-Mr..in Fuzry Sets & Systems l(2) (2006), 131-142
Tb paper is available online a t http://www.pphmj.com
SOLVING POSSIBILISTIC LINEAR PROGRAMMING PROBLEM CONSIDERING MEMBERSHIP FUNCTION OF THE COEFFICIENTS M. ZARAFAT ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI Department of Mathematics, Firouzkooh Branch Islamic Azad University, Firouzkooh, Iran Department of Mathematics, Tehran-North Branch Islamic Azad University, Tehran, Iran e-mail:
[email protected] Department of Industrial Engineering Bu-Ali Sina University, Hamedan, Iran Department of Management, Firouzkooh Branch Islamic Azad University, Firouzkooh, Iran
Abstract In solving Possibilistic Linear Programming (PLP) problems, usually two approaches are used; defuzzification method and a-cut method. By defuzzing the parameters, the information on uncertainty is lost before solving the problem. When using a-cut, the fuzzy parameters are converted into intervals and hence the information on the distribution function of the points in the interval is lost. There is a need for a method which is able to retain this information while modeling and also in the process of solving the possibilistic mathematical programming 2000 Mathematics Subject Classification: 90C70. Keywords and phrases: possibilistic programming, membership function.
linear
programming
problem,
multiobjective
Communicated by K. K. Azad Received J u n e 24, 2006 @ 2006 Pushpa Publishing House
132 bM. Z. ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI problem. This paper develops a methodology for solving such problems by considering the distributions of the parameters. I t is aimed to choose the points with maximum membership values for the fuzzy parameters while maximizing the objective function and satisbing the s e t of constraints. Two numerical examples a r e presented to clarify the proposed method.
1. Introduction
A linear programming problem with imprecise coefficients restricted by possibilistic distributions is called a Possibilistic Linear Progralnrni~~g (PLP) problem. Since Zadeh [18] there has been much research on the possibility theory. Possibilistic decision making models have provided a n important aspect in handling practical decision making problems. Negoita e t al. [lo] were the first who formulated the possibilistic linear programming. Tanaka e t al. [15, 161 provided a symmetric model while the imprecise goal of the objective function is gwen. Inuiguchi e t al. [4, 51 extended the flexible programming into fuzzy coefficients case based on possibility theory. Ramik and Rimanek [ l l ] , and Tanaka and Asai [14] transferred the imprecise constraints into crisp ones by using the ranking concept. Luhandjula [8, 91 provided a-possibly feasible and P-possibly coefficient concepts, and used a utility function to obtain a n auxiliary crisp linear programming problem. Buckley [I, 21 applied the joint and conditional possibilistic distribution, and a-level set concepts to obtain a n auxiliary crisp linear programming. Another method was suggested in Lai and Hwang [7]. They provided a n auxiliary MOLP problem with three objectives. Inuiguchi e t al. [6] discussed possibilistic linear programming with oblique fuzzy vectors. They showed that the possibilistic linear programming problems are reduced to linear programming problems with a special structure. Saati et al. [12, 131 solved the possibilistic linear programming problem based on a-cut. Many authors use the membership functions of the fuzzy objective function and fuzzy constraints but they do not consider the membership functions of fuzzy coefficients [3, 17, 19, 201. One of the most important characteristics of the PLP problems is the distribution function of the parameters. Using the methods mentioned above, a main part gf the information on the uncertainty and the distribution function of the coefficients are ignored. To solve a PLP
SOLVING POSSIBILISTIC LINEAR PROGRAMMING .. .
