TJFS: Turkish Journal of Fuzzy Systems (eISSN: 1309–1190) An Official Journal of Turkish Fuzzy Systems Association Vol.1, No.2, pp. 80-103, 2010
Generalized Simplex Algorithm to Solve Fuzzy Linear Programming Problems with Ranking of Generalized Fuzzy Numbers Amit Kumar Thapar University, School of Mathematics and Computer Applications, 147 004 Patiala, India E-mail:
[email protected] Pushpinder Singh* Thapar University, School of Mathematics and Computer Applications, 147 004 Patiala, India E-mail:
[email protected] *Corresponding author Jagdeep Kaur Thapar University, School of Mathematics and Computer Applications, 147 004 Patiala, India E-mail:
[email protected] Received: May 31, 2010 - Revised: September 12, 2010 – Accepted: October 14, 2010
Abstract In this paper the shortcomings of an existing method for comparing the generalized fuzzy numbers are pointed out and a new method is proposed for same. Also using the proposed ranking method, a generalized simplex algorithm is proposed for solving a special type of fuzzy linear programming (FLP) problems. To illustrate the proposed algorithm a numerical example is solved and the advantages of the proposed algorithm are discussed. Since the proposed algorithm is a direct extension of classical algorithm so it is very easy to understand and apply the proposed algorithm to find the fuzzy optimal solution of FLP problems occurring in the real life situations. Keywords: Fuzzy Linear Programming Problems, Ranking Function, Generalized Trapezoidal Fuzzy Numbers
1. Introduction Linear programming is one of the most frequently applied operations research techniques. Although it is investigated and expanded for more than six decades by many 80
researchers and from the various point of views, it is still useful to develop new approaches in order to better fit the real world problems within the framework of linear programming. Any linear programming model representing real world situations involves a lot of parameters whose values are assigned by experts, and in the conventional approach, they are required to fix an exact value to the aforementioned parameters. However, both experts and the decision makers frequently do not precisely know the value of those parameters. If exact values are suggested these are only statistical inference from past data and their stability is doubtful, so the parameters of the problem are usually defined by the decision makers in an uncertain way or by means of language statement parameters. Therefore, it is useful to consider the knowledge of experts about the parameters as fuzzy data (Zadeh 1965). The concept of fuzzy mathematical programming on general level was first proposed by Tanaka et al. (1974), in the framework of the fuzzy decision of Bellman & Zadeh (1970). The first formulation of FLP was proposed by Zimmermann (1978). Afterwards, many authors (Kacprzyk 1983, Tanaka & Asai 1984, Luhandjula 1987, Campos & Verdegay 1989, Delgado et al. 1989, Inuiguchi et al. 1990, Buckley & Feuring 2000, Inuiguchi & Ramik 2000, Maleki et al. 2000, Ramik 2005, Ganesan & Veeramani 2006, Rommelfanger 2007), considered various types of the FLP problems and proposed several approaches for solving these problems. Fuzzy numbers must be ranked before an action is taken by a decision maker. Real numbers can be linearly ordered by the relation or , however this type of inequality does not exist in fuzzy numbers. Since fuzzy numbers are represented by possibility distribution, they can overlap with each other and it is dificult to determine clearly whether one fuzzy number is larger or smaller than other. An efficient method for ordering the fuzzy numbers is by the use of a ranking function, which maps each fuzzy number into the real line, where a natural order exists. Jain (1976), proposed the concept of ranking function for comparing normal fuzzy numbers. Chen (1985), pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. Since then, tremendous efforts are spent; significant advances are made on the development of numerous methodologies (Campos & Gonzalez 1989, Campos & Verdegay 1989, Liou & Wang 1992, Chang & Lee 1994, Fortemps & Roubens 1996, Cheng 1998, Hsieh & Chen 1999, Chen & Li 2000, Chu & Tsao 2002, Chen & Chen 2003, Yong et al. 2004, Chen & Chen 2007, Chen & Tang 2008, Wang & Lee 2008, Chen & Wang 2009, Chen & Chen 2009), for the ranking of generalized fuzzy numbers. Chen & Chen (2009), pointed out the shortcomings of the existing methods for the ranking of generalized fuzzy numbers and proposed a new method. Ranking function is used in different areas of fuzzy optimization (Okada & Soper 2000, Mishmast Nehi 2004, Nasseri & Ardil 2005, Mahadavi & Nasseri 2006, Mahapatra & Roy 2006, Ganesan & Veeramani 2006, Maleki & Mashinchi 2007, Yu & Wei 2007, Lin 2008, Noora & Karami 2008, Allahviranloo et al 2008, Chen & Hsueh 2008, Dinagar & Palanivel 2009, Pandian & Natarajan 2010). In this paper the shortcomings of the Chen & Chen (2009), method are pointed out and a new method is proposed for the ranking of generalized fuzzy numbers. Also using the proposed ranking method, a generalized simplex algorithm is proposed for solving a
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special type of FLP problem. To illustrate the proposed algorithm a numerical example is solved and the advantages of the proposed algorithm are discussed. Since the proposed algorithm is a direct extension of classical algorithm so it is very easy to understand and apply the proposed algorithm to find the fuzzy optimal solution of FLP problems occurring in the real life situations. This paper is organized as follow: In Section 2, importance of generalized fuzzy numbers are discussed. In Section 3, some basic definitions, arithmetic operations between generalized trapezoidal fuzzy numbers and ranking function are reviewed. In Section 4, shortcomings of Chen & Chen method (2009), are pointed out. In Section 5, a new method is proposed for ranking of generalized fuzzy numbers. In Section 6, formulation and definitions of FLP problems are discussed. In Section 7, a generalized simplex algorithm is proposed for solving a special type of FLP problems. To explain the proposed algorithm a numerical example is solved in Section 8. In Section 9, validation of the proposed ranking method and advantage of proposed algorithm are discussed. In the last section conclusion is discussed. 