Numerical investigations on the MIRUP of the. Jan Riehme, Guntram Scheithauer and Johannes Terno

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1 Working Paper TU Dresden, August 1995 (submitted for publication) Numerical investigations on the MIRUP of the 2-stage guillotine cutting stock problem Jan Riehme, Guntram Scheithauer and Johannes Terno Institute of Numerical Mathematics, Dresden University of Technology, Mommsenstr. 13, D Dresden, Germany Abstract The MIRUP (Modied Integer Round-Up Property) leads to an upper bound for the gap between the optimal value of the integer problem and that of the corresponding continuous relaxation rounded up. This property is known to hold for many instances of the one-dimensional cutting stock problem but there are not known so far any results with respect to the two-dimensional case. In this paper we investigate numerically three variants of the so-called 2-stage guillotine cutting stock problem with respect to the MIRUP. The variants dier by allowing either only horizontal or only vertical or horizontal and vertical guillotine cuts in the rst stage. Within a sample of 900 randomly generated instances there did not arise any instance with gap larger than 1. Moreover, for more than 60% of the instances an optimal solution was found. 1

2 1 Introduction We consider cutting stock problems of the following type. There is given (large, identical) stock material which is unbounded available. From the stock material smaller pieces have to be cut in such way that order quantities are fullled and the number of stock material needed is a minimum. In the one-dimensional case the stock material has a given length L and smaller lengths l i (the pieces) i = 1; : : : ; m, have to be cut. The standard modelling of this problem uses the framework of cutting patterns ([2]). A cutting pattern is a feasible cutting scheme of some smaller pieces from one given length L and is represented by a nonnegative integer vector. The dimension of this vector equals the number m of the small pieces. Minimizing the total number of cutting patterns (i.e. the number of stock materials) needed to full the given order demands of the pieces, leads to the following linear integer optimization model: z = e T x! min s.t. Ax b x 0 ; integer ; where e = (1; : : : ; 1) T. A is the matrix of cutting patterns and b denotes the demand vector. The component x j of the n-dimensional integer vector x gives the number the j-th cutting pattern is used and n is the number of dierent cutting patterns (columns of A). An instance E = (m; l; L; b) of the one-dimensional cutting stock problem is characterized by its input data. The instance E determines the matrix A. In the solution concept of Gilmore/Gomory [2] rst the continuous relaxation (1) z = e T x! min s.t. Ax b ; x 0 ; (2) of (1) is solved, and then, based on an optimal solution of (2) an integer solution is constructed. Let z c (E) denote the optimal value of the continuous problem (2) and let z(e) be the optimal value of the original integer problem (1). In the case of the one-dimensional cutting stock problem the Modied Integer Round-Up Property (MIRUP), that is z(e) dz c (E)e + 1 (3) holds (dae denotes the smallest integer not smaller than a), is proved for a wide range of instances E (see [8], [9] for denitions, [9], [13] for numerical experiments, and [9], [10], [4] for special proofs). This property is very useful for developing ecient algorithms ([11]). In this paper we consider a special two-dimensional cutting stock problem with respect to the gap between the optimal value of the integer problem and that of the corresponding continuous relaxation rounded up. 2

3 2 The 2-stage guillotine cutting stock problem In the 2-stage guillotine cutting stock problem only such cutting patterns are allowed where in the rst stage stripes are cut from the (rectangular) stock sheet by parallel guillotine cuts, and in the second stage the rectangular pieces are cut from the stripes using cuts orthogonal to that of the rst stage. In the case of the exact 2-stage cutting stock problem (E2 CSP) all pieces within a stripe must have the same width as the stripe, i.e. they t exactly in the stripe. In the case of the nonexact 2-stage cutting stock problem (N2 CSP) some pieces may occur with smaller width as that of the stripe (cf. Fig. 1 and 2). Figure 1: Exact 2-stage guillotine pattern with horizontal stripes Figure 2: Nonexact 2-stage guillotine pattern with vertical stripes Applying the cutting pattern concept, the exact and the non-exact 2-stage guillotine cutting stock problem can be modelled as an integer linear optimization problem (1) too. Let E = (m; l; w; L; W; b) denote the input data set with m number of rectangular pieces T i ; i = 1; : : : ; m, l i ; w i length and width of T i ; i = 1; : : : ; m. L; W length and width of the rectangular stock material, b i demand of T i ; i = 1; : : : ; m. As in the one-dimensional case, a common used solution approach consists in solving the continuous relaxation using again a column generation technique, followed by a suitable rounding procedure. Thereby the gap between the objective values of the integer and the continuous problem is of large interest. In this paper we investigate this gap numerically. At rst we give some remarks about the theoretical background and on work connected with this eld. In the following we consider especially the nonexact 2-stage guillotine cutting problem. This problem is of large practical interest and is considered in a lot of papers (cf. [1]). The fundamental approach using a column generation technique was rstly described in [3]. A new solution approach is proposed in [5], [6] which takes into account the assumption that the demands b i have extreme dierent sizes. The new concept (so-called stripe approach) is as follows. In the rst level the demands are fullled using only stripes and then the stripes are composed to patterns. This framework is compared in [6] with the classical framework of Gilmore/Gomory and the 3

