Fuzzy Inf. Eng. (2013) 4: 475-492DOI 10.1007/s12543-013-0159-8

O R I G I N A L A R T I C L E

Applications of Fuzzy Linear Programming withGeneralized LR Flat Fuzzy Parameters

Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma

Received: 22 July 2012/ Revised: 9 October 2013/Accepted: 2 November 2013/© Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the OperationsResearch Society of China

2013

Abstract In this paper, the limitations of existing methods to solve the problemsof fuzzy assignment, fuzzy travelling salesman and fuzzy generalized assignment arepointed out. All these problems can be formulated in linear programming problemswherein the decision variables are represented by real numbers and other parametersare represented by fuzzy numbers. To overcome the limitations of existing methods,a new method is proposed. The advantage of proposed method over existing methodsis demonstrated by solving the problems mentioned above which can or cannot besolved by using the existing methods.

Keywords Fuzzy assignment problem · Fuzzy generalized assignment problem ·Fuzzy travelling salesman problem · Generalized LR flat fuzzy number

1. Introduction

The concept of fuzzy mathematical programming on a general level was first pro-posed by Tanaka et al. [27] in the framework of the fuzzy decision of Bellman andZadeh [1]. The first formulation of fuzzy linear programming (FLP) problem wasproposed by Zimmermann [30]. Thereafter, many authors considered various typesof FLP problems and proposed several approaches to solve them. Tanaka and Asai[28] proposed a possibilistic linear programming formulation where the coefficientsof decision variables are crisp, while decision variables are obtained as fuzzy num-bers. Subsequently, Inuiguchi and Sakawa [11] implemented optimality tests in linearprogramming with possible and necessary measures.

Anila Gupta (�)Shaheed Bhagat Singh State Technical Campus, Ferozepur-152004, Indiaemail: anilasingal@gmail.comAmit Kumar(�) ·Mahesh Kumar Sharma (�)School of Mathematics and Computer Applications, Thapar University, Patiala-147004, Indiaemail: amit rs iitr@yahoo.com

mksharma69@thapar.edu

476 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

In addition, some authors used the concept of comparison of fuzzy numbers tosolve FLP problems. Usually in such methods, authors define a crisp model whichis equivalent to the FLP problem and then use optimal solution of the model as anoptimal one to FLP problem. Based on this idea, Maleki et al. [20] proposed a newmethod to solve the fuzzy number linear programming (FNLP) problem and usedits solution to obtain the fuzzy solution to the fuzzy variable linear programming(FVLP) problem. Mahdavi-Amiri and Nasseri [17] extended the concepts of dualityin FNLP problems as a similar problem leading to the dual simplex algorithm [25]to such problems. However, the method of Maleki et al. [20] has a shortcomingand it is not efficient when the decision variables are bounded in an FNLP problem.Thus, some authors proposed a new approach to overcome this shortcoming basedon dual and primal simplex methods [6, 9]. Moreover, Mahdavi-Amiri and Nasseri[18] used a certain linear ranking function to define the dual of FNLP problems asFVLP problems that lead to an efficient method called the dual simplex algorithm toFVLP problems directly. After that, Ebrahimnejad et al. [8] gave another efficientmethod namely a primal-dual simplex algorithm to obtain a fuzzy solution to FVLPproblems. Nasseri and Ebrahimnejad [24] applied a fuzzy primal simplex method[19] for flexible linear programming problems directly without solving any auxiliaryproblem. Ebrahimnejad and Nasseri [5] used the complementary slackness to solveFNLP and FVLP problems without the need of a simplex table.

Lotfi et al. [16] discussed fully FLP problems where all parameters and vari-able are triangular fuzzy numbers. They used the concept of the symmetric triangu-lar fuzzy number and introduced an approach to defuzzify a general fuzzy quantity.Ganesan and Veeramani [10] introduced a new method for solving a kind of lin-ear programming problems involving symmetric trapezoidal fuzzy numbers withoutconverting them to crisp linear programming problems based on primal simplex algo-rithm. Ebrahimnejad et al. [7] developed their method for situations in which someor all fuzzy decision variables are bounded. Some authors [23, 26] developed the dualof a linear programming problem with symmetric trapezoidal fuzzy numbers withoutconverting them to crisp linear programming problems.

Chen [2] pointed out that in many cases it is not possible to restrict the membershipfunction to normal form and proposed the concept of generalized fuzzy numbers.Since then, tremendous efforts are spent and significant advances are made on thedevelopment of numerous methodologies [3, 4, 14, 15] for comparing generalizedfuzzy numbers. Several authors [12, 13, 21, 22] have used the existing method [17,20, 25] for solving different type of problems.

This paper is organized as follows: In Section 2, some basic definitions and anexisting ranking approach for comparing fuzzy numbers are discussed. Existing for-mulation of FLP problems is considered in Section 3. The method to solve suchproblems is presented in Section 4. In Section 5, applications of existing methodare explained. In Section 6, limitations of existing methods are commented upon.In Section 7, to overcome the limitations of existing methods stated in Section 6, anew method is proposed to find an optimal solution to FLP problems. In Section8, to illustrate the proposed method, a numerical example is solved. The results areanalyzed in Section 9 and the conclusions are dealt with in Section 10.

Fuzzy Inf. Eng. (2013) 4: 475-492 477

2. Preliminaries

In this section, some basic definitions and an existing ranking approach [29] are pre-sented for comparing fuzzy numbers.

2.1. Basic Definitions

In this section, some basic definitions are discussed.

Definition 1 [4] A fuzzy set A, defined on the universal set of real numbers R, issaid to be a generalized fuzzy number if its membership function has the followingcharacteristics:

(i) μA : R −→ [0,w] is continuous, where 0 < w ≤ 1 and w is said to be height ofgeneralized fuzzy set.

(ii) μA(x) = 0 for all x ∈ (−∞, a]⋃

[d,∞).

(iii) μA(x) is strictly increasing on [a, b] and is strictly decreasing on [c, d].

(iv) μA(x) = w for all x ∈ [b, c].

If w = 1, then A is said to be normal fuzzy number.

Definition 2 [4] A fuzzy number A = (m, n, α, β; w)LR is said to be generalized LR flatfuzzy number if

μA(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

wL( m−xα

), for x ≤ m, α > 0,w, for m ≤ x ≤ n,

wR( x−nβ

), for x ≥ n, β > 0.

If m = n, then A = (m, n, α, β; w)LR will be converted into A = (m, α, β; w)LR and is saidto be generalized LR fuzzy number. L and R are called reference functions, which arecontinuous, non-increasing functions that define the left and right shapes of μA(x)respectively and L(0) = R(0) = 1. If w = 1, then A is said to be normal LR flat fuzzynumber.

