Vii ALIO-EURO - Booklet - paginas.fe.up.ptesicup/extern/esicup-8thMeeting/uploads/Conference/8th... · 8thESICUPMeeting 5 Welcome JoséF.Oliveira DavidPisinger DearFriends, Welcometothe8thMeetingofESICUP–EUROSpecialInterestGrouponCuttingandPacking

8th ESICUP Meeting 3

Table of Contents

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

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

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

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

Invited Talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Copenhagen Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Copenhagen, Denmark, May 19-21, 2011

4 8th ESICUP Meeting

Local Organising Committee:

David Pisinger, (chair) Technical University of Denmark

Martin Zachariasen, University of Copenhagen

Jesper Larsen, Technical University of Denmark

Line Blander Reinhardt, Technical University of Denmark

Dina Riis Johannessen, University of Denmark

Program Committee:

José Fernando Oliveira, (chair) University of Porto

A. Miguel Gomes, University of Porto

Gerhard Wäscher, Otto-von-Guericke-Universität Magdeburg

Ramón Alvarez-Valdes, University of Valencia

Julia Bennell, University of Southampton

David Pisinger, Technical University of Denmark

Organised by:

ESICUP— EURO Special Interest Group on Cutting and Packing

University of Copenhagen

Technical University of Denmark

We would like to thank the sponsors:


http://www.diku.dk/

http://www.dtu.dk/

http://en.fi.dk/councils-commissions/the-danish-council-for-strategic-research


Welcome

José F. Oliveira David Pisinger

Dear Friends,

Welcome to the 8th Meeting of ESICUP – EURO Special Interest Group on Cutting and Packing. Since itsformal recognition as an EURO Working Group in 2003, ESICUP has run a series of annual meetings thathave joined researchers and practitioners in the field of cutting and packing. Wittenberg, Southampton,Porto, Tokyo, L’Aquila, Valencia and Buenos Aires have hosted our past meetings and this 8th meetingis now held in Copenhagen. The scientific program that has been put together guarantees that these willbe rather fruitful days.

The number of submissions have, once again, force us to organize a couple of parallel sessions, whichwill force each one of us to choose between excellent presentations. With around 50 participants, mainlyfrom Europe but also from Brazil and Japan, once again the goals of disseminating this field of research,of bringing new people to this scientific community, of being a forum of discussion and intellectualstimulation for all of us interested in these problems, have been fully achieved.

It is a privilege to welcome you to Copenhagen at the best time of the year, in May when the beech getsgreen and the flowers are blossoming. The conference is co-organized by the University of Copenhagenand The Technical University of Denmark. Founded in 1479, the University of Copenhagen is the secondoldest university in Scandinavia, and one of the oldest universities in Northern Europe. It has morethan 37,000 students, the majority of whom are female (59 per cent), and more than 7,000 employees.The university has several campuses located in and around Copenhagen, with the oldest sites located inthe centre of Copenhagen. The Technical University of Copenhagen was founded in 1829 as Denmark’sfirst polytechnic university at the initiative of Hans Christian Ørsted, and today it is ranked as the bestengineering university in Scandinavia and among the leading engineering institutions of Europe.

Copenhagen is a major regional centre of culture, business, media, and science, as indicated by severalinternational surveys and rankings. On several occasions, Copenhagen has been recognized as one of thecities with the best quality of life. It is also considered one of the world’s most eco-friendly cities. Thewater in the inner harbour is so clean that one can swim in it, and 36 per cent of all citizens commute towork by bicycle: every day they cycle a combined 1.2 million km. So don’t forget to take a ride on oneof the free bicycles downtown Copenhagen and to spend some time in some of the many wonderful parksin the city.

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

Our wish is that you may leave Copenhagen looking forward for the 9th ESICUP Meeting.All the best.

José F. Oliveira David PisingerUniversity of Porto, FEUP / INESC Porto Technical University of Denmark

Program Chair Local Organiser



Information for Conference Participants

MEETING VENUEThe meeting takes place at DIKU, Department of Computer Science, University of Copenhagen. Theaddress is: Universitetsparken 1, DK-2100 Copenhagen Ø.DIKU is a former medical anatomical institute which was taken over by Computer Science in 1984. Theauditoriums still have a historic teint.http://www.diku.dk/

REGISTRATIONTakes place on Thursday 19th of May, 18:30 to 20:00, and Friday 20th May, 8:30 to 9:00 at DIKU.

ACCESS TO THE BUILDINGFriday the 20th of May is a national Danish holiday so the university building will be closed. We willdistribute some magnetic cards among the participants, so that you can enter the building.We only have 20 cards so please help each other. Remember to return the cards by the end of theconference.

NOTES ON PRESENTATION• Equipment

All conference rooms are equipped with an overhead projector and with a video projector andlaptop computer.We suggest that you bring your own computer and/or transparencies as a backup.

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

INTERNETTo use the wireless network, you should use Eduroam. We will also try to establish an open networkduring the conference.

GET-TOGETHERTakes place at the conference venue, DIKU, Universitetsparken 1, DK-2100 Copenhagen Ø. There willbe served beer, wine, and snacks.The get-together takes place on Thursday 19th May, 18:30 to 20:00.

CONFERENCE DINNERTakes place at the FIAT restaurant, Kongens Nytorv 18, DK-1050 Copenhagen K. To get there fromDIKU, take any bus to Nørreport station, and then walk 10 minutes to Kongens Nytorv. The conferencedinner starts on Friday 20th May, 19:30.http://www.f-i-a-t.dk/

EXCURSION TO CARLSBERGThe Carlsberg Visitor Centre is located at Carlsbergvej in Valby. To get there from DIKU, take bus 18from Jagtvej/Universitetsparken. Ask the driver to get off at Carlsberg. From the bus stop, it is around5 minutes of walk. At Carlsberg you can walk through the historic buildings where the brewery started,you can see the brewery horses, the old Carlsberg trucks, and the micro brewery Jacobsen. It is possibleto try two free beers in the visitor centre. Additional beer or food can be purchased. Table football isavailable. http://www.visitcarlsberg.dk

THE CITY OF COPENHAGENDoes probably not need any introduction. For an online guide see http://visitcopenhagen.com/.

MOVING AROUND• Busses

The easiest way to get to DIKU is by bus. Busses 43, 150S, 184, 185 all stop almost in front ofthe door. Within a 5-minute walk, you can also get bus 15 (from Vibenshus Runddel) and bus 18(from Jagtvej).


http://www.diku.dk/

http://www.f-i-a-t.dk/

http://www.visitcarlsberg.dk

http://visitcopenhagen.com/


• TaxiTo get a taxi, call (+45) 3535 3535 or (+45) 7025 2525.

• Transport to Copenhagenhttp://visitcopenhagen.com/transport/transport-to-copenhagen

• Transport in Copenhagenhttp://visitcopenhagen.com/transport/transport-in-copenhagen

• Online map (Google maps)http://maps.google.com/maps/ms?source=s_q&hl=en&geocode=&aq=0&ie=UTF8&hq=&hnear=Copenhagen,+Denmark&msa=0&msid=212637035126730735450.0004a337ac2f9ead49982&ll=55.682133,12.566557&spn=0.038131,0.095015&z=14


http://visitcopenhagen.com/transport/transport-to-copenhagen

http://visitcopenhagen.com/transport/transport-in-copenhagen

http://maps.google.com/maps/ms?source=s_q&hl=en&geocode=&aq=0&ie=UTF8&hq=&hnear=Copenhagen,+Denmark&msa=0&msid=212637035126730735450.0004a337ac2f9ead49982&ll=55.682133,12.566557&spn=0.038131,0.095015&z=14




Program Overview



Scientific Program Schedule

Friday May 20th9:00 – 9:45

Opening Session + Invited Talk (Lille Auditorium)

Chair: José Fernado Oliveira

Welcome Adress

Invited Talk – Integrated container loading and transportationDavid Pisinger (joint work with Jens Egeblad, Thomas Gerken)

9:45 – 10:30

Session 1 (Lille Auditorium)Chair: Gerhard Wäscher

1.1 – The practice of load buildingJoaquim Gromicho, Gerhard Post

1.2 – Constraints in Loading ProblemsAndreas Bortfeldt, Gerhard Wäscher

11:00 – 12:30

Session 2 (Lille Auditorium)Chair: Ramón Álvarez-Valdés

2.1 – On computational experimentation in two-dimensional rectangular cutting and packing problemsJosé F. Oliveira, Teresa Costa, Gerhard Wäscher

2.2 – Linear and nonlinear integer models for constrained guillotine two staged two-dimensional cutting patternproblemsHoracio Hideki Yanasse, Reinaldo Morabito

2.3 – Two efficient construction algorithms for the three-dimensional strip packing problemHiroki Kawashima, Yuma Tanaka, Shinji Imahori, Mutsunori Yagiura

2.4 – Lower Bounds for two- and three dimensional multiple bin-size bin packing problemsR. Álvarez-Valdés, F. Parreño, J.M. Tamarit

