a sequential heuristic procedure for one-dimensional cutting

Theory and Methodology

A sequential heuristic procedure for one-dimensional cutting

Miro Gradi�sar a,b,*, Miroljub Kljaji�c a, Gortan Resinovi�c b, Jo�ze Jesenko a

a Faculty of Organizational Sciences, University of Maribor, 4000 Kranj, Kidri�ceva ulica 55 A, Sloveniab Faculty of Economics, University of Ljubljana, 1000 Ljubljana, Kardeljeva plo�s�cad 17, Slovenia

Received 10 September 1997; accepted 23 March 1998

Abstract

The article examines the Sequential Heuristic Procedure (SHP) for optimising one-dimensional stock cutting when

all stock lengths are di�erent. In order to solve a bicriterial multidimensional knapsack problem with side constraints a

lexicographic approach is applied. An item-oriented solution was found through a combination of approximations and

heuristics that minimize the in¯uence of ending conditions leading to almost optimal solutions. The computer program

CUT was developed, based on the proposed algorithm. Two sample problems are presented and solved. A statistical

analysis of parameters that a�ect material utilisation was also made. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Cutting; Heuristics; Optimisation

1. Introduction

The problem of one-dimensional stock cuttingoccurs in many industrial processes [1±3] andduring the past few years it has attracted an in-creasing attention of researchers from all over theworld [4]. Yet attention has been mostly focusedon the solution to the problem in cases with thestock of the same length or with a few di�erentstandard lengths. However, when the stock lengthsare all di�erent, generally acceptable solutions tothis problem have not appeared in the literature sofar. The purpose of this article is to propose a

solution that will generalise and improve our ear-lier solution [5] where a practical problem in theclothing industry is considered.

Most standard problems related to one-di-mensional stock cutting are known to be NP-complete. However, in many cases the problemscan be modelled by means of mathematical pro-gramming and a solution can be found by usingapproximate methods and heuristics. Our objec-tive is to design a plan of one-dimensional cuttingof a certain number of pieces of di�erent lengths(stock lengths), into a large number of short pieces(order lengths), which will minimize the overalltrim loss considering di�erent conditions that mayappear in practice.

Using Dycko�'s typology [6] our integer cuttingstock problem can be described, in the cases with

European Journal of Operational Research 114 (1999) 557±568

* Corresponding author. E-mail: [email protected]

mb.si.

0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 4 0 - 4

enough required material available, as 1/V/D/M,where 1 refers to one-dimensional problem, Vmeans that all items must be produced from a se-lection of large objects, D means that all largeobjects are di�erent and M indicates many smallitems of many di�erent dimensions. In other caseswe can describe it as 1/B/D/M, where B means alllarge objects and a selection of small items.

Dycko� classi®es the solution of integer cuttingstock problems into two groups: item-oriented andpattern-oriented approach. Item-oriented approachis characterised by individual treatment of everyitem to be cut. In the pattern-oriented approach, at®rst, order lengths are combined into cutting pat-terns, for which ± in a succeeding step ± the fre-quencies are determined that are necessary tosatisfy the demands. The literature abounds inpattern-oriented solutions based on a hybrid al-gorithm, which was developed by Gilmore andGomory [7,8]. However, a pattern-oriented ap-proach is possible only when the stock is of thesame length or of the several standard lengths. Oursolution is an item-oriented one because all stocklengths are di�erent and frequencies cannot bedetermined.

Item-oriented solution can be based on exactmethods or on approximation algorithms [6]. Forthe 1/V/D/M or 1/B/D/M type of problem there isno exact method which could ®nd an optimal item-oriented solution within reasonable time limits.The exact algorithm that would provide a solutionwithout the unnecessary trim loss would be uselessdue to high time complexity. Therefore, a solutionin the form of approximation algorithm has to befound.

