approximate algorithms for constrained circular cutting problems

Available online at www.sciencedirect.com

Computers & Operations Research 31 (2004) 675–694www.elsevier.com/locate/dsw

Approximate algorithms for constrained circular cuttingproblems

Mhand Hi+a;b;∗, Rym M’Hallahc

aLaRIA, Laboratoire de Recherche en Informatique d’Amiens, 5 rue du Moulin Neuf, Amiens 80000, FrancebCERMSEM-CNRS UMR 8095, Maison des Sciences Economiques, Universit%e Paris 1 Panth%eon-Sorbonne,

106-112, Boulevard de l’Hopital, Paris 75647, Cedex 13, FrancecDepartment of Quantitative Methods and Information Technologies, Institut Sup%erieur de Gestion de Sousse,

B.P. 763, Sousse 4000, Tunisia

Received 1 June 2002; received in revised form 1 December 2002

Abstract

In this paper, we study the problem of cutting a rectangular plate R of dimensions (L;W ) into as manycircular pieces as possible. The circular pieces are of n di4erent types with radii ri; i = 1; : : : ; n. We solvethe constrained circular problem, where di the maximum demand for piece type i is speci+ed, using twoheuristics: a constructive procedure-based heuristic and a genetic algorithm-based heuristic. Both of theseapproaches search for a good ordering of the pieces and use an adaptation of the best local position proce-dure (Studia. Inform. Univ. 2 (1) (2002) 33) to +nd the “best” layout of this ordered set. This positioningprocedure is speci+cally tailored to circular cutting problems. It acts, for constrained problems, as one ofthe mutation operators of the genetic algorithm. We compare the performance of both proposed approachesto that of existing approximate and exact algorithms on several problem instances taken from the literature.The computational results show that the proposed approaches produce high-quality solutions within reason-able computational times. The genetic algorithm-based heuristic is easily parallelizable; one of its importantfeatures to be investigated in the near future.

Scope and purpose

In many industrial sectors, minimizing waste is a critical issue. This is particularly the case for the packing,textile, naval, and aerospace industries, where minimizing the waste of packing and cutting is a frequentproblem. Our paper studies the constrained circular packing problem where a set of circles needs to be cut ona rectangular stock sheet of +xed width and length. The objective is to maximize the usage of the rectangular

∗ Corresponding author. CERMSEM-CNRS UMR 8095, Maison des Sciences Economiques, UniversitEe Paris 1PanthEeon-Sorbonne, 106-112 Boulevard de l’Hopital, Paris 75647, Cedex 13, France.

E-mail addresses: [email protected] (M. Hi+), [email protected] (R. M’Hallah).

0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0305-0548(03)00020-0

mailto:[email protected]

mailto:[email protected]

676 M. Hi9, R. M’Hallah / Computers & Operations Research 31 (2004) 675–694

sheet while respecting the upper demand value for each circle type. We propose two approximate algorithmsto solve this problem and prove their eLciency.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Best local position; Circular cutting problem; Combinatorial optimization; Genetic algorithms; Heuristics

1. Introduction

Unless P=NP, many interesting problems of operations research and arti+cial intelligence cannotbe solved exactly within a reasonable amount of time. Consequently, heuristics must be used to solvelarge-scale real-world problems. Heuristic algorithms may be divided into two main classes: generalpurpose algorithms designed independently from the optimization problem at hand, and tailoredalgorithms speci+cally designed for a given problem. The aim of this paper is to propose generalalgorithms that approximately solve a variant of the cutting problem.

The cutting problem consists of cutting large stock plates into smaller pieces as to optimize a givenobjective. Di4erent exact and approximate algorithms for the cutting problem have been consideredin the literature over the last 35 years [1–6].

In our study, we investigate the constrained circular cutting problem, denoted CC, where a rect-angular stock plate R, of length L and width W , is cut into as many circular pieces as possible of ndi4erent types. Each piece type i; i = 1; : : : ; n, is characterized by a radii ri, an upper demand valuedi, and a pro+t (or weight) ci. In our case, ci is the area of piece type i. Thus, the objective is tomaximize the usage of the stock plate R.