Definition 3 [4] A function L : [0,∞) → [0, 1] (or R : [0,∞) → [0, 1]) is said to bereference function of fuzzy number iff

(i) L(x) = L(−x) (or R(x) = R(−x)).

(ii) L(0) = 1 (or R(0) = 1).

(iii) L (or R) is non-increasing on [0,∞).

Definition 4 [4] Let A = (m, n, α, β; w)LR be a generalized LR flat fuzzy number andλ be a real number in the interval [0,w]. Then the crisp set Aλ = {x ∈ X : μA(x) ≥λ} = [m − αL−1( λw ), n + βR−1( λw )] is said to be λ-cut of A.

2.2. Comparison of Fuzzy Numbers

In all the existing methods [12, 13, 21, 22], the existing ranking approach [29] is usedfor comparing fuzzy numbers. Using it, the generalized LR flat fuzzy numbers A =(m, n, α, β; w1)LR and B = (m1, n1, α1, β1; w2)LR can be compared as follows:

478 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

Step 1: Calculate �(A) = w12

( ∫ 10 (m − αL−1(λ) + n + βR−1(λ))dλ

)and �(B) =

w22

( ∫ 10 (m1 − α1L−1(λ) + n1 + β1R−1(λ))dλ

).

Step 2: Check that�(A) > �(B),�(A) < �(B) or�(A) = �(B).

(i) If�(A) >�(B), then A � B.

(ii) If�(A) =�(B), then A ≈ B.

(iii) If�(A) <�(B), then A ≺ B.

3. Existing Fuzzy Linear Programming Formulation

Any linear programming problem where decision variables are represented by realnumbers with all other parameters represented by normal LR flat fuzzy numbers, canbe formulated as follows [17, 20, 25]:

Max (or Min)n∑

j=1c j x j

s.t.n∑

j=1ai j x j �,�, bi, i = 1, 2, · · · ,m, (P1)

x j ≥ 0, j = 1, 2, · · · , n,

where, c j, ai j and bi are normal LR flat fuzzy numbers.

Optimal Solution

The optimal solution to FLP problem (P1) is the set of non-negative real numbers {x j}satisfying the following characteristics:

(i)n∑

j=1�(ai j)x j ≤,=,≥ �(bi), i = 1, 2, · · · ,m.

(ii) If there exist any set of non-negative real numbers {x′j}, such thatn∑

j=1�(ai j)x′j ≤,=,≥ �(bi), i = 1, 2, · · · ,m, then

�(n∑

j=1c j x j) ≤ �(

n∑j=1

c j x′j) (in case of minimization),

�(n∑

j=1c j x j) ≥ �(

n∑j=1

c j x′j) (in case of maximization).

4. Existing Method

The optimal solution to FLP problem (P1) can be obtained by using the followingsteps [17, 20, 25]:

Step 1: Assume c j = (mj, n j, α j, β j; 1)LR, ai j = (mi j, ni j, αi j, βi j; 1)LR and bi = (m′i ,n′i , α

′i , β′i ; 1)LR. The resultant FLP problem (P1) can be written as:

Max (or Min)n∑

j=1((mj, n j, α j, β j; 1)LR)x j

Fuzzy Inf. Eng. (2013) 4: 475-492 479

s.t. (n∑

j=1((mi j, ni j, αi j, βi j; 1)LR)x j) �,�, (m′i , n

′i , α′i , β′i ; 1)LR, (P2)

i = 1, 2, · · · ,m,x j ≥ 0, j = 1, 2, · · · , n.

Step 2: Convert the FLP problem (P2) into crisp linear programming problem (P3):

Max (or Min) �(n∑

j=1((mj, n j, α j, β j; 1)LR)x j)

s.t. �(n∑

j=1((mi j, ni j, αi j, βi j; 1)LR)x j) ≤,=,≥ �((m′i , n

′i , α′i , β′i ; 1)LR),

i = 1, 2, · · · ,m, (P3)x j ≥ 0, j = 1, 2, · · · , n.

Step 3: Using the linearity property

�(n∑

j=1((mj, n j, α j, β j; 1)LR)x j) =

n∑j=1�((mj, n j, α j, β j; 1)LR)x j,

the crisp linear programming problem, obtained in Step 2, can be written as:

Max (or Min)n∑

j=1�((mj, n j, α j, β j; 1)LR)x j

s.t.n∑

j=1�((mi j, ni j, αi j, βi j; 1)LR)x j ≤,=,≥ �((m′i , n

′i , α′i , β′i ; 1)LR),

i = 1, 2, · · · ,m,x j ≥ 0, j = 1, 2, · · · , n.

Step 4: Solve the crisp linear programming problem, obtained in Step 3, to findoptimal solution {x j}.Step 5: Put the value of x j in

n∑j=1

c j x j to find the fuzzy optimal value.

5. Applications of Existing Method

Several authors [12, 13, 21, 22] have used the existing method, presented in Section4, for solving different type of problems.

(i) Using the existing method, Mukerjee and Basu [21] proposed a method to findthe solution to following type of fuzzy assignment problems:

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n,

n∑j=1

xi j = 1, i = 1, 2, · · · , n, (S 1)

xi j = 0 or 1,∀ i, j = 1, 2, · · · , n,where

ci j = (mi j, ni j, αi j, βi j; 1)LR : Fuzzy payment to ith person for doing jth job.n∑

i=1

n∑j=1

ci j xi j : Total fuzzy cost for performing all the jobs.

(ii) Using the existing method, Kumar and Gupta [12] proposed a method to find asolution to following type of fuzzy assignment problems:

480 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n,

n∑j=1

xi j = 1, i = 1, 2, · · · , n, (S 2)

xi j = 0 or 1,∀ i, j = 1, 2, · · · , n,where

ci j = (mi j, ni j, αi j, βi j; 1)Li j−Ri j : Fuzzy payment to ith person for doing jth job.n∑

i=1

n∑j=1

ci j xi j : Total fuzzy cost for performing all the jobs.

(iii) Using the existing method, Mukerjee and Basu [22] proposed a method to findthe solution to following type of fuzzy assignment problems:

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n,

n∑j=1

xi j = 1, i = 1, 2, · · · , n, (S 3)

ci j xi j � CPi , ∀ i, j = 1, 2, · · · , n,ci j xi j � CJ j , ∀ i, j = 1, 2, · · · , n,xi j = 0 or 1, ∀ i, j = 1, 2, · · · , n,

where

CPi = (mPi , nPi , αPi , βPi ; 1)LPi−RPi: Maximum fuzzy cost offered to ith person.

CJ j = (mJ j , nJj , αJ j , βJ j ; 1)LJ j−RJ j: Maximum fuzzy cost spent on jth job.

