mehar’s method for solving fuzzy sensitivity analysis problems with lr flat fuzzy numbers

Applied Mathematical Modelling 36 (2012) 4087–4095

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Mehar’s method for solving fuzzy sensitivity analysis problems with LRflat fuzzy numbers

Neha Bhatia ⇑, Amit KumarSchool of Mathematics and Computer Applications Thapar University, Patiala-147004, India

a r t i c l e i n f o

Article history:Received 5 April 2011Received in revised form 3 November 2011Accepted 13 November 2011Available online 20 November 2011

Keywords:Fuzzy linear programming problemsRanking functionSensitivity analysisLR flat fuzzy numbers

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.11.038

⇑ Corresponding author.E-mail addresses: [email protected] (N. B

a b s t r a c t

In published works on fuzzy linear programming there are only few papers dealing withstability or sensitivity analysis in fuzzy mathematical programming. To the best of ourknowledge, till now there is no method in the literature to deal with the sensitivity analysisof such fuzzy linear programming problems in which all the parameters are represented byLR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposedfor the same. To show the advantages of proposed method over existing methods, somefuzzy sensitivity analysis problems which may or may not be solved by the existing meth-ods are solved by using the proposed method.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems.Sensitivity analysis is well-explored area in classical linear programming. Sensitivity analysis is a basic tool for studying per-turbations in optimization problems. There is considerable research on sensitivity analysis for some operations research andmanagement science models such as linear programming and investment analysis.

In most practical applications of mathematical programming the possible values of the parameters required in the mod-eling of the problem are provided either by a decision maker subjectively or a statistical inference from the past data due towhich there exists some uncertainty. In order to reflect this uncertainty, the model of the problem is often constructed withfuzzy data [1].

Fuzzy linear programming provides the flexibility in values. But even after formulating the problem as fuzzy linear pro-gramming problem, one cannot stick to all the values for a long time or it is quite possible that the wrong values got entered.With time the factors like cost, required time or availability of product etc. changes widely. Sensitivity analysis for fuzzy lin-ear programming problems needs to be applied in that case. Sensitivity analysis is one of the interesting researches in fuzzylinear programming problems.

Zimmermann [2] attempted to fuzzify a linear program for the first time, fuzzy numbers being the source of flexibility.Zimmermann also presented a fuzzy approach to multi-objective linear programming problems and its sensitivity analysis.Sensitivity analysis in fuzzy linear programming problem with crisp parameters and soft constraints was first considered byHamacher et al. [3].

Tanaka and Asai [4] proposed a method for allocating the given investigation cost to each fuzzy coefficients by using sen-sitivity analysis. Tanaka et al. [5] formulated a fuzzy linear programming problem with fuzzy coefficients and the value of

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hatia), [email protected] (A. Kumar).

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4088 N. Bhatia, A. Kumar / Applied Mathematical Modelling 36 (2012) 4087–4095

information was discussed via sensitivity analysis. Sakawa and Yano [6] presented a fuzzy approach for solving multi-objec-tive linear fractional programming problems via sensitivity analysis.

Fuller [7] proposed that the solution to fuzzy linear programming problems with symmetrical triangular fuzzy numbers isstable with respect to small changes of centers of fuzzy numbers. Perturbations occur due to calculation errors or just to an-swer managerial questions ‘‘What if . . .’’. Such questions propose after the simplex method and the related research area re-fers to as basis invariance sensitivity analysis.

Dutta et al. [8] studied sensitivity analysis for fuzzy linear fractional programming problem. Verdegay and Aguado [9]proposed that in the case of fuzzy linear programming problems, whether or not a fuzzy optimal solution has been foundby using linear membership functions modeling the constraints, possible further changes of those membership functionsdo not affect the former optimal solution. The sensitivity analysis performed for those membership functions and the cor-responding solutions shows the convenience of using linear functions instead of other more complicated ones.

