An evolutionary algorithm for the one-dimensional cutting stock. problem

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1 International Transactions in Operations Research: aceito para publicação em 2010 An evolutionary algorithm for the one-dimensional cutting stock problem Silvio A. Araujo a, Ademir A. Constantino b and Kelly C. Poldi c a Universidade Estadual Paulista, São José do Rio Preto-SP, Brazil b Universidade Estadual de Maringá, Maringá-PR, Brazil c Universidade de São Paulo-São Carlos-SP, Brazil s: saraujo@ibilce.unesp.br [Araujo]; ademir@din.uem.br [Constantino]; kelly@icmc.usp.br [Poldi] Abstract This paper deals with the one-dimensional integer cutting stock problem, which consists of cutting a set of available objects in stock in order to produce ordered smaller items in such a way as to minimize the waste of material. The case in which there are various types of objects available in stock in limited quantities is studied. A new heuristic method based on the evolutionary algorithm concept is proposed to solve the problem. This heuristic is empirically analyzed by solving randomly generated instances and the results are compared with other methods from the literature. Keywords: Integer Optimization, Cutting Stock Problem, Evolutionary Algorithm.

2 1. INTRODUCTION Over the last four decades, hundreds of papers dealing with cutting and packing problems have been published. The cutting stock problem is a specific class of cutting and packing problems that consists of cutting a set of available objects in stock into smaller pieces (items) by optimizing a certain objective function. According to Wäscher et al. (2007), problems of this category require that a weakly heterogeneous assortment of small items be completely allocated to a selection of large objects that can be identical or heterogeneous and each object has a fixed dimension. This problem arises in various industries such as paper, steel, plastic, wood and many others. This paper is focused on the one-dimensional cutting stock problem where there is a weakly heterogeneous assortment of objects and the quantity of each object in stock is limited, but it is enough to produce all items. The objective is to minimize the waste of material. According to the typology proposed by Wäscher et al. (2007), this problem is classified as a one-dimensional Multiple Stock Size Cutting Stock Problem (MSSCSP {1}- dimensional). The MSSCSP {1}-dimensional problem with limited stock availability can be formulated as an integer linear optimization problem as follows. Data: m: number of types of items; l i : length of item type i, i=1,..., m; d i : demand for item type i, i=1,..., m (d); K: number of different stock lengths available; L k : length of stock object type k, k=1,..., K; e k : availability of stock length type k, k=1,..., K;

3 c jk : cost of stock object type k, cut according to the cutting pattern j, j=1,...,n k, k=1,...,k; a jk : vector which corresponds to the jth cutting pattern of the stock object type k, j=1,..., N k, k=1,..., K. Considering a fixed object type k, there are N k possible cutting α11 k α12 α k 1N kk α 21k α 22 α k 2N kk patterns: a1 k =, a2k =,..., an =, kk M M M α m1k α m2k α mnkk k=1,..., K, where α ijk is the quantity of items type i (i=1,...m) on the jth cutting pattern of the stock object type k. Moreover, a vector a jk is a cutting pattern if and only if l i α ijk L m i= 1 k, with α ijk 0 and integer. Variables: x jk : number of objects type k to be cut according to the cutting pattern j, j=1,..., N k, k=1,..., K. Mathematical Model: K Nk Minimize m f ( x jk ) = Lk liα ijk x jk (1) k = 1j= 1 i= 1 K N k Subject to: a x = d (2) k = 1 j= 1 jk jk N k x j= 1 jk e k, k = 1,...,K (3) x jk 0, and integer, k=1,..., K, j=1,..., N k. (4) The objective function (1) considers the minimization of the total waste of material. The equality constraints in (2) assure that the total amount of items which are produced meets the demand. The constraints (3) assure that the number of each stock object type k cut does not