2. Why generalized fuzzy numbers? In most of the papers generalized fuzzy numbers are converted into normal fuzzy numbers through normalization process (Kaufmann & Gupta 1985), and then obtained normal fuzzy numbers are used to solve the real life problems. Kaufmann & Gupta (1985), pointed out that there is a serious disadvantage of the normalization process. Basically we have transformed a measurement of an objective value to a valuation of a subjective value, which results in the loss of information. Although, this procedure is mathematically correct but it decreases the amount of information that is available in the original data, and we should avoid it. Hsieh & Chen (1999), pointed out that arithmetic operators on fuzzy numbers, presented in Chen (1985), does not only change the type of membership function of fuzzy numbers, but they can also reduce the troublesomeness and tediousness of arithmetic operations. There are several papers (Chen & Chen 2003, Yong et al. 2004, Mahapatra & Roy 2006, Chen & Chen 2007, Chen & Chen 2009, Chen & Wang 2009) in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the FLP problems with fuzzy parameters. 3. Preliminaries In this section some basic definitions and arithmetic operations are reviewed. 3.1. Basic definitions In this section some basic definitions are reviewed (Kaufmann & Gupta 1985). Definition 3.1 The characteristic function A of a crisp set A X assigns a value either 0 or 1 to each member in X . This function can be generalized to a function A such that the value assigned to the element of the universal set X fall within a specified
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range i.e. A : X [0,1] . The assigned value indicates the membership grade of the element in the set A . The function A is called the membership function and the set A ( x, A ( x)); x X defined by A for each x X is called a fuzzy set. Definition 3.2 A fuzzy set A defined on the universal set of real numbers R , is said to be a fuzzy number if its membership function has the following characteristics: (i) A : R [0,1] is continuous. (ii) A ( x) 0 for all x (, a] [d , ). (iii) A ( x) is strictly increasing on [a, b] and strictly decraesing on [c, d ]. (iv) A ( x) 1 for all x [b, c], where a b c d .
Definition 3.3 A fuzzy number A (a, b, c, d ) is said to be a trapezoidal fuzzy number if its membership function is given by
( x a) (b a) , a x b A ( x) 1 ,b x c (x d ) ,c x d (c d ) Definition 3.4 A fuzzy set A , defined on the universal set of real numbers R , is said to be generalized fuzzy number if its membership function has the following characteristics: (i) A : R [0,1] is continuous. (ii) A ( x) 0 for all x (, a] [d , ). (iii) A ( x) is strictly increasing on [a, b] and strictly decreasing on [c, d ]. (iv) A ( x) w, for all x [b, c], where 0 w 1.
Definition 3.5 A generalized fuzzy number A (a, b, c, d ; w) is said to be a generalized trapezoidal fuzzy number if its membership function is given by
w( x a ) (b a) , a x b A ( x) w ,b x c w( x d ) ,c x d (c d )
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Remark 3.1 If the data is collected from an expert and on the basis of collected data the cost (or demand or supply or profit etc.) of the product is represented by a generalized trapezoidal fuzzy number A (a, b, c, d ; w) then it can be explained as follows: (i)
According to decision maker the cost (or demand or supply or profit etc.) of the product will be greater than a units and less than d units.
(ii)
Decision maker is w100% in favour that the cost (or demand or supply or profit etc.) will be greater than or equal to b units and less than or equal to c units.
(iii)
The percentage of the favourness of the decision maker for the remaining values of cost (or demand or supply or profit etc.) can be obtained as follows:
Let x represents the cost (or demand or supply or profit etc.) then the percentage of the favourness of the decision maker for x A ( x) 100 ,
w( x a ) (b a) , a x b where A ( x) w ,b x c w( x d ) ,c x d (c d ) 3.2 Arithmetic operations In this section, arithmetic operations between two generalized trapezoidal fuzzy numbers, defined on universal set of real numbers R , are reviewed (Chen & Chen 2009). Let A (a1 , b1 , c1 , d1; w1 ) and B (a2 , b2 , c2 , d2 ; w2 ) be two generalized trapezoidal fuzzy numbers then (i) A1 A2 (a1 a2 , b1 b2 , c1 c2 , d1 d2 ;minimum(w1 , w2 )) (ii) A1A2 (a1 d2 , b1 c2 , c1 b2 , d1 a2 ;minimum(w1 , w2 )) (iii) A1 A2 (a ', b ', c ', d ';minimum(w1 , w2 )), where a ' minimum(a1a2 , a1d2 , a2d1 , d1d2 )
b ' minimum(b1b2 , b1c2 , c1b2 , c1c2 ), c ' maximum(b1b2 , b1c2 , c1b2 , c1c2 ) , d ' maximum(a 1 a2 , a 1 d2 , a2d1 , d1d2 )
( a1 , b1 , c1 , d1; w1 ), 0 (v) A1 ( d1 , c1 , b1 , a1 ; w1 ), 0
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3.3 Ranking function An efficient method for comparing the fuzzy numbers is by the use of a ranking function (Liou & Wang 1992) : F ( R) R, , where F ( R) is a set of fuzzy numbers defined on set of real numbers, which maps each fuzzy number into the real line, where a natural order exists i.e. (i) A B if ( A) ( B)
(ii) A B if ( A) ( B)
(iii) A B if ( A) ( B)
Remark 3.2. (Wang & Kerre 2001) For all fuzzy numbers A, B, C and D we have (i) A B A C B C
(ii) A B AC BC
(iii) A B A C B C
(iv) A B, C D A C B D
4. Shortcomings of existing method Chen and Chen (2009) pointed out the shortcomings of the existing methods for the ranking of generalized fuzzy numbers and proposed a new method. In this section, the shortcomings of Chen and Chen method, on the basis of reasonable properties of fuzzy quantities and on the basis of height of fuzzy numbers, are pointed out. 4.1. On the basis of reasonable properties of fuzzy quantities In this section shortcomings of Chen & Chen (2009) method on the basis of reasonable properties (Wang & Kerre 2001) of fuzzy numbers are pointed out. Let A and B be any two fuzzy numbers then A B AB BB (Using Remark 3.2)
i.e., ( A) ( B) ( AB) ( BB) In this section, several examples are chosen to prove that the ranking function, proposed by Chen & Chen, does not satisfy the reasonable property, A B AB BB , for
the ordering of fuzzy quantities i.e., according to Chen & Chen method A B AB BB , which is contradiction according to Wang and Kerre (2001).