In the literature the authors have not comeacross an approximation algorithm that would bedirectly applicable to the described problem.Similar problems are solved by Sequential Heu-ristic Procedure (SHP). A similar kind of problemis the classical ``bin packing problem'', which canbe solved for example by the ``First Fit De-creasing'' (FFD) or ``Best Fit Decreasing'' (BFD)SHP [9]. However, the basic feature of this typeof problem is that all stock lengths are the same.Gilmore and Gomory's knapsack method forpattern generation can also not be used directly.Their pattern-oriented method is based on as-

sumption that there are only few standard stocklengths and the optimal combinations for each ofthem are independent. We have to adapt theirmethod to the case when all stock lengths aredi�erent. An input data for the next step are de-pendent on results of the previous step and theoptimal combinations cannot be calculated si-multaneously. An extensive review of the relevantliterature up to 1992 is given in [10].

The primary advantage of SHP is its ability tocontrol factors other than trim loss and to eliminaterounding problems [11,12] by working only withinteger values. The major disadvantage of an SHPis that it may generate a solution that has a greatlyincreased trim loss because of the so-called endingconditions. We have developed such an SHP, whichminimizes the in¯uence of ending conditions.

The objective of the optimisation is materialcost reduction with the plan of stock cutting op-timisation not being too complex and the cost ofcutting as low as possible. So the solution o�ers acompromise between the plan complexity and trimloss by setting parameter Y. Y determines thenumber of di�erent order lengths that will be cutout of one stock length. The statistical analysisgiven in Section 4 reveals that the value of Yshould be between 1 and 4. For Y� 4, the trim lossbecomes so low, that any further reduction wouldprobably not outweigh the extra e�orts of cutting.

The article describes the cutting problem andsolution development in the form of a computerprogram. Two examples of a practical implemen-tation of the program are given. At the end thein¯uence of parameters a�ecting utilisation ofstock is analysed.

2. Problem de®nition and a formal model

For every customer order a certain number ofstock lengths is available. In general all stocklengths are di�erent. We consider the lengths asintegers. If they are not originally integers we as-sume that it is always possible to multiply themwith a factor and transform them to integers. It isnecessary to cut a certain number of order lengthsinto required number of pieces. The followingnotation is used:

558 M. Gradi�sar et al. / European Journal of Operational Research 114 (1999) 557±568

Two cases are possible:Case 1: the order can be ful®lled as the abun-

dance of material is in stock.

For the above model the following functionsare used:

zj �0 if xij � 0 8i;1 otherwise;

�to indicate whether stock length j is used in thecutting plan;

yij �0 if xij � 0;

1 otherwise;

�to indicate whether order length i is cut from roll j;

uj �1 if zj � 1 ^ dj > max si;

0 otherwise;

�to indicate whether the trim loss relating to stocklength j is greater than the longest order length;

tj �dj if zj � 1 ^ dj6UB;

0 otherwise:

�tj indicates the extent of the trim loss relating tostock length j.

Case 2: the order cannot be ful®lled entirely dueto shortage of material in stock.

In case 2 de®nition of the problem depends onthe way in which the uncut pieces are distributedby order lengths. The possibilities that cover mostsituations in practice are explained as follows:

2.1. Distribution of uncut pieces by orderlengths is not important.

2.2. Distribution of uncut pieces by orderlengths is important. This means that some orderlengths are more important and therefore we try toful®l the order for those lengths ®rst. Besides trimloss also distribution of uncut order lengths has tobe considered. This leads to a bicriterial problem.As both criteria, the trim loss and distributioncould be in con¯ict, a decision on priority is to bemade. There are two possibilities:

2.2.1. Trim loss is more important and distri-bution is approximate. We assume that:

(a) uncut pieces belong to all order lengths andare evenly distributed,(b) uncut pieces belong to the shortest orderlengths,(c) uncut pieces belong to the longest orderlengths.

si order lengths; i� 1,. . .,n,bi required number of pieces of order length si,dj stock lengths; j� 1,. . .,m,xij number of pieces of order length si having

been cut from stock length j.

