Downloaded from iors.ir at 10: on Saturday May 12th 2018 Fuzzy Primal Simplex Algorithms for Solving Fuzzy Linear Programming Problems

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1 Iranian Journal of Operations esearch Vol 1, o 2, 2009, pp68-84 Fuzzy Primal Simplex Algorithms for Solving Fuzzy Linear Programming Problems ezam Mahdavi-Amiri 1 Seyed Hadi asseri 2 Alahbakhsh Yazdani 3 Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences We consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems Finally, we give illustrative examples Keywords: Fuzzy linear programming; Trapezoidal fuzzy number; anking function; Fuzzy simplex method 1 Introduction Many application problems, modeled as mathematical programming problems, may be formulated with uncertainty The concept of fuzzy mathematical programming at general level was first proposed by Tanaka et al [13] in the framework of the fuzzy decision of ellman and Zadeh [1] The first formulation of fuzzy linear programming (FLP) was proposed by Zimmermann [17] A review of the literature concerning fuzzy mathematical programming as well as comparison of fuzzy numbers can be seen in Klir and Yuan [6] and also Lai and Hwang [7] Several authors considered various types of the FLP problems and proposed several approaches for solving them [3, 4, 5, 8, 9, 10, 1 Corresponding author, Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, nezamm@ sharifedu 2 Department of Mathematical Sciences, Mazandaran University, nasseri@umzacir 3 Department of Mathematical Sciences, Mazandaran University,yazdani@umzacir

2 Fuzzy Primal Simplex Algorithms 69 14] One convenient method is based on the concept of comparison of fuzzy numbers by use of ranking functions [3, 8, 9, 10, 11, 12] Here, we first review some necessary concepts of fuzzy set theory in Section 2 In Section 3, we explain the notion of comparison of fuzzy numbers by use of a linear ranking function After defining fuzzy linear programming problems in Section 4, we describe fuzzy number linear programming in Section 5 and give a corresponding primal simplex algorithm in Section 6 Section 7 discusses linear programming with trapezoidal fuzzy variables and a corresponding primal simplex algorithm is given in Section 8 Section 9 illustrates the working of the two algorithms through two examples We conclude in Section 10 2 Definitions and otations Here, we give some necessary definitions and results of fuzzy set theory given by ellman and Zadeh [1] (taken from ezdek [2] and Lai and Hwang [7]) A Fuzzy sets Definition 21 Fuzzy sets and membership functions If X is a collection of objects denoted generically by x, then a fuzzy set A in X is defined to be a set of ordered pairs A = {( x, μa( x)) x X}, where μ A ( x ) is called the membership function for the fuzzy set The membership function maps each element of X to a membership value between 0 and 1 emark 21 We assume that X is the real line Definition 22 Support The support of a fuzzy set A is the set of points x in X with μ ( x ) > 0 A Definition 23 Core The core of a fuzzy set is the set of points x in X with μ ( x ) = 1 A Definition 24 ormality A fuzzy set A is called normal if its core is nonempty In other words, there is at least one point x X with μ ( x ) = 1 Definition 25 α cut and strong α cut The α cut or α level set of a fuzzy set A is a crisp set defined by Aα = { x X μa( x) α} The strong α cut is defined to be Aα = { x X μa( x) > α} A Definition 26 Convexity A fuzzy set A on X is convex if for any x, y X any λ [0,1], we have, μ ( λx + (1 λ) y) min{ μ ( x), μ ( y)} A A A and

3 70 Mahdavi-Amiri, asseri and Yazdani emark 22 A fuzzy set is convex if and only if all its α cuts are convex Definition 27 Fuzzy number A fuzzy number A is a fuzzy set on the real line that satisfies the conditions of normality and convexity In fact, any fuzzy number is defined by its corresponding membership function Assume that the membership function of any fuzzy number a% is: L L L a x 1 α, a α x< a L U 1, a x a μa% ( x) = U x a U U 1 β, a < x a + β 0, otherwise The fuzzy number with the above membership function is shown in Fig 1, and is called trapezoidal fuzzy number L L U U a α a a a + β Figure 1 A trapezoidal fuzzy number A trapezoidal fuzzy number can be shown by ( L U a% = a, a, α, β ) The support of a% is L U L U ( a α, a + β), and the core of a% is [ a, a ] Let F( ) denote the set of all trapezoidal fuzzy numbers Arithmetic on trapezoidal fuzzy numbers Let ~ (,,, β ~ a = a L a U α ) and b = ( b L, b U, γ, θ ) be two trapezoidal fuzzy numbers and x Define [6]: L U x 0, xa% = ( xa, xa, xα, x β ), U L x < 0, xa% = ( xa, xa, x β, xα ), ~ ~ L L U U a + b = ( a + b, a + b, α + γ, β + θ ) As an illustration of the above arithmetic, consider two trapezoidal fuzzy numbers as given in Fig 2

