The Modified Integer Round-Up Property of the One-Dimensional Cutting Stock Problem

EJOR 84 (1995) 562 571 The Modified Integer Round-Up Property of the One-Dimensional Cutting Stock Problem Guntram Scheithauer and Johannes Terno Institute of Numerical Mathematics, Dresden University of Technology Mommsenstr. 13, D - 01062 Dresden, Germany Abstract A linear integer minimization problem (IP) has the modified integer round-up property (MIRUP) if the optimal value of any instance of IP is not greater than the optimal value of the corresponding LP relaxation problem rounded up plus one. The aim of this paper is to investigate numerically whether the MIRUP holds for the one-dimensional cutting stock problem. The computational experiments carried out with a lot of randomly generated instances of the one-dimensional cutting stock problem show, that for all instances an integer solution fulfills the MIRUP. Moreover, in most cases the optimal value equals the round-up optimal value of the corresponding LP relaxation. Similarly, the approach proposed to verify the MIRUP is usable as a new heuristic procedure for solving one-dimensional cutting stock problems. This heuristic always leads to very good solutions being optimal in the most cases. Keywords: cutting stock problem, modified integer round-up property 1

1 Introduction In this paper the classical one-dimensional cutting stock problem is investigated with respect to the relation between the optimal value and the corresponding linear programming (LP) relaxation bound. The numerical tests given below show that this difference is smaller than two. Therefore the proposed approach leads to a heuristic procedure for solving one-dimensional cutting stock problems. The starting point of our investigations can be summarized as follows. A frequently used method for solving one-dimensional cutting stock problems involves applying the column generation approach proposed by Gilmore and Gomory [6] and an appropriate rounding of the solution of the continuous relaxation problem [7], [14]. Gilmore and Gomory discuss briefly in [6] the possibility of round-down and using ad hoc -methods to fulfill the unfilled order quantities. In this paper we will take this much more into consideration since many numerical computations show an only small difference between the optimal value of the one-dimensional cutting stock problem and that of its corresponding linear programming (LP) relaxation [2], [15]. On the other hand from the theoretical point of view, Baum and Trotter [1] define an integer programming problem as having the integer round-up property (IRUP) if its optimal value is given by the smallest integer greater than or equal to the optimal value of its LP relaxation. They establish this property for certain classes of matrices arising in the context of the polymatroid theory. In [8] Marcotte investigates the cutting stock problem and integer rounding. She proves that certain classes of cutting stock problems have the IRUP. Furthermore, Marcotte gives in [9] an instance which does not fulfil the IRUP. But the gap between the optimal value and the corresponding LP lower bound always equals 1. Fieldhouse [5] presents an example of the one-dimensional cutting stock problem with a gap equal to 1.0333.... Moreover, an example is given in [11] possessing a gap equal to 1.0378.... This is the largest difference to have been found so far. In this paper we define the modified integer round-up property (MIRUP) of a (linear) integer minimization problem and investigate this property numerically for the classical one-dimensional cutting stock problem. Additionally we refer to [11] where the MIRUP is proven for such one-dimensional cutting stock problems where all cutting patterns fulfill a special density property. The paper is organized as follows. In Section 2 we formulate the one-dimensional cutting stock problem, give some definitions and state the main conjecture. Section 3 contains some theoretical background and the new approach to verify the MIRUP is proposed in Section 4. Results of numerical experiments are summarized in Section 5. Finally a conclusion is given which also includes consequences for practical applications. 2 The one-dimensional cutting stock problem The following cutting stock problem is investigated: 2