(1) minPm

j�1tj (minimize trim loss which issmaller than UB (upper bound for thetrim loss))

(2) s.t.Pn

i�1si � xij � dj � dj 8j (knapsackconstraints),

(3)Pm

j�1xij � bi 8i (demand con-straints ± the numbers of pieces are all®xed),

(4)Pm

j�1uj6 1 (maximum number ofresidual lengths which are larger thanmax si),

(5)Pn

i�1yij6 Y 6 n 8j (maximumnumber of di�erent order lengths fora stock length),

(6) UB6max si, xij P 0, integer "i, j,tj P 0, "j, dj P 0, "j.

(1) minPn

i�1Di � si (minimize sum of uncutorder lengths)

(2) s.t. same as in case 1,(3)

Pmj�1xij � bi ÿ Di 8i (demand

constraints),(4) same as (5) in case 1,(5) Di P 0, integer "i.

(1) minPn

i�1Di (minimize number ofuncut order lengths and satisfycondition (a) or (b) or (c)),

(2) minPm

j�1dj (minimize trim losswhich is smaller than max si),

(3) s.t. same as (2) in case 1,(4) same as (3) in case 2.1,

M. Gradi�sar et al. / European Journal of Operational Research 114 (1999) 557±568 559

2.2.2. Distribution is more important, so thechoice of any a priori distribution is assumed. Inthis case the number of required pieces of a par-ticular order length is being reduced as long asenough material is available. After this a newcutting plan is made. The problem becomes one-criterial again and the minimum trim loss is de-termined, given a precise a priori distribution, withthe following condition:

Condition (6) determines exact distribution ofuncut pieces. Let us consider di�erent possibilitiesof the condition (6) for (a), (b), and (c) of 2.2.1:

(a) 06 �Di ÿ Dk�6 1; k � 1; . . . ; n 8i; k:Set si is divided into two subsets si1 and si2. Sub-set si1 consists of the lengths for which: Di > 0,while subset si2 consists of the lengths for which:Di� 0. The condition (6) for (b) and (c) is de-®ned as:(b) max si16 min si2,(c) min si1 P max si2.Unutilised stock length that is larger than some

UB could be used further and is not considered aswaste. The question is how to determine UB. Theanswer depends on the quantity of available stocklengths.

Let us see case 1 ®rst. If su�cient stock lengthsare available then there will be cutting plans with``no trim loss'' but ever growing stocks. To preventthis an additional condition (case 1, condition (4))has to be set: only one residual length may belonger than the longest order length. UB can be setarbitrarily between 0 and max si. The bigger UBthe greater the cutting problem. UB�min si isfound in practice [5].

In case 2, however, UB�max si. If, for exam-ple, UB is reduced to min si, this would lead to thefollowing problem: As the aim of the algorithm is

minimisation of the overall trim loss, this couldlead to unful®lled requirements for the longestorder lengths, even if the overall trim loss is smalland the aim is achieved according to the logic ofthe algorithm. The trim losses, which would belonger than UB but shorter than the longest orderlengths, could remain unutilised. Because of thissetting UB in the case 2 is not reasonable and isnot included in the model.

3. Solution development

The proposed algorithm was developed on astep by step basis. The number of basic stepsequals to the number of stock lengths necessary forful®llment of an order. At the beginning, all stocklengths belong to the set of unprocessed stocklengths. The set of processed stock lengths isempty. At each step, the set of unprocessed stocklengths is reduced by one and the set of processedstock length increases by one. Also, the number ofcut pieces of particular order lengths changes, aswell as the length of the processed stock length,which becomes equal to trim loss. Algorithm hasthe following steps:

Step 1: Select order lengths.Step 2: Select stock length and cut it with cho-sen order lengths.Step 3: If all stock lengths are not cut yet andthe requirements for order lengths are not ful-®lled, then go back to step 1, else stop.

The algorithm is further developed on the follow-ing assumptions:

It is easier to ®nd a good solution if:1. we can choose from the largest possible set ofpossible solutions.

