ut

Ha

s and

4; acline 1

The number of publications in the area of Cutting and Packing (C&P) has increased considerably over the last two dec-

A typology is a systematic organisation of objectsinto homogeneous categories on the basis of a given

each other).A typology provides a concise view of relevant,

vides the basis for a structural analysis of theunderlying problem types, the identication anddenition of standard problems, the developmentof models and algorithms, problem generators,etc. For the area of cutting and packing (C&P),

erved.

* Corresponding author. Tel.: +49 391 6718225; fax: +49 3916718223.

E-mail address: gerhard.waescher@ww.uni-magdeburg.de (G.Wascher).

European Journal of Operational Resea0377-2217/$ - see front matter 2006 Elsevier B.V. All rights resset of characterising criteria. It is practically ori-ented and meant to deal with important realobjects in the rst place while hypothetical and/orless important real objects might be neglected.A typology resembles a classication, however, withrespect to the above-mentioned focus, unlike the lat-ter, it may not be complete (i.e. not all properties ofa criterion may be considered explicitly), and it may

important objects and, thus, prepares the groundfor practically oriented research. Furthermore, ithelps to unify denitions and notations and,by doing so, facilitates communication betweenresearchers in the eld. Finally, if publications arecategorised according to an existing typology, it willprovide a faster access to the relevant literature.

A typology of OR problems, in particular, pro-ades. The typology of C&P problems introduced by Dyckho [Dyckho, H., 1990. A typology of cutting and packingproblems. European Journal of Operational Research 44, 145159] initially provided an excellent instrument for the orga-nisation and categorisation of existing and new literature. However, over the years also some deciencies of this typologybecame evident, which created problems in dealing with recent developments and prevented it from being accepted moregenerally. In this paper, the authors present an improved typology, which is partially based on Dyckhos original ideas,but introduces new categorisation criteria, which dene problem categories dierent from those of Dyckho. Furthermore,a new, consistent system of names is suggested for these problem categories. Finally, the practicability of the new scheme isdemonstrated by using it as a basis for a categorisation of the C&P literature from the years between 1995 and 2004. 2006 Elsevier B.V. All rights reserved.

Keywords: Cutting; Packing; Typology

1. Introduction be fuzzy (i.e. the categories may not always bedened precisely and properly distinguished fromAn improved typology of c

Gerhard Wascher *, Heike

Otto-von-Guericke-University Magdeburg, Faculty of Economic

Received 30 September 200Available on

Abstractdoi:10.1016/j.ejor.2005.12.047ting and packing problems

uner, Holger Schumann

Management, P.O. Box 4120, D-39016 Magdeburg, Germany

cepted 15 December 20059 June 2006

rch 183 (2007) 11091130

www.elsevier.com/locate/ejor

1110 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130Dyckho (1990) has introduced such a typology,which, unfortunately, has not always found satisfac-tory international acceptance. Furthermore, almostfteen years since its initial publication, it becameobvious that Dyckhos typology is insucient withrespect to the inclusion of current developments.Therefore, the authors decided to present a new,improved typology here, providing a consistent sys-tem of problem types which allows for a completecategorisation of all known C&P problems andthe corresponding literature. Furthermore, it helpsto identify blank spots, i.e. areas, in which noor only very little research has been done. Finally,the suggested typology is also open to new problemtypes to be introduced in the future.

For the problem types included in the typology,a straightforward system of notations is intro-duced. Wherever possible, the problem categoriesare named in accordance with existing notations.By doing so, the authors do not only intend to avoidunnecessary misinterpretations, or even confusion,but also to improve the degree of acceptance amongresearchers in the eld. The system of notations willalso provide the basis for a system of abbrevia-tions which can be used in a database of C&Pliterature for a simple but precise characterisation.The suggested abbreviations may also become partof a more complex coding scheme of C&Pproblems.

The remaining part of this paper is organised asfollows: In Section 2, the general structure of C&Pproblems will be presented, not only because thetypology to be introduced here addresses C&Pproblems, but also because the criteria for the de-nition of the respective problem categories areclosely related to elements of this structure. In Sec-tion 3, Dyckhos typology will be discussed, and,in particular, some severe drawbacks are pointedout which necessitate the development of a new,improved typology. The new typology is outlinedin Section 4. In Section 5, criteria and correspond-ing properties used in the new typology for the def-inition of problem types are introduced. As far asthese criteria have also been used by Dyckho, theywere complemented by additional, potentiallyimportant properties. Then (Section 6), in a rststep, two of the criteria, type of assignment andassortment of small objects, are used to denebasic problem types. The additional application ofa third criterion, assortment of large objects leadsto intermediate problem types. Finally, the applica-

tion of the criteria dimensionality and shape ofsmall items provides rened problem types. In Sec-tion 7 it is demonstrated how well-known problemsdiscussed in the literature t into the suggestedtypology. Furthermore, its usefulness and practica-bility is demonstrated by categorising the literatureon C&P problems from the last decade (19952004) accordingly. An analysis of recent trends inC&P research, based on the categorisation, is givenin Section 8. The paper concludes with an out-look on future work (Section 9), which particu-larly sketches a potential, more rened schemefor the categorisation of (publications on) C&Pproblems.

2. Structure of cutting and packing problems

Cutting and packing problems have an identicalstructure in common which can be summarised asfollows:

Given are two sets of elements, namely

a set of large objects (input, supply) and a set of small items (output, demand),

which are dened exhaustively in one, two, three oran even larger number (n) of geometric dimensions.Select some or all small items, group them into oneor more subsets and assign each of the resulting sub-sets to one of the large objects such that the geomet-ric condition holds, i.e. the small items of each subsethave to be laid out on the corresponding largeobject such that

all small items of the subset lie entirely within thelarge object and

the small items do not overlap,

and a given (single-dimensional or multi-dimen-sional) objective function is optimised. We note thata solution of the problem may result in using someor all large objects, and some or all small items,respectively.

Formally, ve sub-problems can be distin-guished, which, of course, have to be solved simul-taneously in order to achieve a global optimum:

selection problem regarding the large objects, selection problem regarding the small items, grouping problem regarding the selected smallitems,

allocation problem regarding the assignment of

the subsets of the small items to the large objects,

layout problem regarding the arrangement of thesmall items on each of the selected large objectswith respect to the geometric condition.

Special (types of) C&P problems are character-ised by additional properties. In particular, theymay turn out to be degenerated in the sense thatthey do not include all of the above-mentionedsub-problems.

3. Dyckhos typology

Fig. 1 summarises the main features of Dyck-hos typology. The author introduces four criteria(characteristics) according to which C&P problemsare categorised. Criterion 1, dimensionality, cap-tures the minimal number (1,2,3,n > 3) of geomet-ric dimensions which are necessary to describe therequired layouts (patterns) completely. Regardingcriterion 2, the kind of assignment of small itemsto large objects, Dyckho distinguishes two cases,

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1111indicated by B and V. B stands for the GermanBeladeproblem, meaning that all large objectshave to be used. A selection of small items has tobe assigned to the large objects. V (for the GermanVerladeproblem) characterises a situation inwhich all small items have to be assigned to a selec-tion of large objects. Criterion 3 represents theassortment of the large objects. O stands for onelarge object, I for several but identical large objects,

1. Dimensionality(1) one-dimensional(2) two-dimensional(3) three-dimensional(N) N-dimensional with N > 3

2. Kind of assignment(B) all objects and a selection of items(V) a selection of objects and all items

3. Assortment of large objects(O) one object(I) identical figure(D) different figures

4. Assortment of small items(F) few items (of different figures)(M) many items of many different figures(R) many items of relatively few different

(non-congruent) figures(C) congruent figures

Fig. 1. Categorisation criteria, main types, and coding scheme of

Dyckhos typology (cf. Dyckho, 1990, p. 154).and D for several dierent large objects. Criterion 4likewise characterises the assortment of the smallitems. Dyckho distinguishes assortments consist-ing of few items (F), many items of many dierentgures (M), many items of relatively few dierentgures (R), and congruent gures (C; we will beusing the notion shape here, instead of gure).

