IP problem and the optimal value of its LP-relaxation problem is called the

integrality gap.

Two other well know relaxations, besides the LP-relaxation, are the combinato-

rial relaxation and the lagrangian relaxation. In combinatorial relaxation problems,

a set of constraints is removed from the problem, in order to make it easier to solve

2.2. Lower Bounds 11

(for example, the subtour elimination constraints in a shortest path problem). In

lagrangian relaxation problems, a set of constraints is also removed, and the ob-

jective function penalizes solutions that violate the removed constraints.

2.2 Lower Bounds

Let z∗ be the optimal solution of an IP problem P . Values z and z are called,

respectively, a lower bound and an upper bound of P if

z ≤ z∗ ≤ z.

Every feasible solution of P provides an upper bound, if P is a minimization prob-

lem, or a lower bound, if P is a maximization problem.

For a minimization problem P , if dze = bzc = k, z = k is an optimal solution of

P . Lower bounds (and correspondingly, upper bounds for maximization problems)

play a very important role in what concerns the solution of integer problems. They

not only prove optimality but can also assess the quality of both exact and heuristic

methods. Moreover, they often play a crucial role in the effectiveness of some

solution methods, like the branch-and-bound algorithm, whose efficiency greatly

depends on the quality of the lower bounds that limit the search tree. In [45, 37,

84, 145, 42], the authors provide lower bounds for several IP problems.

2.3 Algorithms and Computational Complexity

An important concept when dealing with CO problems is the measure of their

computational complexity. Such a measure can assess how difficult it is to solve

the problem.

Let fA : N → N be a time function for a given algorithm A. This means that

the algorithm running time is at most fA(n) for an input with size n ∈ N. We say

that A is a polynomial time algorithm if, for a given polynomial p, fA(n) ≤ p(n),


∀n ∈ N. Problems that can be solved by a polynomial time algorithm are usually

addressed as easy problems. Problems that cannot be solved by a polynomial time

algorithm are considered to be intractable for instances that are not very small.

A decision problem is a problem that, for any given input, has only two possible

outputs: yes (or 1) and no (or 0). We say that a decision problem Π is polynomially

reducible to a decision problem Π∗ (Π ∝ Π∗) if there is a polynomial time algorithm

which transforms instances i ∈ Π into instances i∗ ∈ Π∗ in such a way that instance

i has an answer yes if and only if instance i∗ has an answer yes.

In 1971, Stephen Cook [52] defined the complexity class NP , laying the foun-

dations of complexity theory. The author proved that any problem in this class

can be converted in polynomial time to the boolean satisfiability problem (SAT),

with the famous Cook’s theorem. With this theorem, the author introduced the

notion of NP -completeness, and SAT became the fist proven NP -complete prob-

lem. Later on, Karp [125] proved other twenty one problems to be NP -complete,

followed by Garey and Johnson [91], who provided a more comprehensive list of

300 NP -complete problems.

Given an optimization problem, for example, mincx : x ∈ X, a corresponding

decision problem (also called recognition problem) would be the problem in the

form of the question: Is there any x ∈ X such that cx ≤ k?, being k a constant.

From a complexity point of view, both the optimization and decision problems are

equivalent in the sense that if we can answer the second one in polynomial time,

the solution for the first one can also be found in polynomial time. Having said

that, we stress the fact that, formally, the NP -completeness theory only addresses

decision problems.

2.3.1 Complexity Classes

Let us now define four classes of problems:

(i) the class of NP (nondeterministic polynomial time) problems is the subset of

2.3. Algorithms and Computational Complexity 13

decision problems for which any instance with an answer yes has a polynomial

proof for it;

(ii) the class of P problems is the subset of decision problems that belong to NP

and for which there is a polynomial algorithm;

(iii) the class of NP -complete problems is the subset of problems Π ∈ NP such

that for every problem ∆ ∈ NP , ∆ is polynomially reducible to Π. This

means that problems Π are at least as hard as every other problem in class

NP ;

(iv) the class of NP -hard problems is the subset of problems Π such that for every

problem ∆ ∈ NP , ∆ is polynomially reducible to Π. Π does not necessarily

belong to class NP . Therefore, the intersection between class NP and class

NP -hard is the class NP -complete. An optimization problem is, therefore,

said to be NP -hard if its recognition version is NP -complete.

It is clear that P ⊆ NP and that NP -complete ⊆ NP . The fundamental

question is P = NP? If someone proves that there is a problem in the class NP -

hard that is polynomially reducible to a problem in the class P , then it would be

proved that all problems could be solved in polynomial time, and thus P = NP .

Neither this nor its contrary has ever been proved. The fact that no one has ever

found a polynomial algorithm for a NP -hard problem may be an empirical evidence

that P 6= NP , although not everyone in the research community is convinced of

that. This is one of the most famous open problems in theoretical computer science,

and it is commonly addressed as the P vs NP problem. The Claus Mathematics

Institute established this problem as one of the seven Millennium Problems, with

a prize of one million dollars for who ever formally proves either that P = NP or

P 6= NP .

NP -complete problems can still be divided into two categories. The weakly

NP -complete and the strongly NP -complete problems. The first are the ones that


can be solved with a pseudo-polynomial algorithm and the second are the ones that

cannot be solved with a pseudo-polynomial algorithm unless P = NP .

Pseudo-polynomial algorithms are polynomial in the dimension of the input and

the size of the numerical data. They differ from strongly polynomial algorithms as

strongly polynomial algorithms have a running time that is independent of the nu-

merical size of the input. An integer sorting algorithm is an example of a strongly

polynomial algorithm. The number os steps required to solve it is independent

of the sizes of the input data, depending only on the number of integers to sort.

The size of an instance is the number of bits required to encode it. Numerical

data are encoded in binary notation. This means that an integer x takes log2x

bits to encode. Technically speaking, pseudo-polynomial algorithms are, therefore,

exponential, but, according to Garey and Johnson [91], they only express an ex-

ponential behavior when the input data contains exponentially large numbers, and

thus can efficiently solve instances where that does not happen. Strongly NP -

complete problems are those that remain NP -complete even if the input data is

encoded in unary notation (an integer number x takes x bits to encode in unary

notation).

2.3.2 Big-O Notation

The efficiency of an algorithm can also be described by the so called Big-O nota-

tion. It measures an algorithm efficiency by its worst-case behavior, i.e., the worst

performance of all inputs possible.

Given two real functions f and g, f(n) is O(g(n)) if |f(n)| ≤ k|g(n)| for all

n > 0 and some k > 0.

The Big-O notation uses an asymptotic analysis to provide an approximate

upper bound on the computational effort for the algorithm to solve an instance of

size n.

2.4. Methods for Linear Programming 15

2.4 Methods for Linear Programming

2.4.1 The Simplex Method

The simplex method is considered as one of the most important algorithms devel-

oped in the last century. It was devised in 1947 by Dantzig, known as the father

of linear programming and, although its worst-case run-time is exponential, it has

proven to be, in practice, a very efficient algorithm.

If the underlying polyhedron of LP problem is nonempty and bounded, the op-

timal solution of the problem is in one of its vertices (extreme points). The main

idea of the simplex method is to visit the vertices of the polyhedron, moving from

one vertex to an adjacent vertex, in a process called pivoting. It is an iterative al-

gorithm that tries to improve the solution, until no further improvement is possible

and, therefore, the optimal solution is found.

In [62], Dantzig provides a comprehensive description of this well known algo-

rithm that revolutionized the way of solving linear programming problems.

2.4.2 Interior Point Methods

In 1984, Karmarkar [124] proposed an algorithm for LP problems that belongs to

the class of interior point methods. These methods had been used for nonlinear

programming problems since the sixties. However, it was only after the revolution-

ary publication of Karmarkar that several of these methods were proposed in the

literature for LP problems.

Unlike the simplex method, interior point methods have polynomial complexity.

Their main concept differs from the simplex algorithm in the sense that the search

for the optimal solution is done through the convergence of points belonging to the

interior of the solution space polyhedron, rather than to its vertices. One of the

most efficient interior point methods is the one proposed in [156], which belongs to


the class of primal-dual path-following interior point methods.

2.5 Branch-and-Bound

One of the most widespread approaches to solve integer programming problems is

the Branch-and-Bound (B&B) algorithm, first proposed to solve an IP problem in

[129]. It is an enumerative technique, whose principle relies on a strategy of dividing

the search space into several subsets, and consecutively solving their relaxations,

until the optimal (integer) solution is found, once certain conditions are verified.

The algorithm is structured as a search tree, being its root node the LP-relaxation

of the original problem. From each node of the tree, two new nodes can be created,

by adding the so called branching constraints. These two nodes partition the search

space represented in the previous node into two subspaces, that are to be solved

independently. This division process continues at each node, unless the node is

pruned (or fathomed). This happens when at least one of three conditions holds:

(i) the optimal solution of the subproblem represented by the node is worse than an

integer solution previously found at another node, (ii) the optimal solution found

at the node is integer or (iii) if the subproblem represented by the node is infeasible.

This pruning process represents an elimination of many possible solutions that are

proved to be either dominated or infeasible.

The main idea on which this technique is based is that, given a MIP problem

and the subsets XMIP1 , . . . , XMIP

d such that XMIP = XMIP1 ∪ . . . ∪XMIP

d ,

zMIP = mincTx+ dTy : (x, y) ∈ XMIP

is equivalent to

zMIP = min1≤i≤d

mincTx+ dTy : (x, y) ∈ XMIPi .

After solving the LP problem at a given node, the question of which node should

be selected next arises. It is known that the search strategy to adopt influences

2.6. Network Flow Models 17

the performance of the algorithm. Although other, eventually more sophisticated,

search strategies can be devised, there are two traditional ones: the Depth-First

Search and the Best-First Search. The former implies that the next node to tackle

is always the one corresponding to the first subset created by the branching at that

node. When there is no such node to explore, the first unexplored node of the

previous levels should be chosen. This strategy typically provides integer feasible

solutions quickly. The latter implies that the next node to go to has been generated

by the node who provides the best lower bound. This strategy typically provides

good lower bounds earlier in the search.

2.6 Network Flow Models

Graphs are very useful to model many combinatorial problems, even some that

have no apparent underlying physical network structure. It is so, not only because

of their intuitive representation, but also because their structure makes them, in

a computational point of view, more efficient than other optimization techniques.

They can represent any problem that may be described by relationships between

pairs of a set of elements. They can model assignment, inventory planning, machine

sequencing or facilities location problems, flows of liquids going through pipes,

current going through electrical circuits or messages going through communications

systems, and even traffic flows going through road networks, just to state a few

examples.

A graph G = (V,A) is a structure that is defined by a set of nodes (or vertices)

V and a set of arcs (or edges) A, each arc (i, j) connecting two nodes i, j ∈ V .

A graph is said to be directed if arc (i, j) ∈ A is considered to be different

from arc (j, i). It is complete if every possible pair of nodes is connected by an

arc. A path between a pair of nodes is a sequence of arcs that connects them. If

all those arcs are either directed or undirected, the path is directed or undirected,


respectively. A graph is connected if there is an undirected path between every

pair of nodes. A cycle is a path that begins and ends in the same node. A tree is

a connected graph with no cycles. A spanning tree of a graph is a subgraph that

is a tree and contains all nodes of the graph, and thus has exactly |V | − 1 arcs.

When nodes or arcs have associated characteristics like a demand or supply, for

the nodes, or a cost, for the arcs, the graph is called a network. In a network flow

model, there is a flow that goes through the arcs, visiting all nodes in V , or even

just a subset of them, in order to satisfy a given set of constraints. The flow in

the network goes from the source node, s, to the sink (or terminal) node, t. For

every node of the graph that is neither the source nor the sink, there must be a

flow conservation, which means that the amount of inflow of a node must equal the

amount of outflow of that same node. Being the source and the sink the beginning

and the end of the network, respectively, there is only flow getting out of the source

node and flow getting in the sink node.

According to the specificities and objective of each optimization problem, the

network that models it may have quite different characteristics. Some of the most

common general classes of network optimization problems are the shortest path,

minimum spanning tree, maximum flow and minimum cost flow problems. The

shortest path problem arises whenever it is necessary to find a path with the small-

est cost or length between two given nodes of a graph. Some of the most known

algorithms to solve this problem are the Dijkstra’s, Ford’s and Floyd’s algorithms.

The minimum spanning tree is the problem of selecting the arcs to include in the

graph, in order to find a spanning tree with the smallest cost or length. This prob-

lem can be solved with the well known Kruskal’s algorithm. The maximum flow

problem arises whenever there is a network whose arcs have capacity constraints

and the objective is to send the maximum amount of flow through the graph. On

the other hand, when it is necessary to send a given amount of flow through the

graph, with the minimum possible cost, we are in the presence of a minimum cost

2.7. Dantzig-Wolfe Decomposition 19

flow problem.

A minimum cost flow problem can be formulated as a LP model (2.1)-(2.3),

where variables xij represent the flow that goes through arc (i, j).

min∑

(i,j)∈A

cijxij (2.1)

s.t.∑

b:(i,j)∈A

xij −∑

j:(i,j)∈A

xji = bi, ∀i ∈ N, (2.2)

0 ≤ xij ≤ ubij ∀(i, j) ∈ A (2.3)

The amount of flow that is absorbed or produced in node i is defined by bi. De-

pending on the value of bi, node i is one of three different kinds of nodes: a supply

node, if this value is positive, a demand node, if it is negative or a transhipment

node, if bi = 0. All minimum flow cost problems must satisfy the so called feasible

solutions property in order to be feasible. It states that∑n

i=1 bi = 0, which ensures

that the total flow supplied to the network equals the flow that is absorbed by it.

In chapters 5 and 7 we use network flow graphs to model a two-dimensional

cutting stock problem and a multi trip vehicle routing problem, respectively, that

are generalizations of the minimum cost flow problem, with additional constraints

on the arcs. Their objective is to minimize the amount of flow that goes through

the network, while satisfying some constraints. All arcs have identical costs and

there is flow conservation in all nodes. All nodes are transhipment nodes, except

for the source, s, which is a supply node, and the sink, t, which is a demand node.

This means that −bs = bt.

2.7 Dantzig-Wolfe Decomposition

Solving integer programming models can be, as discussed before, a hard task and

it often relies on techniques that imply the use of the corresponding LP-relaxation

solution, like the branch-and-bound algorithm (section 2.5). Therefore, it can be


significantly important that the LP-relaxation of the IP model represents a strong

formulation of it.

A method to strengthen an IP formulation is to introduce cutting planes to the

model. These are valid inequalities, i.e., inequalities aTx ≤ d, with a ∈ Rn and

d ∈ R, that are satisfied by every point of the set of feasible solutions XIP of the

IP model. In a general cutting plane algorithm, the inequalities that are violated

by an optimal solution of the LP-relaxation are iteratively introduced in the IP

model, in order to cut off the optimal LP-relaxation solution without removing

any integral solution. This improves the formulation, by reducing the search space

of the LP-relaxation.

Another important technique consists of using reformulation methods, which

usually rely on a decomposition of the model. Examples of reformulation methods

are the Lagrangian decomposition [107], the Benders’ decomposition [31] and the

Dantzig-Wolfe decomposition methods.

The Dantzig-Wolfe decomposition is a problem reformulation technique that

may improve the LP-relaxation bound and can be applied to problems whose con-

straint matrix has a block angular structure.

This means that the constraint matrix can be written as

A01x1 + A0

2x2 + . . . + A0pxp ≤ b0

A1x1 ≤ b1

A2x2 ≤ b2

. . .

Apxp ≤ bp

with A0i ∈ Rn0×mi , Ai ∈ Rni×mi , b0 ∈ Rn0 and bi ∈ Rni , ∀i ∈ 1, . . . , p. Note that

sets Xi = xi ∈ Zmi+ : Aixi ≤ bi, ∀i ∈ 1, . . . , p, are independent except for the

linking constraints∑p

i=1A0ixi ≤ b0, and thus, the IP problem could be expressed

as


zIP = min

cTx :

p∑i=1

A0ixi ≤ b0, xi ∈ Xi,∀i = 1, . . . , p

.

In practice, this block angular structure of the problem happens when there are

subsets of constraints that represent disjoint subsystems within the main system,

and a subset of linking constraints that integrate all the subsystems, dealing with

the common characteristics and coordinating them. Examples of these subsystems

are each depot of a vehicle routing problem with multiple depots, each different

cutting pattern in a cutting stock problem, each different machine in a sequencing

problem, among many others.

This method consists of a reformulation of the variables that allows the prob-

lem to be split into smaller subproblems, to be solved independently. This division

results in a master problem and in a set of pricing subproblems. The master prob-

lem is equivalent to the original problem, but typically has a tighter LP-relaxation

bound (if the subproblems do not have the integrality property, i.e., the polyhedron

that describes its feasible region, Q, is such that Q = conv(Q ∩ Zn+)).

This reformulation of variables implies that the formulation X of the feasible

region of the integer problem XIP is replaced by its convex hull, conv(XIP ), which

is its strongest possible formulation, as mentioned before.

According to Minkowski’s Theorem [166], a nonempty polyhedron X = x ∈

Rn+ : Ax ≤ b can be described asX = x ∈ Rn

+ : x =∑P

p=1 λpxp+

∑Rr=1 δ

rxr,∑

p λp

= 1, δr ≥ 0, being xpp∈P the set of extreme points and xrr∈R the set of extreme

rays of X. If X is bounded, the polyhedron can be described solely by the convex

combination of its extreme points, i.e., X = x ∈ Rn+ : x =

∑Pp=1 λ

pxp,∑

p λp = 1.

Consider the IP problem:

(P ) mincTx : A0x ≤ b0, A1x ≤ b1, x ≥ 0.

Suppose that X1 = x : A1x ≤ b1 is a nonempty polyhedron. It can, therefore,


be reformulated as

X1 =

x ∈ Rn

+ : x =

P1∑p=1

λpxp +

R1∑r=1

δrxr,∑p

λp = 1, δr ≥ 0

,

being P1 and R1 the set of extreme points and extreme rays of X1. All feasible

solutions x of P must belong to set X1, and thus, the IP model can be reformulated

as

min

cT P1∑p=1

λpxp +

R1∑r=1

δrxr

: A0

P1∑p=1

λpxp +

R1∑r=1

δrxr

≤ b0, P1∑p=1

λp = 1, δr ≥ 0

.

This problem, the master problem, has less constraints but typically an expo-

nential number of variables λi, i ∈ P1∪R1, one for each extreme point and ray of the

polyhedron X1. This implies that it may not be possible to explicitly enumerate all

variables, for any reasonably sized practical problem, and thus, this decomposition

technique is normally applied within a column generation framework.

Column generation is basically identical to the simplex method, with the differ-

ence that it only handles a few non-basic variables at each time, being the remaining

ones fixed to zero. The master problem that only considers explicitly some of the

variables is referred to as the restricted master problem (RMP). The algorithm

dynamically adds new variables, according to their attractiveness in what concerns

their reduced costs at the current basis. These new variables are iteratively gener-

ated by solving the so called pricing subproblems, one for each of the blocks of the

problem’s structure. These subproblems should, therefore, not be very difficult to

solve, otherwise this method becomes inefficient for the considered problem. They

should ideally be well known combinatorial algorithms that can be solved with an

efficient algorithm.

The pricing subproblems use the dual information of the master problem to

price out the most attractive (if there is one) new column to be considered in

the RMP. Considering the previous example, the column must represent a feasible


point of X1 and have a negative reduced cost. The subproblem can be formulated

as:

min(c− πA0)x− α) : A1x ≤ b1, x ≥ 0,

being c the cost of the new column and π and α the dual values of the first and

second sets of constraints of the master problem.

In the column generation procedure there is an iterative exchange of information

between the master problem and the pricing subproblems, in order to solve the

global problem. It can be described as follows.

Initialization: A set of columns (variables) must be chosen to belong to the first

RMP to be solved. This set can represent the variables of a heuristic (prefer-

ably good) solution or any other random set, as long as the feasibility of

the problem is ensured. It can be useful to consider some artificial columns,

which are columns that may not be feasible in X1 but always ensure that the

problem remains feasible, so that the iteration does not stop because there

are not enough columns in the RMP to generate a feasible solution. It should

always be guaranteed that those columns only belong to the solution if no

others can be chosen. Typically, this is enforced by setting their costs to very

high values.

Solving the RMP: The LP-relaxation of the RMP is solved. If it is feasible, the

corresponding optimal dual solution (π∗, α∗) is saved.

Solving the pricing subproblems: Each one of the pricing subproblems is solved,

using the previous values (π∗, α∗), as defined above, until (at least) one so-

lution with negative reduced cost is found. If no such solution is found, the

algorithm stops, and the previous optimal solution of the RMP is the optimal

solution of the LP-relaxation of the master problem


Reoptimizing the RMP: If an attractive column is found, it is incorporated in

the RMP, and the LP-relaxation of the problem is reoptimized.

The column generation algorithm only solves the LP-relaxation problem. In

order to solve the integer problem, we need to integrate it in a branch-and-bound

algorithm. The resulting solution scheme is called a branch-and-price (B&P) algo-

rithm [194, 68].

In a B&P algorithm, column generation must be performed in every node of

the branching tree, where a new LP problem is solved. Moreover, the branching

constraints must be such that desirably the structure of the pricing subproblems is

not destroyed.

The B&B algorithm can also be combined with a cutting plane technique. In

this approach, new cuts are added at each node of the branching tree, where a new

LP problem is solved, in order to strengthen its formulation. This methodology is

known as branch-and-cut.

Finally, all the last three methods, column generation, branch-and-bound and

cutting planes, can be combined in a general framework for solving large scale

integer programming problems called branch-and-price-and-cut.

2.8 Constraint Programming

Constraint Programming (CP) is a computational programming approach whose

core is the concept of constraint [151]. The problems are described in terms of the

decision variables, their domains and the relations between them. The idea is to

solve problems by exploring their properties, expressed by constraints. Because of

its natural and declarative way of defining problems through constraints, CP can

be very useful to model optimization problems that are difficult to express, as it

typically allows to define models with fewer decision variables and constraints.

Although the use of constraints has arisen in the sixties [178, 160, 197], in

2.8. Constraint Programming 25

the field of Artificial Intelligence and Computer Graphics, only in the last two

decades has it been considered as an approach to modeling and problem solving.

It has been applied to a wide variety of areas such as combinatorial optimization

problems, numerical analysis, DNA sequencing, graphic systems, computer algebra

and several business applications.

In CP, a problem must be formulated as a constraint satisfaction problem. In

such models, there is a set of variables X1, . . . , Xn, each with a finite domain Di,

i ∈ 1, . . . , n, of possible values and a set of constraints C1, . . . , Cm that restricts

the combinations of values that subsets of variables can simultaneous take. Given

an assignment of variables, it is called consistent if it does not violate any of the

constraints. A solution for the problem is, therefore, a consistent assignment of all

variables. Constraint satisfaction problems are used to find out whether there is a

valid solution for the problem or not, to find one (or more) valid solution or even

to find an optimal (or near optimal) solution (constraint optimization).

The main techniques to solve CP problems consist of constraint propagation and

search methods. The search for a solution to a constraint satisfaction problem can

be made systematically through all the solution space or consistently by removing

inconsistent values from the domains. Usually, a combination of both methods

is used. Constraint propagation algorithms reduce the search space by replacing

the original constraint satisfaction problem by equivalent ones, with smaller search

spaces, obtained by repeatedly reducing domains or even constraints. It works

by propagating the implications of a constraint on a variable onto other variables

[184]. The backtracking algorithm is one of the most common search methods. It

consists of constructing partial assignments of variables and iteratively extending

them by adding another assignment to a variable that is still consistent with the

current partial solution. When no such value is found for a variable, the algorithm

reaches a so-called dead end and goes back to a previous partial solution, until a

consistent solution is found (or the problem proves to be infeasible).


If it is true that CP can be helpful to model complex problems, it is also true

that it may not very efficient to find optimal solutions, specially if the problem

is loosely constrained [119]. On the other hand, optimization methods such as

MIP based algorithms can be efficient when they have strong formulations and

special structures to be exploited. Recently, several authors have proposed hybrid

approaches that integrate CP methods with optimization methods such as MIP

based algorithms, in order to take advantage of both methods’ strengths [110, 119,

113, 2, 157, 67].

2.9 Heuristic Solution Methods

Optimization algorithms can be classified in two main categories: either exact or

heuristic methods. For most NP -hard combinatorial optimization problems, exact

algorithms have very large run times or even become intractable for instances that

are not very small (assuming that P 6= NP ). Despite all the recent advances both

in mathematical programming theory and in the computer technology, in a real-

world context, it may be unrealistic to try to solve these problems with an exact

method. Heuristic algorithms, on the other hand, have typically much smaller

computational times, although their solution is not (proven to be) optimal. This

may also be undesirable in situations where small improvements in the solution are

translated into considerable savings.

The trade between the quality of the solution and computational time must

be contextualized in each particular situation, so that, depending on the reality to

consider, the best method can be chosen.

The heuristic methods [196] may be classified as constructive heuristics, local

search heuristics or metaheuristics. Constructive heuristics build solutions step by

step, incrementing partial solutions iteratively, until a global one is reached. When

2.9. Heuristic Solution Methods 27

the criterion to choose the next part to increment is based on the immediate max-

imization of profit, the heuristic is said to be greedy. Local search heuristics [1]

iteratively try to improve a given solution by searching another one in its neigh-

borhood. This can be achieved by provoking perturbations on the initial solution.

The best solution of a given neighborhood is called a local optimum. Metaheuris-

tics are heuristic solution schemes for optimization problems, whose concept was

introduced in [103]. They represent a heuristic algorithmic framework where the

solution space of a combinatorial optimization problem is explored in a more guided

and efficient way than in the more basic heuristics. They even admit intermedi-

ate non improving or even infeasible solutions in order to escape local optima.

Interestingly, some of the most known metaheuristics are inspired in natural pro-

cesses, like animal behavior, genetic evolution or physical processes. Descriptions

of some of the most important metaheuristics in the literature are provided in

[104, 35]. The following methods are among some of these: tabu search [103, 105],

simulated annealing [127], genetic algorithms [112], ant colony optimization [71],

variable neighborhood search [159]. These methods have shown significant success

in solving difficult combinatorial optimization problems by finding near-optimal,

and sometimes even optimal solutions.

In the last decades, a new approach has been considered, the combination of

metaheuristics and exact algorithms for combinatorial optimization problems. This

combination may be collaborative or integrative [170, 36]. In the former method, the

different algorithms run separately, in a sequential or parallel way, while exchanging

information. In the latter one, there is a main method that embeds the other

in some subroutine. In [121], the authors propose a taxonomy to methods that

hybridize exact and metaheuristic algorithms.

Chapter 3

Cutting & Packing Problems

“To understand is to perceive patterns.”

Isaiah Berlin

Contents3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . 30

3.2 One-dimensional Cutting Stock Problems . . . . . . . . 34

3.3 Two-dimensional Cutting Stock Problems . . . . . . . 39

29

30 3. Cutting & Packing Problems

3.1 Problem Definition

Cutting and Packing (C&P) problems represent a class of combinatorial optimiza-

tion problems, belonging to the geometric combinatorics field, due to their geomet-

ric structure. Generally speaking, in a C&P problem there is a set of small items

that must be placed, without overlapping, in a set of large objects. These problems

are greatly connected to a wide variety of commercial and industrial real-world sit-

uations, including all those that consider the cutting of raw materials, packing of

objects or loading vehicles but also, in a broader sense, other ones concerning the

cutting or packing of abstract objects, like packing time in scheduling problems.

Most extensions and variants of Cutting Stock Problems (CSP) are well known

to be NP-hard. This means, as discussed in chapter 2, that all algorithms cur-

rently known for finding optimal solutions may require a number of computational

steps that grows exponentially with the problem size rather than according to a

polynomial function. Over the years, many different optimization formulations

and solution approaches have arisen in the literature, all including different dimen-

sions, application fields and special constraints and requirements. This originated a

variety of different definitions and notations for problems with similar logical struc-

ture. Therefore, many researchers provided surveys and categorized bibliographies

on this subject (Dowsland and Dowsland [72], Dyckhoff and Finke [75],Dyckhoff

et al. [76], Hopper and Turton [115] Lodi et al. [142, 144]). Moreover, Dyckoff et

al. [74] developed a typology of these problems. In this typology, there is a classi-

fication scheme that combines four main characteristics in the form of four-tuples

α/β/γ/δ, in order to classify any C&P problem as one of 96 possible subproblems.

These characteristics are:

Dimensionality (α): the minimum number of spatial dimensions necessary

to describe a cutting pattern of a feasible solution, that can be (1), (2), (3)

or (N > 3).

3.1. Problem Definition 31

Kind of assignment (β): describes whether all objects and only a selection

of small items must be used (B), or if all small items and only a selection of

objects are considered (V).

Assortment of large objects (γ): describes whether there is one or many,

identical or different large objects to assign. (O) for one large object, (I) for

many identical large objects and (D) for many different large objects.

Assortment of small items (δ): describes whether there are few/many

identical/different small items to assign. (F) if there are few small items of

different dimensions, (M) or (R) if there are many small items of, respectively,

many or relatively few different dimensions or (C) if there are many identical

small items.

In [200], Wascher et al. developed an improved typology of C&P problems,

based on the previous one [74], with new categorization criteria, in order to over-

come some deficiencies detected in the previous work. They considered five main

criteria to classify problems in standard problem types. The first criterion, kind of

assignment, is similar to the one in [74] and distinguishes the problems according to

their objective functions (input minimization, for the cases where all small items can

be assigned to the large objects, or output maximization, for the cases where only

a subset of small items can be assigned to the large objects). In what concerns the

assortment of small items, there can be three different situations: identical small

items or a weakly or strongly heterogeneous assortment of small items. For the

assortment of large objects, there can be one large object or several large objects.

Whenever there is only one large object, it can have fixed or variable dimensions.

If there are several large objects, they can all be identical or have a weakly or

strongly assortment. With respect to the problem’s dimensionality, the authors

only consider one, two or three-dimensional problems. Finally, they introduce a

new criterion for the two or three-dimensional problems that further distinguishes


INTERMEDIATE PROBLEM TYPES

Placement Problem (PP)

Identical Item Packing Problem

Knapsack Problem (KP)

Cutting Stock Problem (CSP)

Open Dimension Problem

Bin Packing Problem (BPP)

Cutting and Packing Problems (C&P)

Output maximization

Input minimization

several identical large objects

heterogeneous assortement

of several large objects

one large object

several identical large objects

heterogeneous assortement

of several large objects

one large object

weakly heterogeneous

assortment of large objects

strongly heterogeneous


identical large objects

weakly heterogeneous


strongly heterogeneous


identical large objects

identical

small items

weakly

heterogeneous

assortment of

small items

strongly

heterogeneous

assortment of

small items

arbitrary variable

dimension(s)

weakly

heterogeneous

assortment of

small items

strongly

heterogeneous

assortment of

small items

BASIC PROBLEM TYPES

Identical Item Packing ProblemIIPP

Single Large Object PP SLOPP

Multiple Identical Large Object PP MILOPP

Multiple Heterogeneous Large Object PP MHLOPP

SKP Single KP

MIKP Multiple Identical KP

MHKP Multiple Heterogeneous KP

ODP Open Dimension Problem

SSSCSP Single Stock Size CSP

Multiple Stock Size CSPMSSCSP

Residual CSPRCSP

Single Bin Size BPPSBSBPP

MBSBPP Multiple Bin Size BPP

RBPP Residual BPP

Figure 3.1: Cutting and packing typology [200]

the shape of small items, which can be regular or irregular. By combining the first

two criteria, kind of assignment and assortment of small items, the authors define

six basic types of C&P problems. These types are then structured further into four-

teen intermediate problem types by the additional application of the third criterion

assortment of large objects. Finally, the application of the criterion dimensional-

ity and, for the two and three-dimensional problems, the criterion shape of small

items, provides the refined problem types. The identification of any C&P prob-

lem should be stated in the form dimensionality-shape-intermediate problem

type. Whenever a problem presents all the properties of its refined problem type,

but it also has some additional characteristics or constraints, it is called a second-

level standard problem. Problems of second-level standard types whose additional

constraints are related with aspects that do not belong to the cutting or packing

3.1. Problem Definition 33

field are said to belong to an extended problem type. Finally, when the assump-

tions of a given problem are different from those of the standard problems, it is

considered as a variant.

Some approaches in the literature integrate the standard versions of the problem

with additional aspects. This is the case of the pattern minimization problem,

where there are setup costs to consider, and thus the objective is to minimize the

number of different cutting patterns, within an optimal solution for the standard

CSP [193, 186, 15]. Some versions also include constraints related with due dates

[139, 120, 173], multiple lengths for the large objects [30, 57, 16], minimization of

open stacks [203, 18, 204], which imply constraints with the sequencing of cutting

patterns, or even the reuse of generated waste in a multiperiod context [183], among

others.

The interest in the field of C&P over the last decades, by practitioners and

researchers in this area, has been so significant that the European Journal of Op-

erations Research (EJOR) has published four special issues on C&P problems over

the last twenty years [77, 34, 198, 168]. These special issues are composed by papers

that have covered both exact and heuristic methods, nesting problems, real-world

application problems, surveys and typologies of C&P problems.

A list of classified articles from 1980 to 2005 (according to [200]) concerning

C&P problems can be found in the web page of ESICUP [78]. ESICUP (EURO

Special Group on Cutting and Packing) is a working group of the association of

European Operations Research Societies ( EURO [79]), composed of researchers

with an interest on this particular field.

In chapters 4 and 5, we propose new lower bounds for the one-dimensional pat-

tern minimization problem and an exact solution algorithm for the two-dimensional

cutting stock problem. These two C&P problems are classified according to this

typology as 1D-SSSCSP and 2D-regular-SSSCSP, respectively.

In this thesis, we are, therefore, specially concerned with cutting stock problems


(CSP), in particular the one-dimensional and two-dimensional cases. The knapsack

problem (KP) is also approached, as a pricing subproblem of a column generation

model framework proposed in chapter 4. A thorough presentation of this last

problem and several of its variants and extensions, as well as a comprehensive

description of its the most relevant algorithmic methods is presented in [152, 126].

The CSP is equivalent to the Bin Packing Problem (BPP), in the sense that

they both have essentially the same logic structure. According to [200], and as

shown in figure 3.1, the only difference between the two problems relies on the fact

that in the CSP there is a weakly heterogeneous assortment of small items, whereas

in the BPP this assortment is strongly heterogeneous. This basically means that

the main difference between these two problems is that in the CSP the demands

are high (there are several copies of each dimension of small items), whereas in the

BPP the demands are very low (there is typically a single copy of each dimension

of small items). Thus, most solution methods devised for one of the problems can

easily be adapted to solve the other, although it can happen that a given algorithm

is more efficient for one of the problems than it is for the other. That is the example

of algorithms that rely on the construction of patterns and the determination of

the number of times those patterns should be used. These algorithms are clearly

more appropriate for the CSP, where there are identical items, and thus patterns to

repeat. A review of LP-based algorithms, sequential heuristics and hybrid solution

procedures for the 1D- and 2D-CSP is provided in [109].