14:00 – 15:30

Session 3A (Lille Auditorium)Chair: Julia Bennell

3A.1 – Mathematical Models and Solution Algorithm for Nesting Problem of Arbitrary Shaped ObjectsT. Romanova, Yu. Stoyan, A. Pankratov

3A.2 – Automatic System of Phi-function Generation for Arbitrary 2D-objectsA. Pankratov, Yu. Stoyan, T. Romanova, M.Zlotnik

3A.3 – Branch and Cut Algorithms to solve Nesting ProblemsAntonio Martínez Sykora, Ramón Álvarez-Valdés, Jose Manuel Tamarit

3A.4 – Convex irregular stock cutting with guillotine cutsJ.A. Bennell, Han Wei, X. Song, J. Gold



Session 3B (Meeting room A+B)Chair: François Clautiaux

3B.1 – The constrained compartmentalized knapsack problemAline A S Leão, Maristela Santos, Robinson Hoto, Marcos Arenales

3B.2 – Extremal and maximal conservative scalesGleb Belov, Vadim M. Kartak

3B.3 – A faster approximation scheme for the multiple knapsack problemK. Jansen

3B.4 – Algorithms and mathematical Models for the Bin Packing problem with Fragile ObjectsFrançois Clautiaux, Mauro Dell’Amico, Manuel Iori, Ali Khanafer

16:00 – 17:30

Session 4A (Lille Auditorium)Chair: J. Valério de Carvalho

4A.1 – Packing Equal Spheres into a Multiconnected Polyhedral ContainerYu. Stoyan, G. Yaskov

4A.2 – Optimizing the empty space exploitation of a partially loaded containerGiorgio Fasano, Maria Chiara Vola

4A.3 – Modelling pattern sequencing problems with interval graphsIsabel Cristina Lopes, J.M. Valerio de Carvalho

4A.4 – A model with symmetry reduced for the MOSP problemIsabel Cristina Lopes, J.M. Valerio de Carvalho

Session 4B (Meeting room A+B)Chair: David Pisinger

4B.1 – Using a closeness model with the sequential-single placing method for the circle placing problemA.V. Kartashov, R.A. Pudlo

4B.2 – Using triangulation to search for a local minimum for a circle placing problemA.V. Kartashov, R.A. Pudlo, A.V. Babkina

4B.3 – A Hierarchical Approach to the Circle Covering ProblemPedro Rocha, A. Miguel Gomes

4B.4 – Cylinder packing with regular patternsJanus Timler Holm, Jakob Lindorff Larsen, David Pisinger

Saturday May 21th9:00 – 10:30

Session 5 (Lille Auditorium)Chair: A. Miguel Gomes

5.1 – A configurable constructive heuristic to solve the rectangle packing area minimization problemMarisa J. Oliveira, Eduarda Pinto Ferreira, A. Miguel Gomes

5.2 – Multi-objetive Solution for 2D Guillotine Cutting Stock ProblemJesica de Armas, Gara Miranda, Coromoto León

5.3 – LP Bounds in Constraint Programming Approaches for Orthogonal PackingMarat Mesyagutov, Gleb Belov, Guntram Scheithauer

5.4 – The Retail Shelf Space Allocation Problem – a ReviewTeresa Bianchi-Aguiar, Maria Antónia Carravilla, José F. Oliveira



11:00 – 12:30

Session 6 (Lille Auditorium)Chair: Andreas Bortfeldt

6.1 – An optimization model for the vehicle routing problem with three-dimensional loading constraintsLeonardo Junqueira, José Fernando Oliveira, Maria Antónia Carravilla, Reinaldo Morabito

6.2 – Reactive GRASP for the Container Loading ProblemMaria Teresa Alonso, R. Álvarez-Valdés, F. Parreño, J.M. Tamarit

6.3 – A biased random key genetic algorithm for for the Container Loading ProblemJosé Fernando Gonçalves

6.4 – A hybrid algorithm for the pickup and delivery problem with three-dimensional loading constraintsAndreas Bortfeldt, Dirk Männel

12:30 – 12:40

Closing Session (Lille Auditorium)

Closing Notes



Invited Talk

Integrated container loading and transportationDavid Pisinger∗ (joint work with Jens Egeblad, Thomas Gerken)

∗ Technical University of Denmark

We consider a real-life container loading problem which occurs at a typical furniture producer. The problem is todetermine an optimal subset from a larger set of furniture which can be loaded into acontainer of given dimen-sions. Each item has an associated profit and a loadable subset of items with maximal total profit is consideredoptimal. In the studied company, the problem arises during the planning of transportation of products to clientshundreds of times daily. The instances may contain more than one hundred of different items with irregularshapes. Large-sized items are combined in specific structures to ensure proper protection of the items duringtransportation and to reduce the complexity of the remaining problem. We have developed a method composedof several heuristics which are applied successively to the problem. The average loading utilization is 91.3% forthe most general instances with average running times around 100 seconds.In the next phase we integrate the container loading with route planning for a fleet of vehicles. The algorithmis based on Adaptive Large Neighborhood Search proposed by Pisinger and Ropke (2007). Several strategies forhandling the packing constraints in the routing problem are discussed, and experimentally compared.Computational experiments on real-life instances from a major furniture company show that the solutions ob-tained by ALNS are better than those currently used. Overall ALNS reduce the transportation expenses for thefurniture manufacturer with around 3.9%.Keywords: Container loading; vehicle routing



Abstracts

1.1The practice of load buildingJoaquim Gromicho∗, Gerhard Post†

∗ ORTEC bv, Groningenweg 6k, 2803 PV Gouda, The Netherlands and Department of Econometrics, FreeUniversity, The Netherlands, † ORTEC bv, Groningenweg 6k, 2803 PV Gouda, The Netherlands and

Department of Applied Mathematics, University of Twente, The Netherlands

We consider the problem of placing load in a 3-dimensional space in different settings like Transport Planning,Warehouse and Stock Management and Offer Calculation. We present an overview of business requirements facedby practitioners in these areas. This concerns placing items on pallets which can be subject to layering constraintsor sequencing and grouping constraints. Pallets are loaded to trucks and also can be subject to sequencing andgrouping constraints combined with load distribution constraints. These load distribution constraints include axleweights constraints with a movable axle, stability constraints like decreasing height, bearing strength constraintsand constraints expressing that there should be a minimal contact between items. Some of these functional issues,extremely relevant in practice, do not seem to appear in the scientific literature.Keywords: Container loading; pallet loading; transport scheduling

1.2Constraints in Loading Problems

Andreas Bortfeldt∗, Gerhard Wäscher†∗ FernUniversität Hagen, † Otto-von-Guericke-Universität Magdeburg

In a seminal paper, Bischoff and Ratcliff (1995) have emphasized that existing approaches to container loadingproblems may only be applicable to a small range of the situations encountered in practice. They remarked thatthis could be attributed to the fact that algorithms are often developed for the solution of standard problems,but do not necessarily take into account for additional practical requirements. In this presentation, the authorswill at first look into constraints relevant for real-world container loading and develop a consistent scheme for thecategorization of such constraints. Whether and to which extent the constraints are considered in the existingliterature on container loading will be dealt with in a second part of the presentation. Based on the typologyof cutting and packing problems by Wäscher, Haussner and Schumann(1997), the authors will point out whichcombinations of problem types and constraints have been dealt with satisfactorily, and which require additionalresearch efforts.Keywords: Three-dimensional Packing, Container Loading, Constraints, Typology

2.1On computational experimentation in two-dimensional rectangular cutting

and packing problemsJosé F. Oliveira∗, Teresa Costa†, Gerhard Wäscher‡

∗ INESC Porto, Faculdade de Engenharia, Universidade do Porto, † ISEP ? Porto Superior Institute ofEngineering, ‡ Otto-von-Guericke-Universität Magdeburg

In this talk we will present some contributes for the development of a new generation of research support tools,concerning the computational experimentation of new algorithms for Cutting and Packing problems, by the eval-uation, testing and building problem generators and proven hard benchmark test instances for every type andclass of cutting and packing problems. These research support tools will strongly contribute for the quality of thecomputational experiments run with cutting and packing problems and for the improvement of the quality of thepapers published in this field.A major limitation that is felt by researchers in cutting and packing is the absence of problem generators widelyand commonly used by all researchers in their computational experiments. For one-dimensional problems a goodproblem generator exists and for pallet loading problems a thorough set of benchmark problems are available,but for all remaining areas computational experiments are run over classical sets, many times without any criticalanalysis of their now-a-days difficulty, adding a few of self-generated instances. This is particular important whenmost of the algorithms developed and published are heuristics, given the computational complexity of cutting and



packing problems.In this talk we will analyse, for each type of two-dimensional rectangular cutting and packing problem, the existentproblem generators and test problem sets, using a specific characterization framework.Keywords: 2D rectangular; date sets; problem generators; computational experimentsPartially supported by Fundação para a Ciência e Tecnologia (FCT) Project PTDC/EIA-CCO/115878/2009 – CPack-BenchFrame through the Programa Operacional Temático Factores de Competitividade (COMPETE) of the Quadro Co-munitário de Apoio III, partially funded by FEDER.