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Example 4.1. Let A (0.1,0.3,0.3,0.5; 1) and B (0.2,0.3,0.3,0.4;1) be two generalized trapezoidal fuzzy numbers then according to Chen & Chen method B A
but BA AA i.e., B A does not implies BA AA.
Example 4.2. Let
A (0.1,0.3,0.3,0.5; 0.8) and B (0.1,0.3,0.3,0.5;1) be two
generalized trapezoidal fuzzy numbers then according to Chen & Chen method B A
but BA AA i.e., B A does not implies BA AA.
Example 4.3. Let A (0.8, 0.6, 0.4, 0.2;0.35) and B (0.4, 0.3, 0.2, 0.1;0.7) be two generalized trapezoidal fuzzy numbers then according to Chen & Chen method A B but AB BB i.e., A B does not implies AB BB .
A (0.2,0.4,0.6,0.8; 0.35) and B (0.1,0.2,0.3,0.4;0.7) be two
Example 4.4. Let
generalized trapezoidal fuzzy numbers then according to Chen & Chen method B A
but BA AA i.e., B A does not implies BA AA.
4.2. On the basis of height of fuzzy numbers In this section, it is shown that, in some cases, Chen & Chen method (2009) states that the ranking of fuzzy numbers depends upon height of fuzzy numbers, while in several cases the ranking does not depend upon the height of fuzzy numbers. Let A (a1 , a2 , a3 , a4 ; w1 ) and B (a1 , a2 , a3 , a4 ; w2 ) be two generalized trapezoidal fuzzy numbers then according to Chen & Chen method (2009)
A B if w1 w2 Case (i) if (a1 a2 a3 a4 ) 0 then A B if w1 w2 B if w1 w2 A
Case (ii) if (a1 a2 a3 a4 ) 0 then A B for all values of w1 and w2 .
Example 4.5. Let A (1,1,1,1; w1 ) and B (1,1,1,1; w2 ) be two generalized trapezoidal fuzzy numbers then according to Chen & Chen method A B if w1 w2 , A B if w1 w2 and A B if w1 w2 .
Example 4.6. Let A (0.4, 0.2, 0.1,0.7; w1 ) and B (0.4, 0.2, 0.1,0.7; w2 ) , be two generalized trapezoidal fuzzy numbers then A B for all values of w1 and w2 .
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According to Chen & Chen (2009) method in first case ranking of fuzzy numbers depends upon height and in second case ranking does not depend upon the height which is contradiction. 5. Proposed ranking method In this section, a new method is proposed for the ranking of generalized trapezoidal fuzzy numbers Let A (a1 , b1 , c1 , d1; w1 ) and B (a2 , b2 , c2 , d2 ; w2 ) be two generalized trapezoidal fuzzy numbers then (i) A B if ( A) ( B)
(ii) A B if ( A) ( B)
(iii) A B if ( A) ( B)
5.1. Method to find the values of ( A) and ( B) Let A (a1 , b1 , c1 , d1; w1 ) and B (a2 , b2 , c2 , d2 ; w2 ) be two generalized trapezoidal fuzzy numbers then use the following steps to find the values of ( A) and ( B) . Step 1 Find w miminum(w1 , w2 )
(b a ) 1 Step 2 Find ( A) { L1 ( x) (1 ) R 1 ( x)}dx, where L1 ( x) a1 1 1 x, 20 w w
R 1 ( x) c1
(d1 c1 ) w w x ( A) (a1 b1 ) (1 )(d1 c1 ) w 2 2
w
1 ( B) { L1 ( x) (1 ) R 1 ( x)}dx, 20
R 1 ( x) c2
where
L1 ( x) a2
and (b2 a2 ) x, w
(d2 c2 ) w w x ( B) (a2 b2 ) (1 )(d 2 c2 ) , where [0,1] . w 2 2
Example 5.1. Let A (6, 2,3,11;0.2), B (7, 4,10,16;0.4) and C (4,7,8,10;0.5) Now to compare A B and C the following steps are used: Step 1 Since A B (13, 2,13, 27;0.2) and C (4,7,8,10;0.5) w miminum(0.2,0.5) 0.2
so
find
Step 2 Now ( A B) 1.45 and (C ) 1.45 Since ( A B) (C ) , so A B C .
Similarly, if we compare
A C (2,5,11, 21;0.2) and B (7, 4,10,16;0.4) then
( A C ) 1.75 and ( B) 1.15 . So, A C B .