4 exceed its availability e k. Finally, (4) assure that the number of objects to be cut is nonnegative and integer. The integer conditions under the variables and the huge amount of possible cutting patterns, make problem (1)-(4) difficult to be solved. There are few papers in the literature dealing with the (MSSCSP {1}-dimensional) problem. Belov and Scheithauer (2002) and Alves and Carvalho (2008) developed exact methods considering an equivalent problem where the objective function (1) is replaced by the minimization of the total length to be cut and the constraints (2) are inequalities instead of equalities. Poldi and Arenales (2009) solved the problem using heuristic methods. Other heuristic procedures were developed for similar problems (Gilmore and Gomory, 1963, Roodman, 1986, Gradisar et al., 1997, Gradisar et al., 1999, Holthaus, 2002, Gradisar and Trkman, 2005). On the other hand, there are various papers dealing with the case in which only one object type is available in an unlimited stock quantity called, according to Wäscher et al. (2007) typology, the Single Stock Size Cutting Stock Problem (SSSCSP {1}-dimensional). A mathematical model for the SSSCSP {1}-dimensional problem can be adapted from the model (1)-(4) by considering K=1 and by eliminating the constraints (3). Regarding the SSSCSP {1}-dimensional problem and considering the minimization of the total waste of material, Degraeve and Peeters (2003), Scheithauer and Terno (1995) and Vance (1998) developed exact methods and Hinxman (1980), Stadtler (1990) and Wäscher and Gau (1996) presented heuristics methods. According to Umetani et al. (2003), objectives other than the waste of material have become important. Yanasse and Limeira (2006) highlighted the importance of bi-criteria decision problems when machine costs are significant when compared to the cost of the material. They presented a bi-criterion function that minimizes the number of objects to be cut and the number of different cutting patterns. Some papers in the literature presented heuristic

5 methods to solve similar problems: Haessler (1975), Farley and Richardson (1984), Haessler and Sweeney (1991), Diegel et al. (1993), Foerster and Wäscher (2000), Vanderbeck (2000), Umetani et al. (2003) and Yanasse and Limeira (2006). In this paper, an Evolutionary Algorithm (EA) is developed to solve the MSSCSP {1}- dimensional problem. The main characteristics of the proposed approach are: simplicity, which allows easy replication; efficiency in terms of computational time, which makes it suitable for solving practical problems; and relatively effective in terms of solution quality. To the best of our knowledge, this is the first heuristic based on Evolutionary Algorithm concepts proposed in the literature for this specific problem. 2. THE PROPOSED ALGORITHM Over the last two decades, Evolutionary Algorithm techniques have received much attention due to their potential as optimization techniques (Michalewicz, 1994, Fogel, 1995, Hertz and Kobler, 2000, Beasley, 2002). These techniques are stochastic optimization methods inspired by Darwin's theory of the survival of the fittest. The EAs use a randomlygenerated initial population of individuals (solutions of the search space) and new individuals are generated using simulated genetic operations. The probability of survival of the newly generated individuals is based on their fitness. Evolutionary Algorithm techniques do not have many mathematical requirements in the optimization problem, and also, they are flexible to hybridize with domain-dependent heuristics. The basic concepts of EA have been adapted in such a way that problem-specific knowledge is considered. The basic characteristics of the EA method used to solve the MSSCSP {1}-dimensional problem are presented below.

6 Representation Consider an individual as a feasible solution for the cutting stock problem, where each gene is represented by a cutting pattern (a jk ) and also the number of times it is cut (x jk ). Therefore, an individual is represented as a matrix where a column represents a cutting pattern (and the number of times it is cut) for a specific object of the individual. The number of columns can be different for each individual because it depends on the number of different cutting patterns. The number of rows is equal to the number of different items plus a row with the number of times the cutting pattern is cut. Initial Population In order to generate the initial population, two aspects were considered, computational time and diversity. An adaptation of the simple Repeat Exhaustion Reduction (RER) Heuristic (Hinxman, 1980) is used. It can be summarized as follows. Step 1: Construct a cutting pattern. Step 2: Apply this cutting pattern as many times as possible (i.e., determine x jk ), do not exceed either the demand or the stock availability. Step 3: The demand and the stock availability are updated and the procedure is repeated until the demand is met (we have a feasible integer solution) or there is no more stock availability (infeasible problem). Individuals with different characteristics should be generated, which is important to diversify the population, therefore four different approaches were used to construct a cutting pattern in Step 1. These approaches are explained in detail as follows: RER 1: Construct one individual using the RER heuristic with the First Fit Decreasing (FFD) Heuristic (Johnson, 1974, Wäscher and Gau, 1996) to construct a cutting pattern in Step 1. A