2.2Linear and nonlinear integer models for constrained guillotine two staged

two-dimensional cutting pattern problemsHoracio Hideki Yanasse∗, Reinaldo Morabito†

∗INPE – National Institute for Space Research, † UFSCar – Federal University of São Carlos

In this work we review some linear and nonlinear models to generate two staged constrained guillotine cuttingpatterns, the exact and non-exact cases. We present new models that are adaptations or extensions of constrainedone-group cutting patterns models. Two staged models arise in different cutting processes, for instance, in furni-ture and hardboard industries. The models are useful for the research and development of more efficient methods,exploring particular structures, decomposition methods, relaxations etc. They are also useful in evaluating theperformance of heuristics, for they allow us (at least for instances of moderate size) to estimate the optimalitygap of solutions obtained by heuristics. To illustrate the application of the models, we analyze the results ofsome computational experiments with instances of the literature and others randomly generated. The resultswere produced using an optimization software and they show that the required computational effort to solve themodels can be quite different.Keywords: Cutting and packing problems; two-stage guillotine cutting; linear and nonlinear models;furniture industry

2.3Two efficient construction algorithms for the three-dimensional strip packing

problemHiroki Kawashima∗, Yuma Tanaka∗, Shinji Imahori∗, Mutsunori Yagiura∗

∗ Nagoya University, Japan

This paper considers the three-dimensional strip packing problem, which is the problem of placing a set of cuboids(boxes) into a three-dimensional bin (the container) of fixed width and height but unconstrained length. Boxesmust be packed so that every edge is parallel to one of the axes of the container. We treat two cases for therotation of boxes: (1) Each box has a fixed orientation, and (2) rotations are allowed.We investigate two heuristic algorithms called the three-dimensional best-fit algorithm (3BF) and the deepest-bottom-beft-fill algorithm (DBLF) for the three-dimensional strip packing problem. These methods are extendedfrom (the two-dimensional) best-fit algorithm and the bottom-left-fill algorithm, respectively, which are repre-sentative heuristic algorithms for the two-dimensional strip packing problem. It is not hard to see that a simpleimplementation of 3BF requires O(tn4) time, where n is the number of boxes and t is the number of box types,and to the best of our knowledge, no theoretical evaluation has been made on the efficiency of existing implemen-tations of 3BF. Algorithm DBLF requires O(n5) time if naively implemented, and there exists an implementationof the DBLF algorithm that requires O(n4) time.This paper presents efficient algorithms that realize the two heuristics. The proposed implementation of 3BFrequires O(tn2 logn) time, and that for DBLF requires O(n3 logn) time. To reduce the computation time further,we incorporate heuristic rules that skip unnecessary calculation processes. Though the worst-case time complexityof the algorithms remains the same even after including such heuristic rules, their actual computation time wasreduced by 90% on average.We performed computational experiments for test instances with up to 10,000 boxes. From the results, we con-firmed that our algorithms consumed very small execution time. They spent about one second to place onethousand boxes, and a few hundred seconds to place ten thousand boxes.Keywords: Three-dimensional strip packing; best-fit heuristic; deepest-bottom-left-fill algorithm; effi-cient implementation



2.4Lower Bounds for two- and three dimensional multiple bin-size bin packing

problemsR. Álvarez-Valdés∗, F. Parreño†, J.M. Tamarit∗

∗University of Valencia, Department of Statistics and Operations Research, Burjassot, Valencia, Spain, †

University of Castilla-La Mancha. Department of Computer Science, Albacete, Spain

The three-dimensional multiple bin-size bin packing problem, MBSBPP, is the problem of packing a set of boxesinto a set of bins when several types of bins of different sizes and costs are available and the objective is tominimize the total cost of bins used for packing the boxes.In this work we develop different types of lower bounds, from fast and simple bounds to more complex procedureswhich obtain more accurate bounds in longer computing times, and compare them on an extensive experimentalstudy.First, we review simple bounds existing in the literature. We have defined two new simple lower bounds, bothbeing improvements of the previous lower bounds proposed by other authors.We also present bounds based on an integer linear relaxation of the problem. We start from a formulation forthe one-dimensional problem, defining variables for the bins used and for the assignment of boxes to bins. Thenumber of bins of each type initially varies from 0 to the number of pieces fitting into this bin type, but we havedeveloped tight lower and upper bounds, reducing the number of variables to use. The original formulations alsoenhanced with logical considerations:

• We have determined some conditions under which a box k can be fixed into a copy j of a bin type i. Fixingpieces means fixing variables to integer values and this is very advantageous when solving the originalinteger or the relaxed linear formulations of the problem.

• One problem for tree-search algorithms solving this problem is that there are many symmetric solutions,that is, solutions with the same configuration of bins and permutations of the boxes packed into the bins ofa given type. In order to avoid most of these permutations, we establish an order among the bins of eachtype, ordering them by non-decreasing volume of the boxes packed into each bin.

• We use dual feasible functions to modify the dimensions of the boxes and then produce new valid constraints.

An alternative way of producing lower bounds is a tree search procedure in which to explore the feasible config-urations of the bins of each type delimited by the upper and lower bounds on the number of bins we describedabove. At each level of the tree, the partial solution is enlarged by adding a new bin type and the pieces fittinginto it. When considering the subproblem at each node, we can use most of the previous improvements for theproblem formulation. In the search process, some partial configurations of bins are found unfeasible and the lowerbound is the minimum solution value associated to the remaining feasible configurations.The proposed bounds are compared with other existing bounds obtaining better results.Keywords: Bin packing; lower bounds; integer formulations

3A.1Mathematical Models and Solution Algorithm for Nesting Problem of

Arbitrary Shaped ObjectsT. Romanova∗, Yu. Stoyan∗, A. Pankratov∗

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

The article considers nesting problem for 2D-objects, whose frontier is formed by circular arcs (convex and con-cave) and line segments. Free homothetic, translation and rotation transformations of objects are allowed. Inparticular allowable rotation constraints (continuous or discrete) may be solitary given for each object. In additionwe consider distance constraints for each pair of objects if any. As an efficient tool of mathematical modeling ofnon-overlapping, containment and distance constraints considering rotations we use the phi-function technique.We prove here that any phi-object may be decomposed with a finite number of basic convex and concave objects.We realise a clear decomposition algorithm for the objects with basic objects. We define a complete class ofphi-functions for a family of basic objects and show a way of deriving phi-functions for all realistic cases of arbi-trary shaped convex and concave objects. We proof that the phi-functions and their derivatives can be describedby quite simple formulas without radicals. Our phi-functions are formed by linear, quadratic and trigonometricfunctions (sins and cosines). A mathematical model of nesting problem is constructed as a nonlinear constraintoptimisation problem. A solution space of the problem is specified by a collection of phi-inequalities. Based onproperties of phi-functions we show how inequality systems may be extracted from the collection. The inequalitysystems involve only smooth functions. That is the crucial issue for gradient methods application in optimisationalgorithms. A general solution strategy and an algorithm using the phi-functions are outlined. A number of



computational results are given.Keywords: Nesting; irregular shapes packing; phi-function; arcs; mathematical model; optimization;algorithm

3A.2Automatic System of Phi-function Generation for Arbitrary 2D-objects

A. Pankratov∗, Yu. Stoyan∗, T. Romanova∗, M.Zlotnik∗∗ Department of Mathematical Modeling, Institute for Mechanical Engineering Problems of the National

Academy of Sciences of Ukraine, Kharkov, Ukraine

We introduce an intelligence system for generation of phi-functions for arbitrary shaped objects (irregular shapes),whose frontiers are formed by circular (convex and concave) arcs and line segments. We allow affine transforma-tions of free object translations and rotations. The only thing we need that is input data on a pair of objects.The object frontier may be given by AutoCAD. A tuple of object input data contains also indicators (check boxes)of translation and rotation allowance. We consider also allowable (minimal or/and maximal) distance for a pairof objects.In order to construct a phi-function for a pair of arbitrary shaped objects we have developed the following pro-gram modules: decomposition algorithm for the objects with basic objects; library of phi-functions for each pairof basic objects; construction of phi-functions for arbitrary shaped objects, considering translations, rotations andallowable distances.The system allow us to generate ready-to-use phi-inequalities for describing non-overlapping, rotation, contain-ment and distance constraints given as systems of linear and non-linear inequalities in real-time mode. Wedemonstrate the first version of the system.Keywords: Irregular shapes; arcs; phi-functions; constraints generator; non-overlapping; containment;rotations; allowable distances

3A.3Branch and Cut Algorithms to solve Nesting ProblemsAntonio Martínez Sykora∗, Ramón Álvarez-Valdés∗, Jose Manuel Tamarit∗

∗ University of Valencia, Department of Statistics and Operations Research, Burjassot, Valencia, Spain