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Remark 5.1. The index of optimism (Liou & Wang 1992) represents the degree of optimism of a decision maker. A larger value of indicates a higher degree of optimism. For 0 and 1 values of ( A) and ( B) represents the view points of a pessimistic and optimistic decision maker respectively while for 0.5 values of ( A) and ( B) represents the view points of a moderate decision maker. Remark 5.2. The arithmetic operations between two fuzzy numbers is obtained using the cut method (Kaufmann & Gupta 1985) and in the cut method the maximum membership value of the resultant is obtained by finding the minimum of the maximum membership values of all fuzzy numbers due to which miminum(w1 , w2 ) w is considered. 6. Fuzzy linear programming problem Linear programming is one of the most frequently applied operations research technique. In the conventional approach value of the parameters of linear programming models must be well defined and precise. However, in real world environment, this is not a realistic assumption. In the real life problems the following situation may occur: If a company want to launch new product in the market then there may exist uncertainty about the profit (or cost) and availability (or demand) of the product. In such a situation the profit (or cost) and availability (or demand) may be represented by generalized fuzzy numbers. In such a case the real life problems, of m fuzzy constraints and n fuzzy variables, may be formulated as follow:
Maximize (or Minimize) z C T X
subject to AX , , , b
X 0
(1)
Where CT [c j ]1n , X [ x j ]n1 , A [aij ]mn , b [bi ]m1 and aij R, c j , x j , bi F ( R).
Definition 6.1. (Mahadavi & Nasseri 2006) Let the i th fuzzy constraint of a FLP n
problem be
a x j 1
n
a x j 1
ij
j
ij
j
bi where bi 0 then a fuzzy variable si such that si 0 and
si bi is called a fuzzy slack variable.
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Definition 6.2. (Mahadavi & Nasseri 2006) Let the i th fuzzy constraint of a FLP n
problem be
a x j 1
ij
j
bi
where bi 0 then a fuzzy variable si such that si 0 and
n
a x s b j 1
ij
j
i
i
is called a fuzzy surplus variable.
Definition 6.3. (Ganesan & Veeramani 2006) Given a system of m fuzzy linear equation involving generalized trapezoidal fuzzy numbers in n unknowns (m n), AX b , where A is a m n real matrix and rank of A is m . Let the
columns of A corresponding to fuzzy variables xk1 , xk2 ,..., xkm are linearly independent then xk1 , xk2 ,..., xkm are said to be fuzzy basic variables and remaining ( n m ) variables are called fuzzy nonbasic variables. Let B the basis matrix formed by linearly independent columns of A . The value of X B ( xk1 , xk2 ,..., xkm )T is obtained by using
X B B 1b, where k j {1, 2,..., n}, ki k j , i, j 1, 2,..., m and the values of non basic variables are assumed to be zero. The combined solution formed by using the values of fuzzy basic variables and fuzzy nonbasic variables are called fuzzy basic solution. Definition 6.4. (Ganesan & Veeramani 2006) Any X which satisfies all the constraints AX , , , b and non-negative restrictions ( X 0 ) of (1) is said to be a fuzzy feasible
solution of (1). Definition 6.5. (Ganesan & Veeramani 2006) Let S F be the set of all fuzzy feasible solution of (1). A fuzzy feasible solution X 0 S F is said to be a fuzzy optimal solution of (1) if CT X 0 C T X (or C T X 0 C T X ) .
7. Generalized fuzzy simplex algorithm There are several papers in the literature (Chen & Chen 2003, Yong et al. 2004, Mahapatra & Roy 2006, Chen & Chen 2007, Chen & Chen 2009, Chen & Wang 2009) in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the linear programming problems with fuzzy parameters. In this section generalized simplex algorithm is proposed to obtain the fuzzy optimal solution for the following type of FLP problem:
Maximize (or Minimize) z C T X
subject to AX , , , b
X 0
Where C [c j ]1n , X [ x j ]n1 , A [aij ]mn , b [bi ]m1 and aij R, c j , x j , bi F ( R). T
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The steps of the proposed algorithm are as follows: Step 1 Check that x j 0 or not j 1, 2,..., n.
Case (i) If x j 0 j , then Go to Step 2.
Case (ii) If there exist any x j such that x j 0 then replace x j by y j in the given
problem so that y j 0 , then Go to Step 2.
Case (iii) If x j a , then replace x j by y j a , where a is a fuzzy number corresponding
to crisp number a , so that y j 0 , then Go to Step 2.
Case (iv) If x j a , then replace x j by ay j , where a is a fuzzy number corresponding
to crisp number a , so that y j 0 , then Go to Step 2.
Case (v) If there exist any x j which is unrestricted, i.e., there is no restriction on the sign of ( x j ) , then replace
x j by x ' j x '' j where x ' j 0 and x '' j 0 in the given
problem then Go to Step 2. Case (vi) If a x j b, i.e., a x j and x j b , then replace x j by y j a so that y j 0 ,
then Go Step 2 Step 2 Check whether the given problem is of maximization or minimization. Case (i) If it is of maximization then Go to Step 3. Case (ii) If it is of minimization then convert it into maximization problem by multiplying 1 in the objective function then Go to Step 3. Step 3 Check that bi 0 or not i 1, 2,..., m.
Case (i) If bi 0 i then Go to Step 4.
Case (ii) If there exist any bi such that bi 0 then multiply corresponding constraint of
bi by 1 so that bi 0 i .
Step 4 Convert all the inequalities of the constraints into equation by introducing fuzzy slack and fuzzy surplus variables. Put the coefficients of these fuzzy slack and fuzzy surplus variables equal to zero in the objective function.
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Step 5 Let Maximize z C T X , subject to AX , , , b , X 0 be converted
into Maximize z C X , subject to AX b , X 0 , where C T [c j ]1N , A [aij ]mN , T
X [ x j ]N 1, b [bi ]m1 and N n . Now check whether an identity sub-matrix of order
m m exist or not in the coefficient matrix A . Case (i) If the identity sub-matrix exist then Go to Step 6. Case (ii) If the identity sub-matrix does not exist, then check is it possible to construct an identity sub-matrix without adding any extra variable and with maintaining the condition bi 0 i .
