An alternative formulation for certain fuzzy set-covering problems

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ELSEVIER

Available online at www.sciencedirect.com M A T H E M A T I C A L

AND

8 C I E N C E ~ ' ~ D I R E C T " C O M P U T E R

MODELLING

Mathematical and Computer Modelling 42 (2005) 363-365

www.elsevier.com/locate/mcm

An Alternative Formulation for Certain

Fuzzy Set-Covering P r o b l e m s

C. I. CHIANG

Department of Marketing and Distribution Management University of Hsuan Chuang, Hsinehu, Taiwan

M. J. HWANG

Department of Information and Finance Management National Chiao Tung University, Hsinchu, Taiwan

Y. H. LIE

Department of Marketing and Distribution Management University of Hsuan Chuang, Hsinchu, Taiwan

(Received February 2004i revised and accepted May 2004)

A b s t r a c t - - T h e fuzzy set-covering model using auxiliary 0-1 variables and a system of inequalities was developed by Hwang et al. [1]. By taking logarithm of the inequalities constraints and using

K e y w o r d s - - S e t covering problems, Fuzzy set-covering problems, 0-1 linear programming, Natural logarithm function.

1. I N T R O D U C T I O N

Given two subsets I = {1, 2 , . . . , m} and J = {1, 2 , . . . , n} of integers. Let

Dj = { ( i , # j ( i ) ) : i e I } be a fuzzy subset of I, and #j(i) e [0,1] be the membership grade

Hwang et al. [1] presented the

of i E I , using the m e m b e r s h i p function pj of fuzzy set Pj. following form of the fuzzy set-covering problem,

(P1) where

Min

~ cjxj

j = l s.t. 1 - ~ I ( 1 - p j ( i ) xj) >_a, j = l xj E {0,1}, j = 1 , 2 , . . . , n ,

i, i f 6 e

Xj = O, otherwise. i : l , 2 , . . . , m ,

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364 C . I . CHIANG et al.

(P2)

Note t h a t a, a given real number (level degree), represents the desired level which for each i E I, the membership grade of i is no less t h a n the level degree 5. According to Theorem 1 in [1], Problem (P1) can be transformed to Problem (P2) by replacing the product of 0-1 variables in constraints (1) with auxiliary variables and a system of linear inequalities. The details can be found in [1].

n

Min

E cjxj

j = l

s.t.

E #jr (i) xjt +

E

(

- 1)k+l ~j~ (i) yJlJ2...J~ - 5,

t = 1 k=2 j l <J2 <...<Jk \ t = 1

i = 1,2,...,m,

2yJlJ>..Jt <-- YJlJ>..Jt-~ q- XJt ~

1 +

YJlJ2...Jt' k

YJlj2...Jk = I I x j t ' t = l

xj C {0,1}, j = 1 , 2 , . . . , n .

t = 1 , 2 , . . . , k ,

The above formulation is mathematically elegant, however, solutions are difficulty to find due to the fact t h a t the number of decision variables is proportion to the number of auxiliary variables and extra constraints, which usually grows exponentially [2] as the number of variables increases.

2 . T R A N S F O R M A T I O N O F C O N S T R A I N T S

In (P1), since

xj E

{0, 1}, obviously that 1 -

#j(i)x 5

= (1 - tzj(i)) xj. It follows that 1 -

n

l-Ij=l(1 - t~j(i), xj) = 1 - I I j = l ( 1 - pj(i))xj. Therefore, the constraint (1) can be rewritten as

1 - a _> lkI (1 - p j ( i ) ) x' . (2)

j = l

By taking the natural logarithm function on both sides of inequality (2) and changing the signs, (2) can be transformed into a linear constraint as,

- i n (1

- a) < ~

xj

[ - In (1 - # j ( i ) ) ] . (3)

j = l

Notice t h a t a = 1, i.e., the crisp cases, is not considered here. Therefore, (P1) can be rewritten as follows. (Pa) n Min

E cjzj

j = l n s.t.

~ x j [ - l n ( 1 - # j ( i ) ) ] _ > - l n ( 1 - a ) ,

j = l i = 1 , 2 , . . . , m , x j E { 0 , 1 } , j = 1 , 2 , . . . , n , where 0 < a < 1.

One can easily see the number of variables, and hence, the number of inequalities, is reduced greatly after (P3) is obtained from (P2) by applying the natural logarithm to the constraints of (P2).

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An Alternative Formulation for Certain 365

3. A N E X A M P L E A N D C O N C L U D I N G R E M A R K S

The same numerical example as in [11 is used to illustrate the advantages of the proposed method. Table 1 shows the matrix of #j(i), i = 1,2,...,5~ j = 1 , 2 , . . . , 4 .

Table 1. The matrix of ttj (i) [1]. ~ G o a l s M e a n s i = 1 i = 2 i = 3 i = 4 i = 5 j = l (cl =4) 0.4 0.1 0.5 0.7 0.8 j = 2 (c2 = 3) 0.1 0.3 0.8 0.2 0.6 j = 3 (c3 = 5) 0.3 0.7 0.2 0.9 0.4 j = 4 (c4 = 2) 0 . 5 0 . 9 0 . 4 0.1 0 . 2

According to (P3), we obtain the following mathematical programming,

Min s.t. 4xl + 3x2 +

5X3

0.51xl + 0.11x2 0.11xl + 0.36x2 0.69Xl + 1.61x2 1.20xl + 0.22x2 1.61Xl + 0.92x2 + 2x4 + 0.36x3 + 0.69x4 > 0.69, + 1.20xa + 2.30x4 ~ 0.69, + 0.22x3 + 0.51x4 > 0.69, + 2.30x3 + 0.11x4 > 0.69, + 0.51xa + 0.22x4 > 0.69, x y E { 0 , 1 } , j = 1 , 2 , . . . , 4 .

Note that the coefficients in the constraints axe obtained by taking the natural logarithm of the corresponding values in the table 1 and a = 0.5 is set for the value of the right-hand sides of the constraints. By using software LINGO, a n optimal solution is obtained,

* = 1 , * = 0 , * = 0 , * = 1 .

X 1 X 2 X 3 X 4

From the example, obviously, we can see the reduction of the formulation. With the proposed technique, the fuzzy set-covering model does not require additional 0-1 variables and constraints which complicate the formulation. Consequently, the model may be easily applied to many large-scale problems in the real world.

R E F E R E N C E S

1. M.J. Hwang, C.I. Chiang and Y.H. Liu, Solving a fuzzy set covering problem, Mathl. Comput. Modelling 40

(7/8), 861-865, (2004).

2. G.L. Nemhauser and L.A. Wolsey, Integer and Combinational Optimization, John Wiley and Sons, New