Cutting and packing problems involving irregular shapes, usually known as Nesting Problems, are common inindustries ranging from clothing and footwear to engineering and shipbuilding. The research publications onthese problems are relatively scarce, compared with other cutting and packing problems with rectangular shapes,and have been mostly focused on heuristic approaches. In this paper we are interested in developing an exactalgorithm for the nesting problem.In our problem, we have a set of two-dimensional pieces, represented by polygons, which can be convex or non-convex. We do not allow pieces to be rotated and our objective is to arrange all the pieces in a strip of constantwidth minimizing the total required length. We propose a MIP formulation, Horizontal Slices Formulation, usingthe Non-Fit-Polygon (NFP) tool to ensure that each pair of polygons does not overlap. We use the Fischetti andLuzzi (2003) approach to write the NFP constraints, but we define the slices in a horizontal way. We also havelifted the bound constraints of the pieces.A Branch and Cut algorithm is proposed to solve optimally the problem. We study different strategies forbranching and we have found several valid inequalities for cutting the relaxed linear problem solutions.In order to study different branching methods we have tested the Fischetti and Luzzi priorities with differentinitial orders of pieces. We also propose a dynamic branching approach to accelerate the separation of the piecesoverlapping in the linear solutions and a branching by constraints approach.We have also found several new valid inequalities. Some of these inequalities are local because they use the boundsof the pieces at each node and others are global. We have developed separation algorithms to identify violatedconstraints and studied different strategies to add the new cuts.The Branch and Cut algorithm has been tested on a set of instances taken from the literature and some preliminaryresults, showing the effect of different branching strategies and new valid inequalities, will be presented.Keywords: Nesting problems; cutting and packing; mixed integer formulation; valid inequalities; branchand cut algorithm



3A.4Convex irregular stock cutting with guillotine cuts

J.A. Bennell∗, Han Wei†, X. Song‡, J. Gold◦∗ School of Management, CORMSIS, University of Southampton, UK, † School of Information Engineering,

Nanjing University of Finance & Economics, China, ‡ School of Mathematics, University of Portsmouth, UK, ◦

Jotika, UK

The problem arises from the glass cutting industry in the manufacture of conservatories (glasshouses). In theUK, conservatories are commonly attached to residential houses and commercial buildings. Each conservatory isa bespoke design for each building; hence, many pieces are unique. In addition to rectangular pieces, there arehigh varieties of irregular convex pieces, commonly for the roof, that have three or four vertices. The materialis expensive, and as a result, efficient use of material is important. Since the stock material is glass, the cuttingprocess constrains layouts to be broken out using guillotine cuts alone. However, these guillotine cuts are notconstrained to be orthogonal, nor is there a limit to the number of turns. Further constraints on the arrangementof the pieces arise from the cutting process. The presentation will describe a number of solution approaches.First, a two-phase approach, which clusters all pairs of pieces into rectangles, selects clusters using a variant ofthe Hungarian algorithm, creates blocks of rectangle clusters using a forest-based search and then constructs aguillotine layout. A second approach extends the number of pieces in each cluster. Our ultimate goal is a single-phase approach that directly packs the irregular pieces and determines the order of the non-orthogonal guillotinecuts. The project is ongoing and therefore the approaches presented will be a work-in-progress. Results will bepresented.Keywords: Irregular shape; guillotine cuts; stock cutting; heuristic

3B.1The constrained compartmentalized knapsack problemAline A S Leão∗, Maristela Santos∗, Robinson Hoto†, Marcos Arenales∗∗ Universidade de São Paulo-Brasil, † Universidade Estadual de Londrina-Brasil

The constrained compartmentalized knapsack problem can be considered as an extension of the constrainedknapsack problem. However, the items have different characteristics and items of the same type are packed intoa compartment separate from other items. The size of a compartment has upper and lower limits. Building acompartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack. The objective is tomaximize the total value of the items loaded in the overall knapsack minus the cost of the compartments. Thisproblem appears in two-stage cutting stock problems, where the roll is cut into intermediate rolls. The need for twostage cutting can be due to the restrictions of production processes and ordered item characteristics, for example,limited number of knives, coating of materials (coated paper and coated steel ribbons), and thickness reduction(steel rolls). And only items with the same characteristic can be cut from an intermediate roll. In the literature,the constrained compartmentalized knapsack problem has been considered in steel roll cutting, where items aregrouped in different classes according to their thickness, i.e., items in the same class have the same thickness. Afirst cut on the original roll in stock produces various intermediate rolls, which have the same thicknesses as theoriginal roll. An intermediate roll is seen as a compartment in the knapsack. Then, the thickness of each sub-rollis reduced to be the same as the item thicknesses of a class. A fixed trim is necessary to eliminate irregularitieson the borders of each intermediate roll, which means a fixed loss of the knapsack capacity. There is a relevantcost of thickness reduction that depends on the percentage of reduction. This cost can be seen as including anew compartment in the knapsack. Finally, the intermediate rolls whose thickness is already reduced are cutinto items. The compartmentalized knapsack problem has been formulated as an integer non-linear program.However, the integer non-linear formulation is difficult to solve. Then, we reformulate it as an integer linearmaster problem with an increased number of variables. Heuristics based on the solution of the restricted masterproblem are investigated and proposed a heuristic that combine some heuristics from the literature. A new andmore compact integer linear model is then proposed, solvable with a commercial optimizer that found most ofthe optimal solutions. The heuristics can find good solutions within low running times. On the other hand, thecommercial solver can provide most of the optimal solutions for the integer linear formulation, but for a few ofexamples it can take a very high running time.Keywords: Cutting problems; knapsack problems; column generation; heuristics



3B.2Extremal and maximal conservative scales

Gleb Belov∗, Vadim M. Kartak†∗ Institute of Numerical Mathematics, Technische Universität Dresden, † Department of Cybernetics, Ufa State

Aviation Technical University

Some packing problems are special cases of several other problems such as resource-constrained scheduling, ca-pacitated vehicle routing, etc.In this paper we consider a bounding technique for one- and higher-dimensional orthogonal packing problems,called conservative scales (CS) (in the scheduling terminology, redundant resources).CS are related to the possible structure of resource consumption: filling of a bin, distribution of the resource tothe jobs, etc. In terms of packing, CS are modified item sizes such that the set of feasible packings is not reduced.In fact, every CS represents a valid inequality for a certain binary knapsack polyhedron.CS correspond to dual variables of the set-partitioning model of a special 1D cutting-stock problem. Alternatively,CS can be constructed by (data-dependent) dual-feasible functions ((D)DFFs). We discuss the relation of CS toDFFs: CS reflect the condition that at most 1 copy of each object can appear in a combination, whereas DFFsallow several copies.The literature has investigated so-called extremal maximal DFFs (EMDFFs) which should provide very strongCS. Analogously, we introduce the notions of maximal CS (MCS) and extremal maximal CS (EMCS) and showthat EMDFFs do not necessarily produce (E)MCS. We propose some greedy methods to “maximalize” a givenCS. Using the fact that EMCS define facets of the binary knapsack polyhedron, we use lifted cover inequalitiesas EMCS. Numerical results compare the new methods with each other and with some Sequential LP methods.Keywords: Resource-constrained problems; capacitated problems; conservative scales; dual-feasiblefunctions; knapsack inequalities; lifting; multilinear programming

3B.3A faster approximation scheme for the multiple knapsack problem

K. Jansen∗∗ University of Kiel

In this talk we propose an improved efficient approximation scheme for the multiple knapsack problem (MKP).Given a set A of n items and set B of m bins with possibly different capacities, the goal is to find a subset S ⊆ Aof maximum total profit that can be packed into B without exceeding the capacities of the bins. Kellerer gavea polynomial time approximation scheme (PTAS) for MKP with identical capacities and Chekuri and Khannapresented a PTAS for MKP with arbitrary capacities with running time nO(1/ε8 log(1/ε)). Recently we found anefficient polynomial time approximation scheme (EPTAS) for MKP with running time 2O(1/ε5 log(1/ε))poly(n).Here we present an improved EPTAS with running time 2O(1/ε log4(1/ε)) + poly(n). If the integrality gap betweenthe ILP and LP values for bin packing with different sizes is bounded by a constant, the running time can beimproved to 2O(1/ε log2(1/ε)) + poly(n).Keywords: Knapsack problem; approximation algorithms

3B.4Algorithms and mathematical Models for the Bin Packing problem with

Fragile ObjectsFrançois Clautiaux∗, Mauro Dell’Amico†, Manuel Iori†, Ali Khanafer‡

∗ University of Lille, † University of Modena and Reggio Emilia, ‡ INRIA Lille Nord Europe