One-dimensional material objects (e.g. paper reels, iron slabs, wooden rods) of a given length L are divided into smaller pieces of desired lengths l 1,..., l m in order to fulfill the order demands b 1,..., b m. The goal is to minimize the total amount of stock material or, equivalently, to minimize the total waste. According to the typology of cutting and packing introduced by Dyckhoff [3] this is a problem of type 1/V/I/R. In the following standard model [6] an integer m-vector a j = (a 1j,..., a mj ) T with mi=1 l i a ij L represents a cutting pattern. The integer variable x j gives the number of times the cutting pattern a j is cut. n z = x j j=1 min s.t. n a ij x j b i, i = 1,..., m, (1) j=1 x j 0, integer, j = 1,..., n, where n denotes the number of cutting patterns. Model (1) is a special case of the model of the standard linear integer optimization (minimization) problem: z = c T x min s.t. Ax b, (2) x 0, integer. Here, an instance is given by the following Input-data: the number of variables n, the number of restrictions m (without non-negativity conditions), the matrix A of coefficients, the objective function c and the right hand side b. Hence to any instance E = (m, l, L, b) of the one-dimensional cutting stock problem an instance (m, n, A, c, b) of type (2) can be assigned. Without loss of generality, all Input-data are assumed to be integer, and to ensure the problem can be solved, we suppose max i=1,...,m l i L. A set P := P (I) = {(m i, n i, A i, c i, b i ) : i I, m i, n i Z +, A i Z m i n i, c i Z n i, b i Z m i } where I is a non-empty set of indices, defines a subproblem (or problem) of type (2). For a given instance E = (m, n, A, c, b) P of a problem P of type (2) let z (E) denote the optimal value of the integer problem and z c (E) the optimal value of the corresponding LP relaxation z = c T x min Then we give the following definitions. s.t. Ax b, x 0. (3) Definition 1 A linear integer minimization problem P has the integer property (IP) if z (E) = z c (E) for all E P. 3

Definition 2 (Baum/Trotter) A linear integer minimization problem P has the integer round-up property (IRUP) if z (E) = z c (E) for all E P. Definition 3 A linear integer minimization problem P has the modified integer roundup property (MIRUP) if z (E) z c (E) + 1 for all E P. Remark: With respect to the integer property we have the well-known result that a problem P has the integer property if and only if for any instance of P the coefficient matrix is totally unimodular (see in [10]). In many practical computations it has been observed that for instances E of onedimensional cutting stock problems P the difference between z (E) and z c (E) is small [2], [15]. The investigations of Marcotte [9], Fieldhouse [5] and ourselves show that the standard one-dimensional cutting stock problem does not possess the IRUP. But there is no counterexample known to the MIRUP. For that reason we formulate the following Conjecture: The standard one-dimensional cutting stock problem has the modified integer round-up property. Remark: In [12] it is shown that the one-dimensional cutting stock problem has the MIRUP when m 5. There the MIRUP is also proven for a wide range of instances of the onedimensional cutting stock problem independently from the number m of small lengths. 3 Theoretical background Using the LP relaxation to obtain feasible integer solutions, one has to overcome the difficulties which arise when rounding the solution of the LP relaxation. Simply rounding up leads always to feasible integer solutions but these are in general worse and can be improved easily by more intelligent rounding procedures. Such methods are proposed in [7] and [14]. In order to investigate the MIRUP we consider a concept which consists in defining and investigating a so-called residual problem which is defined in accordance with a solution of the LP relaxation. The approach is based on the following lemmas. Let E = (m, l, L, b) be an instance of the one-dimensional cutting stock problem and let a j, j = 1,..., n, be the corresponding cutting patterns. Furthermore let A denote the coefficient matrix according to (2). Let x c denote an optimal solution of the corresponding LP relaxation. Rounding down of x c yields an integer vector x with x j = x c j, j = 1,..., n. If x x c then a residual problem can be defined with right hand side b := b Ax. The instance of the residual problem is E := (m, l, L, b). In comparison with the original problem E we have a new (reduced) demand vector b. Lemma 1 Let E be an instance of the one-dimensional cutting stock problem and E a corresponding residual instance. Then it holds that: if z(e) z c (E) + 1 then z (E) z c (E) + 1, i.e. if the residual problem has the MIRUP then the initial problem has this property too. 4

