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Journal of Advanced Research inEvolutionary Algorithms

Vol. 1, Issue. 1, 2009, pp. 1-26Online ISSN: 1943-1414

Multi-objective Strip Packing

Simon Illich and Lyndon While∗

School of Computer Science & Software Engineering, The University ofWestern Australia, Nedlands, Australia 6009.

Abstract. Good algorithms exist for solving the 2D rectangular strip packing prob-lem when the objective is to minimize the amount of wasted material. However, insome applications other criteria are also important. We describe new heuristics forstrip packing that optimize not only for wastage, but also for the efficient use of thecutting equipment, by minimizing the number of independent cuts required by a pack-ing. We describe and evaluate two algorithms for multi-objective strip packing usingthese heuristics: a fast deterministic algorithm that gives good results very quickly;and a multi-objective evolutionary algorithm that gives excellent results across a rangeof benchmark problems, yet still runs in reasonable (and easily controllable) time. Weshow that both algorithms return a set of packings offering a range of trade-offs be-tween the two objectives, and also that by using heuristics that consider cuts, theevolutionary algorithm derives packings with wastage levels that are better than anypreviously-published algorithm that optimizes for wastage alone.

Keywords: Strip packing; Multi-objective optimization; Evolutionary algorithms.

AMS Subject Classifications: 90C29, 68T20.

1 Introduction

Cutting and packing is a significant problem relevant to many major industries worldwide[46]. In two dimensions, the problem is to arrange a set of shapes on a planar surface,which arises in the paper, textiles, sheet metal, wood, leather, and glass industries,among others (e.g. [4, 21–23, 28, 32, 44]). In three dimensions, the problem is to arrangea set of objects within a containing volume, which arises for example in shelf stacking,container and pallet loading, bin packing, and chip design (e.g. [5–7, 20]). Also manynon-spatial problems can be cast as cutting and packing, e.g. job scheduling, project

∗Correspondence to: Lyndon While, School of Computer Science & Software Engineering, The Universityof Western Australia, Nedlands, Australia 6009, Email: [email protected].†Received: 9 March 2009, revised: 26 April 2009, accepted: 9 May 2009.

http://www.i-asr.org/jarea.html 1 c©2009 Institute of Adnanced Scientific Research

2 Multi-objective Strip Packing

planning, timetabling, and other logistics problems (see e.g. [20]). The best packing isusually taken to be the one that wastes the least amount of the underlying substrate.

Solving cutting and packing problems manually is hard, and many people have triedautomation. Many benchmark problems exist (e.g. [9, 10, 33]), along with algorithmsthat can generate good solutions to these problems when minimizing wastage (e.g. [1,8,10,15]). However in some 2D applications, other criteria are also important for assessingpackings. In some applications, the material being cut is either very cheap, or it iseasily recycled, hence a more important criterion for assessing packings may be thespeed at which the shapes can be cut, in order to maximize efficient usage of the cuttingequipment. One way we can quantify this criterion is by minimizing the number ofindependent cuts needed by a packing: an independent cut is one that can be performedwith a single straight movement of a cutting device. Other criteria might include placingtogether shapes which form part of the same “job”, to maximize client satisfaction or tooptimize pricing structures based on turnaround times; or optimizing the placement ofshapes on material of variable quality, e.g. leather.

1.1 Problem specification

The specific problem addressed by this paper is known as 2D rectangular strip packing[46], hereafter abbreviated to strip packing. Given a set of 2D rectangles and a semi-infinite rectangular strip of material of a given width, place the rectangles on the materialsuch that they do not overlap and they occupy the minimum height of the strip. Werestrict ourselves to rectangles with integer dimensions placed with sides parallel tothe edges of the strip, and we assume that all of the shapes fit into the width of thestrip. In general, shapes may be rotated by 90◦ before being placed: however in someapplications (e.g. where the underlying material is decorated or corrugated) this isnot allowed. Another variable is whether guillotine cutting is required [38]. A packing isguillotineable if it can be performed by a sequence of edge-to-edge cuts, e.g. see Figure 1.Combining these two constraints gives us the four subtypes of strip packing: RG (rotation

(a) A guillotineable packing: a sequence of edge-to-edge cuts will suffice.

(b) A non-guillotineable packing: none of thefour cuts runs edge-to-edge.

Figure 1: The guillotine constraint for 2D rectangular strip packing.

allowed and guillotine cutting required), RF (rotation allowed but guillotine cutting notrequired), OG (rotation disallowed but guillotine cutting required), and OF (rotation

S. Illich and L. While 3

disallowed and guillotine cutting not required). In this paper we study the RF and OFsubtypes, although we accept guillotineable packings as solutions.

1.2 Contributions

This paper makes two contributions.

• We describe three new heuristics for constructing strip packing solutions accordingto two objectives: minimizing wastage, by minimizing the height used of the strip;and maximizing efficient use of the cutting equipment, by minimizing the numberof independent cuts in a packing.

• We define two algorithms that use these heuristics: a fast deterministic algorithm(MOSP-D) that gives good results very quickly; and a multi-objective evolution-ary algorithm (MOSP-P) that gives excellent results across a range of benchmarkproblems, yet still runs in reasonable (and easily controllable) time.

When tested across a range of benchmark strip packing problems, MOSP-P generatessolutions with lower wastage levels than any previously-published algorithm, despite thefact that it optimizes for cuts too.