7 cutting pattern for each type of object is constructed and the object related to the minimal waste is chosen. RER 2: Some individuals are constructed using the RER heuristic with a simple heuristic to construct a cutting pattern in Step 1, in which items are randomly chosen. The object is chosen as in RER 1. RER 3: Some individuals are constructed using the RER heuristic with a simple heuristic to construct a cutting pattern in Step 1, in which objects are randomly chosen. The items are chosen as in RER 1. RER 4: Some individuals are constructed using the RER heuristic with a simple heuristic to construct a cutting pattern in Step 1, in which both the items and objects are randomly chosen. Once one cutting pattern is constructed by one of these procedures, the number of times it will be cut x jk is determined by taking the maximum value possible, so that it does not overproduce the demand of any item that belongs to these cutting patterns and not overuse the available stock for object e k. The size of the initial population and the percentage of individuals constructed by using each of the RER 2, RER 3 and RER 4 approaches are parameters to be defined in the EA (notice that one individual is generated by the RER 1 approach). The Initial Population is sorted in non-increasing order of fitness (the best individual is the first one). Selection Process For the selection process, a greedy randomly procedure is used to choose the individual that will co-operate in order to produce a new solution. The better the individual, the bigger the probability is of being chosen. The probability is also a parameter to be set in the EA. In

8 order to facilitate the selection process, the Greedy Randomly procedure uses the fact that the population is sorted in a non-increasing order of fitness, i. e., the best individual (smallest fitness) is the first one and the worst individual (biggest fitness) is the last one. This procedure chooses an individual with a certain probability ( ) that belongs to the first Ω individuals of the sorted population, i. e., the first Ω best individuals. If the value of is too big and the value of Ω too small, it could cause premature convergence and increase the risk of stagnation at a local optimum, while if the value of is too big and the value of Ω is also too big, it would lead to random meandering and a wasteful search. Co-operation Phase In order to create a new individual, after choosing an individual using the above selection process, the procedure tries to randomly choose a cutting pattern from the chosen individual and put this cutting pattern in the new individual. Afterwards, the selection process is executed again to choose another individual. A new cutting pattern is randomly chosen and included in the new individual. This procedure is repeated taking care not to choose a cutting pattern related to an object which is not available in stock or which has an item for which the residual demand has already been met (it explains the way x jk is determined). It stops when a parameter g is reached, where g is the maximum number of cutting patterns tried to be included in the individual. Parameter g must be previously defined and it is based on the population characteristics, i. e., the mean number of cutting patterns of the individuals from the incumbent population plus a certain percentage. If, before g attempts, the residual demand is null, then we have a feasible individual. Otherwise, a self-adaptation phase has to be executed.

9 Self-adaptation Phase If, after g attempts, the residual demand is not null, it is necessary to construct a feasible solution. In this step, the Repeat Exhaustion Reduction Heuristic (RER1) is used again, but considers only the residual demand. Population Replacement Strategy To include the new individual in the population, the Steady-State Population Replacement Strategy is used. If the new individual is better than the worst individual of the population, then the worst one (the last one in this case) is eliminated and the new individual is included in the sorted population in the correct position. Stop Criterion The stop criterion is a parameter to be set. It is defined as the maximum number of iterations, where each iteration represents the generation of an individual. When the stop criterion is reached, the first individual of the sorted population is taken as the final solution. Fitness Function The fitness function considered is the same as the objective function, i. e., minimization of the waste of material (1). Algorithm Overview First of all, an Initial Population of Feasible Individuals (Feasible Solutions) is generated. Afterwards, a co-operation phase is executed where the population exchanges information in order to create a new individual. The new individual can be feasible or not. If it is not feasible, a self-adaptation procedure is applied and feasibility is met. If the new solution is not better