This paper is organized as follows. In Section 2, we present a brief review of existing sequentialalgorithms for some variants of the CC problem. In Section 3, we adapt the best local positionprocedure of [7] to circular pieces, illustrate its application through a small example, show howit can be used as a constructive approach for the CC problem, and explain how the constructiveapproach acts as a feasibility check/mutant operator before providing a constructive heuristic. InSection 4, we detail the genetic algorithm based heuristic. In Section 5, we evaluate, using a setof problem instances taken from the literature, the performance of the proposed approaches andcompare it to that of existing heuristic and exact algorithms. Finally, in Section 6, we summarizethe present work and explain its possible extensions.

2. Related literature

The CC problem, a generalization of the rectangular cutting stock problem [8] which in turn is ageneralization of the single knapsack problem [4], is NP-complete. It has subsequently received verylittle attention. Most published research focuses on packing identical circles. The proposed approachesare heavily inPuenced by the “single-size” constraint [9–11]. To our knowledge, very few papersdealing directly with the problem of packing non identical circles are available. George et al. [12]have designed an approach based upon several building rules simulating the packing operation; thebest rules are a quasi-random algorithm and a genetic algorithm. Hi+ et al. [13] have tailored asimulated annealing-based heuristic whose energy function provides, when assuming small values,

M. Hi9, R. M’Hallah / Computers & Operations Research 31 (2004) 675–694 677

good packings of the circles on the left corner of the stock rectangle. Finally, Stoyan and Yaskov[14] have proposed a closely related interesting work for the so-called dual of the CC problem:the strip packing or the layout problem. The authors have developed a mathematical model and anapproach-based algorithm that searches for a local extremum.

In this paper, we present a genetic algorithm-based heuristic to solve the CC problem. The algo-rithm uses an adapted version of the best local position (BLP) approach [7] to check the feasibilityof a chromosome, to mutate any infeasible chromosome into a feasible one, and to evaluate the+tness of a feasible chromosome.

3. A constructive approach for the CC problem

First, we adapt the BLP procedure, namely BLP, proposed in Hi+ and M’Hallah [7] to circularpieces. Second, we illustrate the application of this adapted version through a small example. Third,we show how this adapted version, namely ABLP, can be used as a constructive approach for the CCproblem. Fourth, we explain how the constructive approach checks the feasibility of a chromosomeand mutates it if infeasible into a feasible chromosome. Finally, we state the constructive heuristic.

3.1. The ABLP approach for circular pieces

The adapted Best Local Position procedure, namely ABLP, constructs a “good” layout of anordered set of circular pieces. Like BLP, it positions circles in the upper left-most available positionbut takes advantage of the circularity of the pieces to explore more interesting positions. It is simplerand faster than BLP since a positioned piece cannot be further translated.

The best position of circle Pj with respect to circle Pi is one that is tangent to Pi. If (xi; yi)and (xj; yj) are, respectively, the coordinates of the center of Pi and Pj, then (xj; yj) must belongto the circle C of center (xi; yi) and radius ri + rj (for more details the reader can refer to Stoyanet al. [6]). The ABLP investigates a subset of this in+nite number of positions. Since the investigatedpositions are tangent to Pj, no further translation of Pj is necessary.

Given an ordered set of pieces, the ABLP proceeds as follows. It starts by placing the referencepoint of the +rst piece on the upper left-most position of the rectangular stock sheet R; i.e., on the(0; 0) coordinates point. It places the second piece on the upper left-most available position avoidingits overlap with the +rst piece and respecting the width constraint. In general, the constructiveapproach chooses for each piece Pj to be packed the best (upper left-most) position among a set ofpossible positions. This set is constructed as follows.

• First, each already placed piece Pi, with reference point (xi; yi), generates a set of four positions.The +rst and second positions are simultaneously adjacent to Pi and to the vertical line x = xi.The third and fourth positions are simultaneously adjacent to Pi and to the horizontal line y= yi.These four positions along with the corner of the rectangular stock sheet are illustrated in Fig. 1.These four positions belong to the circle C de+ned above. Several other particular positions havebeen investigated; in particular, the symmetrical positions of the aforementioned four basic ones.However, experimental testing has shown that they do not improve the solution. Since run timeis of interest, only these four positions are retained.


Fig. 1. Eligible positions of Pj with respect to Pi. Fig. 2. Possible positions of Pj with respect to Pi1 andPi2.

• Second, each pair of adjacent pieces Pi1 and Pi2 generates two possible positions. These positions,illustrated in Fig. 2, make Pj simultaneously adjacent to Pi1 and Pi2.