(iv) Using the existing method, Kumar and Gupta [12] proposed a method to findthe solution to following type of fuzzy travelling salesman problems:

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n and j � i,

n∑j=1

xi j = 1, i = 1, 2, · · · , n and i � j,

xi j + x ji ≤ 1, 1 ≤ i � j ≤ n, (S 4)xi j + x jk + xki ≤ 2, 1 ≤ i � j � k ≤ n,

...

xip1 + xp1 p2 + xp2 p3 + · · · + xp(n−2)i ≤ (n − 2),1 ≤ i � p1 � · · · � p(n−2) ≤ n,

where

ci j = (mi j, ni j, αi j, βi j; 1)Li j−Ri j : Fuzzy travelling cost from ith city to jth city,n∑

i=1

n∑j=1

ci j xi j : The total fuzzy travelling cost of completing the tour,

Fuzzy Inf. Eng. (2013) 4: 475-492 481

xi j = 1 : if the salesman visits city j immediately after visiting city i and xi j =

0, otherwise.

(v) Using the existing method, Kumar and Gupta [13] proposed a method to findthe solution to following type of fuzzy generalized assignment problems:

Minn∑

i=1

m∑j=1

ti j xi j

s.t.m∑

j=1xi j = 1, i = 1, 2, · · · , n,

n∑i=1

ti j xi j � tP j , j = 1, 2, · · · ,m, (S 5)

xi j = 0 or 1,∀ i, j,where

ti j = (mi j, ni j, αi j, βi j; 1)Li j−Ri j : Fuzzy time required to perform ith job by jth

person.n∑

i=1

m∑j=1

ti j xi j : Total fuzzy time for performing all the jobs.

tP j = (mP j , nP j , αP j , βP j ; 1)LP j−RP j: Maximum fuzzy time available with jth

person.

(vi) Mukerjee and Basu [22] modified the fuzzy assignment problems (S 2) into(S 3) by adding two more restrictions. In the same direction, fuzzy generalizedassignment problems (S 5) is modified into (S 6) by adding one more restriction,i.e., due to time constraint, a decision maker does not intend to spend more thanfuzzy time tJi on ith job. So, he imposes the restriction of maximum fuzzy timefor each job. Existing method can be used to find the solution to following typeof fuzzy generalized assignment problems:

Minn∑

i=1

m∑j=1

ti j xi j

s.t.m∑

j=1xi j = 1, i = 1, 2, · · · , n,

n∑i=1

ti j xi j � tP j , j = 1, 2, · · · ,m, (S 6)

ti j xi j � tJi , i = 1, 2, · · · , n; j = 1, 2, · · · ,m,xi j = 0 or 1, ∀ i, j,

where

tJi = (mJi , nJi , αJi , βJi ; 1)LJi−RJi: Maximum fuzzy time restriction on ith job.

6. Limitations of Existing Methods

Chen [2] pointed out that in many cases it is not possible to restrict the membershipfunction to normal form and proposed the concept of generalized fuzzy numbers.Since then, tremendous efforts are spent and significant advances are made on thedevelopment of numerous methodologies [3, 4, 14, 15] for comparing these fuzzynumbers.

482 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

In Step 3 of the existing method, presented in Section 4, the linearity property�(c1x1 ⊕ c2x2 ⊕ · · · ⊕ cnxn) = �(c1)x1 +�(c2)x2 + · · · +�(cn)xn is used to convertan FLP problem into a crisp linear programming one. But the linearity property�(c1x1 ⊕ c2x2 ⊕ · · · ⊕ cnxn) = �(c1)x1 +�(c2)x2 + · · · +�(cn)xn is satisfied only ifc1, c2, · · · , cn are normal LR flat fuzzy numbers or generalized LR flat fuzzy numbersof the same heights. But if c1, c2, · · · , cn are generalized LR flat fuzzy numbers ofdifferent heights, then the linearity property�(c1x1 ⊕ c2x2 ⊕ · · · ⊕ cnxn) � �(c1)x1 +

�(c2)x2 + · · ·+�(cn)xn is not satisfied. Hence, the existing methods [12, 13, 21, 22]cannot be used for solving the following problems:

(i) The existing method [21] can be used to solve such fuzzy assignment prob-lems (S 1) in which the parameters ci j are represented by normal LR flat fuzzynumbers, e.g., fuzzy assignment problem chosen in Example 1.

Example 1 Solve the fuzzy assignment problem for which the fuzzy costs ci j areshown in Table 1.

Table 1: Fuzzy assignment problem with ci j as normal LR flat fuzzy numbers.

Job→ J1 J2 J3 J4

Person↓P1 (5, 6, 2, 1; 1)LR (8, 11, 3, 1; 1)LR (10, 11, 1, 4; 1)LR (8, 10, 3, 1; 1)LR

P2 (8, 10, 1, 1; 1)LR (5, 6, 2, 1; 1)LR (8, 10, 2, 2; 1)LR (8, 9, 3, 1; 1)LR

P3 (4, 5, 2, 1; 1)LR (7, 10, 2, 1; 1)LR (11, 13, 3, 2; 1)LR (6, 7, 2, 3; 1)LR

P4 (8, 10, 2, 2; 1)LR (5, 6, 3, 1; 1)LR (7, 10, 2, 1; 1)LR (4, 5, 2, 2; 1)LR

where L(x) = R(x) = max{0, 1 − x}.But the existing method [21] cannot be used for solving such fuzzy assignment

problems (S 7) in which there is a need to represent parameters ci j by generalized LRflat fuzzy numbers of different heights.

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n,

n∑j=1

xi j = 1, i = 1, 2, · · · , n, (S 7)

xi j = 0 or 1, ∀ i, j = 1, 2, · · · , n,where ci j = (mi j, ni j, αi j, βi j; wi j)LR.

The fuzzy assignment problem, chosen in Example 2, cannot be solved by usingthe existing method [21] as parameters ci j are represented by generalized LR flatfuzzy numbers.

Example 2 Solve the fuzzy assignment problem for which fuzzy costs ci j are shownin Table 2.

Fuzzy Inf. Eng. (2013) 4: 475-492 483

Table 2: Fuzzy assignment problem with ci j as generalized LR flat fuzzy numbers.

