J. Appl. Math. & Computing Vol. 15(2004), No. 1 - 2, pp. 333 - 341

FUZZY NUMBER LINEAR PROGRAMMING: A PROBABILISTIC APPROACH (3) H. R. MALEKI AND M. MASHINCHI

Abstract. In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions AMS Mathematics Subject Classification : 90C05, 90C70 Key words and phrases : Fuzzy number, ranking function, fuzzy linear programming.

1. Introduction Since the pioneering work on the concept of decision making in fuzzy environments by Bellman and Zadeh [2], to solve fuzzy linear programming problems, several approaches proposed by different authors [16, 21, 22, 23]. Lai and Hwang [9], Tong Shaocheng [17], Buckley [4, 5], Negi [10], Campos and Verdegay [6], Cadenas and Verdegay [7], Maleki et. al. [11] among others, considered the situation where all parameters are fuzzy. Some of them used the concept of comparison of fuzzy numbers for solving fuzzy number linear programming problems (FNLP) [6, 11, 15, 19]. One of the most convenient of these methods is based on the concept of comparison of fuzzy numbers by using ranking functions [6, 7, 11, 19]. Usually, in such methods authors define a crisp model which is Received January 14, 2003. Revised June 26, 2003. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

333

334

H. R. Maleki and M. Mashinchi

equivalent with FNLP and then use optimal solution of this model as the optimal solution of FNLP. When the crisp model has alternative optimal solutions these authors conclude that fuzzy problem also has alternative optimal solutions. But unfortunately, the fuzzy value of the objective function for these solutions is not necessarily the same, and it is not evident which optimal solution should be choosen. In this paper, at first we associate a probability density function to each fuzzy number. Then using the expectation of these density functions as a ranking function, we introduce a method for solving FNLP. Finally if FNLP has alternative optimal solutions, in order to choose a solution we use the variance of the density functions associated to the fuzzy values of the objective function corresponding to these solutions to find solution which has minimum variance. This paper extends the method proposed in [12, 13]. 2. Ranking function Many ranking methods can be found in fuzzy literature. Bass and Kwakernaak [1] are among the pioneers in this area. Bortolan and Degani [3] and Wang and Kerre [18] have reviewed different ordering methods. According to Chen and Hwang [8] the methods are categorized into four different groups one of which is a probabilistic approach. To examine the methods of each group see [8]. In spite of the existence of a variety of methods, no one can rank fuzzy numbers satisfactorily in all cases and situations [14]. This is our motivation to use the probabilistic approach to be able to rank the fuzzy numbers for identifying the proper optimal solutions of a fuzzy number linear programming problem discussed in the next section. Now let a ˜ be a fuzzy number, i.e. a convex normalized fuzzy subset of the real line whose membership function is piecewise continuous. In this paper we denote the set of all fuzzy numbers by F (R). There are two important topics in the real world applications of the fuzzy set theory: arithmetic operations on fuzzy numbers and comparison of fuzzy numbers which usually follow arithmetic operations. Thanks to the extension principle we have no problem with the first topic. However there is no common approach for the comparison of fuzzy numbers. Indeed there are different approaches for ranking fuzzy numbers. A simple but efficient approach for ordering of the elements of F (R) is to define a ranking function R : F (R) → R which maps each fuzzy number into the real line, where a natural order exists, and defining order on F (R) by • a ˜ ≥ ˜b iff

R(˜ a) ≥ R(˜b),

R

• a ˜ > ˜b iff R

R(˜ a) > R(˜b),

Fuzzy number linear programming: a probabilistic approach (3)

• a ˜ = ˜b iff

335

R(˜ a) = R(˜b),

R

where a ˜ and ˜b belong to F (R). Also we write

a ˜ ≤ ˜b if and only if R

˜b ≥ a ˜. R

Several ranking functions have been proposed by researchers to suit their requirements of the problems under consideration. Some examples are in [3]. In what follows,we associate a probability density function to a ˜ by the aid of membership function of a fuzzy number a ˜. Then we use the expectation and variance of this density function as the value of the ranking function. One way for converting a membership function into a density function is by the help of a linear transformation fa˜ (x) = c a ˜(x), where c is a proportional constant R satisfying fa˜ (x)dx = 1. For example, if a ˜ = (aL , aU , α, β) is a trapezoidal fuzzy 2 number, then c = 2(aU −aL )+α+β . See [19]. Since we have a density function associated with each fuzzy number, we can use methods of probability theory to determine the expectation and variance of this density function. From now on for simplicity, we use the terms expected value and variance of a ˜ instead of expected value and variance of the probability density function associated with a ˜. In the following we use the Mellin transform to find the expectation and variance [20]. Definition 1. The Mellin transform MX (s) of a probability density function (pdf) f (x), where x is positive, is defined as Z +∞ MX (s) = E(X s−1 ) = xs−1 f (x)dx, 0