• Third and last, the resulting set is reduced by eliminating (i) duplicate positions, (ii) positions thatcause Pj to overlap an already placed piece, and (iii) positions that violate the width constraint.Evidently, the resulting set di4ers according to the piece Pj to be packed.

Our constructive approach places pieces in holes generated by already placed ones without rec-ognizing them as holes; that is, without any geometric computation of the contour of the hole.Furthermore, since the positions are adjacent to already placed pieces, no horizontal or verticalpacking is needed.

3.2. Illustration of the ABLP approach

To illustrate how ABLP proceeds, we consider the +ve piece example of Fig. 3(g), where Pi,i = 1; : : : ; 5, denotes the ith piece. The +ve pieces are considered in their numerical order:

1. The piece P1 is placed on the upper left most position of the stock sheet; i.e., on the (0; 0)coordinates point as shown in Fig. 3(a).

2. Let pij be the jth position generated by Pi. Fig. 3(b) highlights the four potential positions of P2

generated by P1 along with the +fth trivial position (0; 0). Since p11, p14; and p15 are infeasible,the number of potential positions is reduced to two. It turns out that p12 is the best local positionfor P2.

3. Each of P1 and P2 contributes four positions to the set of possible positions of P3. The pair (P1; P2)generates two additional positions for P3, where P1; P2 and P3 are simultaneously adjacent. The(0; 0) position is again annexed to the set of candidate positions. These last three positions arehighlighted in Fig. 3(c). Since it is feasible, the (0; 0) position is preferred as Fig. 3(d) illustrates.

4. P4 cannot +t into R if placed below P2 as Fig. 3(e) exhibits. It follows that the best position ofP4 is that adjacent to P1 and P2, as illustrated by Fig. 3(f). On the other hand, P5 +ts well ontop of P2 as displayed in Fig. 3(g).


Fig. 3. Illustration of the constructive approach on a small example.

3.3. A constructive approach for the CC problem

To be used as a constructive approach for the CC problem, the ABLP approach needs to bemodi+ed as follows. Let P be an ordered set of the n types of pieces, where each piece of type i isduplicated di times. Let S = P and U = ∅. The constructive approach positions the ordered pieces,as ABLP does, as long as the resulting length is less than L, the length of the stock rectangle.


Fig. 4. The main steps of the constructive approach (CA).

If positioning a piece of type i violates the length constraint, this piece and any remaining piecesof type i are removed from S and placed into U , the list of pieces whose placement violates thelength constraint. In this way, the constructive approach yields a feasible solution to the CC problem.A general outline of this constructive approach, called CA, is given in Fig. 4.

The constructive approach, CA, can be used both as a feasibility check operator for the geneticalgorithm-based heuristic and as an approximate algorithm for the CC problem.

3.4. Ordering and feasibility

The CA is used to test the feasibility of every chromosome. A chromosome is assimilated toan ordering P of the set of pieces. Using this ordered set, the constructive approach produces twosubsets of P: (a) a subset S of pieces that +t into R, and (b) a subset U of pieces that cannot+t R. The resulting set S ∪ U is a mutant of the chromosome P. In this way, the constructiveapproach produces feasible solutions to the CC problem. It acts as a mutant operator of the geneticalgorithm-based heuristic (a mutant that swaps infeasible chromosomes into feasible ones).

3.5. A constructive heuristic

To use the CA as an approximate algorithm, we need to identify a good ordering of the pieces.Experimental testing shows that placing the piece types in a nonincreasing order of the ratio ci=riyields good results. In case of equality, the piece type having the highest ci value should be ordered+rst. This favors smaller pieces with higher return.


4. A genetic algorithm-based heuristic approach

The genetic algorithm (GA) is an “intelligent” probabilistic search that can be easily applied toa variety of combinatorial optimization problems [15–17]. It is based on the evolutionary processof biological organisms in nature [18]. During the course of evolution, natural populations evolveaccording to the principles of natural selection. Individuals which are best adapted to the environmenthave a better chance of reproducing; the least +t are eliminated.