4 Fuzzy Primal Simplex Algorithms 71 a % = ( 2,1,1,2) b % = (2,4,1,1) Figure 2 Two trapezoidal fuzzy numbers The results of negation, addition and subtraction are shown in Fig 3 b% = ( 4, 2,1,1) a% + b% = (0,5,2,3) a% b% = ( 6, 1,2,3) Figure 3 esults of negation, addition and subtraction of trapezoidal fuzzy numbers 3 anking Functions anking is a viable approach for ordering fuzzy numbers Various types of ranking functions have been introduced and some have been used for solving linear programming problems with fuzzy parameters [3, 7, 8, 9, 10] A review of some common methods for ranking fuzzy subsets of the unit interval can be seen in [15] Here, we deal with ranking the elements of F( ) In fact, an effective approach for ordering the elements of F( ) is to define a ranking function : F( ) ( ) mapping trapezoidal fuzzy numbers into Consider a% and b % in F( ) Define order on F( ) as follows [8, 9]: a ~ b ~ if ( a% ) ( b % ), (1)

5 72 Mahdavi-Amiri, asseri and Yazdani a ~ > b ~ if ( a% ) > ( b % ), (2) a ~ = b ~ if ( a% ) = ( b % ), (3) where a ~ and b ~ are in F( ) Also we write a ~ b ~ ~ if b a ~ Then, for any linear ranking function we may obtain: a ~ b ~ if and only if a% b % 0 %, or if and only if ~ b a~ Also, if a ~ b ~ and c~ ~ d, then a ~ ~ ~ + c~ b + d emark 31: For a trapezoidal fuzzy number a%, the relation a% 0 % holds, if there exist ε 0 and α 0 such that a% ( ε, εαα,, ) We realize that ( ε, εαα,, ) = 0 (we also consider a% = 0 % if and only if ( a% ) = 0 ) Thus, without loss of generality, throughout the paper we let 0 % = (0,0,0,0) as the zero trapezoidal fuzzy number L U We consider a linear ranking function on a% = ( a, a, αβ, ) F F( ( ) as: L U ( a% ) = cla + cua + cαα + cββ, where cl, cu, c α and c β are constants, at least one of which is nonzero A special version of the above linear ranking function was first proposed by Yager [16]: ~ 1 1 ) L U ( a = ( a + a + ( β α)) (4) 2 2 Thus, using (4), for trapezoidal fuzzy numbers a ~ = ( a L, a U, α, β ) and ~ b = ( b L, b U, γ, θ ), we have: a ~ b ~ L U 1 L U 1 if and only if a + a + ( β α) b + b + ( θ γ) Fuzzy Linear Programming An application of fuzzy set theory to decision making is fuzzy linear programming, first introduced by Zimmermann [17] Here, we introduce fuzzy linear programming (FLP) problems and divide them into two main subclasses: Fuzzy number linear programming (FLP) and linear programming with trapezoidal fuzzy variables (FVLP) problems A crisp linear programming (LP) problem in a standard form is defined as: Max z = cx st Ax = b (5) x 0, where the parameters c ( c1,, c n ), = 1 b= ( b,, b ) T m, m n, and A= [ a ij ] m n are given