10 than the worst solution in the population, it is rejected and the co-operation and selfadaptation steps are repeated. Otherwise, the new individual is included in the population and the worst one is eliminated. If the stop criterion is not satisfied, the steps are repeated, otherwise the procedure stops and the first individual of the sorted population is taken as the final solution. Basically, the following parameters must be set in order to apply the EA: the size of the initial population; how to generate the initial population (percentage of individuals generated by each procedure: RER2, RER 3 and RER 4); the greedy randomly selection of an individual (parameters and Ω); the number of cutting patterns (genes) of an individual that are tried to be inserted by the Evolutionary Algorithm (g) and the stop criterion (maximum number of Iterations). 3. COMPUTATIONAL RESULTS Belov and Scheithauer (2002) developed an exact method for the MSSCSP {1}- dimensional. In this section, the EA method is compared to the Belov and Scheithauer (2002) method in order to evaluate the solution quality and the computational performance of the EA method. The instances from Belov and Scheithauer (2002) were considered. It has 3,100 randomly generated instances grouped into 31 classes (100 instances in each class) according to the following characteristics: Number of types of stock objects: K= 5. Number of types of ordered items: m = 40. Stock object length: L 1 =10,000 and L k {5000; 5100; ; 9900}, k = 2,...,5. Stock availability: The values e k, k = 1,...,K were randomly generated in the interval [2000, 2500].

11 Item length: l i were randomly generated in the interval [v 1 L max, v 2 L max ], where L max =max{l k }, k=1,...,k; v 1 = {0.001; 0.01; 0.05; 0.015; 0.025} and v 2 = {0.2; 0.3; ; 0.9}. Demand: d i, i = 1,...,m were randomly generated in the interval [200, 300[. Extensive computational tests were carried out and some evolutionary parameters were tested. For the computational results, the final parameters used are described in Table 1. Table 1 Parameters for the Evolutionary Algorithm. Parameter Value Size of the Initial population 10 Percentage of individuals generated by the heuristics RER 2, RER 3 and RER 4 33% each (RER 2=3, RER 3=3 and RER 4=3). RER 1=1 individual. Percentage of Greedy Randomly selection of the individual =50% of probability to select one individual among the Ω=50% best ones Maximum Number of genes (g) Average number of genes of the individuals from the initial population plus 20% Stop Criterion (Maximum Number of Iterations (Individuals)) 1500 and 3000 Fitness Function Objective Function (1) Table 2 shows the characteristics of each generated class (v 1 and v 2 ) and the computational results (Gap). The Gap is calculated as follows. ( EA solution ) -( Belov & Scheithauer solution ) Gap = 100 Belov & Scheithauer solution As it is possible to observe in Table 2, the EA results are close to the Belov and Scheithauer (2002) solutions. Considering 1500 iterations the mean Gap is approximately 1.30 %, the maximum is approximately 2.37 % and the minimum is approximately 0.17 % and considering 3000 iterations the mean Gap is approximately 1.13 %, the maximum is approximately 2.22 % and the minimum is approximately 0.16 % Belov and Scheithauer (2002) run their tests in a Pentium III (500 MHz) with a time limit, for each instance, of 60 seconds after the LP bound. Analyzing the EA computational times on an Intel Core 2 Quad (2.4GHz/2.4GHz) and 3.24 GB RAM, considering 1500 iterations the proposed approach spent seconds to solve the 3,100 instances, which means 0.61

12 seconds, on average, for each instance and considering 3000 iterations the proposed approach spent seconds, which means 1.1 seconds, on average, for each instance. Table 2 Characteristics of the instances and Gap*. Gap 1500 Gap 3000 Gap 1500 Class v 1 and v 2 iterations iterations Class v 1 and v 2 iterations Gap 3000 iterations C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and Mean *Belov and Scheithauer (2002) raw values were taken from the web page Other parameters of the EA were tested with different values. Basically, we conclude that: increasing the number of iterations did not cause a significant improvement in the quality of solutions in relation to the increase in the computational time. The same happened when parameter g and the size of the initial population were increased. The greedy procedure to choose an individual (parameters and Ω) and the percentage of utilization of each procedure to construct the initial populations did not have a significant influence on the computational time, but it had some influence on the quality of the solution so that they were set to the best combination found. 3.1 COMPARISON WITH HEURISTIC PROCEDURES In order to generate the instances, a random generator, called CUTGEN1 (Gau and Wäscher, 1995), was adapted for the case with multiple stock lengths and a limited stock.