Case (a) If possible then construct an identity sub-matrix and Go to Step 6. Case (b) If not then generalized fuzzy simplex algorithm can not be applied. Step 6 The fuzzy variables, xB1 , xB 2 ,..., xBm , constituting the identity sub-matrix give the basis X B ( xB1 , xB 2 ,..., xBm ) i.e. columns of xB1 , xB 2 ,..., xBm in the coefficient matrix A are
(1,0,0,...,0)T ,(0,1,0,...,0)T ,...,(0,0,0,...,1)T respectively. The values of xB1 , xB 2 ,..., xBm can be obtained by putting the values of remaining ( N m) fuzzy nonbasic variables equal to zero in AX b . Let cB1 , cB 2 ,..., cBm be coefficients of
xB1 , xB 2 ,..., xBm in the
objective function. Step 7 Construct the fuzzy simplex table in the following format: Table 1. Fuzzy simplex table XB
x1
x2
xr
xN
xB1
a11
a12
a1r
a1N
xB 2
a21
a22
a2r
a2 N
xBi
ai1
ai 2
air
aiN
xBm
am1
am 2
amr
amN
z1c1
z2c2
zr cr
zN cN
m
z cBi xBi
i 1
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Step 8 m
Find z j c j cBi aij c j , j 1, 2,..., N . i 1
Case (i) If z j c j 0 j then the fuzzy basic feasible solution, obtained by using values
of xB1 , xB 2 ,..., xBm of Table 1, is fuzzy optimal solution and z (z ) is the optimal value of maximization (minimization) problem. Case (ii) If there exist any j such that z j c j 0 proceed on to the next Step.
Step 9 If there exist one or more fuzzy variables corresponding to which z j c j 0 then the
fuzzy variable corresponding to which the rank of z j c j is most negative will enter the basis. Let it be zr cr for some j r . Case (i) If aij 0 i then there exist fuzzy unbounded solution to the given FLP problem. Case (ii) If aij 0 for one or more values of i then compute l minimum i air
, air 0, i 1, 2,..., m
where, li is the value of i th fuzzy basic variable. The fuzzy variable, corresponding to which minimum occurs, will leave the basis. Let the minimum occurs corresponding to xBk then the common element akr , which occurs at intersection of k th row and r th column, is known as the leading element.
Step 10 Construct the new fuzzy simplex table and calculate the new entries for the fuzzy simplex table as follows: a a akj kj and aij aij kj air where i 1, 2,..., m, i k , j 1, 2,..., N . akr akr Step 11 Repeat the computational procedure from Step 8 to Step 10 until either Case (i) of Step 8 or Case (i) of Step 9 is satisfied.
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Step 12 If z j c j 0 corresponding to any fuzzy nonbasic variable in the fuzzy optimal table
(simplex table for which z j c j 0 j ) then a fuzzy alternative solution may exist and
to find it enter that fuzzy nonbasic variable into the basis and repeat once Step 8 to 10. The obtained fuzzy optimal solution will be a fuzzy alternative optimal solution.
8. Numerical example A company produces two products P1 and P2 . These products are processed on two different machines M1 and M 2 . The details for the products P1 and P2 are as follows: (i)
The time taken by the machine M 1 to produce the unit quantity of the products P1 and P2 is exactly 3 and 1 hour respectively. Similarly the time taken by the machine M 2 to produce the unit quantity of products P1 and P2 is exactly 2 and 1 hour respectively.
As discussed in Remark 3.1, on the basis of the collected data from an expert: (ii)
The profits on per unit of P1 and P2 are are represented by generalized trapezoidal fuzzy numbers Rs (3,5,8,13;0.6) and Rs (4,6,10,16;0.5) respectively. (iii) The total available time for machines M1 and M 2 are represented b generalized trapezoidal fuzzy numbers (1,3,5,6;0.9) and (1,2,4,7; 0.7) respectively. How many approximate units of P1 and P2 should be produced to maximize the profit. Solution: Let approximate x1 and x2 units of products P1 and P2 should be produced. Then the above problem may be formulated as follow: Maximize z (3,5,8,13;0.6) x1 (4,6,10,16;0.5) x2
subject to 3x1 x2 (1,3,5, 6;0.9)
2 x1 x2 (1, 2, 4, 7; 0.7)
x1 , x2 0
Using Step 1-4 of the proposed algorithm, the formulated problem may be written as Maximize z (3,5,8,13;0.6) x1 (4,6,10,16;0.5) x2
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subject to 3x1 x2 x3 (1,3,5, 6;0.9)
2 x1 x2 x4 (1, 2, 4, 7; 0.7)
x1 , x2 , x3 , x4 0
where x3 , x4 are fuzzy slack variables. Comparing the formulated problem with Maximize z C T X
subject to AX b
X 0
x1 x2 3 1 1 0 T we have C (3,5,8,13;0.6) (4,6,10,16;0.5) 0 0 , X , A and x3 2 1 0 1 x4 (1,3,5, 6;0.9) b . Since the columns of x3 and x4 constitute the identity matrix so (1,2,4,7; 0.7)
x3 and x4 are basic variables. Putting x1 0 and x2 0 , values of x3 and x4
are
(1,3,5,6;0.9) and (1,2,4,7; 0.7) respectively. Using Step 7 the fuzzy simplex table for the above problem is
Table 2 XB
x1
x2
x3
x4
x3 (1,3,5,6;0.9)
3
1
1
0
x4 (1,2,4,7; 0.7)
2
1
0
1
z (0, 0, 0, 0;1)
(13, 8, 5, 3;0.6)
(16, 10, 6, 4;0.5)
0
0
Using Step 9 of the proposed algorithm, fuzzy variable x2 will enter the basis and the fuzzy variable x4 will leave the basis. Using Step 10 of the proposed algorithm new fuzzy simplex table is Table 3 XB
x1
x2
x3
x4
x3 (6, 1,3,5;0.7)
1
0
1
1
x2 (1,2,4,7; 0.7)
2
1
0
1
z (4,12, 40,112;0.5)
(5, 4,15, 29;0.5)
0
0
(4,6,10,16;0.5)
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Using the optimality condition (Step 8, Case (i)) there is no entering variable. Therefore, the obtained fuzzy basic feasible solution is fuzzy optimal solution. Hence x1 0 , x2 (1, 2, 4,7;0.7) and Maximize z (4,12, 40,112;0.5) .