6 Fuzzy Primal Simplex Algorithms 73 with crisp components (members of ) and n is an unknown vector of variables to be found If some parameters are considered to be fuzzy numbers, then we obtain a fuzzy linear programming (FLP) problem We consider the FLP problems in two main classes: (1) Fuzzy number linear programming (FLP), and (2) linear programming with trapezoidal fuzzy variables (FVLP) problems 5 Fuzzy umber Linear Programming A The definition of FLP problem A fuzzy number linear programming (FLP) problem is defined as: Max st ~ z = c~ x Ax = b (6) x 0, m n m n T n where b, x, A, c% ( F( )), and is a linear ranking function Definition 51 Any x satisfying the set of constraints (6) of the FLP problem is called a feasible solution Let Q be the set of all feasible solutions of the FLP problem Then, we say that x 0 Q is an optimal feasible solution for the FLP problem if cx % 0 cx % for all x Q One approach for solving these problems is to use the same ranking function for every equality and inequality For example, we may use the ranking function (4) Applying the ranking function, we obtain a crisp model, equivalent to the FLP problem, the optimal solution of which is the optimal solution of the FLP problem In fact, one can show that problem (5) and (6) are equivalent (see Mahdavi-Amiri and asseri [9]), in the sense that the feasible solution sets of the two problems are the same Then, the two problems either are infeasible and hence have no solution, or are feasible, and hence have the same optimal solutions, or are unbounded asic feasible solution Consider the system Ax= b and x 0, where A is an m n matrix and b is an m vector Suppose that rank( A, b) = rank( A) = m Partition A, after possibly rearranging the columns of A, as [ ] where, m m, is nonsingular It is apparent that x = ( x,, x ) = b, x = 0 is a solution of Ax = b The point T 1 1 m T T T x= ( x, x ) where x = 0 is called a basic solution of the system If x 0, then x is called a basic feasible solution (FS) of the system and the corresponding fuzzy objective value is z% = c% x, where c = ( c,, c ) ow, corresponding to every index 1 m

7 74 Mahdavi-Amiri, asseri and Yazdani 1 j, 1 j n, define: yj = aj and z% j = c% yj Observe that for any basic index j = i, 1 i m, we have: z% c% 1 = c% a c% = c% e c% = c% c% = 0, j j j j i j j j where e i is the ith unit vector ote that is called the basic matrix and is called the nonbasic matrix The components of x are called basic variables, and the components of x are called nonbasic variables The following theorem concerns the so-called nondegenerate FLP problems, where every basic variable corresponding to every feasible basis is positive [9] Theorem 52 Let the FLP problem be nondegenerate A basic feasible solution x = 1 b, x = 0 is optimal to (6) if and only if z% j c% j, for all j, 1 j n T T T Proof: Suppose that x* = ( x, x ) is a basic feasible solution to (6), where 1 1 x = b, x = 0 Then, z% = c% x = c% b On the other hand, for every feasible solution x, we have b = Ax = x + x Hence, we obtain: Then, z% = cx % = c% x + c% x = c% b ( c% a c% ) x 1 1 j j j j i z% = z% * ( z% j c% j) xj (7) j i The proof can now be completed using (7) and Theorem 62 given in Section 6 In the next section, we devise a fuzzy primal simplex algorithm for solving the FLP problems 6 Simplex Method for the FLP Problems A FLP simplex method in tableau format Consider the FLP problem as in (6) We rewrite the FLP problem as: z% = c% x + c% x Max st x + x = b x 0, x 0

8 Fuzzy Primal Simplex Algorithms 75 1 Hence, we have x + x = 1 b Therefore, z% + c% c% x = c% b With x = 0, we have problem as in Table 1 1 x b y0 = =, and 1 c b 1 1 ( ) z% = % Thus, we rewrite the above FLP Table 1 The FLP simplex tableau emark 61: Table 1 gives all the information needed to proceed with the simplex T 1 method The fuzzy cost row in Table 1 is y% = c% A c% where 0, 1 y% 0 j = c% aj c% j = z% j c% j, 1 j n, with y% 0 j = 0 % for j = i, 1 i m According to the optimality conditions (Theorem 52), we are at the optimal solution if y% 0 0 for all j i, 1 i m On the other hand, if y% 0k < 0, for some k i, 1 i m, then the problem is either unbounded or an exchange of a basic variable x and the nonbasic variable xk can be made to increase the rank of the objective value (under nondegeneracy assumption) The following results established in [9] help us to devise the fuzzy primal simplex algorithm Theorem 61 If in an FLP simplex tableau, there is a column k (not in basis) so that y% = z% c% < % and y 0, i = 1,, m, then the FLP problem is unbounded 0k k k 0 ik r j Theorem 62 If in an FLP simplex tableau, a nonbasic index k exists such that y% k = z% k c% k < % and there exists a basic index i such that y ik > 0, then a pivoting row 0 0 r can be found so that pivoting on y rk yields a feasible tableau with a corresponding nondecreasing (increasing under nondegeneracy assumption) fuzzy objective value emark 62 (see [9]): If k exists such that y 0 k = ( y% 0 k) < 0 and the problem is not unbounded, then r can be chosen so that y y min y 0 = y, ik > 1 i m yik r0 i0 rk