The rest of the paper is structured as follows. Section 2 describes previous work onstrip packing, and Section 3 outlines the basics of multi-objective optimization. Section 4defines our heuristics for multi-objective strip packing and their implementation, andSection 5 describes our algorithms MOSP-D and MOSP-P that use these heuristics.Section 6 details our experimental set-up, including the benchmark problems used, andSection 7 gives and analyses the results. Section 8 concludes the paper and outlinessome future work.

2 Previous algorithms for strip packing

The utility of exact algorithms for cutting and packing (e.g. [26, 31, 39, 42]) is limited,as the computational requirement grows exponentially with the number of shapes [25].Approximation algorithms commonly use heuristics to decide how to place shapes. Oneof the first documented was the Bottom Left (BL) algorithm [4], which places eachshape with successive translations downwards and then to the left of the strip. Themore sophisticated Bottom-Left Fill (BLF) algorithm [12] can fill holes in packings bymaintaining a list of bottom-left location points.

A common approach is to combine heuristics with a meta-search [2,13,16,29,30,37,41,48]. [36] describes a genetic algorithm where each solution is represented as a sequenceof shapes and is evaluated using BL. [33] compares several meta-heuristics hybridizedwith BL and BLF: simulated annealing performed best.

The very efficient Best Fit (BF) heuristic [10, 47] does not require an ordering ofshapes to place: it examines the lowest gap in the partial packing and places the shapethat fits it best. [11] presents meta-heuristic enhancements for BF, such as tabu search,


genetic algorithms, and simulated annealing. GRASP [1] also selects a shape based onthe structure of the partial solution. It comprises a randomized construction phase anda local improvement phase.

SPGAL (Strip Packing Genetic Algorithm Layer) [8] performs a genetic search onpackings with a layered structure, searching directly in the space of fully-defined pack-ings: they conclude that SPGAL beats eleven other approaches from the literature.

The heuristic recursive (HR) algorithm [49] takes a divide-and-conquer approach tostrip packing. [15] combines HR with branch-and-bound techniques (HRBB), tuned toproducing guillotineable packings.

Extensive research has produced many novel techniques for strip packing. However,almost all aim only to minimize wastage: there are few multi-objective approaches inthe literature that allow a user to select from a range of solutions to meet their specificrequirements. [17, 40, 45] optimize waste alongside a measure of the number of cutsused in a packing, but they use simple approaches and report only a limited range ofresults. The only other “multi-objective” work done previously is a heuristic approachto optimize two objectives related to waste for metal cutting [43]. However they regardone objective as always more important than the other, so this is really equivalent tousing only one objective: there is still a total ordering on solutions.

3 Background technology

We use an evolutionary algorithm to generate strip packing solutions, and we treatstrip packing as a multi-objective optimization problem. Good introductions to thesetechnologies and precise definitions of the terms used in this paper are given in [3,14,18].

In a multi-objective optimization problem, each solution is assessed according to twoor more independent criteria. The characteristic of a good solution is that improvementin one objective can be achieved only by worsening at least one other objective. Analgorithm thus returns a set of solutions, each offering a different trade-off between theobjectives.

An important concept in multi-objective optimization is domination. A solution xdominates a solution y if and only if x is as good as y in every objective, and is betterin at least one objective. Given a set of solutions X, the non-dominated subset of Xcontains the solutions from X that are not dominated by any other solution in X.

Figure 2 illustrates these concepts for a problem with two objectives, both to bemaximized. Solution a dominates Solution d (i.e. it beats d in both objectives); bdominates e; c dominates e and f ; a, . . . , f all dominate g; and {a, b, c} is the non-dominated subset of the population.

A solution x is Pareto optimal if and only if x is non-dominated with respect to theset of all possible solutions. Such a solution is characterized by the fact that improve-ment in one objective can come only at the expense of some other objective(s). ThePareto optimal set is the set of all Pareto optimal solutions. The goal in multi-objectiveoptimization is to find (or approximate) this Pareto optimal set.


-f1

6f2sa

sbsc

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sesfsg

Figure 2: A population in objective space for a 2-objective maximization problem. The filledcircles represent solutions in the population, each plotted by their values in the objectives.

4 Placement heuristics for multi-objective strip packing

The basic structure of our algorithms follows BF [10]. Given a partial packing and aset of shapes remaining to be placed, we decide heuristically which shape to place next,and where. Iterating this process from an empty strip gives a complete packing. Ouralgorithms employ four heuristics. The process used to choose a heuristic at each stepis described separately for each algorithm in Section 5.

Two definitions are appropriate here. The lowest gaps in a packing are the emptyspaces closest to the “bottom” of the strip, and the neighbors of a gap are the rectanglesimmediately to the left and right of the gap (if any). The algorithms work by tryingto place shapes in the lowest gap of a partial packing: where there are multiple equallowest gaps, they always try the leftmost lowest gap first.

With the RF subtype, “width” or “height” (or related words) can refer to eitherdimension of a rectangle.

4.1 Four heuristics for multi-objective strip packing

Largest-Width heuristic (LW [10]). Place the widest rectangle that fits in the low-est gap. Break ties by placing the rectangle with the largest area.

Nearest-Height heuristic (NH). Place the tallest rectangle which is not taller thanthe lowest gap’s left neighbor. Break ties by placing the rectangle with the largestarea.

Least-Cuts-Large heuristic (LCL). As rectangles are always placed at the bottomand on the side of a gap, placing a rectangle adds a maximum of two cuts to thepartial packing. Place a rectangle that adds no cuts to the packing (or one thatreduces the numbers of cuts): if no such rectangle exists, place one that adds onlyone cut. Break ties by placing the rectangle with the largest area.