Gupta and Bhatia [10] studied the measurement of sensitivity for changes of violations in the aspiration level for the fuzzymulti-objective linear fractional programming problem. Precup and Preitl [11] performed the sensitivity analysis for somefuzzy control systems. Lotfi et al. [12] developed a sensitivity analysis approach for the additive model. Kumar, et al. [13]pointed out the shortcomings of the existing method [14] and proposed a method to find the fuzzy optimal solution of fullyfuzzy linear programming problems with equality constraints.

Kheirfam and Hasani [15] studied the basis invariance sensitivity analysis for fuzzy linear programming problems.Ebrahimnejad [16] generalized the concept of sensitivity analysis in fuzzy number linear programming problems by apply-ing fuzzy simplex algorithms and using the general linear ranking function on fuzzy numbers. Nasseri and Ebrahimnejad [17]proposed a method for sensitivity analysis on linear programming problem with trapezoidal fuzzy variables.

In this paper, the limitations of existing works [16,15,17] are pointed out. To overcome these limitations a new method,named as Mehar’s method, is proposed to deal with the sensitivity analysis of such fuzzy linear programming problems inwhich all the parameters are represented by LR flat fuzzy numbers. To show the advantages of proposed method over exist-ing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solvedby using the proposed method.

This paper is organized as follows: In Section 2, some basic definitions, arithmetic operations and Yager’s ranking ap-proach for comparing LR flat fuzzy numbers are presented. In Section 3, the limitations of existing methods are pointedout. In Section 4, a new method, named as Mehar’s method, is proposed to deal with the sensitivity analysis of such fuzzylinear programming problems in which all the parameters are represented by LR flat fuzzy numbers. Advantages of proposedmethod over the existing methods are discussed in Section 5. Results are discussed in Section 6. Finally we conclude inSection 7.

2. Preliminaries

In this section, some basic definitions, arithmetic operations of LR flat fuzzy numbers and an existing ranking approach forcomparing LR flat fuzzy numbers are presented.

2.1. Basic definitions

In this section, some basic definitions are presented [18].

Definition 2.1. A fuzzy number ~A ¼ ðm;n;a; bÞLR is said to be an LR flat fuzzy number if its membership function is given by:

l~AðxÞ ¼L m�x

a

� �; x 6 m;a > 0;

R x�nb

� �; x P n; b > 0;

1; m 6 x 6 n:

8>><>>:

If m = n then ~A ¼ ðm;n;a; bÞLR will convert into ~A ¼ ðm;a; bÞLR and is said to be an LR fuzzy number. L and R are called refer-ence functions, which are continuous, non-increasing functions that defining the left and right shapes of l~AðxÞ respectivelyand L(0) = R(0) = 1. Two special cases are triangular and trapezoidal fuzzy number, for which L(x) = R(x) = maximum{0,1 � x}, are linear functions. Three commonly used nonlinear reference functions with parameters q, denoted as RFq, aresummarized as follows:

power: RFq = maximum {0,1 � xq}, q P 0,exponential power: RFq ¼ e�xq

; q P 0;rational: RFq ¼ 1

ð1þxqÞ ; q P 0:

Definition 2.2. A fuzzy number ~A ¼ ðm;n;a; bÞLR is said to be a non-negative LR flat fuzzy number if m � a P 0 and is said tobe a non-positive LR flat fuzzy number if n + b 6 0.

N. Bhatia, A. Kumar / Applied Mathematical Modelling 36 (2012) 4087–4095 4089

2.2. Arithmetic operations

In this section, arithmetic operations between two LR flat fuzzy numbers, defined on universal set of real numbers R, arepresented [18].