Note:- 0 where represents a generalized trapezoidal fuzzy number whose rank is zero. 9. Results and discussion In this section the validity of the proposed ranking method is shown and also the advantages of the algorithm, proposed for solving a special type of FLP problem, are discussed. 9.1. Validity of the proposed ranking method In this section the correct ordering of fuzzy numbers, discussed in Section 4, are obtained by applying proposed ranking method. Also, in the Table 4, it is shown that proposed ranking function satisfies the all reasonable properties of fuzzy quantities proposed by Wang and Kerre (2001). Example 9.1. Let A (0.1,0.3,0.3,0.5; 1) generalized trapezoidal fuzzy numbers.
and
B (0.2,0.3,0.3,0.4;1)
be two
Step 1 minimum (1,1) 1 1 Step 2 ( A) ( (0.4) (1 )(0.8)) 0.4 0.2 and 2 1 1 ( B) ( (0.5) (1 )(0.7)) (0.7 0.2 ) 2 2 For a pessimistic decision maker, with 0, ( A) 0.4 and ( B) 0.35 . Since
( A) ( B) so A B .
For optimistic decision maker, with
1, ( A) 0.2 and ( B) 0.25 . Since
( A) ( B) so A B .
For moderate decision maker, with 0.5, ( A) 0.3 and ( B) 0.3 . Since ( A) ( B) so A B .
Example 9.2. Let A (0.1,0.3,0.3,0.5; 0.8) and B (0.1,0.3,0.3,0.5;1) be two generalized trapezoidal fuzzy numbers. Step 1 minimum (1,0.8) 0.8 0.8 ( (0.4) (1 )(0.8)) 0.4(0.8 0.4 ) and Step 2 ( A) 2 0.8 ( B) ( (0.4) (1 )(0.8)) 0.4(0.8 0.4 ) 2
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Since ( A) ( B) so A B .
Example 9.3. Let A (0.8, 0.6, 0.4, 0.2;0.35) and B (0.4, 0.3, 0.2, 0.1;0.7) be two generalized trapezoidal fuzzy numbers. Step 1 minimum (0.35,0.7) 0.35 0.35 (0.35 0.4 ) 2 For a pessimistic decision maker, with 0, ( A) 0.105 and ( B) 0.0525 . Since ( A) ( B) so A B .
Step 2 ( A) 0.35(0.3 0.4 ) and ( B)
For optimistic decision maker, with 1, ( A) 0.245 and ( B) 0.1225 . Since
( A) ( B) so A B .
For moderate decision maker, with 0.5, ( A) 0.175 and ( B) 0.0875 . Since
( A) ( B) so A B .
Example 9.4. Let A (0.2,0.4,0.6,0.8;0.35) and B (0.1,0.2,0.3,0.4;0.7) be two generalized trapezoidal fuzzy numbers. Step 1 minimum (0.35,0.7) 0.35 0.35 (0.7 0.4 ) 2 For a pessimistic decision maker, with 0, ( A) 0.245 and ( B) 0.1225 . Since
Step 2 ( A) 0.35(0.7 0.4 ) and ( B)
( A) ( B) so A B .
For optimistic decision maker, with 1, ( A) 0.105 and ( B) 0.0525 . Since ( A) ( B) so A B .
For moderate decision maker, with 0.5, ( A) 0.175 and ( B) 0.0875 . Since
( A) ( B) so A B .
Example 9.5. Let A (1,1,1,1; w1 ) and B (1,1,1,1; w2 ) be two generalized trapezoidal fuzzy numbers then Step 1 minimum (w1 , w2 ) w w w Step 2 ( A) ( (2) (1 )(2)) w and ( B) ( (2) (1 )(2)) w . Since 2 2 ( A) ( B) so A B .