Least-Cuts-Small heuristic (LCS). As with LCL, but break ties by placing the rec-tangle with the smallest area.


There are four cases where adding a rectangle Z to a packing adds zero cuts. Threecases arise if Z has the same height as its immediate neighbor, and its other verticaledge coincides with an edge below it (Figure 3(a)), or its other neighbor (Figure 3(b)),or the far side of the strip (Figure 3(c)). The fourth case arises if Z covers the full width

Z

(a)

Z

(b)

Z

(c)

Z

(d)

Z

(e)

Figure 3: Adding a rectangle Z to a partial packing while minimizing cuts.

of the strip, and it is placed on top of two or more rectangles with the same height(Figure 3(d)).

There is one case where adding Z reduces the number of cuts by one: if Z has thesame height as both of its neighbors, and each vertical edge coincides with a neighbor(Figure 3(e)). The separate cuts required at the tops of the previously-placed shapesare joined by the placement of Z.

We define two variants of each heuristic, left (L) and right (R), which place therectangle at the leftmost or rightmost position in the gap respectively. Also R heuristicsinspect the right neighbor instead of the left. As an example, consider the partialpacking and the four shapes shown in Figure 4. Some example heuristic applicationsare illustrated in Figure 5. We assume here the RF subtype, where each shape can berotated by 90◦.

a

bc

d

Figure 4: A partial packing, and four shapes remaining to be placed.

4.2 Special cases for placement decisions

There are four special cases for placement decisions.

• If no rectangles meet the criterion for selection by a heuristic, we locate the next


a

(a) LW-L chooses a: biggest width.

b

(b) NH-R chooses b (rotated 90◦): height clos-est to the right neighbor’s.

c

(c) LCL-L chooses c: zero cuts added and morearea than d.

d

(d) LCS-L chooses d (rotated 90◦): zero cutsadded and less area than c.

Figure 5: Adding one shape to the partial packing from Figure 4.

lowest gap and try again. This process occurs iteratively until a rectangle can beplaced.

• Where the lowest gap covers the width of the strip, we always apply LW. Any Rheuristic is applied as LW-R, and any L is applied as LW-L. This prevents thealgorithm from getting stuck where the gap cannot be raised any further. Notethis means that the heuristic used to place the first shape is always applied as LW.

• Where NH is used and the lowest gap has no neighbor on the relevant side, weplace the tallest rectangle.

• Any rectangle which has a dimension bigger than the width of the strip is placedfirst. Clearly such a rectangle could cause problems if it is placed late in theprocess, possibly leading to “towers” [10].

4.3 Implementation of the heuristics

Given a partial packing, a set of rectangles, and a heuristic, an algorithm needs to try(conceptually at least) placing each of the rectangles, in both orientations if necessary,and to choose one based on the definition of the heuristic. This is potentially an expensiveoperation, repeated n times for a problem with n shapes. We optimize it in two mainways.


4.3.1 Locating the gaps

Following [10], we represent a partial packing by an array holding the height of thepacking at each x-coordinate across the strip. The resolution of the array across thestrip can be set as the lowest common denominator of the widths of all the rectangles(for the RF subtype, we need the LCD of both dimensions). Although this representationis pseudo polynomial as opposed to polynomial size, it has the advantage of being simpleand fast in practice.

Figure 6 shows a small example. The partial packing in Figure 6 has four gaps: the

Array 3 4 4 4 4 1 1 5

012345

0 1 2 3 4 5 6 7 8

Figure 6: Array representation of a partial packing.

three spaces bounded below by (0,3)—(1,3), (5,1)—(7,1), and (0,4)—(7,4); and from(0,5)—(8,5) upwards, the part of the strip where no shape has yet been placed. Thelowest gap is the gap with the smallest y-coordinate, i.e. (5,1)—(7,1) in this example.

4.3.2 Testing the rectangles

Each rectangle is represented by a (width, height) pair. We can select a rectangle quicklyby storing those remaining in a pair of lists sorted appropriately for the heuristics.

• NH and LCL examine a list of rectangles sorted by decreasing area. When theyfind a rectangle matching their criteria, they can stop: subsequent rectangles willbe smaller.

• LCS examines the same list from the back, again looking for the first rectanglethat matches its criteria.

• LW examines a list sorted by decreasing width: it simply selects the first rectanglethat fits the lowest gap.

When a rectangle is placed, it is removed from all lists.


4.4 Behavior of the heuristics

Experience shows that different heuristics produce packings with different characteristics.

• LW is good for minimizing wastage, but it is the least effective for minimizingcuts. Placing large rectangles early helps to prevent empty spaces forming. Thisis advantageous as smaller items are useful for filling empty spaces in the latterstages.

• NH tends to create a layered structure with long horizontal cuts running throughthe packing. It fares moderately well in minimizing both wastage and cuts.

• LCS produces a grid-like structure in the packing, which is very effective in mini-mizing cuts. As it delays placing some of the larger objects, however, it generallyproduces packings with more wastage than the other heuristics.

• LCL combines the benefits of LW and LCS, generally performing well in bothobjectives. As well as placing rectangles that add the least cuts to the partialpacking, it also places larger rectangles early, helping to minimize wastage.

5 Two algorithms for multi-objective strip packing

We describe two algorithms for multi-objective strip packing. MOSP-D is a fast de-terministic algorithm that nevertheless gives good results. MOSP-P is an evolutionaryalgorithm that gives excellent results across a range of benchmark problems, yet stillruns in reasonable (and easily controllable) time. Previous versions of these algorithmswere described and evaluated in [34, 35]. Note that the structure of both algorithmslends itself easily to the addition of more heuristics and/or more objectives.