Job→ J1 J2 J3 J4

Person↓P1 (5, 6, 2, 1; .9)LR (8, 11, 3, 1; .8)LR (10, 11, 1, 4; .7)LR (8, 10, 3, 1; .8)LR

P2 (8, 10, 1, 1; .6)LR (5, 6, 2, 1; .7)LR (8, 10, 2, 2; .8)LR (8, 9, 3, 1; .7)LR

P3 (4, 5, 2, 1; .9)LR (7, 10, 2, 1; .8)LR (11, 13, 3, 2; .8)LR (6, 7, 2, 3; .9)LR

P4 (8, 10, 2, 2; .9)LR (5, 6, 3, 1; .7)LR (7, 10, 2, 1; .8)LR (4, 5, 2, 2; .7)LR

where L(x) = R(x) = max{0, 1 − x}.(ii) The existing method [12] can be used to solve such fuzzy assignment problems

(S 2), where parameters ci j are represented by different type of normal LR flatfuzzy numbers, e.g., fuzzy assignment problem, chosen in Example 3.

Example 3 Solve the fuzzy assignment problem for which the fuzzy costs ci j aregiven below:

c11 = (5, 5.6, 1, 1; 1)L11−R11 , c12 = (8, 11, 3, 1; 1)L12−R12 , c13 = (9, 12, 2, 3; 1)L13−R13 ,

c14 = (8, 10, 2, 1; 1)L14−R14 , c21 = (8, 10, 1, 2; 1)L21−R21 , c22 = (5, 6, 2, 2; 1)L22−R22 ,

c23 = (9, 10, 3, 4; 1)L23−R23 , c24 = (8, 9, 3, 2; 1)L24−R24 , c31 = (5, 5, 1, 1; 1)L31−R31 ,

c32 = (11, 13, 3, 4; 1)L32−R32 , c33 = (9, 9, 1, 1; 1)L33−R33 , c34 = (6, 7, 1, 0.5; 1)L34−R34 ,

c41 = (8, 9, 2, 4; 1)L41−R41 , c42 = (4, 4, 2, 2; 1)L42−R42 , c43 = (4, 5, 1, 1; 1)L43−R43 ,

c44 = (7, 8, 1, 3; 1)L44−R44 .But the existing method [12] cannot be used to such fuzzy assignment problems

(S 8) in which there is a need to represent the parameters ci j by different type ofgeneralized LR flat fuzzy numbers of different heights.

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n,

n∑j=1

xi j = 1, i = 1, 2, · · · , n, (S 8)

xi j = 0 or 1, ∀ i, j = 1, 2, · · · , n,where ci j = (mi j, ni j, αi j, βi j; wi j)Li j−Ri j .

The fuzzy assignment problem, chosen in Example 4, cannot be solved by usingthe existing method [12] as parameters ci j are represented by different type of gener-alized LR flat fuzzy numbers.

Example 4 Solve the fuzzy assignment problem for which fuzzy costs ci j are givenbelow:

c11 = (5, 5.6, 1, 1; .9)L11−R11 , c12 = (8, 11, 3, 1; .8)L12−R12 , c13 = (9, 12, 2, 3; .7)L13−R13 ,

c14 = (8, 10, 2, 1; .8)L14−R14 , c21 = (8, 10, 1, 2; .6)L21−R21 , c22 = (5, 6, 2, 2; .7)L22−R22 ,

c23 = (9, 10, 3, 4; .8)L23−R23 , c24 = (8, 9, 3, 2; .7)L24−R24 , c31 = (5, 5, 1, 1; .9)L31−R31 ,

c32 = (11, 13, 3, 4; .8)L32−R32 , c33 = (9, 9, 1, 1; .8)L33−R33 , c34 = (6, 7, 1, .5; .9)L34−R34 ,

c41 = (8, 9, 2, 4; .9)L41−R41 , c42 = (4, 4, 2, 2; .7)L42−R42 , c43 = (4, 5, 1, 1; .8)L43−R43 ,

c44 = (7, 8, 1, 3; .7)L44−R44 .

484 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

(iii) The existing method [22] can be used to solve such fuzzy assignment problems(S 3) where parameters ci j, CPi , CJ j are represented by normal LR flat fuzzynumbers, e.g., fuzzy assignment problem, chosen in Example 5.

Example 5 Solve the fuzzy assignment problem for which the fuzzy costs ci j, CPi , CJ j

are given below:c11 = (5, 5.6, 1, 1; 1)L11−R11 , c12 = (8, 11, 3, 1; 1)L12−R12 ,

c13 = (9, 12, 2, 3; 1)L13−R13 , c14 = (8, 10, 2, 1; 1)L14−R14 ,

c21 = (8, 10, 1, 2; 1)L21−R21 , c22 = (5, 6, 2, 2; 1)L22−R22 ,

c23 = (9, 10, 3, 4; 1)L23−R23 , c24 = (8, 9, 3, 2; 1)L24−R24 ,

c31 = (5, 5, 1, 1; 1)L31−R31 , c32 = (11, 13, 3, 4; 1)L32−R32 ,

c33 = (9, 9, 1, 1; 1)L33−R33 , c34 = (6, 7, 1, 0.5; 1)L34−R34 ,

c41 = (8, 9, 2, 4; 1)L41−R41 , c42 = (4, 4, 2, 2; 1)L42−R42 ,

c43 = (4, 5, 1, 1; 1)L43−R43 , c44 = (7, 8, 1, 3; 1)L44−R44 ,

CP1 = (10, 11, 2, 3; 1)LP1−RP1, CP2 = (8, 9, 1, 1; 1)LP2−RP2

,

CP3 = (10, 12, 2, 1; 1)LP3−RP3, CP4 = (5, 5, 1, 0.25; 1)LP4−RP4

,

CJ1 = (6, 8, 2, 2; 1)LJ1−RJ1, CJ2 = (6, 7, 2, 2; 1)LJ2−RJ2

,

CJ3 = (9, 11, 1, 0.50; 1)LJ3−RJ3, CJ4 = (8, 9, 3, 4; 1)LJ4−RJ4

.

But the existing method [22] cannot be used to solve such fuzzy assignment prob-lems (S 9), wherein there is a need to represent the parameters ci j, CPi and CJ j asgeneralized LR flat fuzzy numbers.

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n,

n∑j=1

xi j = 1, i = 1, 2, · · · , n,ci j xi j � CPi , ∀ i, j = 1, 2, · · · , n, (S 9)ci j xi j � CJ j , ∀ i, j = 1, 2, · · · , n,xi j = 0 or 1, ∀ i, j = 1, 2, · · · , n,

where ci j = (mi j, ni j, αi j, βi j; wi j)Li j−Ri j , CPi = (mPi , nPi , αPi , βPi ; wPi )LPi−RPiand CJ j =

(mJj , nJj , αJ j , βJ j ; wJ j )LJ j−RJ j.

The fuzzy assignment problem, chosen in Example 6, cannot be solved by usingthe existing method [22] as the parameters ci j, CPi and CJ j are represented by gener-alized LR flat fuzzy numbers.

