200 product - In this paper a two-dimensional trim-loss problem connected to the paper-converting industry is considered. The problem is to produce a set of ...
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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
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ELSEVIER
European Journal of Operational Research 104 (1998) 572- 581
Theory and Methodology
Solving a two-dimensional trim-loss problem with MILP Tapio Westerlund
a, *, J o h n n y I s a k s s o n b, I i r o H a r j u n k o s k i
a
a Process Design Laboratory, ,4bo Akaderai University, Biskopsgatan 8, FIN-20500 tlbo, Finland b Wisapak Corporation, FIN-68600 Pietarsaari, Finland
Received 31 May 1996; accepted 24 October 1996
Abstract
In this paper a two-dimensional trim-loss problem connected to the paper-converting industry is considered. The problem is to produce a set of product paper rolls from larger raw paper rolls such that the cost for waste and the cutting time is minimized. The problem is generally non-convex due to a bilinear objective function and some bilinear constraints, which give rise to difficulties in finding efficient numerical procedures for the solution. The problem can, however, be solved as a two-step procedure, where the latter step is a mixed integer linear programming (MILP) problem. In the present formulation, both the width and length of the raw paper rolls as well as the lengths of the product paper rolls are considered variables. All feasible cutting patterns are included in the problem and global optimal cutting patterns are obtained as the solution from the corresponding MILP problem. A numerical example is included to illustrate the proposed procedure. © 1998 Elsevier Science B.V. Keywords: Optimization; Mixed integer non-linear programming; Mixed integer linear programming; Trim-loss problems; Scheduling
problems
I. I n t r o d u c t i o n
Different solutions for solving trim-loss problems in the paper industry have been given during recent decades (Haessler, 1971; Coverdale and Wharton, 1976; Hinxman, 1980; W~ischer, 1990; Westerlund et al., 1995; Harjunkoski et al., 1996; Skrifvars et al., 1996). A good survey of trim-loss and assortment problems is given in Hinxman (1980). In the early papers linear programming formulations were considered, while the latter papers are based on mixed integer and mixed integer non-linear programming formulations. When solving a trim-loss problem where in addition to the lengths of the product paper
* Corresponding author. Fax: + 358-2-265-4791.
rolls, both the widths and lengths of raw paper roils as well as the cutting patterns are considered variables, the resulting problem will be non-convex. The non-convex problem can be transformed into convex form using exponential variable transformations as shown in Skrifvars et al. (1996). The original problem is, in this case, expanded both with respect to new variables and constraints which, however, has a negative impact on the computational work. The non-convex problem can also be transformed into linear form using binary variable transformations as has been shown in Harjunkoski et al. (1996). However, also in this case the original problem has to be expanded both with respect to the variables and constraints, which as in the convex transformation case can burden the computational work substantially.
0377-2217/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0377-2217(97)00005-2
T. Westerlund et al. / European Journal of Operational Research 104 (1998) 572-581
In Westerlund et al. (1995), Harjunkoski et al. (1996) and Skrifvars et al. (1996) trim-loss problems with given lengths of the raw paper and product paper rolls were considered. In the present paper, both the width and length of the raw paper as well as the length of the product paper rolls are considered variables. It is found that the resulting trim-loss problem can be solved quite efficiently with a similar two-step procedure to that given in Westerlund et al. (1995). This is especially the case when the number of feasible cutting patterns is not very large. In the two-step procedure, all feasible cutting patterns are solved as an initial problem using an explicit enumeration procedure given in Westerlund et al. (1995). Thereafter, a resulting mixed integer linear programming problem gives the solution to the original, non-convex MINLP problem. Since all feasible cutting patterns are included in the resulting linear problem, global optimality of the originally non-convex problem can be ensured.