Let ~A1 ¼ ðm1;n1;a1; b1ÞLR;~A2 ¼ ðm2;n2;a2; b2ÞLR be two non-negative LR flat fuzzy numbers and ~A3 ¼ ðm3;n3;a3; b3ÞRL be a

RL flat fuzzy number then,

(i) ~A1 � ~A2 ¼ ðm1 þm2;n1 þ n2;a1 þ a2; b1 þ b2ÞLR

(ii) ~A1 � ~A3 ¼ ðm1 � n3;n1 �m3;a1 þ b3; b1 þ a3ÞLR

(iii) If ~A1 and ~A2 both are non-negative, then
~A1 � ~A2 � ðm1m2;n1n2;m1a2 þ a1m2; n1b2 þ b1n2ÞLR:
(iv) If ~A1 is non-positive and ~A2 is non-negative, then
~A1 � ~A2 � ðm1n2;n1m2;a1n2 �m1b2; b1m2 � n1a2ÞLR:
(v) If ~A1 is non-negative and ~A2 is non-positive, then

~A1 � ~A2 � ðn1m2;m1n2;n1a2 � b1m2;m1b2 � a1n2ÞLR:

(vi) If ~A1 and ~A2 both are non-positive, then

~A1 � ~A2 � ðn1n2;m1m2;�n1b2 � b1n2;�m1a2 � a1m2ÞLR:

ðkm1; kn1; ka1; kb1ÞLRk P 0;8><

k~A ¼ ðkn1; km1;�kb1;�ka1ÞRLk 6 0:>:
(vii)
2.3. Yager’s ranking approach

A number of ranking approaches have been proposed for comparing fuzzy numbers. A relatively simple computationaland easily understandable ranking approach, proposed by Yager [19] and used in existing methods [16,15], is consideredfor comparing fuzzy numbers in this paper. Yager [19] proposed a procedure for ordering fuzzy sets in which a ranking indexRð~AÞ is calculated for an LR flat fuzzy number ~A ¼ ðm;n;a; bÞLR from its k-cut Ak = [m � aL�1(k),n + bR�1(k)] according to thefollowing formula:

Rð~AÞ ¼Z 1

0ðm� aL�1ðkÞÞdkþ

Z 1

0ðnþ bR�1ðkÞÞdk: ð1Þ

Let ~A and ~B be two LR flat fuzzy numbers then

(i) ~APR~B if Rð~AÞP Rð~BÞ

(ii) ~A>R~B if Rð~AÞ > Rð~BÞ

(iii) ~A¼R~B if Rð~AÞ ¼ Rð~BÞ

2 ~ 2a b� �

Lemma 2.1. If L(x) = maximum {0,1 � x } and R(x) = maximum {0,1 � x} then RðAÞ ¼ mþ n� 3 þ 2 :

Proof. Since, L(x) = maximum {0,1 � x2} and R(x) = maximum {0,1 � x} so L�1ðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffi1� kp

and R�1(k) = 1 � k where,k 2 [0,1].

Substituting the values of L�1(k) and R�1(k) in Yager’s ranking formula (1), Rð~AÞ ¼R 1

0 ðm� affiffiffiffiffiffiffiffiffiffiffiffi1� kp

ÞdkþR 10 ðnþ bð1� kÞdk ¼ mþ n� 2a

3 þb2

� �. h

3. Limitations of the existing methods

In this section, the limitations of the existing methods [16,15,17] are pointed out.The existing methods [15,17] can deal with the sensitivity analysis of such fuzzy linear programming problems in which

the decision variables and right-hand side vectors are represented by fuzzy numbers and rest of the parameters are repre-sented by real numbers, i.e.

Maximizeðor MinimizeÞ ðC~XÞSubject to A~X¼R

~b; ðP0Þ~XPR

~0:Similarly;


the existing method [16] can deal with the sensitivity analysis of such fuzzy linear programming problems in which only thedecision variables are represented by real number and rest of the parameters are represented by fuzzy numbers, i.e.

Maximizeðor MinimizeÞ ð~CXÞSubject to ~AX¼R

~b; ðP1ÞX P 0:

Since, for the ranking function, used in the existing methods [16,15,17] the property Rð~A� ~BÞ ¼ Rð~AÞRð~BÞ is not satisfied. So,none of the existing methods [16,15,17] can be used to deal with the sensitivity analysis of the fuzzy linear programmingproblem (P2) in which all the parameters are represented by LR flat fuzzy numbers.