5.1 Deterministic greedy MOSP (MOSP-D)

MOSP-D is a fast deterministic algorithm for constructing strip packing solutions. It gen-erates a set of solutions by two complementary methods, then returns the non-dominatedsubset of those solutions. Experience shows that both types of solutions contribute sig-nificantly to the set returned by MOSP-D.

5.1.1 Simple solutions

MOSP-D generates eight solutions by independently applying the fixed sequences shownin Figure 7. Each sequence uses only one of the heuristics. The symmetry of the heuris-tics and of the shapes means that there is no advantage in duplicating these sequenceson the right.


<LW-L, LW-L, LW-L, LW-L, . . . > <LW-L, LW-R, LW-L, LW-R, . . . ><NH-L, NH-L, NH-L, NH-L, . . . > <NH-L, NH-R, NH-L, NH-R, . . . ><LCL-L, LCL-L, LCL-L, LCL-L, . . . > <LCL-L, LCL-R, LCL-L, LCL-R, . . . ><LCS-L, LCS-L, LCS-L, LCS-L, . . . > <LCS-L, LCS-R, LCS-L, LCS-R, . . . >

Figure 7: The eight fixed sequences used in MOSP-D.

5.1.2 Greedy solutions

MOSP-D generates its other solutions by starting from an empty strip, and at eachstep choosing the heuristic that performs best under a local optimization function eval.Given a heuristic h and a partial packing p:

eval(h, p) = α× normalise(width(h, p)) + (1− α)× normalise(cuts(h, p))

where width(h, p) denotes the width remaining in the lowest gap after the shape chosenby h is placed in p, cuts(h, p) denotes the number of cuts in p after the placement, andnormalise scales both objectives onto [0, 1], to avoid implicit bias due to their differentranges. The heuristic that gives the smallest value of eval(h, p) is used for this step.

The weighting factor α is in [0, 1]: it represents the relative importance of the twoobjectives in generating a given solution. MOSP-D can generate a variety of solutionsby executing the above process with a range of values for α. We have found that α ∈{0.0, 0.1, 0.2, . . . , 0.9, 1.0} offers a good balance between run-time and results, generatingeleven greedy solutions. A good alternative plan might be to adapt the values used forα to a particular problem, recursively splitting the interval [0, 1] until no more distinctsolutions are returned.

5.2 Evolutionary MOSP (MOSP-P)

MOSP-P is a multi-objective evolutionary algorithm for constructing strip packing so-lutions. When creating a multi-objective evolutionary algorithm, the principal designdecisions concern the representation of an individual solution; the objectives againstwhich solutions are measured; and various details in the operation of the algorithm, todo with mutation, selection, population seeding, etc.

5.2.1 Genetic representation

The genotype used in MOSP-P for a problem with n shapes is a sequence of n heuristics.So in a problem with five shapes, one solution would be the sequence

<LW-R, NH-L, LCL-R, LCS-L, NH-R>

Starting with an empty strip, this solution selects and places the first shape according toLW-R (largest fitting rectangle to the right of the gap); then it updates its set of gaps;then it selects and places the next shape using NH-L; then it updates its set of gapsagain; and so on until all five shapes have been placed.


5.2.2 Objectives

MOSP-P optimizes packings on two objectives independently.

Optimizing wastage: the Height objective. Minimizing the height of a packing min-imizes the amount of material wasted. But because we assume shapes with integerdimensions, the actual height is an integer, so many packings have the same height,which causes problems for selection. Thus we add to the actual height a quantityto reflect that empty space at the top of a packing (i.e. connected to the semi-infinite space) is often easier to fill later than empty space in the middle. Thus wedefine Height by

Height = h + (hw − a)/hw

where h is the actual height of the packing, w is the width of the strip, and a isthe free area at the top of the packing. Importantly, 0 ≤ a < hw, so the secondterm is always in (0, 1] and h < Height ≤ h + 1: thus a smaller value of h impliesa smaller value of Height. Note that a bigger value of a implies a better value ofHeight.

Optimizing efficiency: the Cuts objective. Minimizing the number of independentcuts required by a packing maximizes the rate at which the processing equipmentcan operate. The number of cuts is also an integer quantity, but it tends to havea larger range than the height, so this is less of an issue. Thus we define Cuts asthe number of distinct edges within the packing, i.e. excluding cuts at the edgesof the strip.

The values of these two objectives comprise the fitness of a solution.

5.2.3 Mutation

Mutation in MOSP-P randomly selects one heuristic from the genotype and modifiesits value to one of the seven other possibilities, with equal probability. Because we usehighly elitist selection (meaning that the best individuals are guaranteed to survive),mutation is applied with a probability of 1.

Our representation allows multiple genotypes to produce the same packing. To elim-inate duplicate packings, when a solution is mutated its fitness is checked against thefitnesses of the other solutions in the population. If the fitness already exists, the solutionis mutated again until it produces a fitness that is unique in the current population. Thestructure of the genotype means that this process is effectively guaranteed to terminate.Of course two packings with identical fitnesses are not necessarily identical: however itis quite likely, given the definition of the height objective.

MOSP-P does not use recombination of solutions: our experience on other problems(and this one) is that recombination does not contribute to better performance.


