* Corresponding author: Tel.: #886-3-5728709; fax: #886-3-5723792. E-mail address: hlli@cc.nctu.edu.tw(H.-L. Li).
Technical note
A fast algorithm for assortment optimization problems
Han-Lin Li*, Jung-Fa Tsai
Institute of Information Management, National Chiao Tung University Hsinchu, Taiwan 300, Republic of China
Received 1 January 2000; received in revised form 1 March 2000
Abstract
Assortment optimization problems intend to seek the best way of placing a given set of rectangles within a minimum-area rectangle. Such problems are often formulated as a quadratic mixed 0}1 program. Many current methods for assortment problems are either unable to "nd an optimal solution or being computa-tionally ine$cient for reaching an optimal solution. This paper proposes a newmethod which "nds the optimum of assortment problem by solving fewlinear mixed 0}1 programs. Numerical examples showthat the proposed method is more computationally e$cient than current methods.
Scope and purpose
Assortment optimization problems aim at cutting given rectangular pieces from a larger rectangle where the wasteful area is minimized. Current assortment optimization methods (Chen et al., European Journal of Operational Research 1993; 63: 362}67; Li and Chang, European Journal of Operational Research 1998; 105: 604}12) are either unable to "nd optimal solution or being computationally ine$cient for reaching the optimal solution. This paper proposes a fast algorithm which only requires to solve three linear programs. Numerical examples demonstrate that the proposed algorithm is much faster than current methods. By utilizing this algorithm, many practical cutting programs in industries could be solved e$ciently. 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Assortment; Cutting; Optimization
1. Introduction
Assortment problems occur when a number of small rectangular pieces need to be cut from a large rectangle to get minimum area. Assortment optimization techniques have been widely
0305-0548/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 0 0 ) 0 0 0 3 5 - 6
method can "nd the optimal solution. In addition, the computational time of the proposed method is much less than that in current methods.
2. Problem formulation
Given n rectangles with "xed lengths and widths. An assortment optimization problem is to allocate all of these rectangles within an enveloping rectangle, which has minimum area. Denote x and y as the width and the length of the enveloping rectangle (x'0, y'0), the assortment optimization problem is stated brie#y as follows:
Minimize xy
subject to 1. All of n rectangles are non-overlapping.
2. All of n rectangles are within the range of x and y. 3. 0(x
)x)x and 0(y
)y)y (x
, x , y and y are constants).
The related terminologies used in assortment models, referring to Li and Chang [4], are described below(Fig. 1).
(pG, qG): Dimension of rectangle i, pG is the long side and qG is the short side, pG and qG are constants, i3J, J is the set of given rectangles.
(x, y): The top right corner coordinates of the enveloping rectangle, x and y are variables. xG: Distance between center of rectangle i and original point along the x-axis.
yG: Distance between center of rectangle i and original point along the y-axis.
sG: An orientation indicator for rectangle i, i3J. sG"1 if pG (the longer dimension of rectangle i) is parallel to the x-axis; sG"0 if pG is parallel to the y-axis. Take Fig. 1 for example, s"0 and s"1.
The conditions of non-overlapping between rectangles i and k can be reformulated by introducing two binary variables uGI and vGI as follows (Fig. 2).
Condition 1: uGI"0 and vGI"0 if and only if rectangle i is at the right of rectangle k. Condition 2: uGI"1 and vGI"0 if and only if rectangle i is at the left of rectangle k. Condition 3: uGI"0 and vGI"1 if and only if rectangle i is above the rectangle k. Condition 4: uGI"1 and vGI"1 if and only if rectangle i is belowthe rectangle k.
Fig. 1. Graphical illustration of assortment problem.
Fig. 2. Graphical illustration of non-overlapping conditions.
The assortment optimization problem can then be formulated as follows, referring to Li and Chang [4]. Assortment problem P1: Minimize xy subject to (1) (xG!xI)#uGIx#vGIx*[pGsG#qG(1!sG)#pIsI#qI(1!sI)], ∀i,k3J, (2) (xI!xG)#(1!uGI)x#vGIx*[pGsG#qG(1!sG)#pIsI#qI(1!sI)], ∀i,k3J, (3) (yG!yI)#uGIy#(1!vGI)y*[pG(1!sG)#qGsG#pI(1!sI)#qIsI], ∀i,k3J, (4) (yI!yG)#(1!uGI)y#(1!vGI)y*[pG(1!sG)#qGsG#pI(1!sI)#qIsI], ∀i,k3J, (5) x *x*xG#[pGsG#qG(1!sG)], ∀i3J, (6) y *y*yG#[pG(1!sG)#qGsG], ∀i3J, (7) xG![pGsG#qG(1!sG)]*0, ∀i3J, (8) yG![pG(1!sG)#qGsG]*0, ∀i3J, (9)
3. Li and Chang approach
Li and Chang [4] substitutes x and y in (1) as follows: x"V %
E
2E\E#V, y"W & F
2F\F#W,
whereV and W are small positive variables. V and W are the pre-speci"ed constants which are the upper bounds ofV and W, respectively. E and F are 0}1 variables, and G and H are integers which denote the number of required 0}1 variables for representing x or y. The polynomial term xy in (1) is then represented as xy"V % E 2E\Ey#W & F 2F\FV#VW. (10)
A full Li and Chang model is reformulated as a linear mixed 0}1 program below. Model 1: Minimize V % E 2E\zE#W & F 2F\uF subject to (11) zE*y#y(E!1), g"1, 2,2, G, (12) uF*V#V(F!1), h"1, 2,2, H, (13) (2)}(9)
zE*0, uF*0, E, F30, 1.