In this communication, we present new lower and upper bounds for the bin-packing problem with fragile objects.We are given a set of items, each characterized by a weight and a fragility, and a large number of uncapacitatedbins. Our aim is to find the minimum number of bins needed to pack all items, in such a way that in each binthe sum of the item weights is less than or equal to the smallest fragility of an item in the bin. The problem isknown as the Bin Packing Problem with Fragile Objects, and appears in the telecommunication field, when onehas to assign cellular calls to available channels.The literature on the BPPFO is still small. Bansal et al. (2009) present approximation schemes and probabilisticanalysis. They consider approximations both with respect to the number of bins and to the fragility of a bin.They present results for the general BPPFO and for a special case, denoted the frequency allocation problem, inwhich weight and fragility are strictly correlated one to the other. Chan et al. (2007) consider instead the on-line



version of the BPPFO, in which an item arrives only after the previous item has been packed and the decisioncannot be changed. They study the cases in which the ratio between the maximum and the minimum fragility isbounded or unbounded, and present, for both cases, algorithms with asymptotic competitive ratios.To solve the problem practically, we propose two different mathematical formulations and a set of techniques tocompute lower and upper bounds.We designed several combinatorial lower bounds and a column-generation approach. The behavior of the columngeneration is enriched by using ad hoc techniques to price columns (dynamic programming and mathematicalprogramming) and by adding so-called dual cuts to stabilize the process.We also propose several greedy heuristics, adapted from the literature, and a Variable Neighborhood Search (VNS)algorithm. The VNS algorithm we implemented moves in a search space that contains infeasible solutions. Westart by computing a heuristic solution. We then enter a loop in which we modify the number of bins used usinga perturbation method. We then apply a local search algorithm to restore the feasibility of the solution.We conducted extensive computational tests on a large benchmark set. The results we obtained show the effec-tiveness of the algorithms we propose, that are capable of improving the results obtained by an effective compactformulation. In particular, out of 135 instances we can obtain 78 optimal solutions in less than three minuteson average, against the 51 optima found by a compact formulation in a larger computational time. The goodbehavior of the column generation motivates further research for the development of branch-and-price techniques.Keywords: Bin packing problem; fragile objects; variable neighborhood search; column generation

4A.1Packing Equal Spheres into a Multiconnected Polyhedral Container

Yu. Stoyan∗, G. Yaskov∗∗ Department of Mathematical Modeling, Institute for Mechanical Engineering Problems of the National

Academy of Sciences of Ukraine, Kharkov, Ukraine

Let there be an infinite family of congruent spheres of given radius in the 3D Arithmetic Euclidean space and acontainer which is a non-convex polyhedron. We assume that the container is an intersection of a convex polyhe-dron and compliments of convex prisms, dihedral angles and convex polyhedral cones.Problem: Pack a maximal number of spheres into the container.In order to construct a mathematical model of the problem the concept of Phi-functions is used. Phi-functionsfor a convex polyhedral cone and a sphere, a dihedral angle and a sphere and a convex prism and a sphere areconstructed.The problem is reduced to a sequence of non-linear programming subproblems with a linear objective functionconsidering radii of spheres as variables.Subproblem: Search for a maximum of the sum of radii of spheres provided that the spheres are not overlappedand contained in the polyhedron.The mathematical model of a subproblem possesses the following properties: the feasible region is non-convexand specified by linear and quadratic inequalities; a subproblem is multiextremal and NP-hard; local maxima ofa subproblem are reached at extreme points of the feasible region; the feasible region can be presented as a unionof subregions which are specified by inequality systems. Starting from the characteristics for solving the initialproblem a solution approach is offered. The approach consists of three stages: constructing of starting points;searching for global or local maxima of subproblems; searching for an approximation to a global maximum ofthe initial problem. In order to obtain starting points belonging to the feasible region special procedures aredeveloped. For searching for local maxima of subproblems a modification of the Zoutendijk method of feasibledirections and a strategy of active inequalities are used. We offer an approach to search for an approximation toa global maximum of the problem. A number of numerical examples are given including packing spheres into aconvex polytope, a non-convex polytope.Keywords: Optimization; packing; sphere; polyhedron

4A.2Optimizing the empty space exploitation of a partially loaded container

Giorgio Fasano∗, Maria Chiara Vola†∗ CMath FIMA CSci, Thales Alenia Space Italia S.p.A., † Altran Italia S.p.A.

This work has been motivated by the quite challenging issue of the analytical cargo accommodation of spacevehicles and modules. To attain the cargo plan requested, a number of non-trivial three-dimensional packingproblems arise. In this work we discuss the issue of maximizing the volume exploitation of a partially loaded



single container by adding a number of virtual items, repositioning, if necessary, the objects already loaded.Even if this study has originated within the space engineering framework, we expect it could be of some interestalso in different fields such as the ones of industrial logistics (e.g warehouse handling) and transportations ingeneral (e.g. truck loading, ship on-board stowage). More advanced applications could moreover involve theidentification of accessibility zones for autonomous robots.The topic discussed herewith extends a previous MIP-based approach (G. Fasano, MIP-based heuristic for non-standard 3D-packing problems, 4OR 6,291-310) addressed to the three-dimensional orthogonal packing, withrotations, of tetris-like items, i.e. clusters of mutually orthogonal parallelepipeds (components).We thus consider a convex domain D and, inside it, a set of n tetris-like items. We are allowed to add up toN virtual items, consisting of parallelepipeds with sides of variable length (but not less than a given minimalthreshold), so that the total occupied volume (tetris-like items plus virtual items) is maximized, without violatingthe loading rules:

• each virtual item side has to be parallel to an axis of the prefixed orthonormal domain reference frame;

• each virtual item has to be contained within D;

• virtual items cannot overlap either tetris-items or other virtual items.

An exact formulation of the problem, expressed in terms of Mixed Integer Non-linear Programming (MINLP)is introduced first. Then, a simplified approach, based on possible Mixed Integer Linear Programming (MIP)approximations, is pointed out. An MIP-based heuristic approach, aimed at finding quick satisfactory solutionsin practice, is further described. An overview on real-world applications is given, suggesting possible future de-velopments and research objectives.Keywords: Non-standard three-dimensional packing; container loading problem; MIP/MINLP models;linear approximations; heuristics

4A.3Modelling pattern sequencing problems with interval graphs

Isabel Cristina Lopes∗, J.M. Valerio de Carvalho†∗ ESEIG – Polytechnic Institute of Porto, † Department of Production and Systems – University of Minho

In the process of planning industrial cutting operations, there is an important issue, beyond pattern generation,which is the decision of the processing sequence of the set of patterns on the cutting equipment. The PatternSequencing Problems consist in finding a permutation of the predetermined cutting patterns while optimizing agiven objective function. There are several relevant objective functions that give rise to different problems: thenumber of tool changes, the average order spread, the number of discontinuities, and the number of open stacks.For the Minimization of Opens Stacks Problem (MOSP), consider a cutting machine that processes just onecutting pattern at a time. The different items cut from patterns are piled around the machine in separate stacks.The stack of an item type remains near the machine if an item of that type still has to be cut from a forthcomingpattern. A stack is closed and removed from the work area only after all items of that size have been cut, andimmediately before proceeding to the next cutting pattern. As there are often space limitations around the cut-ting machines, it becomes important to minimize the maximum number of simultaneously open stacks, which isachieved by finding an optimal sequence to process the cutting patterns.The Minimization of Order Spread Problem (MORP) also deals with the sequencing of the patterns, but theobjective function aims to optimize the occupation of the stacks and to eliminate unnecessary dispersion.An order from a customer often consists of several items that may be cut from different stock sheets. In somecases it is possible to reduce the handling and storing costs by reducing the time elapsed between the cuttingpieces correspondent to the same order. The order spread is the distance between the first and the last item cutthat belongs to the same order, measured in number of stock sheets. The objective of the MORP is to minimizethe order spread, which is minimizing the time that a stack remains open.The minimization of the number of tool switches problem (MTSP) is a job scheduling problem that involvesflexible manufacturing machines that can hold a set of tools, which can be changed in order to have the adequatetools for each job. As these machines only have a capacity for C tools simultaneously, some tool switching mustbe made between different tasks sometimes. These tool changing operations have a cost which is proportional tothe number of switches. The MTSP problem consists of finding a sequence of the tasks, in order to minimize thenumber of tool switches.There are similarities between MOSP and MTSP: the jobs can act as cutting patterns that need to be sequencedand the tools can act as piece types.In these problems a solution can be modelled by an interval graph exhibiting a set of intervals that match theduration of stacks, or the time that each tool is loaded in the machine.We present an integer programming framework developed for the MOSP, based on the edge completion of a graph



and on a characterization of interval graphs that uses a perfect elimination ordering of the vertices, and exploreits applications to the MORP and MTSP.The computational results for alternative formulations of the model for these pattern sequencing problems arediscussed.Keywords: Pattern sequencing; minimization of opens stacks; interval graphs

4A.4A model with symmetry reduced for the MOSP problem

Isabel Cristina Lopes∗, J.M. Valerio de Carvalho†∗ ESEIG – Polytechnic Institute of Porto, † Department of Production and Systems – University of Minho