whenever the integral exists. Hence E(X s ) = MX (s + 1). Thus, the expectation and variance of a random variable X are E(X) = MX (2) and V ar(X) = MX (3) − (MX (2))2 . Remark 1. The table of Mellin transforms related to some of fuzzy numbers are given in table (2) of [20]. For example if a ˜ = (aL , aU , α, β) is a trapizoidal fuzzy number such that aL −α > 0, then 1 aL (aL − α) − aU (aU + β) L U 2(a + a ) + (β − α) + , E(˜ a) = 3 2(aU − aL ) + (β + α) U 1 (aL )4 − (aL − α)4 (a + β)4 − (aU )4 V ar(˜ a) = − 6(2(aU − AL ) + β + α) β α − [E(˜ a)]2 .

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H. R. Maleki and M. Mashinchi

Note that the expectation ranking function is a linear ranking function, if we take R(˜ a) = E(˜ a). See [19].

3. Fuzzy number linear programming problem

In this section we define fuzzy number linear programming problems and propose a method for solving them.

Definition 2. The model max : z˜ = ˜ cx ,

s.t.

˜, ˜ ≤b Ax

x≥0 ,

(1)

R

˜ = (˜b1 , · · · , ˜bm )0 and a ˜ = (˜ c = (˜ c1 , · · · , c˜n ), b ˜ij , ˜bi and where A aij )m×n , ˜ c˜j ∈ F (R) for i = 1, 2, · · · , m , j = 1, 2, · · · , n , is called a fuzzy number linear programming problem (FNLP) [11, 17, 19].

Definition 3. Any x which satisfies the set of constraints of FNLP is called a feasible solution for FNLP. Let Q be the set of all feasible solutions of FNLP. We shall say that xo ∈ Q is an optimal feasible solution for FNLP if ˜ cx ≤ c˜xo R

for all x ∈ Q. The following theorem which is an extension of Lemma 3.1 in [11] shows that we can reduce any FNLP to a linear programming problem in the classical form. Theorem 1. The following problem and FNLP are equivalent: max : z = cx ,

s.t.

Ax ≤ b ,

x≥0 ,

(2)

where A = (aij )m×n , c = (c1 , · · · , cn ), b = (b1 , · · · , bm )0 and aij , bi , cj are real numbers, respectively corresponding to the fuzzy numbers a ˜ij , ˜bi , c˜j , with respect to a given linear ranking function R; i.e. aij = R(˜ aij ) , bi = R(˜bi ) and cj = R(˜ cj ). Remark 2. It follows that if the model (2) does not have a solution, then FNLP also does not have a solution. If there are alternative optimal solutions for the model (2), then FNLP will also have alternative optimal solutions.

Fuzzy number linear programming: a probabilistic approach (3)

337

Example 1. Consider the following FNLP max : z˜ = s.t.

(1, 3, 1, 1)x1 + (1.5, 2.5, 1, 1)x2 , x1 + x2 ≤ 8 ,

2x1 + x2 ≤ 10 , x1 , x2 ≥ 0,

where (aL , aU , α, β) is a trapezoidal fuzzy number. We may apply the expectation ranking function for solving the above FNLP. Then the problem reduces to the following: max : z = s.t.

2x1 + 2x2

,

x1 + x2 ≤ 8 , 2x1 + x2 ≤ 10 , x1 , x2 ≥ 0. Now solving the above problem, we see that x1 = 2 , x2 = 6 is an optimal basic feasible solution with z˜1 = (11, 21, 8, 8), for FNLP. Moreover, x1 = 0 , x2 = 8 is another optimal basic feasible solution with z˜2 = (12, 20, 8, 8) . Although R(˜ z1 ) = R(˜ z2 ) = 16, we can not say that z˜1 (x) = z˜2 (x) for all x ∈ R. Remark 3. It is important to note that when the model (1) has alternative optimal solutions, the value of ranking function for the fuzzy value of the objective function corresponding to all optimal solutions is the same. However the fuzzy value of the objective function from one optimal solution to another is not necessarily the same. Hence it may be asked if problem (1) has two optimal solutions with different fuzzy values for the objective function, which of these solutions must be preferred ? In this regard consider the problem in the standard form: max : z˜ = c˜x ,