The number of possible solutions is higher if:1.1. there are available as many as possibleorder lengths that are not yet cut to theend;1.2. the proportion between average stocklengths and average order lengths is aslarge as possible;1.3. the number of uncut order lengths is

(1) minPn

i�1Di (minimize number of uncutorder lengths at a determined a prioridistribution)

(2) s.t. same as in case 1,(3) same as in case 2.1,(4) same as (5) in case 1,(5) same as in case 2.1.

(5) same as in case 1,(6) same as (5) in case 2.1.


as high as possible. In the best possiblecase,

min bi ÿXm

j�1

xij

!� si

!P max �dj � �1ÿ zj��:

2. the solutions di�er from each other as muchas possible.

The di�erences between possible solutionsare greatest when:

2.1. the relation between the longest andthe shortest order length is as great as pos-sible;2.2. the relation between the longest andshortest stock length is as great as possi-ble.

The assumptions are statistically proven, whichis shown in the analysis described in Section 4.However, we cannot be quite sure that the as-sumptions are correct in each individual case.

Reduction in the in¯uence of ending conditionsrequires such an algorithm, that at the end of thecutting process it would be possible to choose fromthe largest possible set of solutions di�ering fromeach other as much as possible.

When there is enough material in stock, ormore than needed (case 1) all the assumptionsabove can be taken into consideration more easily.In this case is the last cut stock length mostly onlypartially utilised and there is no additional trimloss. The greater the number of stock lengths leftunutilised after the order requirements have beenful®lled, the greater the possibility to achieve truecutting optimum.

The crucial question is the selection of orderlengths. The following two procedures can beconsidered:

(a) In accordance with 1.1 and 2.1 order lengthswith the greatest number of uncut pieces is chosen.We choose ®rst Y elements from the set of notcompletely cut order lengths, sorted by decreasingnumber of uncut pieces (Di). In this way a greatervariety of order lengths will remain at the end.However, this is not consistent with the assump-tions 1.2 and 1.3, as at the end of the cuttingprocess also longer order lengths could remain,which would mean a smaller number of pieces to

be cut from the rest of the stock. This would beappropriate only when the di�erences between thelongest and shortest order lengths are small.

(b) According to assumptions 1.2 and 1.3 thelongest order lengths, or the ®rst Y elements in theset of not yet entirely cut order lengths sorted bydecreasing lengths, are chosen. All this disagreeswith assumption 1.1, as only the shortest orderlengths remain, and with 2.1 as, because of this,the di�erence in their length is relatively small.Therefore such a choice would be appropriate onlywhen the di�erences between the longest andshortest order lengths are great.

As assumptions 1.1±1.3, 2.1 and also 2.2 cannotbe met at the same time a compromise should beconsidered. Assumption 2.2 is considered as lessimportant and is ignored. There are a few possiblesolutions. A particular number of order lengthscan be chosen in one way and the rest in the other.This number is marked f, where: 06 f6Y. Orderlengths can also be chosen on the basis of the totallength of not yet cut pieces. There are three pos-sibilities:1. f order lengths are chosen using procedure (a),

while Y ) f are chosen using procedure (b);f� 0,. . .,Y.

2. f order lengths are chosen using procedure (b),and Y ) f using procedure (a); f� 1,. . .,Y ) 1.

3. Order lengths can be chosen from the arrange-ment made according to the decreasing value ofsi á Di.

There are 2 á Y + 1 possible solution variants. Thecutting algorithm is designed so that all variantscompete.

The issue of stock length selection can betackled by calculating the solutions for all unpro-cessed stock lengths and selecting among them theone with the lowest trim loss. As it is possible thatmore stock lengths have the lowest trim loss, weapply assumption 1 and choose the shortest stocklength. This can be achieved with the initial ar-rangement of the stock lengths according to in-creasing lengths. By doing so, the longer stocklengths will be processed later. Sometimes only onebest solution exists. Even if there are more of themthey are more likely at longer stock lengths. In thiscase, at the beginning shorter stock lengths aremore preferred, which is achieved by increasing


parameter r. Parameter r is de®ned as a trim lossthat can be neglected. All the solutions with thetrim loss equal or smaller than r can be consideredas equally good. The algorithm for the optimisa-tion of stock length cutting is shown with a ¯ow-chart in Fig. 1.