Recently we proposed a new integer programming formulation for the minimization of the maximum number ofopen stacks (MOSP) based on the completion of a graph with edges. By associating the duration of each stackwith an interval of time, the theory in interval (and perfect) graphs is used to create a model based on the clientgraph, introduced by Yanasse. The structure of this type of graphs admits a linear ordering of the vertices thatdefines an ordering of the stacks, and consequently decides a sequence for the cutting patterns. The polytopedefined by this formulation is full-dimensional and the main inequalities in the model are proved to be facets.Additional inequalities are derived based on the properties of chordal graphs and comparability graphs. Themodeling idea can be applied not only to pattern sequencing problems such as the minimization of open stacks orthe minimization of the order spread (MORP), but also to problems as the minimum interval graph completionproblem. In this talk we present a new model that considers an additional set of O(n2) variables representing thevertex that closes first and the vertex that closes last, where n is the number of vertices in the client graph. Theextended formulation allows the reduction of the symmetry of the previous model by associating a fixed order forthe start of the neighbors of the vertex that closes first. New inequalities are also derived that strengthen theprevious model. Results are relevant, particularly for the Miller graph, leading to a considerable reduction in thenumber of branch-and-bound nodes. Computational tests are presented.Keywords: Pattern sequencing; minimization of open stacks; interval graphs; linear ordering; cuttingstock

4B.1Using a closeness model with the sequential-single placing method for the

circle placing problemA.V. Kartashov∗, R.A. Pudlo∗

∗ Department of computer sciences of National Aerospace University, Kharkov, Ukraine

The problem of optimal placing a set of n circles is considered. It is necessary to place the circles without mutualoverlaps into a strip with fixed width. The objective function is a length of the occupied part of the strip. Thisfunction has to be minimized.One of the methods which are used for solving the problem is the sequential-single placing method. In this methodthe circles are placed sequentially, one after the other. The placed circles are fixed. The current circle is placedto minimize increment of the objective function. It has not to overlap with all the placed circles. This methodallows to find good approximations to the local minima. The complexity of its algorithmic realization is O(n4),because it does not take into account the relative position of the placed circles.The model of closeness, based on the Voronoi diagram and the Delaunay triangulation, was developed. It deter-mines nearest neighbors for each circle. The properties of the model were investigated. It allowed to construct newrealization of the sequential-single placing method with the closeness model. The time complexity was improvedto O(n2).All the developed models and algorithms were realized on C# language. Testing of the algorithms has shown itshigh efficiency.Keywords: Cutting; packing; allocation; optimization; Voronoi diagram; Delaunay triangulation; sequential-single allocation method; circle



4B.2Using triangulation to search for a local minimum for a circle placing

problemA.V. Kartashov∗, R.A. Pudlo∗, A.V. Babkina∗

∗ Department of computer sciences of National Aerospace University, Kharkov, Ukraine

The problem of optimal placing a set of n circles is considered. It is necessary to place the circles without mutualoverlaps into a fixed width strip. The objective function is a length of the occupied part of the strip. This functionhas to be minimized.Mathematical model of the problem consists of a linear objective function, linear and quadratic constraints inR2n+1 space. The problem is multiextremal. It has large number of local minima. The gradient projection methodis used to search for a local minimum. On each step of optimization it is necessary to control non-overlapping foreach pair of circles. In order to exclude checking the conditions for all pairs, it is proposed to store informationabout the neighbors for any circle. For this purpose it invited to use a triangulation on the circle?s centers. Theedges of the triangulation join neighboring circles. Every edge should not cross of any circle, except the pair,which it joins. It allows to control non-overlapping only neighboring circles. When the positions of circles arechanged, triangulation is reconstructed only if it is necessary. In result the time to check all non-overlappingconditions reduced from O(n2) to O(n).Keywords: Cutting; packing; placement; optimization; triangulation; a nonlinear programming; a localminimum; gradient projection method; circle

4B.3A Hierarchical Approach to the Circle Covering Problem

Pedro Rocha∗, A. Miguel Gomes∗∗ INESC Porto, Faculdade de Engenharia, Universidade do Porto

Covering problems aim to cover a certain complex region with a set of simpler geometric forms. The objective isto minimize the number of covering objects while making the best approximation to the initial complex region.A particular case of a problem of this kind is the circle covering problem (CCP), where the goal is to minimizethe radius of circles that can fully cover a given region, with a fixed number of identical circles.This problem isfairly common in a wide range of scientific fields, being debated and researched in wireless networking (cellularcoverage), military applications (known as the bomb problem), collision detection, and several other approaches.The specific problem approached in this work is to cover a multi-connected region represented by a complexpolygon (i.e., irregular polygons with holes).This work presents a hierarchical approach to enclosing a given irregular geometrical form, using non-uniformcircular enclosures. The first step is to enclose all vertices of the irregular geometrical form within a single circle.This is the worst approximation but the fastest to be computed. The second step divides the irregular form intoconvex polygons, and generate for each one a minimum enclosing circle. If the difference from the area of theenclosing circle compared to the area of the enclosed polygon is higher than a certain threshold, the polygonis iteratively divided until the difference of the resulting sub-convex polygons is within specification. This lastapproach has a higher number or circles, and takes significantly more time to compute, but returns a more ap-proximate representation of the enclosed form. Since this algorithm creates several levels of approximation, thefinal result consists of the first hierarchical level (approximation by a single circle), the final hierarchical level withmost detail, and intermediate levels of approximation, depending on the user specification (be it by area, or bynumber of circles).Preliminary computational experiments with complex polygons taken from nesting datasets show promising re-sults.Keywords: Circle covering; irregular polygonsPartially supported by Fundação para a Ciência e Tecnologia (FCT) Project PTDC/EME-GIN/105163/2008 – EaGLeN-est, through the Programa Operacional Temático Factores de Competitividade (COMPETE) of the Quadro Comunitáriode Apoio III, partially funded by FEDER.

4B.4Cylinder packing with regular patterns

Janus Timler Holm∗, Jakob Lindorff Larsen∗, David Pisinger∗∗ DTU Management

We consider the problem of packing uniform cylinders onto a rectangular pallet. First, in order to make thephysical loading easier, we want to restrict the patterns to regular or semi-regular patterns. Various polynomial



and pseudopolynomial algorithms are presented for constructing different regular patterns. Their efficiency ismeasured with respect to solution quality and time consumption. Next, a local search algorithm based on GuidedLocal Search (GLS) is presented. Starting from a regular, but overfilled pattern, the algorithm tries to reach afeasible solution through various horizontal, vertical or diagonally translations. To accelerate the solution ap-proach, Fast Local Search is used to limit the size of the neighborhood.Computational results are reported for both regular and semi-regular patterns, showing that promising resultscan be achieved even when restricted to those patterns. Furthermore, the GLS extension leads to even betterresults, at the expense of the pattern structure.Keywords: Pallet loading; cylinder packing

5.1A configurable constructive heuristic to solve the rectangle packing area

minimization problemMarisa J. Oliveira∗, Eduarda Pinto Ferreira†, A. Miguel Gomes‡

∗ ISEP – Porto Superior Institute of Engineering, † ISEP – Porto Superior Institute of Engineering and Gecad –Knowledge Engineering and Decision Support Research Center, ‡ INESC Porto, Faculdade de Engenharia,

Universidade do Porto

In the rectangle packing area minimization problem one wishes to pack a set of non-overlapping rectangles whileminimizing the enclosing rectangular area. This problem appears in several contexts, such as the placement ofmodules in Very Large Scale Integration (VLSI) circuits for which finding a compact placement for circuit moduleson chips is an important objective and in the design of facility layouts (FL) which is concerned with finding themost efficient non-overlapping arrangement of departments within a facility. This problem is NP-hard, whichlimits the use of exact techniques to find optimal solutions for large instances.To solve the rectangle packing area minimization problem we propose a configurable constructive heuristic withthe aim of obtaining slicing layouts and solutions of good quality in a low computational time. In this heuristic,one successively merges pairs of rectangles and the enclosing rectangle that results from this union is added tothe list of the remaining ones. This process ends when the list has only one rectangle, the one that encloses allthe initial rectangles. The proposed approach has two configuration parameters: one that chooses one of therectangles to merge and another that simultaneously chooses the second one and how to join them. For thevalues of the configuration parameters different options were used based on choices of rectangles with largestor small width, height or area. We tested our algorithm on several different sets of benchmarks with differentcharacteristics (diversity of sizes and total number of rectangles). Statistical tests were used to compare andevaluate the effect of each configuration parameter on the results, to verify if the results are dependent, or not, onthe relationship between the relative dimensions of rectangles (i.e. the rectangles shape) and on the total numberof the considered rectangles.Keywords: Cutting and packing; constructive heuristic; rectangle packingPartially supported by Fundação para a Ciência e Tecnologia (FCT) Project PTDC/EME-GIN/105163/2008 – EaGLeNest,through the Programa Operacional Temático Factores de Competitividade (COMPETE) of the Quadro Comunitário deApoio III, partially funded by FEDER.