The GA simulates natural evolution of species by generating an initial population of individualsand by applying genetic operators in each reproduction. In optimization terms, each individual of thepopulation is represented by a chromosome or a possible solution to a given problem. The +tness ofeach individual is evaluated with respect to a given objective function. Highly +t individuals representsolutions of the problem which are reproduced by exchanging some information, in a crossoverprocedure, with other solutions. In this way, we produce children that inherit some characteristicsfrom the parents. After crossover, the mutation is applied by altering some genes in the chromosome.The o4spring can either replace the whole population (generational method) or replace less +tindividuals (incremental method). This evaluation–selection–reproduction cycle is repeated until asatisfactory solution of the implemented problem is reached. A general outline of the standard GAis provided in Fig. 5.

The performance of the GA depends mainly on the adopted solution con+guration and the asso-ciated +tness function, on the parent selection procedure, on the genetic operators (crossover andmutation), and on the replacement method of populations.

4.1. The solution con9guration and the 9tness function

Each individual of the population is represented by a chromosome P. A chromosome P is theconcatenation of (i) a feasible solution S to the CC problem, and (ii) a set U =P \ S of items thatcannot be positioned in R. An individual is represented by a chromosome.

A chromosome is an ordered structure of ng =∑n

i=1 di nonnegative integer numbers, where ngis the total number of circular pieces. Each nonnegative integer number, or gene, identi+es a piece.The +rst |S| genes constitute a feasible solution to the CC problem, where |S| denotes the numberof items in the set S.

Fig. 5. The standard genetic algorithm.


Fig. 6. Solution con+guration.

For instance, if d1 = d2 = 1 and d3 = d4 = 4, then |P| = d1 + d2 + d3 + d4 = 10: In this case,each chromosome has ten genes with one gene of piece type 1, a second gene of piece type2, another four genes of piece type 3, and the last four genes of piece type 4. For example,P = (P1; P2; P3; P3; P3; P4; P4; P3; P4; P4) is a valid chromosome. This chromosome, as illustrated inFig. 6, is the concatenation of the set S = (P1; P2; P3; P3; P3; P4; P4) of placed pieces into R and ofthe set U = (P3; P4; P4) of pieces that do not +t into R.

Note that the order of a gene in the chromosome is the order in which it is considered bythe constructive phase (see Section 3). Each chromosome corresponds to a unique layout. Sincethe objective of the CC problem is to maximize the usage of the rectangle R, the +tness of achromosome is simply the value

∑i∈S ci it generates. In our speci+c case, this +tness function

reduces to∑

i∈S �r2i .

4.2. Parent selection procedure

Parent selection is the task of assigning reproductive opportunities to each individual in the pop-ulation based on its relative +tness. Frequently, the selection procedure uses one of the followingtransformations: the tournament selection, the proportionate selection and the 9tness scaling. In thiscase, each of the best +t parents are chosen successively to be the +rst parent. The second parent ischosen randomly among the best-+t parents. This selection procedure can be eLciently implemented.In addition, it guarantees to each +t parent a chance to reproduce at least once while allowing acertain diversity of choice. It, therefore, promotes the choice of the best, while minimizing the risksof duplication and stagnation in local minima.

4.3. Crossover operators

The crossover operator [17,19] combines bits from two +t parents to create two new childrenstrings. Theoretically, +t parents have a higher chance of producing +t children. These childrenrepresent new and promising areas of the search space. In our study, we apply a variation of theOX Davis two point crossover, used as follows:

1. Two parents, Parent1 and Parent2, yield two children, Child1 and Child2, with each Childi inher-iting a subsequence [j; : : : ; k] of its genes from its respective Parenti, i=1; 2; with 16 j6 k6 ng.

2. The genes of Parent2 are used in their order of appearance to successively +ll the empty genesof Child1.


Fig. 7. Crossover of Parents 1 and 2 produces Children 1 and 2.

3. If a gene is already in Child1, it is rejected, else it is positioned in the +rst empty gene of Child1.4. This iterative process (steps 1–3) stops when all the ng genes of Child1 are +lled.

Inversely, the empty genes of Child2 are +lled with the genes of Parent1 in their order of appearance.The integer numbers j and k delimiting the subsequence being inherited are chosen randomly fromthe discrete Uniform (1; ng). When j= 1 or k =ng, the two point crossover reduces to a single pointcrossover.

Fig. 7 illustrates the two point crossover of parents 1 and 2 on genes 2–5. In this case, eachChildi inherits from its Parenti the substrings of genes [2; : : : ; 5], i = 1; 2. It is completed with themissing genes in their order of appearance in the mating parent.