4 eciency of the new concept is proved by a lot of computational experiments. A more detailed analysis of the test instances shows an only small dierence between objective value of the integer solution and the continuous lower bound. That gives the idea for the current investigations. In order to investigate the MIRUP conjecture with respect to the N2 CSP let again z(e) and z c (E) denote the corresponding optimal values of the models (1) and (2), respectively, for a given instance E. According to the MIRUP in the one-dimensional case a fundamental lemma was proved in [8], [9]. This lemma holds also in the two-dimensional case. Moreover, this lemma holds for all cutting stock problems which can be modelled as integer linear optimization model (1) using the cutting pattern framework. Because of its importance and since our numerical approach is based on it we repeat here this lemma and its proof. Let x c denote an optimal solution of the LP relaxation (2). Rounding down of x c yields an integer vector x with x j = bx c jc, j = 1; : : : ; n (bac denotes the greatest integer not greater than a). If x 6= x c then a residual problem can be dened with right hand side b := b? Ax. Now let E = (m; l; w; L; W; b) denote an instance of the cutting stock problem and let E = (m; l; w; L; W; b) denote the instance of the corresponding residual problem. Then we have Lemma 1: If z(e) dz c (E)e + 1 then z(e) dz c (E)e + 1, i.e. if the residual problem has the MIRUP then the initial problem has this property too. Proof: It holds z c (E) = e T x c = e T x + e T ( x c? x ) z(e) e T x + z(e) e T x + dz c (E)e + 1 = de T x + z c (E)e + 1 = dz c (E)e + 1: Conjecture: The N2 CSP has the property z(e) d z c (E) e + 2 : (4) This conjecture is based on the following reasons. Firstly, in [7] estimations (lower bounds) of the maximal dierence = z(e)? z c (E) are given for several two-dimensional cutting stock problems. Especially, for the E2 CSP where all stripes have to be horizontal or vertical (RE2 CSP) it is shown that ( RE2 CSP ) > m 3 (5) holds. This means for the RE2 CSP the dierence increases with the number of pieces. But for the N2 CSP only (artical) instances are given with dierence greater than 2 but less than 3. Secondly, considering the 2-stage guillotine cutting problem as two independent one-dimensional cutting stock problems (in similarity to [6]) and taking into account that the greatest dierence known is less than 1.1 in the one-dimensional case, this conjecture is motivated. 4

5 Moreover, the numerical experiments reported below show that within the sample of 900 randomly generated instances there is no instance with gap greater than 1. 3 Solution approach For our numerical tests we use the following solution strategy. At rst, for a given instance E the continuous relaxation (2) is solved. In detail, we terminate the column generation process if there does not exist any pattern with transformed objective coecient?10?9. Then, if necessary, we calculate the corresponding residual instance E and, using two simple heuristics, we generate an integer solution for E, and hence for E. In our tests we consider the (general) N2 CSP (where stripes in length- and width-direction may occur in dierent patterns) and the restricted N2 CSP where all patterns have to have the same direction of the stripes. In dependence of these circumstances the heuristics have to be applied one or two times, respectively. In the following we describe briey the two heuristics used. Since these heuristics are applied to residual problems (where the b i are small in general) they have to take into account the order demands as upper bounds. The two heuristics use the same strategy. They generate a packing pattern, reduce then the order demands and, if necessary, they generate a further pattern until all order demands are fullled. The generation of one packing pattern is done as follows. At rst, for any dierent stripe size (stripe length in case of vertical stripes, or stripe width in case of horizontal stripes) an optimal stripe pattern is computed by solving a knapsack problem with upper bounds (we use a branch and bound algorithm). Then these stripe patterns are evaluated. In Heuristic 1 the utilization area mx l i w i a ik i=1 denes the value of stripe pattern a k = (a1k; : : : ; a mk ) T. In Heuristic 2 the utilization rate used area stripe area = P m i=1 l iw i a ik f k where f k equals the area of the k-th stripe, is used to evaluate the stripe patterns. In both heuristics, that stripe pattern is chosen having the greatest value. This stripe pattern is arranged as often as feasible (no overproduction, t completely on the stock sheet). If there remains a sucient large unused area to arrange a further stripe (of same direction) then this process is repeated starting with the calculation of the stripe patterns. Obviously, some opportunities are involved in the heuristics to improve their eciency but the consideration of more intelligent heuristics is left for future work. 5