9 76 Mahdavi-Amiri, asseri and Yazdani in order to replace x r in the basis by x k, resulting in a new basis ˆ = ( a, K, a,,,, ) 1 a r 1 k a K a r+ 1 m The new basis is primal feasible and the corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy assumption) It can be shown that the new simplex tableau is obtained by pivoting on y rk, that is, doing Gaussian elimination on the k th column using the pivot row r, with the pivot y rk, to transform the k th column to the unit vector e r It is easily seen that yr0 the new fuzzy objective value is: yˆ 00 = y00 y0k y00, where y0j = ( y% 0 j), for all yrk yr0 yr 0 j, since y 0 k < 0 and (if the problem is nondegenerate, then > 0 and hence y y ŷ00 > y00 ) We now describe the pivoting strategy Pivoting and change of basis If x enters the basis and k rk x r simplex tableau is carried out, as follows: 1) Divide row r by y rk 2) For i= 0,1,, m and i r r th row leaves the basis, then pivoting on y rk in the, update the i th row by adding to it yik times the new We now present the simplex algorithm for the FLP problem C The main steps of FLP simplex algorithm Algorithm 1: The fuzzy simplex method for the FLP problem Assumption: A basic feasible solution with basis and the corresponding simplex tableau is at hand rk 1 The basic feasible solution is given by x = y0 and x = 0 The fuzzy objective value is: z% = y% 00 2 Calculate y 0 j =( z% j % cj), j = 1,, n, j, i = 1,, m i Let y 0k = min { y 0 j } If y0k 0, then stop; the current solution is optimal j= 1,, n 3 If yk 0, then stop; the problem is unbounded Otherwise, determine an index r corresponding to a variable x r leaving the basis as follows: yr0 yi0 = min{ yik > 0} y 1 i m y rk ik

10 Fuzzy Primal Simplex Algorithms 77 4 Pivot on y rk and update the simplex tableau Go to step 2 7 Linear Programming with Trapezoidal Fuzzy Variables A The definition of FVLP problem A linear programming with trapezoidal fuzzy variables (FVLP) problem is: Max ~ z = c~ x st Ax ~ ~ = b (8) x% 0, m where b % ( F( )), ( ( )) n x% F, m n A, and T n c ote that an FVLP problem is a linear programming problem in fuzzy environment with the decision making variables being fuzzy numbers Definition 71 We say that a fuzzy vector x% ( F( )) n is a fuzzy feasible solution for (8) if ~ ~ x satisfies the constraints Ax ~ = b and x% 0 % Definition 72 A fuzzy feasible solution ~ x * is a fuzzy optimal solution for (8), if for every fuzzy feasible solution ~ x for (8), we have c ~* x c~ x The fuzzy basic feasible solution Here, we describe fuzzy basic feasible solution (FFS) for the FVLP problem (8) as established by Mahdavi-Amiri and asseri [8] Let A = [ a ij ] m n Assume rank ( A ) = m Partition A, rearranging columns of A, if needed, as [ ], where, m m, is a nonsingular matrix Let y j be the solution to y = aj It is apparent that the basic solution, T 1 x% = ( x%,, x% ) 1 = b% = y% 0, x% 0 m = %, (9) ~ is a solution of A ~ x b T T T = We call x%, accordingly partitioned as ( x%, x% ), a fuzzy basic solution corresponding to the basis If % 0 %, then the fuzzy basic solution is feasible x ow, corresponding to every fuzzy nonbasic variable x% j,1 j n, j i, i = 1,, m, define: 1 zj = cyj = c aj (10) The following result concerns the nondegenerate problems, where every fuzzy basic variable corresponding to every feasible basis is positive [8]

11 78 Mahdavi-Amiri, asseri and Yazdani Theorem 71 Assume the FVLP problem is nondegenerate A fuzzy basic feasible solution x% 1 = b%, x% = 0% is optimal to (8) if and only if zj cj, for every j, 1 j n Proof: See [8] Maleki et al [10] proposed a method for solving FVLP problems by use of an auxiliary problem They discussed some relations between the FVLP problem and the auxiliary problem and used the results for solving the FVLP problem by an algorithm based on the solution of the auxiliary problem ecently Mahdavi-Amiri and asseri [8] developed the duality results for the FVLP problem They showed that the auxiliary problem in [10] is indeed the dual of the FVLP problem ased on the results obtained, they presented a dual simplex algorithm for solving the FVLP problems directly using the primal simplex tableau Here, we discuss and develop the fuzzy primal simplex algorithm for solving the FVLP problems 8 Simplex Method for the FVLP Problems A FVLP simplex method in tableau format Consider the FVLP problem (8), rewritten in the following form: z% = c x% + c x% Max st x% + x% = b (11) x% 0 % 0 x 1 1 We can write x = b% 1 1 % x%, z c( % = b% x% ) + cx%, and 1 1 hence x + x = b% 1 1 % %, and z% + ( c c) x% = c b% Letting ~ ~ 1 x = 0, we have x% = y% 0 = b% and z% = c y% 0 Thus, we write the above FVLP problem in the following tableau format (Table 2) Table 2 The FVLP simplex tableau Table 2 gives us all the information needed to proceed with the fuzzy primal simplex method The cost row in the above tableau is:

