Abstract--This work proposed a set-covering model using the concept of fuzzy set theory to define "fuzzy covers." The proposed fuzzy set-covering model can be ...
MATHEMATICAL AND COMPUTER MODELLING
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Mathematical and Computer Modelling 40 (2004) 861-865 www. elsevier.corn/locate/mere
Solving a Fuzzy Set-Covering P r o b l e m M. J. HWANG Department of Library and Information Science University of Hsuan Chuang, Hsinchu, Taiwan C . I. CHIANG Department of Marketing and Distribution Management University of Hsuan Chuang, Hsinchu, Taiwan
Y. H. L i u Department of Mathematics, University of Nebraska at O m ah a Omaha, NE 68182-0243, U.S.A.
(Received May 2003; revised and accepted September 2003) A b s t r a c t - - T h i s work proposed a set-covering model using the concept of fuzzy set theory to define "fuzzy covers." The proposed fuzzy set-covering model can be reduced to a nonlinear integer programming problem which is easily solvable with modern software. This model is a nature extension of the classical set-covering model, and is able to handle uncertainty. (~) 2004 Elsevier Ltd. All rights reserved. K e y w o r d s - - S e t - c o v e r i n g problems, Fuzzy sets, Fuzzy set-covering problem, Algebraic sum operator.
1. I N T R O D U C T I O N A classical set-covering problem considers the subsets I = {1, 2 , . . . , m} and J = {1, 2 , . . . , n} of integers. A collection p of subset Pj of I is a cover of I, if the union of the members of go is I, i.e., let Pj C I, go = {Pj : j C J0}, J0 C J, go is a cover of I, if U{Pj : j E J0} = I. For each Pj, there is a positive number cj associated with it. The classical set-covering problem finds an optimal cover go* = {Pj : j C J~}, J~ C J, such that, for any cover go = {Pj :j • J0}, J0 C J, we have ~-~J~ep* cj < ~P~e~ cj. The following is the classical set-covering model,
Min E j=l
cjxj
n
s.t.
E a~jxJ > 1, i = 1 , 2 , . . . , m , j=l
xj c {0,1},
j = 1,2,...,n,
This paper is based primarily on research supported by the National Science Council under Contract NSC-912416-H-364-001.
0895-7177/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.10.015
Typeset by JtA~-TEX
862
M.J. HWANGet al.
where xj ----
1,
ifPjEp*,
O,
otherwise,
and aij --
1,
ifi•Pj,
0,
otherwise.
The classical set-covering problem is a well-known combinatorial optimization problem and also known from many applications including facility location, assigning customers to delivery routes, airline crews to flights, and workers to shifts [1-4]. Similarly, a collection of fuzzy subsets of I, /Sj, is called a fuzzy cover of I, denoted by ~5, if I is a fuzzy subset of the union of the members of gS. This problem has been studied before, for example, Zimmermann [5,6] using the max-min operator to define his membership function, or the quality of the fuzzy covering. In this work, we present an alternative of the measurement of the covering, which is defined in next section.
2. F U Z Z Y C O V E R S A N D F U Z Z Y S E T - C O V E R I N G
PROBLEM
Let/Sj be a fuzzy subset o f / a n d be denoted by/Sj = { ( i , # j ( i ) ) : i • I}, where pj(i) • [0,1] is the membership grade of i • I using the membership function #j of fuzzy set/sj. Then, the union of two fuzzy sets is defined as follows. DEFINITION 1. The union of two fuzzy sets Pj U Dk is defined as
Pj u Pk = {(i,,j,k (i)): i • ~}, where ~j,k(i) = ,j(i)
+ ,k(i),j
- (i),k(i).
Note that Definition 1 is also called algebraic sum of two fuzzy sets. There were many discussions on this "sum" in literature; for details, please refer to [7,8]. Since 1 - . j , k (i) = 1 - . j (i) - . k (i) + . j (i) . k (i) = (1 - , j ( i ) ) (1 - . k ( i ) ) ,
the membership grade of i of the union of two fuzzy sets /5i and /Sk can be represented by #j,k(i) = 1 - [(1 - #j(i))(1 - gk(i))]. Consequently, for the union of more than two fuzzy sets, n
= { ( i , . ( 1 , 2 ..... ~) (i)) : i e I } ,
i=1 the membership grade of i of the union can be denoted by n
.(1,2,
,~)(i) = 1 - [I
(1 - . j (i)).
j=l
Let ~5 = {/sj : j E d0} be a collection of fuzzy sets. A fuzzy cover is defined as follows. DEFINITION 2. ~a = {/Sj : j E J0} is called a fuzzy cover with degree ~ E [0, 1], also called (x-cover, i f V i E I, LJjedo PJ = {(i,#(1,2,...,n)(i)) : i E I}, where mini#(1,2 .....n)(i) = c~, j E J0. Now, we are ready to define a fuzzy set-covering problem (FSCP). The FSCP is to find an optimal s-cover gSa with respect to a given set I and the desired degree ~, so that, for each i E I, the membership grade of i is no less than the level degree (~. If for each fuzzy set/Sj of I, there
Solving a Fuzzy Set
863
is a positive number cj associated with it, then, the FSCP can be formulated as the following mathematical programming, n
Min E
cjxj,
j=l n
s.t.
j=l zj E {0,1},
(P1)
i= 1,2,...,m,
_>
1 - H(1
j=l,2,...,n,
where xj =
1,
if/SjEg5 *,
0,
otherwise.
Let's take a closer look at the constraints in (P1). Since
(1-#j(i)xj)=l+E
(-1)k
j=l
= 1+
E
#J'(i)xjt
jx