Example 9.6. Let A (0.4, 0.2, 0.1,0.7; w1 ) and B (0.4, 0.2, 0.1,0.7; w2 ) be two generalized trapezoidal fuzzy numbers then
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Step 1 minimum (w1 , w2 ) w w Step 2 ( A) ( (0.6) (1 )(0.6)) 0.3 and 2 w ( B) ( (0.6) (1 )(0.6)) 0.3 Since ( A) ( B) so A B . 2 In the above examples it can be easily checked that, for a particular value of
(i) A B AB BB i.e., (( AB)( BB)) (( BB)( BB))
(ii) A B AB BB i.e., (( AB)( BB)) (( BB)( BB))
(iii) A B AB BB i.e., (( AB)( BB)) (( BB)( BB))
Wang and Kerre (2001) proposed some reasonable properties that are satisfied by the existing or the proposed ranking functions are denoted by “Y” and the properties that are satisfied by the exsiting or the proposed ranking function are denoted by “N”. It is obvious from Table 4 that the proposed ranking function satisfied the properties proposed by Wang and Kerre (2001). Table 4. Fulfilment of the axioms for the ordering in the first and second class Wang and Kerre (2001) Index
Y1
A1 Y
A2 Y
A3 Y
A4 Y
A '4 Y
A5 Y
A6 N
A '6 N
A7 N
Y2
Y
Y
Y
Y
Y
Y
Y
Y
N
Y3
Y
Y
Y
N
N
Y
N
N
N
Y4 C FR CL LW CM1
Y
Y
Y
Y
Y
Y
Y
N
N
Y Y Y Y Y
Y Y Y Y Y
Y Y Y Y Y
N Y Y Y Y
N Y Y Y Y
Y Y Y Y Y
N Y Y Y Y
N Y Y Y Y
N N N N N
CM 2
Y
Y
Y
Y
Y
Y
Y
Y
N
K W Jk CH k KP k Proposed Methd
Y Y Y Y Y Y
Y Y Y Y Y Y
Y Y Y Y Y Y
N Y Y Y Y Y
N N N Y Y Y
N N N N N Y
N N N N N Y
N N N N N Y
N N N N N N
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9.2. Advantages of the proposed algorithm In this section the advantages of the proposed algorithm, for solving generalized FLP problems, are discussed. 9.2.1. Disadvantages of the normalization process In most of the papers the generalized fuzzy numbers are converted into normal fuzzy numbers through normalization process (Kaufmann & Gupta 1985) and then normal fuzzy numbers are used to solve the real life problems. Kaufmann & Gupta (1985) pointed out that there is a serious disadvantage of the normalization process. Basically we have transformed a measurement of an objective value to a valuation of a subjective value, which results in the loss of information. Although, this procedure is mathematically correct, it decreases the amount of information that is available in the original data, and we should avoid it. To explain the disadvantages of the normalization process the same example is solved, using the proposed method, with and without normalization process and then the obtained results are discussed as follows: 9.2.1.1. Results with normalization process If all the values of the parameters, used in numerical example solved in Section 8, is first normalized and then the problem is solved by using the proposed method then the fuzzy optimal value is z (4,12, 40,112;1) . The membership function for the obtained
result is shown in Fig. 1. From the Fig. 1 it is clear that
z ( x) 1
x 0
4
12
Fig. 1. Membership function of (i) (ii) (iii)
40
112
z
According to decision maker the total profit will be greater than Rs 4 and less than Rs 112. Decision maker is 100% in favour that the total profit will be greater than or equal to Rs 12 and less than or equal to Rs 40. The percentage of the favourness of the decision maker for the remaining values of total profit can be obtained as follows:
Let x represents the value of total profit then the percentage of the favourness of the
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( x 4) , 4 x 12 8 decision maker for x z ( x) 100 , where A ( x) 1 ,12 x 40 ( x 112) , 40 x 112 72 9.2.1.2. Results without normalization process If the values of the parameters of the same example is not normalized and then the problem is solved (as in section 8) by using the proposed method then the fuzzy optimal value is z (4,12, 40,112;0.5) . The membership function for the obtained result is
shown in Fig. 2. From the Fig. 2 it is clear that
z ( x) 0.5
0
4
12
Fig. 2. Membership function of
(i) (ii) (iii)
40
112
x
z
According to decision maker the total profit will be greater than Rs 4 and less than Rs 112. Decision maker is 50% in favour that the total profit will be greater than or equal to Rs 12 and less than or equal to Rs 40. The percentage of the favourness of the decision maker for the remaining values of total profit can be obtained as follows:
Let x represents the value of total profit then the percentage of the favourness of the 0.5( x 4) , 4 x 12 8 ,12 x 40 decision maker for x z ( x) 100 , where A ( x) 0.5 0.5( x 112) , 40 x 112 72 It is obvious from the results explained in Sections 9.2.1.1 and 9.2.1.2 that according to decision maker the range of objective function is same in both cases i.e., total profit will be greater than Rs 4 and less than Rs 112 but due to normalization process the actual
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percentage of favourness for different values of total profit is changed. For example: Results with normalization process represents that the favourness of decision maker about the value of total profit lying between Rs 12 and Rs 40 is 100% while without normalization process the percentage of favourness for the same range is 50%. Similarly it can be found that the percentage of favourness for the same values of total profit are different in both cases i.e., using the normalization process the actual information is lost. 10. Conclusion In this paper, the shortcomings of Chen & Chen (2009) method are pointed out and a new ranking method is proposed for finding the correct ordering of generalized trapezoidal fuzzy numbers. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (2009). A new algorithm is introduced and also the advantages of proposed algorithm are discussed for solving a particular type of FLP problem occurring in real life problems. The proposed method can be extended to propose Big-M and Two phase methods for finding the fuzzy optimal solution of fuzzy linear programming programming problems.
Acknowledgement The authors are extremely thankful to the learned referees and editors for their most valuable comments which have substantially improved the presentation of this paper. References Allahviranloo T, Lotfi T., Kiasary M.K., Kiani N.A., Alizadeh L., Solving fully fuzzy linear programming problem by the ranking function. Applied Mathematical Sciences, 2, 19 – 32, 2008. Bellman R.E., Zadeh L.A., Decision making in a fuzzy environment. Management Science, 17, 141-164, 1970. Buckley J., Feuring T., Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming. Fuzzy Sets and Systems, 109, 35-53, 2000. Campos L., Gonzalez A., A subjective approach for ranking fuzzy number. Fuzzy Sets and Systems, 29, 145-153, 1989. Campos L., Verdegay J.L., Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems, 32, 1-11, 1989. Chang P.T., Lee E.S., Ranking of fuzzy sets based on the concept of existence. Computers and Mathematics with Applications, 27, l-21, 1994.