If not enough material is available and a re-quired distribution of uncut pieces by orderlengths is to be considered, the algorithm (Fig. 1)applies to all distributions, yet it is not carried outcompletely. In the case 2.2.1 (a) only procedure (a)is carried out which means that f� 0 and k� 1. Asin every basic step only the lengths with the highestb are chosen, it can be expected that at the end ofthe procedure the shortage will be equally dis-tributed through all order lengths. In case 2.2.1 (b)only procedure (b) is carried out, which meansonly when f� 0 and k� 0. As in every basic stepthe longest order lengths are chosen it can be ex-pected that only the shortest order lengths shouldremain at the end. In case 2.2.1 (c) the algorithmshould be changed and order lengths sorted byincreasing lengths. However, this would be indiscrepancy with all four basic assumptions (1.1±1.3 and 2.1) and would cause a great trim loss.Therefore this case is considered in the same wayas 2.2.2, where the problems are transformed inthe form described in case 1.

The ¯owchart indicates that it is necessary tosolve a series of knapsack problems for each basicstep of algorithm. The dynamic programmingscheme of the procedure KNAPSACK in theprocedure CUT which selects and cuts the stocklength with the lowest trim loss Rmin according to®rst optimal combination of four (Y� 4) selectedorder lengths s1, s2, s3, and s4, can be summarizedas follows:

In the knapsack procedure a sequence of vec-tors (xj)Y is generated in a lexicographically de-creasing order to ®nd the ®rst optimalcombination. The int function converts a numericexpression to an integer, all digits to the right ofthe decimal place are ignored.

3.1. Complexity analysis

Step 3 takes

O�m � int�dj=s1��; �1�step 5 takes

O m �Xint�dj=s1�

i�0

int��dj ÿ s1 � i�=s2� !

; �2�

and steps 7±19 take

O m �Xint�dj=s1�

i�0

Xint��djÿs1�i�=s2�

k�0

int��dj ÿ s1 � iÿ s2 � k�=s3�!

�3�

time.In case s1� s2� s3� s and Nj� int(dj/s) step 3

takes O(m á Nj), step 5 takes O�m � �N2j � 3Nj��;

and steps 7±19 take O�m � �N3j � 6N 2

j � 11Nj��time.

Procedure KNAPSACK

0. initialize Rmin�maxint1. for j� 1 to m do

2. for i�min{b1, int(dj/s1)} to 0 step-1 do

3. D1 ¬ dj ) s1 á i4. for k�min{b2, int(D1/s2)} to 0 step-1 do

5. D2 ¬ D1 ) s2 á k6. for l�min{b3, int(D2/s3)} to 0 step-1 do

7. D3 ¬ D2 ) s3 á l8. t ¬ min{b4, int(D3/s4)}

9. R ¬ D3 ) s4 á t10. if R < Rmin

11. then

12. Rmin ¬ R13. jmin ¬ j14. x1j ¬ i, x2j ¬ k, x3j ¬ l, x4j ¬ t15. if R6 r16. then

17. stop18. endif

19. endif

20. endfor

21. endfor22. endfor

23. endfor


Fig. 1. Flowchart of the cutting algorithm.


Time complexity depends on m, si and dj. Theprocedure KNAPSACK itself depends on Y. ForY� 3 steps 6, 7, and 20 should be left out and timecomplexity can be calculated with expression (1)and (2). For Y� 2 steps 4±7, 20, and 21 should beleft out and time complexity can be calculated withexpression (1). For Y� 1 steps 2±7, 20±22 should beleft out and time complexity is O(m). The greatesttime complexity is for Y� 4 but for common values,for example m� 50 and Nj� 30, the algorithm isvery fast and a short response time can be expected.