Maximizeðor MinimizeÞ ð~CT � ~XÞSubject to ~A� ~X6R or ¼R or PR

~b; ðP2Þ

where, ~b ¼ ½~bj�m�1;~X ¼ ½~xj�n�1;

~A ¼ ½~aij�m�n;~CT ¼ ½~cj�1�n;R is a linear ranking function, ~xj ¼ ðmj;nj;aj; bjÞLR is a non-negative LR

flat fuzzy number, ~cj ¼ ðm00j ;n00j ;a00j ; b00j ÞLR;

~bj ¼ ðm0j;n0j;a0j; b0jÞLR and ~aij ¼ ðmij;nij;aij; bijÞLR are either non-negative or non-positive

LR flat fuzzy numbers.

4. Mehar’s method

In this section, to overcome the limitations of the existing methods [16,15,17], discussed in Section 3, a new method,named as Mehar’s method, is proposed to deal with the sensitivity analysis of such fuzzy linear programming problems(P2) in which all the parameters are represented by LR flat fuzzy numbers. The same method can be used to deal with thesensitivity analysis of any type of fuzzy linear programming problems (P0) and (P2).

The steps of proposed method are as follows:

Step 1 Convert the fuzzy linear programming problem (P2) into the crisp linear programming problem (P3):

Maximizeðor MinimizeÞ Rð~CT � ~XÞSubject to Rð~A� ~XÞ 6 or ¼ or P Rð~bÞ; ðP3Þ

mj � aj P 0; mj 6 nj;aj P 0; bj P 0 j ¼ 1;2;3; . . . ;n:

Step 2 Solve the crisp linear programming problem, obtained in Step 1, to find the optimal solution {mj,nj,aj,bj}.Step 3 Find the fuzzy optimal solution ~X ¼ ½~xj�n�1; by putting the values of mj, nj, aj and bj, obtained from Step 2, in

~xj ¼ ðmj;nj;aj; bjÞLR and the fuzzy optimal value by putting the values of ~X in ~CT � ~X.Step 4 Check that which of the following case is to be considered:

1. Change in the fuzzy cost vector,2. Change in the fuzzy requirement vector ~b;3. Addition of a new fuzzy variable,4. Change in the fuzzy coefficient matrix,5. Addition of new fuzzy constraint.

4.1. Case 1: change in the fuzzy cost vector

If the cost vector ~CT changes to ~C0T in the given fuzzy linear programming problem (P2) then replace Rð~CT � ~XÞ byRð~C0T � ~XÞ in crisp linear programming (P3) to obtain (P4):

Maximizeðor MinimizeÞ Rð~C 0T � ~XÞSubject to Rð~A� ~XÞ 6 or ¼ or P Rð~bÞ; ðP4Þ

mj � aj P 0; mj 6 nj; aj P 0; bj P 0 j ¼ 1;2;3; . . . ;n:Nowapply

the existing sensitivity analysis technique to find the optimal solution of (P4) with the help of optimal solution of (P3) and useStep 3 of the proposed method to find the fuzzy optimal solution of the resulting fuzzy linear programming problem.

4.2. Case 2: change in fuzzy requirement vector ~b

If the change in fuzzy requirement vector is made i.e., ~b is changed to ~b0 in (P2) then, replace Rð~bÞ by Rð~b0Þ in crisp linearprogramming problem (P3) to obtain (P5):


Maximizeðor MinimizeÞ Rð~CT � ~XÞSubject to Rð~A� ~XÞ 6 or ¼ or P Rð~b0Þ; ðP5Þ

mj � aj P 0; mj 6 nj; aj P 0bj P 0 j ¼ 1;2;3; . . . ;n:Nowapply


4.3. Case 3: addition of a new fuzzy variable

Suppose a new non-negative fuzzy variable, say ~xnþ1; is added in (P2). Assume that if ~cnþ1 is cost and ~Anþ1 is the columnassociated with ~xnþ1; then replace Rð~A� ~XÞ by Rð~A� ~X � ~Anþ1 � ~xnþ1Þ and Rð~CT � ~XÞ by Rð~CT � ~X � ~cnþ1 � ~xnþ1Þ in (P3) to ob-tain the crisp linear programming problem (P6):