Example 6 Solve the fuzzy assignment problem for which the fuzzy costs ci j, CPi , CJ j

are given below:c11 = (5, 5.6, 1, 1; .9)L11−R11 , c12 = (8, 11, 3, 1; .8)L12−R12 ,

c13 = (9, 12, 2, 3; .7)L13−R13 , c14 = (8, 10, 2, 1; .8)L14−R14 ,

c21 = (8, 10, 1, 2; .6)L21−R21 , c22 = (5, 6, 2, 2; .7)L22−R22 ,

c23 = (9, 10, 3, 4; .8)L23−R23 , c24 = (8, 9, 3, 2; .7)L24−R24 ,

c31 = (5, 5, 1, 1; .9)L31−R31 , c32 = (11, 13, 3, 4; .8)L32−R32 ,

c33 = (9, 9, 1, 1; .8)L33−R33 , c34 = (6, 7, 1, 0.5; .9)L34−R34 ,

Fuzzy Inf. Eng. (2013) 4: 475-492 485

c41 = (8, 9, 2, 4; .9)L41−R41 , c42 = (4, 4, 2, 2; .7)L42−R42 ,

c43 = (4, 5, 1, 1; .8)L43−R43 , c44 = (7, 8, 1, 3; .7)L44−R44 ,

CP1 = (10, 11, 2, 3; .9)LP1−RP1, CP2 = (8, 9, 1, 1; .9)LP2−RP2

,

CP3 = (10, 12, 2, 1; .9)LP3−RP3, CP4 = (5, 5, 1, 0.25; .9)LP4−RP4

,

CJ1 = (6, 8, 2, 2; .9)LJ1−RJ1, CJ2 = (6, 7, 2, 2; .9)LJ2−RJ2

,

CJ3 = (9, 11, 1, 0.50; .9)LJ3−RJ3, CJ4 = (8, 9, 3, 4; .9)LJ4−RJ4

.

(iv) The existing method [12] can be used to solve such fuzzy travelling salesmanproblems (S 4), where parameters ci j are represented by normal LR flat fuzzynumbers, e.g., fuzzy travelling salesman problem, chosen in Example 7.

Example 7 Solve the fuzzy travelling salesman problem for which the fuzzy costs ci j

are given below:c12 = (8, 11, 3, 1; 1)L12−R12 , c13 = (9, 12, 2, 3; 1)L13−R13 , c14 = (8, 10, 2, 1; 1)L14−R14 ,

c21 = (8, 10, 1, 2; 1)L21−R21 , c23 = (9, 10, 3, 4; 1)L23−R23 , c24 = (8, 9, 3, 2; 1)L24−R24 ,

c31 = (5, 5, 1, 1; 1)L31−R31 , c32 = (11, 13, 3, 4; 1)L32−R32 , c34 = (6, 7, 1, 0.5; 1)L34−R34 ,

c41 = (8, 9, 2, 4; 1)L41−R41 , c42 = (4, 4, 2, 2; 1)L42−R42 , c43 = (4, 5, 1, 1; 1)L43−R43 .

But the existing method [12] cannot be used to such fuzzy travelling salesmanproblems (S 10), in which there is a need to represent the parameters ci j as generalizedLR flat fuzzy numbers.

Minn∑

i=1

n∑j=1

ci j xi j

s.t.n∑

i=1xi j = 1, j = 1, 2, · · · , n and j � i,

n∑j=1

xi j = 1, i = 1, 2, · · · , n and i � j,

xi j+x ji ≤ 1, 1 ≤ i � j ≤ n, (S 10)xi j + x jk + xki ≤ 2, 1 ≤ i � j � k ≤ n,

...

xip1 + xp1 p2 + xp2 p3 + · · · + xp(n−2)i ≤ (n − 2),1 ≤ i � p1 � · · · � p(n−2) ≤ n,

where ci j = (mi j, ni j, αi j, βi j; wi j)Li j−Ri j .

The fuzzy travelling salesman problem, chosen in Example 8, cannot be solved byusing the existing method [12] as parameters ci j are represented as generalized LRflat fuzzy numbers of different heights.

Example 8 Solve the fuzzy travelling salesman problem for which the fuzzy costs ci j

are given below as generalized LR flat fuzzy numbers:c12 = (8, 11, 3, 1; .8)L12−R12 , c13 = (9, 12, 2, 3; .7)L13−R13 ,

c14 = (8, 10, 2, 1; .8)L14−R14 , c21 = (8, 10, 1, 2; .6)L21−R21 ,

c23 = (9, 10, 3, 4; .8)L23−R23 , c24 = (8, 9, 3, 2; .7)L24−R24 ,

c31 = (5, 5, 1, 1; .9)L31−R31 , c32 = (11, 13, 3, 4; .8)L32−R32 ,

c34 = (6, 7, 1, 0.5; .9)L34−R34 , c41 = (8, 9, 2, 4; .9)L41−R41 ,

c42 = (4, 4, 2, 2; .7)L42−R42 , c43 = (4, 5, 1, 1; .8)L43−R43 .

(v) The existing method [13] can be used to such fuzzy generalized assignment

486 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

problems (S 5), where all the parameters ti j, tP j are represented by normal LRflat fuzzy numbers, e.g., fuzzy generalized assignment problem, chosen in Ex-ample 9.

Example 9 Solve the fuzzy generalized assignment problem for which the parametersti j, tP j are given below:

t11 = (5, 5.6, 1, 1; 1)L11−R11 , t12 = (6, 9, 2, 3; 1)L12−R12 , t21 = (2, 3, 1, 2; 1)L21−R21 ,

t22 = (6, 7, 1, 2; 1)L22−R22 , t31 = (9, 11, 3, 4; 1)L31−R31 , t32 = (5, 5, 1, 1; 1)L32−R32 ,

t41 = (9, 9, 1, 1; 1)L41−R41 , t42 = (8, 9, 2, 4; 1)L42−R42 , t51 = (4, 4, 2, 2; 1)L51−R51 ,

t52 = (10, 11, 3, 4; 1)L52−R52 , tP1 = (18, 19, 3, 4; 1)LP1−RP1, tP2 = (11, 13, 3, 4; 1)LP2−RP2

.

But the existing method [13] cannot be used to such fuzzy generalized assignmentproblems (S 11), in which there is a need to represent the parameters ti j, tP j by gener-alized LR flat fuzzy numbers.

Minn∑

i=1

m∑j=1

ti j xi j

s.t.m∑

j=1xi j = 1, i = 1, 2, · · · , n,

n∑i=1

ti j xi j � tP j , j = 1, 2, · · · ,m, (S 11)

xi j = 0 or 1, ∀ i, j,where ti j = (mi j, ni j, αi j, βi j; wi j)Li j−Ri j , tP j = (mP j , nP j , αP j , βP j ; wP j )LP j−RP j

.

