Discrete Optimization
Approximately global optimization for assortment problems
using piecewise linearization techniques
Han-Lin Li
a,*, Ching-Ter Chang
b, Jung-Fa Tsai
aa
Institute of Information Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC
b
Business Department of Education, National Changhua University of Education, Hsinchu 30050, Taiwan, ROC Received 25 January 2000; accepted 23 May 2001
Abstract
Recently, Li and Chang proposed an approximate model for assortment problems. Although their model is quite promising to find approximately global solution, too many 0–1 variables are required in their solution process. This paper proposes another way for solving the same problem. The proposed method uses iteratively a technique of piecewise linearization of the quadratic objective function. Numerical examples demonstrate that the proposed method is computationally more efficient than the Li and Chang method. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Cutting; Assortment; Global 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. Recently, Li and Chang [1] developed a method for finding the global optimal solution of the assortment problem. Li and Chang’s method, however, requires to use numerous 0–1 variables to linearize the polynomial objective function in their model, which would cause heavy computational burden. This paper proposes instead a piecewise linearization method. The major advantage of this method is that it uses
much less number of 0–1 variables to linearize the quadratic objective function than used in Li and Chang’s model. The computational efficiency can therefore be improved significantly. The numerical examples demonstrate that the computation time of the proposed method is much less than that in Li and Chang’s model.
2. Problem formulation
Given n rectangles with fixed lengths and widths. An assortment optimization problem is to allocate all of these rectangles within an envelop-ing 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 optimiza-tion problem is stated briefly as follows:
www.elsevier.com/locate/dsw
*
Corresponding author. Tel.: 5728709; fax: +886-3-5723792.
E-mail address: hlli@cc.nctu.edu.tw (H.-L. Li).
0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 1 9 4 - 1
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 6 x and 0 < y 6 y (x and y are constants). An assortment optimization problem can be formulated below. The related notations are those of Li and Chang [1]. 2.1. Assortment problem Minimize xy ð1Þ subject to ðx0 i x 0 kÞ þ uikxxþ vikxx P1 2½pisiþ qið1 siÞ þ pkskþ qkð1 skÞ 8i; k 2 J ; ð2Þ ðx0 k x 0 iÞ þ ð1 uikÞxx þ vikxx P1 2½pisiþ qið1 siÞ þ pkskþ qkð1 skÞ 8i; k 2 J ; ð3Þ ðyi0 y 0 kÞ þ uikyyþ ð1 vikÞyy P1 2½pið1 siÞ þ qisiþ pkð1 skÞ þ qksk 8i; k 2 J ; ð4Þ ðyk0 y 0 iÞ þ ð1 uikÞyy þ ð1 vikÞyy P1 2½pið1 siÞ þ qisiþ pkð1 skÞ þ qksk 8i; k 2 J ; ð5Þ xP x P x0iþ1 2½pisiþ qið1 siÞ 8i 2 J ; ð6Þ yP y P yi0þ1 2½pið1 siÞ þ qisi 8i 2 J ; ð7Þ x0i1 2½pisiþ qið1 siÞ P 0 8i 2 J ; ð8Þ yi01 2½pið1 siÞ þ qisi P 0 8i 2 J ; ð9Þ
where uik, vik, si, sk are 0–1 variables, and x, y, x0i,
x0k, y0
i and yk0 are bounded continuous variables, pi
and qi are length and width of ith object.
Constraints (2)–(5) are the non-overlapping conditions and constraints (6)–(9) ensure that all rectangles are within the enveloping rectangle.
Li and Chang [1] proposed an approach for solving this problem to obtain a global optimum. The basic idea of their method is to approximately
substitute x and y continuous variables in (1) by a set of 0–1 variables thus to linearize the prod-uct term xy. Problem (1)–(9) can then be refor-mulated as a linear mixed 0–1 problem which can be solved to reach a global optimum within a tolerable error.
3. Li and Chang approach
Li and Chang [1] substitute x and y by the following 0–1 representation: x¼ eex XG g¼1 2g1hgþ ex; ð10Þ y ¼ eey XH h¼1 2h1dhþ ey; ð11Þ
where exand ey are small positive variables. eexand
eey are the pre-specified constants which are the
upper bounds of ex and ey, respectively. hg and dh
are 0–1 variables, and G and H are integers which denote the number of required 0–1 variables for representing x and y.
Then the Li and Chang model is reformulated as a linear mixed 0–1 program below.
3.1. Model 1 [1] Minimize eex XG g¼1 2g1zgþ eey XH h¼1 2h1uh ð12Þ subject to zgP yþ yyðhg 1Þ; g¼ 1; 2; . . . ; G; ð13Þ uhP exþ eexðdh 1Þ; h¼ 1; 2; . . . ; H ; ð14Þ constraints in (2)–(9) zgP0; uhP0; hg;dh2 f0; 1g: ð15Þ
The major difficulty of Model 1 is that it involves
Gþ H additional 0–1 variables. The smaller the
tolerable errors (i.e., ex and ey), the larger the size
of G and H and the longer the CPU time for solving the problem.