Ax = b ,

s.t.

n

0

n

x≥0 ,

(3)

m

where x ∈ R , A is a m × n matrix, ˜ c ∈ (F (R)) and b ∈ R . Let B be a basis matrix corresponding to A. Then every column aj of A can be written as a linear combination of columns of B, i.e. aj = Byj . Moreover B determines a basic solution of Ax = b. This basic solution, defined by a m-component vector xB , is xB = [xB1 , · · · , xBm ]0 = B−1 b and the fuzzy value of the objective function z˜ for xB is given by z˜ = c˜B xB , where c˜B = (˜ cB1 , · · · , c˜Bm ). Define the fuzzy variable z˜j as follow: z˜j = ˜ cB yj

,

j = 1, · · · , n.

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Now suppose that a basic feasible solution for model (3) is given by xB = B−1 b. The fuzzy value of the objective function for this solution is z˜ = ˜ cB xB . In addition, we assume that for this solution yj = B−1 aj and z˜j = ˜ cB yj are known for every column aj of A not in B. Then the following theorems, which are extension of those in [11], propose a method to find the optimal solution of model (3) if it exists. The proofs are straightforward and omitted. Theorem 2. If for any column ak in A, but not in B, the condition z˜k < c˜k R

holds, and if at least one yik > 0 , i = 1, · · · , m, then it is possible to obtain a new basic feasible solution by replacing one of the columns in B by ak , and the ˜ ˆ B satisfies zˆ˜ ≥ z˜. new fuzzy value of the objective function ˆ z˜ = ˆ cB x R

Theorem 3. If for any basic feasible solution to model (3) there is some column ak not in the basis for which z˜k < c˜k and yik ≤ 0 , i = 1. · · · , m, then the model R

(3) has an unbounded solution. Theorem 4. If xB = B−1 b is a feasible solution for the model (3) such that z˜j ≥ c˜j for all aj in A, then xB is an optimal basic feasible solution for the R

model (3). We now investigate conditions in which the alternative optimal solutions for the model (3) can be obtained. Theorem 5. If x1 , · · · , xk are k different extreme points of the set of feasible solutions of the model (3) which are optimal, then any convex combination of these solutions is an optimal solution for the model (3). Pk Pk Proof. Let x = i=1 µi xi where µi ≥ 0, i = 1, · · · , k and i=1 µi = 1. It is clear that x is a feasible solution of the model (3). If ˜ cxi = ˜ cx1 , i = 2, · · · , k, R

then the fuzzy value of the objective function for x is: c˜x =

k X

µi c˜xi =

k X

R

i=1

R

i=1

µi ˜ cx1 = c˜x1 R

Hence x is an optimal solution for the model (3).

k X i=1

µi = c˜x1 . R

Fuzzy number linear programming: a probabilistic approach (3)

339

Theorem 6. Let xB be an optimal basic feasible solution for model (3) with z˜j ≥ c˜j for each aj in A and suppose that for some ak not in the basis, z˜k = c˜k R R Pm and yik ≤ 0 for all i , then for each θ > 0 we have i=1 (xBi −θyik )bi +θak = b, which is an optimal nonbasic solution.

4. Identifying the optimal solution Now we are ready to study the problem of alternative optimal solutions for model (3) raised in Remark 3. When the model (3) has alternative optimal solutions, as mentioned in Remark 3, the value of the ranking function for the fuzzy value of the objective function corresponding to all optimal solutions is equal. However the fuzzy value of the objective function may not be equal from one optimal solution to another. If we apply the expectation ranking function, it is natural to identify the optimal solution as one with least variance of the fuzzy value of the objective function. Especially, when we have x1 , · · · , xk as k different optimal extreme points, using Theorem 5 and the above idea we may solve the following model to determine optimal solution of (3) with least variance of the fuzzy value of the objective function: hP i k ˜ min : V ar , λ c x j j j=1 s.t.

Pk

j=1

λj = 1 ,

(4)

λj ≥ 0 . Note that

Pk

j=1

λj c˜xj in (4) is a fuzzy number.