3.2 One-dimensional Cutting Stock Problems

The standard one-dimensional version of the CSP (1D-CSP) can be stated as fol-

lows: a given set of small items, each item i ∈ 1, ...m of size li and demand of

bi pieces, has to be cut out of a virtually infinite supply of large objects, called

rolls, of size L (where 0 < li ≤ L, ∀i ∈ 1, ...,m), usually in order to minimize the

3.2. One-dimensional Cutting Stock Problems 35

number of rolls to be used (figure 3.2).

CUTTING PATTERNS

ROOLS

SMALL ITEMS

+

+

+

Figure 3.2: 1D-CSP representation

The mathematical model proposed by Kantorovich [123] in 1939 (although it

was only translated to English in 1960) for the 1D-CSP is considered to be the first

one to be proposed for this problem. It appears in a section called Minimization

of scrap, in a monograph whose aim was to describe mathematical methods to

improve the organization and planning for production, for the Soviet system.

Let R be an upper bound for the number of available rolls, zj, j ∈ 1, . . . , R,

the binary variable that defines whether or not roll j is used and xij the integer

variable defining the number of items i ∈ 1, . . . ,m in the roll j. The model can

be described as follows:

minR∑j=1

zj (3.1)

s.t.m∑i=1

lixij ≤ Lzj, ∀j ∈ 1, . . . , R, (3.2)

R∑j=1

xij ≥ bi, ∀i ∈ 1, . . . ,m, (3.3)


xij ∈ Z+, zj ∈ 0, 1, ∀i ∈ 1, . . . ,m, ∀j ∈ 1, . . . , R. (3.4)

The objective is to minimize the number of rolls to be used, and thus the

generated waste (3.1). The first set of constraints (3.2) ensure that the items fit

in the rolls they are assigned to, and constraints (3.3) are concerned with the

fulfillment of demands.

In 1961, Gilmore and Gomory [100] proposed an integer programming model

and a column generation approach, to solve the standard one-dimensional version of

the CSP. The Dantzig-Wolfe decomposition results in a master problem (3.5)-(3.7)

whose variables are feasible cutting patterns, and a pricing problem (3.8)-(3.9) that

is a knapsack problem (denoted as a single knapsack problem (SKP), according to

[200]), to determine, at each step, promising patterns (columns) that might improve

the current solution.

Let us consider J as the set of all feasible valid cutting patterns, i.e., all vectors

(a1j, ..., amj)T such that

∑mi=1 liaij ≤ L, being aij,∀i ∈ 1, ...m,∀j ∈ J , nonneg-

ative integers that represent the number of items i ∈ 1, . . . ,m in the cutting

pattern j ∈ J . The variables λj, ∀j ∈ J , represent the number of times pattern j

is used. The master problem states as follows:

min∑j∈J

λj (3.5)

s.t.∑j∈J

aijλj ≥ bi ∀i ∈ 1, . . . ,m, (3.6)

λj ≥ 0 and integer ∀j ∈ J. (3.7)

The sum of the frequencies of each cutting pattern represents the number of rolls

to be used, and thus, its minimization represents the objective of minimizing waste

(3.5). The other constraints included in the master (3.6) are concerned with the

fulfillment of demands. The feasibility of the cutting patterns is dealt with in the

pricing problem. It represents the following knapsack problem:

maxm∑i=1

πiyi (3.8)

3.2. One-dimensional Cutting Stock Problems 37

s.t. liyi ≤ L ∀i ∈ 1, . . . ,m (3.9)

yi ≥ 0 and integer ∀i ∈ 1, . . . ,m (3.10)

where πi, ∀i ∈ 1, . . . ,m, represents the dual variable associated with constraints

(3.6) and yi is the decision variable which denotes the number of times item i is

selected in a new pattern.

This model has a stronger formulation than the one proposed by Kantorovich.

In fact, the difference between the integer optimal solution and the LP-relaxation

for this model (gap) is conjectured to be always smaller than 2, being it smaller

than 1 for most instances (integer round-up property) [149].

Valerio de Carvalho [188] proposed an IP arc-flow model for the 1D-CSP (3.11)-

(3.14), in which every cutting pattern corresponds to a path in an acyclic directed

graph G = (V,A), with V = 0, 1, ..., L as its set of L + 1 vertices, which define

positions in the stock sheet, and A = (a, b) : 0 ≤ a < b ≤ L and b − a =

li,∀i = 1, ...,m as its set of arcs. It is formulated as a minimum flow problem,

where variables xab correspond to the flow in arc (a, b), i.e., the number of items of

width b− a placed at a distance of a units from the beginning of a given roll, and

variable z corresponds to the total flow through the graph, and can be seen as the

return flow from vertex L to vertex 0.

min z (3.11)

s.t.∑

(a,b)∈A

xab −∑

(b,c)∈A

xbc =

−z , if b = 0

0 , if b = 1, 2, ..., L− 1

z , if b = L

, (3.12)

∑(c,c+li)∈A

xc,c+li ≥ bi, ∀i ∈ 1, ...,m, (3.13)

xab ≥ 0 and integer, ∀(a, b) ∈ A. (3.14)

Constraints (3.12) are related to flow conservation and constraints (3.13) ensure

that the demands are fulfilled. Valerio de Carvalho [188] defined criteria to reduce


the number of allowable arc-variables, reducing the size of the model and the sym-

metry of the solution space. Symmetry should be avoided, as much as possible,

as it increases the computational effort, because the same physical solution can be

explored in different nodes of the branching tree. For any pattern, the items are

sorted by their decreasing sizes and the arcs corresponding to its waste (unit arcs)

always appear in the last positions of the stock sheet. Furthermore, the number of

items of a given size in a cutting pattern can not be greater than the demand of

those items.

This model belongs to the family of position-indexed models [189], as the deci-

sion variables are indexed according to the position of their associated items in the

rolls. Its formulation is equivalent to the one of the previous model (3.5)-(3.7).

In [189], these and other models for the 1D-CSP are reviewed, including one-

cut models, where decision variables are associated to single cuts on the rolls, and

models associating the bin packing problem as a special case of the vehicle routing.

A wide variety of both exact and heuristic approaches for the 1D-CSP have been

published in the literature. In what concerns heuristic methods, some simple well

known greedy heuristics (first fit, next fit and best fit algorithms, among others)

[51], LP based heuristics [199], and several metaheuristic methods [111, 80, 172,

140, 87, 138] have been devised to tackle this problem.

Exact solution methods for the 1D-CSP are mainly based on branch-and-price

techniques. Several have been proposed in the literature [191, 190, 188, 192, 66,

65, 16].

In [94], the authors propose a problem generator for the 1D-CSP, so that re-

searchers in this area can easily generate appropriate test instances and properly

benchmark their results.

3.3. Two-dimensional Cutting Stock Problems 39

3.3 Two-dimensional Cutting Stock Problems

The standard version of the 2D-CSP can be stated as follows: a given set of small

items, each item i ∈ 1, ...m of width wi, height hi and demand of bi pieces, has to

be cut out of a virtually infinite supply of large objects, called stock sheets, of width

W and height H (where 0 < wi ≤ W and 0 < hi ≤ H, ∀i ∈ 1, ...,m), usually

in order to minimize the number of stock sheets to be used. The 2D-CSP can be

14

1111

222

33333

444

55555

111 1

222

344

33

554

33

555

Stock sheets

Small items

Cutting patterns

Figure 3.3: 1D-CSP representation

further classified into several categories, depending on specific constraints. It can be

regular, if the shapes of the items can be described by few parameters, or irregular,

otherwise. Cutting irregular shapes is also known as nesting. Regular cuts can

be rectangular or non-rectangular, for the cases where the items are rectangles or

have a different shape, respectively. The rectangular cutting can be oriented, if

an item of width w and height h is considered to be different from another one of


width h and height w, or non-oriented, otherwise. If all cuts must be made straight

from one edge to the opposite edge of the stock sheet (or of one of its already cut

fragments), dividing it in two, the cutting patterns produced are of guillotine type.

Non-guillotine patterns are not restricted by this rule, being the corresponding

problems much harder to solve. A staged pattern is a guillotine pattern whose

items are cut in a limited number of phases. The direction of the first stage cuts

may be either horizontal or vertical (parallel to one side of the stock sheet), and

the cuts of the same stage are in the same direction. The cut directions of any

two adjacent stages must be perpendicular to each other. If the maximum number

of stages is not allowed to exceed a value n, the problem is called n-stage. When

there is no such restriction, the problem is called non-stage. Whenever a final stage

for separating items from waste areas is allowed, the problem is called non-exact ;

otherwise, it is called exact.

Gilmore and Gomory [102] proposed the first model for the 2D-CSP, by extend-

ing their column generation approach to the 1D-CSP [100, 101]. They solve the

two-dimensional guillotine version of the problem as a two-stage one-dimensional

problem, with the first stage corresponding to the cutting of the stock sheets into

strips, and the second stage corresponding to the cutting of those strips into the

demanded items.

In the first stage they consider all the patterns corresponding to cutting items

of heights hi, ∀i ∈ 1, . . . ,m′, where m′ ≤ m is the number of different items’

heights, out of stock sheets of height H, i.e., all the possible combinations of items

i ∈ 1, . . . ,m′ such that∑m′

i=1 aihi ≤ H. In this stage the items’ widths are not

considered. In the second stage there are m′ sets of patterns. Each set s ∈ 1, ...m′

considers patterns that include only items i such that hi ≤ hs and that satisfy the

inequality∑m

i=1 aiwi ≤ W . They formulate the problem as follows.

min∑j∈J0

λ0j (3.15)


s.t. M ′.λ = 0 (3.16)

M ′′.λ ≥ B (3.17)

λ ≥ 0 and integer (3.18)

where J0 is the set of feasible cutting patterns corresponding to the first stage,

λ = (λ01, . . . , λ

11, . . . , λ

m′

1 , . . .)T , λ0j is the jth pattern associated to the first stage

and λsj is the jth pattern associated to the sth set of patterns of the second stage

and B = (b1, . . . , bm)T . M ′ and M ′′ correspond, respectively, to the first m′ rows

and last m rows of the matrix

M =

M0

−1 . . . −1

0...

0

−1 . . . −1

0...

. . .

...

0

−1 . . . −1

0 M1 M2 ... Mm′

,

where submatrix M0 includes all the possible patterns corresponding to the first

stage, and submatrixes Ms include all the patterns that belong to set s ∈ 1, ...m′.

Constraints 3.16 guarantee that the strips cut in the first stage are the ones

that are used in the second stage, to be cut into the demanded items, whereas

constraints 3.17 guarantee that the items’ demands are fulfilled.

Example 3.1 Consider an instance with stock sheets of height H = 20 and width

W = 30 and a set of items i such that (hi, wi) : i ∈ 1, . . . , 5 = (5, 7), (5, 10),

(7, 12), (10, 8), (12, 10) with respective demands b = (4, 3, 5, 3, 5).

Table 3.1 represents the LP formulation for this instance. The integer optimum

solution has a value of 3, with a set of pattern frequencies associated to the first

stage of 1 and 2 for λ03 and λ0

7, respectively. This means that three stock sheets are

used, with one of them being cut into two strips of height 5 and a strip of height

10 and the other two being cut into two strips of heights 7 and 12 (figure 3.4).


Op

timal

int.

sol.

00

10

00

20

10

01

00

20

00

01

00

00

00

00

01

00

00

01

λ01λ02λ03λ04λ05λ06λ07λ08λ11λ12λ13λ14λ21λ22λ23λ31λ32λ33λ34λ35λ36λ37λ38λ39λ41λ42λ43λ44λ45λ46λ47λ48λ49λ410λ411λ412

(hi ,w

i )=

(5,[7,10])

42

21

1-1

-1-1

-1=

0

(7,1

2)1

21

1-1

-1-1

=0

(10

,8)

11

2-1

-1-1

-1-1

-1-1

-1-1

=0

(12

,10)1

1-1

-1-1

-1-1

-1-1

-1-1

-1-1

-1=

0

(hi ,w

i )=

(5,7)

42

12

13

21

12

11

11

≥4

(5,10

)1

23

12

11

12

11

1≥

3

(7,12

)1

12

11

11

1≥

5

(10,8

)1

21

22

11

13

21

11

1≥

3

(12,1

0)

12

12

12

11

11

13≥

5

O.F

.1

11

11

11

1

Tab

le3.1:

LP

formulation

(exam

ple

3.1)


In the second stage, the variables’ values are associated with the cutting of the

seven strips of the first stage into the demanded items. For example, λ35 = 1 means

that the strip of height 10 was cut into two items (10, 8) and one item (7, 12).

Many heuristics based on this column generation formulation where proposed in

5 x 7 5 x 7 5 x 7 5 x 7

5 x 10 5 x 10 5 x 10

10 x 8 10 x 8

7 x 12

7 x 12 7 x 12

12 x 10 12 x 1010 x 8

7 x 127 x 12

12 x 10 12 x 10 12 x 10

Figure 3.4: Solution representation (example 3.1)

the literature [81, 61, 12, 11]. Moreover, Lodi et al. [143] proposed a general

framework of heuristic and metaheuristic approaches for several variants of the 2D-

BPP. Other greedy heuristics [46, 33], local search algorithms [117] and evolutionary

algorithms [128, 116, 114] were also proposed. Several exact algorithms have also

been described [154, 195, 17, 59, 171, 47]. Reviews on the 2D-BPP problem are

provided in [142, 144, 29].

Chapter 4

New Lower Bounds based on

Column Generation and

Constraint Programming for the

Pattern Minimization Problem 1

“Each problem that I solved became a rulewhich served afterwards to solve other problems.”

René Descartes


4.2 Mathematical Programming Models: State-of-the-art 52

4.2.1 A Nonlinear Compact Formulation . . . . . . . . . . . . 52

4.2.2 The Gilmore and Gomory Model with Setup Variables . 53

4.2.3 The IP Model of Vanderbeck [193] . . . . . . . . . . . . 54

4.2.4 An Arc-flow Formulation . . . . . . . . . . . . . . . . . 55

4.3 Exploring a New IP Formulation . . . . . . . . . . . . . 56

4.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2 Strengthening the Model . . . . . . . . . . . . . . . . . 61

4.4 A CP Model . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.2 Deriving Lower Bounds from the CP Model . . . . . . . 69

4.4.3 Using the CP to Derive Valid Inequalities for SP1 . . . 69

4.5 Comparative Computational Results . . . . . . . . . . . 71

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 77

1The work presented in this chapter has been published in [14]

45

46 4. New Lower Bounds based on CG and CP for the PMP

4.1 Introduction

The Pattern Minimization Problem (PMP) is a cutting and packing problem, con-

sidered as a 1D-SSSCSP (one-dimensional single stock size cutting stock problem)

according to the recent typology proposed in [200]. Its goal is to determine a cut-

ting plan with the minimum number of different patterns. This objective may be

relevant when changing from one pattern to another involves a cost for setting up

the cutting machine. As in the standard Cutting Stock Problem (CSP), the input

of the problem is given by a set of item sizes and demands, and by a set of roll

sizes. The PMP can be solved in a single stage or in two stages. In the former, one

tries to find a cutting plan with the best balance between waste and the number of

different patterns. In the latter, the PMP is solved by assuming that no more than

a given number of rolls can be used. Usually, this number is obtained by solving a

standard CSP. In this chapter, we address the latter case and intend to contribute

to the exact resolution of the one-dimensional PMP with new results (figure 4.1).

McDiarmid [155] proved that the PMP is strongly NP-hard, as he showed that

it is so even for the simplest nontrivial restricted problem that only considers items

such that any two fit in a stock roll but no three do. Thus, most of the methods

described in the literature to solve this problem are heuristic approaches. Solving

the problem exactly has been a real challenge, and only very few exact solution

methods have been reported so far in the literature.

Haessler [108] presented one of the first methods to handle the PMP. He pro-

posed a sequential heuristic procedure, where the cutting patterns are determined

successively by an algorithm that tries to find patterns with small wastes and high

usage levels, so that there is a balance between minimizing the waste and the num-

ber of different patterns. This tradeoff can be adjusted by changing the values of

the parameters.

Teghem et al. [180] analyzed a real problem concerning book coverings in the

4.1. Introduction 47

CSP PMP

Number of Rolls: 8

Number of different patterns: 4

Number of Rolls: 8

Number of different patterns: 5

Maximum number of rolls: 8

Minimize number of different patterns

1

3

1

3

1

3

1

3

1

3

1

3

1

5

4

1

5

4

2

3

5

2

3

5

2

3

5

1

3

3

3

4

2

2

1

5

4

2

4

4

12

3 45

b1 = 5 b3 = 6b2 = 3 b5 = 3b4 = 3

rolls

items

Figure 4.1: One-dimensional pattern minimization problem

publishing industry, where all the covers are grouped in subsets of homogeneous

characteristics and one pattern corresponds to a plate where four covers of the same

subset can be placed. To solve this PMP the authors used a simulated annealing

procedure. Chen et al. [44] also proposed a simulated annealing algorithm to solve

the CSP with a model whose objective function considers both material and setup

costs.

Foerster and Wascher [88] defined their approach as a generalization of Diegel

et al.’s [69] heuristic method of combining two different patterns into a single

pattern with a frequency equal to the sum of the frequencies of the previous ones,

maintaining the number of rolls used. The authors extended Diegel et al.’s concept

to combinations of p different patterns into q different ones, only considering p to


(p-1 ) combinations and limiting p to 4. They proposed a class of methods called

KOMBI, each defining a way and sequence of combining different cutting patterns,

and performed computational experiments in the particular methods KOMBI23

(2 to 1 and 3 to 2 combinations) and KOMBI234 (2 to 1, 3 to 2 and 4 to 3

combinations).

Umetani et al. [186] proposed a metaheuristic approach for the PMP. Their

method begins by fixing a constant number p of different cutting patterns and

then it uses a local search algorithm to find a solution with a small deviation of the

item demands. This process is repeated for different values of p, so as to choose the

most convenient one. To define a neighborhood for the local search, one cutting

pattern is replaced by another one from a set of cutting patterns generated by an

adaptive pattern generation algorithm. To evaluate each solution, the authors used

an heuristic based on the nonlinear Gauss-Seidel method.

In [187], Umetani et al. improved the metaheuristic proposed in [186]. They

proposed a local search algorithm for the 1D-CSP with an additional constraint

on the maximum number of different patterns. They call it the pattern restricted

problem (PRP). This new heuristic uses two different types of local search proce-

dures, in an iterated local search framework. The search of better solutions is made

in a neighborhood alternatively constructed by two different local search processes,

the 1-add neighborhood and the shift neighborhood. The metaheuristic process

consists of repeating the local search algorithm with different initial solutions: the

first one generated by a variant of the first-fit algorithm, and the following ones

randomly selected from the best current solution neighborhood (the shift neighbor-

hood) or by a randomized first-fit algorithm, in the cases where the previous one is

infeasible. To compute the pattern frequencies and evaluate a new solution, they

solve heuristically an auxiliary integer programming (IP) model. To further im-

prove the metaheuristic, some linear programming (LP) techniques are used, both

to reduce the number of candidate solutions in the neighborhood and to accelerate


the resolution of the auxiliary problems.

Yanasse and Limeira [205] proposed a three-phase hybrid procedure to solve

the PMP of any dimension. The first phase consists of finding “good” patterns

and corresponding activation levels by applying the Repeated Pattern Exhaustion

Technique (RPET). When RPET is no longer able to find good patterns, a residual

cutting problem is solved as a standard CSP. This residual problem deals with the

remaining items that these patterns do not consider. Finally, when the whole set of

cutting patterns is defined, reduction techniques from the literature are applied. In

their implementation, Yanasse and Limeira solved the residual problem rounding

to the nearest integer the fractional solution of the LP relaxation. The reduction

technique used for the third phase was KOMBI234 of Foerster and Wascher [88].

Cui et al. [60] proposed a sequential heuristic procedure (SHPC) for the PMP.

This procedure iteratively generates cutting patterns by solving a bounded knap-

sack problem, using a selection of items that have not yet been assigned. The best

cutting plan is also determined iteratively, by choosing the one with the minimum

number of different cutting patterns within the ones with minimum wastes. Ac-

cording to the authors, a strong point of this algorithm is the fact that it is simple

to implement and does not need other algorithms to generate cutting plans, as

other approaches do.

Very few publications in the literature report on results for exact solution pro-

cedures or lower bounds for the PMP. Vanderbeck [193] proposed the first exact

approach. The author presented a branch-and-price-and-cut algorithm that solved

to optimality 12 out of 16 real-world instances. His formulation is a Dantzig-Wolfe

decomposition of a compact nonlinear program, where each column represents a fea-

sible cutting pattern associated with a possible multiplicity. As the model implies

the enumeration of an exponentially growing number of columns and associated

binary variables, it is solved using a column generation technique, that implies

solving a quadratic integer subproblem. The author overcame this nonlinearity


by solving a set of bounded knapsack problems, one for each possible value of

the variable associated with the multiplicity of the pattern. In practice, as the

author described, not all the problems have to be solved since a given knapsack

solution may remain optimal for many successive multiplicities. To strengthen his

model, the author used a cutting plane scheme based on applying a superadditive

function to single rows of the restricted master problem. Vanderbeck proved that

the cuts are equivalent or dominate the rank 1 Chvtal-Gomory cuts. Futhermore,

they have no impact on the complexity of the pricing subproblems. The branching

scheme consists of enforcing a branching rule that is related with a special subset

of columns, selected from a list of seven possible subsets derived by the author.

In [7], Aloisio et al. showed that the model proposed by Vanderbeck may be

weaker than the model based on an extension of the Gilmore and Gomory [100]

for the CSP, which was formulated explicitly by Vanderbeck [193]. In [8], the same

authors addressed the PMP for special instances where no more than two items

fit in a stock roll. They explored two formulations for the problem, and derived

various results concerning the existence of specific solutions.

In [15], Alves and Carvalho proposed a branch-and-price-and-cut algorithm to

solve exactly the PMP, which is based on the model of Vanderbeck [193]. They

added a new constraint to the model bounding the total waste, which not only

strengthened the LP bound but also reduced the number of knapsack problems to

be solved in the column generation process. The authors used three dual-feasible

functions described by Fekete and Schepers [82] to derive valid inequalities. They

proved that the cuts given by these functions are equivalent to or dominate the

function used by Vanderbeck. In a further study, Clautiaux et al. [50] proved that

the function of Vanderbeck is not maximal, which means that it may be dominated

by other superadditive functions. Alves and Carvalho devised an algorithm in which

branching is performed on the variables of an arc-flow model (similar to the model

introduced in [188]), which is equivalent to the original model. This branching


scheme avoids symmetry in the branch-and-bound tree.

Belov [29] proposed a formulation for the PMP that considers a tradeoff be-

tween waste and pattern minimization by allowing the use of more rolls than the

ones used in the optimal solution of the corresponding CSP. This tradeoff is ob-

tained by introducing not only setup costs but also material costs in the objective

function. The author used the Gilmore and Gomory model for the CSP with setup

variables. To handle the huge number of constraints, he simplified it, turning it

into a nonlinear model, which he linearized by approximation. To solve the integer

problem, Belov described a branch-and-price algorithm, where the column gener-

ation procedure is identical to the one performed in [193]. The author devised a

branching scheme based on variables, and he enforced branching constraints on

single fractional variables. With this method, Belov solved exactly only 7 out of

the 16 instances presented by Vanderbeck, but he got, on average, better results

than KOMBI234 of Foerster and Wascher [88], on tests with 12 classes of instances.

In this chapter, we explore a different integer programming model that can be

solved using column generation, and describe different strategies to strengthen it,

among which are constraint programming (CP) and new families of valid inequal-

ities. We also describe new lower bounds for the PMP, derived from the new IP

model and also from the CP model. Our computational experiments performed

on a set of real instances from the literature show that these approaches provide

good lower bounds in a short amount of time for these instances. Additionally, we

compare the quality of our new bounds with the continuous bound given by the

model of Vanderbeck [193] for a large set of random benchmark instances also used

in the literature. This chapter is organized as follows. In section 4.2, we briefly

recall the models for the PMP that were described in the literature. In section 4.3,

we introduce a new IP model, and describe different strategies to strengthen it.

The CP model that we use to derive lower bounds and strengthen the new IP

model is described in section 4.4. Our computational experiments are reported in


section 4.5, while some conclusions are drawn in section 4.6.

4.2 Mathematical Programming Models: State-

of-the-art

The PMP consists of finding a cutting plan with the minimum number of distinct

patterns for a given optimal solution of the standard CSP, with a corresponding

value zCSP . We assume that only zCSP rolls can be used. We address the one-

dimensional version of the PMP (1D-PMP), where an instance of the problem is

defined by a set of m item sizes li with a respective demand of bi units, and a

set of rolls with identical sizes L. The total waste in the optimal solution of the

corresponding CSP is denoted by wCSP . Overproduction is not allowed, and hence

the item demands must be satisfied exactly. From this point forward, we will

assume the items to be sorted by decreasing order of their sizes.

Let z and z be, respectively, a lower and an upper bound on the number of

different patterns. A lower bound z can be computed by solving a bin packing

problem with the same item sizes as in the PMP. An upper bound z can be obtained

from the number of different patterns in the optimal solution of the corresponding

CSP.

Four distinct formulations were reported in the literature. In this section, we

briefly recall them. In the next section, we will present a new model that dominates

these formulations.

4.2.1 A Nonlinear Compact Formulation

The PMP can be formulated in a classical way using variables that represent the

assignment of items to patterns. The resulting model is compact but nonlinear. It

4.2. Mathematical Programming Models: State-of-the-art 53

was formally stated for the first time in [193], and it is defined as follows.

min

zCSP∑k=1

yk (4.1)

s.t.

zCSP∑k=1

zkxik = bi, ∀i ∈ 1, . . . ,m, (4.2)

zCSP∑k=1

zk ≤ zCSP , (4.3)

zk ≤ zCSPyk, ∀k ∈ 1, . . . , zCSP, (4.4)m∑i=1

lixik ≤ Lyk, ∀k ∈ 1, . . . , zCSP, (4.5)

xik ∈ N, ∀i ∈ 1, . . . ,m, ∀k ∈ 1, . . . , zCSP, (4.6)

yk ∈ 0, 1, ∀k ∈ 1, . . . , zCSP, (4.7)

zk ∈ N, ∀k ∈ 1, . . . , zCSP. (4.8)

The number of times an item i appears in a pattern k is represented by xik, while

the activation level of a pattern k is denoted by zk. The nonlinearities of this

model come from the demand constraints (4.2). The sum of all the activation

levels is equal to the number of rolls that are used. This value shall not exceed

zCSP (constraint (4.3)). The yk binary variables indicate if a pattern k is used or

not. They are defined through constraints (4.4), and are used both in the knapsack

constraints (4.6) and in the objective function (4.1).

4.2.2 The Gilmore and Gomory Model with Setup Vari-

ables

The well-known Gilmore and Gomory model for the CSP [100], where columns

represent feasible cutting patterns, can easily be extended to the PMP by adding

a setup variable for each column, to indicate if the pattern is used or not. This

model can be obtained by applying a Dantzig-Wolfe decomposition to (4.1)-(4.8)


in which (4.2)-(4.4) are dualized. The resulting master problem states as follows.

min∑k∈K

λk (4.9)

s.t.∑k∈K

aikλk = bi, ∀i ∈ 1, . . . ,m, (4.10)∑k∈K

xk ≤ zCSP , (4.11)

xk ≤ ukλk, ∀k ∈ K, (4.12)

xk ∈ N, ∀k ∈ K, (4.13)

λk ∈ 0, 1, ∀k ∈ K. (4.14)

The set of patterns is denoted by K. If a pattern k ∈ K is used, the variable λk

will be set equal to 1. The number of times a pattern k is used is denoted by xk.

Coefficients aik indicate the number of items i in pattern k. Coefficients uk, for k ∈

1, . . . , K, are upper bounds on pattern multiplicities. They strongly determine

the quality of the bounds given by the linear programming (LP) relaxation of

(4.9)-(4.14). When uk = zCSP , for k ∈ 1, . . . , K, the model is very weak: its

continuous bound is never greater than 1 ([193]). In [29], Belov showed that when

the uk = mini=1,...,m

⌊biaik

⌋, the model is as strong as the column generation model of

[193]. In a recent work, Aloisio et al. [7] showed that, in fact, for a given set of

parameters uk, formulation (4.9)-(4.14) may even be stronger than the model of

[193].

Model (4.9)-(4.14) has an exponential number of columns and constraints. In

[29], Belov solved a relaxation where constraints (4.12) are dropped, but he did not

get better results than [193].

4.2.3 The IP Model of Vanderbeck [193]

A different decomposition of (4.1)-(4.8) was explored in [193]. The author defined

his new model by dualizing only constraints (4.2) and (4.3). The resulting model

4.2. Mathematical Programming Models: State-of-the-art 55

has an exponential number of columns that represent patterns with an associated

multiplicity. The master problem is defined as follows.

min∑k∈K

uk∑n=1

λkn (4.15)

s.t.∑k∈K

uk∑n=1

naikλkn = bi, ∀i ∈ 1, . . . ,m, (4.16)

∑k∈K

uk∑n=1

nλkn ≤ zCSP , (4.17)

λkn ∈ 0, 1, ∀k ∈ K, ∀n ∈ 1, . . . , uk. (4.18)

The binary variables λkn indicate if a pattern k with multiplicity n is used or

not. The bounds on the uk parameters are set implicitly in this model. Given the

demand constraints (4.16), the maximum multiplicity uk of a pattern k will always

be smaller than or equal to mini=1,...,m

⌊biaik

⌋.

The pricing subproblem is a nonlinear knapsack problem. In [193], each sub-

problem was solved as a sequence of linear knapsack problems, one for each value

of the pattern multiplicity. For that purpose, an upper bound (nmax) on the value of

the multiplicities is necessary. In [193], the author set nmax = min zCSP − z + 1,maxi bi.

In practice, a complete enumeration of the multiplicities is not always necessary,

since a knapsack solution may remain optimal for many successive multiplicities.

Some improvements to model (4.15)-(4.18) were proposed in [15]. For example,

by considering the waste generated in an optimal solution of the corresponding

CSP, one may reduce the value of the uk parameters to

min

min

i=1,...,m

⌊biaik

⌋,

⌊wCSP

L−∑m

i=1 liaik

⌋.

4.2.4 An Arc-flow Formulation

In [15], the authors described a pseudo-polynomial formulation for the PMP based

on arc-flow variables.


The model is defined on a acyclic graph G = (V,A) with L + 1 nodes, each

one representing a discrete position within the rolls. For each multiplicity, there

is a set An of arcs (i, j) representing the placement of an item of size j − i at a

position i of the leftmost border of the roll in a pattern of multiplicity n. In this

model, a pattern of multiplicity n is modeled as a path from vertex 0 to vertex

L. Solving the PMP consists of finding the minimum flow over G, such that the

demand constraints are satisfied:

minnmax∑n=1

zn (4.19)

s.t.−∑

(r,s)∈An

xnrs +∑

(s,t)∈An

xnst

=

zn, if s = 0

−zn, if s = L

0, otherwise

, ∀n ∈ 1, . . . , nmax, (4.20)

nmax∑n=1

∑(r,r+li)∈An

nxnr,r+li = bi , ∀i ∈ 1, . . . ,m, (4.21)

xnrs ∈ N, n = 1, . . . , nmax, ∀(r, s) ∈ An, (4.22)

zn ∈ N, ∀n ∈ 1, . . . , nmax. (4.23)

The xnrs variables are general integer variables representing the flow over the arc

(r, s) of An. Variables zn, n ∈ 1, . . . , nmax, indicate the number of patterns of

multiplicity n that are used. The flow conservation constraints are stated in (4.20),

while (4.21) are the demand constraints. The authors use this model to derive a

robust branching scheme for their branch-and-price-and-cut algorithm.

4.3 Exploring a New IP Formulation

In this chapter, we elaborate on a new model for the PMP. The model can be

obtained by applying a different Dantzig-Wolfe decomposition to (4.1)-(4.8). We

4.3. Exploring a New IP Formulation 57

begin by introducing the model, and then we describe the different strategies that

may be used to strengthen it.

4.3.1 The Model

By dualizing only the demand constraints (4.2) of (4.1)-(4.8), and by treating

independently constraints (4.4)-(4.5) (with yk = 1) and constraint (4.3), we get

a reformulation of the problem. The decomposition is based on two independent

subproblems: the first one consists of choosing a combination of multiplicities,

while the second one consists of selecting patterns using these multiplicities. In the

master, the columns defined by these two subproblems are interconnected. The

new model states as follows:

min∑p∈P

dpδp (4.24)

s.t.∑p∈P

amultnp δp −∑k∈K

λkn = 0, ∀n ∈ 1, . . . , nmax, (4.25)

∑k∈K

uk∑n=1

naitemsikn λkn = bi,∀i ∈ 1, . . . ,m, (4.26)∑p∈P

δp = 1, (4.27)

λkn ∈ 0, 1,∀ k ∈ K, ∀n ∈ 1, . . . , uk, (4.28)

δp ∈ 0, 1,∀ p ∈ P. (4.29)

Variables λkn have the same meaning as in (4.15)-(4.18). We denote by P the set of

all the feasible combinations of multiplicities. The binary variables δp represent the

selection of a specific combination of multiplicities. A multiplicity n corresponds to

the usage of n rolls, and each combination of multiplicities must use exactly zCSP

rolls. Hence, only one combination can be used in a feasible solution (constraint

(4.27)). Constraints (4.25) are the linking constraints: for each multiplicity that is

used in the selected combination, there must be a corresponding cutting pattern

associated to it. Coefficients aitemsikn represent the number of times an item is con-


sidered in the cutting pattern k of multiplicity n. Coefficients amultnp indicate the

number of times the multiplicity of value n is considered in the combination p of

multiplicities. Coefficients dp represent the number of multiplicities that are in a

combination p.

Example 4.1 To illustrate the structure of (4.24)-(4.29), we use the kT05 instance

of [193]. In this instance, there are 10 different item sizes, the roll sizes are equal to

47244, and we have zCSP = 47 and nmax = 15. The set of item sizes and demands

(li, bi) is the following: (22098, 10); (21336, 10); (20574, 10); (19812, 5); (19050, 5);

(18288, 15); (16764, 10); (17018, 15); (13716, 25); (12192, 20). A restricted master

problem for this instance is given in table 4.1. The last row in this table represents

the objective function 4.19. Only the columns that correspond to combinations of

multiplicities have a positive coefficient in this row, representing the number of

multiplicities used in the corresponding combination.