13 Basically, CUTGEN1 was designed to generate instances for the SSSCSP-{1}dimensional, however, Poldi and Arenales (2009) adapted the generator for the MSSCSP-{1} dimensional, after extensive computational tests in order to ensure the feasibility of the instances. There are 360 randomly generated instances grouped into 18 classes (20 instances in each class) according to the following characteristics (based on Poldi and Arenales, 2009): Number of types of stock objects: K=3, 5 and 7. Number of types of ordered items: m = 5, 10, 20 and 40. Stock object length: The values of L k, k = 1,...,K were randomly generated in the interval [10, 1000]. Stock availability: The values e k, k = 1,...,K were randomly generated in the interval [1,100 m/2]. Item length: l i were randomly generated in the interval [v 1 L, v 2 L], where L is the average value among the L k, k=1,...,k; v 1 = 0.01 and, v 2 = 0.2 (small) and 0.8 (medium). Demand: d i, i = 1,...,m were randomly generated in the interval [1, 10]. The EA results are compared to Poldi and Arenales (2009) residual heuristics (PA1, PA2 and PA3) which are based on the idea that rounding up all the frequencies of cutting patterns might be infeasible, but some of them could be rounded up. Thus, at each iteration, these heuristics basically solve a relaxed linear optimization problem and sort the solution vector (which represents the frequency of the cutting patterns) in a specified way (they analyzed three sortings-pa1, PA2 and PA3). Each component of the solution vector, sorted as specified, is rounded up and the solution feasibility is tested (no overproduction is allowed). If it is not feasible (overproduction occurred), this frequency is decreased by one until all overproduction is eliminated. When the last cutting pattern is examined, the demand and the

14 stock availability are updated and a new residual problem is solved. For details, see Poldi and Arenales (2009). Table 3 shows the results related to the total waste of the PA1, PA2 and PA3 heuristics compared with the proposed Evolutionary Algorithm (EA). The column GG (Gilmore and Gomory) is the linear relaxation of the problem solved by the simplex method with column generation (Gilmore and Gomory, 1961, Gilmore and Gomory, 1963) which is a lower bound. The first four columns describe the characteristics of the generated instances. Table 3 Comparing the waste of material for each method. Class K M v 1 and v 2 GG PA1 PA2 PA3 EA C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and C and Total The EA approach obtained the best results in 15 classes. In those classes where the lower bound (GG) is zero (which means that there is no waste of material in the cutting patterns), the EA results were much closer to the lower bound than the other heuristics, for example, in classes C9, C11, C13, C15 and C17. For the EA, the total waste of material for the 360 instances is , while for the approaches from Poldi and Arenales (2009), it is about and the lower bound (GG) is

15 In Table 4, the computational times (in seconds) for the three heuristics from Poldi and Arenales (2009) and for the proposed EA are shown. The methods were run on the same machine: Intel Core 2 Quad (2.4GHz/2.4GHz) and 3.24 GB RAM. The last three columns in Table 4 analyze the convergence of the EA approach. The column EA-AV shows the average (among the 20 instances of each class) of the iteration for which the best solution was obtained. Columns EA-MIM and EA-MAX show, respectively, the minimum and the maximum iteration for the 20 instances of each class. Table 4 Total Computational time (in seconds) for each method for 18 classes. Class PA1 PA2 PA3 EA EA-AV EA-MIN EA-MAX (Seconds) (Seconds) (Seconds) (Seconds) (Iterations) (Iterations) (Iterations) C C C C C C C C C C C C C C C C C C Average By analyzing the computational time, it is possible to see that the EA approach is very fast and robust. While for some instances, such as C5, C11 and C17 seem to be very difficult for the residual heuristics PA1, PA2 and PA3, the EA approach kept a low computational time. In the last three columns in Table 4, the convergence of the EA approach is analyzed. On average the method needs iterations to find the best solution. Column EA-MAX shows

16 that for some instances of some classes, it needs more than 1,300 iterations and column EA- MIN shows that the best solutions for some instances were obtained by the FFD heuristics (EA-MIN = 1). 4. CONCLUSION AND FUTURE RESEARCH This paper dealt with the one-dimensional integer cutting and stock problem with multiple stock lengths with limited availability. In practice, the number of items can be hundreds and the number of cutting patterns can be thousands, which makes the problem difficult to be solved. In this paper a straightforward Evolutionary Algorithm was developed and produced good solutions in terms of quality and computational time. Considering future research, as highlighted in Section 1, others objectives apart from minimization of the waste of material are important, such as the minimization of the number of different cutting patterns. An important characteristic of the proposed EA is the flexibility regarding the objective function to be minimized. Some computational experiences with the proposed Evolutionary Algorithm pointed out that it can obtain good results considering different fitness functions. For future research, a bi-objective function (minimization of the waste of material and the number of different cutting patterns) will be considered by developing a multi-objective Evolutionary Algorithm. Acknowledgements: The authors would like to thank the Brazilian agencies CNPq and FAPESP for financial support, the referees for their very helpful feedback, resulting in a better paper, and are also grateful to Gleb Belov for making the data and the results from his paper available. REFERENCES