When it was initially published, Dyckhos workrepresented a milestone in C&P research as it high-lighted the common underlying structure of cuttingproblems on one hand and packing problems on theother. By doing so, it supported the integration andcross-fertilisation of two largely separated researchareas. Unfortunately, on an international level histypology has not always been accepted as widelyas was desirable, most probably due to the fact thatthe provided coding scheme was not self-explana-tory from the view point of an international (Eng-lish-speaking) community of researchers. Thisbecomes particularly evident for the problem typesB and V.

Obviously, this deciency could be overcome byintroducing more appropriate (English) names forthe respective problem categories/types. However,some drawbacks have to be taken more seriously,which became evident in the light of some recentdevelopments in the eld of C&P.

Not necessarily all C&P problems (in the narrowsense) can be assigned uniquely to problem types

In Dyckhos paper already, a well-known standardproblem, the Vehicle Loading Problem, has beencoded both as 1/V/I/F and 1/V/I/M (Dyckho,1990, p. 155). F characterises a situation in whichfew items of dierent shapes (i.e. items which aredierent with respect to shape, size and/or orienta-tion) are to be assigned. M represents a situationwith many items of many dierent shapes. Apartfrom the fact that it is denitely desirable to haveonly one option available for the assignment of awell-known standard problem like the VehicleLoading Problem, it does not become clear how thisdierentiation will provide a set of problem catego-ries which is more homogeneous (with respect tomodel building and the development of algorithms)than a single type, which includes both categories.(It is interesting to note that problem category Fonly appears once in Dyckhos examples of com-bined types (1990, p. 155). One may take this asan additional indicator that this kind of dierentia-

tion might not prove very promising in the end.)

1112 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 Dyckhos typology is partially inconsistent; itsapplication might have confusing results

In the literature, a particular two-dimensional pack-ing problem, the so called Strip-Packing Problem,namely the packing of a set of small items (often rect-angles) of dierent sizes into a single rectangle withxed width and minimal (variable) length (Jacobs,1996; Martello et al., 2003), has attracted consider-able attention. Most probably, this problem wouldbe coded both by researchers and by practitionersas 2/V/O/M, while inDyckhos typology it has beenassigned the notation 2/V/D/M. Dyckho justiesthis notation rather articially from our point ofview by saying that this problem . . . is equivalentto an assortment selection problem where the stockis given by an innite number of objects of this widthand of all possible lengths and where only one objecthas to be chosen from stock, namely that of minimallength (1990, p. 155). What makes Dyckhos char-acterisation of the problem even more confusing isthe fact that he calls it a Two-Dimensional Bin Pack-ing Problem (Dyckho, 1990, p. 155). Not only in ourunderstanding (also see Lodi et al., 2002a; Miyazawaand Wakabayashi, 2003; Faroe et al., 2003) it wouldhave been more obvious to reserve this name for thenatural extension of the Classic (One-Dimensional)Bin Packing Problem, i.e. the packing of a set of smallitems of dierent sizes into a minimum number ofrectangles (large objects) of identical size. The nota-tion becomes evenmore questionable for two-dimen-sional problems where both width and length arevariables, and, likewise, for three-dimensional prob-lems, where width, length and/or height are vari-ables. We conclude that in order to avoidconfusion about the contents of problem notationsand denitions it is advisable to distinguish prob-lem types with respect to xed and with respect tovariable dimensions.

Application of Dyckhos typology does notnecessarily result in homogeneous problemcategories

Gradisar et al. (2002) notice that in the case of one-dimensional cutting, where a large number of smallitems of relatively few dierent shapes has to be pro-duced from standard material in unlimited supply,two situations should be distinguished if the stan-dard material comes in dierent sizes. In the rst sit-uation, the large objects can be separated into to a

few groups of identical size, while in the second sit-uation all large objects are entirely dierent. InDyckhos coding scheme both situations wouldbe included in the problem type 1/V/D/R. Gradisaret al. (2002, p. 1208) now argue that the inclusion inthe same problem category is not very usefulbecause the two situations require dierent solutionapproaches, namely a pattern-oriented approach forcutting problems with few groups of identical largeobjects and an item-oriented approach for suchproblems with entirely dierent large objects. Wejust would like to add that the same can be observedfor problems of higher dimensions and for packingproblems, as well. Therefore, we conclude that, inorder to develop categories of homogeneous prob-lems, the properties (main types) of Dyckhos thirdcriterion, the assortment of large objects, should befurther dierentiated.

The latter aspect turns out to represent the mostsevere limitation of Dyckhos typology, as it gener-ally questions whether one of the central goals ofthe introduction of a problem typology is achieved,namely to provide a homogeneous basis for thedevelopment of models and algorithms.

4. Outline of the new typology and overview of

problem categories

Problem types can generally be dened as ele-mentary types or combined types. Types whichcan be used as a basis for the development of mod-els, algorithms, and problem generators, and for thecategorisation of literature must be relatively homo-geneous. Consequently, it is very likely that theymust be introduced as combined types, which stemfrom the subsequent or simultaneous applicationof dierent categorisation criteria. Five criteriawill be used here for the denition of combinedproblem types of C&P problems, namely dimen-sionality, kind of assignment, assortment oflarge objects, assortment of small items, andshape of the small items.

It goes without saying that each typology of C&Pproblems should oer an option to characterise agiven problem with respect to the number of prob-lem-relevant dimensions. Consequently, this crite-rion will be adopted directly from Dyckhostypology.

Dyckhos criterion kind of assignment hasproven to be useful in distinguishing between dier-ent kinds of C&P problems in the past. Therefore, it

will also be used here; however, in order to avoid the

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1113German notations Verladeproblem and Belade-problem, the authors will refer to the correspond-ing problem categories/elementary types in ageneral sense as input minimisation and out-put maximisation, respectively.

On the other hand, in order to overcome theabove-sketched limitations of Dyckhos typology,the two criteria assortment of large objects andassortment of small items will be redened and/or supplemented with new properties.

Unlike in Dyckhos typology, the two criteriakind of assignment and assortment of smallitems will not be taken for the denition of two dif-ferent elds in a classication scheme for C&P prob-lems. Instead, they will be used in combination inorder to dene basic problem types. These basicproblem types (which already represent combinedtypes in the sense of Dyckho; see Dyckho, 1990,p. 154) provide the core objects for the introductionof a new, more widely accepted nomenclature.Existing names have been adopted as far as possible,in particular, wherever there was no or only a smallprobability that their use would result in misinter-pretations of their contents.

The subsequent application of the criterionassortment of large objects will provide intermedi-ate problem types. Further characterisation withrespect to the number of problem relevant dimen-sions (dimensionality) and in the case of prob-lems of two and more dimensions with respectto the shape of small items will provide rened(combined) problem types. The name of each ofthese rened types consists of the name of theunderlying intermediate type and one or two addi-tional adjectives which indicate the respective prop-erties. Rened problem types will be used for thecategorisation of publications, here.

An instance of a specic rened problem type willexhibit all the properties which have been used forthe denition of the respective category and proba-bly additional constraints and/or characteristics. Aproblem instance which only exhibits the deningproperties, but no additional constraints or charac-teristics could be interpreted as being (an instanceof) a (rst-level) standard problem (type). First-levelnon-standard problems (problem types) are charac-terised by the properties dening the respectiveproblem category and additional constraints and/or characteristics. Of course, in particular whenconsidered in scientic research, they may representwell-known standard problems as well. In such case,

in order to distinguish them from the previouslymentioned ones, we would call them second-levelstandard problems whenever necessary. The identi-cation of standard problems (types) represents amajor goal of our typology. These problem typesprovide the basis of scientic research, in particularfor the development of (standard) models, algo-rithms, and problem generators.