Example 2. In order to choose an optimal solution from those found in Example 3.1, we may solve the following problem: min :

V ar((11, 21, 8, 8)λ1 + (12, 20, 8, 8)λ2) ,

s.t. λ1 + λ2 = 1 , λ1 , λ2 ≥ 0, now solving the above problem, we find λ1 = 0 and λ2 = 1. Thus we can choose x = (0, 8)0 as optimal solution of our problem.

5. Conclusion

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H. R. Maleki and M. Mashinchi

In this paper we associate a probability density function to each fuzzy number. This association gives the possibility of using expectation as a linear ranking function to solve a fuzzy number linear programming problem. If the solution is not unique, then the fuzzy value of the objective function for these solutions are usually different. The notion of the variance which makes it possible to identify the optimal solution with the least variance of the fuzzy value of the objective function, and we propose this as the ”best” solution.

Acknowledgement We would like to thank the referee for his very useful comments.

References 1. S. M. Bass and H. Kwakernaak, Rating and Ranking of Multiple Aspent Alternative Using Fuzzy Sets, Automatica 13 (1977), 47-58. 2. R. E. Bellman and L. A. Zadeh, Decision Making in a Fuzzy Environment, Management Sci. 17 (1970), 141-164. 3. G. Bortolan and R. Degani, A Review Of Some Methods For Ranking Fuzzy Numbers, Fuzzy Sets and Systems 15 (1985), 1-19. 4. J. J. Buckley, Possibilistic linear programming with triangular fuzzy numbers, Fuzzy Sets and Systems 26 (1988), 135-138. 5. J. J. Buckley, Solving possibilistic linear programming problems, Fuzzy Sets and Systems 31 (1989), 329-341. 6. L. Campos and J. L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems 32 (1989), 1-11. 7. J. M. Cadenas and J. L. Verdegay, Using ranking functions in multiobjective fuzzy linear programming, Fuzzy Sets and Systems, 111 (2000), 47-53. 8. S. J. Chen and C. L. Hwang, Fuzzy Multiple Attribute Decision Making, Methods and Applications, Springer, Berline, 1992. 9. Y. J. Lai and C. L. Hwang, A new approach to some possibilistic linear programming problem, Fuzzy Sets and Systems 49 (1992), 121-133. 10 Y. J. Lai and C. L. Hwang, Fuzzy Mathematical Programming Methods and Applications, Springer-Verlag, 1992. 11 H. R. Maleki, M. Tata and M. Mashinchi, Linear Programming With Fuzzy Variables, Fuzzy Sets and Systems 109 (2000), 21-33. 12 H. R. Maleki, H. Mishmast N. and M. Mashinchi, Fuzzy Number Linear Programming: A Probabilistic Approach, Nikhil R. Pal and Michio Sugeno, Eds., Advances in Soft Computing AFSS 2002, Proceedings of 2002 AFSS International Conference on Fuzzy Systems, Calcutta, India, Feb. 2002, Springer-Verlag, Berlin, 2002, 491-496. 13 H. R. Maleki, H. Mishmast N. and M. Mashinchi, Fuzzy Number Linear Programming: A Probabilistic Approach (2), N. Mastorakis and V. Mladenov, Eds., Recent Advances in Computers, Computing and Communications WSEAS 2002, 214-218.

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14 M. Modarres and S. Sadi-Nezhad, Ranking Fuzzy Numbers by Preference Ratio, Fuzzy Sets and Systems 118 (2001), 429-436. 15. M. Roubens, Inequality contraints between fuzzy numbers and their use in mathematical programming, R. Slowinski and J. Teghem, Eds., Stochastic Versus Fuzzy Approachs To Multiobjective Mathematical Programming Under Uncertainty, Kluwer Academic Publishers, 1991, 321-330. 16. H. Tanaka, T. Okuda and K. Asai, On fuzzy mathematical programming, J. Cybernetic 3 (1974), 37-46. 17. Tong Shaocheng, Interval number and fuzzy number linear programming, Fuzzy Sets and Systems 66 (1994), 301-306. 18. X. Wang and E. Kerre, On the classification and the dependencies of the ordering methods, Da Ruan, Ed., Fuzzy Logic Foundations and Industrial Applications, Kluwer, 1996, 73-90. 19. E. Yazdani Peraei, H. R. Maleki and M. Mashinchi, A Method For Solving A Fuzzy Linear Programming, J. Appl. Math.& Computing (old:KJCAM) Vol. 8 (2001), No.2, 347-356. 20. K. P. Yoon, A Probabilistic Approach To Rank Complex Fuzzy Numbers, Fuzzy Sets and Systems 80 (1996), 167-176. 21. H. J. Zimmermann, Optimization in fuzzy environment, Presented at 21th Int. TIMS and 46th ORSA Conference, San Juan, Puerto Rico, 1974. 22. H. J. Zimmermann, Applications of Fuzzy Sets Theory to Mathematical Programming, Inform. Sci. 36 (1985), 29-58. 23. H. J. Zimmermann, Fuzzy Set Theory and its Applications , Kluwer Academic Publishars, Dordrecht, 3nd edition, 1996. H. R. Maleki received his B.Sc from Shiraz University in Iran, M.Sc and Ph.D (in 1999) both from Shahid Bahonar University of Kerman in Iran. He is Assistant Professor of mathematics in Shahid Bahonar University of Kerman. His research interest is fuzzy operation research and graph theory. Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran. Corresponding author: Tel. and Fax: ++98-341-3221080 e-mail: [email protected] M. Mashinchi received his B.Sc. and M.Sc in Iran from Ferdowsi University and Shiraz University, respectively and his Ph.D in Japan from Waseda University (in 1978). He is Professor of mathematics in Shahid Bahonar University of Kerman. His research interest is in fuzzy mathematics, especially on decision making and algebric systems. Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran. e-mail: [email protected]

