different ways of describing the problem as an ILP/MILP problem. ... ways to formulate and solve it as an ILP or MILP problem and compare the efficiency of.
Computers chem. Engng Vol.20, Suppl.,pp. S121-SI26, 1996 Copyright© 1996 ElsevierScience Ltd S0098-1354(96)00031-2 Printed in Great Britain.All rightsreserved
( ~ Pergamon
0098-1354196 $15.00+ 0.130
DIFFERENT FORMULATIONS FOR SOLVING TRIM LOSS PROBLEMS IN A PAPER-CONVERTING MILL WITH ILP IIRO HARJUNKOSKI,
TAPIO WESTERLUND*
JOHNNY
and HANS
ISAKSSON**
SKRIFVARS
Process Design Laboratory at Abo Akademi University, Biskopsgatan 8, FIN-20500/~BO, Finland **Wisapak Corporation, FIN-66900 Pietarsaaxi, Finland A b s t r a c t - In the present paper, trim loss problems connected to the paper-converting industry are analyzed and solved. The objective is to produce a set of paper rolls from storage rolls such that a cost function including the minimization of the trim loss as well as the time for cutting is considered. The problem is a non-convex integer non-linear programming (INLP) problem, due to its bilinear constraints. The problem can, however, be written in an expanded lineax form and can, thus, be solved as an integer linear programming (ILP) or a mixed integer linear programming (MILP) problem. The linear formulation is attractive from the point of view of formality. One drawback of linear formulations is the increased number of vaxiables and constraints they give rise to. It is, though, of interest to compare different ways of describing the problem as an ILP/MILP problem. There has previously been some academic interest in solving trim loss problems as linear programming problems. In this paper, we will present a general INLP formulation, some ways to formulate and solve it as an ILP or MILP problem and compare the efficiency of these different approaches. The examples considered are taken from real daily trim optimization problems encountered at a Finnish paper-converting mill with a capacity of 100,000 tons/yeax.
K e y w o r d s - Optimization, Integer Linear Programming, Integer Non-Linear Programming, Trim Loss Problems, Scheduling Problems.
INTRODUCTION The trim loss problem arises when a wide raw paper roll is to be cut into smaller ones. Since the width of a product is fully independent of the width of the raw paper, we are dealing with a complicated combinatorial problem. Trim loss problems in the paper-converting industry have, in recent years, mainly been solved manually using certain heuristic rules. The practical problem formulation has, therefore, in most eases been restricted by the fact that the solution methods ought to be able to handle the entire problem. Consequently, only a suboptimal solution to the original problem has been obtained. The insufficient computer programs presently available have not been able to reduce the trim loss. That is the most likely reason why this rather mechanic work has been left to a manual stage. This means, in reality, that there has been an upper limit of raw paper widths considered in order to diminish the number of possible combinations. This upper limit then gives a lower limit for the total time needed, because the effective use of wider paper rolls naturally leads to shorter processing time. There axe a number of articles on the trim loss problem, especially from the late sixties and the seventies.
Gilmore ~ Gomory (1971) used linear programming approximations for the real cutting stock problem. Efforts to solve the non-linear problem with heuristic methods were made by, for example, Haessler (1971), Coverdale and Wharton (1976) and Johnston (1979). Mixed integer linear programming has also been a common tool for solving trim loss problems. A good overview of different formulations and solution methods is given by ttinxman (1980). * Author to whom all correspondence should be addressed
SI21
S122 FORMULATION
European Symposium on Computer Aided Process Engineering--6. Part A OF THE TRIM LOSS PROBLEM
In the following, the index i refers to products that have been ordered and will be produced, i.e., cut from wider paper rolls. The index j refers to a specific cutting pattern, a trim set. The problem is treated as a 1½-dimensional trim problem, since the length is assumed to be fixed and the width is variable. Consider a problem of cutting "I" different types of sheets. Each type of sheet corresponds to a certain width, Bi, and also corresponds to a certain number, Ni of sheets. The length of the sheets are all assumed to be similar, but the total width, at each type of cut, "j", is between Bt,mi n and B j , m ~ and the total number of sheets at each type of cut, is at most Nj,ma~. The Nj,m,z is a physical restriction which makes it possible to separate the sheets from each other. The cost of the material and the cuts should be minimized. Let J be the total number of different types of cuts (trim set types) and nit the corresponding number of sheets i in the j t h trim set type (see Figure 1.). Define also the number ms, the multiple for each cut type j and a binary variable, yj, defining if the cutting type j is to be used or not. We also need a weight factor, f , to give a weight for such products that exceed the demand. If f = 0 then all extra products produced are considered as trim loss. If, on the other hand, f = 1 then the extra products (naturally within specified constraints) are considered as a part of the production. By changing the value of f we can give a weight for extra products e.g. f = 0.8 means that the extra products can be sold with a reduced price of 80 % of the original price. We can now write a general non-linear formulation. The problem can thus be formulated as follows,
rain nlj,mj,yj
(1) j=l
i=1
subject to I
Z
B, . n # - Bj,,n~, _< 0
(2)
i----1 I
-
B,.
+ Bj,m , _< 0
(3)
i----1 I
Z
nit - Nj,m~,~ < 0
(4)
i=1
mj - M . y j < O
(5)
j=l,...,J
J
We -
n,t < o
(0)
j=l J
Z
mr" nit - Ni - Ai < 0
(7)
t=l
i = 1,...,I
nij, mj 6 Z +
{0,1} The cost-coefficient cj stands for the cost of raw material and Cj for the cost of a knife change. Due to the bilinear objective function and inequalities (6-7), the problem is non-linear (and even non-convex). M is a sufficiently large positive number (e.g. M = max {Ni}) and A, is the number of extra products i that can i
be sold with a reduced price. The variables are illustrated in the following figure, where all the rnj variables = 1, except m4 = 9.. We have six products and five different cutting patterns. One can observe that, e.g., in pattern 1, n l , = 1 and n21 = 2. The spill is shaded and the question mark indicates that the spill is wide enough to be eliminated by producing some extra product.
European Symposium on Computer Aided Process Engineering--6. Part A
.~
~i ~
Bj ,max
" £ :;
I
lj
S123
~'L
i=1
i=l
i=l
i=5
i=5
i=3
i=4
i=5
i=5
i=3
i=2 i=2
i=3
j=l j=2 j=3 j=4 j--4 F i g u r e 1. An example with six products and five cutting patterns
j=5
A Linear Formulation The problem can be expanded and rewritten in linear form using a set of binary variables, ~jk, corresponding to the variables, m j , as well as some slack variables, sijk, corresponding to the variables nij. We obtain the following MILP-formulation:
rrfin
nlj ~Yj~]Jj~~sijle
{~-~cj. j=l
2k-l'/3j,k'Bj,ma~--f'EE2k-l'sijk'Bi k=l
+Cj'yj}
(8)
i=1 kin1
subject to I
Z B, .n,j - Bj,mo~ < o
(9)
i=1 I
- ~ B~. n~j + Bj,m~. < 0
(10)
i=1 I
En,j
- N j , m ~