17 Alves, C., Carvalho, J. M. V., A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem. Computers and Operations Research 35, Beasley, J. E., Population heuristics. In: Pardalos, P.M., Resende, M.G.C. (Eds.), Handbook of Applied Optimization. Oxford University Press, Oxford. pp Belov, G., Scheithauer, G., A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths. European Journal of Operational Research 141, Degraeve, Z., Peeters, M., Optimal integer solution to industrial cutting stock problems: part 2, benchmark results. INFORMS Journal on Computing 15, Diegel, A., Chetty, M., Van Schalkwyk, S., Naidoo, S., Setup combining in the trim loss problem-3to2 & 2to1. Working paper, Business Administration, University of Natal, Durban, First Draft. Farley A. A., Richardson, K. V., Fixed charge problems with identical fixed charges. European Journal of Operational Research 18, Foerster, H., Wäscher, G., Pattern reduction in one-dimensional cutting stock problem. International Journal of Production Research 38, Fogel, D. B., Evolutionary Computation. Toward a New Philosophi of Machine Intelligence. IEEE Press. Gau, T., Wäscher, G., CUTGEN1: A problem generator for the standard one dimensional cutting stock problem. European Journal of Operational Research 84, Gilmore, P. C., Gomory, R. E., A linear programming approach to the cutting stock problem. Operations Research 9,

18 Gilmore, P. C., Gomory, R. E., A linear programming approach to the cutting stock problem - Part II. Operations Research 11, Gradisar M., Jesenko J., Resinovic G., Optimization of roll cutting in clothing industry. Computers and Operations Research 24, Gradisar M., Kljajic M., Resinovic G., Jesenko J., A sequential heuristic procedure for one-dimensional cutting. European Journal of Operational Research 114, Gradisar, M., Trkman, P., A combined approach to the solution to the general onedimensional cutting stock problem. Computers and Operations Research 32, Haessler, R. W., Controlling cutting pattern changes in one-dimensional trim problems. Operations Research 23, Haessler, R. W., Sweeney, P.E., Cutting stock problem and solution procedures. European Journal of Operational Research 54, Hertz, A., Kobler, D., A framework for the description of evolutionary algorithms. European Journal of Operation Research 126, Hinxman, A., The trim-loss and assortment problems: a survey. European Journal of Operational Research 5, Holthaus, O., Decomposition approaches for solving the integer one-dimensional cutting stock problem with different types of standard lengths. European Journal of Operational Research 141, Johnson, D. S., Fast algorithms for bin packing. Journal of Computer and System Sciences 8, Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, New York, 2 nd edition, Poldi, K. C., Arenales, M. N., Heuristics for the one dimensional cutting stock problem with limited multiple stock lengths. Computers and Operations Research 36,

19 Roodman G. M., Near-optimal solutions to one-dimensional cutting stock problems. Computers and Operations Research 13, Scheithauer, G., Terno, J., The modified integer round-up property of the onedimensional cutting stock problem. European Journal of Operational Research 84, Stadtler, H., A one-dimensional cutting stock problem in the aluminium industry and its solution. European Journal of Operational Research 44, Umetani, S., Yagiura, M., Ibaraki, T., One-dimensional cutting stock problem to minimize the number of different patterns. European Journal of Operation Research 146, Vance, P., Branch-and-price algorithms for the one-dimensional cuttings stock problem. Computational Optimization and Applications 9, Vanderbeck, F., Exact algorithm for minimizing the number of setups in the onedimensional cutting stock problem. Operations Research 48, Wäscher, G., Gau, T., Heuristics for the integer one-dimensional cutting stock problem: a computational study. OR Spektrum 18, Wäscher, G., Haußner, H., Schumann, H., An improved typology of cutting and packing problems. European Journal of Operational Research 183, Yanasse, H. H., Limeira, M. S., A hybrid heuristic to reduce the number of different patterns in cutting stock problems. Computers and Operations Research 33,