5.2.4 Selection

MOSP-P selects solutions for survival and reproduction based on their Pareto rank [24],where the rank of a solution x is defined as the number of solutions in the population thatdominate x. A lower rank implies a superior solution. We distinguish equally-rankedsolutions using crowding distance [19]. The dispersion of a solution in an objective isthe distance to the closest solution in that objective, and the crowding distance cd of thesolution is the sum of its dispersions in each objective. Larger values of cd are preferred:this helps to maintain diversity and to prevent premature convergence.

MOSP-P uses an elitism rate of 50% [3]. In each generation, each of n solutionsgenerates one child, and the best n of these 2n solutions survive.

5.2.5 Population seeding

MOSP-P seeds an initial population of size n with three groups of solutions:

• 32 fixed solutions, being four copies of each of the eight sequences from Figure 7;

• the m non-dominated solutions from a set of greedy runs of MOSP-D(α ∈ {0.0, 0.1, 0.2, . . . , 0.9, 1.0}, so 1 ≤ m ≤ 11);

• the remaining n− 32−m solutions constructed randomly.

Experience shows that this seeding provides a diverse set of solutions early in the evo-lution and decreases the overall convergence time.

Solutions comprising just one heuristic are sometimes very good in one objective,but they are seldom good in both objectives. Solutions good in both objectives usuallyuse a mixture of heuristics, which can be discovered only by a search process.

6 Experimental set-up

We applied the MOSP algorithms to the RF and OF subtypes of the strip packingproblem, on a range of benchmark data. This section describes this data and the set-upused for each algorithm. The results are presented in Section 7. All programs werewritten in Java and all runs were performed on a 2.4GHz Pentium 4 running WindowsXP.

6.1 Benchmark problems

We used three sets of test data: the 21 C problems from [33]; the 13 N problemsfrom [10]; and the 360 “large” problems from [9] (we call these the B problems). Table 1gives the details for these problems.

Each of the C and N problems is constructed by slicing up a known rectangle, so theoptimal height of the packing is known. For each problem, n is the number of shapes,w is the width of the strip, and opt.h is the optimal height.


n w opt.h n w opt.h Class nt clb1 16 N1 10 40 40 B1 10 39,398

C1 2 17 20 20 N2 20 30 50 B2 10 40,4783 16 N3 30 30 50 B3 10 2,4781 25 N4 40 80 80 B4 10 2,292

C2 2 25 40 15 N5 50 100 100 B5 100 39,7313 25 N6 60 50 100 B6 100 40,1451 28 N7 70 80 100 B7 100 2,498

C3 2 29 60 30 N8 80 100 80 B8 100 2,3113 28 N9 100 50 150 B9 1,000 39,9751 49 N10 200 70 150 B10 1,000 39,985

C4 2 49 60 60 N11 200 70 150 B11 1,000 2,5063 49 N12 500 100 300 B12 1,000 2,3031 72 N13 3,152 640 960

C5 2 73 60 903 721 97

C6 2 97 80 1203 971 196

C7 2 197 160 2403 196

Table 1: Statistics for the C, N , and B problems.

The optimal height is not known for the B problems, so height results are presentedrelative to a lower bound derived by dividing the sum of the areas of the shapes by thewidth of the strip. Every problem has 1,000 shapes with a strip width of 1,000; for eachproblem, nt is the number of distinct shapes and clb is the average lower bound on theheight for the thirty problems in that class. Following [9], we present all results for theB problems as averages for twelve sets of thirty problems each.

6.2 Settings and timings for the greedy algorithm

We ran MOSP-D with α ∈ {0.0, 0.1, 0.2, . . . , 0.9, 1.0}, thus it generates nineteen solutionsfor each problem and returns the non-dominated subset of these. We have found thatthis range offers a good balance between run-time and results. Approximately 40% of thesolutions returned for the benchmark problems were “simple solutions” as described inSection 5.1.1: the rest were generated by the greedy method described in Section 5.1.2.

Each run took 0.5–2.5s (average 1.1s) for the C problems, 0.4–8.8s (average 1.8s) forN1–12, 6 minutes for N13, and 72–85s (average 79s) for the B problems.

6.3 Settings and timings for the evolutionary algorithm

We ran MOSP-P as an (n + n) evolutionary strategy with a population of 100 for250 generations, which gives good results in reasonable time. The same settings were


used for all of the experiments. Approximately 55% of the solutions returned for thebenchmark problems originated from single-heuristic seeds, approximately 33% origi-nated from MOSP-D seeds, and the rest originated from randomly-constructed seeds.The latter tended to generate more solutions for the smaller problems and fewer for thebigger problems.

Each run took 3–39s (average 11s) for the C problems, 4–124s (average 23s) forN1–12, 62 minutes for N13, and 6–17 minutes (average 10 minutes) for the B problems.

Evolutionary algorithms are by nature non-deterministic (mostly due to the random-ness in the mutation process), so a common approach to get good results is to run thealgorithm multiple times and to combine the results. The results presented for the Cand N problems are the combined non-dominated sets from five runs on each problem.Note however that excellent solutions are often available with fewer runs, or early in arun, and that even better solutions are certainly available with more runs. The resultspresented for the B problems are derived from one run on each problem.

For comparison, BF+ and GRASP run for 60s on all problems; HRBB averages 1.9sfor the C problems; and SPGAL averages 139s for the C problems and 13 minutes forthe B problems (for the results reported here).

7 Results and discussion

We discuss the performance of the MOSP algorithms in three ways.

• We compare their best height solutions with several previous algorithms.

• We discuss their best cuts solutions, although comparison with previous algorithmsis necessarily limited.

• We illustrate the trade-offs offered by the fact that MOSP returns a set of solutionsfor each problem.