5.2Multi-objective Solution for 2D Guillotine Cutting Stock Problem

Jesica de Armas∗, Gara Miranda∗, Coromoto León∗∗ University of La Laguna. Dpto. Estadística, I.O. y Computación., Av. Astrofísico Fco. Sánchez, s/n, 38271

La Laguna, Spain

A multi-objective approach to solve the Constrained 2D Guillotine Cutting Stock Problem is presented. Thisproblem targets the cutting of a large rectangle of fixed dimensions in a set of smaller rectangles using orthogonalguillotine cuts. Usually, the main goal is to find a feasible cutting pattern maximising the total profit. However,in some industrial fields, the raw material is either very cheap or can be easily recycled, so in such cases, amore important criterion for the pattern generation may be the speed at which the pieces can be obtained, thusminimising the production times and maximising the usage of the cutting equipment. This cutting process isspecifically limited by the features of the machinery available but, in general, it is determined by the number ofcuts involved in the packing pattern. Moreover, the number of cuts required for the cutting process is also crucialto the life of the industrial machines. Since the number of cuts is an important aspect in determining the cost andefficiency of the production process, a comprehensive optimization methodology should take also this criterioninto consideration. Therefore, in this study, the number of cuts to achieve the final demanded pieces is taken as



a second design objective.So, we want to optimise the layout of rectangular parts on the sheet of raw material so as to maximise the totalprofit, as well as minimise the number of cuts to achieve the final demanded pieces. For this, we have appliedMulti-Objective Evolutionary Algorithms given its great effectiveness when dealing with other types real-worldmulti-objective problems. Finally, we have selected the NSGAII algorithm, and an encoding scheme which isbased on a complete representation of pattern layouts using a postfix notation. According to the two differentoptimisation criteria the approach provides a set of solutions offering a range of trade-offs between the two ob-jectives, from which clients can choose according to their needs. Although the multi-objective approach doesn’treach the profit values provided by the single-objective method, the obtained solutions are very close to suchvalues and involve quite lower values for the second objective. This way, we have designed an approach whichprovides a wide range of solutions with a fair compromise between the two objectives. Moreover, we have achievedgood quality solution without having to implement an exact algorithm which involves an important associateddifficulty and cost, and is just focused on one objective without considering the possible negative effects on otherfeatures of the solutions.Keywords: 2D cutting stock problem; multi-objective optimisation; evolutionary algorithms

5.3LP Bounds in Constraint Programming Approaches for Orthogonal Packing

Marat Mesyagutov∗, Gleb Belov∗, Guntram Scheithauer∗∗ Dresden University, Germany

Let us consider a set of m rectangular items (wi, hi), with i ∈ I := {1, . . . ,m}. The 2-dimensional orthogonalpacking feasibility problem (OPP-2) [S. P. Fekete and J. Schepers. A combinatorial characterization of higher-dimensional orthogonal packing. Mathematics of Operations Research, 29(2):353–368, 2004.] asks whether allitems (wi, hi) with i ∈ I can be orthogonally packed into the given container (W,H) without rotations. Theguillotine constraint is not considered. OPP-2 is a subproblem in solution methods for orthogonal bin-packing,i.e., 2D strip packing problem (SPP-2).Suppose we have a coordinate system with point (0, 0) matched with the left bottom point of the container whosex, y-axis are associated with its W , H-sides correspondingly. Let us introduce sets of variables X = {xi : i ∈ I}and Y = {yi : i ∈ I}, which represent the start-positions for items in packing layouts of (0, x)-, (0, y)-directionscorrespondingly. The assignment of variables to the certain values is feasible in the meaning of OPP-2, if thefollowing constraints are satisfied.

xi + wi ≤ xj ∨ xj + wj ≤ xi ∨yi + hi ≤ yj ∨ yj + hj ≤ hi, i, j ∈ I : i < j; (1)

0 ≤ xi ≤W − wi, i ∈ I; (2)0 ≤ yi ≤ H − hi, i ∈ I; (3)

It assures that items do not overlap within the container, constraints (1), and lie within the container edges,constraints (2), (3). If there exist values of variables so that constraints (1)-(3) are satisfied, then the problem iscalled feasible, otherwise infeasible.In order to solve the OPP-2 we investigate the known state-of-the-art constraint programming (CP) approach ofClautiaux at al. [François Clautiaux, Antoine Jouglet, Jacques Carlier, and Aziz Moukrim. A new constraintprogramming approach for the orthogonal packing problem. Computers & Operations Research, 35(3):944–959,2008.]. We compare some known branching strategies and propose a new dichotomy branching strategy. CP-solvers employ special procedure, constraint propagation, which decides whether the set of constraints (1)-(3) isconsistent. In other world, it tries to prove the infeasibility of the current partial solution when a particular setof items is fixed or their allocation area is bounded. If inconsistence of the set of constraints can not be proven,then the procedure tries to reduce the set of possible values for variables. We propose advanced pruning rulesbased on the 1D relaxation bounds of the corresponding 2D instance. The latter problem is further relaxed bythe linear programming (LP) and solved by the column generation. The input date for the LP is formed fromthe partial solution and the information from the constraint propagation. Application of the above modificationsand CP with LP brings better results compared to those from the literature.Keywords: Constraint programming; constraint propagation; linear programming



5.4The Retail Shelf Space Allocation Problem – a Review

Teresa Bianchi-Aguiar∗, Maria Antónia Carravilla∗, José F. Oliveira∗∗ Faculty of Engineering, University of Porto

In this talk we will discuss shelf space allocation and present a literature review on this problem.The Shelf Space Allocation Problem arises when retailers have a large number of products to display and limitedshelf space available at disposal. Experimental studies have shown that the product choice of customers maybe influenced by in-store factors such as the shelf space allocated to the products. Therefore, well designedplanograms can attract customers and improve the financial performance of an outlet, fundamental in the highlycompetitive retail environment of today.The shelf space allocation is a decision problem which involves distributing the scarce shelf space among differentproducts held within a retail store. It is considered an extension of the multi-knapsack problem, a well-knownNP-Hard Problem, with additional marketing and retailing variables. The difficulty is further increased whendifferent types of fixtures are considered.Commercial software products gained many customers within the retailing industry due to their simplicity andpracticability. These products provide the retailer with a realistic view of the shelves and help allocating productsaccording to simple heuristics such as turnover or gross margin. However, they fail to incorporate the effect ofshelf space on product sales and the results are clearly suboptimal. Consequently, retailers use them mainly forvisual and insertion purposes, to reduce the amount of time spent on manually manipulating the shelves.Improving the decisions of shelf space allocation has gained considerable research attention in both marketing anddecision support literatures. This attention has focused both on measuring the effects of shelf space allocation onthe demand and sales of products as well as on building decision models and optimisation algorithms.Experimental studies have consistently shown the positive effect of shelf space allocation on the demand of anitem. This effect is largely attributed to the consumers’ unplanned purchases and to out-of-stocks. Three mainfactors have been studied: (1) impact of shelf space in the demand (defined as space elasticity), (2) impactof display location in the demand and (3) interdependency between items (defined as cross elasticity). Severalspace allocation models have been proposed in the literature. At first, models appeared as a polynomial formusing space elasticity parameters. Later, some researchers proposed non-linear models by considering additionalfactors such as cross elasticity, and linearisation techniques for approximating them into linear MIP models alsoappeared. The difficulty in obtaining reliable estimation of so many parameters led recently to simpler models butthe simplifications used so far make retail application unrealistic. Due to the differences in company’s strategy,management style and characterization of products and stores, the difficulty in developing a generic model thatcan represent all real world shelf allocation problems has also been stated.The computational complexity of the resulting models (NP-Hard) turns impractical to work out polynomial timemethods that can solve every problem instance to optimality. Dynamic programming has already been proposedto optimise this problem but with extremely high computational times for large problem instances. Alternativeheuristics and metaheuristics methods have been used to solve this problem, such as simulated annealing, geneticalgorithms and tabu search. However, some authors point out the lack of academic work on both exact andheuristic methods.Keywords: Shelf space allocation; retail; multi-knapsackThe first author is grateful to the Portuguee Foundation for Science and Technology for awarding her the grant SFRH /BD / 74387 / 2010

6.1An optimization model for the vehicle routing problem with

three-dimensional loading constraintsLeonardo Junqueira∗, José Fernando Oliveira†, Maria Antónia Carravilla†, Reinaldo Morabito∗

∗ Departamento de Engenharia de Produção, Universidade Federal de São Carlos, † Faculdade de Engenharia,Universidade do Porto

The vehicle routing literature has been recently merged with the container loading literature to treat cases wherethe goods required by customers are wrapped up in discrete items, such as boxes. This effort arises from theattempt to avoid expressing the demands of the customers simply as their weights or volumes. In other words,if the demand constraints are seen from a one-dimensional point of view, it is assumed that each demand fillsa certain section of the vehicle or that the cargo shapes up smoothly according to the vehicle shape. However,when dealing with rigid discrete items, their geometry may lead to losses of space or even to infeasible solutions ifthe vehicle has not enough capacity. If other practical constraints are also considered, the coupling of the routingand loading structures becomes even more complex. In this paper, we present an integer linear programming



model for the vehicle routing problem that considers real-world three-dimensional loading constraints. In thisproblem, a set of customers make requests of goods, that are wrapped up in boxes, and the objective is to findminimum cost delivery routes for a set of identical vehicles that, departing from a depot, visit all customers onlyonce and return to the depot. Apart of the usual 3D container loading constraints that ensure that the boxesare packed completely inside the vehicles and that they do not overlap each other, the problem also takes intoaccount constraints related to the vertical stability of the cargo, multi-drop situations and load bearing strengthof the boxes (including fragility). The practical importance of incorporating these constraints to the problem isto avoid loading patterns where boxes are floating inside the vehicles, where an unnecessary additional handlingis incurred when each drop-off point of the route is reached, or where products are damaged due to deformationof the boxes that contain them. Computational tests with the proposed model were performed using an optimiza-tion solver embedded into a modeling language. The results validate the model but show that it is only able tohandle problems of a moderate size. However, this model might be useful to motivate future research exploringother solution approaches to solve this problem, such as decomposition methods, relaxation methods, heuristics,among others, as well as to treat other variants of the vehicle routing problem, such as when time windows or aheterogeneous fleet are present, among others.Keywords: Capacitated vehicle routing problem; three-dimensional loading; combinatorial optimiza-tion; mathematical modelling