Given the large number of columns in (4.24)-(4.29), the model shall be tackled

by enumerating implicitly its columns, and by generating dynamically the columns

that may improve the master. As referred to above, two independent subproblems

must be solved in order to price the attractive columns. Let SP1 refer to the

subproblem defined by constraint (4.3), and SP2 be the subproblem that results

from the discretization of (4.4)-(4.5). Problem SP2 is the same as the pricing

subproblem related to (4.15)-(4.18). In the next section, we will describe some

strategies to further improve its resolution.

Problem SP1 consists of finding a combination of multiplicities from a set of

successive integer values up to nmax, such that their sum is equal to zCSP . Let π be

the vector of dual variables associated to (4.25), and ρ be the dual variable related

to (4.27). To strengthen the LP-relaxation of the master, some constraints can be

enforced on the SP1 subproblem, in order to force combinations to have at least z

multiplicities and no more than z multiplicities. The optimization problem related


Table 4.1: Restricted master problem (example 4.1).

δ1 δ2 δ3 δ4 δ5 λ1,11 λ2,5 λ3,10 λ4,10 λ5,11 λ6,12 λ7,7 λ8,5 λ9,9 λ9,10 λ10,5 λ10,10 λ11,8

n = 1 = 02 = 03 = 04 = 05 1 2 2 3 1 -1 -1 -1 = 0

6 = 07 2 3 -1 = 08 4 1 -1 = 09 1 -1 = 010 1 2 2 1 -1 -1 -1 -1 = 0

11 1 1 -1 -1 = 012 1 1 -1 = 013 = 014 = 015 = 0

li= 22098 10 10 = 1021336 10 10 9 10 = 1020574 10 10 = 1019812 5 5 = 519050 5 5 = 518288 11 12 8 = 1517018 11 14 = 1516764 8 = 1013716 11 22 24 10 9 10 10 = 2512192 11 7 9 10 8 = 20

1 1 1 1 1 = 1

O.F. 6 6 6 6 6

to SP1 states as follows:

minnmax∑n=1

qn −nmax∑n=1

πnqn − ρ (4.30)

s.t.nmax∑n=1

nqn = zCSP , (4.31)

nmax∑n=1

qn ≥ z, (4.32)

nmax∑n=1

qn ≤ z, (4.33)

qn ∈ N. (4.34)


Variables qn represent the number of times a multiplicity of value n is considered

in the combination. Note that SP1 can be solved by using dynamic programming

in O(nmaxzCSP z) time. To reduce its complexity, one may consider to solve SP1

without constraints (4.32) and (4.33). In this case, the problem can be solved

in O(nmaxzCSP ) time, but the continuous bound provided by (4.24)-(4.29) may

become weaker.

Model (4.24)-(4.29) is at least as strong as (4.15)-(4.18), since it is obtained by

dualizing less constraints and SP1 does not have the integrality property. It also

has a pseudo-polynomial number of constraints since its size depends on the value

of nmax, the maximum multiplicity among all the cutting patterns. In [193, 15], the

authors showed how an upper bound on this value could be computed. In the next

section, we will propose new approaches to further reduce this upper bound. An

interesting characteristic of (4.24)-(4.29) is that we clearly know which combination

of multiplicities is being used by a fractional solution, and that allows us to derive

schemes to exclude infeasible combinations of multiplicities.

Note that model (4.24)-(4.29) can be easily extended to the case where there

are multiple stock sizes. For each stock size, there will be a set of constraints of

type (4.25). Instead of using the minimum number of rolls (zCSP ), one would have

to consider the solution of the multiple size CSP, which is the minimum total size

of stock rolls that is necessary to cut all the items. This value would be used in

SP1. Some of the results presented below can be directly extended to the case of

multiple stock sizes. This is the case of inequalities (4.37), (4.38) and (4.40). Other

aspects, like some of the upper bounds on the amultnp coefficients would not apply

directly. Clearly, for large instances, with a large number of different stock sizes,

the model might become very large in terms of the number of rows and columns.

Hence, in this chapter, we focus on the case of identical stock sizes.


4.3.2 Strengthening the Model

Reducing the Set of Columns Generated by SP1

The LP relaxation of (4.24)-(4.29) can be tightened by reducing the size of the

solution space of SP1. We report on bounds on the amultnp coefficients, and on

bounds on the number of rolls and patterns to be used up to a given multiplicity.

One way of strengthening this formulation is to bound the number of times a

given multiplicity can appear in a combination. Let this upper bound be denoted

by ub(amultnp ). A trivial upper bound on amultnp is given by⌊zCSP

n

⌋, ∀n, p. This bound

is implicitly enforced in SP1. Another obvious bound is the following: amultnp ≤ z,

∀n, p. These two bounds can easily be strengthened for some multiplicities using the

following arguments. Clearly, a combination of multiplicities cannot be composed

only by a single value n ≤ bmin, with bmin being the smallest item demand, if all

the item demands are not divisible by n. Hence, if n does not divide all the item

demands, then amultnp ≤⌊zCSP

n

⌋− 1, ∀p. Furthermore, since an item with demand bi

can only appear in patterns of multiplicity smaller than or equal to bi, for a given

multiplicity n, if there are items i with bi < n, then amultnp ≤ z − 1, ∀p.

Let In be the set of items i such that bi ≥ n, for a given multiplicity n. The

cutting patterns with multiplicity n can only have items from In. In the worst case,

each cutting pattern of multiplicity n will be defined by a single item which belongs

to In. In that case, there will be no more than∑

i∈In

⌊bin

⌋patterns of multiplicity

n, and hence amultnp ≤∑

i∈In

⌊bin

⌋, ∀n, p.

For the highest multiplicities, it may be possible to set the amultnp equal to 0 if

one takes into account the minimum waste that will be generated in the respective

patterns. As a consequence, the value of nmax may also decrease. Let i1 and i2 be

the items with the highest and second highest demand respectively. If bi2 ≤ nmax,

then for bi2 < n ≤ bi1 , we have that |In| = 1, which means that only items with

the same size li1 can be in patterns of multiplicity n. Let ain = min⌊

Lli

⌋,⌊bin

⌋.


If n × (L − ai1nli1) > wCSP , then amultnp = 0, ∀p. This result can be extended to

other multiplicities, but at the expense of an increase in complexity. Indeed, for

n ≤ bi2 , since |In| ≥ 2, we have to compute the minimum waste of patterns with

multiplicity n by solving, for example, the following knapsack problem:

max znmax =∑i∈In

lixi

s.t.∑i∈In

lixi ≤ L,

xi ≤⌊bin

⌋, ∀i ∈ In,

xi ∈ N.

The set In is defined as above for a given multiplicity n, while the xi variables

indicate the number of items i in the knapsack. The minimum waste of a pattern

with multiplicity n (say wnmin) will be given by L−znmax. Again, if n×wnmin > wCSP ,

then amultnp = 0, ∀p.

Note that if one computes the wnmin for n = 1 up to nmax, by using the minimum

among all these values (say wmin = minn=1,...,nmax wnmin), we can enforce constraints

on the cutting patterns generated by SP2 that may be stronger than constraint

(17) proposed in [15]:

L−m∑i=1

liaitemsikn ≤

⌊w − (wmin(zCSP − n))

n

⌋, ∀k, n, (4.35)

being w = zCSPL −∑m

i=1 libi the maximum waste of a given solution. Another

bound may be derived from the idea that a multiplicity should not be used so many

times that it will not be possible to complete the combination (whose values must

sum zCSP ) with no more than z multiplicities, without using a multiplicity greater

than nmax. If ub(amultnp ) is the best upper bound available for amultnp , ∀p, then we will

have

zCSP − ub(amultnp )n

z − ub(amultnp )≤ nmax,


and hence, the following upper bound applies:

amultnp ≤⌊nmaxz − zCSPnmax − n

⌋, ∀n, p. (4.36)

This bound is useful only if one does not explicitly enforce an upper bound (z) on

the dp coefficients in SP1 for a given pattern p. Otherwise, this bound is modeled

implicitly.

The final value for the upper bound ub(amultnp ) consists in the minimum value

among all the bounds described above.

As referred to above, since overproduction is not allowed, the demand bi of an

item i can only be satisfied from patterns with a multiplicity n ≤ bi. Let ri be a

lower bound on the number of rolls that are necessary to cut all the items with

a demand smaller than or equal to bi, and m′ be the number of different item

demands. These demands are assumed to be ordered in nonincreasing order. A

combination p of multiplicities will be feasible only if it satisfies

minbi,nmax∑n=1

namultnp ≥ ri, ∀i ∈ 1, . . . ,m′. (4.37)

Note that enforcing these constraints in SP1 does not increase its complexity. The

problem can still be solved as a knapsack problem with some additional constraints

in O(nmaxzCSP ) time, using dynamic programming.

Similarly, if pi is a lower bound on the number of distinct patterns necessary to

cut all the items with a demand smaller than or equal to bi, then a combination p

of multiplicities will be feasible only if

minbi,nmax∑n=1

amultnp ≥ pi, ∀i ∈ 1, . . . ,m. (4.38)

If we consider a whole combination p of multiplicities, constraint (4.38) stands as

follows:

nmax∑n=1

amultnp ≥ z. (4.39)


These constraints force to impose a set of cardinality constraints on SP1. In that

case, SP1 can be solved in O(nmaxzCSP z) time.

Bounds on the aitemsikn Coefficients

It is possible to bound from below the value of the aitemsikn coefficients. Indeed, in

each roll, we know that there are at most ai =⌊Lli

⌋items i. As a consequence, to

satisfy the demand of items i, we will need at least⌈biai

⌉rolls. This means that if a

pattern k of multiplicity n = zCSP −⌈biai

⌉is used, it must contain at least one item

i (aitemsikn ≥ 1, ∀k). Whenever there is a positive lower bound on one or more aitemsikn

coefficients for a given multiplicity n, the right-hand side of the knapsack constraint

in SP2 can be decreased accordingly. On the other hand, this lower bound must

be checked when computing the highest multiplicity for which a solution to the

knapsack problems related to SP2 remains optimal [193].

We can also strengthen the upper bound ub(aitemsikn ) proposed in [193] for aitemsikn ,

∀k, and for a given multiplicity n. This bound is given by

ub(aitemsikn ) = min

⌊L

li

⌋,

⌊bin

⌋.

For a given multiplicity n and an item i, when aitemsikn is set equal to its upper

bound, the rolls divide into two subsets: zCSP −n rolls of size L, and n rolls of size

L−ub(aitemsikn )li. The remaining items can also be separated in two groups: one with

the item sizes that fit on both types of rolls and a second with the item sizes that

only fit in the largest rolls. By computing a bound on the number of rolls necessary

to cut the items from the second group, we can check if the upper bound ub(aitemsikn )

is set too high or not. Indeed, if this bound is greater than zCSP − n, then there

will be no feasible solution with aitemsikn = ub(aitemsikn ). In this case, the bound can

be decreased by at least one unit. Clearly, updating these upper bounds for each

item and multiplicity may be computationally expensive in some cases. However,

this method can always be applied to a restricted subset of items. As above, this

4.4. A CP Model 65

upper bound must be checked whenever we look for the highest multiplicity that

guarantees the optimality of a knapsack solution for SP2.

A Family of Valid Inequalities

When the demand of an item is odd, in any integer solution for (4.24)-(4.29), there

will be at least one pattern k of some multiplicity n such that n × aitemsikn is odd.

However, when the integrality constraints are relaxed in (4.24)-(4.29), the item

demands with odd values may be satisfied in (4.26) through a linear combination

of coefficients n×aitemsikn with even values. Let Codd be the set of cutting patterns k of

multiplicity n in (4.24)-(4.29) such that n×aitemsikn is odd. The following inequalities

are valid for (4.24)-(4.29)

∑Codd

λkn ≥ 1, ∀i : bi mod 2 = 1. (4.40)

Constraints (4.40) shall be enforced in the LP-relaxation of (4.24)-(4.29) for each

item with a demand that is odd. As we will see in section 4.5, these constraints

effectively cut infeasible solutions, leading to a tighter formulation. Even when we

enforce constraints (4.40), the knapsack problems related to SP2 remain solvable

in O(mW ) time by using a dynamic programming procedure.

Forcing the item demands to be satisfied through a combination of appropriate

n× aitemsikn terms is an important issue that will be explored below.

4.4 A CP Model

In this section, we describe a CP model that can be used to derive both lower

bounds for the PMP, and valid inequalities for SP1 in model (4.24)-(4.29). The

elements of the model are presented in section 7.3.2. The procedure for computing

lower bounds for the PMP using this CP model is described in section 4.4.2, and

the valid inequalities for the SP1 are discussed in section 4.4.3.


4.4.1 The Model

Let X be the set of multiplicities considered in a given combination, and let Xj,

∀j ∈ 1, . . . , z, denote the value of the jth multiplicity in X. We assume the

multiplicities to be ordered in nondecreasing order, except for the last ones which

may be 0 if less than z multiplicities are considered. With this scheme, we know

that X1 is the first multiplicity in the combination, X2 the second, and so on. The

constraints of our CP model state as follows:

Xj ≤ Xj+1 ∨Xj+1 = 0, ∀j ∈ 1, . . . , z − 1, (4.41)

Xj = 0⇒ Xj+1 = 0, ∀j ∈ 1, . . . , z − 1, (4.42)z∑j=1

AijXj = bi, ∀i ∈ 1, . . . ,m, (4.43)

z∑j=1

Xj = zCSP , (4.44)

count(X,n) ≤ ub(amultnp ), ∀n ∈ 1, . . . , nmax, (4.45)

minbi,nmax∑n=1

n count(X,n) ≥ ri, ∀i ∈ 1, . . . ,m′, (4.46)

minbi,nmax∑n=1

count(X,n) ≥ pi, ∀i ∈ 1, . . . ,m. (4.47)

Constraints (4.41) and (4.42) define the ordering of the Xj variables. Con-

straints (4.43) state that the combination X of multiplicities must be such that

each item demand can be expressed as a linear integer combination of the corre-

sponding Xj, ∀j ∈ 1, . . . , z. The variables Aij are general integer variables that

represent the number of times a multiplicity j should be used so as to recover the

value of the demand bi. A valid upper bound on the Aij variables is⌊Lli

⌋, ∀j. Con-

straint (4.44) forces the use of exactly zCSP rolls, i.e., the sum of all the selected

multiplicities, which represents the total number of rolls used, must be equal to

zCSP . We also incorporate in the CP model the bounds on the amultnp coefficients

4.4. A CP Model 67

and the constraints (4.37) and (4.38) described in section 4.3.2. Given that the

values of ub(amultnp ) are independent of p, constraints (4.45) hold. The counting ex-

pression count(X,n) accounts for the number of elements in X that are equal to n.

This expression is also used to express constraints (4.37) and (4.38), respectively,

through constraints (4.46) and (4.47).

By definition, the domain of the Xj variables is in [1, nmax] for j ∈ 1, . . . , z,

and in [0, nmax] for ∀j ∈ z + 1, . . . , nmax. To improve the solution of the model,

we may reduce the domain of these variables by taking into account constraints

(4.37). For example, given that the items with the smallest demand will only

appear on patterns with a multiplicity smaller than or equal to that demand, the

first multiplicity X1 will never be greater than this smallest demand, and hence,

we have X1 ≤ mini=1,...,m

bi. Generally, we have

Xj ≤ bi, ∀i ∈ 1, . . . ,m′, j ∈

1, . . . ,

⌈ribi

⌉.

Our CP model can be seen as a relaxation of the compact model (4.1)-(4.8) with

additional constraints. The demand constraints (4.2) and (4.3) are enforced in our

CP model, while the remaining constraints of (4.1)-(4.8) are relaxed. In addition,

we enforce the bounds on the amultnp coefficients together with constraints (4.37)

and (4.38) described above. These constraints are easily enforced in the CP model,

while integrating them in (4.1)-(4.8)would increase the size of the model, leading to

a pseudo-polynomial number of variables and constraints. In particular, to model

exactly constraints (4.37) in (4.1)-(4.8), we would need to define additional binary

variables ykn (one for each multiplicity n up to nmax) that take the value 1 only if

pattern k is used n times. The following constraints would complete the model:

zk = nykn, ∀k ∈ 1, . . . , zCSP,∀n ∈ 1, . . . , nmax,bi∑n=1

nykn ≥ ri, ∀i ∈ 1, . . . ,m,

ykn ∈ 0, 1, ∀k ∈ 1, . . . , zCSP,∀n ∈ 1, . . . , nmax.


Constraints (4.43) may be linked to the demand constraints (4.2), while the vari-

ables Aij may be interpreted as the number of times an item i is considered in the

jth pattern of multiplicity Xj. However, note that we relax the knapsack constraints

(4.5) of (4.1)-(4.8), which are the strongest constraints on the assignment of items

to rolls. That allows us to focus on the relation between the values of the item

demands and the combination of multiplicities, and to apply different procedures

to reduce the number of constraints (4.43) and variables Aij. First, we only need

to consider in (4.43) the largest item among those with the same demand. Without

loss of generality, assume that two items s and t with ls ≥ lt have the same value

of demand (bs = bt). Constraints (4.43) for these two items are as follows.

As1X1 + As2X2 + As3X3 + . . .+ AszXz = bs (4.48)

At1X1 + At2X2 + At3X3 + . . .+ AtzXz = bt (4.49)

Clearly, given that Asj ≤⌊Lls

⌋≤⌊Llt

⌋, if (4.48) is satisfied, the same will happen

with (4.49). The item demands that are equal to the sum of two (or more) other

item demands may also be excluded from constraints (4.43) provided some condi-

tions are fulfilled. For example, assume that we are given three items r, s and t

such that br = bs + bt. For these items, constraints (4.43) stand as follows.

Ar1X1 + Ar2X2 + Ar3X3 + . . .+ ArzXz = br (4.50)

As1X1 + As2X2 + As3X3 + . . .+ AszXz = bs (4.51)

At1X1 + At2X2 + At3X3 + . . .+ AtzXz = bt (4.52)

If⌊Llr

⌋≥⌊Lls

⌋+⌊Llt

⌋, then any combination X that satisfies (4.51) and (4.52) will

satisfy also (4.50). The values of the Arj variables can be obtained by summing

Asj with Atj, ∀j. Hence, we do not have to consider constraint (4.50) and the

corresponding Arj variables in our CP model.

4.4. A CP Model 69

4.4.2 Deriving Lower Bounds from the CP Model

As we mentioned above, a lower bound for the PMP can be computed by solving

a corresponding bin packing problem (with all the demands equal to one) with

the same set of item sizes, and demands that are equal to 1. It is so because

the solution to a bin packing problem corresponds also to the minimum number

of different patterns that are needed to pack the items. These bounds can be

computed efficiently (see for example [50]). We will denote this specific bin packing

lower bound by z, as we already did above.

We can improve this lower bound by using the CP model described above. For

this purpose, we solve a sequence of constraint satisfaction problems. The first step

consists of solving a constraint satisfaction problem based on the previous model

with the additional constraint Xz+1 = 0. By solving this problem, we try to find

out if there is a solution with at most z multiplicities (positive elements in X) that

satisfies all the constraints of the model. If there is no such a solution, then the

lower bound can be increased by one unit, i.e., z will take the value z + 1. The

process is repeated until a feasible solution is found for the constraint satisfaction

problem. In that case, the procedure ends with a lower bound given by z.

4.4.3 Using the CP to Derive Valid Inequalities for SP1

Many combinations of multiplicities in (4.24)-(4.29) are clearly not interesting,

namely those from which we cannot express all the item demands as a linear in-

teger combination of the multiplicities. In example 4.1, a possible combination

of multiplicities could correspond to using 5 times the multiplicity with a value

equal to 9, and once the multiplicity with a value equal to 2. In table 4.1, the

corresponding column for the rows associated to the multiplicities will be defined

as (0, 1, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0)T . The issue is that there is no integer solution

for the problem of determining how many times the item with size 13716 should


appear in each one of these six different patterns so as to fulfill the correspond-

ing demand, which is of 25 units. Clearly, this demand cannot be expressed as

a linear integer combination of these multiplicities. However, this combination of

multiplicities is feasible for the LP-relaxation of (4.24)-(4.29).

It is difficult to force the combinations of multiplicities in (4.24)-(4.29) to be

such that each item demand can be expressed as a linear integer combination of

the multiplicities. In fact, these constraints are typically nonlinear, and hence they

cannot be considered explicitly in SP1. Our CP model can be used to handle

these hard constraints, and to derive valid inequalities for SP1. The development

of hybrid methods combining CP approaches with IP based techniques has been a

major topic of research in the recent years [119, 157]. These techniques complement

each other, CP being traditionally used to deal with the complicated constraints

that cannot be expressed in a linear form. In our case, solving the related constraint

satisfaction problems will allow us to derive bounds on the variables of SP1.

We used the ILOG CP Optimizer 1.0 to solve this CP model with the objective

of further reducing the domain of the X variables. These domains translate into

valid inequalities in SP1. For this purpose, we enforce additional constraints on

the X variables, and test the satisfiability of the problem. Assume that the domain

of Xj is in [ldj, udj], j = 1, . . . , z. We add a constraint with the form Xj ≥ h, for

some j. If the problem is unsatisfiable, we can update the upper bound on Xj

accordingly setting udj = h − 1. Of course, a complete search may be computa-

tionally expensive. It would require solving O(z log2 nmax) constraint satisfaction

problems. However, the computational experiments that we conducted show that

formulation (4.24)-(4.29) may be strengthened even with a few inequalities on a

subset of the X variables.

4.5. Comparative Computational Results 71

4.5 Comparative Computational Results

To evaluate the quality of the lower bounds given by the LP-relaxation of (4.24)-

(4.29) and by the procedure described in section 4.4.2, we conducted a set of

computational experiments on benchmark instances from the literature. A first set

of experiments were conducted on the real instances used in [193]. We discuss the

corresponding results in the first part of this section. In a second set of experiments,

we compare the lower bounds obtained with the different approaches described in

this chapter for 360 random instances used in [88, 205, 187, 60]. The results from

these experiments are presented in the second part of this section.

The algorithms were coded in C++, and the experiments were performed on

a PC with a 2.20GHz Intel Core Duo processor, and 2GB of RAM. We used the

CPLEX 10.2 Callable Library to implement some optimization subroutines. As

referred to above, the CP model was solved using ILOG CP Optimizer 1.0.

In table 4.2, we list the names of the instances together with the number m of

different item sizes and the value z of the corresponding bin-packing lower bound

for each instance used in [193]. The entry Inst stands for the name of the instance.

We report on three sets of experiments. In the first one, we solved the LP-

relaxation of (4.24)-(4.29) by using dynamic column generation with all the strate-

gies described in section 4.3.2, except the cutting planes (4.40). These cuts are

considered in the second set of experiments. Finally, in the third set, we assess

the performance of the procedure described in section 4.4.2. In all the cases, our

results are compared with the results obtained with the IP model (4.15)-(4.18).

The entries of the forthcoming tables have the following meaning:

zCG continuous lower bound given by the LP-relaxation of (4.15)-

(4.18);

tCG solution time for the LP-relaxation of (4.15)-(4.18)(in seconds);


zCGnew continuous lower bound given by the LP-relaxation of (4.24)-

(4.29);

tCGnew solution time for the LP-relaxation of (4.24)-(4.29)(in seconds);

zCGcutsnew

continuous lower bound given by the LP-relaxation of (4.24)-(4.29)

with the cutting planes (4.40);

tCGcutsnew

solution time for the LP-relaxation of (4.24)-(4.29)(in seconds)

with the cutting planes (4.40);

zCP lower bound given by the procedure described in section 4.4.2;

tCP computing time for the procedure described in section 4.4.2(in sec-

onds);

best a * in this column identifies an instance for which the new bound

is better than the continuous bound obtained with model (4.15)-

(4.18).

Table 4.4 describes the results of the first set of experiments. We enforced a

minimum number z and maximum number z of multiplicities on the combinations

generated by SP1. The scheme presented in section 4.4.3 was used to derive bounds

for the first multiplicities of these combinations. In each column generation itera-

tion, we generate the most attractive column from SP1 and the set of attractive

columns of SP2. The values of the ri coefficients in (4.37) are computed using the

L2 bound of Martello and Toth [153].

In most of the cases, the continuous bound of (4.24)-(4.29) is greater than the

bound given by the model of [193]. For six instances, the integer bound is also

improved. Furthermore, despite the fact that model (4.24)-(4.29) is larger than

(4.15)-(4.18), convergence seems to be improved since the former takes on average

nearly less than 28% of the time to be solved.

The valid inequalities for SP1 derived by using the procedure described in

section 4.4.3 may have a nonnegligible impact on the quality of the lower bounds.


Table 4.2: Benchmark instances [193].

Inst. m z

1 kT03 7 32 kT05 10 43 kT01 5 14 kT02 24 135 kT04 16 66 d16p6 16 67 7p18 7 28 d33p20 23 59 12p19 12 210 d43p21 32 711 kT06 9 112 kT07 11 213 14p12 14 214 kT09 14 215 11p4 11 116 30p0 26 4

To evaluate this impact, we repeated the first set of experiments on the instances of

[193] without deriving these inequalities. The bounds of two instances (kT06 and

11p4) decreases when these inequalities are not used. The continuous bound for

kT06 is 2.0 (instead of 3.0 with the inequalities), while the bound of 11p4 becomes

3.0 (instead of 3.003). In both cases, the improvement was enough to reach the next

integer. Note that these results were achieved by deriving only a few inequalities

on the first multiplicities. These inequalities allow to cut many combinations of

multiplicities that would never be used in any integer solution. By reducing the

number of columns in the master problem, they may lead to stronger models.

When the set of inequalities (4.40) is considered, one has to solve explicitly

SP2 for each value of the multiplicities. Hence, in our implementation, we solved

first the model (4.24)-(4.29) without these cuts. The cuts are added to the LP

master once there are no more attractive columns. The results obtained are given

in table 4.5.


Table 4.4: Comparing the new IP model with the model of [193].

Inst. zCG tCG zCGnew tCGnew Best

1 kT03 4.77 0.06 5.00 0.032 kT05 5.65 0.61 5.65 0.253 kT01 2.10 0.05 2.10 0.014 kT02 15.93 0.12 15.93 0.255 kT04 6.74 1.08 7.00 0.646 d16p6 6.74 1.19 7.00 0.757 7p18 3.74 0.69 4.00 1.208 d33p20 6.18 2.85 7.00 1.789 12p19 2.89 4.54 3.10 1.93 *10 d43p21 7.86 39.39 8.41 19.09 *11 kT06 1.75 18.38 3.00 5.19 *12 kT07 2.86 11.43 3.06 19.72 *13 14p12 3.75 8.20 4.07 5.65 *14 kT09 3.65 47.18 4.00 50.4315 11p4 2.48 21.63 3.003 41.93 *16 30p0 5.51 120.21 6.00 49.56

Avg. 17.35 12.40

With the cuts (4.40), the integer lower bound is improved for 10 instances.

The average computing time increases, but it remains slightly under the average

computing time necessary to solve the model of [193].

We report in table 4.6 the last set of results, which are related to the procedure

described in section 4.4.2.

The results obtained with this procedure are quite impressive. For 12 instances,

the lower bound is at least equal to the bound given by the LP-relaxation of model

(4.15)-(4.18). In three cases, the integer bound is improved. These results were

obtained almost 8 times faster than the time it takes to solve the model of [193].

The second set of experiments were conducted on instances from the literature

[88, 205, 187, 60]. These instances were randomly generated using CUTGEN1 [94].

There are 18 classes of 100 instances each. In our experiments, we used the first

20 instances in each class. Table 4.7 describes the main characteristics of these


Table 4.5: Comparing the new IP model with cuts (4.40) with the model of [193].

Inst. zCG tCG zCGcutsnew

tCGcutsnew

Best

1 kT03 4.77 0.06 5.00 0.052 kT05 5.65 0.61 6.71 0.75 *3 kT01 2.10 0.05 2.50 0.014 kT02 15.93 0.12 16.16 0.34 *5 kT04 6.74 1.08 7.02 1.22 *6 d16p6 6.74 1.19 7.02 1.34 *7 7p18 3.74 0.69 4.00 1.378 d33p20 6.18 2.85 7.00 1.869 12p19 2.89 4.54 3.20 3.06 *10 d43p21 7.86 39.39 8.42 25.48 *11 kT06 1.75 18.38 3.00 6.70 *12 kT07 2.86 11.43 3.08 28.43 *13 14p12 3.75 8.20 4.33 8.93 *14 kT09 3.65 47.18 4.00 62.0915 11p4 2.48 21.63 3.01 49.70 *16 30p0 5.51 120.21 6.00 80.08

Avg. 17.35 16.96

instances. The entries m and b are the average number of items and the average

demand in each class, respectively. The entries v1 and v2 determine, respectively,

the size of the smallest and largest item compared to the size of the rolls. The size

of the items will be in the interval [v1L, v2L].

In table 4.8, we compare the lower bounds obtained with the three strategies

used in our first set of experiments. The entries zCG, zCGnew and zCGcutsnew

have the

same meaning as in the previous tables. They correspond now to the average values

of the 20 instances in each class. Column zCSP represents the average number of

rolls used. The average lower bound over the 20 instances in each class is given

in column BPP . This bound corresponds to the L2 bound of Martello and Toth

[153] for the corresponding bin packing instance. The percentage improvement of

the lower bound for each approach compared to the continuous bound provided by

model (4.15)-(4.18) is given in columns imp.%. The number of instances for which


Table 4.6: Deriving lower bounds using the CP model.

Inst. zCG tCG zCP tCP Best

1 kT03 4.77 0.06 5.00 0.472 kT05 5.65 0.61 5.00 0.193 kT01 2.10 0.05 2.00 0.014 kT02 15.93 0.12 14.00 0.195 kT04 6.74 1.08 7.00 0.016 d16p6 6.74 1.19 7.00 0.027 7p18 3.74 0.69 4.00 0.458 d33p20 6.18 2.85 6.00 0.119 12p19 2.89 4.54 4.00 0.06 *10 d43p21 7.86 39.39 8.00 0.2311 kT06 1.75 18.38 3.00 0.17 *12 kT07 2.86 11.43 3.00 0.4513 14p12 3.75 8.20 4.00 0.3714 kT09 3.65 47.18 4.00 0.8715 11p4 2.48 21.63 4.00 3.78 *16 30p0 5.51 120.21 6.00 26.66

Avg. 17.35 2.13

the value of the integer bound was improved is reported in the columns imp.lb.

For the CP approach, we also report the number of instances for which the integer

bound was equal to the bound given by the LP relaxation of (4.15)-(4.18).

The results listed in table 4.8 show a clear improvement of the results for the

first six classes. The percentage of improvement of the lower bounds is up to

almost 27% when the new IP is considered. The CP model also performs well on

these instances. The corresponding bounds are never smaller than the continuous

bounds given by the model of Vanderbeck [193]. There are also improvements for

the remaining instances but to a smaller extent. For large values of zCSP , the

difficulty of solving the new IP model increases. The number of combinations of

multiplicities may become larger, thus impacting both on the convergence and the

quality of the lower bounds.

4.6. Conclusions 77

Table 4.7: Benchmark instances randomly generated using CUTGEN1.

m v1 v2 b1 10 0.01 0.2 102 10 0.01 0.2 1003 20 0.01 0.2 104 20 0.01 0.2 1005 40 0.01 0.2 106 40 0.01 0.2 100

7 10 0.01 0.8 108 10 0.01 0.8 1009 20 0.01 0.8 1010 20 0.01 0.8 10011 40 0.01 0.8 1012 40 0.01 0.8 100

13 10 0.02 0.8 1014 10 0.02 0.8 10015 20 0.02 0.8 1016 20 0.02 0.8 10017 40 0.02 0.8 1018 40 0.02 0.8 100

4.6 Conclusions

The PMP is a very challenging problem that has been approached in several pub-

lications in the literature, although most of them rely on heuristic methods. The

exact solution algorithms proposed so far still have difficulties in solving medium

size instances. With the methods described in this chapter, we intend to contribute

with new results concerning the computation of lower bounds for the PMP. We pro-

pose a new IP model, and different strategies to strengthen it, among which are

the use of a CP model and new families of valid inequalities. Our computational

experiments were conducted on two sets of benchmark instances from the litera-

ture. The results show an improvement in the quality of the lower bounds, and in

the convergence of the methods for some instances.


Tab

le4.8:

Com

parin

gth

elow

erb

ounds

forth

eran

dom

lygen

eratedin

stances.

zCSP

BP

PzCG

zCG

new

imp.%

imp.

lbzCG

cuts

new

imp.%

imp.

lbzCP

imp.%

imp.

lbeq

.lb

111.30

2.002.26

2.8626.40

22.86

26.453

2.9028.35

218

2108.60

2.102.47

3.0824.67

113.14

27.0312

3.9559.83

182

323.15

2.853.33

4.0020.09

34.01

20.133

4.0019.95

218

4226.05

2.803.69

4.2314.43

54.24

14.766

4.9032.70

146

542.15

4.205.00

5.6713.44

45.71

14.124

5.8015.95

416

6417.55

4.205.41

5.868.43

115.88

8.7312

6.2515.57

1010

748.35

4.955.87

6.327.57

66.36

8.347

6.256.39

312

8482.75

4.956.05

6.395.66

46.52

7.696

6.8012.36

78

998.80

9.4011.02

11.231.91

311.33

2.835

11.00-0.20

010

10983.75

9.4011.44

11.641.73

011.72

2.481

10.85-5.13

07

11171.80

16.3018.80

18.950.81

319.72

1.707

18.15-3.45

05

121715.05

16.3019.60

19.740.71

219.79

0.983

17.55-10.45

00

1362.15

6.106.91

7.427.27

57.49

8.286

7.305.58

115

14620.35

6.107.05

7.536.79

57.60

7.675

7.709.15

414

15125.15

11.9513.62

13.801.32

113.91

2.152

13.45-1.25

07

161248.25

11.9513.80

14.071.95

214.10

2.203

13.05-5.41

04

17219.55

20.7523.65

23.740.36

124.02

1.567

22.85-3.39

03

182191.10

20.7524.12

24.320.82

124.32

0.831

22.10-8.39

01

Chapter 5

Models and Bounds for the

Two-Dimensional Guillotine

Cutting Stock Problem and its

Variants 1

“Divide each difficulty into as many parts

as is feasible and necessary to resolve it.”