Fig. 2 gives an overview of the previously intro-duced types and their relationships. In this paper,we concentrate on pure C&P problem types inthe above-dened sense (see Section 2), i.e. problemsin which the solution consists of information on theset of patterns according to which the (selected)small items have to be cut from/packed into (a sub-set of) the large objects and the corresponding objec-tive function value. The inclusion of additionalaspects which extend the view of the planning prob-lem beyond the core of cutting or packing will giverise to an extended problem type or a problem exten-sion. Apart from information on the patterns, a solu-tion to a problem of this kind includes additionalinformation on other problem-relevant aspects suchas the number of dierent patterns (as in the patternminimisation problem; cf. Vanderbeck, 2000), pro-cessing sequences (as in the pattern sequencing prob-lem; cf. Foerster and Wascher, 1998; Yanasse, 1997;Yuen, 1995; Yuen and Richardson, 1995) or lot-sizes(cf. Nonas and Thorstenson, 2000). We would like topoint out, however, that the C&P-related part ofproblems of this type can be categorised in the sameway as pure problems.

When dening basic, intermediate and renedproblem types, it will be necessary to introduce cer-tain assumptions. Some of these assumptions arerelated to general properties of the problem andrestrict the view to single-objective, single-periodand deterministic problems. Replacing the assump-tions by dierent ones leads to problem types whichwill be considered as problem (type) variants, here.Problems with multiple objectives (cf. Wascher,1990), stochastic problems (in which, e.g., the sizesof the large objects are random variables; cf. Dasand Ghosh, 2003), fuzzy problems (where thecosts of the large objects are fuzzy coecients; cf.Katagiri et al., 2004), or on-line problems (inwhich, e.g., parcels arrive one after another at apacking station and a decision has to be madeimmediately on their arrival where they should bepacked into a container; cf. Hemminki et al., 1998;Abdou and Elmasry, 1999) belong to this category.A second class of assumptions also leading

to problem variants arises from C&P-specic

kind o

sortme

ortmen

as

as

as

dimas

1114 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130C&P-RelatedProblem

Types

ProblemExtensions Pure C&P Problem Types

additional aspects aspects of C&P only

output value max,input value min

as

uniformly structured demands

Basic Problem Types

assrectangular, homogeneousmaterial

Intermediate Problem Types

1-, 2-, 3- dimensionalaspects. They will be explained in greater detaillater.

5. (Modied) criteria for the denition of problem

types

5.1. Dimensionality

We distinguish between one-, two-, and three-dimensional problems. In the literature, occasion-ally, also problems with more than three geometricdimensions are considered (e.g., Lins et al., 2002).Problems of this type (n > 3) are looked upon asvariants, here.

5.2. Kind of assignment

Again, as in Dyckho (1990), we introduce twobasic situations, of which we prefer to speak of out-

no mixes, orthogonal layout

Refined ProblemTypes

First-LevelStandard Problems

First-LevelNon-Standard Prob

no further constraints additional constraints

Second-LevelStandard Problems

shapea

Fig. 2. Overview of problem typef assignmentothers, e.g. multi-dimensional objectivefunctions

nt of small itemsothers, e.g. varying demands

t of large objectsothers, e.g. inhomogeneous material

ProblemVariants

differentassumptions

sumptions

sumptions

sumptions

ensionalitysumptions others, e.g. n-dimensionalput (value) maximisation and input (value) minimi-sation, respectively.

output (value) maximisation

In the case of output (value) maximisation, a set ofsmall items has to be assigned to a given set of largeobjects. The set of large objects is not sucient toaccommodate all the small items. All large objectsare to be used (in other words: there is no selectionproblem regarding the large objects), to which aselection (a subset) of the small items of maximalvalue has to be assigned.

input (value) minimisation

Again, a given set of small items is to be assigned toa set of large objects. Unlike before, in the case ofinput (value) minimisation the set of large objects

others, e.g. non-orthogonal layout

lems

Special Problems

of small itemsssumptions

s related to C&P problems.

tional to their size such that the objective function

strongly heterogeneous assortment

The set of small items is characterised by the factthat only very few elements are of identical shapeand size. If that occurs, the items are treated asindividual elements. Consequently, the demandof each item is equal to one. This categoryincludes Dyckhos elementary types M (manyitems of many dierent shapes) and F (fewitems (of dierent shapes); cf. Dyckho, 1990,p. 154).

For the denition of standard problems weassume that the set of small items is uniformly struc-tured, i.e. that it does not contain items with large

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1115considers length (one-dimensional problems), area(two-dimensional problems), or volume (three-dimensional problems) maximisation (output) orminimisation (input). In such cases, it might alsobe possible to translate both output (value) maxi-misation and input (value) minimisation intowaste minimisation, i.e. the minimisation of thetotal size of unused parts of the (selected) largeobjects. In the environment of cutting problemsoften the term trim-loss minimisation is used.

We would also like to point out that in order todene basic problem types only these two situa-tions will be considered here. Of course, problemsencountered in practice and/or discussed in theliterature may be characterised by the fact that aselection problem exists with respect to both largeobjects and small items. This requires an extendedobjective function which combines revenuesand costs (prot maximisation). Also situationsexist in which more than one objective functionmay have to be taken into account. Again, problemsof this type will be considered as problem variantshere.

5.3. Assortment of small items

With respect to the assortment of the small itemswe distinguish three cases, namely identical smallitems, a weakly heterogeneous assortment of smallitems, and a strongly heterogeneous assortment ofsmall items (cf. Fig. 3):

identical small items

Regarding their problem-relevant dimensions (i.e.their extension in the problem-relevant number ofdimensions length, width, and height), allis sucient to accommodate all small items. Allsmall items are to be assigned to a selection (asubset) of the large object(s) of minimal value.There is no selection problem regarding the smallitems.

Here, output (value) maximisation and input(value) minimisation are used in a general, non-specic manner. When treating specic problems,the value of objects/items has to be dened moreprecisely and may be represented by costs, revenues,or material quantities. Often, the value of theobjects/items can be assumed to be directly propor-items are of the same shape and size. In the outputmaximisation case, it can be assumed that the (sin-gle) item type has an unlimited demand. This prob-lem category is identical with Dyckhoselementary type C (congruent shapes; cf. Dyck-ho, 1990, p. 154).

weakly heterogeneous assortment

The small items can be grouped into relativelyfew classes (in relation to the total number ofitems), for which the items are identical withrespect to shape and size. By denition, smallitems of identical shape and size which requiredierent orientations are treated as dierentkinds of items. The demand of each item type isrelatively large, and may or may not be limitedby an upper bound. This category correspondsto Dyckhos elementary type R (many itemsof relatively few dierent (non-congruent)shapes; cf. Dyckho, 1990, p. 154).

identical small items

weakly heterogeneous assortment

strongly heterogeneous assortment

assortment of small items

Fig. 3. Cases to be distinguished with regard to the assortment ofsmall items.

demands and others with small demands. A prob-lem with strongly varying demands (cf. Riehmeet al., 1996) will be considered as a variant, here.