Maximizeðor MinimizeÞ Rð~CT � ~X � ~cnþ1 � ~xnþ1ÞSubject to Rð~A� ~X � ~Anþ1 � ~xnþ1Þ 6 or ¼ or P Rð~bÞ; ðP6Þ

mj � aj P 0; mj 6 nj;aj P 0; bj P 0 j ¼ 1;2;3; . . . ;nþ 1:Nowapply

ply the existing sensitivity analysis technique to find the optimal solution of (P6) with the help of optimal solution of (P3) anduse Step 3 of the proposed method to find the fuzzy optimal solution of the resulting fuzzy linear programming problem.

4.4. Case 4: addition of a new fuzzy constraint

Suppose if a new fuzzy constraint is added in the original fuzzy linear programming problem (P2) then, replaceRð~A� ~XÞ 6 or ¼ or P Rð~bÞ by Rð~A0 � ~XÞ 6 or ¼ or P Rð~b0Þ in (P3) to obtain crisp linear programming problem (P7):

Maximizeðor MinimizeÞ Rð~CT � ~XÞSubject to Rð~A0 � ~XÞ 6 or ¼ or P Rð~b0Þ; ðP7Þ



4.5. Case 5: change in fuzzy constraint matrix

Suppose if the column of the constraint matrix, corresponding to the fuzzy variable ~xj, is changed from ~Aj to ~A0j in the ori-ginal fuzzy linear programming problem (P2) then, replace Rð~A� ~XÞ 6 or ¼ or P Rð~bÞ by Rð~A0 � ~XÞ 6 or ¼ or P Rð~bÞin (P3) to obtain new crisp linear programming problem (P8):

Maximize ðor MinimizeÞ Rð~CT � ~XÞSubject to Rð~A0 � ~XÞ 6 or ¼ or P Rð~bÞ; ðP8Þ



Remark 4.1. The other cases i.e., deletion of fuzzy variables, deletion of fuzzy constraints, simultaneous change incoefficients of the fuzzy decision variables in objective function and requirement vectors etc. can also be solved by usingproposed method.

5. Advantage of proposed method over the existing methods

The main advantage of proposed method over the existing methods [16,15,17], is that there may exist several fuzzy sen-sitivity analysis problems, as discussed in Section 3, which may or may not be solved by the existing methods but the pro-posed method can be used to solve all such fuzzy sensitivity analysis problems. To show the advantage of the proposedmethod over existing methods [16,15,17], the fuzzy sensitivity analysis problem, chosen in Example 5.1 which cannot besolved by using any of the existing methods, is solved by using the proposed method.


Example 5.1. Consider the following fully fuzzy linear programming problem,

Maximize ½ð2;3;2;5ÞLR � ~x1 � ð1;2;1;3ÞLR � ~x2 � ð3;4;3;4ÞLR � ~x3�

Subject to 2;3;1;12

� �LR� ~x1 � 1;3;

12;32

� �LR� ~x2 � ð3;4;1;1ÞLR � ~x36Rð5;7;2;2ÞLR; ðE1Þ

ð2;3;2;5ÞLR � ~x1 � 1;3;1;12

� �LR� ~x26Rð3;5;1;1ÞLR ðE1Þ;

~x1; ~x2; ~x3 are non-negative LR flat fuzzy numbers

where L(x) = maximum {0,1 � x2}, R(x) = maximum {0,1 � x}

(i) Discuss the effect of changing the cost coefficients (2,3,2,5)LR, (1,2,1,3)LR and (3,4,3,4)LR of the LR flat fuzzy decisionvariables ~x1; ~x2 and ~x3 to (2,3,1,4)LR, (2,3,1,4)LR and (3,4,1,5)LR respectively on the fuzzy optimal solution and fuzzyoptimal value of resulting fuzzy linear programming problem.

