EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 105 (1998) 594-603
Theory and Methodology
Different transformations for solving non-convex trim-loss problems by MINLP Iiro Harjunkoski a, Tapio Westerlund a'*, Ray Pi3rn b, Hans Skrifvars a a Process Design Laboratory Jtbo Akademi University Biskopsgatan 8, FIN-20500 Turku, Finland b Deparmzent of Mathentatics, Jlbo Akademi University Fiinriksgatan 3 B, FIN-20500 Turku, Finland
Received 19 August 1996; accepted 21 January 1997
Abstract In the present paper trim-loss problems, often named the cutting stock problem, connected to the paper industry are considered. The problem is to cut out a set of product paper rolls from raw paper rolls such that the cost function, including the trim loss as well as the costs for the over production, is minimized. The problem is non-convex due to certain bilinear constraints. The problem can, however, be transformed into linear or convex form. The resulting transformed problems can, thereafter, be solved as mixed-integer linear programming problems or convex mixed-integer non-linear programming problems. The linear and convex formulations are attractive from a formal point of view, since global 6ptimal solutions to the originally non-convex problem can be obtained. However, as the examples considered will show, the numerical efficiency of the solutions from the different transformed formulations varies considerably. An example based on a trim optimization problem encountered daily at a Finnish paper converting mill is, finally, presented in order to demonstrate differences in the numerical solutions. (~) 1998 Elsevier Science B.V. Keywords: Optimization; Mixed integer non-linear programming; Integer linear programming;Trim-loss problems
1. Introduction Trim-loss problems in the paper industry have in recent years predominantly been solved by means of linear programming combined with heuristic rules in order to handle non-linearities and discrete decisions (Haessler, 1971; Coverdale et al., 1976; W~ischer, 1990). A good survey o f available solution methods for trim-loss and assortment problems is given in Hinxman (1980). The solution o f trim-loss problems with a mixed integer linear programming (MILP) algorithm is considered in Harjunkoski et al. (1996). *Corresponding author. Fax: +358-2 2654 791; e-mail:
[email protected]
In the present paper, it is demonstrated that nonconvex trim-loss problems can be transformed into convex or linear form and, thereafter, efficiently solved by MILP algorithms or mixed integer non-linear programming ( M I N L P ) algorithms for convex M I N L P problems. The resulting transformed problems are, however, expanded both in terms of the number o f variables and constraints. Thus, selecting the most efficient transformation and solution strategy from the numerical point o f view is not an altogether straightforward matter. In the paper, results are given to demonstrate the differences between the numerical solutions. Efficient algorithms for solving ILP problems (Crowder, Johnson and Padberg, 1983; Van Roy and Wolsey, 1987), MILP problems (Land and Doig,
0377-2217/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S0377-2217(97)00066-0
L Harjunkoski et aL/European Journal of Operational Research 105 (1998) 594-603
1960) and even for solving mixed integer non-linear programming problems are presently available. In the case of MINLP, efficient methods are available especially for convex MINLP problems; see Geoffrion (1972), Duran and Grossmann (1986), Yuan, Pibouleanu and Domenech (1989), Viswanathan and Grossmann (1990), Westerlund, Pettersson and Grossmann (1994), Fletcher and Leyffer (1994), and Westerlund and Pettersson (1995).
problem. For each type, we introduce the variables nij, which define how many times each product paper roll, i, is included in the cutting pattern, j, and a multiple, mj, defining how many times the corresponding pattern is repeated. Thus, the problem of minimizing the total cost of the trim loss and the knife changes can be formulated as follows:
mj'yj'nij ~ j=l
2. Formulation of the trim problem
595
cj . my + Cj . yj
( 1)
subject to I
The problem of cutting different product paper rolls from raw paper rolls is considered. The width of a raw paper roll is given by Bmax. There are I different types of product rolls to be cut. Each type of product roll corresponds to a certain width, bi, and the total number of a specific product roll that is to be cut is given by a number,///,order, specified in a product order. The length of the product paper rolls is assumed to be equal to the length of the raw paper rolls. The sum of the widths of the product paper rolls at each type of cut 'j' must be between Bmax and Bmax - A, where A is a given width tolerance. The total number of product paper rolls at each cut is, due to practical constraints in cutting and winding, limited to a certain number, Nmax. The limit, Nmax, is equal to five in the considered paper converting mill. Exceeding the limit would give rise to difficulties in separating the cut paper rolls from each other after the winding. Cutting the rolls generally results in waste, which is called trim-loss, and in over-production. The cost of the trim loss and the over-production of paper should be minimized. There are several different strategies to handle the over-production but in this paper we maintain that over-production is somewhat minimized when the total production is minimized. In order to restrict the over-production, an upper bound for production could be introduced to determine the number of extra products that can be sold at a lowered price. This problem is considered in Westerlund et al. (1997). Let the cost of the raw material be cj. The change of a cutting pattern involves a cost since the cutting machine has to be stopped before repositioning the knives. Let the cost coefficient, Cj, denote this cost. A total number, J, of different types of cutting patterns (trimset types) must, further, be defined for the
bi" nii - Bmax