4.4. Mutation operators

Two types of mutation operators are used. The +rst type is applied to every newly created child,be it a result of crossover or mutation. It swaps the child into a feasible mutant. For the exampleof Fig. 6, it replaces the infeasible chromosome (P1; P2; P3; P3; P3; P3; P4; P4; P4; P4) by the feasiblechromosome (P1; P2; P3; P3; P3; P4; P4) ∪ (P3; P4; P4).

The second type of mutation expands the search space by reintroducing valuable genetic informa-tion lost because of premature convergence. This second type of mutation is applied systematicallyto feasible mutants. It consists in swapping two subsequences of genes, or inverting the order of asubsequence of genes.

4.5. Replacement population method

Once constructed through the GA operators, new children solutions replace “less-+t” membersof the population. The average +tness of the population improves since the children solutions havebetter +tnesses than those solutions being replaced. This method, called “incremental replacement,”guarantees that the best solutions are always in the population, and the newly created ones areimmediately available for selection and reproduction.


We choose the best m +t individuals among a population of 5m distinct individuals: m parents,2m children from crossover, and 2m chromosomes from the second type of mutation of the 2mchildren. Our computational experiments show that our GA implementation using these parameterswith a population of 20 individuals produces satisfactory solutions for the CC problem.

The initial population consists of nonidentical items. The +rst individual is obtained using theconstructive heuristic. The remaining m−1 individuals are randomly generated. These m individualsare subject to the second type of mutation discussed above (we recall that any created individual issystematically subject to the +rst type of mutation). The best m chromosomes among these distinct3m individuals constitute the initial population.

5. Computational results

The purpose of this section is twofold: (i) to evaluate the constructive heuristic (CH) and thegenetic algorithm-based heuristic (GA-BH), and (ii) to assess how critical the nonincreasing ci=riordering is.

This section is organized as follows. First, we evaluate the performance of both CH and GA-BH.For a set of six problems extracted from the literature, we compare the results obtained by bothheuristics to the optimal solution, and to the results of a simulated annealing-based heuristics. Second,we assess the importance of the nonincreasing ci=ri ordering on the performance of GA-BH and showhow critical this ordering is. Finally, we study the degree of improvement provided by GA-BH overCH. Based on the above analyses, we provide few recommendations for solving the constrainedcircular cutting problem.

CH and GA-BH are coded in Fortran, and run on a Pentium III, 733 MHz and 128 MB ofRAM.

5.1. Performance of CH and GA-BH

To evaluate the performance of CH and GA-BH, we use the six test problems 1 of Stoyan andYaskov [14]. The optimal solution for each of these six problems, referred to as SY1; : : : ; SY6 inTable 1, is known. For each problem, we report (L;W ), the dimensions of the initial stock rectangleR; and n the number of circles to be cut.

The CH and the GA-BH solutions for SY1; : : : ; SY6 are reported in Table 2. The results of CHare provided in columns 3–5. Column 3 contains the usage (denoted Usage). Column 4 shows thegap (denoted Gap) between the usage of the CH yielded solution and the optimal usage, denotedusage∗: The gap is equal to 100 × usage−usage∗

usage∗ . Column 5 displays the CH run time (denoted T andmeasured in seconds). Columns 6, . . . ,11 summarize the results of three di4erent runs of GA-BH.Columns 6, 8 and 10 provide the percent usage for each of the three runs while columns 7, 9 and11 tally the corresponding gap from the optimal solution.

1 We have made these instances publicly available from ftp://panoramix.univ-paris1.fr/pub/CERMSEM/hifi/OR-Benchmark.html, hoping to aid further development of exact and approximate algorithms for the CC problem.

ftp://panoramix.univ-paris1.fr/pub/CERMSEM/hifi/OR-Benchmark.html



Table 1Test problem details

Inst L W n

SY1 17.491 9.5 30SY2 14.895 8.5 20SY3 14.930 9 25SY4 24.355 11 35SY5 38.047 15 100SY6 38.647 19 100

Table 2Performance of CH and GA-BH on constrained circular cutting problems

Inst The constructive heuristic Three trials of GA-BH

Usage Gap T (s) Usage Gap Usage Gap Usage Gap

SY1 79.582 4.25 ¡1 80.605 3.02 80.605 3.02 80.960 2.59SY2 77.535 4.98 ¡ 1 78.916 3.29 79.846 2.15 79.474 2.60SY3 79.756 2.62 ¡ 1 81.898 0.00 81.898 0.00 81.164 0.90SY4 80.307 1.70 ¡ 1 80.307 1.70 80.307 1.70 80.549 1.41SY5 82.220 0.00 13 — 0.00 — 0.00 — 0.00SY6 82.042 0.24 14 82.042 0.24 82.243 0.00 82.243 0.00