René Descartes


5.2 Network Flow Model . . . . . . . . . . . . . . . . . . . . 84

5.3 Arc Reduction . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 A New Family of Cutting Planes . . . . . . . . . . . . . 89

5.6 A New Lower Bound . . . . . . . . . . . . . . . . . . . . 90

5.7 Variants of the Problem . . . . . . . . . . . . . . . . . . 90

5.7.1 The Non-Oriented Case . . . . . . . . . . . . . . . . . . 90

5.7.2 Orientation of the First Stage’s Cuts . . . . . . . . . . . 91

5.8 Computational Results . . . . . . . . . . . . . . . . . . . 94

5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 114

1The work presented in this chapter has been published in [147]

79

80 5. Models and Bounds for the 2D-CSP and its Variants

5.1 Introduction

In this chapter, we describe an exact model for the 2D-CSP with two stages and

the guillotine constraint. It is an IP network flow model, formulated as a minimum

flow problem, which is an extension of a model proposed by Valerio de Carvalho

for the one-dimensional case [188], which we have described in chapter 3. In this

chapter, we explore the behavior of this model when it is solved with a commercial

software, explicitly considering all its variables and constraints. We also derive a

new family of cutting planes and a new lower bound, and consider some variants

of the original problem. The model was tested on a set of real instances from the

wood industry, with very good results. Furthermore, the lower bounds provided by

the LP-relaxation of the model compare favorably with the lower bounds provided

by a model based on assignment variables.

We focus on the exact solution of the two-stage, non-exact 2D-CSP with the

guillotine constraint (figure 5.1). Some exact solution methods concerning this

problem are described in the literature.

Gilmore and Gomory [102] proposed the first model for the 2D-CSP, by ex-

tending their column generation approach to 1D-CSP. They proposed an integer

programming approach [100, 101], with a column generation technique, to solve the

one-dimensional version of the CSP, as described in section 3.2. They introduce a

variable for each feasible cutting pattern, but, as they increase exponentially, not all

of them are explicitly considered. Instead, they use a column generation approach

to determine, at each step, promising patterns (columns) that might improve a cur-

rent solution. This auxiliary problem, a knapsack problem, uses the shadow prices

as the coefficients of the objective function. Later on, they extended this approach

to solve the two-dimensional guillotine version of the problem [102]. They approach

it as a two-stage one-dimensional problem, with the first stage corresponding to

the cutting of the stock sheets into strips, and the second stage corresponding to


the cutting of those strips into the requested heights (see section 3.3).

Amossen [17] described a branch-and-bound algorithm for dBPP (where d = 2

is a particular case), with the guillotine cutting constraint. It is based on the

exact branch-and-bound algorithm presented by Martello et al. [152] for the 3BPP,

but generalized to use arbitrary lower bounds and Orthogonal Packing Problems

OPP solvers, so that it is no longer restricted to the three-dimensional case. The

3BPP algorithm proposed by Martello et al. [152] follows a depth-first strategy.

Its main branching tree assigns items to the different bins without specifying their

actual position. The feasibility of the assignments is checked, at each decision node,

first by the computation of a lower bound for the sub-instance, then by applying

an heuristic and finally, when these two methods fail, by using an exact branch-

and-bound algorithm (ONEBIN), which also finds the best pattern. Amossen’s

approach also consists of an outer algorithm that assigns items to bins and an

inner sub-algorithm for solving an OPP for each bin. The OPP problem is solved

by extending the graph theory framework by Fekete and Schepers [83, 85] to handle

guillotine cuttings, as the original version does not take this constraint into account.

Puchinger and Raidl [171] presented an algorithm for three-stage 2D-BPP,

where a feasible solution consists of a set of bins, being each bin partitioned (hor-

izontally) into a set of strips, each strip consisting of a set of slacks, each slack

consisting of a set of items having equal width. They developed two polynomial-

size integer IP models for three-stage 2BPP: a restricted model, particularly useful

for quickly obtaining near-optimal solutions, and the original model, computa-

tionally more expensive. In the restricted version, the highest stack of each strip

always consists of a single item that, consequently, defines its height, helping the

determination of the total height of all strips contained in a bin. Both IP models

only involve polynomial numbers of variables and constraints and effectively avoid

symmetries. The restricted IP model uses O(n2) variables and O(n) constraints,

and the unrestricted one uses O(n3) variables and constraints and is considered,


to the authors’ knowledge, as the first polynomial-size IP for three-stage 2D-BPP.

Puchinger and Raidl [171] proposed, as an alternative approach, a branch-and-

price framework for a set covering formulation with column generation, based on

a Dantzig-Wolfe decomposition of the previous IP models. They also apply dual

subset inequalities in order to stabilize the column generation process. Column

generation is performed by applying a four level hierarchy of pricing algorithms,

each one being used only if the previous one fails in finding a pattern with negative

reduced costs:

1. the first level, a greedy heuristic, first fit heuristic respecting branching con-

straints (FFBC), is the fastest of the four methods, but can only solve easy

pricing problems.

2. as a second level pricing strategy, they use a more sophisticated metaheuris-

tic, an evolutionary algorithm, that is slower but can solve harder pricing

problems.

3. the third level is another heuristic approach, defined analogously to the re-

stricted 2D-BPP model as the restricted version of the pricing problem, which

they called restricted three-stage 2D-KP.

4. finally, when all pricing heuristics fail, the complete unrestricted pricing prob-

lem, a three-stage 2D-KP, is solved exactly. This algorithm is more time

consuming.

Vanderbeck [195] proposed an approximate solution method for the two-dimensional

three-stage CSP based on a nested decomposition of the problem, with a recursive

use of the column generation technique. Although this method is only approxi-

mate, it could, according to the author, be adapted to generate exact solutions.

The author described a particular industrial real-world problem, with specific ad-

ditional properties, where the main objective is to minimize waste. Additionally,


other issues such as aging stock sheets, urgent or optional orders, and fixed setup

costs are also considered. Stock sheets may have different dimensions and items

have lower and upper bounds on order productions instead of specific demands.

The three stages involved in the problem use only orthogonal guillotine cuts and

consist of the cutting of the stock pieces into sections, the sections into slits and

finally, the slits into items. The solution approach involves three layers of decom-

position: a first layer considering the subproblem associated with the generation

of feasible cutting patterns for a given stock sheet, a second layer considering the

decomposition of the cutting pattern generation problem into subproblems asso-

ciated with the generation of sections, and a third layer considering the section

generation as the selection of horizontal combinations of items and the selection of

the section length. There are constraints imposed on the cutting process, which

make the combinatorial structure of the problem more complex but limit the size of

the solution space. The sections generation subproblem is solved exactly, whereas

the cutting patterns generation subproblem is only solved approximately, although

it could, in principle, be solved to optimality if the presented approach was used

at each node of a branch-and-bound tree.

Cui [59] analyzed the problem of cutting circular blanks for the manufacturing

of electric motors. He considered multi-segment patterns, which are guillotine two-

stage patterns. Parts cut in the first stage are denoted as segments, and the ones

cut in the second stage as strips. Each strip contains identical blanks that can

appear in a given maximum number of rows. The linear programming model is

solved with column generation, being the multi-segment patterns generated with

a knapsack algorithm. This problem could be approached as a two-stage 2D-CSP

with the guillotine constraint and rectangular items if we considered as a different

rectangular item each possible type of strip resulting from the combination of the

heights of all the possible number of rows of each blank size with all their possible

widths, calculated according to the author’s definition of break points. For each


different item, the type and number of circular blanks should be considered, in

order to properly fulfill the demands.

Cintra et al. [47] proposed a column generation algorithm for the 2D-CSP with

the guillotine constraint. They introduced two new rectangular knapsack algo-

rithms to generate new columns, and described how to find integer solutions.

Alves et al. [13] presented an exact algorithm for the two-stage 2D-CSP with

the guillotine constraint, and computational experiments with real-world instances

from the furniture industry. It is a plain exact algorithm that does not use any

heuristics or other strategies to accelerate its converge. It consists of a branch-and-

price procedure whose branching scheme is compatible with the pricing subprob-

lems.

5.2 Network Flow Model

Valerio de Carvalho [188] proposed a IP network flow model for the 1D-CSP (3.11)-

(3.14). We extended this formulation for the two-stage 2D-CSP with the guillotine

constraint. We consider that, in the first stage, the stock sheets are cut into

horizontal strips, which are, in the second stage, cut into the demanded items. We

handle the two-dimensional problem as a set of two one-dimensional problems. In

the first stage, we cut strips out of the stock sheets. This means that we only need

to consider the items’ and the stock sheets’ heights. In the second stage, the strips

generated are cut into the demanded items (with vertical cuts). Again, this means

that we only have to consider the items’ and stock sheets’ widths, because we have

defined the set of items that can be cut out of a given strip (the ones whose height

is smaller than or equal to the strip height).

We consider H∗ = h∗1, ..., h∗mh as the set of mh different heights ordered by

their increasing values, a graph G0 = (V 0, A0), with V 0 = 0, 1, ..., H and A0 =

(a, b) : 0 ≤ a < b ≤ H and b − a = h∗i ,∀i ∈ 1, ...,mh, for the first stage,

5.2. Network Flow Model 85

111 1

222

344

33

554

111 1

222

344

111 1

222

443

33

554

33

55 5

3 3

554

3 3

555

33

555

First stageGuillotine patterns Second stage

Figure 5.1: Non-exact two-stage guillotine cutting patterns

and a set of graphs Gs = (V s, As), with V s = 0, 1, ...,W and As = (d, e) : 0 ≤

d < e ≤ W and e− d = wi, ∀i ∈ 1, ...,m : hi ≤ hs, for each of the mh sets of

patterns of the second stage. Every set As, ∀s ∈ 0, 1, ...,mh, includes unit arcs

that represent waste, which are strips of waste of width W when s = 0, and waste

within the strip when s > 0.

min z0 (5.1)

s.t.∑

(a,b)∈A0

x0ab −

∑(b,c)∈A0

x0bc =

−z0 , if b = 0

0 , if b = 1, 2, ..., H − 1

z0 , if b = H

, (5.2)

∑(c,c+h∗s)∈A0

x0c,c+h∗s

− zs = 0, ∀s ∈ 1, ...,mh, (5.3)


∑(d, e) ∈ As

h∗ ∈ H∗

xsdeh∗ −∑

(e, f) ∈ As

h∗ ∈ H∗

xsefh∗

=

−zs , if e = 0

0 , if e = 1, 2, ...,W − 1

zs , if e = W

, ∀s ∈ 1, ...,mh, (5.4)

mh∑s=1

∑(f,f+wi)∈As

xsf,f+wi,hi≥ bi, ∀i ∈ 1, ...,m, (5.5)

x0ab ≥ 0 and integer, ∀(a, b) ∈ A0, (5.6)

xsdeh∗ ≥ 0 and integer, ∀(d, e) ∈ As,

∀s ∈ 1, ...,mh, ∀h∗ ∈ H∗. (5.7)

In this formulation, variable z0 represents the number of stock sheets used and

variables zs, ∀s ∈ 1, ...,mh, denote the number of strips of height hs cut in

the first stage. Variables x0ab represent the flow in arc (a, b), on graph G0, and

variables xscdh∗ represent the flow in arc (c, d) ∈ Gs corresponding to items of width

(d− c) and height h∗ ∈ H∗. The third index h∗ differentiates items with the same

width but different heights within the same graph Gs. Constraints (5.3) make the

connection between the two stages of the problem: the number of strips of height

h∗i , ∀i ∈ 1, ...,mh, cut in the first stage must be equal to the number of strips of

height h∗i cut in the second stage into the demanded items. Constraints (5.2) and

(5.4) are related with flow conservation in the first and second stages, respectively,

and constraints (5.5) ensure that all the demands are fulfilled.

5.3 Arc Reduction

The model has a pseudo-polynomial number of variables and constraints. Variables

represent arcs in the network flow model; reducing the number of arcs reduces the

5.3. Arc Reduction 87

size of the model, increasing its efficiency. Three reduction criteria were applied in

order to accomplish this:

(i) only maximal cutting patterns are considered: in the first stage, we only con-

sider combinations of strips whose heights sum Sh is such that H−Sh < hmin,

being hmin the smallest item height, and in the second stage, we only consider

combinations of items whose width sum Sw is such that W − Sw < wmin,

being wmin the smallest item width. This is possible because the demand

constraints (5.5) are inequalities.

(ii) all the strips should contain at least one item with height equal to the strip

height. We guarantee this by enforcing that all arcs leaving node 0, in all

graphs of the second stage, correspond to items whose height is equal to the

strip height.

(iii) in the first or the second stages, the strips and items, respectively, to some

extent, are sorted by their decreasing values of size. This is accomplished

by constructing the graphs in a way such that an arc (b, c), with b 6= 0

(and taking into account reduction criterion (iii) for the graphs of the second

stage), is only considered if there is another arc (a, b) such that b− a ≥ c− b.

None of these criteria eliminates cutting patterns that cannot be replaced by equiv-

alent ones, in what concerns finding the minimum number of used stock sheets.

Example 5.1 Consider an instance with stock sheets of height H = 20 and width

W = 30 and a set of items I = (hi, wi, bi) : i ∈ 1, . . . , 5 = (5, 7, 4), (5, 10, 3),

(7, 12, 5), (10, 8, 3), (12, 10, 5).

By applying the previous criteria, the graph corresponding to the first stage,

G0, and the graph of the second stage, corresponding to strips of height 7, G2,

would be the ones represented in figures 5.2 and 5.3, respectively.


r r r r r r r r r r r r r r r r r r r r r0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

I I I I I II

I I I I

I I

I

J

I I I

Figure 5.2: Graph G0 of the network flow model (example 5.1).

r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

I I I

I

I I

J

I I I I I I

Figure 5.3: Graph G2 of the network flow model (example 5.1).

5.4 Symmetry

In this model, symmetry occurs whenever different paths corresponding to an iden-

tical cutting pattern are allowed. Reduction criterion (iii), previously described,

reduces some symmetry, although it does not totally prevent it. To reduce it fur-

ther, additional constraints can be added to the model:

∑(a,b)∈A0

x0ab(b− a) ≥

∑(b,c)∈A0

x0bc(c− b), (5.8)

∑(d, e) ∈ As : d > 0

h∗ ∈ H∗

xsdeh∗(e− d) ≥∑

(e, f) ∈ As

h∗ ∈ H∗

xsefh∗(f − e) ∀s ∈ 1, ...,mh. (5.9)

Constraints (5.8) and (5.9) reinforce the reduction criterion (iii), in the first and

second stage, respectively. At each node k ∈ 1, . . . ,W − 1, constraints (5.2) and

(5.4) ensure that the flow into a node is equal to the flow out of it. With (5.8) and

5.5. A New Family of Cutting Planes 89

(5.9) we force the sum of the lengths of the arcs into a node k ∈ 1, . . . ,W − 1

to be greater than or equal to the sum of lengths of the arcs out of it, in order to

avoid that any unit of flow goes from a given arc to another one with a greater

length.

5.5 A New Family of Cutting Planes

In order to strengthen the model, we derived a new family of cutting planes. They

rely on the fact that any set of items with height greater than or equal to hj must be

cut out of strips with height greater than or equal to hj. To compute a minimum

number of strips needed to cut all items of set Ij = i ∈ 1, ...,m : hi ≥ hj,

∀j ∈ 1, . . . ,mh, two sets of lower bounds, LB Strip 1j and LB Strip 2j are

considered. LB Strip 1j is the continuous lower bound, i.e.,

LB Strip 1j =

⌈∑i∈Ij wibi

W

⌉∀j ∈ 1, . . . ,mh. (5.10)

This lower bound can be tightened using of dual feasible functions (see for example

[82]).

To derive lower bound LB Strip 2j, we consider the maximum number of items

that can be cut of a given strip. To compute this, we consider the item, from the

set of items that could be cut from that strip, with minimum width.

LB Strip 2j =

∑

i∈Ij bi⌊W

mini∈Ij wi

⌋ ∀j ∈ 1, . . . ,mh. (5.11)

The new family of cutting planes is, therefore, described as follows:

mh∑l=j

zl > max(LB Strip 1j, LB Strip 2j) ∀j ∈ 1, . . . ,mh. (5.12)


5.6 A New Lower Bound

We derived a new lower bound for the number of required stock sheets. This lower

bound is based on the lower bounds described in section 5.5, but now applied to

the vertical dimension.

In order to obtain a lower bound for the sum of heights of all strips of a

given solution, we use the lower bounds (5.10) and (5.11). Thus, let lb auxj =

max(LB Strip 1j, LB Strip 2j), ∀j ∈ 1, . . . ,mh. Remember that H∗ = h∗1, ...,

h∗mh is the set of mh different heights ordered by their increasing values. Let us

consider the maximum height, i.e., h∗mh. We know that at least lb auxmh

strips of

this height will be used. We also know that at least lb aux(mh−1) strips of heights

h∗mhor h∗(mh−1) will be required. If we are considering the minimum possible sum

of heights, we can say that we would have at least lb auxmhheights h∗mh

and

(lb aux(mh−1) − lb auxmh) heights h∗(mh−1), and so on. Therefore, (5.13) is a lower

bound for the sum of strip heights in the solution.

h strip =

mh∑j=1

lb auxj − ∑i∈Ij\j

lb auxi

× h∗j . (5.13)

To compute our lower bound LB, we consider two lower bounds, LB1 =⌈hstripH

⌉and LB2 =

⌈lb aux1⌊

Hh∗1

⌋⌉

, being LB = max(LB1, LB2).

5.7 Variants of the Problem

5.7.1 The Non-Oriented Case

If the material of the stock sheets has any kind of pattern, rotation of items is

not generally allowed (oriented case). For example, if the stock sheets are made

5.7. Variants of the Problem 91

of wood, the items may not be allowed to be rotated, as the direction of the wood

grain would be modified (figure 5.4). This means that an item of width w and

1

2

1

2

2

Figure 5.4: Rotating items in a wood stock sheet

height h is considered to be different from an item of width h and height w, as

they are meant to have a different orientation. When this does not happen, and

rotation is allowed, the number of feasible cutting patterns increases.

The previous model does not consider item rotation. However, it can easily

be considered by defining a different item i′ for each item i, such that hi′ = wi,

wi′ = hi, bi′ = 0 and the sum of items i and i′ is equal to the demand of item i.

5.7.2 Orientation of the First Stage’s Cuts

So far, we considered that, in the first stage, cuts are horizontal (generating hori-

zontal strips) and, in the second stage, cuts are vertical. Of course, we may begin

with vertical cuts and proceed with horizontal ones. Allowing the first cut to be

either horizontal or vertical can obviously reduce the total number of stock sheets

in the cutting plan, which may have some sheets with horizontal strips and others

with vertical ones.


Although this can potentially improve the optimal solution, it increases con-

siderably the size of the model (5.14)-(5.25). Let us consider H∗ = h∗1, ..., h∗mh

and W ∗ = w∗1, ..., w∗mv as the set of mh and mv different heights and widths,

respectively, ordered by their increasing values. We would have to consider two

graphs for the first stage, G0h = (V 0

h , A0h), with V 0

h = 0, 1, ..., H and A0h = (a, b) :

0 ≤ a < b ≤ H and b − a = h∗i ,∀i ∈ 1, ...,mh and G0v = (V 0

v , A0v), with

V 0v = 0, 1, ...,W and A0

v = (c, d) : 0 ≤ c < d ≤ W and d − c = w∗i ,∀i ∈

1, ...,mv, and two sets of graphs for the second stage, Gsh = (V s

h , Ash), with

V sh = 0, 1, ...,W and Ash = (e, f) : 0 ≤ e < f ≤ W and f − e = wi,∀i ∈

1, ...,m : hi ≤ hs,∀s ∈ mh, and Guv = (V u

v , Auv), with V u

v = 0, 1, ..., H and

Auv = (g, h) : 0 ≤ g < h ≤ W and h− g = hi,∀i ∈ 1, ...,m : wi ≤ wu, ∀u ∈

mv.

min z0h + z0

v (5.14)

s.t.∑

(a,b)∈A0h

xh0ab −

∑(b,c)∈A0

h

xh0bc =

−z0

h , if b = 0

0 , if b = 1, 2, ..., H − 1

z0h , if b = H

, (5.15)

∑(c,c+h∗s)∈A0

h

xh0c,c+h∗s

− zsh = 0, ∀s ∈ 1, ...,mh, (5.16)

∑(g,k)∈A0

v

xv0gk −

∑(k,l)∈A0

v

xv0kl =

−z0

v , if k = 0

0 , if k = 1, 2, ...,W − 1

z0v , if k = W

, (5.17)

∑(l,l+w∗u)∈A0

v

xv0l,l+w∗u

− zuv = 0, ∀u ∈ 1, ...,mv, (5.18)

∑(d, e) ∈ As

h

h∗ ∈ H∗

xhsdeh∗ −∑

(e, f) ∈ Ash

h∗ ∈ H∗

xhsefh∗

=

−zsh , if e = 0

0 , if e = 1, 2, ...,W − 1

zsh , if e = W

,∀s ∈ 1, ...,mh, (5.19)

5.7. Variants of the Problem 93

∑(p, q) ∈ Au

v

w∗ ∈W ∗

xvupqw∗ −∑

(q, r) ∈ Auv

w∗ ∈W ∗

xvuqrw∗

=

−zuv , if q = 0

0 , if q = 1, 2, ..., H − 1

zuv , if q = H

,∀u ∈ 1, ...,mv, (5.20)

mh∑s=1

∑(f,f+wi)∈As

h

xhsf,f+wi,hi+

mv∑u=1

∑(r,r+hi)∈Au

v

xvur,r+hi,wi≥ bi,

∀i ∈ 1, ...,m, (5.21)

xh0ab ≥ 0 and integer, ∀(a, b) ∈ A0

h, (5.22)

xhsdeh∗ ≥ 0 and integer, ∀(d, e) ∈ Ash, ∀s ∈ 1, ...,mh

∀h∗ ∈ H∗, (5.23)

xv0gk ≥ 0 and integer, ∀(g, k) ∈ A0

v, (5.24)

xvupqw∗ ≥ 0 and integer, ∀(p, q) ∈ Auv , ∀u ∈ 1, ...,mv

∀w∗ ∈ W ∗. (5.25)

This formulation is similar to formulation (5.1)-(5.7). Variables z0h and z0

v represent

the number of used stock sheets with horizontal and vertical first cuts, respectively,

and variables zsh, ∀s ∈ 1, ...,mh and zuv , ∀u ∈ 1, ...,mv, denote the number of

horizontal and vertical strips of height hs and width wu, respectively, cut in the first

stage. Variables xh0ab and xv0

cd represent the flow that goes through arcs (a, b) and

(c, d) on graphs G0h and G0

v, respectively. Variables xhsdeh∗ and xvupqv∗ represent the

flow on graphs Gsh and Gu

v that goes through arcs (d, e) and (p, q), corresponding to

items of width (e− d) and height h∗ ∈ H∗ and height (q − p) and width w∗ ∈ W ∗,

respectively. The objective function (5.14) minimizes the total number of stock

sheets used (the sum of stock sheets with horizontal cuts, z0h, and vertical cuts, z0

v ,

in the first stage). Constraints (5.15) and (5.17) are related with flow conservation

of the first stage and constraints (5.19) and (5.20) with flow conservation of the


second stage. Constraints (5.16) and (5.18) make the connection of the two stages,

ensuring that the number of horizontal or vertical strips cut in the first stage is

the same as the number of horizontal or vertical strips used in the second stage.

Finally, constraints (5.21) guarantee that all the demands are fulfilled.

5.8 Computational Results

The exact network flow model was tested with two sets of real instances from the

furniture industry, set A and set B. The procedure for generating arcs was coded

in C++, and the model was solved with ILOG CPLEX 10.2. The computational

tests were run on a PC with a 1.87 GHz Intel Core Duo processor and a 2GB RAM.

Tables 5.1 and 5.2 characterize the set A and set B instances, respectively.

They show, for each instance, the number of different items, n, the total number

of items, nt, the area of all items, ai, the width, W , and height, H, of the stock

sheets, the area of one stock sheet, as, the minimum, amin, maximum, amax, and

average, aav, percentage of occupied area, in the stock sheet, by one item, and the

minimum, bmin, maximum, bmax, and average, bav, demand.

5.8. Computational Results 95

Tab

le5.1

:D

escr

ipti

on

of

the

setA

inst

an

ces.

Nam

en

nt

ai

WH

as

am

inam

ax

aav

b min

b max

b av

A-1

224

7.45

E+

062750

1220

3.3

6E

+06

5.3

321.0

39.2

56

18

12.0

0A

-24

387.

51E

+07

2750

1220

3.3

6E

+06

3.4

965.6

958.9

01

28

9.5

0A

-32

171.

63E

+07

2750

1220

3.3

6E

+06

5.3

329.9

728.5

21

16

8.5

0A

-42

167.

31E

+06

2750

1220

3.3

6E

+06

0.8

726.3

513.6

18

88.0

0A

-58

138

5.73

E+

072470

2080

5.1

4E

+06

2.2

844.7

38.0

88

32

17.2

5A

-62

173.

96E

+06

2750

1220

3.3

6E

+06

6.8

09.1

16.9

41

16

8.5

0A

-75

583.

15E

+07

2750

1220

3.3

6E

+06

6.0

232.9

216.1

98

16

11.6

0A

-81

162.

41E

+06

2750

1220

3.3

6E

+06

4.4

94.4

94.4

916

16

16.0

0A

-930

770

2.93

E+

082550

2100

5.3

6E

+06

0.7

035.3

27.1

16

68

25.6

7A

-10

344

9.88

E+

062550

2100

5.3

6E

+06

0.7

08.6

34.1

912

16

14.6

7A

-11

2072

42.

32E

+08

2550

2100

5.3

6E

+06

2.1

123.2

25.9

88

140

36.2

0A

-12

344

5.82

E+

072550

2100

5.3

6E

+06

8.6

334.4

924.6

88

20

14.6

7A

-13

830

46.

35E

+07

2550

2100

5.3

6E

+06

0.7

017.0

33.9

08

120

38.0

0A

-14

3180

93.

38E

+08

2550

2100

5.3

6E

+06

1.4

125.1

47.8

08

52

26.1

0A

-15

1233

91.

88E

+08

2550

2100

5.3

6E

+06

2.8

725.1

410.3

48

80

28.2

5A

-16

2774

44.

17E

+08

2550

2100

5.3

6E

+06

0.6

625.1

410.4

84

96

27.5

6A

-17

313

52.

39E

+07

2550

2100

5.3

6E

+06

2.9

63.6

73.3

028

57

45.0

0A

-18

2055

93.

16E

+08

2550

2100

5.3

6E

+06

0.6

625.1

410.5

65

71

27.9

5A

-19

2750

72.

73E

+08

2550

2100

5.3

6E

+06

1.0

837.4

410.0

53

50

18.7

8A

-20

1021

51.

20E

+08

2550

2100

5.3

6E

+06

2.1

937.4

410.4

33

60

21.5

0A

-21

2145

01.

36E

+08

2550

2100

5.3

6E

+06

2.1

014.6

95.6

34

70

21.4

3A

-22

124

1.04

E+

072550

2100

5.3

6E

+06

8.0

68.0

68.0

624

24

24.0

0A

-23

824

86.

32E

+07

2550

2100

5.3

6E

+06

2.9

39.7

84.7

68

60

31.0

0A

-24

107

217

1.65

E+

082550

2100

5.3

6E

+06

1.2

442.3

214.2

41

12

2.0

3A

-25

7515

68.

54E

+07

2550

2100

5.3

6E

+06

1.2

140.0

610.2

21

72.0

8A

-26

3461

3.42

E+

072550

2100

5.3

6E

+06

1.0

733.5

210.4

71

81.7

9A

-27

7918

09.

62E

+07

2550

2100

5.3

6E

+06

0.6

629.6

29.9

81

15

2.2

8A

-28

5410

65.

30E

+07

2550

2100

5.3

6E

+06

1.4

123.6

69.3

51

61.9

6A

-29

8221

81.

35E

+08

2550

2100

5.3

6E

+06

0.5

242.8

211.6

01

18

2.6

6A

-30

2439

1.76

E+

072550

2100

5.3

6E

+06

1.7

619.8

08.4

31

71.6

3A

-31

3664

3.64

E+

072550

2100

5.3

6E

+06

3.0

141.4

710.6

21

61.7

8


Tab

le5.

1:D

escr

ipti

on

of

the

setA

inst

an

ces

(conti

nu

ed).

Nam

en

nt

ai

WH

as

am

inam

ax

aav

b min

b max

b av

A-3

299

184

1.30

E+

082550

2100

5.3

6E

+06

1.4

557.4

913.1

71

91.8

6A

-33

134

309

1.75

E+

082550

2100

5.3

6E

+06

1.0

552.6

810.5

61

15

2.3

1A

-34

2646

2.46

E+

072550

2100

5.3

6E

+06

1.5

041.3

59.9

91

10

1.7

7A

-35

6814

48.

09E

+07

2550

2100

5.3

6E

+06

1.4

551.6

510.4

91

14

2.1

2A

-36

1652

3.48

E+

072550

2100

5.3

6E

+06

1.2

956.2

812.4

81

83.2

5A

-37

878

2.22

E+

072550

2100

5.3

6E

+06

1.9

649.5

45.3

11

40

9.7

5A

-38

4216

01.

08E

+08

2550

2100

5.3

6E

+06

1.0

756.2

812.6

11

40

3.8

1A

-39

1122

1.46

E+

072550

2100

5.3

6E

+06

1.6

430.9

812.4

21

42.0

0A

-40

4016

34.

81E

+07

2750

1220

3.3

6E

+06

1.5

661.4

78.7

91

20

4.0

8A

-41

3271

4.73

E+

072750

1220

3.3

6E

+06

1.1

069.6

919.8

71

92.2

2A

-42

813

1.48

E+

072750

1220

3.3

6E

+06

10.1

861.4

733.9

51

31.6

3A

-43

1122

1.59

E+

072750

1220

3.3

6E

+06

1.4

365.6

921.5

41

42.0

0

Average

28.7

419

8.72

9.49

E+

072599.3

01874.4

24.8

4E

+06

2.5

334.7

112.3

14.7

932.3

012.6

8


Tab

le5.2

:D

escr

ipti

on

of

the

setB

inst

an

ces.

Nam

en

nt

ai

WH

as

am

inam

ax

aav

b min

b max

b av

B-1

975

76.

14E

+08

4250

2120

9.0

1E

+06

3.6

923.7

39.0

055

190

84.1

1B

-213

1040

1.35

E+

094250

2120

9.0

1E

+06

3.6

934.1

614.4

128

165

80.0

0B

-34

1342

6.95

E+

084250

1860

7.9

1E

+06

2.5

015.0

36.5

5122

820

335.5

0B

-46

123

1.41

E+

084250

1860

7.9

1E

+06

4.0

137.2

114.4

910

44

20.5

0B

-54

118

7.06

E+

074250

1860

7.9

1E

+06

2.6

822.4

77.5

711

80

29.5

0B

-63

427.

03E

+07

4250

1860

7.9

1E

+06

16.3

837.2

121.1

74

30

14.0

0B

-74

3000

2.22

E+

092440

1860

4.5

4E

+06

3.3

823.1

516.3

4600

1200

750.0

0B

-86

2545

1.46

E+

092800

1860

5.2

1E

+06

4.5

424.0

211.0

3225

1360

424.1

7B

-921

2629

6.45

E+

084250

1860

7.9

1E

+06

1.1

95.4

23.1

128

330

125.1

9B

-10

2128

385.

80E

+08

4250

1860

7.9

1E

+06

1.0

216.5

12.5

811

660

135.1

4B

-11

2861

342.

04E

+09

4250

1860

7.9

1E

+06

1.4

927.7

04.2

227

1300

219.0

7B

-12

665

702.

89E

+09

3760

1860

6.9

9E

+06

1.9

416.7

16.3

0110

2480

1095.0

0B

-13

1158

711.

50E

+09

3760

1860

6.9

9E

+06

1.7

56.7

73.6

5103

2060

533.7

3B

-14

731

82.

67E

+08

4250

2120

9.0

1E

+06

3.4

114.4

29.3

123

84

45.4

3B

-15

510

61.

13E

+08

4250

2200

9.3

5E

+06

5.2

221.6

511.3

85

38

21.2

0B

-16

831

62.

03E

+08

4250

2200

9.3

5E

+06

3.2

815.6

16.8

715

74

39.5

0B

-17

257

6.84

E+

074250

2120

9.0

1E

+06

11.1

115.0

413.3

225

32

28.5

0B

-18

314

71.

08E

+08

4250

2120

9.0

1E

+06

6.0

211.1

18.1

344

55

49.0

0B

-19

1152

94.

06E

+08

4250

2200

9.3

5E

+06

3.2

815.6

28.2

028

97

48.0

9B

-20

610

806.

67E

+08

4250

1860

7.9

1E

+06

2.7

115.9

77.8

150

600

180.0

0B

-21

730

001.

50E

+09

4250

1860

7.9

1E

+06

2.2

615.9

76.3

3110

1420

428.5

7B

-22

624

41.

36E

+08

4250

2120

9.0

1E

+06

3.4

110.8

36.1

828

77

40.6

7B

-23

1314

035.

67E

+08

3770

1860

7.0

1E

+06

2.1

148.6

35.7

69

220

107.9

2B

-24

216

56.

54E

+07

4250

1860

7.9

1E

+06

2.2

010.6

55.0

155

110

82.5

0B

-25

570

81.

94E

+08

2600

1860

4.8

4E

+06

3.2

111.7

15.6

722

334

141.6

0B

-26

836

778.

14E

+08

3760

1860

6.9

9E

+06

1.7

54.4

43.1

7106

1260

459.6

3B

-27

534

131.

39E

+09

3760

1860

6.9

9E

+06

2.2

416.7

15.8

153

1260

682.6

0B

-28

688

02.

84E

+08

4250

1860

7.9

1E

+06

2.1

18.6

24.0

9110

220

146.6

7B

-29

544

01.

34E

+08

4250

1860

7.9

1E

+06

2.8

25.6

13.8

655

110

88.0

0B

-30

1413

074.

34E

+08

4250

1860

7.9

1E

+06

2.1

18.6

24.2

022

220

93.3

6B

-31

539

62.

03E

+08

4250

1860

7.9

1E

+06

3.2

216.4

46.4

822

220

79.2

0


Tab

le5.

2:D

escr

ipti

on

of

the

setB

inst

an

ces

(conti

nu

ed).

Nam

en

nt

ai

WH

as

am

inam

ax

aav

b min

b max

b av

B-3

227

2600

1.38

E+

094250

1860

7.9

1E

+06

1.6

221.0

76.7

220

240

96.3

0B

-33

668

93.

54E

+08

4250

1860

7.9

1E

+06

2.2

718.7

66.5

055

220

114.8

3B

-34

1410

855.

36E

+08

4250

1860

7.9

1E

+06

1.6

218.7

66.2

522

165

77.5

0B

-35

447

65.

97E

+08

3760

1860

6.9

9E

+06

4.8

730.2

717.9

252

180

119.0

0B

-36

474

06.

11E

+08

3760

1860

6.9

9E

+06

5.3

915.6

811.8

140

280

185.0

0B

-37

655

03.

50E

+08

4250

1860

7.9

1E

+06

3.9

221.0

78.0

420

180

91.6

7B

-38

236

51.

42E

+08

2820

1870

5.2

7E

+06

6.2

614.3

57.3

750

315

182.5

0B

-39

231

51.