5.4. Assortment of large objects

With respect to the assortment of the largeobjects we introduce the following cases (cf. Fig. 4):

one large object

In this case the set of large objects consists of asingle element. The extension of the large objectmay be xed in all problem-relevant dimensions(all dimensions xed), or its extension may bevariable in one or more dimensions (one or more

of small items either consists of regular or irregular

Problems of the output maximisation type have

1116 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130variable dimensions). The rst category is identi-cal with Dyckhos type O, while the second cat-egory represents an extension of Dyckhos set ofelementary types (cf. Dyckho, 1990, p. 154).

several large objects

With respect to the kind of problems which aredescribed in the literature, in the case of severallarge objects it does not appear necessary to dis-tinguish between xed and variable dimensions;only xed dimensions will be considered. In anal-ogy to the categories which have been introducedfor the assortment of the small items, we distin-guish between identical large objects, a weaklyand a strongly heterogeneous assortment of largeobjects. By doing so, we again extend Dyckhostypology, who only identies large objects withidentical (type I) and dierent shapes (type D).

one large object

all dimensions fixed

one or more variable dimensions

several large objects (all dimensions fixed)

identical large objects

weakly heterogeneous assortment

strongly heterogeneous assortment

assortment of large objects

Fig. 4. Cases to be distinguished with regard to the assortment of

large objects.in common that the large objects are only suppliedin limited quantities which do not allow for accom-modating all small items. As the value of the accom-modated items has to be maximised, all large objectswill be used. In other words, generally there is aselection problem regarding the small items, butnone regarding the large objects.

According to Fig. 5, we distinguish the followingelements. Problems which allow for non-orthogonallayouts and/or mixes of regular and irregularsmall items will be looked upon as problem variantsagain.

6. Basic, intermediate and rened problem types

6.1. Basic problem types

Basic types of C&P problems are developed bycombination of the two criteria type of assign-ment and assortment of small items. Fig. 5depicts the relevant combinations and the corre-sponding basic problem types.

In the following sections these problem types willbe characterised in greater detail.

6.1.1. Output maximisation typesIn the case of two- and three-dimensional prob-lems, for the denition of rened problem typeswe further distinguish between regular small items(rectangles, circles, boxes, cylinders, balls, etc.)and irregular (also called: non-regular) ones. In thetwo-dimensional case, the former are sometimes fur-ther distinguished into rectangular items, circularitems, and others.

In accordance with what is usually considered inthe literature, we assume that rectangular items areto be laid out orthogonally. Furthermore, the setFor the denition of basic problem types weassume that in the two- and three-dimensionalcase all large objects are of rectangular shape(rectangles, cuboids) and consist of homogeneousmaterial. Non-rectangular large objects (e.g., circu-lar objects such as discs) and/or non-homogeneouslarge objects (e.g., stock material including defects)give rise to problem variants, again.

5.5. Shape of small items(basic, output maximisation) problem types:

glyneou

acklem

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1117weaklyheterogeneous

stronheteroge

outputmaximisation

PlacementProblem

KnapsProb

C&Pproblemtypes

assortmentofsmall items

kind ofassignment

identical

IdenticalItem Packing

Problem

all dimensions fixed Identical Item Packing Problem

This problem category consists of the assignmentof the largest possible number of identical smallitems to a given, limited set of large objects. Wenote that, due to the fact that all the small itemsare identical, there is in fact no real selectionproblem regarding the small items, and, further-more, neither a grouping nor an allocation prob-lem occurs. In other words, the general structureof C&P problems (cf. Section 2) is reduced to alayout problem regarding the arrangement ofthe (identical) small items on each of the largeobjects with respect to the geometric condition.

Placement Problem

In the literature, problems of this category areknown under many dierent names. In order toavoid additional confusion, here we have intro-duced a somewhat more neutral notion. Here,the term Placement Problem denes a problemcategory in which a weakly heterogeneous assort-ment of small items has to be assigned to a given,

Fig. 5. Basic prosweakly

heterogeneousstrongly

heterogeneousarbitrary

Cutting StockProblem

Bin PackingProblem

inputminimisation

variable dimension(s) all dimensions fixed

OpenDimensionProblemC&P problemslimited set of large objects. The value or the totalsize (as an auxiliary objective) of the accommo-dated small objects has to be maximised, or,alternatively, the corresponding waste has to beminimised.

Knapsack Problem

According to our interpretation, the KnapsackProblem represents a problem category which ischaracterised by a strongly heterogeneous assort-ment of small items which have to be allocated toa given set of large objects. Again, the availabilityof the large objects is limited such that not allsmall items can be accommodated. The value ofthe accommodated items is to be maximised.

6.1.2. Input minimisation types

Problems of the input minimisation type are char-acterised by the fact that the supply of the largeobjects is large enough to accommodate all smallitems. Their demands have to be satised completely,so that no selection problem regarding the smallitems exists. The value of the large objects necessaryto accommodate all small items has to be minimised.

blem types.

objects (or another corresponding auxiliaryobjective) has to be minimised.

6.2. Intermediate problem types

In order to dene more homogeneous problemtypes the above-developed basic problem types arestructured further into intermediate problem types.This is achieved by taking into account the assort-ment of the large objects as an additional dierenti-ating criterion. Figs. 6 and 7 depict the intermediateproblem types related to output maximisation, Figs.8 and 9 the intermediate types related to input min-imisation. Figs. 69 also present our suggestions fornaming these types.

We note at this stage that due to its simpleproblem structure it is not necessary to further dif-ferentiate the Identical Item Packing Problem. Inorder to solve a problem of this type, it can be splitinto a set of independent sub-problems where eachsub-problem is related to a particular large object(or a particular type of large objects, if the largeobjects are at least partially identical) to which the

one large object Single Large ObjectPlacement Problem

Placement Problem

1118 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 Open Dimension Problem

The Open Dimension Problem denes a problemcategory in which the set of small items has to beaccommodated completely by one large object orseveral large objects. The large objects are given,but their extension in at least one dimension canbe considered as a variable. In other words, thisproblem involves a decision on xing the exten-sion(s) in the variable dimension(s) of the largeobjects. Only the part(s) of the large object(s) nec-essary to accommodate the items completely rep-resents input within the meaning of the generalstructure of C&P problems (cf. Section 2). Thevalue of the input (or a corresponding auxiliarymeasure like length, size, or volume) is to be mini-mised. For the denition of basic problem typeswe restrict ourselves to large objects which before and after having xed their extension(s) inthe variable dimension(s) are rectangles (two-dimensional problems) or cuboids (three-dimen-sional problems). By doing so, in particular thoseproblems are excluded from our analysis in whichthe small items have to be enclosed in non-rectan-gular large objects of minimal size (e.g., when cir-cles have to be packed into another circle ofminimal radius; cf. Birgin et al., 2005, pp. 27.),or in which the density of the packing has to bemaximised (e.g., as in the case of two-dimensionallattice packing; cf. Stoyan and Patsuk, 2000).

Cutting stock problem

Problems of this category require that a weaklyheterogeneous assortment of small items is com-pletely allocated to a selection of large objects ofminimal value, number, or total size. The exten-sion of the large objects is xed in all dimensions.We point out that we do not make any assump-tions on the assortment of the large objects. Itmay consist of identical objects, but it could alsobe aweakly or strongly heterogeneous assortment.

Bin packing problem

In contrast to the previously described problemcategory, this one is characterised by a stronglyheterogeneous assortment of small items. Again,the items have to be assigned to a set of identicallarge objects, a weakly heterogeneous or stronglyheterogeneous assortment of large objects. The

value, number, or total size of the necessary largeseveral large objects

identical large objects

(Multiple) IdenticalLarge ObjectPlacement Problem

heterogeneous assortment (Multiple)HeterogeneousLarge ObjectPlacement Problem

Fig. 6. Intermediate types of the placement problem.

one large object SingleKnapsack Problem

several large objects

identical large objects (Multiple) IdenticalKnapsack Problem

heterogeneous assortment (Multiple) HeterogeneousKnapsack Problem

Knapsack Problem

Fig. 7. Intermediate types of the knapsack problem.

largest possible number of small items has to beassigned. Consequently, in this typology the nameIdentical Item Packing Problem will also be reservedfor a (combined) problem type in which the largestpossible number of identical small items has to beassigned to a single given large object. Furthermore,with respect to the fact that the existing literature onthe Open Dimension Problem concentrates on avery limited number of standard problems, we alsorefrained from structuring this basic problem typein greater detail at this stage.