4. Proposed new linear strategy
This paper proposes another strategy for lin-earizing the quadratic objective function xy in (1).
Define K be a set expressed as
K¼ f0 < x 6 x 6 x; 0 < y 6 y 6 y; x; y 2 F ;
F is a feasible setg:
It is clear that an optimization program P1: fMinimize Obj1 ¼ xy j x; y 2 Kg is equivalent to the program below.
P2 :fMinimize Obj2 ¼ ln x þ ln y j x; y 2 Kg:
Following propositions discuss the proposed approach of linearizing the logarithmic terms ln x and ln y.
Proposition 1 [3]. A logarithm function ln x; 0 <
a16x 6 am; as shown in Fig. 1, can piecewise
lin-early be approximated ln x _¼¼ ln ^xx¼ ln a1þ s1ðx a1Þ þX m1 j¼2 sj sj1 2 x aj þ x aj ; ð16Þ
where aj; j¼ 1; 2; . . . ; m; are the break points of
ln x, aj< ajþ1 and sjare the slopes of line segments
between aj and ajþ1;
sj¼
ln ajþ1 ln aj
ajþ1 aj
for j¼ 1; 2; . . . ; m 1:
Proposition 2. Since ln x is concave function, it is clear that the approximation bounds ln x from
be-low. That means ln x P ln ^xx.
Proposition 3 (Lower bound). Consider the follow-ing two programs:
P2 :fMinimize Obj2 ¼ ln x þ ln y j x; y 2 Kg:
P3 :fMinimize Obj3 ¼ ln ^xxþ ln ^yyj x; y 2 Kg:
Program P3 provides a lower bound on Program P2 due to Proposition 2.
Now we discuss the way to linearize ln ^xx.
Con-sider the following proposition.
Proposition 4 (Linearization). ln ^xx in (16) can be
linearized as follows. ln ^xx¼ ln a1þ s1ðx a1Þ þX m1 j¼2 ðsj sj1Þðajujþ x aj wjÞ ð17Þ where ðiÞ amuj6x aj6amð1 ujÞ for j¼ 2; 3; . . . ; m; ðiiÞ amuj6wj6amuj for j¼ 2; 3; . . . ; m; ðiiiÞamðuj 1Þ þ x 6 wj6amð1 ujÞ þ x for j¼ 2; 3; . . . ; m; ðivÞ ujP uj1 for j¼ 2; 3; . . . ; m;
ðvÞ ujare 0–1 variables and wjP0
for j¼ 2; 3; . . . ; m:
Proof. If x ajP0 then uj¼ 0 and wj¼ 0 based
on (i) and (ii); which results in
ajujþ x aj wj¼ x aj þ x aj 2 ¼ x aj:
If x aj<0 then uj¼ 1 and wj¼ x based on (i)
and (iii); which results in
ajujþ x aj wj¼ x aj þ x aj 2 = ¼ 0:
Therefore, ln ^xx in (16) is equivalent to (17). Now
we consider condition (iv).
Since aj1< aj, if x < aj (i.e. uj¼ 1) then
x < ajþ1 and we have ujþ1¼ 1.
If x > ajþ1 (i.e. ujþ1¼ 0) then x > aj, and we
have uj¼ 0.
Therefore, it is true that ujP uj1, for j¼
2; 3; . . . ; m.
Condition (iv) is used to accelerate the
compu-tational speed of solving the problem.
5. Solution algorithm
From the above discussion, the proposed
algo-rithm is as follows: Let Sr and Tr be respectively a
set of break points of ln x and ln y at the rth iter-ation. Denote e as a tolerable error. Then the lin-earization (17) is built up iteratively as follows.
Let Sr¼ Sr1UfxðrÞg, Tr¼ Tr1UfyðrÞg, where
‘U’ means union. Denote the number of elements
in Sras mr, and the number of elements in Tras nr.
Solving the following linear mixed 0–1 program: Minimize
ðx;yÞ Objðxðr þ 1ÞÞ þ Objðyðr þ 1ÞÞ
¼ ln a1þ s1ðx a1Þ þX mr1 j¼2 ðsj sj1Þðajujþ x aj wjÞ þ ln b1þ t1ðy b1Þ þX nr1 j¼2 ðtj tj1Þðbjvjþ y bj qjÞ subject to Restrictions part 1: xxuj6x aj6xxð1 ujÞ; xxuj6wj6xxuj; xxðuj 1Þ þ x 6 wj6xxð1 ujÞ þ x; ujP uj1; for j¼ 2; 3; . . . mr ðvariable xÞ; a1; a2; . . . ; amr 2 Sr; a1¼ x < a2< < amr ¼ xx; Restrictions part 2: yyvj6y bj6yyð1 vjÞ; yyvj6qj6yyvj; yyðvj 1Þ þ y 6 qj6yyð1 vjÞ þ y; vjP vj1;
for j¼ 2; 3; . . . nr ðvariable yÞ;
b1; b2; . . . ; bnr 2 Tr;
b1¼ y < b2< < bnr ¼ yy;
uj, vjare 0–1 variables, wj; qjP0.