12 Fuzzy Primal Simplex Algorithms 79 y 0 j = 0, j i =, 1 i m, y0 j = cyj cj = zj cj, 1 j n, j i, 1 i m According to the optimality conditions for these problems, we are at the optimal solution if y0 j 0, for all j i, 1 i m On the other hand, if y 0k < 0, for some k i, 1 i m, then the problem is either unbounded or an exchange of a basic variable x% r, for some r, and the nonbasic variable x% k can be made to increase the rank of the objective value (under nondegeneracy assumption) The following theorems taken from [8] state the conditions for unboundedness of the FVLP problem and the conditions permitting the update of the tableau to a new tableau having a nondecreasing (increasing under nondegeneracy assumption) rank of the objective value Theorem 81 If in an FVLP simplex tableau, there is a column k (not in basis) for which z c < 0 and y 0, i = 1,, m, then the FVLP problem is unbounded k k ik Theorem 82 If in an FVLP simplex tableau, a nonbasic index k exists such that zk ck < 0 and there exists a basic index i such that y ik > 0, then a pivoting row r can be found so that pivoting on y rk yields a fuzzy feasible tableau with a corresponding nondecreasing (increasing under nondegeneracy assumption) objective value Pivoting and change of basis If x% k enters the basis and x% r leaves the basis, then pivoting on y rk in the simplex tableau is carried out, as follows: 1) Divide row r by y rk 2) For i= 0,1,, m and i r, update the i th row by adding to it yik times the new r th row ote: The pivoting results in the simplex tableau corresponding to the new basis C The main steps of FVLP simplex algorithm Algorithm 2: The fuzzy simplex method for the FVLP problem Assumption: A basic feasible solution with basis and the corresponding simplex tableau is at hand

13 80 Mahdavi-Amiri, asseri and Yazdani 1 The basic feasible solution is given by x% = y% 0 and x% = 0 The fuzzy objective value % % % is: z= y00 = c y0 2 Let y0k = min{ y0 j}, j = 1,, n, j i, i = 1,, m If y0 0, then stop; the current solution is optimal j k 3 If yk 0, then stop; the problem is unbounded Otherwise, determine the index r of the variable x% r leaving the basis as follows: yr0 yi0 = min{ yik > 0}, y 1 i m rk yik where yi0 = ( y% i0), i= 1,, m 4 Pivot on y rk and update the simplex tableau Go to Step 2 9 umerical Illustrations For illustrations of the FLP and FVLP simplex methods, we solve an FLP problem and an FVLP problem by use of FLP and FVLP simplex algorithms, respectively Example 91 Consider the FLP problem, Max z% = (5,8, 2,5) x1+ (6,10, 2,6) x2 st 2x1+ 3x2 6 5x1+ 4x2 10 x, x Adding the slack variables, we have the new constraints: 2x1+ 3x2 + x3 = 6 5x1+ 4x2 + x4 = 10 x, x, x, x The FLP simplex tableau corresponding to = I, 1 = 3, 2 = 4 is given in Table 3

14 Fuzzy Primal Simplex Algorithms 81 Table 3 The first simplex tableau of the FLP problem Since ( y% 01, y % 02) = (( 8, 5,5, 2),( 10, 6,6, 2)), and ( ( y% 01), ( y % 02)) = ( 725, 9), then x 2 enters the basis and the leaving variable is x 3 Pivoting on y 32 = 3 results in the next tableau as in Table 4 Table 4 The next feasible simplex tableau From ( y% 01, y% 03) = (( 4, 3, 3,6),(2, 3, 3, 2)), ( y 01, y 03) = ( ( y% 01), ( y% 03)) = ( 125,3), it follows that x 1 is an entering and x 4 is a leaving variable The last tableau is shown in Table 5 Table 5 The optimal simplex tableau of the FLP problem ote that Table 5 is optimal, because y% 03, y% 04 > 0 %, as shown below: c% b= (,,, ) % % % % 1 ( y 03, y 04 ) = c c = ((,,, ),(,,, )) ,