100
Chen S.H., Operations on fuzzy numbers with function principal. Tamkang Journal of Management Sciences, 6, 13-25, 1985. Chen S.J., Chen S.M., A new method for handling multicriteria fuzzy decision making problems using FN-IOWA operators. Cybernetics and Systems, 34, 109-137, 2003. Chen S.J., Chen S.M., Fuzzy risk analysis on the ranking of generalized trapezoidal fuzzy numbers. Applied Intelligence 26, 1-11, 2007. Chen S.M., Chen J.H., Fuzzy risk analysis based on the ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications, 36, 6833-6842, 2009. Chen S.P., Hsueh Y.J., A simple approach to fuzzy critical path analysis in project networks. Applied Mathematical Modelling, 32, 1289-1297, 2008. Chen S.H., Li G.C., Representation, ranking and distance of fuzzy number with exponential membership function using graded mean integration method. Tamsui Oxford Journal of Mathematical Sciences, 16, 125-131, 2000. Chen C.C., Tang H.C., Ranking nonnormal p-norm trapezoidal fuzzy numbers with integral value. Computers and Mathematics with Applications, 56, 2340-2346, 2008. Chen S.M., Wang C.H., Fuzzy risk analysis based on ranking fuzzy numbers using cuts, belief features and signal/noise ratios. Expert Systems with Applications, 36, 5576-5581, 2009. Cheng C.H., A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95, 307-317, 1998. Chu T.C., Tsao C.T., Ranking fuzzy numbers with an area between the centroid point and original point. Computers and Mathematics with Applications, 43, 111-117, 2002. Delgado M., Verdegay J.L., Vila M.A., A general model for fuzzy linear programming. Fuzzy Sets and Systems, 29, 21-29, 1989. Dinagar D.S., Palanivel K., The transportation problem in fuzzy environment, International Journal of Algorithms. Computing and Mathematics, 2, 65-71, 2009. Fortemps P., Roubens M., Ranking and defuzziffication methods based on area compensation. Fuzzy Sets and Systems, 82, 319-330, 1996. Ganesan K., Veeramani P., Fuzzy linear programs with trapezoidal fuzzy numbers. Annals of Operations Research, 143, 305-315, 2006. Hsieh C.H., Chen S.H., Similarity of generalized fuzzy numbers with graded mean integration representation, in: Proceedings 8th International Fuzzy System Association World Congress, Vol-2, Taipei, Taiwan, Republic of China, 551-555, 1999.
101
Inuiguchi M., Ichihashi H. Kume Y., A solution algorithm for fuzzy linear programming with piecewise linear membership function. Fuzzy Sets and Systems, 34, 15-31, 1990. Inuiguchi M., Ramik J., Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems, 111, 3-28, 2000. Jain R., Decision-making in the presence of fuzzy variables. IEEE Transactions on Systems Man and Cybernetics, 6, 698-703, 1976. Kacprzyk J., A generalization of fuzzy multistage decision making and control via linguistic quantifiers, Internaternational Journal of Control, 38, 1249-1270, 1983. Kaufmann A., Gupta M.M., Introduction to Fuzzy Arithmetics: Theory and Applications, New York: Van Nostrand Reinhold (1985). Lin F.T., Time-cost tradeoff in fuzzy critical path analysis based on (1 ) 100% confidence-interval estimates. IEEE International Conference on Systems, Man and Cybernetics, 601-606, 2008. Liou T.S. Wang M.J., Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50, 247-255, 1992. Luhandjula M.K., Linear programming with a possibilistic objective function. European Journal of Operational Research, 31, 110-117, 1987. Mahadavi-Amiri N., Nassseri S.H., Duality in fuzzy number linear programming by the use of a certain linear ranking function. Applied Mathematics and Computation, 180, 206- 216, 2006. Mahapatra G.S., Roy T.K., Fuzzy multi-objective mathematical programming on reliability optimization model. Applied Mathematics and Computation, 174, 643-659, 2006. Maleki H.R., Mashinchi M., Multiobjective geometric programming with fuzzy parameters. International Journal of Information Science and Technology, 5, 35-45, 2007. Maleki H.R., Tata M., Mashinchi M., Linear programming with fuzzy variables. Fuzzy Sets and Systems, 109, 21-33, 2000. Mishmast Nehi H., Maleki H.R., Mashinchi M., Solving fuzzy number linear programming problem by lexicographic ranking function. Italian Jornal of Pure and Applied Mathematics, 15, 9-20, 2004.
102
Nasseri S.H., Ardil E., Simplex method for fuzzy variable linear programming problems. Proceedings of World Academy of Science, Engineering and Technology, 8, 1307-6884, 2005. Noora A. A., Karami P., Ranking functions and its application to fuzzy DEA. International Mathematical Forum, 3, 1469-1480, 2008. Okada S., Soper T., A shortest path problem on a network with fuzzy arc lengths. Fuzzy Sets and Systems, 109, 129-140, 2000. Pandian P., Natarajan G., A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences, 4, 79 – 90, 2010. Ramik J., Duality in fuzzy linear programming: some new concepts and results. Fuzzy Optimization and Decision Making, 4, 25-39, 2005. Rommelfanger H., A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values. Fuzzy Sets and Systems, 158, 18921904, 2007. Tanaka H., Asai K., Fuzzy linear programming with fuzzy numbers. Fuzzy Sets and Systems, 13, 1-10, 1984. Tanaka H., Okuda T., Asai K., On fuzzy mathematical programming. Journal of Cybernetics, 3, 37-46, 1974. Wang X., Kerre E. E., Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems, 118, 375-385, 2001. Wang Y.J., Lee H.S., The revised method of ranking fuzzy numbers with an area between the centroid and original points. Computers and Mathematics with Applications, 55, 2033-2042, 2008. Yong D., Wenkang S., Feng D., Qi L., A new similarity measure of generalized fuzzy numbers and its application to pattern recognition. Pattern Recognition Letters, 25, 875883, 2004. Yu J.R., Wei T.H., Solving the fuzzy shortest path problem by using a linear objective programming, Journal of the Chinese Institute of Industrial Engineers, 24, 360-365, 2007. Zadeh L.A., Fuzzy sets. Information and Control, 8, 338-353, 1965. Zimmermann H.J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 45-55, 1978.
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