Table 3Summary results of GA-BH

Inst Average

Usage Ratio R.T. (s) T.B.S. (s)

SY1 80.723 2.88 100 25SY2 79.412 2.68 35 9SY3 81.653 0.30 85 20SY4 80.388 1.60 74 15SY5 82.220 0.00 0 0SY6 82.176 0.08 1388 287

Table 3 summarizes the results obtained by GA-BH. For each instance, we report the averagepercentage usage (denoted Usage), the average gap between the obtained solution and the optimalone (denoted Ratio), the average run time (R.T. measured in seconds) and the average time it takesGA-BH to +rst reach the +nal best solution (T.B.S.).

From Tables 2 and 3, we observe that:

1. CH gives good quality results. On average, it is 2.30% from the optimum, with a worst caseof 4.98%. It yields excellent solutions when the number of pieces is very large. For instance, it


Fig. 8. The Stoyan and Yaskov’s example (SY5): an optimal solution of usage 82.220 reached by CH.

gets the optimal solution for problem instance SY5 (cf. Fig. 8). In addition, CH is very fast. Itsaverage run time is less than 1 s for small problems and less than 14 s for the larger problemswith 100 pieces. CH is a valuable starting point for more complex procedures. It is a particularlygood heuristic for large size problems.

2. GA-BH, as observed in Table 2, improves the results of CH, but within a slightly larger com-putational time. Indeed, it reduces the average deviation from optimum to 1.26%. The observedpercent deviation varies from 0.00 to 3.29.We observe from Table 3 that the average T.B.S., time to +nd the best solution, is under +ve minfor the largest problem SY6, which is reasonable considering the high quality of the results andthe proximity of the constructive heuristic solution to the exact one. GA-BH reached the optimalsolution in several occasions.

3. For all treated problems, we notice that GA-BH converges generally rather rapidly. Indeed, allaverage T.B.S. are reasonably small. As expected, all run times are less than their correspondingT.B.S. The genetic algorithm is stopped if +ve complete generations bring no improvement of thecurrent best solution. It follows that we can reduce the run time by altering this stopping criterionwithout a4ecting the solution quality.

The seemingly high computational time for SY6 can be explained as follows. First, all computationsare double precisions. Second, the number of possible positions gets larger as the number of piecesincreases. The number of dominated positions is relatively small since these positions are not insidealready placed pieces. Consequently, we have to preprocess each position to check if it would yieldan overlap with already placed pieces. Third, the sizes of the pieces are very diverse, which impliesthat smaller pieces have a large number of feasible positions. Fourth, all pieces are distinct.


Fig. 9. The CH solution for the Stoyan and Yaskov’s example (SY1) with an approximate usage of 79.582% correspondingto a 4.25% deviation from the optimum.

Fig. 10. A GA-BH approximate solution of the Stoyan and Yaskov’s example (SY1) (third trial of Table 2) with an80.960% usage corresponding to a 2.59% deviation from the optimum.

Figs. 9 and 10 illustrate the CH and the GA-BH improved solutions obtained for SY1 whileFigs. 11 and 12 illustrate the CH solution and the optimal solution reached by GA-BH or Sy6.

Fig. 9 shows the structure of the solution of the instance SY1 produced by CH. This solutioncorresponds to a 4.25% gap from optimum. Finding this solution requires less than a second. Thissolution is a good starting solution to GA-BH. GA-BH reduces in less than 100 s the gap to 2.59%as illustrated by Fig. 10.


Fig. 11. The CH solution of the Stoyan and Yaskov’s example (SY6) with an 82.042% approximate usage correspondingto a 0.24% deviation from optimum.

Fig. 12. A GA-BH optimal solution of the Stoyan and Yaskov’s example (SY6) (second trial of Table 2) of usage82.243%.