Restrictions part 3: constraints in (2)–(9).
Let the solution beðxðr þ 1Þ; yðr þ 1ÞÞ.
IfjObjðxðr þ 1ÞÞ ln xðr þ 1Þj < e and jObjðyðr þ
1ÞÞ ln yðr þ 1Þj < e then terminate the process,
andðxðr þ 1Þ; yðr þ 1ÞÞ is the optimal solution.
Otherwise, let r¼ r þ 1 and resolve the above
program. (A precise procedure is available by re-quest, etc. from the authors.)
Proposition 5 (Convergence). The above algorithm
(run with e _¼¼0) terminates with the incumbent
solu-tion ð^xx; ^yyÞ being optimum to the assortment
prob-lem (1)–(9) when r! 1.
Proof. By the concavity of ln x (and ln y) in (16)
and the mean value theorem, ln ^xx (and ln ^yy) are
the lower bounds of ln x (and ln y); ð^xx; ^yyÞ
there-fore is the optimal solution.
6. Numerical examples
Consider the following assortment optimization problems adopted from Li and Chang [1]: 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. Here we solve the same problem using Model 1 [1] and proposed model by LINGO [2], a common-used optimization package, running in a Pentium III 1000 personal computer.
Model 1 solves problem 1 by specifying eex ¼
eey ¼ 0:1, and obtains the global optimal solution
ðx; yÞ ¼ ð31; 38Þ with the objective value 1178.
Pro-posed model solves Problem 1 by specifying eex ¼
eey ¼ 0:1 and obtains the same solution as found by
Model 1. Table 1 shows that for Problem 1 con-taining four rectangles, the proposed model only spends 1/7 of CPU time as spent in Model 1. For Problem 2 containing five rectangles, the proposed model uses much less CPU time than Model 1. The associated graphs are presented in Figs. 2 and 3.
To compare the capability of the two models in treating larger sizes of assortment problems, two
other problems are examined as shown in Table 2. All rectangles in these two problems are squares. For Problem 3 where the number of squares is 8,
the proposed model spends less than 1/7 CPU time as spent in Model 1 to obtain a global optimum. The result is depicted in Fig. 4. For Problem 4 with
Fig. 2. Result for Problem 1 (four rectangles). Table 1
Computational comparison of models with rectangles Problem Number of
rectangles
pi qi CPU time (hh:mm:ss) Objective value
Model 1 Proposed model Model 1 Proposed model Problem 1 4 24 20 00:05:12 00:00:44 1178 Global optimum 1178 Global optimum 18 16 16 14 21 7 Problem 2 5 33 10 >10:00:00 00:16:16 NA 1518 Global optimum 30 11 25 15 18 14 18 10
Fig. 3. Result for Problem 2 (five rectangles).
Table 2
Computational comparison of models with squares Problem Number of
squares
Side pi CPU time (hh:mm:ss) Objective value
Model 1 Proposed model Model 1 Proposed model Problem 3 4 1 1 00:04:48 00:00:37 25 Global optimum 25 Global optimum 3 2 2 1 3 3 Problem 4 4 1 1 >10:00:00 00:08:26 NA 30 Global optimum 4 2 2 1 3 3
nine squares, Model 1 cannot find the solution within 10 hours while the proposed model takes eight and half minutes to find the global solution. The solution is depicted in Fig. 5. The reported CPU times can be improved if these problems are run in a workstation computer instead of running in a personal computer.
7. Conclusions
This paper proposes a piecewise linearization method to solve the assortment problem. By piece-wisely linearizing the quadratic objective function in the assortment problem, the proposed method reformulates the original problem as a linear mixed 0–1 program. Solving the linear mixed 0–1 problem iteratively, the proposed method can fi-nally find a global optimum. Numerical examples demonstrate that the proposed method uses much less CPU time than that in [1] for reaching the global optimum.
References
[1] H.L. Li, C.T. Chang, An approximately global optimization method for assortment problems, European Journal of Operational Research 105 (1998) 604–612.
[2] LINGO Hyper 6.0: Lindo Systems Inc., 1999.
[3] H.L. Li, C.S. Yu, A fuzzy multiobjective program with quasiconcave membership functions and fuzzy coefficients, Fuzzy Sets and Systems 109 (2000) 59–81.
Fig. 4. Result for Problem 3 (eight squares).