(ii) Discuss the effect of changing the requirement vector from (5,7,2,2)LR, (3,5,1,1)LR to (5,8,2,2)LR, (3,6,2,2)LR on thefuzzy optimal solution and fuzzy optimal value of resulting fuzzy linear programming problem.

(iii) Find the effect of addition of a new non-negative LR flat fuzzy number ~x4 with cost (2,3,2,5)LR and column vectors[(2,4,1,3)LR, (1,3,1,1)LR]Ton the fuzzy optimal solution and fuzzy optimal value of resulting fuzzy linear programmingproblem.

(iv) Discuss the effect of changing column of the constraint matrix corresponding to the LR flat fuzzy variable ~x2 by[(2,4,1,3)LR, (3,4,1,5)LR]T on the fuzzy optimal solution and fuzzy optimal value of resulting fuzzy linear programmingproblem.

(v) Find the effect of addition of a new fuzzy constraint ð2;3;1;4ÞLR � ~x1 � ð1;2;1;3ÞLR � ~x26Rð2;4;1;1ÞLR on the fuzzyoptimal solution and fuzzy optimal value of resulting fuzzy linear programming problem.

Solution: The solution of the fuzzy sensitivity analysis problem, chosen in Example 5.1, by using the proposed methodcan be obtained as follows:

Assuming ~x1 ¼ ðm1;n1;a1; b1ÞLR; ~x2 ¼ ðm2;n2;a2; b2ÞLR and ~x3 ¼ ðm3;n3;a3; b3ÞLR the fuzzy linear programming problem,chosen in Example 5.1, can be written as:

Maximize ½ð2;3;2;5ÞLR�ðm1;n1;a1;b1ÞLR�ð1;2;1;3ÞLR�ðm2;n2;a2;b2ÞLR�ð3;4;3;4ÞLR�ðm3;n3;a3;b3ÞLR�;

Subject to 2;3;1;12

� �LR�ðm1;n1;a1;b1ÞLR� 1;3;

12;32

� �LR�ðm2;n2;a2;b2ÞLR�ð3;4;1;1ÞLR�ðm3;n3;a3;b3Þ6Rð5;7;2;2Þ; ðE2Þ

ð2;3;2;5ÞLR�ðm1;n1;a1;b1Þ� 1;3;1;12

� �LR�ðm2;n2;a2;b2Þ6Rð3;5;1;1ÞLR

ðm1;n1;a1;b1ÞLR; ðm2;n2;a2;b2ÞLR and ðm3;n3;a3;b3ÞLR are non�negative LR flat fuzzy numbers:UsingStep

1 of the proposed method the fuzzy linear programming problem (E2) is converted into the following crisp linear program-ming problem:

Maximize2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 7n2

2þ 5b2

2þm3 þ 6n3 þ 4b3

� �

Subject to4m1

3þ 13n1

4þ 7b1

4� 2a1

3þ 2m2

3þ 15n2

4þ 9b2

4� a2

3þ 7m3

3þ 9n3

2þ 5b3

2� 4a3

36

353

ðE3Þ

2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 13n2

4þ 7b2

46

476

m1 � a1 P 0; m2 � a2 P 0; m3 � a3 P 0m1 6 n1; m2 6 n2; m3 6 n3

a1 P 0; a2 P 0; a3 P 0b1 P 0; b2 P 0; b3 P 0:

The optimal solution of the crisp linear programming problem (E3) is:m1 = 0, m2 = 0, m3 = 0, n1 = 0, n2 = 0, n3 = 0, a1 = 0, a2 = 0, a3 = 0, b1 = 1.957, b2 = 0, b3 = 3.293and the optimal value is 21.005Using Step 3 of the proposed method the fuzzy optimal solution is given by ~x1 ¼ ð0;0;0;1:957ÞLR; ~x2 ¼ ð0;0;0;0ÞLR; ~x3 ¼

ð0;0;0;3:293ÞLR and the fuzzy optimal value is (0,0,0,42)LR.