The fuzzy generalized assignment problem, chosen in Example 10, cannot besolved by using the existing method [13] as all the parameters ti j, tP j are representedby generalized LR flat fuzzy numbers of different heights.

Example 10 Solve the fuzzy generalized assignment problem for which the parame-ters ti j, tP j are given below:

t11 = (5, 5.6, 1, 1; .9)L11−R11 , t12 = (6, 9, 2, 3; .8)L12−R12 ,

t21 = (2, 3, 1, 2; .8)L21−R21 , t22 = (6, 7, 1, 2; .8)L22−R22 ,

t31 = (9, 11, 3, 4; .7)L31−R31 , t32 = (5, 5, 1, 1; .9)L32−R32 ,

t41 = (9, 9, 1, 1; .9)L41−R41 , t42 = (8, 9, 2, 4; .8)L42−R42 ,

t51 = (4, 4, 2, 2; .7)L51−R51 , t52 = (10, 11, 3, 4; .8)L52−R52 ,

tP1 = (14, 16, 3, 4; .9)LP1−RP1, tP2 = (11, 13, 3, 4; .9)LP2−RP2

.

(vi) The existing method can be used to such fuzzy generalized assignment prob-lems (S 6), where all parameters ti j, tP j and tJi are represented as normal LR flatfuzzy numbers, e.g., fuzzy generalized assignment problem, chosen in Exam-ple 11.

Example 11 Solve the fuzzy generalized assignment problem for which parametersti j, tP j , tJi are given below:

t11 = (5, 5.6, 1, 1; 1)L11−R11 , t12 = (8, 11, 3, 1; 1)L12−R12 , t21 = (9, 12, 2, 3; 1)L21−R21 ,

t22 = (8, 10, 2, 1; 1)L22−R22 , t31 = (8, 10, 1, 2; 1)L31−R31 , t32 = (9, 10, 3, 4; 1)L32−R32 ,

t41 = (8, 9, 3, 2; 1)L41−R41 , t42 = (9, 10, 1, 0.5; 1)L42−R42 , t51 = (6, 8, 2, 2; 1)L51−R51 ,

t52 = (5, 5, 1, 0.25; 1)L52−R52 , tP1 = (18, 19, 2, 4; 1)LP1−RP1,

Fuzzy Inf. Eng. (2013) 4: 475-492 487

tP2 = (25, 26, 3, 1; 1)LP2−RP2, tJ1 = (6, 7, 1, 0.5; 1)LJ1−RJ1

,

tJ2 = (10, 12, 2, 1; 1)LJ2−RJ2, tJ3 = (10, 11, 2, 3; 1)LJ3−RJ3

,

tJ4 = (8, 10, 2, 2; 1)LJ4−RJ4, tJ5 = (8, 9, 3, 2; 1)LJ5−RJ5

.

But the existing method [13] cannot be used to such fuzzy generalized assignmentproblems (S 12), wherein there is a need to represent all the parameters ti j, tP j and tJi

as generalized LR flat fuzzy numbers.

Minn∑

i=1

m∑j=1

ti j xi j

s.t.m∑

j=1xi j = 1, i = 1, 2, · · · , n,

n∑i=1

ti j xi j � tP j , j = 1, 2, · · · ,m, (S 12)

ti j xi j � tJi , i = 1, 2, · · · , n; j = 1, 2, · · · ,m,xi j = 0 or 1, ∀ i, j,

where tJi = (mJi , nJi , αJi , βJi ; wJi )LJi−RJi.

The fuzzy generalized assignment problem, chosen in Example 12, cannot besolved by using the existing method [13] as all the parameters ti j, tP j are representedby generalized LR flat fuzzy numbers of different heights.

Example 12 Solve the fuzzy generalized assignment problem for which the param-eters ti j, tP j , tJi are given below:

t11 = (5, 5.6, 1, 1; .9)L11−R11 , t12 = (8, 11, 3, 1; .8)L12−R12 ,

t21 = (9, 12, 2, 3; .8)L21−R21 , t22 = (8, 10, 2, 1; .8)L22−R22 ,

t31 = (8, 10, 1, 2; .6)L31−R31 , t32 = (9, 10, 3, 4; .8)L32−R32 ,

t41 = (8, 9, 3, 2; .7)L41−R41 , t42 = (9, 10, 1, 0.5; .9)L42−R42 ,

t51 = (6, 8, 2, 2; .9)L51−R51 , t52 = (5, 5, 1, 0.25; .9)L52−R52 ,

tP1 = (18, 19, 2, 4; .9)LP1−RP1, tP2 = (25, 26, 3, 1; .9)LP2−RP2

,

tJ1 = (6, 7, 1, 0.5; .9)LJ1−RJ1, tJ2 = (10, 12, 2, 1; .9)LJ2−RJ2

,

tJ3 = (10, 11, 2, 3; .9)LJ3−RJ3, tJ4 = (8, 10, 2, 2; .9)LJ4−RJ4

,

tJ5 = (8, 9, 3, 2; .9)LJ5−RJ5,

where in Examples 3 to 12,L11(x) = L13(x) = L23(x) = L42(x) = L51(x) = L52(x) = LP1 (x) = R41(x) =

RP1 (x) = RP2 (x) = LJ1 (x) = RJ3 (x)= max {0, 1 − x2}.L12(x) = L14(x) = L22(x) = L31(x) = L34(x) = L43(x) = L44(x) = LP2 (x) =

LP3 (x) = LP4 (x) = LJ2 (x) = LJ4 (x) = R11(x) = R12(x) = R22(x) = R23(x) = R34(x) =R43(x) = R52(x) = RP3 (x) = RJ2 (x) = RJ4 (x)= max {0, 1 − x}.

L21(x) = L24(x) = L32(x) = L33(x) = L41(x) = LJ5 (x) = R24(x) = R31(x) =R42(x) = R51(x) = RP4 (x) = LJ3 (x)= max {0, 1 − x4}.

R13(x) = R14(x) = R44(x) = RJ1 (x) = RJ5 (x) = e−x2.

R21(x) = R32(x) = R33(x) = e−x.

7. Proposed Method

In this section, to overcome the limitations of the existing methods and discussed inSection 6, a new method based on existing ranking approach [15] is proposed to findoptimal solutions of such linear programming problems in which decision variables

488 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

are represented by real numbers and other parameters are represented by differenttype of generalized LR flat fuzzy numbers.

The steps of the proposed method are as follows:

Step 1: Assume c j = (mj, n j, α j, β j; wj)Lj−R j , ai j = (mi j, ni j, αi j, βi j; wi j)Li j−Ri j andbi = (m′i , n

′i , α′i , β′i ; wi)Li−Ri , the FLP problem (P1) can be written as:

Max (or Min)n∑

j=1((mj, n j, α j, β j; wj)Lj−R j )x j

s.t. (n∑

j=1((mi j, ni j, αi j, βi j; wi j)Li j−Ri j )x j) �,�, (m′i , n

′i , α′i , β′i ; wi)Li−Ri ,

i = 1, 2, · · · ,m, (P4)x j ≥ 0.