FUZZY NUMBER LINEAR PROGRAMMING: A PROBABILISTIC APPROACH (3) H. R. MALEKI AND M. MASHINCHI

Abstract. In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions AMS Mathematics Subject Classification : 90C05, 90C70 Key words and phrases : Fuzzy number, ranking function, fuzzy linear programming.

1. Introduction Since the pioneering work on the concept of decision making in fuzzy environments by Bellman and Zadeh [2], to solve fuzzy linear programming problems, several approaches proposed by different authors [16, 21, 22, 23]. Lai and Hwang [9], Tong Shaocheng [17], Buckley [4, 5], Negi [10], Campos and Verdegay [6], Cadenas and Verdegay [7], Maleki et. al. [11] among others, considered the situation where all parameters are fuzzy. Some of them used the concept of comparison of fuzzy numbers for solving fuzzy number linear programming problems (FNLP) [6, 11, 15, 19]. One of the most convenient of these methods is based on the concept of comparison of fuzzy numbers by using ranking functions [6, 7, 11, 19]. Usually, in such methods authors define a crisp model which is Received January 14, 2003. Revised June 26, 2003. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

333

334

H. R. Maleki and M. Mashinchi

equivalent with FNLP and then use optimal solution of this model as the optimal solution of FNLP. When the crisp model has alternative optimal solutions these authors conclude that fuzzy problem also has alternative optimal solutions. But unfortunately, the fuzzy value of the objective function for these solutions is not necessarily the same, and it is not evident which optimal solution should be choosen. In this paper, at first we associate a probability density function to each fuzzy number. Then using the expectation of these density functions as a ranking function, we introduce a method for solving FNLP. Finally if FNLP has alternative optimal solutions, in order to choose a solution we use the variance of the density functions associated to the fuzzy values of the objective function corresponding to these solutions to find solution which has minimum variance. This paper extends the method proposed in [12, 13]. 2. Ranking function Many ranking methods can be found in fuzzy literature. Bass and Kwakernaak [1] are among the pioneers in this area. Bortolan and Degani [3] and Wang and Kerre [18] have reviewed different ordering methods. According to Chen and Hwang [8] the methods are categorized into four different groups one of which is a probabilistic approach. To examine the methods of each group see [8]. In spite of the existence of a variety of methods, no one can rank fuzzy numbers satisfactorily in all cases and situations [14]. This is our motivation to use the probabilistic approach to be able to rank the fuzzy numbers for identifying the proper optimal solutions of a fuzzy number linear programming problem discussed in the next section. Now let a ˜ be a fuzzy number, i.e. a convex normalized fuzzy subset of the real line whose membership function is piecewise continuous. In this paper we denote the set of all fuzzy numbers by F (R). There are two important topics in the real world applications of the fuzzy set theory: arithmetic operations on fuzzy numbers and comparison of fuzzy numbers which usually follow arithmetic operations. Thanks to the extension principle we have no problem with the first topic. However there is no common approach for the comparison of fuzzy numbers. Indeed there are different approaches for ranking fuzzy numbers. A simple but efficient approach for ordering of the elements of F (R) is to define a ranking function R : F (R) → R which maps each fuzzy number into the real line, where a natural order exists, and defining order on F (R) by • a ˜ ≥ ˜b iff