Figs. 10 and 11 summarise the system of interme-diate problem types which has been introducedhere. They can be interpreted as the landscapeof C&P problems. Also, for each intermediate prob-lem type, an abbreviation derived from the corre-sponding name is given which allows for a uniqueidentication of each type.

6.3. Rened problem types

In a nal step, rened problem types are obtainedby application of the criterion dimensionality and

identical large objects Single Stock SizeCutting StockProblem

weakly heterogeneous Multiple Stock Sizeassortment of large objects Cutting Stock

Problem

strongly heterogeneous Residualassortment of large objects Cutting Stock

Problem

Cutting Stock Problem

Fig. 8. Intermediate types of the cutting stock problem.

identical large objects Single Bin SizeBin PackingProblem

weakly heterogeneous Multiple Bin Sizeassortment of large objects Bin Packing

Problem

strongly heterogeneous Residualassortment of large objects Bin Packing

Problem

Bin Packing Problem

Fig. 9. Intermediate types of the bin packing problem.

assortment of the small items

characteristicsof the large objects

identical

onelarge object

Identical ItemPacking Problem

IIPP

identicalalldimensions

fixed

heterogeneous

Fig. 10. Landscape of intermediate pro

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1119 for two- and three-dimensional problems of thecriterion shape of the small items. The resultingsubcategories are characterised by adjectives whichare added to (the names of) the intermediate prob-lem types (IPT) according to the following system:

weaklyheterogeneous

stronglyheterogeneous

SingleLarge ObjectPlacementProblem

SLOPP

SingleKnapsack Problem

SKP

Multiple IdenticalLarge Object PlacementProblem

MILOPP

Multiple IdenticalKnapsack Problem

MIKPMultiple

HeterogeneousLarge Object PlacementProblem

MHLOPP

MultipleHeterogeneous

Knapsack Problem

MHKPblem types: output maximisation.

ly gene

Stocktock

SCS

Stoctock

SCS

siduatockP

CSP

O

1120 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 assortment of small items

characteristicsof large objects

weakhetero

identicalSingle

Cutting S

SS

weaklyheterogeneous

MultipleCutting S

MS

alldimensions

fixed

stronglyheterogeneous

ReCutting S

R

one large object

variable dimension(s) {1,2,3}-dimensional {rectangular, circular, . . . ,irregular} {IPT}.

7. Specic problem categories

In order to make the consequences of the sug-gested typology clearer, we will now explain the dif-ferent categories in further detail by giving examplesof specic cutting and packing problems (describedin the literature) which belong to the problem typesintroduced above. In particular, well-known stan-dard problems will be pointed out as representativesof the respective categories.

7.1. Identical Item Packing Problem

A well-known (regular/rectangular) representa-tive of this problem type is the (Classic) Manufac-turers Pallet Loading (Packing) Problem (MPLP;Dowsland, 1987; Morabito and Morales, 1998), inwhich a single pallet has to be loaded with a maxi-mal number of identical boxes (the problem itself without this particular name has been describedeven earlier in the literature, see, e.g., Steudel, 1979;

Fig. 11. Landscape of intermediate prous

stronglyheterogeneous

SizeProblem

P

Single Bin Size Bin Packing Problem

SBSBPP

k SizeProblem

P

Multiple Bin SizeBin Packing Problem

MBSBPP

lroblem

ResidualBin Packing Problem

RBPP

pen Dimension Problem

ODPSmith and de Cani, 1980). It is usually assumed thatthe boxes are loaded in layers, in which all boxeshave the same vertical orientation. By means of thisassumption, the problem is actually reduced to atwo-dimensional one (cf. Bischo and Ratcli,1995, p. 1322; also see Dyckhos (1990, p. 155)characterisation of the problem as being of type 2/B/O/C), namely to the problem of assigning a max-imal number of small identical rectangles (represent-ing the bottom surfaces of the boxes) to a givenlarge rectangle (representing the pallet). In otherwords: The MPLP is a two-dimensional, rectangu-lar Identical Item Packing Problem (IIPP). The Cyl-inder Packing Problem (Correia et al., 2000, 2001;Birgin et al., 2005) possesses an almost identicalproblem structure, in which instead of identicalrectangles a maximal number of non-overlappingcircles of the same size has to be assigned to a givenrectangle (two-dimensional, circular IIPP; also seeIsermann, 1991). Among others, it can be found inthe production of cans where the circles representthe lids and bottoms of (cylindrical) cans to be cutfrom tin plates, or in logistics management, wherecylindrical cans have to be packed on a pallet.

oblem types: input minimisation.

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1121The Single-Box-Type Container Packing Problem(as discussed in George, 1992) is an example of athree-dimensional special case of the Rectangular(i.e. regular) IIPP. It requires the loading of a singlecontainer (large object) with a maximum number ofidentical boxes (small items).

7.2. Single Large Object Placement Problem

The Bounded Knapsack Problem and theUnbounded Knapsack Problem (as described in Mar-tello and Toth, 1990, pp. 8291, and pp. 91103,respectively) are one-dimensional representativesof the SLOPP type. Both require that a (single)knapsack (large object) of given limited weightcapacity has to be packed with a selection (subset)from a given set of small items of given weightsand values, such that the value of the packed itemsis maximised. The assortment of the small items isweakly heterogeneous, only relatively few dierentitem types can be identied. The number of timesan item of a particular type can be packed maybe limited (bounded problem), or unlimited(unbounded problem).

Haims and Freeman (1970) introduce a two-dimensional problem of the SLOPP type which theyname Template-Layout Problem. According to theirgeneral denition, a weakly heterogeneous set ofregular or non-regular small items (forms) hasto be cut from a single large rectangle such thatthe value of the cut items is maximised. In the sub-sequent discussion, the authors concentrate on rect-angular small items (i.e. the discussed problem is ofthe two-dimensional, rectangular SLOPP type). InChristodes and Whitlock, 1977; Beasley, 1985a;Christodes and Hadjiconstantinou, 1995, e.g., theauthors also restrict themselves to rectangular smallitems. They call it the Two-Dimensional CuttingProblem (more precisely, we would rather call itthe two-dimensional, rectangular SLOPP). Thenumber of times an item of a particular type canbe cut from the large rectangle may be constrainedexplicitly by an upper bound (as in Christodesand Whitlock, 1977; Wang, 1983; Beasley, 1985b;Christodes and Hadjiconstantinou, 1995; Beasley,2004). In Scheithauer and Sommerwei (1998) theunbounded version of this problem (also see Herz,1972; Beasley, 1985c) is discussed under the nameRectangle Packing Problem. In the (Constrained)Circular Cutting Problem (Hi and MHallah,2004; according to the suggested typology, the prob-

lem would be characterised as a two-dimensional,circular SLOPP), a rectangular plate (large object)of given size has to be cut down into circular items,which are of m dierent types (radii). The number ofsmall items which may be cut is bounded, and theobjective is to minimise the unused space of theplate.

What has been called the Single Container Load-ing (Packing) Problem in the literature (Georgeand Robinson, 1980; Bortfeldt et al., 2003) is anexample of the three-dimensional, rectangularSLOPP. It requires loading a fairly large, weaklyheterogeneous consignment of boxes into a givencontainer such that the volume or value of thepacked boxes is maximised, or, equivalently, theunused space of the container or the value ofthe unpacked boxes is minimised (also see Ratcliand Bischo, 1998; Bortfeldt and Gehring, 1998).A set of weakly heterogeneous cargo has to bepacked on a pallet in the Distributors (Single)Pallet Loading Problem (Hodgson, 1982; Bischoet al., 1995). Unlike the Manufacturers PalletLoading Problem, this is a truly three-dimensionalrectangular problem, as due to the dierent boxsizes it may not be sucient to concentrate onlayered packing if the space available above thepallet is to be used in the best possible way. Inreal-world Single Container and Pallet LoadingProblems, the number of boxes which are to bepacked of a particular box type will be limited byan upper bound. Hi (2004), on the other hand,considers the unconstrained case.