15 82 Mahdavi-Amiri, asseri and Yazdani Example 92 Consider the FVLP problem, Max z% = 3x% 1+ 4x% 2 st 3 x% 1+ x% 2 (2, 4,1,3) x% 1+ x% 2 (3, 2,3, 2) x%, x% 0 % 1 2 Adding the slack variables, we rewrite the constraints of the problem as: 3 x% + x% + x% = (2,4,1,3) x% 1+ x% 2 + x% 4= (3,,3, ) x%, x%, x%, x% 0 % ow, we rewrite the above problem as the FVLP simplex tableau (Table 6) Table 6 The first simplex tableau of the FVLP problem Since ( y 01, y 02) = ( 3, 4), then x% 2 enters the basis and the leaving variable is x% 4, by the fact that y 20 = min{ ( y% 10) =( 2,1, 2,6), ( y% 20) =(3, 2,3, 2)} Then, pivoting on y 22 = 1, we obtain the next tableau given as Table 7 Table 7 The optimal simplex tableau of the FVLP problem Since y0 j 0, for all j i, 1 i 2, then the basis is optimal and the optimal fuzzy objective value is: z% = (12,18,12,2)

16 Fuzzy Primal Simplex Algorithms Conclusions We considered two classes of fuzzy linear programming problems: (1) Fuzzy number linear programming (FLP), and (2) linear programming with trapezoidal fuzzy variables (FVLP) problems We made use of trapezoidal fuzzy numbers and a linear ranking function to describe a fuzzy concept of the basic feasible solutions for both problems We then used the optimality conditions for the FLP and the FVLP problems and developed fuzzy primal simplex algorithms for solving these problems Finally, we solved illustrative examples using the proposed simplex algorithms Acknowledgements The first author thanks the esearch Council of Sharif University of Technology for its support and the second author thanks the esearch Center of Algebraic Hyperstructures and Fuzzy Mathematics for its partial support eferences [1] ellman, E and Zadeh, LA (1970), Decision making in a fuzzy environment, Management Science, 17, [2] ezdek, JC (1993), Fuzzy models What are they, and why?, IEEE Transactions on Fuzzy Systems, 1, 1-9 [3] Campos, L and Verdegay, JL (1989), Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems, 32, 1-11 [4] Delgado, M, Verdegay, JL and Vila, MA (1989), A general model for fuzzy linear programming, Fuzzy Sets and Systems, 29, [5] Fang, SC and Hu, CF (1999), Linear programming with fuzzy coefficients in constraint, Comp Math App, 37, [6] Klir, GJ and Yuan, (1995), Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, PT, ew Jersey [7] Lai, YJ and Hwang, CL (1992), Fuzzy Mathematical Programming Methods and Applications, Springer, erlin [8] Mahdavi-Amiri, and asseri, SH (2007), Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables, Fuzzy Sets and Systems, 158, [9] Mahdavi-Amiri, and asseri, SH (2006), Duality in fuzzy number linear programming by use of a certain linear ranking function, Applied Mathematics and Computation, 180, [10] Maleki, H, Tata, M and Mashinchi, M (2000), Linear programming with fuzzy variables, Fuzzy Sets and Systems, 109, [11] Ozelkan, E and Duckstein, L (1999), Optimal fuzzy counterparts of scheduling rules, European Journal of Operational esearch, 113, [12] Tanaka, H, Ichihashi, H and Asai, K (1984), A formulation of fuzzy linear programming problem based on comparison of fuzzy numbers, Control and Cybernetics, 13,

17 84 Mahdavi-Amiri, asseri and Yazdani [13] Tanaka, H, Okuda, T and Asai, K (1974), On fuzzy mathematical programming, The Journal of Cybernetics, 3(4), [14] Verdegay, JL (1984), A dual approach to solve the fuzzy linear programming problem, Fuzzy Sets and Systems, 14, [15] Wang, X and Kerre, E (2001), easonable properties for the ordering of fuzzy quantities (2 parts), Fuzzy Sets and Systems, 118, [16] Yager, (1981), A procedure for ordering fuzzy subsets of the unit interval, Inform Science, 24, [17] Zimmermann, HJ (1978), Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 45-55