Step 2: On the basis of existing ranking approach [15], the optimal solution of FLPproblem (P4) can be obtained by solving the following crisp linear programmingproblem:

Max (or Min) �(n∑

j=1((mj, n j, α j, β j; w)Lj−Rj )x j)

s.t. �(n∑

j=1((mi j, ni j, αi j, βi j; wi)Li j−Ri j )x j �,�, ((m′i , n

′i , α′i , β′i ; wi)Li−Ri )),

i = 1, 2, · · · ,m, (P5)x j ≥ 0,

where w = min(w1,w2, · · · ,wn) and wi = min(wi1,wi2, . . . ,win,wi).

Step 3: Using the linearity property�(n∑

j=1((mj, n j, α j, β j; w)L j−Rj )x j) =

n∑j=1�((mj, n j,

α j, β j; w)L j−Rj )x j, the crisp linear programming problem (P5) can be written as:

Max (or Min)n∑

j=1�((mj, n j, α j, β j; w)Lj−R j )x j

s.t.n∑

j=1�((mi j, ni j, αi j, βi j; wi)Li j−Ri j )x j ≤,=,≥ �((m′i , n

′i , α′i , β′i ; wi)Li−Ri ),

i = 1, 2, · · · ,m,x j ≥ 0.

Step 4: Solve the crisp linear programming problem, obtained in Step 3, to find the

optimal solution {x j} and Yager’s ranking index (n∑

j=1�(c j)x j) corresponding to fuzzy

optimal value (n∑

j=1c j x j).

8. Advantages of the Proposed Method

The advantage of proposed method is its applicability to all fuzzy assignment, fuzzytravelling salesman and fuzzy generalized assignment problems which could be orcould not be solved by using the existing methods [12, 13, 21, 22] and is amplydemonstrated by solving fuzzy assignment problem chosen in Example 6 as follows:(For the sake of convenience, we drop LR sign.)

Step 1: The FLP formulation of fuzzy assignment problem, chosen in Example 6, is:

Min [(5, 5.6, 1, 1; .9)x11 ⊕ (8, 11, 3, 1; .8)x12 ⊕ (9, 12, 2, 3; .7)x13 ⊕ (8, 10, 2, 1; .8)x14

Fuzzy Inf. Eng. (2013) 4: 475-492 489

⊕ (8, 10, 1, 2; .6)x21 ⊕ (5, 6, 2, 2; .7)x22 ⊕ (9, 10, 3, 4; .8)x23 ⊕ (8, 9, 3, 2; .7)x24

⊕ (5, 5, 1, 1; .9)x31 ⊕ (11, 13, 3, 4; .8)x32 ⊕ (9, 9, 1, 1; .8)x33 ⊕ (6, 7, 1, 0.5; .9)x34

⊕ (8, 9, 2, 4; .9)x41 ⊕ (4, 4, 2, 2; .7)x42 ⊕ (4, 5, 1, 1; .8)x43 ⊕ (7, 8, 1, 3; .7)x44]s.t.

x11 + x12 + x13 + x14 = 1, x11 + x21 + x31 + x41 = 1,x21 + x22 + x23 + x24 = 1, x12 + x22 + x32 + x42 = 1,x31 + x32 + x33 + x34 = 1, x13 + x23 + x33 + x43 = 1,x41 + x42 + x43 + x44 = 1, x14 + x24 + x34 + x44 = 1,(5, 5.6, 1, 1; .9)x11 � (10, 11, 2, 3; .9), (8, 11, 3, 1; .8)x12 � (10, 11, 2, 3; .9),(9, 12, 2, 3; .7)x13 � (10, 11, 2, 3; .9), (8, 10, 2, 1; .8)x14 � (10, 11, 2, 3; .9),(8, 10, 1, 2; .6)x21 � (8, 9, 1, 1; .9), (5, 6, 2, 2; .7)x22 � (8, 9, 1, 1; .9),(9, 10, 3, 4; .8)x23 � (8, 9, 1, 1; .9), (8, 9, 3, 2; .7)x24 � (8, 9, 1, 1; .9),(5, 5, 1, 1; .9)x31 � (10, 12, 2, 1; .9), (11, 13, 3, 4; .8)x32 � (10, 12, 2, 1; .9),(9, 9, 1, 1; .8)x33 � (10, 12, 2, 1; .9), (6, 7, 1, .5; .9)x34 � (10, 12, 2, 1; .9),(8, 9, 2, 4; .9)x41 � (5, 5, 1, .25; .9), (4, 4, 2, 2; .7)x42 � (5, 5, 1, .25; .9),(4, 5, 1, 1; .8)x43 � (5, 5, 1, .25; .9), (7, 8, 1, 3; .7)x44 � (5, 5, 1, .25; .9),(5, 5.6, 1, 1; .9)x11 � (6, 8, 2, 2; .9), (8, 10, 1, 2; .6)x21 � (6, 8, 2, 2; .9),(5, 5, 1, 1; .9)x31 � (6, 8, 2, 2; .9), (8, 9, 2, 4; .9)x41 � (6, 8, 2, 2; .9),(8, 11, 3, 1; .8)x12 � (6, 7, 2, 2; .9), (5, 6, 2, 2; .7)x22 � (6, 7, 2, 2; .9),(11, 13, 3, 4; .8)x32 � (6, 7, 2, 2; .9), (4, 4, 2, 2; .7)x42 � (6, 7, 2, 2; .9),(9, 12, 2, 3; .7)x13 � (9, 11, 1, .5; .9), (9, 10, 3, 4; .8)x23 � (9, 11, 1, .5; .9),(9, 9, 1, 1; .8)x33 � (9, 11, 1, .5; .9), (4, 5, 1, 1; .8)x43 � (9, 11, 1, .5; .9),(8, 10, 2, 1; .8)x14 � (8, 9, 3, 4; .9), (8, 9, 3, 2; .7)x24 � (8, 9, 3, 4; .9),(6, 7, 1, .5; .9)x34 � (8, 9, 3, 4; .9), (7, 8, 1, 3; .7)x44 � (8, 9, 3, 4; .9),xi j = 0 or 1, ∀ i = 1, 2, 3, 4 and j = 1, 2, 3, 4.