Proof: Let e = (1,..., 1) T R n. Then it holds: z c (E) = e T x c = e T x + e T {x c } z (E) e T x + z(e) e T x + z c (E) + 1 = e T x + z c (E) + 1 = z c (E) + 1. It is easy to verify that a similar statement is true if x c is replaced by any feasible solution x of the LP relaxation fulfilling e T x = z c (E). Therefore in some cases the LP bound does not need to be computed exactly. In order to verify whether a solution x with e T x = z c (E) is found, a dual-like problem has to be solved. For more details we refer to [13]. In order to decide whether an instance or a residual instance has the IRUP or the MIRUP we use the following two lemmas which are proven in [12]. Lemma 2 Let E be an instance of the one-dimensional cutting stock problem and E a corresponding residual instance. Then it holds that: if z c (E) 1.5 then E has the IRUP, if 1.5 < z c (E) 3 then E has the MIRUP. Lemma 3 Let E be an instance of the one-dimensional cutting stock problem and E a corresponding residual instance. Furthermore let x c denote an optimal solution of the LP relaxation problem of E. Then it holds that: if z c (E) > κ 1 then E has the IRUP, if z c (E) > κ 2 then E has the MIRUP, where κ := j sign(x c j). Remark: Using Lemma 1 we can verify that if the residual instance possesses the MIRUP, then the initial instance possesses the MIRUP too. On the other hand, such a conclusion cannot be stated in the case that if E does not have the IRUP then neither does E, since instances E of the one-dimensional cutting stock problem exist whereby z c (E) = z (E) < z + z(e) = z c (E) + 1. Such an instance is L = 396, l 1 = 132, l 2 = 99, l 3 = 44, l 4 = 36 with b 1 = 2, b 2 = 3, b 3 = 9 and b 4 = 6. The continuous solution is 3 0 0 0 2 2 0 3 0 + 3 4 4 0 + 1 0 9 + 6 0 11 0 = 3 9 0 0 0 11 6 with z c (E) = 391 since the applied patterns contain no waste, and z = 1. The residual 132 problem E with b = (2, 3, 0, 6) T has the optimal value z(e) = 3 but z (E) = 3. A further helpful tool for estimations and bound computations in a branch and bound algorithm is based on the next lemma which is simple to verify. 5

Lemma 4 Let E be an instance of the one-dimensional cutting stock problem. Then there exists an optimal solution x which fulfills the additional constraint j with a j b = x j = 0. (4) In comparison to the integer problems, the restriction (4) often leads to a stronger lower LP bound in comparison to the bound obtained from the LP relaxation of (1) especially in the case of residual instances. 4 The new approach In the following, we describe the concept for computational investigations with respect to the MIRUP for the one-dimensional cutting stock problems. Let E = (m, l, L, b) be an instance of the one-dimensional cutting stock problem. Then E is investigated with respect to the MIRUP as follows. If the LP relaxation does not lead to an integer solution we define a residual instance E as described above. Then because of Lemma 2 or 3 we are either able to conclude the MIRUP for E or further investigations are necessary. In the latter case we try to find an appropriate integer solution for E using two heuristics. If this also fails a stronger lower bound for E is computed in accordance to Lemma 4. The new bound may lead to a termination of the process in some cases or to a further (second) residual instance Ê. Investigating Ê analogously, may lead the termination. If all this fails Ê or/and E has to be solved using a branch&bound-algorithm until a solution is found which proves the MIRUP of E, or until an instance is found which does not possess the MIRUP. Moreover, during the investigations we distinguish between the knowledge of exact solution and the MIRUP. In the description of the procedure we consider only that case when the LP relaxation problems are solved exactly. The modifications with respect to the remark of Lemma 1 are omitted for the sake of simplicity. The two heuristics are described briefly below. At first we give the general algorithm for testing an instance with respect to the MIRUP. Algorithm MIRUP test for instance E 1. Computation of an optimal solution x c of the corresponding LP relaxation. If an integer solution is found - STOP. Otherwise build the residual instance E. Let z := e T x c. Then z c (E) = z c (E) z. If E has the MIRUP because of Lemma 2 or 3 - STOP. 2. Heuristic solution of E yields the value z h. If z h = z c (E) then an optimal solution to E is found - STOP. If z h = z c (E) + 1 then E has the MIRUP - STOP. 3. Computation of an optimal solution x r of the LP relaxation of E in consideration of the condition i {1,..., m} : a ij > b i x j = 0. If x r is integer and e T x r = z c (E) then an optimal solution of E is found - STOP. If x r is integer and e T x r = z c (E) + 1 then E has the MIRUP - STOP. 6