6.2Reactive GRASP for the Container Loading ProblemMaria Teresa Alonso ∗, R. Álvarez-Valdés†, F. Parreño∗, J.M. Tamarit†

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

We deal with the three dimensional packing problem, specifically, the container loading problem. Thus, givena container with dimensions (W,L,H) and a number of boxes of different types, qi, i ∈ [1 . . . n]. Each boxhas certain dimensions (di1, di2, di3) and an associated weight, Wi as well as a number of allowed orientations,Oi,j , i ∈ [1 . . . n], j = 1, 2, 3. Let us note that each type of box can support a given weight in each possibleorientation, bi,j , where i ∈ [1 . . . n], j = 1, 2, 3 if the box is placed in the orientation Oi,j .Our aim is to determine a packing in order to maximize the volume of the boxes introduced in the container, butdealing with the following constraints:

• The boxes must be full supported.

• The boxes have load bearing constraints, i.e., do not support an unlimited weight on them, then the weightdepends on the box orientation, one orientation may support more weight than another and there may becases that have a side that supports more weight than others.

• Each box has a set of allowed orientations, this may be due to the material contained or because a particularface is stronger and supports more weight.

According to the typology of Wäscher, the container loading problem can be classified as three-dimensional prob-lem with only a large object, the container, whose denomination would be 3D SLOPP, if the box set is weaklyheterogeneous, or the three-dimensional knapsack problem 3D SKP, if the box set is strong heterogeneous.Our proposal for solving the problem is to use a GRASP algorithm. The constructive approach is an iterativeprocess that consists of three steps, choice of the placement, to made up the blocks to place and to update the cor-responding data structure. The data structure used is the matrix of Ngoi, this representation is a two-dimensionalmatrix that reproduces the overhead view of the container, where the first row represents the horizontal projec-tions of the boxes on the X axis and the first column represents the transverse projections of the boxes on theY axis. The rest of the cells corresponds to the height of the surfaces and the weights that supports the uppersurface.To randomize the constructive, in the elections of the box and block, we have applied a reactive GRASP in whichthe parameter δ is adjusted depending on the quality of the solutions. When the constructive phase is finished,we have got a solution which we want to improve in a second phase with movements based on removing blocks,that was placed, and it fill again with a different objective function and deterministically.We present a computational study with the different alternatives in each phases, justifying our election and wepresent a comparative between our algorithm with the previous works in the problem.Keywords: 3D packing; container loading; GRASP



6.3A biased random key genetic algorithm for for the Container Loading

ProblemJosé Fernando Gonçalves∗

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

This paper addresses a three-dimensional container loading problem (3D-CLP), where a subset of a given set ofrectangular boxes is to be loaded into a large rectangular container such that the packed volume is maximized.The proposed approach hybridizes a placement procedure with a multi-population genetic algorithm based onrandom keys (MPGA).The genetic algorithm is used to evolve:

• the order in which the boxes are placed in the container;

• the orientation used for each box.

The different populations of the MPGA are evolved independently for a predetermined number of generationsafter which there is an exchange of chromosomes (solutions) between all the populations.The approach uses the concept of maximal-space and the difference process of Lai and Chan (1997) to managethe set of empty maximal-spaces in the container.To drive the optimization to better solutions a novel fitness function is also introduced. The new fitness function isconstructed in a way such that it assigns higher quality to solutions where the empty spaces are more concentrated(i.e. a large empty space is better than a many small empty spaces).A novel procedure is developed for joining free maximal-spaces in the case where full support from below isrequired.The approach is extensively tested on the complete set of test problem instances of Bischoff and Ratcliff (1995)and Davies and Bischoff (1999) and is compared with 13 other 3D-CLP approaches. The test set consists of 1500instances from weakly to strongly heterogeneous cargo. The computational experiments demonstrate not onlythat the approach performs very well in all types of instance classes but also that it obtains the best overall resultswhen compared with other approaches published in the literature.Keywords: Packing; container loading problem; three-dimensional knapsack problem; genetic algorithm;biased random keysSupported by Fundação para a Ciência e Tecnologia (FCT) project PTDC/GES/72244/2006

6.4A hybrid algorithm for the pickup and delivery problem with

three-dimensional loading constraintsAndreas Bortfeldt∗, Dirk Männel∗∗ University of Hagen, Germany

In the capacitated vehicle routing problem with pickup and delivery (VRPPD) a set of transportation requestshas to be satisfied by some vehicles. For each request a pickup point and a corresponding delivery point is givenas well as a demand to be transported between these places. This contribution considers the 3L-VRPPD, i.e. acombination of the VRPPD and three-dimensional (3D) loading with additional packing constraints frequentlyoccurring in freight transportation. Each of the given (homogeneous) vehicles has a 3D loading space and thedemand of a transportation request consists of a set of 3D rectangular items (boxes).A hybrid algorithm is proposed including a large neighborhood search algorithm for routing and a tree searchalgorithm for packing boxes into a loading space. Twenty-eight instances of the 3L-VRPPD were derived fromwell-known VRPPD instances from the literature. The numerical experiments show that the hybrid algorithm isable to provide good quality results in relative short computation times.Keywords: Pickup and delivery problem; three-dimensional loading; packing constraints; large neigh-borhood search; tree search



List of ParticipantsBennell, JuliaUniversity of [email protected]

Parreño, FranciscoUniversidad Castilla-La [email protected]

Alonso Martinez, Maria TeresaUniversidad Castilla-La [email protected]

Martínez Sykora, AntonioUniversity of [email protected]

Alvarez-Valdes, RamonUniversity of [email protected]

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

Fasano, GiorgioThales Alenia Space [email protected]

Jansen, KlausUniversität [email protected]

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

Yagiura, MutsunoriGraduate School of Information Science, [email protected]

Clautiaux, FrançoisUniversity of [email protected]

Imahori, ShinjiGraduate School of Information Science, [email protected]

Bortfeldt, AndreasUniversity of [email protected]

Romanova, TatianaIMEP of the National Academy of Sciences of [email protected]

Pankratov, AlexandrIMEP of the National Academy of Sciences of [email protected]

Stoyan, YuriIMEP of the National Academy of Sciences of [email protected]

Post, GerhardORTEC bv and University of [email protected]

Gromicho, JoaquimORTEC and Free University, The [email protected]

Kawashima, HirokiGraduate School of Information Science, [email protected]

Gomes, A. MiguelUniversidade do Porto - Faculdade de Engenharia,INESC [email protected]

Oliveira, José FernandoUniversidade do Porto - Faculdade de Engenharia,INESC [email protected]

Pudlo, RostyslavDepartment of computer sciences of National AerospaceUniversity, [email protected]

Rocha, Pedro FilipeINESC [email protected]



Leão, AlineUniversidade de São [email protected]

Yanasse, Horacio HidekiINPE - Instituto Nacional de Pesquisas [email protected]

Kartashov, A.V.Department of computer sciences of National AerospaceUniversity, [email protected]

Bianchi-Aguiar, TeresaUniversidade do Porto - Faculdade de [email protected]

Leon, CoromotoUniversidad de La [email protected]

Junqueira, LeonardoDepartamento de Engenharia de Produção, Universi-dade Federal de São [email protected]

Lopes, Isabel CristinaESEIG ? Polytechnic Institute of Porto / University [email protected]

Belov, GlebDresden University of TechnologyGermany–-

Oliveira, MarisaInstituo Superior de Engenharia do [email protected]

Valério de Carvalho, J.M.Universidade do [email protected]

Morabito, ReinaldoUniversidade Federal de Sao Carlos, [email protected]

Mesyagutov, MaratDresden University of [email protected]

Svensson, PatrikIGEMS Software [email protected]

Pisinger, DavidDTU [email protected]

Zachariasen, MartinDIKU University of [email protected]

Gerken, ThomasMaersk [email protected]

Larsen, JesperDTU [email protected]

Reinhard, Line BlanderDTU [email protected]

Villumsen, Jonas ChristofferDTU [email protected]

Larsen, Jakob LindorffDTU [email protected]

Holm, Janus TimlerDTU [email protected]

Reck, HanneDTU [email protected]



Notes






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