Similarly, Fig. 11 shows the structure of the CH solution of the instance SY6. This solutioncorresponds to a 0.24% gap. Finding this solution requires 14 s. This is obviously a high qualitysolution that can be used if run time is an issue. GA-BH converges to the optimal solution, displayedin Fig. 12, in less than 5 min of computational time.


Table 4Comparison of the performance of both CH and GA-BH to that of the simulated annealing-based approach of Hi+ et al.[13]

Inst Average usage Improvement (%)

SA CH GA-BH CH GA-BH

SY1 74.357 79.582 80.273 7.03 7.96SY2 69.908 77.535 79.412 10.91 13.60SY3 65.385 79.756 81.653 21.98 24.88SY4 71.796 80.307 80.388 11.85 11.97SY5 80.208 82.220 82.220 2.51 2.51SY6 79.453 82.042 82.176 3.26 3.43

Fig. 13. Improvements brought by CH and GA-BH to the results of the Simulated Annealing approach of [13].

Since both proposed approaches are heuristics, we compare their yielded results to those reachedby the simulated annealing-based heuristic (SA) developed in [13]. The proposed CH consistentlyyields better results than SA. The percent improvement, displayed in Table 4, for problem instancesSY1; : : : ; SY6 is on average 9.59%, and reaches 21.98% for SY3. This improvement gets even largerwhen GA-BH is used. It becomes 10.26% on average, reaching 24.88% for SY3 with GA-BH. Fig. 13shows the di4erent percentage improvements realized by CH and GA-BH over SA for all consideredproblems.


Table 5Data for additional test problems: these problems are obtained by combining Stoyan and Yaskov’s examples [14]

Inst L W n

SY12 17.491 9.5 50SY13 17.491 9.5 55SY14 24.355 11 65SY23 14.930 9 45SY24 24.355 11 55SY34 24.355 11 60SY56 38.647 19 200SY123 17.491 9.5 75SY124 24.355 11 85SY134 24.355 11 90SY234 24.355 11 80SY1234 24.355 11 110

5.2. Impact of the nonincreasing ci=ri ordering

To assess the impact of the nonincreasing ci=ri ordering on GA-BH, we use in addition to theoriginal six problems (SY1; : : : ; SY6) whose optimal solutions are known, a set of 12 problems 2

whose optimal solutions are unknown. Table 5 summarizes the data for each of these additional 12problems.

Each of the 18 problems is run twice: once with a totally random initial population, and once withan initial population including the solution yielded by the heuristic (i.e., with the circles ordered ina nonincreasing order of the ratio ci=ri). Table 6 summarizes the results of both runs as well asthe percent improvement due to the use of this particular ordering. The results of the +rst runare tallied in the second column labeled GA-BHr while those of the second run are tallied inthe third column labeled GA-BH. The last column displays the percent improvement, computed as100 × (GA-BH−GA-BHr)

GA-BH .Table 6 shows that including the CH solution in the initial population improves the performance

of GA-BH. This improvement equals 2.36% on average, and reaches 4.69% for SY56. Hence, theinitial ordering of the pieces according to the nonincreasing ratio ci=ri is critical for the performanceof the genetic algorithm-based heuristic.

To further assess the importance of this ordering, we apply the ABLP approach to the nonin-creasing ci=ri order and to a hundred random permutations of the circles. Table 7 displays the CHsolution and the average usage of the solutions yielded by the ABLP approach when applied tothe random orders. The CH solution is 6:07% better on average with the average percent deviationreaching 11:64% for SY3.

2 Publicly available from ftp://panoramix.univ-paris1.fr/pub/CERMSEM/hifi/OR-Benchmark.html



Table 6Impact of the nonincreasing ci=ri ordering on the performance of GA-BH

Inst Average usage

GA-BHr GA-BH Improvement (%)

SY1 78.241 80.960 3.36SY2 77.891 79.846 2.45SY3 78.151 81.898 4.58SY4 79.138 80.549 1.75SY5 80.498 82.220 2.09SY6 80.044 82.243 2.67SY12 81.330 81.711 0.47SY13 81.451 82.908 1.76SY14 81.709 83.154 1.74SY23 80.998 82.783 2.16SY24 81.321 82.380 1.29SY34 81.952 83.627 2.00SY56 81.798 85.821 4.69SY123 81.257 82.953 2.05SY124 81.145 83.290 2.58SY134 82.194 83.392 1.44SY234 81.074 83.766 3.12SY1234 82.258 84.060 2.14

Table 7Importance of nonincreasing ci=ri order.