4. Results

The proposed algorithm is written in FOR-TRAN programming language. The programconsists of 2500 lines of code. The data input andthe printout of the results are made in 4GL, as it ismost suitable for this purpose. The program canbe run on a personal computer. It was called CUT.CUT is intended for general use and is an im-proved and generalised version of program COLA[5]. COLA was designed for cutting rolls in

Fig. 2. Two examples of the CUT program printout.


clothing industry and is in use for more than 12years.

Creating a cutting plan for an extensive orderconsisting of 50 stock lengths and 1000 patternpieces takes less than 20 s on a personal computer(Pentium, 133 MHz). The time spent on creatingthe plan is negligible. The speed enables the usersto carry out a ``what-if'' analysis by changing theparameters of the order. For instance, a possible

question could refer as to what extent the solutionwould be di�erent if we change Y from 3 to 2 or rfrom 0 to 5.

Two examples of computational results withrandomly generated data are seen in Fig. 2. Thecustomer orders are rather small, containing only®ve di�erent order lengths and only four and ®vestock lengths. In the ®rst case we are short ofmaterial so only 120 out of 125 anticipated pieces

Fig. 2. (continued).


can be carried out. The di�erence is approximatelyequally distributed according to order lengths.Total trim loss is 2 cm, which makes 0.0046% oftotal length. The saving of the material is 7.23 m.In the second case there is no shortage of material.The devised cutting plan assumes a complete cut-ting of four stock lengths and partial cutting ofone stock length. The anticipated trim loss is 0 cm.The saving of the material in this order is 8.1 m. Inboth cases creating a cutting plan takes less than1 s.

The column ``the following order length'' (foll.o.l.) contains order length reference numberswhich are about to be cut from the stock lengthsgiven in the column ``stock length''. This item ofinformation is helpful for the worker at the cuttingmachine.

4.1. Parameters a�ecting utilisation of stock

With the program CUT basic assumptions 1and 2 in Section 3 are statistically tested. Let usde®ne two parameters p1, and p2 á p1, is the aver-age number of pieces cut out of each stock length.The initial number of possible solutions dependson p1 á p2 is the average di�erence between thelongest and the shortest order length. The initialdi�erences between possible solutions are deter-mined with parameters p2 and Y.

The in¯uence of p1, p2 and Y has been analysedby calculating a large number of random gener-

ated orders with di�erent values of parameters inaccordance with the sample case and thus ob-taining the average values of trim loss. The samplecase presents to the situation of cutting rolls inclothing industry so that the results could becompared with those obtained with the programCOLA. The sample case can be described as fol-lows:· order lengths, number of pieces and stock

lengths are generated by sampling an integerfrom the uniform distributions [300, 500], [10,50] and [9000, 15 000],

· Y� 4, r� 0,· material in excess is approximately 5%.

Let us consider ®rst the impact of parameter p1.If p1 is higher there is more room for manoeuvrewhen seeking the optimal combination, and theresult can be better. Very good results can beachieved if p1 is more than 30. Fig. 3 shows theconditions.

Fig. 4 shows dependence of the trim loss on p2

expressed as a percentage. It can be observed thatp2 between 30% and 40% can su�ce for the opti-mal utilisation of material.

A crucial parameter, which is set as a parameterof the program, is the number of di�erent orderlengths, cut out of a stock length. A higher Ymeans more possible combinations and, therefore,also lower trim loss. Analysis shows that in case ofa combination of two order lengths in a stocklength the average trim loss accounts for 0.1143%.In combinations of three the average trim loss is

Fig. 3. Dependence of the trim loss on the average number of pieces cut out of a stock length.


reduced to 0.0121%. In combinations of four theaverage trim loss is 0.0034%. It is noteworthy thateven when cutting only one order length out of astock length as in the case of the traditional pro-cedure without a cutting plan, implementing theCUT program can lead to the trim loss reductionsof up to 70% as shown in Fig. 5. This can beachieved by merely placing the stock lengths to becut into a sequential arrangement proposed in theprogram.