4. If ẑ := e T x r b > 0 then build a further residual instance Ê with right hand side := b A x r, otherwise go to step 5. Heuristic solution of Ê yields ẑh. If ẑ + ẑ h = z c (E) then an optimal solution to E is found - STOP. If ẑ + ẑ h = z c (E) + 1 then E has the MIRUP - STOP. Exact solution of Ê by using a branch and bound algorithm yields ẑ. If ẑ + ẑ = z c (E) then an optimal solution to E is found - STOP. If ẑ + ẑ = z c (E) + 1 then E has the MIRUP - STOP. 5. Exact solution of E by using a branch and bound algorithm yields z. If z = z c (E) then an optimal solution to E is found - STOP. If z = z c (E) + 1 then E has the MIRUP - STOP. 6. If this point is reached then E is an instance of the one-dimensional cutting stock problem which does not have the MIRUP. Remark: The last step takes into consideration the fact that the MIRUP is not proven for the standard one-dimensional cutting stock problem so far. In order to decide in such a situation whether E has the MIRUP or not, further investigations are required. For the description of the heuristics used in Step 2 and 4 let E = (m, l, L, b) denote an instance of the one-dimensional cutting stock problem. Without loss of generality we may assume L l 1 >... > l m > 0. These heuristics are used to get a feasible cutting pattern a Z m taking into consideration the right hand side b. In the first heuristic the cutting pattern is constructed by using the greedy method. There the pieces are packed in the given sequence if there is sufficient length but no more than required. Heuristic 1 (L, b, a). L := L, b := b; for i := 1 to m do a i := min{ b i, L l i }; b i := b i a i ; L := L a i l i ; In the second heuristic the cutting pattern is constructed by using a modified greedy method. Let β := z c (E) e T x c (E). That means, β is the sum of all fractional parts of the optimal solution x c of the corresponding linear relaxation problem. Using the weight 1/β a more equalized cutting pattern is constructed in comparison to Heuristic 1. Heuristic 2 (L, b, a); L := L; b := b, β := max{1, β}; for i := 1 to m do a i := min{ b i, L }; b β l i := b i a i L := L a i l i ; i if L l T a l m then heuristic1 ( L, b, a), a := a + a. 7

In order to get a feasible solution for the instance E the first or second heuristic are repeatedly applied until b = 0. As the computational experiments show it is not necessary to use the branch&boundalgorithm and therefore we will not go into the details of a description. Instead we refer to [13]. 5 Computational results In order to investigate the one-dimensional cutting stock problem with respect to the MIRUP, we solved series of randomly generated instances. Thereby the Input-data are chosen from a uniform distribution on some ranges given below. For a given material length L and a chosen m [m, m] the lengths l i are in [ L/(m 2), L/2] and the order quantities b i are in [2m, 10m]. The LP relaxation problems are solved using the simplex method with column generation where the new pattern is obtained by the greedy algorithm, and if this fails the corresponding knapsack problem is solved exactly using a dynamic programming forward state algorithm. The generation process is terminated (columns termination) if the optimality condition is fulfilled (column opt) or, secondly, if a given maximum number of solved knapsack problems is exceeded (column iter) or, thirdly, if the decrease of the objective function value is smaller then 0.1 within the last m/2 iteration steps (column eps). In the cases iter and eps it is ensured that the current objective function value z fulfills the condition z c (E) = z in accordance to the remark to Lemma 1. The columns value characterize the best found integer solution. exact gives the number of instances which are exactly solved because IRUP is fulfilled. MIRUP gives the number of instances where a solution proving the MIRUP is found but optimality is not ensured. The column E characterizes the first residual problem. lb counts the number of terminations because of Lemma 3 and h1 gives the number of instances where Heuristic 1 leads to a termination due to a solution which proves the MIRUP. Since Heuristic 1 is used before Heuristic 2 and since in any case after applying Heuristic 1 a solution is found proving the MIRUP, Heuristic 2 is not needed. But in the reverse order, Heuristic 1 has to be used in some cases after Heuristic 2. The last column describes the average time per instance in seconds on a PC 486 DX, 50 MHz. The following tables summarize the results for L = 1000 (Table 1), L = 2000 (Table 2) and L = 3000 (Table 3). For each range [m, m] 20 instances were generated. The average times for the different L-values reflect the dependence of the knapsack procedures on the absolute size of the Input-data. Since the solution of the instances with more than m = 110 pieces requires a large amount of computer time and otherwise, since the instances with m 110 make clear the dependence of the computer time and the stock length L, we omit the solution of instances with L = 2000, L = 3000 and m > 110. 8