7.1 Height results

Table 2 shows the height performance of MOSP and four other leading algorithms onthe C and N problems. Each entry gives the height over the optimum, so a value of0 indicates an optimal solution. The data show that MOSP-D performs solidly, it iscompetitive with BF+ [10] and HRBB [15], but it is generally beaten by the sloweralgorithms. Also we observe that:

• MOSP-P generates optimal solutions to 11/34 problems for OF, and to 16/34problems for RF: for the remaining problems it is almost always 1 over optimal;

• MOSP-P beats BF+ on 23/34 problems, drawing on the rest;

• MOSP-P beats GRASP [1] on 7/34 problems, losing only 1;


OF OF OF OF RF RF RF RF RFSPGAL GRASP MOSP-D MOSP-P BF+ HRBB SPGAL MOSP-D MOSP-P

1 0 0 0 0 1 0C1 2 0.3 0 3 0 0 0.3 0.3 1 0

3 0 1 0 0 1 01 0 1 0 1 1 0

C2 2 0.3 0 1 0 1 0.0 0.0 2 03 0 1 0 1 0 01 0 3 0 1 2 0

C3 2 1.0 1 3 1 1 0.3 0.7 2 13 0 2 0 1 2 01 1 6 1 1 2 1

C4 2 1.7 1 5 1 1 1.3 0.0 3 03 1 3 1 1 1 11 1 3 1 1 2 1

C5 2 1.3 1 8 1 1 1.7 0.0 2 03 1 6 1 1 1 11 1 6 1 2 3 1

C6 2 2.0 1 5 1 1 1.7 0.3 2 13 1 4 1 2 2 11 4 9 2 4 3 1

C7 2 3.0 2 9 2 4 3.0 0.7 3 13 3 8 2 5 3 1

N1 0 0 0 0 4 0N2 0 2 1 0 3 0N3 1 4 0 1 2 0N4 1 12 0 2 3 1N5 2 6 1 3 4 1N6 1 4 1 2 2 1N7 1 6 1 4 6 1N8 1 4 1 2 3 1N9 1 6 1 2 5 1N10 1 4 1 2 1 0N11 1 2 1 3 2 0N12 3 5 2 6 3 1N13 3 2 1 4 1 0

Table 2: Height performance of MOSP and comparison with previous algorithms on the C andN problems. Each entry gives the height over optimal, so 0 indicates an optimal solution.


• MOSP-P beats HRBB on 5/7 C problem sets, drawing on the others (althoughHRBB returns guillotineable packings);

• for OF, MOSP-P beats SPGAL [8] on all 7 C problem sets;

• for RF, MOSP-P beats SPGAL on 2/7 C problem sets and loses on 4, althoughthe difference between the two algorithms is always small.

Table 3 shows the height performance of MOSP and SPGAL on the B problems. SPGAL

OF OF OF RF RF RFClass SPGAL MOSP-D MOSP-P SPGAL MOSP-D MOSP-P

B1 4.43 4.04 1.58 2.44 2.88 0.84B2 3.79 3.78 1.51 1.86 1.89 0.82B3 3.07 2.41 1.88 2.61 1.84 1.11B4 2.85 1.75 1.27 2.34 1.41 0.84B5 2.08 2.48 1.11 1.27 1.46 0.57B6 1.68 1.74 0.92 1.04 0.80 0.45B7 2.39 2.46 1.81 1.87 1.86 1.22B8 1.63 1.73 1.27 1.18 1.19 0.63B9 4.34 2.38 0.97 3.03 1.37 0.49B10 1.67 1.35 0.76 1.78 0.57 0.29B11 2.45 2.25 1.56 1.87 1.66 1.23B12 2.12 1.37 0.93 1.83 0.78 0.44avg 2.71 2.31 1.30 1.93 1.48 0.75

Table 3: Height performance of MOSP and comparison with SPGAL on the B problems. Eachentry gives the percentage excess over the (theoretical) lower bound.

returns guillotineable packings, but [9] states that it makes only negligible improvementswhen exploiting non-guillotineability. The data show that:

• MOSP-D beats SPGAL on 16/24 problem sets, sometimes by a substantial amount;

• MOSP-P beats SPGAL on all 24 problem sets, usually by a substantial amount;

• MOSP-P always gets within 2% of the lower bound, and usually within 1% (re-member that the lower bound is probably unattainable for most problems).

Occasionally an algorithm operating in OF mode beats itself operating in RF mode(e.g. MOSP-D on N1 and MOSP-P on N4), which seems counter-intuitive: RF offersa superset of the options of OF. However if an optimal solution can be derived in OFmode, then operating in RF mode simply expands the search space, complicating thealgorithm’s task.

7.2 Cuts results

Tables 4 and 5 show the cuts performance of MOSP and BF. Although the literaturedoes not provide any solutions that were optimized for cuts, we present two sets of figuresfor comparison: default cuts def. c, and results from BF.