42E

+08

2820

1870

5.2

7E

+06

8.2

49.2

18.5

6105

210

157.5

0B

-40

722

61.

37E

+08

4250

1860

7.9

1E

+06

4.0

911.6

87.6

417

63

32.2

9B

-41

529

41.

36E

+08

4250

2200

9.3

5E

+06

3.2

412.1

44.9

512

163

58.8

0B

-42

718

81.

15E

+08

4250

1860

7.9

1E

+06

4.2

911.8

67.7

114

44

26.8

6B

-43

258

3.53

E+

074250

1860

7.9

1E

+06

6.9

99.2

97.7

018

40

29.0

0B

-44

1142

11.

74E

+08

4250

1860

7.9

1E

+06

1.8

811.6

85.2

34

84

38.2

7B

-45

215

67.

69E

+07

4250

1860

7.9

1E

+06

4.0

911.6

86.2

344

112

78.0

0B

-46

620

91.

75E

+08

4250

1860

7.9

1E

+06

5.0

518.4

810.5

87

108

34.8

3B

-47

494

5.87

E+

074250

1860

7.9

1E

+06

4.2

911.8

67.9

012

34

23.5

0B

-48

627

81.

24E

+08

4250

2200

9.3

5E

+06

2.1

49.2

44.7

512

98

46.3

3B

-49

316

11.

26E

+08

4250

1860

7.9

1E

+06

6.4

014.5

79.9

134

67

53.6

7B

-50

198

54.

79E

+08

4200

1860

7.8

1E

+06

6.2

26.2

26.2

2985

985

985.0

0B

-51

1039

04.

67E

+08

4200

2130

8.9

5E

+06

3.7

826.7

713.3

920

62

39.0

0B

-52

123

908.

90E

+08

4250

2070

8.8

0E

+06

4.2

34.2

34.2

32390

2390

2390.0

0B

-53

410

01.

86E

+07

4250

1860

7.9

1E

+06

1.6

13.5

92.3

510

50

25.0

0B

-54

615

35.

76E

+07

2820

1870

5.2

7E

+06

4.0

915.0

77.1

39

46

25.5

0B

-55

995

03.

72E

+08

4250

1860

7.9

1E

+06

1.1

016.6

64.9

522

330

105.5

6B

-56

1737

811.

42E

+09

3760

1860

6.9

9E

+06

1.8

018.8

35.3

832

1050

222.4

1B

-57

1411

443.

30E

+08

4250

1860

7.9

1E

+06

1.1

76.5

83.6

522

275

81.7

1B

-58

646

51.

42E

+08

4250

1860

7.9

1E

+06

1.8

25.3

13.8

660

165

77.5

0B

-59

223

17.

21E

+07

4250

1860

7.9

1E

+06

2.8

86.9

93.9

560

171

115.5

0B

-60

424

28.

89E

+07

4250

1860

7.9

1E

+06

2.8

88.8

24.6

529

109

60.5

0B

-61

214

83.

71E

+07

4250

1860

7.9

1E

+06

2.3

65.3

43.1

740

108

74.0

0B

-62

434

91.

29E

+08

4250

1860

7.9

1E

+06

2.8

88.8

24.6

738

142

87.2

5B

-63

839

42.

69E

+08

4250

1860

7.9

1E

+06

3.1

917.4

08.6

212

80

49.2

5


Tab

le5.

2:D

escr

ipti

on

of

the

setB

inst

an

ces

(conti

nu

ed).

Nam

en

nt

ai

WH

as

am

inam

ax

aav

b min

b max

b av

B-6

49

295

1.45

E+

084250

1860

7.9

1E

+06

1.7

432.2

96.2

310

83

32.7

8B

-65

2020

131.

27E

+09

4250

1860

7.9

1E

+06

1.8

219.8

17.9

744

198

100.6

5B

-66

1050

22.

85E

+08

4250

1860

7.9

1E

+06

1.7

420.2

77.1

710

138

50.2

0B

-67

3125

001.

87E

+09

4250

1860

7.9

1E

+06

1.8

220.8

89.4

49

280

80.6

5B

-68

1771

03.

65E

+08

4250

1860

7.9

1E

+06

1.7

432.2

96.5

010

130

41.7

6B

-69

320

91.

44E

+08

4250

1860

7.9

1E

+06

1.8

414.2

58.7

455

88

69.6

7B

-70

1613

938.

56E

+08

4250

1860

7.9

1E

+06

1.8

419.8

17.7

826

176

87.0

6B

-71

1090

94.

80E

+08

4250

1860

7.9

1E

+06

1.7

419.1

06.6

822

138

90.9

0B

-72

850

12.

55E

+08

4250

1860

7.9

1E

+06

1.8

414.2

56.4

544

88

62.6

3B

-73

634

71.

34E

+08

4250

1860

7.9

1E

+06

1.7

410.7

04.9

022

88

57.8

3B

-74

610

185

2.56

E+

094050

2200

8.9

1E

+06

1.6

36.0

12.8

2165

6000

1697.5

0B

-75

836

058.

01E

+08

3760

1860

6.9

9E

+06

1.7

54.4

43.1

8106

1260

450.6

3B

-76

534

131.

39E

+09

3760

1860

6.9

9E

+06

2.2

416.7

15.8

153

1260

682.6

0B

-77

1553

591.

16E

+09

2820

2070

5.8

4E

+06

1.6

86.8

23.7

1158

841

357.2

7B

-78

617

343.

87E

+08

2820

2070

5.8

4E

+06

1.7

15.0

63.8

2158

526

289.0

0B

-79

610

522.

37E

+08

2820

1870

5.2

7E

+06

2.4

76.4

34.2

6105

316

175.3

3B

-80

715

753.

17E

+08

2820

2070

5.8

4E

+06

2.2

34.6

53.4

5210

315

225.0

0B

-81

1838

022.

94E

+09

4250

1860

7.9

1E

+06

1.9

019.5

59.7

840

770

211.2

2B

-82

1012

408.

23E

+08

4250

2120

9.0

1E

+06

1.6

717.1

67.3

760

230

124.0

0B

-83

1311

607.

07E

+08

4250

1860

7.9

1E

+06

1.9

016.3

37.7

150

150

89.2

3B

-84

1214

568.

51E

+08

4250

2120

9.0

1E

+06

1.6

712.9

16.4

950

280

121.3

3B

-85

720

658.

04E

+08

4250

1860

7.9

1E

+06

2.1

210.1

74.9

270

1050

295.0

0B

-86

720

658.

04E

+08

4250

1860

7.9

1E

+06

2.1

210.1

74.9

270

1050

295.0

0B

-87

836

788.

15E

+08

3760

1860

6.9

9E

+06

1.7

54.4

43.1

7106

1260

459.7

5B

-88

536

271.

43E

+09

3760

1860

6.9

9E

+06

2.2

416.7

15.6

5267

1260

725.4

0B

-89

1090

902.

30E

+09

3760

1860

6.9

9E

+06

1.7

56.7

73.6

2320

3150

909.0

0B

-90

712

813

5.23

E+

093760

1860

6.9

9E

+06

2.2

416.7

15.8

3155

3708

1830.4

3B

-91

866

952.

30E

+09

3760

2120

7.9

7E

+06

2.0

910.0

94.3

1206

3090

836.8

8B

-92

738

251.

31E

+09

4250

1860

7.9

1E

+06

2.1

410.1

74.3

4110

1900

546.4

3B

-93

843

161.

49E

+09

4250

1860

7.9

1E

+06

2.1

210.1

74.3

8158

1890

539.5

0B

-94

443

08.

15E

+07

4250

1860

7.9

1E

+06

2.1

24.2

52.4

025

200

107.5

0B

-95

313

244.

13E

+08

4250

1860

7.9

1E

+06

2.8

54.2

03.9

5144

880

441.3

3


Tab

le5.

2:D

escr

ipti

on

of

the

setB

inst

an

ces

(conti

nu

ed).

Nam

en

nt

ai

WH

as

am

inam

ax

aav

b min

b max

b av

B-9

66

2090

7.67

E+

083760

1860

6.9

9E

+06

2.4

011.5

25.2

5110

660

348.3

3B

-97

212

603.

83E

+08

3760

1860

6.9

9E

+06

2.9

07.2

44.3

4420

840

630.0

0B

-98

811

884.

53E

+08

4250

1860

7.9

1E

+06

2.6

47.0

64.8

355

440

148.5

0B

-99

449

354.

62E

+09

2700

1860

5.0

2E

+06

9.4

629.8

918.6

6615

1860

1233.7

5B

-100

1512

184.

49E

+08

4250

1860

7.9

1E

+06

1.4

212.0

24.6

613

300

81.2

0B

-101

1710

274.

10E

+08

4250

1860

7.9

1E

+06

1.4

212.0

25.0

517

200

60.4

1B

-102

1382

13.

41E

+08

4250

1860

7.9

1E

+06

1.4

132.5

65.2

522

165

63.1

5B

-103

757

82.

55E

+08

3760

1860

6.9

9E

+06

3.2

311.2

56.3

25

136

82.5

7B

-104

135

1.83

E+

074250

2200

9.3

5E

+06

5.6

05.6

05.6

035

35

35.0

0B

-105

140

02.

09E

+08

3760

1860

6.9

9E

+06

7.4

87.4

87.4

8400

400

400.0

0B

-106

1312

225.

11E

+08

4250

1860

7.9

1E

+06

2.7

112.9

95.2

927

206

94.0

0B

-107

3156

542.

19E

+09

4250

1860

7.9

1E

+06

1.5

614.6

34.9

021

1628

182.3

9B

-108

2414

281.

53E

+09

4250

2120

9.0

1E

+06

1.8

627.3

811.8

521

105

59.5

0B

-109

1432

01.

97E

+08

4250

2200

9.3

5E

+06

1.2

632.5

16.5

93

66

22.8

6B

-110

132

2.09

E+

074250

2120

9.0

1E

+06

7.2

57.2

57.2

532

32

32.0

0B

-111

2312

134.

04E

+08

4250

1860

7.9

1E

+06

1.5

614.6

34.2

18

381

52.7

4B

-112

1016

96.

46E

+07

4200

2200

9.2

4E

+06

1.6

15.6

74.1

43

104

16.9

0B

-113

710

710

1.99

E+

094250

2120

9.0

1E

+06

0.8

84.3

62.0

6335

4405

1530.0

0B

-114

6054

991.

89E

+09

3760

1860

6.9

9E

+06

1.1

525.2

94.9

26

1078

91.6

5B

-115

314

29.

32E

+07

4250

2200

9.3

5E

+06

6.0

18.4

17.0

222

63

47.3

3B

-116

2544

701.

76E

+09

4250

2200

9.3

5E

+06

1.6

010.2

44.2

127

526

178.8

0B

-117

153

3.78

E+

074250

1860

7.9

1E

+06

9.0

39.0

39.0

353

53

53.0

0B

-118

444

42.

98E

+08

4200

2200

9.2

4E

+06

4.3

215.7

97.2

722

296

111.0

0B

-119

1320

561.

29E

+09

4250

2200

9.3

5E

+06

3.1

315.6

06.7

331

788

158.1

5B

-120

492

3.30

E+

074250

1860

7.9

1E

+06

3.0

85.2

24.5

310

42

23.0

0B

-121

233

1.41

E+

074200

2200

9.2

4E

+06

4.4

65.0

94.6

39

24

16.5

0

Average

8.92

1758

.67

7.19

E+

0840

29.2

61934.6

37.8

0E

+06

3.1

314.5

06.6

497.0

5606.1

3249.8

2


Tables 5.3 and 5.4 describe the computational results for the set A and set B

instances, respectively. The columns present the results for the original version of

the network flow model, without considering the cutting planes or lower bound, AF ,

and considering them, AFcut+lb, respectively, and also for the ILP model proposed

in [145], Lodi et al., and the branch-and-price algorithm proposed in [13], Alves et

al.. ZRL and Z stand for the values of the linear relaxation and integer solutions,

respectively, and t represents the total computational time, in seconds. An asterisk

(*) represents an instance which was not solved exactly within the time limit of

7200 seconds.

As mentioned in section 5.5, the cutting planes in AFcut+lb can be improved

by using dual feasible functions. We computed them considering function φ(ε), as

described in [82, 49]. This function, derived for the 1D-BPP, does not consider

items that are smaller than a given parameter ε, and separates the remaining items

in two different classes, according to whether they are larger than half of the bin

size, or not. It gives a constant size to the first ones, and computes the others

considering an estimation of the number of the first ones that could be placed in

one bin with them.

Table 5.3: Results for the set A instances.

AF AFcut+lb Lodi et al. Alves et al.

Name Z ZLR t(s) ZLR t(s) ZLR t(s) ZLR t(s)

A-1 4 3.30 0.20 3.30 0.17 2.22 0.06 3.30 0.02A-2 36 36.00 0.33 36.00 0.36 22.42 0.02 36.00 0.02A-3 8 8.00 0.16 8.00 0.16 4.86 0.02 8.00 0.05A-4 3 2.67 0.19 3.00 0.22 2.19 0.02 2.67 0.00A-5 13 12.53 0.75 12.71 0.75 11.16 32.00 12.53 0.75A-6 2 1.89 0.09 2.00 0.09 1.18 0.02 1.89 0.02A-7 14 13.13 0.64 13.13 0.69 9.39 * 13.13 0.03A-8 2 1.07 0.08 2.00 0.09 1.00 0.05 1.07 0.01A-9 61 60.67 27.34 60.67 228.94 54.74 * 60.67 *A-10 3 1.97 0.34 3.00 0.30 1.84 48.13 2.00 0.06A-11 46 45.76 7.72 45.79 224.17 43.33 * 45.76 54.80A-12 14 14.00 0.33 14.00 0.38 10.86 0.14 14.00 0.01A-13 14 13.53 0.91 13.53 0.86 11.86 * 13.53 0.75A-14 67 66.82 183.53 66.82 52.11 63.08 * 66.82 *


Table 5.3: Results for the set A instances (continued).



A-15 39 38.94 1.11 38.94 1.16 35.06 * 38.94 3.03A-16 83 82.26 12.03 82.26 108.95 77.94 * 82.26 15.61A-17 5 4.70 0.34 4.73 0.28 4.46 13.83 4.70 *A-18 65 64.39 2.97 64.65 3.38 59.34 * 64.68 *A-19 58 57.23 17.33 57.23 11.39 50.93 * 57.23 15.58A-20 27 26.10 1.22 26.10 1.11 22.42 * 26.10 1.28A-21 28 27.24 57.63 27.29 23.38 25.33 * 27.24 10.08A-22 3 2.40 0.24 3.00 0.09 1.93 0.05 2.40 0.01A-23 14 12.92 0.77 13.12 0.78 11.80 * 12.92 2093.86A-24 35 34.32 1012.61 * * 30.94 * 34.36 1739.19A-25 18 17.02 2595.83 17.29 * 16.26 * 17.33 *A-26 8 7.05 19.98 7.09 59.61 6.44 0.83 7.08 24.38A-27 20 19.04 3965.55 19.18 * 17.98 * 19.20 *A-28 12 10.81 118.88 11.13 338.98 10.25 18.11 11.17 985.02A-29 28 27.28 3155.83 27.28 * 25.32 * 27.29 *A-30 4 3.59 15.50 3.66 8.81 3.41 0.22 3.68 14.08A-31 8 7.47 23.39 7.47 62.20 6.84 2.33 7.49 81.97A-32 27 26.85 83.91 26.85 702.58 24.32 * 26.90 2257.09A-33 35 34.63 4793.70 34.63 * 32.71 * 34.67 *A-34 6 5.20 3.13 5.24 2.63 4.63 0.20 5.22 6.99A-35 17 16.36 44.72 16.37 126.81 15.17 * 16.40 358.70A-36 9 8.88 1.72 8.88 1.73 6.55 0.38 8.88 0.70A-37 5 4.63 0.86 4.63 0.88 4.17 16.48 4.63 *A-38 23 22.11 11.83 22.11 17.64 20.21 * 22.12 54.73A-39 4 3.02 1.00 3.03 1.03 2.78 0.05 3.06 0.38A-40 17 15.81 79.20 15.81 663.89 14.39 * 15.90 *A-41 19 18.50 4.72 18.50 4.06 14.18 0.28 18.50 11.97A-42 8 7.17 0.95 7.38 0.99 4.53 0.00 7.17 0.08A-43 7 6.38 1.00 6.38 1.08 4.80 0.03 6.42 5.03

Table 5.4: Results for the set B instances.



B-1 76 74.95 0.59 74.95 0.64 68.16 * 74.95 8.98B-2 171 170.79 1.08 170.79 1.11 149.82 * 170.79 0.22B-3 110 108.91 0.24 109.06 0.22 87.95 * 108.91 0.11B-4 21 20.70 0.98 20.70 1.00 17.82 334.58 20.70 0.11B-5 11 10.53 0.45 10.53 0.47 8.94 8.42 10.53 0.14B-6 10 9.67 0.38 10.00 0.42 8.89 0.08 9.67 0.01B-7 537 536.25 0.34 536.25 0.39 * * 536.25 0.01


Table 5.4: Results for the set B instances (continued).



B-8 334 333.89 0.30 333.89 0.30 * * 333.89 0.13B-9 85 84.19 63.14 84.19 89.36 * * 84.19 *B-10 78 77.74 388.45 77.80 244.70 * * 77.75 *B-11 271 270.05 20.11 270.05 82.70 * * * *B-12 465 464.96 0.34 464.96 0.36 * * 464.96 0.08B-13 230 229.11 1.30 229.11 0.69 * * 257.95 *B-14 35 34.33 0.78 34.33 0.83 29.60 * 34.33 0.98B-15 16 15.44 1.42 15.44 1.42 12.06 15.92 15.44 0.31B-16 24 23.25 0.85 23.25 0.88 21.71 * 23.25 0.94B-17 12 11.13 0.17 11.13 0.17 7.59 0.25 11.13 0.08B-18 14 13.10 0.27 13.21 0.30 11.95 10.80 13.10 *B-19 47 46.21 1.09 46.21 1.13 43.37 * 46.21 2.17B-20 95 95.00 0.84 95.00 0.83 84.37 * 95.00 0.03B-21 229 228.33 0.42 228.33 0.50 * * 228.33 0.11B-22 17 16.50 0.61 16.50 0.62 15.08 * 16.50 0.23B-23 85 84.33 1.11 84.33 1.13 80.82 * 84.33 3.11B-24 9 8.83 0.20 9.00 0.25 8.27 * 8.83 0.11B-25 47 46.75 0.34 46.75 0.36 40.11 * 46.75 0.03B-26 128 127.71 0.66 127.71 0.70 * * 127.71 *B-27 221 220.51 0.34 220.51 0.34 * * 220.51 0.16B-28 38 36.98 0.30 37.00 0.35 35.99 * 36.98 0.16B-29 19 18.24 0.39 18.26 0.41 16.97 * 18.24 0.41B-30 57 56.77 16.52 56.77 17.64 54.93 * 56.77 5.69B-31 30 29.33 0.61 29.33 0.63 25.64 * 29.33 0.05B-32 192 191.07 3.56 191.07 2.50 * * 191.07 *B-33 52 50.89 0.75 50.89 0.78 44.80 * 50.89 0.78B-34 83 82.67 1.61 82.67 1.63 67.85 * 82.67 1.64B-35 96 95.83 0.39 95.83 0.42 85.32 226.53 95.83 0.08B-36 96 95.83 0.41 95.83 0.44 87.39 * 95.83 0.03B-37 53 52.22 0.39 52.22 0.42 44.23 * 52.22 0.03B-38 35 34.58 0.17 34.63 0.20 26.89 705.75 34.58 0.00B-39 35 35.00 0.19 35.00 0.22 26.96 35.88 35.00 0.00B-40 19 18.64 2.47 18.64 2.53 17.27 * 18.64 0.06B-41 17 16.67 4.22 16.67 4.61 14.55 * 16.67 0.66B-42 16 15.39 0.28 15.39 0.28 14.50 * 15.39 0.09B-43 6 5.33 0.69 5.39 0.70 4.47 0.56 5.33 0.55B-44 24 23.32 12.38 23.32 12.69 22.02 * 23.32 *B-45 11 10.17 2.88 10.17 2.91 9.72 * 10.17 0.11B-46 28 26.99 1.06 26.99 1.09 22.12 * 26.99 0.03B-47 8 7.92 0.13 8.00 0.19 7.43 4.06 7.92 0.06B-48 14 13.45 10.41 13.45 11.13 13.22 * 13.45 1.31B-49 21 20.07 0.44 20.14 0.45 15.96 1.31 20.07 0.03B-50 71 70.36 0.11 71.00 0.13 61.28 5293.47 70.36 0.00B-51 63 62.14 1.05 62.14 1.09 52.23 * 62.14 0.19B-52 120 119.50 0.14 120.00 0.16 * * 119.50 0.00B-53 3 2.80 0.47 2.80 0.47 2.35 6.70 2.80 0.03





B-54 13 12.20 0.27 12.40 0.27 10.92 * 12.20 4.72B-55 51 49.94 0.86 49.94 0.86 47.06 * 49.94 26.33B-56 214 213.83 4.05 213.83 3.73 * * 213.83 16.63B-57 44 43.86 34.55 43.86 32.91 41.75 * 43.86 *B-58 20 19.12 0.36 19.12 0.31 17.97 * 19.12 0.14B-59 11 10.39 0.22 10.39 0.27 9.12 1648.48 10.39 0.01B-60 13 12.17 0.53 12.19 0.56 11.25 * 12.17 0.09B-61 6 5.33 0.24 5.33 0.23 4.69 * 5.33 0.01B-62 18 17.55 0.53 17.57 0.55 16.30 * 17.55 0.06B-63 38 37.76 1.06 37.76 1.08 33.98 * 37.76 0.31B-64 20 19.28 1.16 19.28 1.27 18.38 * 19.28 2.78B-65 172 171.56 3.17 171.56 2.45 * * 171.56 10.17B-66 41 40.00 1.17 40.05 1.30 36.00 * 40.00 1.52B-67 262 260.96 21.86 261.01 18.03 * * 260.96 3049.28B-68 49 48.94 2.28 48.94 2.30 46.14 * 48.94 *B-69 20 19.94 0.22 19.94 0.25 18.27 * 19.94 0.16B-70 117 116.16 2.02 116.16 2.03 108.34 * 116.16 *B-71 66 65.08 1.34 65.08 1.39 60.68 * 65.08 17.50B-72 34 33.78 0.55 33.78 0.56 32.32 * 33.78 1.58B-73 19 18.26 0.64 18.27 0.67 17.00 * 18.26 0.48B-74 309 308.59 1.55 308.64 1.50 * * 308.59 *B-75 126 125.86 0.70 125.86 0.70 * * 125.86 1.36B-76 221 220.51 0.33 220.51 0.38 * * 220.51 0.16B-77 208 207.00 1.61 207.02 1.41 * * 207.00 67.67B-78 73 72.53 0.28 72.54 0.27 * * 72.53 0.13B-79 50 49.53 0.50 49.53 0.58 44.86 * 49.53 2.81B-80 57 56.91 0.50 56.91 0.66 * * 56.91 1.24B-81 425 424.04 1.83 424.04 1.84 * * 424.04 62.45B-82 105 104.83 1.19 104.84 1.23 91.38 * 104.83 4.44B-83 97 96.55 1.64 96.59 2.56 89.38 * 96.55 9.70B-84 103 102.17 1.27 102.23 1.33 94.45 * 110.50 *B-85 106 105.05 0.58 105.05 0.52 * * 105.05 0.53B-86 106 105.05 0.56 105.05 0.50 * * 105.05 0.53B-87 128 127.73 0.67 127.73 0.67 * * 127.73 0.94B-88 226 225.62 0.33 225.62 0.36 * * 225.62 0.61B-89 353 352.44 0.66 352.44 0.72 * * 352.44 4.14B-90 841 840.42 0.50 840.42 0.50 * * 840.42 0.13B-91 317 316.50 0.81 316.50 0.88 * * 316.50 *B-92 176 175.24 0.34 175.26 0.39 * * 175.24 0.25B-93 201 200.07 0.78 200.11 0.80 * * 200.07 4.70B-94 12 11.08 0.44 11.08 0.44 10.30 * 11.08 0.05B-95 60 59.60 0.34 59.66 0.39 52.31 * 59.60 *B-96 124 123.75 0.63 123.75 0.66 * * 123.75 *B-97 60 60.00 0.20 60.00 0.23 * * 60.00 0.01B-98 68 67.02 1.27 67.02 0.45 57.34 * 67.02 3.28B-99 1095 1095.00 0.78 1095.00 0.81 * * 1095.00 0.01





B-100 59 58.57 15.86 58.60 4.19 56.81 * 58.57 13.64B-101 59 58.34 3.81 58.55 8.14 51.90 * 58.34 *B-102 46 45.22 2.13 45.22 2.17 43.11 * 45.22 *B-103 41 39.86 3.27 40.02 3.27 36.52 * 39.86 *B-104 3 2.33 0.14 3.00 0.16 1.96 0.05 2.33 0.02B-105 34 33.33 0.11 34.00 0.13 29.92 73.41 33.33 0.00B-106 71 70.18 2.05 70.18 2.02 64.67 * 70.18 *B-107 300 299.00 586.72 299.00 417.27 * * 299.00 *B-108 185 183.72 * 183.72 16.73 169.27 * 183.72 *B-109 26 25.14 4.61 25.14 3.44 21.09 * 25.14 9.23B-110 4 3.20 0.13 4.00 0.14 2.32 0.08 3.20 0.00B-111 56 55.17 79.60 55.17 1022.19 51.11 * 55.17 186.77B-112 9 8.10 1.17 8.11 1.17 6.99 * 8.10 1.38B-113 231 230.44 0.42 230.44 0.25 * * 230.44 0.66B-114 * 279.98 * 279.98 * * * 279.99 *B-115 15 14.19 9.03 14.19 8.47 9.97 116.97 14.19 0.01B-116 192 191.27 1765.78 191.28 6396.39 * * 191.27 *B-117 6 5.30 0.66 6.00 0.66 4.79 0.22 5.30 0.00B-118 36 35.25 4.19 35.25 4.12 32.30 * 35.25 0.02B-119 144 143.32 8.14 143.32 11.08 * * 143.32 *B-120 5 4.65 0.61 4.68 0.59 4.17 10.16 4.65 *B-121 3 2.10 0.27 2.11 0.27 1.53 * 2.10 0.27

The cutting planes and the lower bound applied to the network flow model improved

its linear relaxations in 46.5% of the set A instances, and 32.2% of the set B

instances. In three instances, for the set A, and four instances, for the set B, the

lower bound provided by the linear relaxation increased one unit. With the cuts

and lower bounds, only one instance of the set A and six instances of the set B

have a lower bound that is not equal to the optimal integer solution. Although

the linear relaxations improved considerably with the cutting planes and the lower

bound, the computational times did not. We were able to solve one extra instance

of the set B, but we could not solve five of the set A instances that were solved

otherwise.

The IP model proposed in [145] was not able to solve to optimality about 49%


of the set A instances and about 83% of the set B instances. In fact, it did not

even solve the linear relaxations of about 32% of the set B instances. Moreover,

the lower bounds provided by its linear relaxations are considerably weaker than

the ones provided by the network flow model.

The branch-and-price algorithm described in [13] solved 77% of the instances

within the time limit of 7200 seconds, while the network flow model was able to

solve all the instances, according to table 5.3 (column AF ). Moreover, considering

the set of instances solved in both methods, the network flow model took 19% of

the time spent by the branch-and-price algorithm. We also tested the algorithm

proposed by Alves et al. using the set B instances. It solved 79% of the instances

within the time limit of 7200 seconds, while the network flow model was only not

able to solve two instances, according to table 5.4 (column AF ). Again, considering

the set of instances that both methods solved, the network flow model took 6% of

the time spent by the branch-and-price algorithm.

The set A and set B instances were also tested for the non-oriented case, AFRot,

for the horizontal or vertical first cut case, AFH/V , and finally, for the non-oriented

case with horizontal or vertical first cut, AFH/V+Rot.

Tables 5.5 and 5.6 describe the computational results for the set A and set B

instances, respectively, for the original version of the network flow model, AF , and

for the three other variants AFRot, AFH/V and AFH/V+Rot. Again, the values of

Z marked with an asterisk (*) represent instances in which the algorithm did not

find the optimal solution within the time limit of 7200 seconds.


Tab

le5.5

:V

ari

ants

for

the

setA

inst

an

ces.

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

A-1

3.30

40.

203.

30

40.3

13.0

03

0.7

03.0

03

1.0

0A

-236

.00

360.

3336

.00

36

0.5

036.0

036

0.4

236.0

036

0.8

6A

-38.

008

0.16

8.00

80.2

58.0

08

0.2

78.0

08

0.4

4A

-42.

673

0.19

2.67

30.2

22.6

73

0.2

82.6

73

0.4

1A

-512

.53

130.

7511

.87

12

7.0

312.2

113

2.0

311.8

712

8.3

8A

-61.

892

0.09

1.55

20.2

81.8

92

0.1

71.7

02

0.4

4A

-713

.13

140.

6410

.81

11

1.0

913.0

013

0.8

010.8

111

2.3

6A

-81.

072

0.08

0.89

10.1

61.0

72

0.1

40.8

91

0.2

7A

-960

.67

6127

.34

57.2

3*

*58.0

359

442.1

4*

**

A-1

01.

973

0.34

1.93

31.8

11.9

73

0.5

91.9

12

2.3

1A

-11

45.7

646

7.72

44.5

045

2929.4

545.6

146

137.5

544.3

3*

*A

-12

14.0

014

0.33

14.0

014

0.9

714.0

014

0.7

514.0

014

2.4

4A

-13

13.5

314

0.91

12.5

813

7.6

113.2

014

1.5

912.3

513

48.8

9A

-14

66.8

267

183.

5365

.51

**

66.2

167

1015.0

5*

**

A-1

538

.94

391.

1137

.31

38

7.0

638.5

139

2.6

937.1

838

18.9

5A

-16

82.2

683

12.0

381

.66

**

81.9

282

750.8

480.5

7*

*A

-17

4.70

50.

344.

53

51.3

84.7

05

0.6

64.5

35

3.6

6A

-18

64.3

965

2.97

62.8

263

1006.9

264.0

565

12.8

162.7

263

2068.1

9A

-19

57.2

358

17.3

351

.83

**

54.7

2*

*51.7

5*

*A

-20

26.1

027

1.22

24.3

225

15.7

325.0

226

2.7

323.7

624

175.8

9A

-21

27.2

428

57.6

325

.42

26

3171.9

726.3

827

328.6

125.2

526

5726.9

1A

-22

2.40

30.

242.

40

30.8

62.4

03

0.5

02.4

03

1.7

3A

-23

12.9

214

0.77

12.2

513

6.4

512.5

213

1.5

512.1

813

16.0

5A

-24

34.3

235

1012

.61

**

**

**

**

*A

-25

17.0

218

2595

.83

**

**

**

**

*A

-26

7.05

819

.98

**

*6.8

87

14.7

0*

**

A-2

719

.04

2039

65.5

5*

**

**

**

**

A-2

810

.81

1211

8.88

**

*10.3

2*

**

**

A-2

927

.28

2831

55.8

3*

**

**

**

**


Tab

le5.5

:V

ari

ants

for

the

setA

inst

an

ces

(conti

nu

ed).

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

A-3

03.

594

15.5

03.

43

45295.1

63.5

14

409.3

43.3

9*

*A

-31

7.47

823

.39

7.28

84665.3

47.3

78

3875.4

9*

**

A-3

226

.85

2783

.91

**

**

**

**

*A

-33

34.6

335

4793

.70

**

**

**

**

*A

-34

5.20

63.

134.

78

5311.6

35.0

86

13.9

14.7

55

3666.5

6A

-35

16.3

617

44.7

2*

**

**

**

**

A-3

68.

889

1.72

7.17

88.5

88.8

89

2.8

67.0

58

19.1

1A

-37

4.63

50.

864.

39

52.8

84.5

35

1.4

54.3

15

6.5

9A

-38

22.1

123

11.8

321

.44

22

2337.3

621.7

7*

**

**

A-3

93.

024

1.00

2.90

48.1

32.9

54

1.9

12.8

93

23.8

6A

-40

15.8

117

79.2

015

.28

16

603.2

815.3

916

51.8

414.9

6*

*A

-41

18.5

019

4.72

18.1

719

18.6

317.2

018

6.2

717.2

018

33.0

8A

-42

7.17

80.

956.

50

72.0

07.1

38

1.3

36.5

07

3.4

5A

-43

6.38

71.

006.

18

73.8

06.3

87

1.6

35.8

06

6.5

5


Tab

le5.6

:V

ari

ants

for

the

setB

inst

an

ces.

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

B-1

74.9

576

0.59

71.8

772

4.4

171.8

072

1.8

070.5

871

12.5

5B

-217

0.79

171

1.08

161.

89

162

6.1

1159.5

5160

2.5

2157.0

0158

10.7

8B

-310

8.91

110

0.24

99.1

0100

0.6

1107.9

0108

0.5

897.7

698

1.2

7B

-420

.70

210.

9819

.79

20

1.9

118.6

719

1.4

718.5

119

3.8

4B

-510

.53

110.

4510

.53

11

0.5

29.8

110

0.5

99.8

110

0.6

6B

-69.

6710

0.38

9.33

10

0.8

69.0

09

0.7

89.0

09

1.8

6B

-753

6.25

537

0.34

531.

43

532

0.6

7533.4

4534

0.5

5530.0

0530

1.4

4B

-833

3.89

334

0.30

313.

88

314

1.3

8333.8

9334

0.9

1302.0

0302

3.2

8B

-984

.19

8563

.14

**

*84.1

885

1229.2

8*

**

B-1

077

.74

7838

8.45

**

*77.5

678

1318.5

5*

**

B-1

127

0.05

271

20.1

1*

**

269.7

4270

521.1

3*

**

B-1

246

4.96

465

0.34

463.

49

464

2.1

4462.8

7463

0.7

0443.2

6444

4.2

5B

-13

229.

1123

01.

3021

8.05

219

119.0

0229.1

1230

1.6

9217.6

7218

118.5

8B

-14

34.3

335

0.78

31.3

132

3.0

834.3

335

1.3

131.2

032

6.5

2B

-15

15.4

416

1.42

15.4

416

2.6

115.0

516

1.7

714.1

815

4.1

7B

-16

23.2

524

0.85

22.4

523

4.6

923.0

524

1.6

122.4

423

9.4

1B

-17

11.1

312

0.17

11.1

312

0.3

011.1

312

0.3

39.1

310

0.6

1B

-18

13.1

014

0.27

13.1

014

0.9

713.1

014

0.5

212.6

113

1.7

2B

-19

46.2

147

1.09

45.3

746

7.3

346.1

847

2.0

945.1

346

15.3

3B

-20

95.0

095

0.84

93.0

494

0.9

195.0

095

0.8

090.7

091

2.0

6B

-21

228.