7.3. Multiple Identical Large Object Placement

Problem

Straightforward extensions of the single largeobject placement problem could take into accountmultiple identical large objects. This could give rise,e.g., to the (Bounded) Multiple Knapsack Problem,the Multiple Stock Sheet Cutting Problem, etc.Problems of this type (MILOPP), however, havenot yet been discussed in the literature, at leastnot to the knowledge of the authors of this paper.

7.4. Multiple Heterogeneous Large Object

Placement Problem

Another extension of the Single Large ObjectPlacement Problem which takes into account multi-ple, non-identical large objects also appears tohaving been treated very rarely in the literature. A

one-dimensional cutting problem is described in

Schepers (1997) nally look at the n-Dimensional

dimensional (regular) special case of this problem

1122 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130Gradisar et al. (1999, p. 559, case 2), in which aweakly heterogeneous assortment of order lengthshas to be cut from a set of input lengths which areall dierent (one-dimensional MHLOPP). The sup-ply of input lengths is not sucient to satisfy allorders. Therefore, the objective is to minimise thetotal size of the uncut order lengths.

Eley (2003) considers three-dimensional con-tainer loading problems, in which a weakly hetero-geneous consignment of boxes has to be packedinto containers available in dierent sizes. In oneof the discussed situations, the available containerspace is not sucient to accommodate all boxes.Therefore, a selection of boxes has to be determinedthat maximises the volume utilisation, or, alterna-tively, the value of the packed boxes (three-dimen-sional, rectangular MHLOPP). An exact algorithmfor the unconstrained version of this problem isintroduced in Hi (2004).

7.5. Single Knapsack Problem

The Classic (One-Dimensional) Knapsack Prob-lem, also called 0-1-Knapsack Problem (Martelloet al., 2000; Martello and Toth, 1990, p. 13) whichrequires packing a given set of (dierent) items ofgiven weights and values into a (single) knapsackof given limited weight capacity such that the valueof the packed items is maximised (cf. Martello andToth, 1990, pp. 3, 1377) obviously is a special caseof this problem type (i.e. the one-dimensional SKP).The Subset-Sum Problem (cf. Martello and Toth,1990, pp. 105.) is a 0-1-knapsack problem in whichthe weight of an item is identical with its value. Astraightforward extension of this problem, namelythe Multiconstraint (01) Knapsack Problem(Gavish and Pirkul, 1985; Drexl, 1988; Thiel andVoss, 1994, the problem is also called m-ConstraintKnapsack Problem, cf. Schilling, 1990), also belongsto this category. Apart from the capacity constraint,m 1 additional constraints have to be satised bythe packed items.

Extensions of the Classic (One-Dimensional)Knapsack Problem into two and more geometricdimensions give rise to the Two-Dimensional (SingleOrthogonal) Knapsack Problem (Caprara and Mon-aci, 2004; also see e.g., Healy and Moll, 1996), inwhich a set of small, distinct rectangles has to becut from a single large rectangle (two-dimensional,rectangular SKP), and the Three-Dimensional (Sin-gle Orthogonal) Knapsack Problem (also called

Knapsack Container Loading Problem; cf. Pisinger,category, in which a given number of identical con-tainers has to be lled with a (strongly heteroge-neous) set of items of given weights and values.The total value of the packed items has to bemaximised (three-dimensional, rectangular SKP).According to the authors knowledge, this is theonly representative of this problem type discussedin the literature so far.

7.7. Multiple Heterogeneous Knapsack Problem

Martello and Toth (1990, pp. 157187), also seePisinger (1999) consider the 0-1 Multiple KnapsackProblem, which is a one-dimensional representativeof this problem type (i.e. a one-dimensionalMHKP). A strongly heterogeneous set of smallitems, each of which is characterised by a specicweight and prot, has to be packed into a set of(Single Orthogonal) Knapsack Problem, whichextends the Classic Knapsack Problem into n geo-metric dimensions.

George et al. (1995) describe a two-dimensional,circular SKP, in which (the bottom of) a container(rectangle) has to be lled with a set of distinct pipes(circles) such that the value of the selected pipes ismaximised.

7.6. Multiple Identical Knapsack Problem

In the literature, a specic one-dimensional caseof the Multiple Identical Knapsack Problem isknown as the Maximum Cardinality Bin PackingProblem, in which a xed number of large objectswith a given, identical capacity and a (strongly het-erogeneous) set of small, indivisible items of givenweights are given. The objective is to maximise thenumber of packed items (Labbe et al., 2003).

The Multiple Container Packing Problem (asdescribed in Raidl and Kodydek, 1998) is a three-2002), in which rectangular-shaped boxes have tobe packed into a container (three-dimensional, rect-angular SKP; also see Bischo and Marriott, 1990;Scheithauer, 1999; Bortfeldt and Gehring, 2001).In both cases usually the value of the cut/packedsmall items is to be maximised. If their value canbe assumed to be proportional to their size/volume,then equivalently the unused space of the large rect-angle or the container can be minimised. Fekete andknapsacks of distinct capacities. For each knapsack,

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1123the packed items must not exceed the availablecapacity, and the total prot of the small itemswhich are packed has to be maximised.

7.8. Open Dimension Problem

In the literature, Open Dimension Problems areusually discussed for a single large object. Obvi-ously, problems of this kind are only possible intwo and more dimensions.

The (Two-Dimensional) Strip Packing Problem isan Open Dimension Problem in which a set of two-dimensional small items has to be laid out on a rect-angular large object; the width of the large object isxed, its length is variable and has to be minimised.In case the small items are rectangles (cf. Kroger,1995; Jacobs, 1996; Hopper and Turton, 2001a;Martello et al., 2003) one may also refer to thisproblem as the Rectangular Strip Packing Problem,or even as the Orthogonal Rectangular Strip PackingProblem, if the small rectangles have to be laid outon the large object orthogonally. If the rectangleshave to be packed in levels, this problem is alsocalled Level Packing Problem (cf. Lodi et al., 2004).

In case the small items are non-regularly shapedobjects (Oliveira and Ferreira, 1993; Bennell andDowsland, 2001), like in the shoe-manufacturingindustry, where they may represent pieces of shoesto be cut from a roll of leather, the problem (i.e.in case of the two-dimensional, irregular ODP,according to the notation introduced here) is alsoreferred to as the Irregular Strip Packing Problem(Hopper and Turton, 2001b, p. 257) or the NestingProblem (cf. Oliveira and Ferreira, 1993; Oliveiraet al., 2000; Carravilla et al., 2003). With respectto applications in specic areas, the same problemmay also be known under dierent names (e.g., asthe Marker-Making Problem in the apparel indus-try; cf. Li and Milenkovic, 1995).

A three-dimensional, rectangular Open Dimen-sion Problem with a single variable dimension(length) occurs in distribution planning, when agiven set of cargo (small items) has to be tted intoa container in such a way that the least space interms of container length is used (Scheithauer,1991; Miyazawa and Wakabayashi, 1997).

In the Minimal Enclosure Problem (Milenkovicand Daniels, 1999), also called Rectangular PackingProblem (Hi and Oua, 1998), a set of two-dimen-sional items has to be laid out such that it can beincluded in a rectangular (large) object of minimal

area. In this case, the extension of the large objectsin both dimensions (width and length) is variable.The small items may be of rectangular (cf. Hi andOua, 1998), circular (Stoyan and Yaskov, 1998),or irregular shape (Milenkovic and Daniels, 1999).