R(˜ a) ≥ R(˜b),

R

• a ˜ > ˜b iff R

R(˜ a) > R(˜b),

Fuzzy number linear programming: a probabilistic approach (3)

• a ˜ = ˜b iff

335

R(˜ a) = R(˜b),

R

where a ˜ and ˜b belong to F (R). Also we write

a ˜ ≤ ˜b if and only if R

˜b ≥ a ˜. R

Several ranking functions have been proposed by researchers to suit their requirements of the problems under consideration. Some examples are in [3]. In what follows,we associate a probability density function to a ˜ by the aid of membership function of a fuzzy number a ˜. Then we use the expectation and variance of this density function as the value of the ranking function. One way for converting a membership function into a density function is by the help of a linear transformation fa˜ (x) = c a ˜(x), where c is a proportional constant R satisfying fa˜ (x)dx = 1. For example, if a ˜ = (aL , aU , α, β) is a trapezoidal fuzzy 2 number, then c = 2(aU −aL )+α+β . See [19]. Since we have a density function associated with each fuzzy number, we can use methods of probability theory to determine the expectation and variance of this density function. From now on for simplicity, we use the terms expected value and variance of a ˜ instead of expected value and variance of the probability density function associated with a ˜. In the following we use the Mellin transform to find the expectation and variance [20]. Definition 1. The Mellin transform MX (s) of a probability density function (pdf) f (x), where x is positive, is defined as Z +∞ MX (s) = E(X s−1 ) = xs−1 f (x)dx, 0

whenever the integral exists. Hence E(X s ) = MX (s + 1). Thus, the expectation and variance of a random variable X are E(X) = MX (2) and V ar(X) = MX (3) − (MX (2))2 . Remark 1. The table of Mellin transforms related to some of fuzzy numbers are given in table (2) of [20]. For example if a ˜ = (aL , aU , α, β) is a trapizoidal fuzzy number such that aL −α > 0, then 1 aL (aL − α) − aU (aU + β) L U 2(a + a ) + (β − α) + , E(˜ a) = 3 2(aU − aL ) + (β + α) U 1 (aL )4 − (aL − α)4 (a + β)4 − (aU )4 V ar(˜ a) = − 6(2(aU − AL ) + β + α) β α − [E(˜ a)]2 .

336

H. R. Maleki and M. Mashinchi

Note that the expectation ranking function is a linear ranking function, if we take R(˜ a) = E(˜ a). See [19].

3. Fuzzy number linear programming problem

In this section we define fuzzy number linear programming problems and propose a method for solving them.

Definition 2. The model max : z˜ = ˜ cx ,

s.t.

˜, ˜ ≤b Ax

x≥0 ,

(1)

R

˜ = (˜b1 , · · · , ˜bm )0 and a ˜ = (˜ c = (˜ c1 , · · · , c˜n ), b ˜ij , ˜bi and where A aij )m×n , ˜ c˜j ∈ F (R) for i = 1, 2, · · · , m , j = 1, 2, · · · , n , is called a fuzzy number linear programming problem (FNLP) [11, 17, 19].

Definition 3. Any x which satisfies the set of constraints of FNLP is called a feasible solution for FNLP. Let Q be the set of all feasible solutions of FNLP. We shall say that xo ∈ Q is an optimal feasible solution for FNLP if ˜ cx ≤ c˜xo R

for all x ∈ Q. The following theorem which is an extension of Lemma 3.1 in [11] shows that we can reduce any FNLP to a linear programming problem in the classical form. Theorem 1. The following problem and FNLP are equivalent: max : z = cx ,

s.t.

Ax ≤ b ,

x≥0 ,

(2)

where A = (aij )m×n , c = (c1 , · · · , cn ), b = (b1 , · · · , bm )0 and aij , bi , cj are real numbers, respectively corresponding to the fuzzy numbers a ˜ij , ˜bi , c˜j , with respect to a given linear ranking function R; i.e. aij = R(˜ aij ) , bi = R(˜bi ) and cj = R(˜ cj ). Remark 2. It follows that if the model (2) does not have a solution, then FNLP also does not have a solution. If there are alternative optimal solutions for the model (2), then FNLP will also have alternative optimal solutions.

Fuzzy number linear programming: a probabilistic approach (3)

337

Example 1. Consider the following FNLP max : z˜ = s.t.