3322

90.

4221

9.75

220

4.7

5228.3

3229

1.4

5214.2

5215

8.2

2B

-22

16.5

017

0.61

15.7

316

2.1

116.3

817

1.0

815.7

316

4.6

6B

-23

84.3

385

1.11

82.4

883

51.8

884.2

985

2.4

982.4

783

89.5

2B

-24

8.83

90.

208.

73

90.5

28.8

39

0.3

68.7

39

0.9

5B

-25

46.7

547

0.34

46.1

347

1.1

946.7

547

0.7

443.6

744

2.4

2B

-26

127.

7112

80.

6612

0.02

121

163.6

9126.9

2127

1.8

9119.1

2120

58.2

8B

-27

220.

5122

10.

3421

9.05

220

1.2

5219.7

3220

0.6

1210.8

9211

3.5

9B

-28

36.9

838

0.30

36.3

737

4.4

236.7

537

1.0

236.3

737

10.5

9B

-29

18.2

419

0.39

17.3

118

2.0

517.6

618

0.7

517.2

618

4.5

2


Tab

le5.6

:V

ari

ants

for

the

setB

inst

an

ces

(conti

nu

ed).

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

B-3

056

.77

5716

.52

55.4

656

336.8

056.1

057

6.5

555.3

856

1420.7

8B

-31

29.3

330

0.61

28.8

129

1.1

328.5

529

0.8

927.6

928

2.1

4B

-32

191.

0719

23.

5618

5.98

186

969.1

3187.4

8188

7.8

3182.4

5183

453.2

8B

-33

50.8

952

0.75

47.9

949

1.7

249.7

850

0.9

547.9

048

5.0

6B

-34

82.6

783

1.61

78.4

579

12.9

776.1

777

2.8

875.1

776

22.2

8B

-35

95.8

396

0.39

94.1

795

1.2

295.8

396

0.7

492.7

293

2.5

6B

-36

95.8

396

0.41

95.5

696

0.8

895.5

696

0.6

695.5

696

2.2

7B

-37

52.2

253

0.39

50.9

551

0.9

250.9

851

0.8

350.0

050

2.8

0B

-38

34.5

835

0.17

34.5

835

0.3

432.5

033

0.3

432.5

033

0.7

2B

-39

35.0

035

0.19

35.0

035

0.3

435.0

035

0.2

829.1

730

0.8

9B

-40

18.6

419

2.47

18.6

419

2.3

918.6

419

1.1

618.2

719

5.2

8B

-41

16.6

717

4.22

15.2

616

2.8

115.6

216

1.2

315.2

616

5.3

6B

-42

15.3

916

0.28

15.1

516

2.4

515.0

716

1.0

314.9

215

4.8

0B

-43

5.33

60.

695.

33

60.6

75.3

36

0.4

45.1

36

1.2

2B

-44

23.3

224

12.3

822

.86

23

215.9

922.8

123

3.0

222.5

623

2051.2

0B

-45

10.1

711

2.88

10.1

711

0.2

510.1

711

0.2

510.1

711

0.5

5B

-46

26.9

928

1.06

24.3

525

1.6

626.6

327

0.9

424.3

525

3.0

2B

-47

7.92

80.

137.

80

80.6

17.9

28

0.4

57.7

58

1.2

0B

-48

13.4

514

10.4

113

.44

14

5.3

113.4

514

1.1

113.4

414

10.7

8B

-49

20.0

721

0.44

17.4

718

0.7

520.0

721

0.6

317.4

718

1.4

5B

-50

70.3

671

0.11

70.3

671

0.2

570.3

671

0.2

065.6

766

0.4

7B

-51

62.1

463

1.05

61.6

662

2.6

662.1

463

1.4

154.2

855

7.6

9B

-52

119.

5012

00.

1411

3.81

114

0.2

7119.5

0120

0.2

0113.8

1114

0.4

5B

-53

2.80

30.

472.

46

31.1

42.8

03

0.5

82.4

53

4.0

0B

-54

12.2

013

0.27

11.4

912

1.2

212.1

013

0.5

911.4

412

2.5

3B

-55

49.9

451

0.86

49.8

550

110.8

649.6

250

1.6

149.3

650

83.3

0B

-56

213.

8321

44.

0521

0.37

211

471.7

7213.4

7214

9.5

6209.1

0210

834.1

9B

-57

43.8

644

34.5

542

.55

43

737.0

943.8

344

123.7

742.4

7*

-B

-58

19.1

220

0.36

18.7

619

7.6

318.9

719

0.9

418.4

319

9.1

3B

-59

10.3

911

0.22

9.88

10

0.5

310.3

911

0.4

29.8

810

1.1

6


Tab

le5.6

:V

ari

ants

for

the

setB

inst

an

ces

(conti

nu

ed).

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

B-6

012

.17

130.

5311

.93

12

1.7

812.0

813

0.8

911.9

312

3.9

4B

-61

5.33

60.

244.

91

50.5

85.3

36

0.3

64.9

15

1.0

2B

-62

17.5

518

0.53

17.2

618

1.7

517.4

218

0.8

317.2

418

4.1

9B

-63

37.7

638

1.06

36.9

537

4.6

637.3

638

1.8

135.8

736

10.3

4B

-64

19.2

820

1.16

19.2

520

13.3

919.1

720

1.9

718.7

419

27.4

2B

-65

171.

5617

23.

1716

8.93

169

2291.1

6170.6

1171

9.2

0166.3

0*

*B

-66

40.0

041

1.17

37.9

438

14.8

838.6

339

2.2

237.7

338

26.4

4B

-67

260.

9626

221

.86

259.

78

**

252.1

1253

84.6

7*

**

B-6

848

.94

492.

2847

.56

48

1909.6

648.4

049

6.0

547.5

548

2869.8

0B

-69

19.9

420

0.22

19.6

720

0.3

719.8

020

0.4

519.6

720

0.7

5B

-70

116.

1611

72.

0211

4.14

115

454.9

8115.2

7116

4.1

1113.3

9114

843.5

3B

-71

65.0

866

1.34

63.5

964

21.8

163.6

864

2.3

862.6

963

46.5

9B

-72

33.7

834

0.55

33.6

234

8.0

233.7

834

1.3

033.5

234

17.9

2B

-73

18.2

619

0.64

17.6

418

7.2

418.1

819

1.0

017.6

118

12.6

7B

-74

308.

5930

91.

5530

8.59

309

4.5

8308.5

9309

1.0

0308.5

9309

7.7

7B

-75

125.

8612

60.

7011

8.09

119

31.0

9124.9

2125

1.0

8117.2

5118

101.8

0B

-76

220.

5122

10.

3321

9.05

220

1.2

4219.7

3220

0.6

6210.8

9211

3.5

9B

-77

207.

0020

81.

6120

1.88

**

206.3

8207

3.6

6201.4

9202

3235.3

3B

-78

72.5

373

0.28

68.4

269

3.0

671.8

172

0.7

768.4

269

7.5

9B

-79

49.5

350

0.50

46.8

747

3.3

149.5

250

0.8

946.7

047

8.9

5B

-80

56.9

157

0.50

55.0

756

6.4

556.2

557

0.9

255.0

656

39.5

8B

-81

424.

0442

51.

8341

8.34

419

143.6

6409.9

6411

4.1

6396.6

7397

129.9

5B

-82

104.

8310

51.

1910

3.97

104

12.1

999.1

8100

2.2

797.2

598

33.8

8B

-83

96.5

597

1.64

95.3

496

57.5

895.2

796

3.2

894.4

495

96.7

2B

-84

102.

1710

31.

2798

.41

99

20.8

8100.1

9101

2.2

597.5

198

318.0

8B

-85

105.

0510

60.

5810

4.19

105

9.3

8104.0

1105

1.6

4104.0

0104

26.5

2B

-86

105.

0510

60.

5610

4.19

105

9.3

9104.0

1105

1.6

3104.0

0104

26.4

7B

-87

127.

7312

80.

6712

0.04

121

29.1

3126.9

4127

0.9

9119.1

5120

308.7

4B

-88

225.

6222

60.

3322

4.32

225

1.1

9225.0

8226

0.6

2216.4

4217

2.4

2B

-89

352.

4435

30.

6633

4.43

335

25.7

3352.4

4353

1.0

2334.1

9335

118.3

0


Tab

le5.6

:V

ari

ants

for

the

setB

inst

an

ces

(conti

nu

ed).

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

B-9

084

0.42

841

0.50

802.

98

803

3.2

7830.3

9831

0.8

6781.8

7782

7.2

4B

-91

316.

5031

70.

8130

0.42

301

6.4

7316.5

0317

1.2

7299.9

3300

21.7

2B

-92

175.

2417

60.

3417

1.51

172

6.3

3171.1

8172

0.8

8169.4

3170

9.7

3B

-93

200.

0720

10.

7819

4.68

195

11.8

0195.1

5196

2.0

9192.9

7193

49.9

2B

-94

11.0

812

0.44

10.7

811

0.6

910.5

411

0.6

110.5

411

1.1

1B

-95

59.6

060

0.34

54.0

054

0.8

359.6

060

0.5

253.2

654

1.4

7B

-96

123.

7512

40.

6311

4.13

115

1.9

1123.7

5124

0.7

7111.8

2112

5.4

2B

-97

60.0

060

0.20

58.3

359

0.5

060.0

060

0.3

458.3

359

1.1

3B

-98

67.0

268

1.27

60.6

761

6.7

566.6

867

0.9

759.7

960

21.1

6B

-99

1095

.00

1095

0.78

1077

.75

1078

1.0

61095.0

01095

0.9

21076.2

91077

2.1

1B

-100

58.5

759

15.8

657

.86

58

525.3

458.0

759

2.6

457.8

558

1796.4

4B

-101

58.3

459

3.81

52.7

153

698.5

058.2

959

4.3

352.6

953

2263.5

6B

-102

45.2

246

2.13

43.8

244

195.5

944.2

345

3.9

743.8

244

1147.6

4B

-103

39.8

641

3.27

37.8

238

7.4

439.7

540

1.5

337.6

138

17.7

2B

-104

2.33

30.

142.

33

30.2

32.3

33

0.2

42.1

93

0.4

9B

-105

33.3

334

0.11

33.3

334

0.2

233.3

334

0.2

233.3

334

0.4

5B

-106

70.1

871

2.05

66.0

267

530.6

967.2

068

3.2

865.9

266

2496.7

0B

-107

299.

0030

058

6.72

**

*298.8

8299

5400.1

3*

**

B-1

0818

3.72

**

177.

41

178

29.0

0178.8

9179

4.6

3174.2

1175

94.6

3B

-109

25.1

426

4.61

23.6

024

346.5

022.7

123

74.5

922.4

623

4123.8

6B

-110

3.20

40.

133.

20

40.1

33.2

04

0.2

23.2

04

0.3

0B

-111

55.1

756

79.6

0*

**

55.0

856

2287.8

6*

**

B-1

128.

109

1.17

7.29

8154.7

78.0

79

2.5

27.1

28

567.8

6B

-113

230.

4423

10.

4222

9.44

230

20.8

6230.4

4231

0.6

3228.8

5229

7.7

0B

-114

279.

98*

**

**

278.7

7*

**

**

B-1

1514

.19

159.

0311

.04

12

0.5

914.1

915

0.3

911.0

412

1.1

3B

-116

191.

2719

217

65.7

8*

**

190.9

7*

**

**

B-1

175.

306

0.66

5.30

60.1

35.3

06

0.2

25.3

06

0.2

2B

-118

35.2

536

4.19

33.7

634

1.1

635.2

536

0.7

033.7

634

2.1

6B

-119

143.

3214

48.

1414

1.47

142

116.6

9142.4

7143

3.3

1141.3

5142

294.0

2


Tab

le5.6

:V

ari

ants

for

the

setB

inst

an

ces

(conti

nu

ed).

AF

AFRot

AFH/V

AFH/V+Rot

Nam

eZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

ZLR

Zt(s)

B-1

204.

655

0.61

4.42

52.0

84.5

95

0.9

54.3

85

5.5

8B

-121

2.10

30.

271.

66

20.6

22.0

02

0.5

01.6

22

1.4

2


In what concerns these variants, AFRot, AFH/V , AFH/V+Rot, the network flow

model solved, respectively, about 70%, 77% and 58% of the set A instances, and

93%, 98% and 92% of the set B instances, within the time limit of 7200 seconds.

There was a reduction in the number of stock sheets used of about 4%, 2% and 7%

for the set A instances, and 3%, 1% and 5% for the set B instances.

5.9 Conclusions

In this chapter, we presented an exact network flow model for the 2D-CSP, with

two stages and the guillotine constraint. We solved it with a commercial software

(CPLEX), considering explicitly all its variables and constraints. Reduction cri-

teria were applied to reduce the size and symmetry of the model, and increase

its efficiency. We derived a new family of cutting planes and a new lower bound.

We also explored some variants of the problem such as the rotation of items, and

the possibility of considering the horizontal or vertical orientation for the first cut.

Finally, we presented computational results based on real instances from the fur-

niture industry. For the sets of tested instances, the model proved to be stronger

and more efficient than other methods presented in the literature.

Chapter 6

The Vehicle Routing Problem

“He who loves practice without theoryis like the sailor who boards ship

without a rudder and compassand never knows where he may cast.”

Leonardo da Vinci

Contents6.1 Problem Definition and Some of its Variants . . . . . . 116

6.1.1 Depot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.2 Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1.3 Customers . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1.4 Planning Period . . . . . . . . . . . . . . . . . . . . . . 120

6.1.5 Type of Input Data . . . . . . . . . . . . . . . . . . . . 120

6.1.6 Objective Function . . . . . . . . . . . . . . . . . . . . . 121

6.2 Mathematical Formulations . . . . . . . . . . . . . . . . 121

6.2.1 Two-index Network Flow Model . . . . . . . . . . . . . 121

6.2.2 Multicommodity Flow Model . . . . . . . . . . . . . . . 122

6.2.3 Set Partitioning Model . . . . . . . . . . . . . . . . . . . 123

6.3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . 124

6.3.1 Exact Approaches . . . . . . . . . . . . . . . . . . . . . 124

6.3.2 Heuristic and Meta-Heuristic Approaches . . . . . . . . 125

115

116 6. The Vehicle Routing Problem

6.1 Problem Definition and Some of its Variants

The Vehicle Routing Problem (VRP) belongs to the class of routing problems

and is related with the issue of finding paths along the edges of a graph. This

problem was formally stated for the first time in [64]. The authors introduced it

as a generalization of the Traveling Salesman Problem (TSP), another well know

routing problem. In this chapter, we give an overview of the VRP, we define it,

discuss several of its variants and present some models and solution methods from

the literature.

The VRP is a combinatorial optimization problem whose application area is

very wide. It is of major practical relevance for every activity that implies any

kind of route design, whether we are talking about a transportation system where

there is a fleet of vehicles to manage (public transports, courier services, trans-

portation of materials, distribution services) or of less intuitive fields as robotics

or microprocessors. It is well known that the transportation costs represent a big

percentage of the total logistics costs, and the great growth in the number of soft-

ware packages concerning the VRP, available in the market, illustrates the fact

that industries understand the economical savings that come with a more efficient

use of their resources. A more efficient planning of routes may also translate into

environmental, social and energy benefits. Therefore, and not surprisingly, this is

one of the most studied problems in the Operations Research field.

Generally speaking, the VRP is the problem of scheduling a fleet of vehicles to

visit a set of customers, to whom (or from whom) they must deliver (or collect)

a demanded quantity of supplies. The problem consists of finding the best set of

routes, according to a given objective function, such that all operational constraints

of the vehicles are respected, and the set of customers is covered. The classical

version of the VRP is commonly called the Capacitated Vehicle Routing Problem

(CVRP). For this version, there is a single depot, o, which is the beginning and the

6.1. Problem Definition and Some of its Variants 117

end of all vehicle routes, and a fleet of homogeneous vehicles, with a capacity of

Q units. It is assumed that there are K available vehicles in the fleet. The set of

customers is represented by N = 1, ..., n. There is a distance, dij, and a traveling

time, tij, associated to every pair i, j ∈ N ∪ o, and a non-negative demand, di,

associated to every customer i ∈ N .

Naturally, any real-world system requires more flexibility than the overly sim-

plified classical version of the problem. There are several specific constraints and

particularities that have to be taken into account in a real system. In order to

deal with this, many variants and extensions of this model have been presented in

the literature, featuring some aspects that occur in real-world applications. These

versions are usually called Rich Vehicle Routing Problems (RVRP). They consider

additional constraints to the basic model, in order to adapt it to specific scenarios.

Of course, considering additional assumptions in the model implies increasing its

complexity. Therefore, these additional constraints are often considered separately.

We can say that there are six fundamental aspects to consider in the VRP: the

depots, the vehicles, the customers, the planning period, the type of input data and

the objective function. Each of these entities may have different assumptions that

originate several variants of the problem. Annotated bibliographies of the VRP

and its variants can be found in [136, 131, 150].

6.1.1 Depot

There can be one single depot, to which all vehicles are associated and from which

they all must begin their routes and return to, or multiple depots. In the latter case,

there are several aspects to consider. Each depot may have identical or different

capacities, there can be customers that must be served by a certain depot, or not.

When the depots to be open must be chosen from a set of available ones, the variant

is known as the Location Routing Problem (LRP). In the multiple depot version,

the problem becomes harder, as it is necessary not only to define the set of routes


to perform, but also the depot to which each customer is associated. In the Open

Vehicle Routing Problem (OVRP), vehicles are not required to return to the depot

at the end of the routes, which means that a route corresponds to a hamiltonian

path rather than to a hamiltonian cycle.

6.1.2 Vehicles

The fleet of vehicles can be homogeneous or heterogeneous. This means that differ-

ent vehicles may have different characteristics, like different capacities, the ability

of reaching or not some of the customers locations, the presence of some specific

equipments as refrigerators or other containers or even different operational costs.

Depending on the kind of goods, the capacity constraint of the vehicles may not

be enough to ensure that the vehicle can carry its assigned load. Vehicles must

be packed with the customers’ loads. Some authors have combined the problem of

selecting routes with the problem of loading the vehicles that perform them. This

is a combination of two NP -hard problems, the VRP and the Bin Packing Problem

(BPP). This problem is known as the Capacitated Vehicle Routing Problem with

n-Dimensional Loading Constraints (nL-CVRP). Some heuristic approaches for the

n = 2 and n = 3 cases have been proposed in the literature [70, 96, 161, 89], but

only one exact solution method has been presented. In [118], the authors propose

a branch-and-cut algorithm to solve exactly the 2L-CVRP.

Operating a vehicle implies the use of driver. In practice, this means that there

can be additional operational constraints, like a maximum number of driving hours,

drivers breaks or other scheduling aspects. The variant that combines these aspects

is called the Vehicle Routing and Crew Scheduling Problem (VRCSP). There can

also be additional constraints on the routes. The most common is the existence of

a maximum route duration. This means that if a route takes more than a given

time, it is not considered as being feasible. This can be very relevant if we consider

legal restrictions on the drivers’ working hours or even in the delivery of perishable

6.1. Problem Definition and Some of its Variants 119

supplies. Similarly, a maximum route length can be considered. Some authors

have approached a VRP variant that considers that a vehicle can be assigned to

more than one route per planning period. This variant has been addressed as the

Multi Trip Vehicle Routing Problem or Vehicle Routing Problem with multiple

routes (MVRP). It was first approached in [86]. Some heuristic solution methods

[179, 39, 169, 174, 167, 9] are described in the survey provided in [58].

6.1.3 Customers

Different additional constraints concerning customers are often considered in the

VRP, giving rise to different variants of the problem. In the Pickup and Delivery

VRP, it is assumed that the vehicles may deliver or pickup supplies from customers.

The VRP with Backhauls is similar but with the additional constraint that all

deliveries must be performed before the pickups begin. In the Split Delivery VRP,

customers can be served by more than one vehicle.

An also very known and studied variant of the problem is the VRP with Time

Windows. For this problem, there is a time period associated to each customer,

and also to the depot. Each customer must be served within its established period

of time, and all routes must be performed within the depot’s time window. Along

with the distances between all customers and depots, there are also traveling and

service times. Sometimes it is admissible to start the service at a customer after

its time window has closed, with some penalty. In this case, the time windows are

said to be soft. A survey on the Vehicle Routing Problem with Time Windows

is presented in [40, 41], where the authors describe heuristic and metaheuristic

solution methods from the literature.

The visit to all customers may or may not be mandatory. It can happen that

the available fleet of vehicles does not have the capacity to serve the complete set

of customers, or that the effort to do so is not compensated by the revenue that

would come from it. When this is the case, usually there is a penalty associated to


each customer whose service is not performed. This penalty may be different for

different customers.

The Arc Routing Problem (ARP) is a variant of the VRP that considers that

customers are located along the arcs of the graph rather than at its nodes [73].

This variant of the VRP arises in areas like garbage collection or mail delivery.

6.1.4 Planning Period

The Periodic VRP (PVRP) considers planning periods of multiple days, in opposi-

tion to the classical version that considers planning periods of a single day. There

is a frequency associated to each customer, which represents the number of times

the customer needs to be visited during the planning horizon. The visits to the

customer must be made according to an allowable combination. Let us consider the

case where the planning horizon is equal to seven days D1, D2, D3, D4, D5, D6, D7.

If a given customer has a visiting frequency of three, it means that during those

seven days it must be visited three times. Suppose it does not want to be visited in

consecutive days nor in D7, and demands to be visited in D1. In that case, the only

allowable combinations of visiting days for that customer would be D1, D3, D5,

D1, D3, D6 or D1, D4, D6. In [162], the authors provide a classification of prob-

lems concerning this variant.

6.1.5 Type of Input Data

When there is any component of the problem with a random behavior, i.e., if there

is uncertainty aspects like the customer’s demands, the costs or the service times,

the classic deterministic problem becomes a Stochastic Vehicle Routing Problem

(SVRP). Some examples of applications of the SVRP are the refuse collection or

snow removal. A summary of the literature concerning this variant is given in [97].

6.2. Mathematical Formulations 121

6.1.6 Objective Function

The most common objective of VRPs is the minimization of costs, whether these

costs only represent the traveled distances or a combination of several direct or

indirect operational expenses and revenues. But there can be other objectives, like

the maximization of the service level or customer satisfaction, the minimization of

the number of vehicles to be used, the maximization of the balance between the

different assigned routes to each vehicle or the minimization of the environmental

impact, among others. In [122], the authors present a survey of multi-objective

algorithms for routing problems.

6.2 Mathematical Formulations

Over the years, several formulations for the CVRP have been proposed in the

literature. In this section we present a Two-index Network Flow Model, with an

integer variable associated to each arc, a Multicommodity Flow Model, and a Set

Partitioning Model, which is usually used in a column generation framework, as all

feasible routes are implicitly enumerated.

Let us consider graph G = (V,A), being V = N ∪ o its set of nodes and

A = (i, j) : i, j ∈ V its set of arcs. The cost of selecting arc (i, j) for the solution

is cij and binary variables xij are equal to 1 if and only if arc (i, j) belongs to the

solution.

6.2.1 Two-index Network Flow Model

This formulation is an extension of the model proposed by Dantzig et al. [63] for

the TSP, and was originally presented by Laporte et al. [134]. It was the base of

the formulation of the branch-and-cut algorithms proposed by Naddef and Rinaldi

[163] and Lysgaard et al. [146].


Let r(S) be the minimum number of vehicles needed to serve all customers in

set S ⊆ N . Computing this number is NP-hard (it requires solving a BPP) and

thus, it can be replaced by a lower bound, for example⌈∑

i∈S diQ

⌉. The model states

as follows.

min∑

(i,j)∈A

cijxij (6.1)

s.t.∑j∈V

xoj = K, (6.2)∑j∈V

xjo = K, (6.3)∑i∈V

xij = 1 ∀j ∈ V, (6.4)∑j∈V

xij = 1 ∀i ∈ V, (6.5)∑i∈S

∑j∈S

xij ≤ |S| − r(S), ∀S ⊆ V \o, S 6= ∅, (6.6)

xij ∈ 0, 1 ∀i, j ∈ V. (6.7)

Constraints (6.2)-(6.5) ensure that exactly K routes are created, and all customers

are visited once and only once (degree constraints). Constraints (6.6) ensure that

no subtours are created and that the capacity constraints are not violated.

6.2.2 Multicommodity Flow Model

A multicommodity flow formulation was originally proposed by Garvin et al. [92]

and later extended by Gavish and Graves [95]. Baldacci et al. [22] proposed a

branch-and-cut algorithm based on a formulation of this type for the CVRP.

For this formulation, it is necessary to consider an extended graph G′ = (V ′, A′),

where V ′ = V ∪o′, being o′ a copy of the depot, to where vehicles must return at

the end of the routes, and A′ = (i, j) : i, j ∈ V ′. Additional variables yij related

to the load of a vehicle on arc (i, j) must also be considered. Variable yji represents

the empty space of the vehicle on arc (i, j) and thus, yji = K − yij (6.13).

6.2. Mathematical Formulations 123

min∑

(i,j)∈A′cijxij (6.8)

s.t.∑j∈V ′

(yji − yij) = 2di ∀i ∈ V ′\o, o′, (6.9)∑j∈V ′\o,o′

yoj =∑

i∈V \o,o′

di, (6.10)

∑j∈V ′\o,o′

yjo = KQ−∑

i∈V \o,o′

di, (6.11)

∑j∈V ′\o,o′

yo′,j = KQ, (6.12)

yij + yji = Qxij ∀(i, j) ∈ A′, (6.13)∑i∈V ′

xik +∑j∈V ′

xkj = 2 ∀k ∈ V ′\o, o′, (6.14)

yij ≥ 0 ∀(i, j) ∈ A′, (6.15)

xij ∈ 0, 1 ∀(i, j) ∈ A′. (6.16)

Constraints (6.9) are flow conservation constraints for the commodity flow vari-

ables and constraints (6.14) are degree constraints. Consistent flows between o and

o′ are imposed by constraints (6.10)-(6.12).

6.2.3 Set Partitioning Model

This formulation was originally proposed by Balisty and Quandt [25], and it implies

the use of column generation to dynamically generate feasible routes. Fukasawa et

al. [90] combine both two-index and set partitioning formulations in a branch-and-

cut-and-price method for the CVRP. Let R be the set of all feasible routes, xj a

binary variable equal to 1 if and only if route j is in the solution and aij a binary

coefficient equal to 1 if and only if customer i belongs to route j. The model states


as follows.

min∑

i∈1,...,|R|

cjxj (6.17)

s.t.

|R|∑j=1

aijxj = 1 ∀i ∈ V \o, (6.18)

|R|∑j=1

xj = K, (6.19)

xj ∈ 0, 1 ∀j = 1, . . . , |R|. (6.20)

Constraints (6.18) ensure that every customer is visited once, while constraints

(6.19) define that K feasible routes are created.

6.3 Solution Methods

Several exact and heuristic methods for the VRP and its variants have been pre-

sented in the literature for the past decades. General surveys on these problems are

provided in Toth and Vigo [182] and Cordeau et al. [56]. Laporte [132] reports on

the main solution methods for the CVRP, in what concerns exact algorithms and

mathematical formulations, classical heuristics (namely, constructive and improve-

ment heuristics) and metaheuristics (local search, population search and learning

mechanisms).

In this section, we give a general overview of the state-of-art publications con-

cerning the VRP.

6.3.1 Exact Approaches

Chapters 2, 3 and 4 in the book edited by Toth and Vigo [182] provide an up to

date (2002) overview of exact algorithms for the CVRP, namely, branch-and-bound

algorithms (Toth and Vigo [181]), branch-and-cut algorithms (Naddef and Rinaldi

[163]) and set-covering-based algorithms (Bramel and Simchi-Levi [38]). This work

6.3. Solution Methods 125

was updated in the survey of Cordeau et al. [56], for both heuristics and exact

methods. Baldacci et al. [24] compares three exact approaches for the CVRP

[146, 90, 21] in an updated version of the authors’ previous survey [23]. These are,

according to the authors, the most effective exact methods currently available for

the CVRP.

Lysgaard et al. [146] proposed a branch-and-cut algorithm based on a two-

index vehicle flow model. The authors use several separation algorithms to derive

classes of valid inequalities. These valid inequalities work as cutting planes and are

used, according to the authors, to strengthen the very weak linear programming

relaxation of the original model.

Fukasawa et al. [90] proposed an exact algorithm for the CVRP that con-

sistently solves instances with up to 135 customers. It combines both two-index

vehicle flow and set partitioning formulations.

Baldacci et al. [21] proposed an exact solution method for the CVRP that,

according to the authors, can be adapted to other variants of the VRP. It is based

on a set partitioning formulation with additional capacity and clique inequalities.

They use a dynamic route generation algorithm, named GENROUTE, to generate

feasible routes, and a bounding procedure that consists of computing a lower bound

for the linear relaxation based on a near-optimal solution for the dual problem.

6.3.2 Heuristic and Meta-Heuristic Approaches

When dealing with large-scale problems, it is clear that heuristic methods are

more suitable than the state-of-the-art exact methods, which until now cannot

consistently deal with CVRP instances with much more than 100 customers.

Chapters 5 and 6 of the book edited by Toth and Vigo [182] provide an up

to date (2002) overview of, respectively, classical heuristics (Laporte and Semet

[137]) and metaheuristics (Gendreau et al. [98]) for the CVRP. Cordeau et al.

[55] described and compared several heuristics and metaheuristics for the CVRP,


in what concerns their accuracy, speed, simplicity and flexibility. Gendreau et al.

[99] presented a categorized bibliography of metaheuristics for the VRP and its

variants.

Classical heuristics for the VRP are often classified in one of two categories:

constructive heuristics, where routes are constructed by sequentially adding arcs

until a feasible solution is found, or improvement heuristics, where efficient solutions

are found by the exchange of sets of arcs from an initial solution. One of the first

and most know classical heuristics for the VRP is the savings heuristic proposed

by Clarke and Wright [48]. Several enhancements of this algorithm have been

proposed by other authors [93, 206, 10, 27].

Metaheuristics are among the most efficient methods for the VRP, and have

been applied to this problem for the last two decades. Among these are simulated

annealing, tabu search, VNS and genetic algorithms. Some methods are hybridiza-

tions of two or more of these algorithms.

Chapter 7

An Exact Algorithm for the

Vehicle Routing Problem with

Time Windows and Multiple

Routes1

“If you don’t know where you are goingany road will get you there.”

Lewis Carroll


7.2 Integer Programming Models: State-of-the-art . . . . 129

7.2.1 Problem Definition and Notation . . . . . . . . . . . . . 129

7.2.2 A Compact Formulation . . . . . . . . . . . . . . . . . . 130

7.2.3 A Column Generation Model . . . . . . . . . . . . . . . 132

7.3 A New Pseudo-Polynomial Network Flow Model . . . 133

7.3.1 Vehicle Routes . . . . . . . . . . . . . . . . . . . . . . . 133

7.3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.3.3 Arc Reduction . . . . . . . . . . . . . . . . . . . . . . . 137

7.4 Iteratively Refining the Discretization . . . . . . . . . . 141

7.4.1 Initial Rounding Strategy . . . . . . . . . . . . . . . . . 142

7.4.2 Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . 144


7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 153

1The work presented in this chapter has been submitted [148]

127

128 7. An Exact Algorithm for the MVRPTW

7.1 Introduction

In this chapter, we address the Vehicle Routing Problem with Time Windows

and Multiple Routes (MVRPTW). It is a variant of the VRP with time windows

that considers that a vehicle can be assigned to more than one route per planning

period. Despite its apparent practical relevance (delivering perishable goods, for

example), this variant of the classical VRP has not been the subject of a large

number of studies. It was first approached in [86] and some heuristic solution

methods [179, 39, 169, 167, 174, 9] are described in the survey provided in [58]. In

what concerns exact methods, to our knowledge there is only one contribution in

the literature [20]. It is a generalization of a previous method [19], which considers

the same problem but with a single vehicle available. It is a branch-and-price

algorithm, with a master problem that is a set-covering problem with variables

representing workdays, i.e., sequences of routes assigned to one vehicle for one

planning period. The pricing problem is an elementary shortest path problem

with resource constraints, formulated in a graph whose nodes represent vehicle

routes. The vehicle routes are generated a priori. This is possible because there

is an additional constraint on their duration, which makes the number of feasible

routes decrease drastically. With this exact solution method, the authors were

able to solve instances with up to 40 customers. We present a new exact solution

approach for the MVRPTW. As in [20], we consider the additional route duration

constraint and generate all feasible vehicle routes a priori. We propose a new

algorithm that is based on a pseudo-polynomial network flow model, whose nodes

represent discrete time instants and whose solution is composed of a set of paths,

each representing a workday. An issue of this model is that its size depends on the

duration of the workdays. The time instants we consider in the model are integer,

and so, when non integer traveling times occur, we use rounding procedures that

allow us to obtain a (strong) lower bound. Our model is then embedded in an exact

7.2. Integer Programming Models: State-of-the-art 129

algorithm that iteratively adds new time instants to the network flow model, and re-

optimizes it, until the solution found is proved to be feasible. Practically speaking,

the number of iterations is generally rather small (one in most of the cases). We

tested our algorithm on the benchmark used in [20] to compare our results with

theirs. Our method outperforms the column-generation based algorithm of [20]

in many cases. In the cases where the two methods find a solution, our method

drastically reduces the computational time needed. The chapter is organized as

follows. In section 7.2, we define our problem, along with its notation, and briefly

recall some integer programming models from the literature. We contribute to

the solution of this problem with a new network flow formulation, where variables

represent feasible vehicles routes. This model is described in section 7.3, as well as

some reduction criteria to eliminate some of the vehicle routes, in order to improve

its efficiency. The network flow model is embedded in an exact solution algorithm

that is thoroughly described in section 7.4. This algorithm was tested on a set of

benchmark instances from the literature. In section 7.5 we present the results of

those computational tests. Finally, some conclusions are drawn in section 7.6.

7.2 Integer Programming Models: State-of-the-

art

7.2.1 Problem Definition and Notation

In the MVRPTW, there is a single depot, denoted by o, which is the beginning

and the end of all the vehicle routes. The fleet of vehicles is homogeneous. All

the vehicles have a capacity of Q units. It is assumed that there are K available

vehicles in the fleet. The set of customers is represented by N = 1, ..., n. There

is a distance, dij, and a traveling time, tij, associated to every pair i, j ∈ N ∪ o.

Each customer i has a demand qi, a revenue gi, a service time si and a time window


[ai, bi], where ai is the earliest time and bi the latest time to start the service at

customer i. This means that if a vehicle arrives at customer i earlier than ai, it

must wait. We assume, without loss of generality, that a vehicle starts the service

at a customer as soon as possible. The service time for the depot is defined as

so = 0, and all the vehicle routes must respect the depot’s time window, [ao, bo],

which means that no vehicle can leave the depot before ao, nor access it after

bo. This time window represents the duration W of a workday. We assume that

bi + si + di0 ≤ b0, ∀i ∈ N .