133
problem, i t is better to develop a method which can keep the uncertainty until a n optimal solution is achieved. I t is known that a fuzzy number is a real interval with a membership function. Each point of the interval h a s a membership degree. Using this notion and keeping the uncertainty, we develop a method to solve the PLP problem. The proposed method while optimizing the objective function and satisfying the set of constraints, searches a point in the support of each parameter having maximum possible membership degree. We use this philosophy to solve a PLP problem with fuzzy resources. I t is evident that using the dual problem, a PLP problem with fuzzy objective coefficients can be converted to a PLP problem with fuzzy resources. Therefore, the proposed method can be used to solve it. When the technical coefficients or the objective coefficients with resources are fuzzy numbers, this method obtains a nonlinear programming problem. I n the second section, suggested method is introduced. I n the third section a numerical example is solved using the developed method i n Section 2. The fourth section is assigned to PLP with all fuzzy coefficients. A brief conclusion is introduced i n Section 5. 2. Suggested Method to Solve PLP Problem
Consider the following PLP problem:
max
C zjx; ;=1
For simplification of notations, we consider the coefficients as triangular fuzzy numbers ( n 1 u). The main idea of the suggested method is based on the membership functions of the coefficients. The membership functions of the coefficients can be defined as
134 M. Z. ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI
I n this paper, we will consider a PLP problem in two cases; PLP with fuzzy resources and PLP with all fuzzy coefficients. Then, we will develop a method to solve them. 2.1. PLP with fuzzy resources
Linear programming problem with fuzzy resources is a s follows:
max
C
cjx,
;=1
I n the suggested method, the objective function of (3) a s well a s the membership functions of resources are maximized. Therefore, the following model is proposed:
SOLVING POSSIBILISTIC LINEAR PROGRAMMING ...
135
which is a multiobjective linear programming problem. Theorem 1. The optimal solution of (4) will be attained by the second
condition of (2.3), i.e., p- (bi) = bi
b. - bU
"p
'
by
, bi
E
[ b y , by].
Proof. Let br be the optimal solution for bi. It is clear that there exist two values in the interval [ b f ,b:] with same membership function say, b;
E
[bf , b:'] and b:2
E
[ b y ) b y ] . Since the problem is maximization,
by shifting these values towards blu, the value of the objective function is maximized. So, b L will be the point which makes the objective function maximum. Hence, the second condition of p-. ( b i ) is sufficient.
4
Since the maximum value of membership function is 1, the following model is developed to solve (4):
136 M. Z. ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI
where
(5) is a bounded variable linear programming problem.
Theorem 2. The optimal value of (5) belongs to [0, 11.
Proof. Proof is evident. Since
p
0, the upper bound of
2
b, (i = 1, ..., m )
is redundant and
can be removed. With omitting these bounds and a slight variable *
-
substitution bi = bi - brL(i = 1, ..., m ) , (5) is reduced to the following model:
SOLVING POSSIBILISTIC LINEAR PROGRAMMING . ..
137
2.2. PLP with fuzzy objective coefficients
Linear programming problem with fuzzy objective coefficients is as follows:
max
C
F;X;
Using the dual of (B), it can be converted to a PLP with fuzzy right hand side vector and is solved by the proposed method i n t h e previous section. 3. Numerical Example
Consider the following linear programming problem with fuzzy resource:
rnax xl
+0.5~2
138 M. Z. ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI
Figure 1. Feasible region of example in the worst and best. To obtain Z* , the following problem should be solved: max Z = xl
+0.5~2
The optimal value a n d solution of (10) are Z* = 4 and (x;, 24)
=
(4, O), respectively. With these results and using (7), the following model
is obtained: max s.t.
P
xl+x2-h I 3 XI - ~2 -
b2 I 2
3x2 - b3 I 4
SOLVING POSSIBILISTIC LINEAR PROGRAMMING ...
The optimal value and solution of (11) are Z*
=
139
3.09 and (x;, x ; )
(2.95, 0.27), respectively. This solution is obtained by
P
=
=
0.77. I t should
be mentioned t h a t after solving this problem, the achieved right hand side vector is ( 4 , b2, b3) = (3.23, 2.68, 4). This vector is specified in a way that not only the objective function of (9) is maximized but also the selected points from right hand side of the constraints have maximum membership degree. By P = 0.77, the suggested method by Saati e t al. [12] yields the same result. For better clarification, Figure 1 illustrates the behavior of this problem. Lines Li, Li and Lh are corresponding to the constraints of ' (10). The optimal solution for this case is the point A. When the
membership degree of the parameters of the second objective functions are being maximized, then these lines move towards the lower bounds say L;', L s and L;, respectively, I n this case, the point A' will be the optimal solution. Hence, the lines L1, L2 and L3 are the hyperplanes with
p
=
0.77. 4. A11 Fuzzy Coefficients
Consider the linear programming problem (1) with all fuzzy coefficients. To solve it, by considering the proposed method i n the next section, nonlinear programming problem (12) is suggested a s follows:
140 M. Z. ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI
i
1 < aij a.. LJ
5
a!?L LJ
= 1, ..., rn,
i = l , ..., In, j = l , ...,
i
= 1, ...,
?L,
rn,
I n (12), Z* is obtained by solving the following problem:
Z* = max
C qxj
I n a similar manner to PLP with fuzzy resources, the bounds of bounded variables in (12) can be eliminated. This case is out of the scope of this study. 5. Conclusion
Most methods for solving PLP problems, during transformation into
SOLVING POSSIBILISTIC LINEAR PROGRAMMING
...