The major di$culty of Model 1 is that it involves G#H additional 0}1 variables. The smaller the tolerable errors (i.e., V and W), the larger the size of G and H and the longer the CPU time for solving the problem.
Fig. 3.
4. Proposed method
Denote F as a feasible set of Problem P1, F"(2), (3), (4),2, (9). De"ne a local optimum of Problem P1 below.
De5nition 1. A point (xH, yH), (xH, yH)3F, is a local optimum of Problem P1 if xHyH)(xH$)(yH$) for all (xH$, yH$)3F, where is a noticeable small positive value. Remark1. By referring to De"nition 1, there are eight neighborhood points for a given reference (x, y) as shown in Fig. 3:
(x#, y#), (x#, y), (x#, y!), (x, y#), (x, y!), (x!, y#), (x!, y) and (x!, y!).
Consider a linear mixed 0}1 program below. P2: Minimize z"x#y
subject to (x, y)3F, x*y.
where the constraint x*y is to denote x as the larger side of the rectangle. Let the obtained objective value of solving P2 be z.
P2 may have multiple optimal solutions (x, y). What we are interested in is to"nd one of these solutions, which has minimal xy value.
Consider the following proposition.
Proposition 1. Let S be a set of solutions of P1. S"(x, y) x#y"z,(x, y)3F, x*y. If there is a point (x, y)3F in which x*x for all (x, y)3F, then xy)xy.
Proof. Since x*x, there exist a *0 such that x"x! and y"y#. Because x!*y#, it is clear that x!y!2*0. We then have
Table 1
Computational comparison of two models Problem No. No. of
rectangles pG qG
CPU time (hh : mm : ss) Objective value
Model 1 Proposed method Model 1 Proposed method 1 4 24 20 18 16 00 : 05 : 12 00 : 00 : 03 1178 1178 16 14 21 7 2 5 33 10 '10 : 00 : 00 00 : 01 : 18 NA 1518 30 11 25 15 18 14 18 10
Proof. Examining the eight neighborhood points of (x, y) in Remark 1 as follows.
(i) Some of these points may not be the feasible points of P1. Since the minimal value of x#y for (x, y)3F is x#y"z, it is clear that three points (x, y!),(x!, y) and (x!, y!) are not feasible points of P1.
(ii) Since x"Max x (x, y)3F, x*y, x#y"z, point (x#, y!) is also an infeasible point of P1 (x#'x).
(iii) It is unclear that whether these points (x#, y#), (x#, y) and (x, y#) are feasible or not for P1. However, since (x#, y#)'xy, (x#)y'xy and x(y#)'xy, (x, y) is better than these three points.
(iv) By referring to Proposition 1, we have, (x!)(y#)'xy.
Since each neighborhood point of (x, y) is either infeasible or inferior to (x, y). Therefore, (x, y) is a local optimum of P1 following De"nition 1. 䊐
Fig. 5. Result for "ve rectangles. Fig. 4. Result for four rectangles.
5. Numerical examples
Consider the following assortment optimization problem adopted from Chen et al. [3]: Some given rectangles are required to be placed within a rectangle which has minimum area. The sizes of pieces of rectangles are given in Table 1 and Figs. 4 and 5. Here we solve the same problem using Chen et al. [3] model, Model 1 and the proposed model by LINGO 5.0 (LINDO SYSTEMS INC., 1998, a common-used optimization package) [5,6] running in a personal computer.
Chen et al.'s model treats Problem 1 by "xing the value of y as y"36, then they solved a nonlinear mixed 0}1 program to obtain a local optimal solution with an objective value that equals 1224. Model 1 solves the problem by specifying V"W"0.1, and obtains the optimal solution which has the objective value 1178. Proposed method solves the problem and obtains the same solution as found by Model 1. Table 1 shows that for the cases with four and "ve rectangles, the proposed method spends much less time than Model 1 for "nding the optimal solution.
[2] Beasley JE. An algorithm for the two-dimensional assortment problem. European Journal of Operational Research 1985;19:253}61.
[3] Chen CS, Sarin S, Balasubramanian R. A mixed-integer programming model for a class of assortment problems. European Journal of Operational Research 1993;63:362}7.
[4] Li H-L, Chang C-T. An approximately global optimization method for assortment problems. European Journal of Operational Research 1998;105:604}12.
[5] Schrage L. LINDO Release 5.3 Chicago: LINDO SYSTEMS INC, 1994.
[6] Schrage L. Optimization modeling with LINGO, 2nd ed. Chicago: LINDO SYSTEMS INC, 1998.
Han-Lin Li is professor of Information Management at National Chiao-Tung University, Taiwan, ROC. He received his Ph.D. from the University of Pennsylvania in 1983. In his research, Dr. Li investigates information management systems and optimization. Currently, his main interest is in the areas of global optimization, supply chain management, and competence set analysis. His publications include 45 journal articles and two books.
Jung-Fa Tsai is a research sta! at Institute of Information Management where he received his master degree in 1997. His research interests are global optimization, nonlinear integer programs and geographic information system.