Each vehicle can perform several routes during a workday. It means that it can

perform one route, reload at the depot and leave to the following route, until the

end of the workday. A route r is defined by a sequence of visits to a subset of

customers Nr ⊆ N . It is feasible if the sum of the demands of all customers that

belong to Nr does not exceed the vehicle capacity and if its sequence of visits is

such that it is possible to visit every customer within its time window. We also

consider that the service of all customers in the route cannot start later that tmax

time units after the route begins. We denote by R the set of all feasible routes.

For each route, there is also a setup time to consider. Before leaving the depot to

perform route r, the vehicle needs β∑

i∈Nrsi time units to load, β ∈ R+. Note that

it may not be possible to visit all customers due to the limitation on the number

of available vehicles. However, it is always desirable to visit as many customers as

possible.

7.2.2 A Compact Formulation

The problem can be formulated in a complete directed graph G = (V,A), being

V = N ∪ o its set of nodes and A = (i, j) : i, j ∈ V its set of arcs. This

compact formulation, where binary variables assign customers to routes and define

7.2. Integer Programming Models: State-of-the-art 131

consecutive pairs of routes, is defined in [20], and states as follows:

min∑r∈R

∑(i,j)∈A

dijxrij − α

∑r∈R

∑i∈N

giyri (7.1)

s.t.∑j∈V

xrij = yri , ∀i ∈ N, ∀r ∈ R, (7.2)∑r∈R

yri ≤ 1, ∀i ∈ N, (7.3)∑i∈V

xrih −∑j∈V

xrhj = 0, ∀h ∈ N, ∀r ∈ R, (7.4)∑i∈V

xroi = 1, ∀r ∈ R, (7.5)∑i∈V

xrio = 1, ∀r ∈ R, (7.6)∑i∈N

qiyri ≤ Q, ∀r ∈ R, (7.7)

tri + si + tij −M(1− xrij) ≤ trj , ∀(i, j) ∈ A, ∀r ∈ R, (7.8)

aiyri ≤ tri ≤ biy

ri , ∀i ∈ N, ∀r ∈ R, (7.9)

tro ≥ β∑i∈N

siyri , ∀r ∈ R, (7.10)

tri ≤ tro + tmax, ∀i ∈ N, ∀r ∈ R, (7.11)

tso +M(1− zrs) ≥ t′ro + β

∑i∈N

siysi , ∀r, s ∈ R, r < s, (7.12)∑

r∈R

∑s∈R|r<s

zrs ≥ |R| −K, (7.13)

xrij ∈ 0, 1, ∀(i, j) ∈ A, ∀r ∈ R, (7.14)

yri ∈ 0, 1, ∀i ∈ N, ∀r ∈ R, (7.15)

zrs ∈ 0, 1, ∀r, s ∈ R, r < s, (7.16)

tri ≥ 0, ∀i ∈ N, r ∈ R. (7.17)

The binary variables xrij and yri define, respectively, if arc (i, j) and customer i

belong to route r (7.2), whereas the binary variables zrs define if there is a vehicle

that performs route r followed by route s in its workday. Notation r < s means

that a same vehicle is assigned to perform route s after having performed route r.


Variables tri (7.15) represent the starting instant of service at customer i, if it is

served by route r, and tro and t′ro represent the starting and ending times of route

r, respectively. The objective function (7.1) translates the fact that it is always

desirable to visit as many customers as possible. Note that for the model to be

valid, constant α has to be set to a value that ensures that. Constraints (7.4)-(7.6)

are flow conservation constraints, and (7.7) and (7.13) define the vehicles’ capacity

and the size of the fleet, respectively. The fact that the visits to customers must

respect their time windows is expressed in (7.9). Every two clients with consecutive

visits in a same route must have compatible visit times (7.8), the same happening

with two consecutive routes performed by a same vehicle (7.12). Finally, the setup

time for every route must always be considered (7.10), (7.12).

7.2.3 A Column Generation Model

In [20], the authors propose a branch-and-price algorithm to solve the MVRPTW.

The master problem of the column generation scheme is a set covering model, where

each column represents a vehicle’s workday, w. A workday is a sequence of routes

assigned to a vehicle, to be performed during the stipulated planning period.

Let Ω be the set of all feasible workdays, dw and gw the total traveled distance

and revenue of workday w ∈ Ω, and aiw a binary coefficient that indicates whether

or not customer i is served in workday w. The formulation of [20] states as follows.

min∑w∈Ω

(dw − αgw)xw (7.18)

s.t.∑w∈Ω

aiwxw ≤ 1, i ∈ V, (7.19)∑w∈Ω

xw ≤ K, (7.20)

xw ∈ 0, 1, w ∈ Ω. (7.21)

Binary variables xw (7.21) define if workday w ∈ Ω belongs to the solution. The

objective function (7.18) must ensure that a customer may only not be visited if

7.3. A New Pseudo-Polynomial Network Flow Model 133

it is not possible due to time or vehicle constraints. Parameter α must be set to a

value that guarantees that. Constraints (7.19) state that each customer is visited

at most once. The number of workdays in a solution can never exceed the number

of available vehicles in the fleet (7.20).

The pricing problem generates feasible workdays, and it is formulated as an

elementary shortest path problem with resource constraints, defined in a graph

whose nodes represent vehicle routes, and whose arcs represent pairs of consecutive

routes. This means that all feasible routes must be generated a priori.

7.3 A New Pseudo-Polynomial Network Flow Model

In this section, we present a new network flow model for the MVRPTW, whose

variables represent feasible vehicle routes. As in [20], all vehicle routes are pre-

viously generated. The integer model is then solved with a commercial software

(CPLEX), explicitly considering all its variables. Because the nodes of the graph

represent time instants, a discretization of time is required. The discretization will

be discussed in the next section.

7.3.1 Vehicle Routes

A route r ∈ R may remain feasible when it begins at different time instants.

Therefore, for every route r, we consider that there are several routes rt, one for

each possible departure instant t. The duration of a route r, σr, may be different

for different departure instants, as the waiting times to serve customers may vary.

Let (i1, . . . , i|Nr|) be the sequence of customers visited in route r ∈ R. The first

possible time instant to end route r is T′−r = θri|Nr |

+si|Nr |+ti|Nr |o

, being θri|Nr |the first

possible instant to start service at the last customer i|Nr| in route r. It is possible

to recursively calculate T′−r , considering that θrih = maxθrih−1

+ sih−1+ tih−1ih , aih,

for h ∈ 1, . . . , |Nr|, with θri0 = ao. This means that beginning route r at any


instant t∗r ≤ T−r = θri1 − toi1 implies ending it at instant T′−r . Therefore, it is clear

that such a route is dominated by the route r that begins at instant T−r , and thus

it may not be considered. Figure 7.1 illustrates this situation.

Similarly, the last possible time instant to end route r is T′+r = φri|Nr |

+ si|Nr |+

ti|Nr |o, being φri|Nr |

the last possible instant to start service at customer i|Nr| in route

r and φirh = minφirh−1+ sih−1

+ tih−1ih , bih, for h ∈ 1, . . . , |Nr| with φri0 = bo.

This means that beginning a route in any instant after T+r = φri1 − toi1 implies that

the route is not feasible, as it does not respect at least one of the customers’ time

windows.

Note that if route r begins within the time interval [T−r , T+r ], it will have the

minimum duration, as the waiting times are minimized.

For each r ∈ R, the interval [T−r , T+r ] is computed as described. The number of

different feasible routes we consider is, therefore, equal to∑

r∈R

⌈T+r −T−r +1

U

⌉, being

U the time unit. We consider U = 1.

0 W

Figure 7.1: First and last beginning and ending instants of route r

7.3.2 The Model

In our network flow model, one workday of every vehicle corresponds to a path

in an acyclic directed graph Π = (∆,Ψ) (Figure 7.2). Its set of vertices ∆ =


0, 1, ...,W represents discrete time instants from 0 to the workday length W and

Ψ = (u, v)r : 0 ≤ u < v ≤ W,u ∈ [T−r , T+r ], v = u + σr, r ∈ R

⋃(u, v)o :

0 ≤ u < v ≤ W, v = u + 1 represents its set of arcs. Arcs correspond either to

feasible vehicle routes or to waiting time periods (unit arcs). These waiting time

arcs represent the instants of time, in a workday, that are spent by the vehicle at

the depot. Note that in this model, we adjust the beginning time instant of each

route r ∈ R to β∑

i∈Nrsi time instants before, in order to consider the loading

time of the vehicle. The model is formulated as a minimum flow problem, and

DEPOT

8

3

7

12

4

r

s

t

0

r

Wa b c d e

s t

11

10

1

2

9

6

5

(e,W)o

(a,b)o, (c,d)

o,

Figure 7.2: Example of one path on the network flow model

it has a pseudo-polynomial number of variables and constraints. Its variables λruv

correspond to the flow in arc (u, v)r, i.e., to the number of vehicles that go through

route r, leaving the depot at instant u and arriving at instant v of their workday.

Variable z corresponds to the total flow through the graph, and can be seen as the


return flow from vertex W to vertex 0. Coefficient dr represents the cost of route r,

i.e., the sum of the total traveled distance in r. The model states as follows.

min∑

(u,v)r∈Ψ

(dr − α

∑i∈Nr

gi

)λruv (7.22)

s.t.∑

(u,v)r∈Ψ|i∈Nr

λruv ≤ 1, ∀i ∈ N, (7.23)

−∑

(u,v)r∈Ψ

λruv +∑

(v,y)s∈Ψ

λsvy =

z , if v = 0

0 , if v = 1, . . . ,W − 1,

−z , if v = W

(7.24)

z ≤ K, (7.25)

λruv ≥ 0 and integer, ∀(u, v)r ∈ Ψ, (7.26)

z ≥ 0 and integer. (7.27)

The objective is to minimize the total traveled distance of all vehicles during

one workday (7.22). It may not be possible to visit all customers due to the

limited number of available vehicles (7.25). The inequalities in constraints (7.23)

express that. It is, however, always desirable to visit as many customers as possible.

Constraints (7.24) are the flow conservation constraints of the network. They

ensure that the amount of flow that goes into a node is equal to the amount of flow

that goes out of it.

The following example illustrates the structure of our model.

Example 7.1 Consider an instance of MVRPTW with five customers (n = 5),

two available vehicles (K = 2) with a capacity of Q = 10 units, tmax = 5 and

β = 0.2. Table 7.1 describes the coordinates (xi, yi), time window [ai, bi], demand

bi and service time si for node i ∈ N = 1, . . . , 5⋃o. The distance between

two nodes i and j is equal to the Euclidian distance between them. Table 7.2 lists

all feasible routes, their beginning intervals and durations, as described in section

7.3.1, and all arcs to consider in the model. Figure 7.3 represents the network flow


graph generated for this instance, and the corresponding optimal solution is shown

in Figure 7.4. In this solution, two vehicles are required, each with two routes to

perform in the workday. One of the vehicles would perform routes b and e, visiting

customers 4, 2 and 3. It would start loading to perform route b at time instant 3.44,

arrive at the depot at time instant 12.16, wait for 0.04 time instants, start loading

for route e at time instant 12.20, arriving at the depot at time instant 21.65, where

it would remain until the end of the workday. The second vehicle would perform

routes f and a, visiting customers 1 and 5. It would start loading to perform route

f at time instant 3.60, arrive at the depot at time instant 8.00, wait for 2.99 time

instants, start loading for route a at time instant 10.99, arriving at the depot at

time instant 20.60, where it would remain until the end of the workday. In this

solution, all customers are visited.

Table 7.1: Instances description (example 7.1)

Customer Coordinates Time Windows Demand Service

i (xi, yi) [ai, bi] bi si

o (0, 0) [0, 25] 0 01 (1, 0) [5, 6] 1 22 (0, 1) [12, 15] 7 23 (1, 2) [15, 18] 1 24 (3, 1) [7, 9] 2 25 (2, 3) [10, 15] 3 2

7.3.3 Arc Reduction

The variables represent feasible vehicle routes in our network flow model. Clearly,

reducing the number of arcs reduces the size of the model, increasing its efficiency.

The next four propositions define dominance rules that allow us to discard some

routes that are never interesting when compared to some other one, and also to


Table 7.2: Feasible routes (example 7.1)

Route Customers Beginning Interval Duration Arcs

r [T−r , T+r ] σr (u, v)r

a (5) [5.99, 10.99] 9.61 (6, 15)a; (7, 16)a; (8, 17)a;(9, 18)a; (10, 19)a; (11, 20)a

b (4) [3.44, 5.44] 8.72 (4, 12)b; (5, 13)b; (6, 14)b

c (3) [12.36, 15.36] 6.88 (13, 19)c; (14, 20)c;((15, 21)c; (16, 22)c

d (2) [10.60, 13.60] 4.40 (11, 15)d; (12, 16)d;(13, 17)d; (14, 18)d

e (2, 3) [10.20, 12.79] 9.45 (11, 19)e; (12, 20)e; (13, 21)ef (1) [3.60, 4.60] 4.40 (4, 8)f ; (5, 9)f

0 15 25105 20

Figure 7.3: Complete network flow graph (example 7.1)

reduce the number of different beginning instants for some of the routes.

The first dominance rule states that if two routes serve the same customers, a

route can be dropped if it has a larger or equal cost and if it does not begin after

nor end before the other.

Proposition 1 Let r, s ∈ R be two vehicle routes such that Nr = Ns. Let t ∈

[T−r , T+r ] and t′ ∈ [T−s , T

+s ]. If dr ≥ ds, t ≤ t′ and t + σr ≥ t′ + σs, route st′


0 15 25105 20

Figure 7.4: Optimal solution (example 7.1)

dominates route rt.

Proof. It is never better to consider route rt over route st′, as the first one

does not begin after nor end before the second one, both routes visit the same set

of customers and the cost of r is not smaller than the cost of s.

The second dominance procedure states that if all replications of a given route r end

after the beginning of any other route, then the latest replication of r dominates

the other replications of r.

Proposition 2 Let r ∈ R be a route such that T−r +σr ≥ maxT+s : s ∈ R, s 6= r.

All routes rt, t ∈ [T−r , T+r [, are dominated by route rT+

r.

Proof. Let T = maxT+s : s ∈ R. No route can begin after time instant T

and route r will always end after time instant T . Consequently, if route r belongs to

a vehicle’s workday, it is always the last one. Therefore, any route rt, t ∈ [T−r , T+r [,

can be replaced by the route rT+r

without losing the optimal solution, as t ≤ T+r and

no other route will be performed after r.

The third dominance rule states that if no routes begin in the interval of possible

ends for a route s, then the latest replication of s dominates the other replications

of s.

Proposition 3 If⋃r∈R[T−r , T

+r ]⋂

[T−s + σs, T+s + σs] = ∅, route sT+

sdominates


all routes st, with t ∈ [T−s , T+s [.

Proof. Let (λ∗, z∗) be an optimal solution for the flow model of an instance of

MVRPTW, where the route s−t− (respectively s+t+) precedes (resp. follows) the route

st, in the same workday. This means that we have λ∗s−

t−,t′− = λ∗st,t′ = λ∗s+

t+,t′+ = 1.

Since the the solution (λ∗, z∗) is feasible, T+s ≥ t ≥ t′−. If

⋃r∈R[T−r , T

+r ]⋂

[T−s +

σs, T+s + σs] = ∅, we have that t+ > T+

s + σs. Therefore, a solution (λ′, z∗), where

λ′ equals λ∗ by changing only the two components λ′st,t′ = 0 and λ′sT+s ,T

+s +σs

= 1,

remains feasible and has the same cost as (λ∗, z∗), and thus is an optimal solution.

The fourth dominance rule is similar to the third. It states that if no routes end

in the interval of possible beginning of a route s, then the earliest replication of s

dominates the other replications of s.

Proposition 4 If⋃r∈R[T−r + σr, T

+r + σr]

⋂[T−s , T

+s ] = ∅, route sT−s dominates

all routes st, t ∈]T−s , T+s ].

Proof. Let (λ∗, z∗) be an optimal solution for the flow model of an instance of

MVRPTW, where the route s−t− (respectively s+t+) precedes (resp. follows) the route

st, in the same workday. This means that we have λ∗s−

t−,t′− = λ∗st,t′ = λ∗s+

t+,t′+ = 1.

Since the solution (λ∗, z∗) is feasible, T−s + σs ≤ t′ ≤ t+. If⋃r∈R[T−r + σr, T

+r +

σr]⋂

[T−s , T+s ] = ∅, we have that t′− < T−s . Therefore, a solution (λ′, z∗), where

λ′ equal to λ∗ by changing only the two components λ′st,t′ = 0 and λ′sT−s ,T

−s +σs

= 1,

remains feasible and has the same cost as (λ∗, z∗), and thus is an optimal solution.

Example 7.2 Consider an instance of the MVRPTW for which eight feasible ve-

hicle routes, a, b, c, d, e, f , g and h, can be generated. The first graph in Figure

7.5 represents the complete graph with all the replications of those eight routes, as

described in section 7.3.1. In the second graph, there are only the routes that need to

be considered, after the arc reduction criteria described in section 7.3.3 are applied.

7.4. Iteratively Refining the Discretization 141

T sets represent the beginning time instants of the routes and T′

the ending time

Tb T’bTc

Td

T’c

T’dTe T’e

b

c

d e

0 W

T

Tf

Ta T’a

a

T’f

f

Th T’hhg

Tc T’c

b

c

d e

0 W

T

Tf

Ta T’a

a

T’f

f

Tg T’gTh T’h

hg

Tb T’b

Td Te T’e T’d

T’gTg

Figure 7.5: Arc reduction (example 7.2)

instants. Set Tb was reduced to its first point as it does not intersect any ending

set (Proposition 4). Sets Td and Te were reduced to their last points, as all points

in T′

d and T′e are greater than T (Proposition 2). Finally, Tg was also reduced to

its last point, as set T′g does not intersect any beginning set (Proposition 3).

7.4 Iteratively Refining the Discretization

Our approach to solve an instance of a MVRPTW problem consists of enumer-

ating all feasible routes, as described in sections 7.3.1 and 7.3.3, and solving the

corresponding network flow model.

The nodes of model (7.22)–(7.27) represent time instants. The distances (and


thus the time) in the benchmarks we use are not integer, and thus we have two

alternatives. Either we use a finely grained discretization (for example, each time

unit would be 0.01), or we use some rounding procedures to use an integer time unit.

The former alternative would lead to an network flow model with a huge number

of variables and constraints, and would not allow a fast solution. Consequently, we

used the latter alternative.

We discuss in the following the different ways of rounding the values. We chose

a rounding strategy which slightly relaxes the problem. In many cases, the solution

found remains feasible. However, it may happen that the solution found by model

(7.22)–(7.27) is not feasible. Since our solution approach intends to be exact, we

developed an algorithm that iteratively refines the discretization. It allows us to

achieve the optimal value, despite our initial coarse discretization of time.

7.4.1 Initial Rounding Strategy

Recall that an arc (u, v)r in model (7.22)–(7.27) is related to a route r beginning at

time instant u and ending at time instant v. Given that the set of vertices of graph

Π is defined as a discrete set of values ∆ = 0, ...,W, it might be necessary for some

arcs (u, v)r ∈ Ψ, to round u and v to a value that belongs to set ∆. As mentioned

before, we initially consider time units equal to 1, i.e., ∆ = 0, 1, 2, ...,W − 1,W.

Several rounding procedures are possible, which are commented below.

• u = buc and v = dve. In this case, in the model, a route begins slightly before

and ends slightly after it actually does. It means that we may miss some

solutions, but each feasible solution of (7.22)–(7.27) using this relaxation will

be feasible for the initial problem. Consequently, this relaxation leads to a

heuristic.

• u = due and v = bvc. In this case, the model is slightly relaxed. It may

happen that the solution found is not feasible. However it leads to a valid


lower bound.

• (u = due and v = dve) or (u = buc and v = bvc). In this case, a valid

solution of (7.22)–(7.27) can be infeasible, and yet some parts of the problem

are relaxed. This technique neither leads to a lower bound, nor to an upper

bound.

In our algorithm, we used the second rounding procedure: considering u = due and

v = bvc. We chose this technique for two reasons: the relaxation is expected to be

tight, and infeasibilities are local and can be corrected, as we will explain below.

Note that infeasible solutions are only related to paths of the flow model (work-

days) including two consecutive routes r and r′ with one or less units of waiting

time between them. For example, if a replication of route r ends at time instant

15.35, and a replication of route r′ begins at time instant 15.15, our rounding pro-

cedure will make route r end at time instant 15 and route r′ start at time instant

16, allowing r and r′ to belong to the same working day even though this is not

possible in the initial problem.

A first idea is to correct the solution by shifting route r backward or route

r′ forward to avoid this problem. If it makes the solution feasible, we prove the

optimality, because the feasible solution will have the same set of routes and thus

a cost equal to the lower bound. Practically speaking, after having obtained a

solution x∗, we try to build a feasible solution, using the same routes found in

solution x∗.

The algorithm works as follows. For each workday, we try to build a new

path, only maintaining the sequence of routes. Let (r1, . . . , rp) be the sequence of

routes in the workday, and Tri the new beginning and T′ri

the new end of route ri,

∀i ∈ 1, . . . , p. We set Tri = max(T−ri , T′ri−1

). If Tri ≤ T+ri,∀i ∈ 1, . . . , p, the

solution is feasible. If not, we cannot prove feasibility, and another algorithm has

to be used.


7.4.2 Iterative Algorithm

Our solution approach consists of iteratively fixing the infeasibilities due to dis-

cretization issues. At each step of the algorithm, we detect all the time instants

where infeasibilities occur. For each of these time instants, we locally modify the

discretization, adding the fractional values needed to refine the relaxation entailed

by the initial discretization.

The initial relaxation can be seen as an aggregation of many time instants into

an unique integer one. Our refinement technique is equivalent to a disaggregation

of some nodes of the current graph. For each pair of conflicting arcs (u, v)r and

(u′, v′)r′, we consider the fractional values of v and u′ in order to correct the solution.

Our solution approach is summarized in Algorithm 1. The node disaggregation

Algorithm 1: Iterative Disaggregation Algorithm

Input: Instance I of the MVRPTW

Output: Optimal solution x∗

optimal=False;

while optimal=False do

Solve I with model (7.22)–(7.27), possibly obtaining an infeasible

solution x∗;

Check if x∗ is feasible;

if x∗ is feasible then

optimal=True;

elseApply node disaggregation;

process works as follows. We check all the arcs that belong to the solution. If

there are at least two conflicting arcs (u, P )r, (P, v)s in the solution, we decompose

the integer node P . In order to do so, we check all arcs of the model (w, z)r ∈ Ψ

such that w = P or z = P (except the waiting arcs), setting them to their original


values w′ = P − 1 + ξa or z′ = P + ξb, with 0 ≤ ξa, ξb < 1. We also check for all

arcs (x, y)r ∈ Ψ such that y = P − 1 or x = P + 1, setting them to their original

values y′ = P − 1 + ξc or x′ = P + ξd, with 0 ≤ ξc, ξd < 1. When all the new

η values are calculated, we sort them by their increasing values, and add the new

nodes Pq, q ∈ 1, . . . , η, to the graph, always respecting their order.

Example 7.3 Consider an instance of the MVRPTW with an optimal solution

with two incident arcs in node P , (a, b)r, (c, d)s with b = c = P , an arc (e, f)t with

e = P , an arc (g, h)u with h = P − 1 and an arc (i, j)v with i = P + 1. Let Pb, Pc,

Pe, Ph and Pi be the original non rounded values of b, c, e, h and i respectively.

Figure 7.6 illustrates the graph transformation.

PP-1 P+1

PP-1 P+1

Pc Pe Pb

(a,P)r (P,d)

s

(P,f)t

(a,Pb)r

(Pc,d)s

(Pe,f)t

(g,P-1)u

(P+1,j)v

(g,Ph)u (Pi,j)

v

Ph Pi

Figure 7.6: Node disaggregation (example 7.3)

We also need to check if there are pairs of arcs in the solution, (u, v)r and (x, z)s,


such that v = P and x = P + 1 and their original non rounded values v′ = P + ξv

and x′ = P + ξx are such that ξv > ξx. In such cases, all beginning points incident

in P + 1 or ending points incident in P must be set to their non rounded values.

Example 7.4 Consider an instance of the MVRPTW for which arcs (a, b)s and

(c, d)t, with b = P and c = P + 1, belong to the optimal solution. All ending points

incident in P or beginning points incident in P + 1 must be set to their original

values. Figure 7.7 illustrates the disaggregation procedure for this case.

PP-1 P+1

PP-1 P+1

Pc Pe Pb

(a,P)r

(P,d)s

(P,f)t

(a,Pb)r

(Pc,d)s

(Pe,f)t

(g,P-1)u

(P+1,j)v

(g,Ph)u (Pi,j)

v

Ph Pi

Figure 7.7: Node disaggregation (example 7.4)


Our algorithm was tested and compared with the branch-and-price algorithm de-

scribed in [20]. In order to do so, we used the same set of instances and the same


values of parameters.

The comparisons are performed on the well known Solomon instances for the

VRP with time windows [177] that were used by [20]. The instances are divided

into three different categories, RC with 8 instances, R with 11 instances and C

with 8 instances, according to the distribution of the customers’ locations. Each

category is divided into two groups (1 and 2). We are only considering the second

group instances, as the first group ones have small time windows, which would

compromise the existence of workdays with more than one route. Originally, each

instance has 100 customers but we only consider the n first ones, similarly to [20].

The value n is specified for each set of computational results.

The algorithm was implemented in C++ and the network flow model was solved

with ILOG CPLEX 12.1. Note that the computational tests were run on a PC with

a 2.66 GHz Quad-Core processor and 4GB of RAM, whereas the tests in [20] were

run on an AMD Opteron 3.1 GHz with 16GB of RAM.

The tests were run for K = 2, α = 2maxi,j∈Ndij + 1, β = 0.2, gi = 1,∀i ∈ N

and tij = dij,∀i, j ∈ V . All instances were run considering the first 25 and 40

clients. Each of these instances was tested for two different values of tmax. First for

a smaller one (75 for instances of groups RC and R, and 220 for group C) which

results in less routes to consider, making the problem easier to solve, and then for

a larger one (100 for instances of groups RC and R, and 250 for group C).

Table 7.3 shows the impact of the arc reduction described in section 7.3.3. Each

line represents the average values of a group of instances, belonging to one of the

three different categories, RC, R, or C, with n clients and a given tmax. The other

columns represent the following: |R| is the number of different routes and |Υ| is

the number of all routes for all their beginning instants, as described in section

7.3.1; |Rred| and |Υred| represent these same values, after the reduction procedure

has been applied; finally, DifR% = |R|−|Rred||R| % and DifΥ% = |Υ|−|Υred|

|Υ| %. We

were able to reduce approximately 29% of the number of variables to include in the


Table 7.3: Arc reduction

Inst n tmax |R| |Υ| |Rred| |Υred| DifR% DifΥ%

RC 25 75 516.75 116631.13 419.13 71673.63 18.90 34.13R 25 75 702.64 216311.18 618.82 149259.09 12.64 28.89C 25 220 336.00 183614.25 318.86 128034.00 16.55 28.58

RC 40 75 641.13 154921.25 520.75 96329.75 19.07 33.44R 40 75 3661.82 1247141.73 3266.73 899976.00 11.36 25.62C 40 220 1530.63 1153261.75 804.57 313353.29 16.83 26.40

RC 25 100 4463.88 746806.00 3374.88 411120.38 21.48 37.63R 25 100 4705.91 1226177.91 4154.18 816231.18 12.15 29.98C 25 250 1308.25 610079.88 1237.29 452880.29 16.34 26.84

RC 40 100 5648.88 1096099.38 4357.13 614915.63 21.51 37.41R 40 100 5995.50 424145.00 5057.50 342092.50 14.38 16.80C 40 250 2891.00 835131.67 2291.83 576158.67 16.92 22.53

Average 2700.20 667526.76 2201.80 406002.03 16.51 29.02

model .

We now report the results obtained by our algorithm on the instances used

by [20]. For these results, the reduction procedures are applied before running

the algorithm. Tables 7.4-7.7 describe the computational results for the instances

that were solved to optimality by at least one of the two methods tested. Several

combinations of the number of customers and values tmax are tested. We ran the

instances for a maximum time of 7200 seconds. The number of iterations of the

iterative disaggregation algorithm is represented by nit and zIDA stands for the

optimal value obtained with our approach. Columns tIDA and tAGP show the time,

in seconds, required to solve the iterative disaggregation algorithm and the branch-

and-price algorithm in [20], respectively . The percentage of visited customers is

represented by %Cust, and the last column tRed represents [(tAGP − tIDA)/tAGP ]%.


Table 7.4: Computational results for 25 customers and tmax 75 and 220

Inst n tmax nit %Cust zIDA tIDA tAGP tRed

RC201 25 75 1 100% 988.2 0.3 3.1 91.29%RC202 25 75 1 100% 881.6 37.2 - -RC203 25 75 1 100% 749.26 54.2 - -RC204 25 75 1 100% 744.83 171.0 - -RC205 25 75 1 100% 840.47 1.6 28.8 94.48%RC206 25 75 1 100% 761.14 2.0 7156.8 99.97%

R201 25 75 1 100% 762.53 0.5 68.3 99.22%R202 25 75 1 100% 645.86 3.1 205.2 98.51%R203 25 75 1 100% 622.04 10.6 1333.2 99.21%R204 25 75 1 100% 579.75 106.2 30983.3 99.66%R205 25 75 1 100% 634.17 1.5 354.1 99.59%R206 25 75 1 100% 596.81 4.7 318.4 98.52%R207 25 75 1 100% 585.81 19.4 2853.5 99.32%R208 25 75 1 100% 579.75 66.0 9270.3 99.29%R209 25 75 1 100% 602.47 4.9 262.6 98.12%R210 25 75 1 100% 636.24 11.8 5094.1 99.77%R211 25 75 1 100% 575.97 64.5 5648.6 98.86%

C201 25 220 1 100% 659.15 10.6 40361.2 99.97%C202 25 220 1 100% 653.5 212.4 - -C203 25 220 1 100% 646.51 233.9 - -C204 25 220 1 100% 602.58 423.0 - -C205 25 220 1 100% 636.52 34.7 - -C206 25 220 1 100% 636.52 40.2 - -C207 25 220 1 100% 603.34 29.5 - -C208 25 220 1 100% 613.34 12.9 - -

SolvedAF : 92.59% SolvedAGP : 55.56% Average tRed: 98.39%




RC201 40 75 4 77.50% 1292.35 29.4 14.6 -50.37%RC202 40 75 1 92.50% 1458.09 40.2 6823.2 99.41%RC205 40 75 3 85% 1290.75 6992.6 1904.2 -72.77%

R201 40 75 1 95% 1130.73 2358.8 2979.5 20.83%R203 40 75 1 100% 962.42 436.0 - -R205 40 75 4 100% 1019.89 3263.7 244494.0 98.67%R206 40 75 1 100% 931.94 209.9 - -R209 40 75 1 100% 935.95 771.3 - -R210 40 75 4 100% 963.45 1803.9 - -

C201 40 220 1 100% 1169.04 25.5 19978.9 99.87%C202 40 220 1 100% 1111.34 79.4 - -C203 40 220 1 100% 1089.24 342.3 - -C205 40 220 1 100% 1084.02 63.6 - -C206 40 220 1 100% 1081.57 109.3 - -C207 40 220 1 100% 1055.24 659.0 - -C208 40 220 1 100% 1072.22 112.7 3221.7 96.5%





RC201 25 100 1 100% 849.45 2.0 46.3 95.7%RC202 25 100 1 100% 679.95 11.6 1096.3 98.94%RC203 25 100 1 100% 593.63 47.0 - -RC205 25 100 1 100% 702.61 8.2 262.8 96.86%RC206 25 100 1 100% 604.23 8.0 222.7 96.42%RC207 25 100 2 100% 514.9 91.7 - -

R201 25 100 1 100% 698.26 1.3 43.6 96.95%R202 25 100 1 100% 617.6 32.6 25249.9 99.87%R203 25 100 1 100% 577.8 64.1 75729.3 99.92%R205 25 100 1 100% 559.21 9.4 1202.3 99.22%R206 25 100 1 100% 523.7 40.0 28498.1 99.86%R209 25 100 1 100% 517.74 47.7 11173.9 99.57%R210 25 100 1 100% 547.29 58.9 26690.2 99.78%

C201 25 250 1 100% 541.02 0.4 1.3 68.46%C202 25 250 1 100% 533.55 167.9 - -C205 25 250 1 100% 530.05 3.7 116.6 96.87%C206 25 250 1 100% 527.95 20.7 1987.2 98.96%C207 25 250 1 100% 525.57 44.2 - -C208 25 250 1 100% 525.57 63.2 - -





RC201 40 100 1 85% 1157.65 3.6 77.8 95.36%RC202 40 100 4 97.5% 1322.08 1013.8 - -RC205 40 100 1 95% 1195.51 35.7 4733.3 99.25%

R201 40 100 - - - - 127424.0 -

C201 40 250 1 100% 966.89 6.4 659.2 99.03%C205 40 250 1 100% 921.37 88.5 - -C206 40 250 1 100% 919.24 290.5 - -C208 40 250 1 100% 915.61 491.5 - -


We solved around 62% of all instances, whereas in [20] only about 37% of the

instances were solved to optimality. We were able to solve 28 instances that were

not solved by [20]. On the contrary, one instance is solved by this latter method

and not by ours. Note that the computational time spent to solve that instance

in [20] was much superior to our time limit of 7200 seconds. The difference in the

number of instances solved is larger for the instances with n = 40 and with smaller

tmax. For this set of instances, we were able to solve 16 instances, whereas the

method of [20] only solved 7 instances.

If we consider the instances solved by both methods, ours solved them in ap-

proximately 2% of the time reported in [20]. Our method is faster for every case

with a reduction of at least 91%, except for two instances where the reduction is

smaller, and the notable exception of two other instances for which our method

is slower than the method of [20]. These two instances involve 40 customers and

a tmax equal to 75. In one of these two instances, the number of disaggregation

iterations is 4, and in the other, the model ran three times.

7.6. Conclusions 153

In what concerns the number of iterations of the algorithm, only in six cases

out of the sixty seven solved was it necessary to run the model more than once.

The model was solved twice and thrice in one case each and four times for four test

cases. This means that our algorithm always converged in four or less iterations.

As expected, when the number of customers increases it becomes harder to

solve the problem to optimality. This also happens when we increase the value of

tmax, as the bigger this parameter is, the larger is the number of feasible routes,

and thus variables, to consider.