Table 1: L = 1000 m termination value E average m m opt iter eps exact MIRUP lb h1 time 11 20 20 0 0 19 1 0 20.2 21 30 20 0 0 14 6 0 20 1.4 31 40 20 0 0 12 8 0 20 5.8 41 50 15 5 0 16 4 0 20 13.0 51 60 18 0 2 18 2 0 20 24.5 61 70 19 0 1 18 2 0 20 39.2 71 80 20 0 0 17 3 0 20 59.2 81 90 20 0 0 18 2 0 20 91.6 91 100 20 0 0 19 1 0 20 138.0 101 110 20 0 0 19 1 0 20 192.1 111 120 20 0 0 15 5 0 20 266.7 121 130 20 0 0 16 4 0 20 335.3 131 140 20 0 0 14 6 0 20 492.9 141 150 20 0 0 17 3 0 20 667.0 Table 2: L = 2000 m termination value E average m m opt iter eps exact MIRUP lb h1 time 11 20 20 0 0 20 0 1 19.3 21 30 20 0 0 17 3 0 20 2.3 31 40 19 0 1 12 8 0 20 8.4 41 50 11 0 9 15 5 0 20 21.7 51 60 10 0 10 17 3 0 20 41.6 61 70 11 0 9 14 6 0 20 75.9 71 80 11 0 9 18 2 4 16 116.7 81 90 14 1 5 16 4 2 18 168.0 91 100 12 0 8 18 2 1 19 250.5 101 110 16 4 0 11 9 0 20 336.8 Table 3: L = 3000 m termination value E average m m opt iter eps exact MIRUP lb h1 time 11 20 20 0 0 17 3 1 19.5 21 30 19 0 1 15 5 0 20 2.6 31 40 17 0 3 15 5 0 20 13.0 41 50 5 0 15 15 5 0 20 35.2 51 60 3 0 17 16 4 0 20 65.7 61 70 5 2 13 15 5 0 20 112.1 71 80 5 0 15 13 7 0 20 142.7 81 90 4 2 14 16 4 0 20 241.8 91 100 4 8 8 18 2 1 19 349.3 101 110 7 7 6 17 3 2 18 540.7 9