OF OF OF RF RF RFdef.c MOSP-D MOSP-P BF MOSP-D MOSP-P

1 11 11 11 19 17 11C1 2 17 20 15 21 17 14

3 11 13 11 20 15 101 16 22 16 33 22 17

C2 2 23 25 18 30 24 183 16 20 14 34 18 161 18 24 18 36 25 20

C3 2 29 31 26 34 30 243 17 26 17 39 29 211 33 50 42 66 45 38

C4 2 49 50 44 62 45 363 32 45 42 63 45 371 47 67 57 95 61 57

C5 2 72 69 60 82 62 583 47 68 57 92 56 491 62 91 78 123 70 66

C6 2 96 91 84 113 79 753 63 89 82 120 73 721 131 183 165 252 141 138

C7 2 188 179 163 243 152 1463 127 180 171 251 156 149

N1 8 8 18 13 8N2 21 19 30 24 15N3 31 25 38 28 21N4 44 33 56 37 33N5 61 44 73 52 43N6 59 43 69 46 41N7 71 56 88 64 51N8 72 65 112 68 60N9 72 69 123 79 67N10 120 108 205 107 95N11 158 137 281 130 125N12 218 218 464 228 203N13 1,839 1,632 3,413 1,468 1,376

Table 4: Cuts performance of MOSP and comparison with BF on the C and N problems.


OF OF RF RF RFClass MOSP-D MOSP-P BF MOSP-D MOSP-P

B1 751 630 893 800 668B2 771 648 842 779 670B3 512 491 688 528 476B4 583 556 734 606 561B5 1,121 1,108 1,296 1,049 1,042B6 1,115 1,054 1,168 1,076 1,013B7 911 906 1,424 827 825B8 1,023 1,017 1,378 935 909B9 1,147 1,144 1,519 1,082 1,081B10 1,216 1,213 1,511 1,120 1,118B11 892 888 1,617 721 719B12 941 938 1,509 830 828avg 915 883 1,215 863 826

Table 5: Cuts performance of MOSP and comparison with BF on the B problems.

As described earlier, each of the C and N problems is constructed by slicing upa known rectangle. [33] provides the layout of each C problem as it was initially con-structed: we refer to the number of cuts in each of these height-optimal solutions as thedefault cuts. We do not present default cuts for the N problems or the B problems asthe authors do not provide the original layouts.

Each of the Ck.1 and Ck.3 problems was constructed as a guillotineable packing,while the Ck.2 problems were constructed in non-guillotineable form. For this reasonthe default cuts for the Ck.2 problems are 45–65% higher than the others. Despite this,the MOSP algorithms usually produce similar results for all three problems in each set.

BF cuts denotes the number of cuts in a packing produced by the BF algorithm.Of course BF minimizes only for wastage, so we do not expect it to be competitive forminimizing cuts, but it is probably indicative of cuts results for algorithms that ignorecuts.

The data show that:

• both MOSP algorithms beat BF on all problems, usually by a substantial amount;

• MOSP-D usually and MOSP-P almost always beat BF across their entire fronts(i.e. the best height/worst cuts solution for each algorithm has fewer cuts thanBF: this solution can be identified by combining Tables 4 and 6 or Tables 5 and7);

• MOSP-P outperforms MOSP-D, as expected;

• both algorithms usually generate packings with fewer cuts than shapes, whichseems to indicate a high level of performance. MOSP-P in particular is able topack over 90% of the problems with a cuts:shapes ratio of less than 1.00 for RF,and over 50% of the problems with a ratio of less than 0.75 for both subtypescombined.


7.3 Trade-offs

Tables 6 and 7 show the trade-offs offered by MOSP. For each algorithm on eachproblem in each subtype, we give the number of non-dominated solutions returned, andthe differences between the minimum and maximum height and cuts values. The datashow that:

• for most C and N problems, the algorithms produce 2–5 non-dominated solutions;

• for the B problems, MOSP-P produces many more distinct solutions, probablybecause the algorithm is run only once, so it has less opportunity to generatesolutions that excel in both objectives;

• both algorithms tend to produce more solutions on bigger problems;

• on some smaller problems, the algorithms produce only one solution. For MOSP-P in particular, this is usually when it produces a solution that excels in bothobjectives;

• on many problems, the range in height of the solutions is considerably less thanthe range in cuts. This indicates cases where a substantial gain in efficiency canbe achieved with only a small sacrifice in wastage.

To illustrate the trade-offs available with the multi-objective approach, Figure 8 plotsthe solutions generated for problem N10 in RF mode: the three solutions generated byMOSP-P, the four solutions generated by MOSP-D, and the solution generated by BF.The plot also highlights the savings in wastage achieved by using heuristics that considercuts.

-

Cuts

6

Heightoveroptimal

0

2

4

6

8

95 115 135 155 175 195 215s\

sHHs

c\c

JJJ

c

ll

ll

l

c

×

t MOSP-Pd MOSP-D× BF

Figure 8: Solutions from MOSP and BF for problem N10 in RF mode, in objective space.

Figures 9 and 10 show some example packings for the C and N problems, showingthe savings in cuts available by sacrificing height. We do not include example packingsfor the B problems because their size makes them hard to depict clearly. Broadly, weobserve that:


MOSP-D (OF) MOSP-P (RF) MOSP-D (RF) MOSP-P (RF)