6 4 Computational experiments In our numerical tests we consider 9 series of randomly generated instances. They are characterized by the pairs (m; L) with m 2 f10; 20; 30g and L 2 f2000; 3000; 4000g. The width of the stock sheets is always set to be 1000 and the piece lengths and widths are randomly chosen from uniform distributions on the ranges [100,500] and [100,300], respectively. All pieces are allowed to be rotated by 90 degrees. The b i -values are also randomly chosen from [5,2000]. Within the column generation process several knapsack problems (KP) have to be solved. In order to compare various KP solver we implemented a shortest path (SPM) algorithm, a forward state strategy (FSS) algorithm and a branch and bound (B&B) algorithm for solving the KP. As noted above, we distinguish between the (general) N2 CSP (where lengthand width-stripes are allowed) and the restricted N2 CSP where only lengthstripes are allowed in the rst stage or only width-stripes are allowed. We denote these three dierent problems by N2 CSP, VN2 CSP and HN2 CSP, respectively. Table 1 shows the main results with respect to the gap between the objective values of the integer solution computed and the LP lower bound rounded up. Every horizontal block of three lines corresponds to one KP solver. The entries give the numbers the gap "=0" or "=1" or " 2" occur within the corresponding series. Of course, every KP solver leads to the same LP bound but, in general, to dierent solutions. For that reason dierent residual problems arise and hence, dierent integer solutions may be obtained. It is remarkable that although for a single KP solver instances with gap 2 are found (more precicely: the approach leads to an integer solution with gap = 2), but the best solution with respect to the KP solver has always a gap 1 for all instances but one (for N2CSP and HN2CSP). This is shown in the horizontal blocks in Table 1 denoted by BEST. It is to note that a more detailed inspection of the solutions of the residual problem with gap =2 leads in both cases also to an integer solution with gap =1. The main consequence of these circumstances consists in the necessity of considering of various (optimal) solutions of the LP relaxation (2) when an integer solution is computed with gap 2 or, to improve the solution concept for the residual problem. Further investigations in these directions are required. Some more detailed informations are given in the tables 2 to 4. In dependence of the corresponding KP solver the rows contain the following data: 6

7 n number of test instances, a average number of dierent cutting patterns of the LP solution used for the integer solution, b average number of dierent cutting patterns in the integer solution, c dierence b-a, d average dierence between z c (E) and e T x, e minimal dierence between z c (E) and e T x, f maximal dierence between z c (E) and e T x, g number, heuristic 1 leads to a better solution than heuristic 2, h number, heuristic 2 leads to a better solution than heuristic 1, i average area utilization of the LP solution, j average area utilization of the computed integer solution, k average running time (in seconds, PC 486 DX2, 66 MHz), l average number of simplex steps. Not surprisingly, the number of dierent cutting patterns (row b) is relatively large in comparison to the number m of pieces because of the used solution approach. This may be nonadvantageous for practical applications. As the rows g and h show there is no signicant dierence between the two heuristics but since their computational amount is very small both should be used. Since the column generation process is only terminated when the minimal transformed objective function coecient is not smaller than?10?9, relative many simplex steps have to be carried out in average. Probably an essential decrease of this number can be reached if a more suitable termination criterion is used. Another interesting aspect results from the comparison of the (average) area utilization of N2 CSP, HN2 CSP and VN2 CSP (rows i and j). The more restricted problem HN2 CSP leads to nearly the same results as N2 CSP does but the corresponding solution is probably more suitable for practical applications and can be computed with fewer computational eort. Last not least we remark, the simple heuristics produce very good integer solutions even for instances with only small order demands. This can be seen in the rows c and d because row d gives the average LP bound of the residual problems and row c shows the (average) number of cutting patterns in the corresponding integer solution. Moreover, Table 1 gives also the gaps which arise for the residual problems. 5 Conclusional remarks The computational experiments reported for the N2 CSP shows an always small gap between the integer solution found and the corresponding LP lower bound. These numerical tests constitute a strong argument for the validity of our conjecture. But there remains a lot of questions, e.g. theoretical foundations of the conjecture; a decision criterion whether a solution with gap 1 is optimal or not; or, constructing of an ecient algorithm for computing always integer solutions with gap not greater than 1. 7