(1, 3, 1, 1)x1 + (1.5, 2.5, 1, 1)x2 , x1 + x2 ≤ 8 ,

2x1 + x2 ≤ 10 , x1 , x2 ≥ 0,

where (aL , aU , α, β) is a trapezoidal fuzzy number. We may apply the expectation ranking function for solving the above FNLP. Then the problem reduces to the following: max : z = s.t.

2x1 + 2x2

,

x1 + x2 ≤ 8 , 2x1 + x2 ≤ 10 , x1 , x2 ≥ 0. Now solving the above problem, we see that x1 = 2 , x2 = 6 is an optimal basic feasible solution with z˜1 = (11, 21, 8, 8), for FNLP. Moreover, x1 = 0 , x2 = 8 is another optimal basic feasible solution with z˜2 = (12, 20, 8, 8) . Although R(˜ z1 ) = R(˜ z2 ) = 16, we can not say that z˜1 (x) = z˜2 (x) for all x ∈ R. Remark 3. It is important to note that when the model (1) has alternative optimal solutions, the value of ranking function for the fuzzy value of the objective function corresponding to all optimal solutions is the same. However the fuzzy value of the objective function from one optimal solution to another is not necessarily the same. Hence it may be asked if problem (1) has two optimal solutions with different fuzzy values for the objective function, which of these solutions must be preferred ? In this regard consider the problem in the standard form: max : z˜ = c˜x ,

Ax = b ,

s.t.

n

0

n

x≥0 ,

(3)

m

where x ∈ R , A is a m × n matrix, ˜ c ∈ (F (R)) and b ∈ R . Let B be a basis matrix corresponding to A. Then every column aj of A can be written as a linear combination of columns of B, i.e. aj = Byj . Moreover B determines a basic solution of Ax = b. This basic solution, defined by a m-component vector xB , is xB = [xB1 , · · · , xBm ]0 = B−1 b and the fuzzy value of the objective function z˜ for xB is given by z˜ = c˜B xB , where c˜B = (˜ cB1 , · · · , c˜Bm ). Define the fuzzy variable z˜j as follow: z˜j = ˜ cB yj

,

j = 1, · · · , n.

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Now suppose that a basic feasible solution for model (3) is given by xB = B−1 b. The fuzzy value of the objective function for this solution is z˜ = ˜ cB xB . In addition, we assume that for this solution yj = B−1 aj and z˜j = ˜ cB yj are known for every column aj of A not in B. Then the following theorems, which are extension of those in [11], propose a method to find the optimal solution of model (3) if it exists. The proofs are straightforward and omitted. Theorem 2. If for any column ak in A, but not in B, the condition z˜k < c˜k R

holds, and if at least one yik > 0 , i = 1, · · · , m, then it is possible to obtain a new basic feasible solution by replacing one of the columns in B by ak , and the ˜ ˆ B satisfies zˆ˜ ≥ z˜. new fuzzy value of the objective function ˆ z˜ = ˆ cB x R

Theorem 3. If for any basic feasible solution to model (3) there is some column ak not in the basis for which z˜k < c˜k and yik ≤ 0 , i = 1. · · · , m, then the model R

(3) has an unbounded solution. Theorem 4. If xB = B−1 b is a feasible solution for the model (3) such that z˜j ≥ c˜j for all aj in A, then xB is an optimal basic feasible solution for the R

model (3). We now investigate conditions in which the alternative optimal solutions for the model (3) can be obtained. Theorem 5. If x1 , · · · , xk are k different extreme points of the set of feasible solutions of the model (3) which are optimal, then any convex combination of these solutions is an optimal solution for the model (3). Pk Pk Proof. Let x = i=1 µi xi where µi ≥ 0, i = 1, · · · , k and i=1 µi = 1. It is clear that x is a feasible solution of the model (3). If ˜ cxi = ˜ cx1 , i = 2, · · · , k, R

then the fuzzy value of the objective function for x is: c˜x =

k X

µi c˜xi =

k X

R

i=1

R

i=1

µi ˜ cx1 = c˜x1 R

Hence x is an optimal solution for the model (3).

k X i=1

µi = c˜x1 . R

Fuzzy number linear programming: a probabilistic approach (3)

339

Theorem 6. Let xB be an optimal basic feasible solution for model (3) with z˜j ≥ c˜j for each aj in A and suppose that for some ak not in the basis, z˜k = c˜k R R Pm and yik ≤ 0 for all i , then for each θ > 0 we have i=1 (xBi −θyik )bi +θak = b, which is an optimal nonbasic solution.