As the tables show, all randomly generated instances of the one-dimensional cutting stock problem fulfill the MIRUP. Moreover, a corresponding integer solution was found by the greedy heuristic. Applying the branch and bound algorithm was not necessary. In many cases (see column exact) an optimal solution was obtained with this approach. For 80.4 percent of the instances the optimality is proved for the solution found. Hence, solving one-dimensional cutting stock problems via column generation and computing of a greedy integer solution for the residual problem leads to an integer solution whose value does not diverse more than one unit in comparison to the optimal solution. Additionally to the investigations with respect to the MIRUP, we have paid attention to the exact difference between the optimal value and the LP bound. For that reason we define the MAXGAP problem as follows. Let P be a problem of type (2). Definition 4 The maximum gap problem (MAXGAP) of a problem P consists in determining the maximum difference between the optimal value z (E) and the LP lower bound z c (E) with respect to all instances E of P : (P ) := max E P {z (E) z c (E)}. In the case of the standard one-dimensional cutting stock problem we have (s. [11]) (1D CSP) 137 132 = 1.0378... This gap arises, e.g., in the residual problem E given in Section 3. However, our conjecture is (1D CSP) = 1.0378.... In order to investigate the one-dimensional cutting stock problem numerically with respect to MAXGAP a similar concept as in Section 4 can be used. But all the numerical experiments did not lead to an instance with a larger gap. Therefore our conjecture seems to be still more realistic. We refer to [13] for a branch&bound algorithm for solving onedimensional cutting stock problems exactly. 6 Conclusional Remarks The extensive computational tests support the conjecture that the one-dimensional cutting stock problem fulfills the MIRUP. If this conjecture can be proven there arises some new insights in Integer Programming, especially for problems with submodular objective functions. Based on the optimistic numerical results, investigations are motivated with respect to theoretical statements and for practical applications in the case of one-dimensional cutting stock problems as well as for higher dimensional cutting stock problems. Moreover, the proposed approach can be used directly as a heuristic procedure for solving instances of the one-dimensional cutting stock problem. And the objective function value of the obtained solution differs from the optimal value by 1 at the most. Although the algorithm is not polynomial the obtained results show that instances of medium size can be solved within an acceptable running time. 10

Acknowledgement The authors wish to thank Uta Sommerwei and Jan Riehme for implementing the algorithm and doing the extensive computational tests. Furthermore we thank the editors and the anonymous referees for their helpful comments. References [1] Baum, S., and Trotter, L.E., Jr., Integer rounding for polymatroid and branching optimization problems, SIAM J. Alg. Disc. Meth. 2 (1981) 4, 416-425. [2] Diegel, A., Integer LP solution for large trim problem, Working Paper, University of Natal, South Africa, 1988. [3] Dyckhoff, H., A typology of cutting and packing problems, it EJOR 44 (1990) 145-159. [4] Dyckhoff, H., and Finke, U., Cutting and packing in production and distribution, Physica Verlag, Heidelberg, 1992. [5] Fieldhouse, M., The duality gap in trim problems, SICUP-Bulletin No. 5, 1990. [6] Gilmore, P.C., and Gomory, R.E., A linear programming approach to the cutting stock problem, Operations Res. 9 (1961) 849-859. [7] Johnston, R.E., Rounding algorithms for cutting stock problems, Asia-Pacific J. of OR 3 (1986) 166-171. [8] Marcotte, O., The cutting stock problem and integer rounding, Mathematical Programming 33 (1985) 82-92. [9] Marcotte, O., An instance of the cutting stock problem for which the rounding property does not hold, Oper. Res. Lett. 4 (1986) 5, 239-243. [10] Nemhauser, G.L., and Woolsey, L.A., Integer and Combinatorial Optimization, John Wiley & Sons, New York 1988. [11] 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, 1992. [12] Scheithauer, G., and Terno, J., Theoretical investigations on the MIRUP (modified integer round-up property) for the one-dimensional cutting stock problem, Preprint MATH-NM12-1993, TU Dresden, 1993 (submitted for publication). [13] Scheithauer, G., and Terno, J., A branch&bound algorithm for solving onedimensional cutting stock problems exactly, Working Paper, TU Dresden, 1994 (submitted for publication). 11

[14] Terno, J., Lindemann, R., und Scheithauer, G., Zuschnittprobleme und ihre praktische Lösung, Verlag Harry Deutsch, Thun und Frankfurt/Main, und Fachbuchverlag Leipzig 1987. [15] Wscher, G., and Gau, T., Two approaches to the cutting stock problem, IFORS 93 Conference, Lisboa 1993. 12