At least to our knowledge, Open DimensionProblems with more than one large object havenot been discussed frequently in the area of C&P.Benati (1997) considers a two-dimensional problemwith several large objects of dierent widths whichall have an innite length. However, we would liketo point out that the well-studied (Classic) Multi-processor Scheduling Problem could be interpretedas a one-dimensional problem of this kind. In thisproblem, a given set of indivisible jobs (small items)with given processing times (length) has to be allo-cated to a given number of identical processors(large objects) such that the maximal completiontime (also called schedule length) is minimised (cf.Heuer, 2004; Brucker, 2004, pp. 107154; Ba _zewiczet al., 2001, pp. 137203). The completion time of aprocessor is the time necessary to process all thejobs which have been assigned to this processor.In other words, an identical, minimal time-capacity(extension in the variable dimension) has to beassigned to each of the processors that is largeenough that all the jobs can be completed. Due tothe fact that the Multiprocessor Scheduling Prob-lem and other related problems are hardly ever dis-cussed with respect to C&P problems, we excludeOpen Dimension Problems with more than onelarge object from the following considerations.

7.9. Single Stock-Size Cutting Stock Problem

Problems of this type include the Classic One-Dimensional Cutting Stock Problem (Gilmore andGomory, 1963; Wascher and Gau, 1996), in whichstandard (or: stock) material of a specic, singlelength has to be cut down into a weakly heteroge-neous set of order lengths (small items). In the Clas-sic Two-Dimensional Cutting Stock Problem

(Gilmore and Gomory, 1965), a weakly heteroge-neous set of order rectangles has to be cut fromstock plates of a specic, single size (length andwidth). In both problems, the number or the valueof the necessary large objects (stock lengths or stockplates) has to be minimised.

Examples for a three-dimensional (rectangular)problem of this type include the Multi-Pallet Load-ing Problem (as discussed in Terno et al., 2000)and the Multi-Container Loading Problem (Scheit-

hauer, 1999), in which a weakly heterogeneous

1124 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130assortment of cargo (i.e. a set of boxes) is to bepacked on a minimum number of pallets or into aminimum number of containers (also see Bortfeldt,2000).

7.10. Multiple Stock-Size Cutting Stock Problem

Problems of this kind include the natural exten-sions of the one-dimensional and two-dimensionalcutting stock problems to more than one stock size(see Gilmore and Gomory, 1961; Rao, 1976; Dyck-ho, 1981; Scheithauer, 1991; Belov and Scheit-hauer, 2002 for the one-dimensional and Riehmeet al., 1996; Morabito and Arenales, 1996 for thetwo-dimensional, rectangular case). The One-Dimensional Multiple Stock-Size Cutting StockProblem has also been considered under the namePaper Trim Problem in the literature (Golden,1976, p. 265f.). For a two-dimensional application(i.e. a two-dimensional, rectangular MSSCSP)from furniture manufacturing (see Carnieri et al.,1994).

One of the container-loading problems discussedin Eley 2003 represents a three-dimensional (rectan-gular) case of this problem type. Both containersand boxes can be grouped into classes. Associatedwith each container type are specic costs, the totalcosts of the containers necessary to accommodateall boxes are to be minimised.

7.11. Residual Cutting Stock Problem

For a Cutting Stock Problem with a strongly het-erogeneous assortment of large objects we have cho-sen the name Residual Cutting Stock Problemhere, because in practice this case comes aboutwhenever large objects are to be used which repre-sent unused parts of input material (left-overs)from previous C&P processes. Under the name ofHybrid One-Dimensional Cutting Stock Problema one-dimensional case of this problem type hasbeen introduced in the literature (cf. Gradisaret al., 2002, p. 1212; Gradisar et al., 1999). Theauthors introduce a one-dimensional cutting prob-lem, in which a weakly heterogeneous assortmentof order lengths has to be cut from a set of inputlengths which are all dierent. The supply of inputlengths is sucient to satisfy the demands, and theobjective is to minimise the trim loss of the inputlengths which are to be used (also see Gradisaret al., 1999, p. 559, case 1). They argue that tradi-

tional, pure item-oriented or pure pattern-orientedsolution approaches are not appropriate. Instead,they suggest a new solution approach which is acombination of both. The two-dimensional case isconsidered in Vanderbeck, 2001. Extensions ofthis problem type into three or even more dimen-sions have not been discussed in the literature sofar.

7.12. Single Bin-Size Bin Packing Problem

The Classic (One-Dimensional) Bin PackingProblem is a representative of this problem type. Itrequires packing a given set of distinct small itemsof given weights to a minimal number of largeobjects (bins) of identical size (capacity) such thatfor each bin the total capacity of the small itemsdoes not exceed its capacity (cf. Martello and Toth,1990; Scholl et al., 1997; Schwerin and Wascher,1997). We also note that this problem type has alsobeen named Vehicle Loading Problem (cf. Golden,1976, p. 266) and Binary Cutting Stock Problem(cf. Vance et al., 1994) in the literature. The k-ItemBin Packing Problem (cf. Babel et al., 2004) addi-tionally requires the assignment of at most k itemsto each bin.

What is called the Two-Dimensional (Orthogo-nal) Bin Packing Problem (Lodi et al., 1999,2002b, p. 379; Lodi et al., 2002a, p. 242; Martelloand Vigo, 1998 also refer to this problem type asthe Two-Dimensional Finite Bin Packing Problemin order to distinguish it from the Two-DimensionalStrip Packing Problem, in which the large object hasan innite extension in one dimension) consists ofassigning a set of distinct rectangles orthogonallyto a minimum number of rectangular bins. Accord-ing to the suggested typology this is a two-dimen-sional, rectangular SBSBPP type. George et al.(1995, p. 693) mention the Cylindrical Bin PackingProblem. This is a two-dimensional circularSBSBPP, in which the large objects are rectanglesand the small items are circles. Real problems of thiskind arise in logistics when (a minimum number of)containers are to be loaded with pipes.

In the Three-Dimensional (Orthogonal) BinPacking Problem the items are assumed to be rect-angular boxes which are to be tted orthogonallyinto a minimal number of rectangular containersof identical size (cf. Lodi et al., 2002c). The CubePacking Problem is a special case of the three-dimensional rectangular Bin Packing Problem, inwhich all boxes and bins are cubes (Miyazawa and

Wakabayashi, 2003).

C&P problems (Dyckho, 1990, p. 148) have onlybeen considered as far as these problems (like thoseof the knapsack type) have been introduced in thesuggested typology, i.e. we refrained from includingother papers from this category not addressing C&Pdirectly, like papers on ow-line balancing (Talbotet al., 1986; Scholl, 1999), multi-processor schedul-ing (Brucker, 2004, pp. 107154; Ba _zewicz et al.,2001, pp. 137203), or capital budgeting (Lorie

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 11257.13. Multiple Bin-Size Bin Packing Problem

The (One-Dimensional) Variable-Sized Bin Pack-ing Problem (Chu and La, 2001; Kos and Duhovnik,2002) is an extension of the Classic One-Dimen-sional Bin Packing Problem in which several bintypes are introduced (i.e. it is a one-dimensionalMBSBPP). Each bin type is in unlimited supplyand characterised by specic costs and size. All thesmall items have to be assigned to bins, and the totalcosts of the used bins have to be minimised (cf.Kang and Park, 2003). A two-dimensional, rectan-gular case is considered in Tarasova et al., 1997,in which the large objects are in limited supplyand where the total area of the material necessaryto cut all small items is to be minimised.

7.14. Residual Bin Packing Problem

In analogy to the cutting stock problem, for thisproblem type which is characterised by a set ofstrongly heterogeneous large objects we have chosenthe name Residual Bin Packing Problem.