7.6 Conclusions

The MVRPTW is a variant of the classical vehicle routing problem that has re-

ceived little attention in the literature. In this chapter, we described a new network

flow model, and an exact solution algorithm to solve this problem. Our algorithm

is iterative and it relies on a pseudo-polynomial network flow model whose nodes

represent time instants, and whose arcs represent feasible vehicle routes. We con-

ducted some computational experiments on a set of benchmark instances and com-

pared the results with the only other exact method in the literature. Our approach

proved to be more efficient, by solving more instances and clearly outperforming

the other method in terms of computational time, for most of the cases.

Chapter 8

A Network Flow Model for the

Location Routing Problem with

Multiple Routes

“There in no higher or lower knowledge,but only one, flowing out of experimentation.”

Leonardo da Vinci


8.2 Problem Definition . . . . . . . . . . . . . . . . . . . . . 160

8.3 Mathematical Programming Models . . . . . . . . . . . 161

8.3.1 Three-index Commodity Flow Model . . . . . . . . . . . 161

8.3.2 Set-partitioning Model . . . . . . . . . . . . . . . . . . . 162

8.4 Network Flow Model . . . . . . . . . . . . . . . . . . . . 163

8.4.1 Arc Reduction . . . . . . . . . . . . . . . . . . . . . . . 165


8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 171

155

156 8. A Network Flow Model for the MLRP

8.1 Introduction

In this chapter, we analyze the Location Routing Problem (LRP). It is an inte-

grated problem, combining two difficult ones, the vehicle routing problem and the

location problem. The integration of these two important optimization problems

to distribution networks can translate into considerable savings. Its standard ver-

sion represents an interrelation of a vehicle routing problem, which determines the

optimal set of routes to fulfill the demands of a set of customers, and a location

problem, as the depots from which the vehicles performing the routes can be as-

sociated must be chosen from a set of possible locations. Being a generalization of

the VRP it is also a NP -hard problem.

Laporte [130] describes application areas, different formulations and solution

methods in a survey of deterministic LRP. Based on the work published up to

1988, the author concludes by proposing promising future research areas for this

problem. In [158], the authors propose a taxonomy for the LRP and a classification

scheme that categorizes the up to date exact and heuristic approaches in the liter-

ature, according to their specific solution method and to eleven other features that

characterize the problem’s assumptions, in what the authors call a two-way clas-

sification. They also provide an annotated literature review, organized according

to the proposed classification. A more recent survey of LRPs is provided in [165].

The authors also propose a classification scheme and a review of deterministic exact

and heuristic methods, and also of stochastic and dynamic solution approaches to

the problem. Laporte et al. [133] applied an exact algorithm based on an integer

programming formulation to the LRP. The procedure consists of a branch-and-cut

algorithm where subtour eliminating constraints and chain barring constraints are

introduced. It solved instances with up to 20 customers and 8 depots. Although

the depots have no capacity, there are lower and upper bounds on the number of

depots to consider and on the number of capacitated vehicles to be associated to


each depot. Laporte et al. [135] solve an asymmetrical LRP with lower and upper

bounds on the number of vehicles based at each depot, and a constraint on the

total cost of a route. They reformulate the problem as a constrained assignment

problem through the use of a graph transformation. They use a variant of the

branch-and-bound algorithm proposed in [43] for the traveling salesman problem

to solve the integer problem. The authors report on computational results with up

to 3 possible depots and 80 customers Berger et al. [32] proposed a branch-and-

price algorithm, based on a set-partitioning formulation, with distance constraints

and uncapacitated vehicles and depots. They also introduce a set of valid inequal-

ities in order to strengthen the proposed formulation. Akca et al. [4] proposed an

exact solution method for the LRP with capacities on both vehicles and depots.

It is a branch-and-price algorithm based on a set-partitioning formulation. The

proposed model is an adaptation of a previously presented formulation [3] for the

location routing and scheduling problem. The authors presented four variants of

the algorithm, based on different heuristic approaches to solve the pricing problem

of their column generation scheme. Belenger et al.[28] provided a branch-and-cut

algorithm for the LRP with capacitated facilities and capacitated vehicles. The

integer programming model uses only binary variables and is based on the one

proposed in [133], with the additional depots’ capacity constraints. Both heuris-

tic and exact separation algorithms are proposed, in order to find violated valid

inequalities in the embedded cutting plane scheme.

According to the classification scheme presented in [165], heuristic methods

for the LRP can be clustering-based, iterative or hierarchical. Clustering-based

heuristics are composed by two phases. In a first phase, the set of customers is

grouped in clusters, which are, in the second phase, allocated either to depots or

to a vehicle route. In the first case, the algorithm concludes by solving a VRP

(or TSP) for each depot and in the second case a TSP is solved for each cluster

which is posteriorly associated to a depot. In iterative methods, the subproblems


of location and routing are solved separately, in an iterative scheme that exchanges

information between these two phases, in order to improve the solution. Finally,

hierarchical heuristics tackle the problem as a location problem, incorporating the

routing problem as a subproblem in some phases of the routine.

Nagy and Salhi [164] propose a hierarchical heuristic method for the LRP, based

on a tabu search algorithm. It is a nested method, where the routing problem is

considered in the evaluation of possible moves, which can be add moves (opening

a closed depot), drop moves (closing an open depot) or shift moves (closing an

open depot and opening a closed one). Tuzun et Burke [185] propose a hierar-

chical two-phase tabu search for the LRP with capacitated vehicles. These two

coordinated phases correspond either to a location or a routing problem. Every

time a neighborhood move is performed in the location phase, the routing phase is

started. The search in the routing phase is not global. Only the routing sections

affected by the previous location move are explored, in order to, according to the

authors, explore the solution space efficiently. Wu et al. [202] decompose the LRP

into a location-allocation problem and a vehicle routing problem and solve it with

an iterative heuristic, on a combined tabu search and simulated annealing frame-

work. The authors consider the problem with capacitated heterogeneous vehicles.

Albareda-Sambola et al. [5] approached the LRP with capacitated depots with one

single vehicle associated to each depot. The authors solve the LP-relaxation of a

compact model and use it as a lower bound and as a base for the first solution in a

tabu search framework. The problem is solved with a hierarchical heuristic where

the intensification and diversification phases are concerned, respectively, with the

location and the routing problems. They allow infeasible solutions in the intensifi-

cation phase, which are controlled with a penalty in the objective function. They

propose an additional lower bound that presents a better quality than the one pro-

vided by the LP-relaxation. Barreto et al. [26] used a clustering-based heuristic for

the LRP with capacity constraints both for the vehicles and the depots. Customers


are grouped in clusters with a capacity limit, by using one of four grouping meth-

ods. A TSP is then solved for each of these groups, by using an exact method, if the

group has 40 or less elements, or a two stages heuristic procedure, otherwise. After

improving the routes with a local search procedure, a location problem is solved

in order to assign them to the selected depots. By combining the four clustering

methods and six proximity measures, the authors present several versions of this

routing-first, location-second heuristic.

We address a variant of this problem, the location routing problem with multiple

routes (MLRP), where each vehicle can perform several routes in the same planning

period. This variant has been explored exactly in [3], and heuristically in [141],

although in the latter the authors also propose a branch-and-bound algorithm for

the problem. In [3], the authors propose a three-index commodity flow model

and also a branch-and-price algorithm based on a set-partitioning formulation of

the problem. The branch-and-price algorithm is based on a column generation

framework, where each column represents what the authors define as pairings. The

pricing problem is an elementary shortest path problem with resource constraints.

Finally, the authors propose valid inequalities in order to strengthen the model. In

[6], the authors define several formulations for the case where only one customer

is visited in each route. The problem is addressed as the capacity and distance

constrained plant location problem. For one of the formulations, they propose

families of valid inequalities in order to strengthen it, and assess its effectiveness

when it is directly solved with a commercial solver. As they do not devise any

column generation scheme, they propose an extension of the problem, which can

have practical applicabilities, as they describe, where there is a constraint on the

number of customers that each vehicle can visit during the planning period. This

constraint is addressed as the cardinality constraint.

The remainder of this chapter is organized as follows. In section 8.2, we define

our problem, along with its notation. Some mathematical formulations from the


literature are briefly recalled in section 8.3. In section 8.4 we describe our network

flow model, as well as some reduction criteria derived for it. Computational ex-

periments are reported in section 8.5, while some conclusions are drawn in section

8.6.

8.2 Problem Definition

In this section, we define our problem, along with its notation. There is a set

D of available depots that can be selected to be open, and a set of customers

N = 1, . . . , n to visit. Customers are served by a single depot and a single

vehicle. Each depot d ∈ D has a finite capacity Ld and has several associated

vehicles, which perform routes that begin and end at that same depot.

The fleet of vehicles is homogeneous, and each vehicle has a capacity Q. A

same vehicle can perform several routes in the same planning period, although

it can never travel for more than W units of time. A route r corresponds to a

sequence of customers to be visited by a vehicle, always beginning and ending at

the same depot. Its duration is represented by tr. Given that vehicles have finite

capacities, a route r can never consider the visit to a set of customers Nr ⊆ N

whose sum of demands is greater than Q. The load of route r is represented by lr.

When a depot d is selected to be open, there is a fixed cost Cdf to consider.

Similarly, the decision of using a given vehicle incurs in a fixed cost Cv. Apart

from these fixed costs, there are also variable costs to contemplate, corresponding

to the traveled distances to serve customers. The subsequent cost of a route r is

represented by Cr. No service or setup times are considered.

Finally, we consider an adaptation of the cardinality constraint considered in

[6]. For each route, there is an upper bound α on the number of customers that

can be visited.

8.3. Mathematical Programming Models 161

8.3 Mathematical Programming Models

In [3], the authors propose two different formulations for the MLRP, a three-index

commodity flow formulation (8.1)-(8.12) and a set partitioning formulation (8.13)-

(8.17). We briefly recall them in this section.

8.3.1 Three-index Commodity Flow Model

This model is represented in a graph G = ((N ∪ D), A), where A = (D × N) ∪

(N ×N). Let H be the set of available vehicles and Hd the set of vehicles assigned

to depot d ∈ D. The time it takes to go through arc (i, j) ∈ A is represented

by tij, and the operational cost of one unit of traveled time is represented by Co.

Variables xikh (8.9) define if vehicle h ∈ H goes through arc (i, k) ∈ A and variables

yikh (8.10) represent the flow that vehicle h ∈ H carries on arc (i, k) ∈ A. Binary

variables λd (8.11) and vh (8.12) define whether depot d ∈ D opens or not and

whether vehicle h ∈ H is used or not, respectively. The model states as follows.

min∑d∈D

Cdfλd + Cv

∑h∈H

vh + Co∑h∈H

∑(i,k)∈A

tikxikh (8.1)

s.t.∑h∈H

∑k∈(N∪D)

xikh = 1 ∀i ∈ N, (8.2)

∑k∈(N∪D)

xikh −∑

k∈(N∪D)

xkih = 0 ∀i ∈ N ∪D, ∀h ∈ H, (8.3)

∑h∈Hd

∑k∈N

ydkh − Ldλd ≤ 0 ∀d ∈ D, (8.4)

yikh −Qxikh ≤ 0 ∀(i, k) ∈ A, ∀ h ∈ H, (8.5)∑k∈N

yikh −∑k∈N

ykih + bi∑

k∈(N∪D)

xikh = 0 ∀i ∈ N,∀h ∈ H, (8.6)

∑(i,k)∈A

tikxikh −Qvh ≤ 0 ∀h ∈ H, (8.7)

xdkh = 0 ∀d ∈ D, ∀k ∈ (N ∪D),∀h ∈ Ht,∀t ∈ D\d, (8.8)


xikh ∈ 0, 1, ∀(i, k) ∈ A,∀h ∈ H, (8.9)

yikh ≥ 0, ∀(i, k) ∈ A, ∀h ∈ H, (8.10)

λd ∈ 0, 1 ∀d ∈ D, (8.11)

vh ∈ 0, 1 ∀h ∈ H. (8.12)

The objective function (8.1) considers the minimization of all fixed and variable

costs. Each customer is visited exactly once, which is guaranteed by constraints

(8.2)-(8.3). Constraints (8.4) and (8.5)-(8.6) state that the capacities associated to

depots and to vehicles, respectively, must be respected. The time limit W is taken

into account with constraints (8.7). Finally, if a vehicle is associated to a depot d,

it can only travel through arcs associated to that depot (8.8).

8.3.2 Set-partitioning Model

The authors define the concept of pairing, which is the schedule of one vehicle for

the planning period, i.e., the set of routes one vehicle is scheduled to perform. Let

Pd be the set of all feasible pairing associated to facility d ∈ D and Cp the associated

cost of a pairing p, including the fixed cost of one vehicle and the operational cost

of the route. Coefficients aip define whether or not customer i ∈ N is visited in

pairing p ∈ Pd. Binary variables λd (8.16) define whether depot d ∈ D opens or

not and binary variables δd determine if a pairing p ∈ Pd is associated to depot d.

The model states as follows

min∑d∈D

Cdfλd +

∑d∈D

∑p∈Pd

Cpδp (8.13)

s.t.∑d∈D

∑p∈Pd

aipδp = 1, ∀i ∈ N, (8.14)∑p∈Pd

aip∑i∈N

aipbiδp − Cdfλd ≤ 0, ∀d ∈ D, (8.15)

λd ∈ 0, 1, ∀d ∈ D, (8.16)


δp ∈ 0, 1, ∀p ∈ Pd, d ∈ D. (8.17)

The objective function (8.13) considers the minimization of all fixed and variable

costs. Each customer is visited exactly once, and therefore belongs to one and

only one pairing of the solution (8.2). Constraints (8.3) state that the capacities

associated to depots must be respected.

The generation of pairings is done by the solution of a pricing problem that

is an elementary shortest path problem with resource constraints. The authors

propose two heuristic algorithms to solve it and only solve it exactly when no more

attractive columns are identified by the heuristic methods.

8.4 Network Flow Model

We propose a pseudo-polynomial network flow model whose variables are explicitly

generated and represent feasible vehicle routes. The integer model is solved with a

commercial software (CPLEX).

Let Rd, ∀d ∈ D, be the set of all feasible routes for depot d ∈ D, i.e., all

routes r such that lr ≤ Q and whose duration is not greater than the maximum

pre-defined amount of time W . We consider a set of acyclic directed graphs Πd =

(∆,Ψd), ∀d ∈ D. The set of vertices ∆ = 0, 1, ...,W represents discrete time

instants from 0 to the time limit W . Each set of arcs Ψd, ∀d ∈ D, is defined as

Ψd = (u, v)r : 0 ≤ u < v ≤ W, r ∈ Rd⋃(u, v)o : 0 ≤ u < v ≤ W, v = u + 1.

Arcs correspond either to feasible vehicle routes or to waiting time periods (unit

arcs). These waiting time arcs represent the instants of time spent by the vehicle

at the depot. An arc (u, v)r ∈ Ψd represents route r beginning at time instant u

and ending at time instant v. This means that tr = v − u for depot d.

Binary variables λd (8.25) define whether depot d ∈ D opens or not. Each arc

(u, v)r corresponds to route r ∈ R starting at instant u and ending at instant v.

Variables xduvr (8.23) represent the flow of arc (u, v)r on graph Ψd. Finally, variables


zd (8.24) represent the total flow that goes through graph d ∈ D, which equals the

number of vehicles that are used in the corresponding depot. The model states as

follows.

min∑d∈D

∑(u,v)r∈Ψ

Crxduvr + Cv

∑d∈D

zd +∑d∈D

Cdfλd (8.18)

s.t.∑d∈D

∑(u,v)r∈Ψ|i∈Nr

xduvr = 1, ∀i ∈ N, (8.19)

−∑

(u,v)r∈Ψ

xduvr +∑

(v,y)t∈Ψ

xdvyt (8.20)

=

zd if v = 0

0 if v = 1, . . . ,W − 1

−zd if v = W

, ∀d ∈ D,

∑(u,v)r∈Ψ

lrxduvr ≤ Ld, ∀d ∈ D, (8.21)

Mλd ≥ zd, ∀d ∈ D, (8.22)

xduvr ≥ 0 and integer, ∀(u, v)r ∈ Ψd, ∀d ∈ D, (8.23)

zd ≥ 0 and integer, ∀d ∈ D, (8.24)

λd ∈ 0, 1. (8.25)

The objective function (8.18) contemplates the minimization of all fixed and

variable costs regarding vehicles, depots and routes, as it has been previously de-

scribed. Every customer must be visited exactly once (8.19). Constraints (8.21)

stipulate that there is flow conservation in all graphs, and constraints (8.21) state

that the capacities associated to depots must be respected. Let M be a sufficiently

large number, for example, an upper bound for the total number of used vehicles.

Constraint (8.22) certify that a depot must open if there are routes, and therefore

vehicles, associated to it in the solution.


8.4.1 Arc Reduction

Given that we are explicitly considering all feasible routes in our model, it is im-

portant to eliminate the ones that, although feasible, do not have any potential

interest to the optimal solution. By reducing the number of routes to take into

account, we need to consider less arcs in the model, increasing its efficiency. In

order to do this, we defined two reduction criteria.

Each path of a graph Πd corresponds to a set of routes to be performed by one

vehicle throughout the planning period, which has a maximum duration of W units

of time. The sum of the durations of the routes associated to a same path cannot,

therefore, exceed W . Given that there are no time windows to consider, the order

on which the routes are performed is irrelevant, as well as the specific time instants

at which routes begin, providing that the time limit is respected. Given that, it

is possible to only consider paths with routes sorted by their decreasing values of

duration. By doing so, the symmetry of the model is greatly reduced. On the

other hand, it is also possible to consider sequences of routes without waiting times

between them.

This considerations allow us to define the following reduction criterion:

(i) For every possible route r in Rd, ∀d ∈ D, there is an arc (0, tr)r in graph d.

For every other nodes i ∈ ∆\0,W of graph d, there is an arc (i, i + tr)r if

and only if there is an arc (a, i)s in that same graph, with a ∈ ∆ and s ∈ Rd,

such that i− a ≥ tr and i+ tr <= W .

The same set of n customers originates n! different routes, by exchanging the

sequence on which customers are visited. A different sequence of clients may have

a different cost but it always has the same load. From the list of n! different

routes associated to a depot d, let us consider two, r and s, with costs Cr and Cs,

respectively. If Cr < Cs, it is not necessary to consider route r for depot d, as they

both visit the same customers, have the same load, and the sequence defined in r


represents a greater cost than the one defined in s. Therefore, the second reduction

criterion is the following:

(ii) From each set of routes visiting the same set of customers, we choose, for each

depot, the one with the lowest cost and ignore all the others. This means

that for each graph corresponding to each depot, there are no two different

routes visiting the same set of clients.


To evaluate the quality of model (8.18)-(8.25), with the implementation details

described in section 8.4, we conducted a set of computational experiments. The

algorithm was implemented in C++ and the network flow model was solved with

ILOG CPLEX 12.0. The computational tests were run on a PC with Intel Core i3,

CPU with 2.27 GHz and 4GB of RAM.

The considered test instances are an adaptation of the instances with 20 and

25 customers described in [3], with Cdf = 1900 and Cv = 225. These instances

were generated from instances p01, p03 and p07 described in [54]. There are five

available depots, whose coordinates are the same ones proposed in [3]. For every

instance Ca,b, a stands for the number of customers and b identifies the instance in

[54] from which Ca,b is derived, as well as the considered customers, according to

table 8.1. The duration of every route equals its traveled distance . As in [3],

distances between customers and depots are rounded. Each of these instances was

tested for two different values of W , 100 and 140, and two different values of α, 3

and 4.

Tables 8.2 and 8.3 report the results obtained for the 20 and 25 customers in-

stances, respectively. Column nArcs describes the number of arcs that correspond

to routes, without considering the unit waiting arcs. The computational times, in

seconds, required to generate the problem and to solve it are represented by tr and


Table 8.1: Instances description.

Ca,b customers Ca,b customers

p01C20,1 1 - 20 C25,1 1 - 25C20,2 31 - 50 C25,2 26 - 50

p03C20,3 1 - 20 C25,3 1 - 25C20,4 28 - 47 C25,4 26 - 50C20,5 56 - 75 C25,5 51 - 75

p07C20,6 1 - 20 C25,6 1 - 25C20,7 21 - 40 C25,7 26 - 50C20,8 31 - 50 C25,8 51 - 75

tNFM , respectively. Columns #routes, #vehic and #dep represent the number of

routes, vehicles and selected depots in the optimal solution, and in column z the

optimal value of the solution is given.

Table 8.2: Computational results for 20 customers.

Inst α W Q nArcs tr (s) tNFM (s) z #routes #vehic #dep

C20,1 3 100 50 5916 0.14 11.46 5059 9 4 2C20,1 3 100 70 8310 0.27 23.40 5050 8 4 2C20,2 3 100 50 4098 0.23 9.23 5347 7 5 2C20,2 3 100 70 4697 0.31 18.14 5347 7 5 2C20,3 3 100 60 5955 * * * * * *C20,3 3 100 80 7573 * * * * * *C20,4 3 100 60 5275 0.25 14.90 5063 8 4 2C20,4 3 100 80 6926 0.49 34.91 5030 7 4 2C20,5 3 100 60 3650 * * * * * *C20,5 3 100 80 3806 0.41 9.44 5357 7 5 2C20,6 3 100 50 5692 0.27 574.41 5304 7 5 2C20,6 3 100 70 6095 0.34 974.44 5300 7 5 2C20,7 3 100 50 3368 0.19 69.37 5619 8 6 2C20,7 3 100 70 3875 0.30 2.38 5406 9 5 2C20,8 3 100 50 1605 0.16 3.97 5643 8 6 2C20,8 3 100 70 1953 0.25 10.74 5599 7 6 2

C20,1 3 140 50 21991 0.20 981.08 4836 9 3 2C20,1 3 140 70 33045 * * * * * *C20,2 3 140 50 16498 * * * * * *C20,2 3 140 70 19019 * * * * * *C20,3 3 140 60 22457 0.22 188.96 4863 9 3 2


Table 8.2: Computational results for 20 customers (continued).


C20,3 3 140 80 30081 0.42 851.03 4847 7 3 2C20,4 3 140 60 18032 0.20 299.99 4842 9 2 2C20,4 3 140 80 24702 0.44 1637.93 4814 7 2 2C20,5 3 140 60 16321 0.31 5486.65 5149 7 4 2C20,5 3 140 80 17358 * * * * * *C20,6 3 140 50 23586 0.33 84.43 4864 8 3 2C20,6 3 140 70 25572 0.48 220.54 4860 7 3 2C20,7 3 140 50 14789 0.25 359.57 5170 8 4 2C20,7 3 140 70 17499 0.41 2743.39 5151 7 4 2C20,8 3 140 50 8212 0.23 47.89 5218 9 4 2C20,8 3 140 70 10295 0.33 130.79 5168 7 4 2

C20,1 4 100 50 6778 1.48 25.41 5050 8 4 2C20,1 4 100 70 11496 5.43 173.63 5020 7 4 2C20,2 4 100 50 4658 2.76 15.26 5336 7 5 2C20,2 4 100 70 6946 7.84 143.22 5312 5 5 2C20,3 4 100 60 6246 * * * * * *C20,3 4 100 80 10604 5.44 60.02 5039 6 4 2C20,4 4 100 60 5429 0.66 25.19 5063 8 4 2C20,4 4 100 80 8838 * * * * * *C20,5 4 100 60 4637 4.59 11.61 5351 7 5 2C20,5 4 100 80 5406 7.91 300.48 5306 6 5 2C20,6 4 100 50 7392 4.27 15.39 5051 6 4 2C20,6 4 100 70 9345 8.80 66.67 5044 6 4 2C20,7 4 100 50 3976 2.91 2.08 5375 7 5 2C20,7 4 100 70 5634 7.14 19.60 5342 6 5 2C20,8 4 100 50 1745 2.42 3.18 5597 7 6 2C20,8 4 100 70 2244 5.76 6.47 5339 6 5 2

C20,1 4 140 50 25207 2.11 2618.32 4827 8 3 2C20,1 4 140 70 47463 * * * * * *C20,2 4 140 50 19777 4.26 55.40 4891 7 3 2C20,2 4 140 70 31155 11.86 120.92 4853 5 2 2C20,3 4 140 60 23897 1.27 171.42 4863 9 3 2C20,3 4 140 80 43763 * * * * * *C20,4 4 140 60 18997 0.99 161.92 4842 9 3 2C20,4 4 140 80 36865 * * * * * *C20,5 4 140 60 23227 * * * * * *C20,5 4 140 80 28531 11.19 117.59 4845 6 3 2C20,6 4 140 50 34062 6.15 1042.68 4819 6 3 2C20,6 4 140 70 46913 * * * * * *C20,7 4 140 50 18519 3.82 1136.23 5144 8 4 2C20,7 4 140 70 28601 * * * * * *C20,8 4 140 50 10366 3.63 99.91 5157 7 4 2C20,8 4 140 70 15409 8.54 2163.29 5114 6 4 2


Table 8.3: Computational results for 25 customers.


C25,1 3 100 50 9178 0.36 260.90 5354 11 5 2C25,1 3 100 70 13117 * * * * * *C25,2 3 100 50 7164 0.84 19.99 5676 9 6 2C25,2 3 100 70 8192 1.21 45.64 5655 9 6 2C25,3 3 100 60 7868 0.47 343.86 5660 11 6 2C25,3 3 100 80 9941 * * * * * *C25,4 3 100 60 11135 0.51 565.45 5366 11 5 2C25,4 3 100 80 14436 * * * * * *C25,5 3 100 60 6255 0.89 12.55 5718 10 6 2C25,5 3 100 80 6732 1.16 11.84 5713 10 6 2C25,6 3 100 50 7622 0.85 314.07 5640 9 6 2C25,6 3 100 70 8201 1.19 523.33 5629 9 5 2C25,7 3 100 50 4328 0.50 97.79 5970 10 7 2C25,7 3 100 70 5365 1.02 7.74 5715 11 6 2C25,8 3 100 50 7640 0.73 706.79 5646 9 6 2C25,8 3 100 70 8730 1.17 4.51 5405 11 5 2

C25,1 3 140 50 35102 * * * * * *C25,1 3 140 70 52120 * * * * * *C25,2 3 140 50 29235 1.36 186.43 5217 9 4 2C25,2 3 140 70 33443 1.74 793.46 5206 9 4 2C25,3 3 140 60 31402 0.71 1132.22 5209 10 4 2C25,3 3 140 80 41637 * * * * * *C25,4 3 140 60 37655 * * * * * *C25,4 3 140 80 51036 * * * * * *C25,5 3 140 60 28880 * * * * * *C25,5 3 140 80 31706 * * * * * *C25,6 3 140 50 32691 1.34 5277.95 5187 9 4 2C25,6 3 140 70 35990 * * * * * *C25,7 3 140 50 19877 * * * * * *C25,7 3 140 70 25142 * * * * * *C25,8 3 140 50 30993 1.04 4032.81 5196 9 4 2C25,8 3 140 70 36295 * * * * * *

C25,1 4 100 50 10983 4.92 1756.64 5343 11 5 2C25,1 4 100 70 20304 16.13 59.47 5078 8 4 2C25,2 4 100 50 9059 9.02 108.02 5634 8 6 2C25,2 4 100 70 14027 22.59 38.91 5394 7 5 2C25,3 4 100 60 8487 3.42 732.26 5660 11 6 2C25,3 4 100 80 14903 15.54 373.65 5409 8 5 2C25,4 4 100 60 11671 3.12 937.88 5348 10 5 2C25,4 4 100 80 21694 * * * * * *C25,5 4 100 60 8385 12.44 31.13 5677 8 6 2C25,5 4 100 80 10729 22.86 418.45 5635 7 6 2C25,6 4 100 50 10175 11.81 33.87 5369 7 5 2C25,6 4 100 70 13227 23.43 63.27 5350 8 5 2C25,7 4 100 50 5007 6.60 2670.68 5931 10 7 2


Table 8.3: Computational results for 25 customers (continued).


C25,7 4 100 70 7408 17.29 214.84 5638 8 6 2C25,8 4 100 50 10507 9.70 2600.30 5638 9 6 2C25,8 4 100 70 14805 21.14 40.23 5332 7 5 2

C25,1 4 140 50 45404 * * * * * *C25,1 4 140 70 91802 * * * * * *C25,2 4 140 50 40584 12.78 1051.81 5178 8 4 2C25,2 4 140 70 68122 * * * * * *C25,3 4 140 60 36751 4.87 2385.73 5209 10 4 2C25,3 4 140 80 72567 * * * * * *C25,4 4 140 60 42567 * * * * * *C25,4 4 140 80 94182 * * * * * *C25,5 4 140 60 45778 17.95 386.67 5213 8 4 2C25,5 4 140 80 61318 31.67 660.31 5185 7 4 2C25,6 4 140 50 54060 * * * * * *C25,6 4 140 70 78969 * * * * * *C25,7 4 140 50 26783 * * * * * *C25,7 4 140 70 45041 24.67 403.39 5195 8 4 2C25,8 4 140 50 47802 * * * * * *C25,8 4 140 70 72464 * * * * * *

We ran the instances for a maximum time of 7200 seconds. In the instances

marked with an asterisk (*), we could not find the optimal solution, either because

of the time limit or because they had memory problems when being solved with

CPLEX. Note, however, that in the vast majority of the cases, it is the memory

problem that occurs.

The percentage of instances solved within the time limit was around 68%. Con-

sidering the instances with 20 customers, 77% were solved, while only 59% of the

instances with 25 customers were solved. When parameter α increases from 3 to

4, the percentage of solved instances decreases from 67% to 66%. Considering pa-

rameter W , 86% of the instances with W = 100 are solved to optimality, while

only 50% are solved for W = 140.

As expected, by increasing parameter W from 100 to 140, the number of vari-

ables increases considerably. This increase on the number of variables is much

8.6. Conclusions 171

more significant than the one that occurs when α goes from 3 to 4, or even the one

that occurs when the number of customers goes from 20 to 25. Not surprisingly,

the number of instances solved always decreases when parameter W increases and

the other parameters remain the same (81% to 75% and 88% to 63% for the in-

stances with 20 customers and 81% to 31% and 94% to 31% for the instances with

25 customers), which does not always happen when parameters α and number of

customers increase.

Only about 17% of the solved instances took more than 1000 seconds to solve,

and 44% took less than 100 seconds.

8.6 Conclusions

Location routing is an integrated problem that combines two difficult ones, the

vehicle routing problem and the location problem. In this chapter, we explored a

variant of this problem in which vehicles can perform several consecutive routes

during one planning period. This is relevant for the many situations where the

vehicle routes have smaller durations than, for example, one workday. We also

considered an additional constraint on the maximum number of customers per

route and capacities both on the depots and vehicles.

Some reduction criteria were applied to reduce the size and symmetry of the

model, whose nodes of the corresponding graph represent time instants and whose

arcs represent feasible routes. The model is pseudo-polynomial and was solved with

a commercial solver (CPLEX). It was tested with a set of instances with up to 25

customers. For the solved instances, the computational times were reasonable.

This model can easily be adapted to other variants of the considered problem.

Chapter 9

Conclusions

“We can only see a short distance ahead,but we can see plenty there that needs to be done”.

Alan Turing

Contents9.1 Contributions of the Thesis . . . . . . . . . . . . . . . . 174

9.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . 175

173

174 9. Conclusions

9.1 Contributions of the Thesis

In this thesis, several variants of the cutting stock problem and vehicle routing

problem were addressed. New column generation, constraint programming and

pseudo-polynomial models were described, as well as several exact solution algo-

rithms, lower bounds and valid inequalities. To assess the effectiveness of the pro-

posed methods, we performed extensive computational experiments using instances

from the literature and also from the real-world.

For the challenging pattern minimization problem addressed in chapter 4, we

contributed with new results concerning the computation of lower bounds. For that,

we proposed an integer programming model and different strategies to strengthen

it. These strategies include new families of valid inequalities and the use of a new

constraint programming model.

For the remainder of the studied problems, we explored pseudo-polynomial

network flow models, and assessed their efficiency to solve the proposed problems.

The results obtained with the pseudo-polynomial model for the two-dimensional

cutting stock problem, analyzed in chapter 5, outperformed the other methods

presented in the literature, for the set of tested instances. This problem was greatly

inspired in a practical real-world situation and, therefore, it was tested with a

set of real-world instances. We proposed several reduction criteria in order to

eliminate most of the model’s symmetry and strengthened it with a new family of

cutting planes. We also derived a new lower bound. Moreover, we presented the

reformulations of the model for some variants of the problem such as the rotation

of items, and the possibility of considering the horizontal or vertical orientation for

the first cut. For those, we also presented the results of computational experiments,

and assessed the savings that could be accomplished when considering them.

For the vehicle routing problem with time windows and multiple use of vehicles,

we proposed an iterative exact solution algorithm. We formulated the problem as a

9.2. Future Research 175

network flow model for two main reasons. On the one hand, the time windows and

constraints in the duration of routes allowed us to explicitly generate all feasible

routes. On the other hand, the multiple use of each vehicle within a given workday

resembles a packing problem, which we suspected could be efficiently solved with a

network flow formulation, due to the results presented in chapter 5. This variant of

the classical vehicle routing problem has received little attention in the literature.

To our knowledge, there is only other exact method proposed for it. Our model

proved to be more efficient, by solving more instances and clearly outperforming

the other method in terms of computational time, for most of the cases.

Finally, we intended to explore how an adaptation of the network flow model

proposed in chapter 7 for the vehicle routing problem with time windows and

multiple use of vehicles behaved for the location routing problem with multiple use

of vehicles. The results are presented in chapter 8. For this problem, there are no

time windows for the clients and depots and most part of the network flow model is

repeated, with some variation, for each depot. We derived some reduction criteria

and tested the model with some adapted instances from the literature. For the set

of tested instances, it did not prove to be as efficient as the model described in the

previous chapter, but it is important to emphasize that the second problem is a

combination of two hard problems, rather than one. Even so, the model seems to

be promising to be further explored.

9.2 Future Research

Throughout the development of all the work presented in this thesis, several ideas

for future research have occurred. In this section, we expose some of them, which

will certainly be tackled in the near future.

Given some encouraging results obtained with the network flow models for the

studied problems, we intend to explore algorithmic frameworks based on them that

176 9. Conclusions

could be applied to generalized integer problems. In particular, we hope to do it

with the iterative algorithm described in chapter 7, for the vehicle routing problem

with time windows and multiple routes.

One of the weaknesses of the network flow models is the huge number of arcs

that they may require, even for not very large instances. This often translates into

memory problems. An interesting investigation course would be to try to generate

arcs dynamically, instead of explicitly considering them all from the beginning,

even if some reduction criteria is devised. This would probably make it possible to

solve larger instances than the ones presented in this thesis.

In what concerns the model considered in chapter 8, it is our intention to further

explore it, and to address other variants, namely the one considering time windows

for the vehicles and depots.

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