Inst CH Random Improvement (%)

SY1 79.582 75.482 5.15SY2 77.535 70.205 9.45SY3 79.756 70.475 11.64SY4 80.307 74.630 7.07SY5 82.220 78.829 4.12SY6 82.042 76.832 6.35SY12 80.615 77.670 3.65SY13 82.578 78.481 4.96SY14 82.902 77.369 6.67SY23 82.745 76.310 7.78SY24 81.560 77.463 5.02SY34 82.952 75.458 9.03SY56 85.024 81.022 4.71SY123 81.789 78.334 4.22SY124 82.492 78.786 4.49SY134 83.214 80.474 3.29SY234 83.728 76.684 8.41SY1234 83.602 80.874 3.26


Table 8Improvements brought up by GA-BH

Inst Usage Average usageCH GA-BH Improvement (%)

SY12 80.615 81.711 1.34SY13 82.578 82.908 0.40SY14 82.902 83.154 0.30SY23 82.745 82.783 0.05SY24 81.560 82.380 1.00SY34 82.952 83.627 0.81SY56 85.024 85.821 0.03SY123 81.789 82.953 1.40SY124 82.492 83.290 0.96SY134 83.214 83.392 0.21SY234 83.728 83.766 0.45SY1234 83.602 84.060 0.55

5.3. Usefulness of GA-BH

The analysis of Tables 2 and 3 could suggest that the need for GA-BH disappears as the problemgets larger. To investigate this matter, we study the improvements brought up GA-BH to the set of12 additional examples. Table 8 displays for each problem the CH solution, the GA-BH solution,and the corresponding percent improvement brought up by GA-BH. The improvements equal onaverage 0.875% and vary from 0:00 to 2:615. If runtime is not an issue, GA-BH should be used toimprove the solution yielded by CH.

Based upon the above observations, we conclude that:

1. if fast solutions are needed, CH should be used;2. if high quality solutions are preferred to speed, GA-BH should be chosen;3. if intermediate solutions within reasonable computing time are needed, GA-BH can be used with

(i) a slight modi+cation of the stopping criterion, and (ii) a reduction of some feasible pointsusing hill-climbing strategies (see Morabito et al. [20]).

6. Conclusion

We have solved the constrained circular cutting problem using two approximate algorithms: aconstructive procedure and a genetic algorithm-based heuristic. Both approaches search for a goodordering of the pieces and use a new constructive approach to +nd the best layout of this orderedset. The constructive approach is also used as a mutation operator that modi+es any infeasiblesolution to a feasible one. Extensive testing of di4erent problem instances taken from the literatureshows that the constructive heuristic yields good solutions within a very short computing time.The genetic algorithm-based heuristic yields high quality solutions, reaching the optimal in severalinstances, within a reasonable run time. The proposed heuristic can be easily implemented in a


parallel environment. The use of parallel approaches will increase the size of the problems that canbe solved, and should converge rapidly.

Acknowledgements

The authors thank two anonymous referees for their helpful comments and suggestions whichimproved the presentation of this paper. Similarly, the authors thank Professor V. Zissimopoulos forkindly providing the data sets.

References

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Mhand Hi% is Professor of Computer Science in the University of Picardie Jules Verne, Amiens, France. He got hisB.S. in Computer Engineering from USTHB, Algiers, Algeria. He got his M.S. in Modelling and Mathematical Methods inEconomics from the University of Paris 1 Pantheon-Sorbonne. He got his Ph.D. in Computer Science from the Universityof Versailles St Quentin en Yvelines. His research interest is NP Hard combinatorial optimisation (sequential and parallelapproaches) applied to cutting, packing, knapsacking and other OR problems.

Rym M’Hallah is associate professor in the Department of Quantitative Methods and Information Technologies, InstitutSupEerieur de Gestion, Sousse, Tunisia. She has a B.S. in Industrial Engineering, an M.S. and a Ph.D. in IndustrialEngineering and Operations Research, all from Penn State University. Previously, she was employed by the RegionalInstitute for Research on Informatics and Telecommunication, Tunis, Tunisia. Her research interests are cutting, packing,and scheduling.

approximate algorithms for constrained circular cutting problems

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