2. The trim-loss problem We will consider a problem of cutting ' I ' different types of paper product rolls from ' R ' different types of raw paper rolls. Each type, i, of product paper corresponds to a certain width, b,, and to a certain amount, wi.o,de~, (in weight) given by an order. The length of the product paper rolls produced should be within given tolerances specified by a lower and an upper limit, l~i n and lmax, respectively. The raw paper rolls from which the product rolls are cut have widths defined by the variables, Br, as well as lengths specified by lower and upper limits, Lr.min and Lr,max, respectively. The total length of each raw paper is defined by the variables, L~, and the number of raw paper rolls of each width is defined by the integer variables, M r. The index, r, corresponds to the raw paper rolls of type r. The sum of the widths of the product rolls at each cutting pattern ' j ' for each raw paper ' r ' must be within a width range Br,mi n t o Br,ma x . The raw paper width, B r should of course be greater or equal to the width of the cutting pattern with the maximum width. The number of product rolls in each cutting pattern should not exceed a given number, N~.~ax. The total number of different cutting patterns is
573
defined by the parameter, Jr, for each raw paper, r. Jr is obtained when solving the feasible cutting patterns with the explicit enumeration procedure in Westerlund et al. (1995). For each product roll of type i the cutting patterns are defined by the numbers, hi,j, r. ni.s. r is the number of product paper rolls of type i in the jth cutting pattern for the rth raw paper. We also define integer variables, mj, r, c o r r e s p o n d i n g to the multiple of the cut, j, of raw paper, r. The total length of the paper in the cutting pattern, j, for the rth raw paper is defined by real variables, j,r"
Also some binary variables are defined. Yr defines if the raw paper type, r, will be used or not. The variables, Yr, can then be used when defining costs for the format changes (change to another raw paper width). The binary variables, Yj, r, define if the cutting pattern, j, of the rth raw paper is used or not. The binary variables, Ys.r, can then be used when defining costs for knife changes. Using the above set of variables, both the loss function and the constraints for the problem can be defined. We will first consider the part of the loss function that corresponds to the trim-loss and overproduction. First, as in Westerlund et al. (1995) we define five different areas of paper: the total area, the ordered area, the produced area, the area of trim-loss and the area of over-production. The total area, A T , corresponding to all raw paper used, can be expressed by the widths, Br, as well as the total lengths, Lr, of the different raw papers used. The total area of each raw paper, At, is given by A r = B r . L r , and the total area A T of all raw papers is R
AT= E Lr'Br.
(!)
r=l
The ordered area, A o, corresponds to the area of the product paper in the original order. A o can be expressed in terms of the 'ordered' lengths, /i.or~er' as well as the widths, b~, of the ordered product paper according to I
Ao = ~
bi'li,orae r.
(2)
i=l
The 'ordered' product paper length, //.order, can
T. Westerlundet al. / European Journal of Operational Research 104 (1998) 572-581
574
simply be expressed in terms of the ordered amount, Wi.order (in weight), according to Wi.order
1
K= (C'-
//.order: aw " b-----~'
C) "a o = ( C ' -
C ) . E bi'li.o,der"
(3)
where A W is the area weight and b~ is the width of the corresponding product paper roll. Since it is not always possible to produce exactly the amount of paper that is ordered, an area of production, Ap, is defined. The produced area, Ap, can be expressed in terms of the lengths, lj, r, of the trimsets, the numbers of different product paper rolls in each cutting pattern, hi,j, r, and the widths, b~, of the product paper rolls according to
R Jr I AP = E E lj.," E bi'ni,j.~. rffil j = l i=1
(4)
ATL = A T -- A p .
(5)
As was mentioned above the produced area, Ap, does not need to be equal to the ordered area, A o. Thus it is convenient also to define an area of over-production, Aop, which is simply given by A o p = A p -- A o.
(6)
Although the over-production consists of full quality paper, it cannot be treated in the same way as the original order. Nor can it be treated in the same way as the trim-loss. However, the over-production can, in most paper-converting mills, be sold to the customer as full quality paper if a reduced price is offered. We now define an area price for the product paper, C, as well as an area price for the over-production, C'. Then the economic loss because of the over-production is ( C - C ' ) . A o p . The total economic loss, E L , corresponding to the trim-loss and over-production can be expressed as
C') "Aoe.
(7)
By inserting Expressions (5) and (6) into (7) we find that the economic loss because of the trim-loss and the over-production can be expressed as
EL = C . AT - C' . Ap + ( C' - C ) " A o.
(8)
(9)
iffil
The total price, P, of the delivery (the order including the cost of the paper rolls corresponding to the over-production (which are sold at a reduced price)) is given by
P = C.A o + C"Aop I = ( C - C')" E bi "li.orde, i=1 R
+ C'' E
Jr
1
E lj.r" E bi'ni,j,r"
r = l jffil
Now the area of the trim-loss, ATE, can be defined. We find that the trim-loss is given by
EL = C'ATL + (C-
The last term in (8) is a constant, which is given by
(10)
i=1
A total loss function for the cutting should be minimized. The loss function should not only include the cost of the lost paper but costs of other economic losses too. Using different raw paper widths gives rise to costs because of so-called format changes (changes to another raw paper width). We therefore define the cost of the format change as C'r. Moreover, costs for the cutting time including the knife changes should be included in the loss function. We therefore define c).r as the cost for the knife change at the jth cutting pattern for the rth raw paper. The cost of the machine time is included in the cost function by multiplying the length of the used raw paper by the hourly cost, C m, for the cutting machine divided by the machine speed, vm. Including all cost terms discussed above the trimloss problem can now be formulated as follows
(.(
Cm
min K + Y'~ C, A~ + - Ar Lr lj.r Yr~) r ni j r Mr mJ r r~ l Urn
Jr •L r - E C j , r . l j , r + C ; . r r + jr1
Jr )) EC'j,r'Yj,r /=l
(11)
subject to
-A~+
bi'ni,j,
r
"L~-H'.(1-yj.,)