4. Identifying the optimal solution Now we are ready to study the problem of alternative optimal solutions for model (3) raised in Remark 3. When the model (3) has alternative optimal solutions, as mentioned in Remark 3, the value of the ranking function for the fuzzy value of the objective function corresponding to all optimal solutions is equal. However the fuzzy value of the objective function may not be equal from one optimal solution to another. If we apply the expectation ranking function, it is natural to identify the optimal solution as one with least variance of the fuzzy value of the objective function. Especially, when we have x1 , · · · , xk as k different optimal extreme points, using Theorem 5 and the above idea we may solve the following model to determine optimal solution of (3) with least variance of the fuzzy value of the objective function: hP i k ˜ min : V ar , λ c x j j j=1 s.t.

Pk

j=1

λj = 1 ,

(4)

λj ≥ 0 . Note that

Pk

j=1

λj c˜xj in (4) is a fuzzy number.

Example 2. In order to choose an optimal solution from those found in Example 3.1, we may solve the following problem: min :

V ar((11, 21, 8, 8)λ1 + (12, 20, 8, 8)λ2) ,

s.t. λ1 + λ2 = 1 , λ1 , λ2 ≥ 0, now solving the above problem, we find λ1 = 0 and λ2 = 1. Thus we can choose x = (0, 8)0 as optimal solution of our problem.

5. Conclusion

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In this paper we associate a probability density function to each fuzzy number. This association gives the possibility of using expectation as a linear ranking function to solve a fuzzy number linear programming problem. If the solution is not unique, then the fuzzy value of the objective function for these solutions are usually different. The notion of the variance which makes it possible to identify the optimal solution with the least variance of the fuzzy value of the objective function, and we propose this as the ”best” solution.

Acknowledgement We would like to thank the referee for his very useful comments.

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14 M. Modarres and S. Sadi-Nezhad, Ranking Fuzzy Numbers by Preference Ratio, Fuzzy Sets and Systems 118 (2001), 429-436. 15. M. Roubens, Inequality contraints between fuzzy numbers and their use in mathematical programming, R. Slowinski and J. Teghem, Eds., Stochastic Versus Fuzzy Approachs To Multiobjective Mathematical Programming Under Uncertainty, Kluwer Academic Publishers, 1991, 321-330. 16. H. Tanaka, T. Okuda and K. Asai, On fuzzy mathematical programming, J. Cybernetic 3 (1974), 37-46. 17. Tong Shaocheng, Interval number and fuzzy number linear programming, Fuzzy Sets and Systems 66 (1994), 301-306. 18. X. Wang and E. Kerre, On the classification and the dependencies of the ordering methods, Da Ruan, Ed., Fuzzy Logic Foundations and Industrial Applications, Kluwer, 1996, 73-90. 19. E. Yazdani Peraei, H. R. Maleki and M. Mashinchi, A Method For Solving A Fuzzy Linear Programming, J. Appl. Math.& Computing (old:KJCAM) Vol. 8 (2001), No.2, 347-356. 20. K. P. Yoon, A Probabilistic Approach To Rank Complex Fuzzy Numbers, Fuzzy Sets and Systems 80 (1996), 167-176. 21. H. J. Zimmermann, Optimization in fuzzy environment, Presented at 21th Int. TIMS and 46th ORSA Conference, San Juan, Puerto Rico, 1974. 22. H. J. Zimmermann, Applications of Fuzzy Sets Theory to Mathematical Programming, Inform. Sci. 36 (1985), 29-58. 23. H. J. Zimmermann, Fuzzy Set Theory and its Applications , Kluwer Academic Publishars, Dordrecht, 3nd edition, 1996. H. R. Maleki received his B.Sc from Shiraz University in Iran, M.Sc and Ph.D (in 1999) both from Shahid Bahonar University of Kerman in Iran. He is Assistant Professor of mathematics in Shahid Bahonar University of Kerman. His research interest is fuzzy operation research and graph theory. Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran. Corresponding author: Tel. and Fax: ++98-341-3221080 e-mail: [email protected] M. Mashinchi received his B.Sc. and M.Sc in Iran from Ferdowsi University and Shiraz University, respectively and his Ph.D in Japan from Waseda University (in 1978). He is Professor of mathematics in Shahid Bahonar University of Kerman. His research interest is in fuzzy mathematics, especially on decision making and algebric systems. Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran. e-mail: [email protected]