Chance constrained fuzzy goal programming with right-hand side uniform random variable coefficients

Fuzzy Sets and Systems 109 (2000) 107–110

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Chance constrained fuzzy goal programming with right-hand side uniform random variable coecients W. Mohammed Ministry of Higher Education, King Faisal University, College of Management Sciences & Planning, AL-AHSA, P.O. Box 1760, Saudi Arabia Received January 1996; received in revised form April 1998

Abstract We present a basic idea about stochastic goal programming and the approach of chance constrained linear goal programming (CCLGP). We developed the method to transform probabilistic goal programming into deterministic goal programming and consider the stochastic fuzzy goal programming problem when the right-hand side coecients are random variables distributed according to uniform distribution. A chance constrained goal programming in the case of having uncertain aspiration levels is considered. However, these aspiration levels are random variables with known probability distribution functions. The c 2000 Elsevier Science B.V. All equivalent deterministic goal program is developed and an illustrative example is given. rights reserved.

1. Introduction In this paper we present a basic idea about stochastic linear goal programming and the approach of chance constrained linear goal programming (CCLGP). We developed the method to transform probabilistic goal programming into deterministic linear goal programming when the right-hand side coecients are random variables. This problem can be formulated as Pr (Gi (Xj )6bi )¿i ;

xj ¿0

for j;

(1)

where xj are decision variables, j = 1; : : : ; n: bi are constants representing available resources, i = 1; : : : ; m. We now present our contributions to the area of CCGP. To this end we assume that, bi is a ran-

dom variable having the distribution function Fi , i = 1; : : : ; m. We deÿne F −1 (x) = inf {y: F(y)¿x} it is interesting to note that the basic idea behind the transformation of the stochastic constraints (1) into an equivalent Gi (xj )6Ki ;

(2)

where Ki = Fi (i ); i = 1; : : : ; m and the resulting problem is solved, as a deterministic one, then we call the obtained solution a “conservative” solution of the original chance constrained program. Here the word conservative means that the extent by which the ith constraint violated is at most, i In the following subsections, the Fi will be speciÿed to be a uniform distribution [9].

c 2000 Elsevier Science B.V. All rights reserved. 0165-0114/00/$ – see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 1 5 1 - 1

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2. Chance constrained linear programming when the right-hand side coecients have uniform distributions [3,4] We assume that aij , j = 1; : : : ; n; i = 1; : : : ; m are constants and bi ; i = 1; : : : ; m; are mutually independent random variables, having uniform distributions, bi ∼ U (ci ; di ), i.e., the density function of bi is   1 ci 6bi 6di ; (3) gi (bi ) = di − ci 0 otherwise with the cumulative distribution function given by   bi − ci ci 6bi 6di ; (4) Gi (bi ) = di − ci  1 di 6bi : Thus the stochastic problem (1) is equivalent to Gi (xj )6Ki ;

06i 61; for i = 1; : : : ; m;

(5)

where Ki is the solution of i = (Ki − ci )=(di − ci ) i.e., Ki = ci + i (di − ci )

(6)

or Kÿi = ci + ÿi (di − ci ): 3. A chance constrained fuzzy goal programming [6] Consider the GP as follows:

x¿0; 


i = ki − kÿi

(7)

where x is the vector of the decision variables, P stands for probability, Gi (xj ) is the ith goal constraint, bi is the aspiration level which is assumed to be a uniform random variable with a known probability distribution function, Fi (y) = P(bi 6y), i is a predetermined tolerance probabilistic measure to satisfy the  are system constraints. The symith goal, and Ax ¿C, bol refers to the fuzziÿcation of the tolerance measures i which means that the decision maker is satisÿed

and

Bi =

ki − kÿi : 2

The method employed by Huey et al. depends on Tiwari’s algorithm [2,8]. The proposed algorithm is to solve the same FGP problem by treating the modiÿed linear membership function, which is represented by a single line function, rather than the original triangular membership function, which is represented by a piecewise linear function. By minimizing the largest deviation of the highest membership value to 1, the best solution is then easily derived. The problem to be solved by this algorithm is as follows [2,5].    Gi (xj )  ; Min  = max
i (9) Kÿi 6Gi (xj )6Ki or alternatively,

s.t.

Gi (xj )6ci + i (di − ci )  Ax ¿C;

where

Max

Find X = (x1 ; x2 ; : : : ; xn ) to satisfy (according to speciÿc priority ranking) s.t.

with a solution even though it makes the probability in (6) less than i to a certain limit. Let these limits be ÿi , (ÿi ¡i ). Then according to Zimmermann [9]:  if Gi (xj ) = Bi ;  1    0 if Gi (xj )6kBi ;    (8) i (x) = Gi (xj ) − kBi ; kBi 6Gi (xj )6Bi ; 
i     k − Gi (xj )    i ; kBi 6Gi (xj )6ki ;
i

 = 1 −   Gi (xj ) − 1; ¿
i Gi (xj )  ; ¿1 −
i

(10)

Kÿi 6Gi (xj )6Ki : 4. Numerical example (1) A manufacturer produces two products. The unit proÿt of product 1 is $80, that of product 2 is $40. The plant manager wants to earn a proÿt of b1 ∼ U (320; 376). He seeks to sell b2 ∼ U (3; 3:9) of product 1 and b3 ∼ U (2; 2:8) of product 2. Here x1 ; x2

W. Mohammed / Fuzzy Sets and Systems 109 (2000) 107–110

represent the unit of products 1 and 2. The fullest possible extent with probability is greater than or equal to 1 = 0:8; 2 = 0:7; 3 = 0:9; ÿ1 = 0:763; ÿ2 = 0:5 and ÿ3 = 0:5, respectively. The manager assigns ÿrst priority to the proÿt goal G1 and the remaining goals are in the second priority.