#solns Hrng Crng #solns Hrng Crng #solns Hrng Crng #solns Hrng Crng

1 1 1 1 1C1 2 1 2 6 1 2 4 1 1

3 1 1 1 11 2 1 2 1 2 3 1 2 1 3

C2 2 2 1 1 2 1 1 2 3 2 13 2 1 3 1 1 11 3 3 11 1 4 7 11 2 2 2

C3 2 2 3 5 3 4 4 1 3 2 33 1 1 2 2 6 2 1 31 2 3 24 4 8 9 2 1 13 3 4 3

C4 2 2 7 14 4 3 6 3 5 7 13 2 6 12 3 2 4 2 8 5 11 3 7 18 2 2 1 3 3 20 3 2 2

C5 2 2 1 15 3 2 4 3 4 21 13 5 12 20 5 6 10 2 1 18 11 2 3 27 2 1 2 1 2 2 2

C6 2 3 4 24 3 5 8 2 1 20 2 1 23 4 10 37 5 17 12 3 4 25 2 1 31 1 5 5 11 4 16 73 5 49 5

C7 2 3 63 56 8 19 40 2 5 48 4 5 163 2 2 56 5 12 13 3 4 57 3 12 4

N1 1 1 2 1 1 1N2 2 2 6 3 3 3 1 1N3 3 8 9 2 2 1 2 7 2 1N4 3 23 13 1 1 3 2 5N5 2 1 5 3 2 11 5 9 17 2 1 7N6 2 1 4 3 3 9 2 1 7 3 3 3N7 4 29 9 4 11 5 3 14 11 3 5 10N8 4 22 25 6 23 11 2 7 12 5 17 8N9 3 6 42 5 5 20 3 8 38 1N10 4 24 39 6 28 18 4 7 35 3 2 13N11 3 11 4 4 7 9 2 4 13 5 5 25N12 3 20 69 7 23 45 3 5 114 4 3 15N13 3 5 44 5 4 319 2 1 149 3 2 32avg 2.4 8.2 17.5 3.4 6.3 18.1 2.3 4.0 21.4 2.3 3.7 5.2

Table 6: Trade-offs for the C and N problems.


(a) N5 MOSP-P: min height (101) and max cuts (55) vs 2.0% more height (103) and 20% fewercuts (44).

(b) N13 MOSP-P: min height (961) and max cuts (1,951) vs 0.4% more height (965) and 16% fewercuts (1,632).

Figure 9: Some example packings for the OF subtype.


(a) C7.3 MOSP-D: min height (243) and max cuts (213) vs 1.6% more height (247) and 27% fewercuts (156).

(b) N12 MOSP-D: min height (303) andmax cuts (342) vs 1.7% more height (308)and 33% fewer cuts (228).

(c) N10 MOSP-P: optimal height (150) and max cuts(108) vs 1.3% more height (152) and 12% fewer cuts(95).

Figure 10: Some example packings for the RF subtype.


MOSP-D (OF) MOSP-P (OF) MOSP-D (RF) MOSP-P (RF)

Class #solns Hrng Crng #solns Hrng Crng #solns Hrng Crng #solns Hrng Crng

B1 3.0 1.76 125 15.0 3.15 310 2.8 1.17 121 14.1 1.92 248B2 3.1 1.68 177 15.2 3.28 276 2.6 1.41 138 13.6 2.15 225B3 2.3 0.59 89 4.9 1.28 78 2.5 0.60 105 4.7 0.86 85B4 2.4 1.16 126 6.3 2.08 118 2.3 0.49 89 5.0 0.70 101B5 2.9 1.10 240 17.4 2.16 125 3.4 1.15 259 16.0 1.80 134B6 2.7 1.07 148 8.8 0.64 81 3.5 1.19 168 10.8 0.48 113B7 2.1 0.41 222 10.5 2.05 219 1.7 0.25 85 9.3 1.23 195B8 2.6 1.07 269 10.3 1.86 156 2.2 0.41 128 7.8 0.96 71B9 3.4 1.22 454 23.3 2.50 265 3.2 0.78 354 21.6 1.65 219B10 4.1 1.26 421 19.6 1.70 318 4.6 0.85 462 23.7 1.09 307B11 2.5 0.60 293 15.2 2.03 338 1.9 0.31 161 8.9 0.87 250B12 2.2 0.30 250 11.3 1.66 243 1.8 0.12 37 7.4 0.50 86avg 2.8 1.02 235 13.2 2.03 211 2.7 0.73 176 11.9 1.18 170

Table 7: Trade-offs for the B problems.

• to minimize height, an algorithm places large shapes first and uses smaller shapesto fill gaps between them, typically using LW, NH, and LCL;

• to minimize cuts, an algorithm organizes similarly-sized shapes (big or small) intogrid layouts, typically using LCL and LCS. These heuristics try to line up bothhorizontal and vertical cuts in a packing. The grid organization often serves wellto minimize height too, because shapes are arranged in layers across the strip.

8 Conclusions

We have presented new heuristics for strip packing which optimize for two objectives:minimizing wastage, by minimizing the height of the strip used by a packing; and max-imizing the efficient use of the cutting equipment, by minimizing the number of in-dependent cuts in a packing. We have presented and evaluated two algorithms thatuse these heuristics: a fast deterministic algorithm MOSP-D that generates good solu-tions very quickly; and a multi-objective evolutionary algorithm MOSP-P that generatesexcellent solutions across a range of benchmark problems in reasonable (and easily con-trollable) time. In fact MOSP-P generates solutions with better wastage levels than anypreviously-published algorithm, despite the fact that it optimizes for cuts too.

The structure of both MOSP algorithms lends itself easily to the addition of moreheuristics and/or more objectives. We plan to extend this work with other objectiveswhich are important in some industries. In some applications, shapes are grouped asbelonging to different jobs or clients. The relative placement of the shapes in a groupthen becomes important, maybe for optimizing turnaround times (e.g. delaying thestart of a job as long as possible, while still meeting an agreed completion time) or foroptimizing pricing structures (e.g. offering different rates for different completion times).


Quantifying and including such objectives will increase the utility of this work. We alsoplan to investigate new heuristics, both for the current objectives and for any new ones.

Acknowledgments

Luigi Barone, Graham Kendall, and Phil Hingston contributed to some of these ideas.

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