Chen et al. (1995) consider a three-dimensionalcontainer loading problem, in which a weakly heter-ogeneous consignment of boxes has to be packedinto containers available in dierent sizes. In oneof the discussed situations, the available containerspace is not sucient to accommodate all boxes.Therefore, a selection of boxes has to be determinedthat maximises the volume utilisation, or, alterna-tively, the value of the packed boxes (three-dimen-sional, rectangular RBPP).

8. Categorisation of recent literature (19952004)

In order to demonstrate the practicability andusefulness of the suggested typology, the recentC&P literature has been reviewed and categorisedaccording to the scheme introduced above. We con-centrated on papers which are publicly availableand have been published in English in internationaljournals, edited volumes, or conference proceedingsduring the decade between 1995 and 2004. Mono-graphs and working papers have not been consid-ered in our investigation.

In order to exclude publications from the periph-ery of C&P from our analysis, which are only ofmarginal interest to researchers and practitionersin the eld, we restricted ourselves to papers relatedto C&P problems in a narrow sense (Dyckho,

1990, p. 148) in the rst place. Papers on abstractand Savage, 1955).Furthermore, only papers dealing with C&P

problems in the sense of rened problem types havebeen included in our analysis. Articles dealing withproblem extensions and problem variants were nottaken into account for. We note again that we haveexcluded papers of this kind only in order to keepthe number of papers to be considered to a manage-able size and to nd a denition of a paper clusterthat is of interest to researchers and practitionersin the eld of C&P. Focussing our investigation inthis particular way by no means limits the usefulnessand the value of the suggested typology. In fact, alsothe problems discussed in papers which have beenexcluded here can be categorised according to ourtypology.

Finally we remark that only such papers havebeen categorised which strictly satisfy the above-given denition of C&P problems and subsequentspecications. That means that papers, e.g. dealingwith divisable small items and/or large objects(Kang and Park, 2003), or allowing the overpack-ing of bins (Coman and Lueker, 2001) do notappear in our analysis.

As far as for December 2005, 413 papers havebeen identied containing material relevant in theabove-described sense. These papers are listed athttp://www.uni-magdeburg.de/mansci/rm/cp_typo-logy, together with the corresponding problemtypes to which they have been assigned. Table 1shows what problem types and number of dimen-sions are dealt with in these papers. The total num-bers given in this and the following tables is larger

Table 1Number of problem-relevant dimensions and assignment types ofproblems dealt with in the literature

Kind of assignment 1D 2Dregular

2Dirregular

3D Total

Input minimisation 108 79 52 24 263Output maximisation 64 71 12 35 182Total 172 150 64 59 445

than 413 because in some papers more than oneproblem type is addressed. Consequently, thosepapers had to be counted more than once.

Research on input minimisation problems (dealtwith in 263 papers, 59%) clearly dominates researchon output maximisation problems (182 papers,41%). One-dimensional and two-dimensional prob-

lems (172 papers, 39%, and 214 papers, 48%) areconsidered signicantly more often than three-dimensional ones (59 papers, 13%).

Figs. 12 and 13 give a more detailed analysis ofhow the papers are distributed over the dierentproblem categories. Among research on output max-imisation problems, papers on problems with a sin-

assortment of the small items

characteristicsof the large objects

identical weaklyheterogeneousstrongly

heterogeneous

onelarge object

IIPP23

SLOPP56

SKP86

identical MILOPP1MIKP

6

alldimensions

fixed

heterogeneous MHLOPP4

MHKP6

Fig. 12. Distribution of publications: output maximisation.

assortment of small items

characteristicsof large objects

weakly heterogeneous

stronglyheterogeneous

identical SSSCSP38

SBSBPP89

weaklyheterogeneous

MSSCSP18

MBSBPP4

alldimensions

fixed

CSP10

1126 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130stronglyheterogeneous

R

one large object

variable dimension(s) Fig. 13. Distribution of publicaRBPP2

ODP102tions: input minimisation.

gle large object are prevailing (165 papers, 37% of allpapers and 91% of those belonging to the output-maximisation type). With respect to input minimisa-tion, most of the publications deal with problems inwhich the assortment of large objects either consistsof identical large objects of a single given size (127papers, 29% of all papers, and 48% of those belong-ing to the input-minimisation type), or, alternatively,of a single large object with one variable dimension(102 papers, 23 and 39%).

As becomes further evident, published researchconcentrates on ve problem types, namely on the

been extensively studied in the 1980s is only repre-

access to the literature of each specic problem type.

G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130 1127ODP (102 papers, 23%), SBSBPP (89 papers,20%), SKP (86 papers, 19%), SLOPP (56 papers,13%) and the SSSCSP (38 papers, 9%). Papers onthese ve problem types account for 371 out of445 publications (83%). In Table 2 the numbers ofpapers which consider these problem types are fur-ther dierentiated with respect to the number ofproblem-relevant dimensions.

Table 2 reveals that research on C&P still israther traditionally oriented. It stresses areas whichinclude clearly-dened (classic) standard prob-lems, well-studied for three decades or an evenlonger period of time, such as the one-dimensionalSBSBPP (including the Classic Bin Packing Prob-lem), the two-dimensional ODP (including the Reg-ular and the Irregular Strip-Packing Problem), theone-dimensional SSSCSP (including the ClassicCutting Stock Problem), the one-dimensional SKP(including the Classic Knapsack Problem), and thetwo-dimensional SLOPP (including the Distribu-tors Pallet Loading Problem). As far as researchdeparts from these traditional areas, it is devotedto straightforward extensions of these standardproblems into a higher number of dimensions,e.g., to the two-dimensional Bin Packing Problem,the two-dimensional Knapsack Problem, etc. Otherkinds of problem extensions (e.g., considering dier-

Table 2Number of papers on selected problem types, dierentiatedaccording to the number of problem-relevant dimensions

Problem types 1D 2Dregular

2Dirregular

3D Total

ODP 46 49 7 102SBSBPP 61 17 2 9 89SKP 49 18 7 12 86SLOPP 4 32 1 19 56SSSCSP 29 2 1 6 38Other 29 35 4 6 74Total 172 150 64 59 445Fig. 2 also outlines the area of future work. Inorder to structure the rened problem types further,it will be necessary to compile, analyse and groupthe respective additional constraints. Furthermore,problem extensions and problem variants and thecorresponding standard problems have to be inves-tigated in greater detail.

Acknowledgements

The basic ideas of this typology have been pre-sented twice at meetings of the EURO WorkingGroup ESICUP, namely at Lutherstadt Witten-berg in 2004 and at Southampton in 2005. Theauthors are thankful to the participants for theirvarious critical comments which helped to elaboratethe basic structure of the typology more clearly.sented in 23 papers. This seems to indicate that thecentral standard problem of this type, the classicManufacturers Pallet Loading Problem, has beensolved satisfactorily and that research has shifted tomore complex problem situations recently, takingcare, in particular, of a (weakly) heterogeneousassortment of small items (boxes to be packed).

9. Outlook

The suggested typology should be sucient foran initial, brief orientation in a particular area ofC&P. To practitioners and researchers confrontedwith particular C&P problems Fig. 2 points outthe relevant problem parameters, allowing them toassign their problems to the relevant problem cate-gories. An extensive database of C&P publications(also including those concerning problem extensionsand problem variants), in which papers are classiedand organised in accordance with the suggestedtypology, is available at the ESICUP webpage(http://www.apdio.pt/sicup/). It provides directent assortments of large objects and small items)which are probably more relevant to the solutionof real-world C&P problems can only be found farless commonly in the literature, even though afew, more recent papers considering a heteroge-neous assortment of large objects (51) seem to indi-cate that the preferences of researchers might bechanging.

Wenally note (cf. Fig. 12) that the IIPPwhich hasJose Fernando Oliveira deserves particular thanks

1128 G. Wascher et al. / European Journal of Operational Research 183 (2007) 11091130for his continuous encouragement and availabilityfor discussions throughout the preparation of thispaper.

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