141
crisp equivalent model miss a part of information described in fuzzy numbers. It is ideal to maintain all the information till the problem is solved. This paper attempts to solve PLP problem by retaining in fuzziness of the model by maximizing the membership function of the parameters. The value of the objective function is equivalent to the a-cut of interval programming problem with maximum possible value of the objective function. Numerical example provides the solution procedure. Comparing of the results with the method of Saati e t al. [12] validates the solution. In case of fuzzy resources, the proposed method converts a PLP problem with I L variables and In constraints into a linear programming problem with ( I L + rn + 1 ) variables and (2m + 1) constraints. A PLP problem with all fuzzy coefficients, and rn constraints is converted to a nonlinear programming problem with (rnn + 2n + rn + 1) variables and
(ma + 2m + 11 + 1 ) constraints. As further studies, it is suggested to investigate: (i) reducing the size of the converted (crisp equivalent) problem, (ii) possible linearization of the nonlinear model.
References 111 J. J. Buckley, Possibilistic linear programming with triangular fuzzy numbers, Fuzzy Sets and Systems 26 (1988), 135-138. [2] J. J. Buckley, Solving possibilistic linear programming problems, Fuzzy Sets and Systems 31 (1989), 329-341.
[3] D. Dubois, Linear programming with fuzzy data, Analysis of Fuzzy Information, Vol. 111: Applications in Engineering and Science, J. C. Bezdek, ed., pp. 241-263, CRC Press, Boca Raton, Florida, 1987.
[I] M. Inuiguchi, H. Ichihashi and H. Tanaka, Possibilistic linear programming with measurable multiattribute value functions, ORSA J. Comput. l(3) (1989), 146-158. [5] M. Inuiguchi and H. Ichihashi, Relative modalities and their use in possibilistic linear programming, Fuzzy Sets and Systems 35 (1990), 97-109. [6] h4. Inuiguchi, J. Ramik and T. Tanino, Oblique fuzzy vector and their use in possibilistic linear programming, Fuzzy Sets and Systems 135 (2003), 123-150.
[q
Y. J. Lai and C. L. Hwang, Interactive fuzzy linear programming, Fuzzy'Sets and Systems 45 (1992), 169-183.
142 M. Z. ANGIZ, S. SAATI, A. MEMARIANI and M. M. MOVAHEDI [8] M. K. Luhandjula, On possibilistic linear programming, European j.Oper. Res. 31 (1987),110-117. [9] M. K. Luhandjula, Multiple objective programming problem with possibilistic coefficients, Fuzzy Sets and Systems 21 (1987), 135-145. [lo] C. V. Negoita, S. Minoiu and E. Stan, On considering imprecision in dynamic programming, Econ. Comput. Econ. Cybernet. Stud Res. 3 (1976),83-95.
[ll] J. Ramik and I. Rimanek, Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems 16 (1985),123-138. [12] M. S. Saati, A. Memariani and G. R. Jahanshahloo, Efficiency analysis and ranking of DMUs with fuzzy d a b , Fuzzy Optim. Decision Making 1 (2002),255-267. [13] S. Saati and A. Memariani, Possibilistic programming with trapezoidal fuzzy numbers, Modarres Tech. Engg. J. 11 (2003),129-132. [14] H.Tanaka and K. Asai, Fuzzy solutions in fuzzy linear programming problems, IEEE Trans. Syst. Man and Cybernet. 14 (1984),325-328. [15] H.Tanaka and K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems 13 (1984), 1-10. [16] H. Tanaka, H. Ichihashi and K. Asai, A formulation of fuzzy linear programming probIems based on comparison of fuzzy numbers, Control and Cybernetics 13 (1984), 186-194. [17] H. Tanaka, T. Okuda and K Asai, On fuzzy mathematical programming, J. Cybernetics 3 (1974),37-46. [l8] L.A.Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978),13-28. [19] H. J. Zimmermann, Description and optimization of fuzzy systems, Int. J. Gen. Sys. 2 (1976),209-215. [20] H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978),45-55.