Step 2: Using Steps 2 and 3 of the proposed method, the formulated FLP problem isconverted into the following crisp linear programming problem:

Min [�(5, 5.6, 1, 1; .6)x11 +�(8, 11, 3, 1; .6)x12 +�(9, 12, 2, 3; .6)x13 +

�(8, 10, 2, 1; .6)x14 +�(8, 10, 1, 2; .6)x21 +�(5, 6, 2, 2; .6)x22 +

�(9, 10, 3, 4; .6)x23 +�(8, 9, 3, 2; .6)x24 +�(5, 5, 1, 1; .6)x31 +

�(11, 13, 3, 4; .6)x32 +�(9, 9, 1, 1; .6)x33 +�(6, 7, 1, .5; .6)x34 +

�(8, 9, 2, 4; .6)x41 +�(4, 4, 2, 2; .6)x42 +�(4, 5, 1, 1; .6)x43 +

�(7, 8, 1, 3; .6)x44]s.t.

x11 + x12 + x13 + x14 = 1, x11 + x21 + x31 + x41 = 1,x21 + x22 + x23 + x24 = 1, x12 + x22 + x32 + x42 = 1,x31 + x32 + x33 + x34 = 1, x13 + x23 + x33 + x43 = 1,x41 + x42 + x43 + x44 = 1, x14 + x24 + x34 + x44 = 1,�(5, 5.6, 1, 1; .9)x11 � �(10, 11, 2, 3; .9),�(8, 11, 3, 1; .8)x12 � �(10, 11, 2, 3; .9),�(9, 12, 2, 3; .7)x13 � �(10, 11, 2, 3; .9),�(8, 10, 2, 1; .8)x14 � �(10, 11, 2, 3; .9),

490 Anila Gupta · Amit Kumar ·Mahesh Kumar Sharma (2013)

�(8, 10, 1, 2; .6)x21 � �(8, 9, 1, 1; .9),�(5, 6, 2, 2; .7)x22 � �(8, 9, 1, 1; .9),�(9, 10, 3, 4; .8)x23 � �(8, 9, 1, 1; .9),�(8, 9, 3, 2; .7)x24 � �(8, 9, 1, 1; .9),�(5, 5, 1, 1; .9)x31 � �(10, 12, 2, 1; .9),�(11, 13, 3, 4; .8)x32 � �(10, 12, 2, 1; .9),�(9, 9, 1, 1; .8)x33 � �(10, 12, 2, 1; .9),�(6, 7, 1, .5; .9)x34 � �(10, 12, 2, 1; .9),�(8, 9, 2, 4; .9)x41 � �(5, 5, 1, .25; .9),�(4, 4, 2, 2; .7)x42 � �(5, 5, 1, .25; .9),�(4, 5, 1, 1; .8)x43 � �(5, 5, 1, .25; .9),�(7, 8, 1, 3; .7)x44 � �(5, 5, 1, .25; .9),�(5, 5.6, 1, 1; .9)x11 � �(6, 8, 2, 2; .9),�(8, 10, 1, 2; .6)x21 � �(6, 8, 2, 2; .9),�(5, 5, 1, 1; .9)x31 � �(6, 8, 2, 2; .9),�(8, 9, 2, 4; .9)x41 � �(6, 8, 2, 2; .9),�(8, 11, 3, 1; .8)x12 � �(6, 7, 2, 2; .9),�(5, 6, 2, 2; .7)x22 � �(6, 7, 2, 2; .9),�(11, 13, 3, 4; .8)x32 � �(6, 7, 2, 2; .9),�(4, 4, 2, 2; .7)x42 � �(6, 7, 2, 2; .9),�(9, 12, 2, 3; .7)x13 � �(9, 11, 1, .5; .9),�(9, 10, 3, 4; .8)x23 � �(9, 11, 1, .5; .9),�(9, 9, 1, 1; .8)x33 � �(9, 11, 1, .5; .9),�(4, 5, 1, 1; .8)x43 � �(9, 11, 1, .5; .9),�(8, 10, 2, 1; .8)x14 � �(8, 9, 3, 4; .9),�(8, 9, 3, 2; .7)x24 � �(8, 9, 3, 4; .9),�(6, 7, 1, .5; .9)x34 � �(8, 9, 3, 4; .9),�(7, 8, 1, 3; .7)x44 � �(8, 9, 3, 4; .9),xi j = 0 or 1, ∀ i = 1, 2, 3, 4 and j = 1, 2, 3, 4.

Step 3: Solving the crisp linear programming problem, obtained in Step 2, the op-timal solution is x11 = 1, x22 = 1, x34 = 1, x43 = 1 and Yager’s ranking indexcorresponding to minimum total fuzzy cost is 12.957.

9. Results and Discussion

Table 3 summarizes the results of Examples 1 to 12 solved by existing and proposedapproaches. It is obvious irrespective of whether we use existing methods or proposedmethod, the same results are obtained for Examples 1, 3, 5, 7, 9 and 11. WhileExamples 2, 4, 6, 8, 10 and 12 can be solved only by using the proposed method.Hence, the proposed method has wider applicability.

Fuzzy Inf. Eng. (2013) 4: 475-492 491

Table 3: Comparison of results obtained by using existing and proposed methods.

Examples Existing methods Proposed method Solution[12, 13, 21, 22]

1 Only [21] applicable Applicable P1 → J3, P2 → J2, P3 → J1, P4 → J4

3 Only [12] applicable Applicable P1 → J1, P2 → J2, P3 → J4, P4 → J3

5 Only [22] applicable Applicable P1 → J1, P2 → J2, P3 → J4, P4 → J3

7 Only [12] applicable Applicable 1→ 2→ 4→ 3→ 19 Only [13] applicable Applicable J1 → P1, J2 → P2, J3 → P2, J4 → P1, J5 → P1

11 Only [13] applicable Applicable J1 → P1, J2 → P2, J3 → P2, J4 → P1, J5 → P1

2 None applicable Applicable P1 → J3, P2 → J2, P3 → J1, P4 → J4

4 None applicable Applicable P1 → J1, P2 → J2, P3 → J4, P4 → J3

8 None applicable Applicable 1→ 2→ 4→ 3→ 110 None applicable Applicable J1 → P1, J2 → P2, J3 → P2, J4 → P1, J5 → P1

12 None applicable Applicable J1 → P1, J2 → P2, J3 → P2, J4 → P1, J5 → P1

10. Conclusion

This paper addresses the limitations of existing methods [12, 13, 21, 22] to solvefuzzy assignment, fuzzy travelling salesman and fuzzy generalized assignment prob-lems. To overcome these limitations, a new method is proposed. By comparing theresults of the proposed method and existing methods, the superiority of the proposedmethod is exemplified.

Acknowledgments

The authors would like to thank the Editor-in-Chief and anonymous referees for thevarious suggestions which have led to an improvement in both the quality and clarityof the paper.

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