For the above FGP problem, Huey et al. formulated a single linear programming problem as given below: Max

  = 1 − ; − + 1:66x1 610;

Solution: Step 1: According to the information provided in this example it is easy to ÿnd that k1 = 632; k2 = 6; k3 = 4; kÿ1 = 620; kÿ2 = 5:4 and kÿ3 = 3:8 Step 2: The transformed deterministic goal program is

 + 1:66x1 610; − + 5x2 620;  + 5x2 620; 80x1 + 40x2 = 632;

Find x = (x1 ; x2 ) so as to a = [(d+ 2

109

x1 66:6; − d− 3 ); (d1 )]

+ Lexico-min s.t. G1 : 80x1 + 40x2 ≈ 626; G2 : x1 ≈ 5; G3 : x2 ≈ 4:6; x1 ; x2 ¿0; i = 1; 2; 3: 1 (G1 ) =  1; 80x1 + 40x2 = 632;     0; 80x  1 + 40x2 6632 − 12;     80x1 + 40x2 − 620 ; 620680x1 + 40x2 6632; 12    644 − (80x1 + 40x2 )   ; 632680x1 + 40x2 6644;   12   0; 80x1 + 40x2 ¿644;  1; x1 = 6;      0; x1 66 − 0:6;      x1 − 5:4 ; 5:46x1 66; 2 (G2 ) = 0:6    6:6 − x1     0:6 ; 66x1 66:6;    0; x1 ¿6:6;  1; x2 = 4;      0; x2 64 − 0:2;      x2 − 3:8 ; 3:86x2 64; 3 (G3 ) = 0:2    4:2 − x2   ; 46x2 64:2;    0:2   0; x2 ¿4:2:

x1 ¿5:4; x2 64:2; x2 ¿3:8;  x1 ; x2 ¿0: ; Optimal solution:  = 0:17667; x1 = 5:91767;

 = 0:82233; x2 = 3:96467;

G1 = 632:

But by using Tiwari algorithm we formulated the following four subproblems and solved them analytically: Subproblem (1) Max:



s.t.

61:66x1 − 9; 65x2 − 19; 80x1 + 40x2 = 632; 5:46x1 66; 3:86x2 64; ; x1 ; x2 ¿0:

Solution:  = 0:82233; x1 = 5:91767; x2 = 3:96467; G1 = 632:

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Subproblem (2) Max:



s.t.

611 − 1:66x1 ; 621 − 5x2 ; 80x1 + 40x2 = 632; 66x1 66:6; 46x2 64:2; ; x1 ; x2 ¿0:

Solution: Infeasible solution. Subproblem (3) Max: s.t.

 61:66x1 − 9; 621 − 5x2 ; 80x1 + 40x2 = 632; 5:46x1 66; 46x2 64:2; ; x1 ; x2 ¿0: Solution:  = 0; x1 = 5:8; x2 = 4:2; G1 = 632: Subproblem (4) Max: s.t.

 611 − 1:66x1 ; 65x2 − 19; 80x1 + 40x2 = 632; 66x1 66:6; 3:86x2 64; ; x1 ; x2 ¿0: Solution: Infeasible solution.

Tiwari et al. claimed that subproblem 1 gives the highest membership value ( = 0:823), therefore it is the solution for the original problem. It can be seen that our solution is exactly the same as the ÿrst problem solution obtained by the Huey et al. method.

5. Conclusion In this paper the PFLGP is solved by using the chance-constrained approach and the linear membership function. The equivalent deterministic program is A LGP with the same size as the original PLGP. We discussed in detail only fuzzy goal programming and left it to future researchers to generalize to other fuzzy stochastic goal programming. It seems natural to look at the joint solution (fuzzy vector solution). In multivariate statistics we ÿrst require the joint distribution from which we obtain the marginals. The same is true in fuzzy equations, fuzzy optimization, etc. where we ÿrst require the joint solution from which we get the other way from marginals to joint without knowing all the interactions between the variables. Also, many solution methods for fuzzy optimization attempt to generate only the marginals. Here we proposed to ÿrst obtain the joint solution. References [1] A. Charnes, W. Cooper, Deterministic equivalent for optimization and statistic under chance constraints, O.R. 11 (1) (1963) 18–39. [2] Huey-Kuo Chen, A note on a fuzzy goal programming algorithm by Tiwari, Dhamar, and Rao, Fuzzy Sets and Systems 62 (1994) 287–290. [3] M.K. Gharraph, W. Mohammed, Chance constrained linear programming when the coecients are uniform random variables, The Egyptian Comput. J. ISSR, Cairo Univ. 22 (1) (1994) 1–16. [4] J. Lusk, Wrigt, Deriving the probability density for sums of Uniform Random Variables, The Amer. Statist. 36 (2) (1982) 128 –130. [5] R. Narasimhan, Goal programming in a fuzzy environment, Decision Sci. 17 (1980) 325 –336. [6] M. Ramadan, A chance-constrained fuzzy goal program, Fuzzy Sets and Systems 47 (1992) 183 –186. [7] J. Sengupta, On the active approach of stochastic linear programming, Metrika 15 (1970) 59 –70. [8] R.N. Tiwari, S. Dharmar, J.R. Rao, Priority structure in fuzzy goal programming, Fuzzy Sets and Systems 19 (1986) 251–259. [9] H.-J. Zimmermann, Fuzzy Sets, Theory – and Its Applications, 2nd rev. ed., Kluwer, Boston, 1991.