8 References [1] Dyckho, H., and Finke, U., Cutting and packing in production and distribution, Physica Verlag, Heidelberg, [2] Gilmore, P.C., and Gomory, R.E., A linear programming approach to the cutting stock problem, Operations Res. 9 (1961) 849{859. [3] Gilmore, P.C., and Gomory, R.E., Multistage cutting stock problems of two and more dimensions, Operations Res. 13 (1965) 94{120. [4] Nica, V.: A general counterexample to the round-up property. Working Paper 1994 (submitted for publication, personal communication). [5] Riehme, J.: Guillotine { Zuschnittprobleme mit extrem unterschiedlichen Stckzahlforderungen. Diplomarbeit, TU Dresden, [6] Riehme, J., G. Scheithauer, J. Terno: The Solution of Two-stage Guillotine Cutting Stock Problems Having Extremly Varying Order Demands. Preprint, MATH{NM{05{1995, TU Dresden (to appear in EJOR). [7] Scheithauer, G.: On the MAXGAP Problem for Cutting Stock Problems. J. Inform. Process. Cybernet. - EIK 30 (1994) 2, 111{117. [8] Scheithauer, G., and Terno, J., About the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem, Operations Research Proceedings 1991, Springer-Verlag, Berlin, Heidelberg, [9] Scheithauer, G., and Terno, J., The Modied Integer Round-Up Property of the One-Dimensional Cutting Stock Problem, European J. of Operational Research 84 (1995) [10] Scheithauer, G., and Terno, J., Theoretical investigations on the MIRUP (modied integer round-up property) for the one-dimensional cutting stock problem, Preprint MATH{NM12{1993, TU Dresden, 1993 (submitted for publication). [11] Scheithauer, G., and Terno, J., A branch&bound algorithm for solving onedimensional cutting stock problems exactly, applicationes mathematicae 23.2 (1995) [12] Terno, J., R. Lindemann, G. Scheithauer: Zuschnittprobleme und ihre praktische Lsung. Verlag Harry Deutsch, Thun und Frankfurt/Main, und Fachbuchverlag Leipzig [13] Wscher, G., and Gau, T., Two approaches to the cutting stock problem, IFORS'93 Conference, Lisboa

9 L=2000 L=3000 L=4000 sum m vertical stripes only (VN2 CSP) = SPM = = FSS = = B&B = = BEST = horizontal stripes only (HN2 CSP) = SPM = = FSS = = B&B = = BEST = vertical and horizontal stripes (N2 CSP) = SPM = = FSS = = B&B = = BEST = Table 1: Number of instances with gap 0, 1 or 2 9

10 L=2000 L=3000 L= vertical stripes only (VN2 CSP) a b c d e f g h i j k l horizontal stripes only (HN2 CSP) a b c d e f g h i j k l vertical and horizontal stripes (N2 CSP) a b c d e f g h i j k l Table 2: Statistical informations Shortest Path Method 10

11 L=2000 L=3000 L= vertical stripes only (VN2 CSP) a b c d e f g h i j k l horizontal stripes only (HN2 CSP) a b c d e f g h i j k l vertical and horizontal stripes (N2 CSP) a b c d e f g h i j k l Table 3: Statistical informations Forward State Strategy 11

12 L=2000 L=3000 L= vertical stripes only (VN2 CSP) a b c d e f g h i j k l horizontal stripes only (HN2 CSP) a b c d e f g h i j k l vertical and horizontal stripes (N2 CSP) a b c d e f g h i j k l Table 4: Statistical informations Branch & Bound 12