As we can see the values of all three parametersp1, p2 and Y of the sample case are great enough,so a small trim loss can be expected. The questionis how great is the trim loss and what is theprobability of an optimal solution when all threeparameters have favourable values. In our case the

values are: p1� 30, p2� 40% and Y� 4. We car-ried out an analysis of 200 randomly generatedsamples of the sample case. In 100 cases theshortage of material was about 5%, in other 100cases there was about 5% surplus.

We obtained the following results: in the ®rst100 cases the average trim loss was 0.0098%(0.1121% in the worst case). In the other 100 casesthe average trim loss was 0.0034% (0.0351% in theworst case). In all 200 cases the sum of all trimlosses was lower than the shortest order length.This means that we have certainly found an opti-mal solution. Comparison with the program CO-LA shows an about two times smaller trim loss.Two typical examples from these 200 cases areshown in Fig. 2.

Fig. 5. Dependence of trim loss on the number of di�erent order lengths cut out of the same stock length.

Fig. 4. Dependence of trim loss on di�erences in order lengths.


5. Conclusion

The paper analyses the problem of reducingtrim loss in one-dimensional stock cutting when allstock lengths are di�erent. The item-oriented so-lution in the form of an SHP through a combi-nation of approximation and heuristics is found.The algorithm is designed so that the conditionsand restrictions that could appear in practice aretaken into account. Such an algorithm is supposedto be applicable as generally as possible in a vari-ety of real life situations. On the basis of theproposed algorithm the computer program CUT isdeveloped. This program is an improved andgeneralised version of the program COLA, whichhas been used in clothing industry for more than12 years. The program is very fast. Its speed en-ables the users to carry out a ``what-if'' analysis bychanging the parameters. When all the parametersare similar to or greater than those in the samplecase, the optimal solution can be expected in alllikelihood.

References

[1] J.S. Ferreira, M.A. Neves, P. Fonseca, A two-phase roll

cutting problem, European Journal of Operational Re-

search 44 (1990) 185±196.

[2] R.W. Haessler, P.E. Sweeney, Cutting stock problems and

solution procedures, European Journal of Operational

Research 54 (1991) 141±150.

[3] R.W. Haessler, M.A. Vonderembse, A procedure for

solving the master slab cutting stock problem in the stell

industry, AIIE Transactions 11 (1979) 160±165.

[4] E.E. Bisho�, G. Waesher, Cutting and packing, European

Journal of Operational Research 84 (1995) 503±505.

[5] M. Gradi, J. Jesenko, G. Resinovi�c, Optimization of roll

cutting in clothing industry, Computers & Operations

Research 24 (1997) 945±953.

[6] H. Dyckho�, A typology of cutting and packing problems,

European Journal of Operational Research 44 (1990) 145±

159.

[7] P.C. Gilmore, R.E. Gomory, A linear programming

approach to the cutting stock problem, Operations

Research 9 (1961) 849±859.

[8] P.C. Gilmore, R.E. Gomory, A linear programming

approach to the cutting stock problem, Part II, Operations

Research 11 (1963) 863±888.

[9] E.G. Co�man Jr., M.R. Garey, D.S. Johnson, Approxi-

mation algorithms for bin packing ± An updated survey,

in: G. Ausiello, M. Lucertini, P. Sera®ni (Eds.), Algorithm

Design for Computer Systems Design, Springer, New

York, 1984, pp. 49±106.

[10] P.E. Sweeney, E.R. Paternoster, Cutting and packing

problems: A categorised, application-orientated research

bibliography, Journal of the Operational Research Society

43 (1992) 691±706.

[11] P.E. Sweeney, R.W. Haessler, One-dimensional cutting

stock decisions for rolls with multiple quality grades,

European Journal of Operational Research 44 (1990) 224±

231.

[12] G. Waesher, T. Gau, Generating almost optimal solutions

for the integer one-dimensional cutting stock problem,

Working Paper No. 94/06, Institut f�ur Wirtschafts-

wissenschaften, Technische Universitat Braunschweig,

1994.


a sequential heuristic procedure for one-dimensional cutting

Documents