(i) Since, the cost coefficients corresponding to the LR flat fuzzy variables ~x1; ~x2 and ~x3 changes to (2,3,1,4)LR, (2,3,1,4)LR

and (3,4,1,5)LR respectively in the original fuzzy linear programming problem so replacing crisp linear programmingproblem (E3) by (E4):


Maximize4m1

3þ 5n1 �

2a1

3þ 7b1

2þ 4m2

3þ 5n2 �

2a2

3þ 7b2

2þ 7m3

3þ 13n3

2þ 9b3

2� 4a3

3

� �

Subject to4m1

3þ 13n1

4þ 7b1

4� 2a1

3þ 2m2

3þ 15n2

4þ 9b2

4� a2

3þ 7m3

3þ 9n3

2þ 5b3

2� 4a3

36

353

ðE4Þ

2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 13n2

4þ 7b2

46

476


a1 P 0; a2 P 0; a3 P 0b1 P 0; b2 P 0; b3 P 0:

Now apply the existing sensitivity analysis techniques, the optimal solution of crisp linear programming problem (E4)is: m1 = 0, m2 = 0, m3 = 0, n1 = 0, n2 = 0, n3 = 0, a1 = 0, a2 = 0, a3 = 0, b1 = 1.957, b2 = 0, b3 = 3.293 and the optimal value is21.673.Using Step 3 of the proposed method the fuzzy optimal solution is given by ~x1 ¼ ð0;0;0;1:957ÞLR; ~x2 ¼ð0;0;0;0ÞLR; ~x3 ¼ ð0;0;0;3:293ÞLR and the fuzzy optimal value is (0,0,0,43.336)LR.

(ii) Since, the requirement vector is changed from (5,7,2,2)LR, (3,5,1,1)LR to (5,8,2,2)LR, (3,6,2,2)LR in the original fuzzylinear programming problem (E1) so replacing Rð5;7;2;2ÞLR;Rð3;5;1;1ÞLR by Rð5;8;2;2Þ;Rð3;6;2;2Þ i.e., 35

3 ;476 by

383 ;

263 respectively in (E3).

Maximize2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 7n2

2þ 5b2

2þm3 þ 8n3 þ 8b3

� �

Subject to4m1

3þ 13n1

4þ 7b1

4� 2a1

3þ 2m2

3þ 15n2

4þ 9b2

4� a2

3þ 7m3

3þ 9n3

2þ 5b3

2� 4a3

36

383

ðE5Þ

2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 13n2

4þ 7b2

46

263


a1 P 0; a2 P 0; a3 P 0b1 P 0; b2 P 0; b3 P 0:

Now apply the existing sensitivity analysis techniques, the optimal solution of crisp linear programming problem (E5)is: m1 = 0, m2 = 0, m3 = 0, n1 = 0, n2 = 0, n3 = 0, a1 = 0, a2 = 0, a3 = 0, b1 = 2.165, b2 = 0, b3 = 3.548 and the optimal value is22.854.Using Step 3 of the proposed method the fuzzy optimal solution is given by ~x1 ¼ ð0;0;0;2:165ÞLR;~x2 ¼ ð0;0;0; 0ÞLR; ~x3 ¼ ð0;0;0;3:548ÞLR and the fuzzy optimal value is (0,0,0,45.705)LR.

(iii) Since, a new non-negative LR flat fuzzy variable ~x4 with cost (2,3,2,5)LR and column vectors [(2,4,1,3)LR, (1,3,1,1)LR]T isadded in the original fuzzy linear programming problem (E1) so replacing crisp linear programming problem (E3) by(E6):

Maximize2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 7n2

2þ 5b2

2þm3 þ 6n3 þ 4b3 þ

2m4

3þ 11n4

2þ 4b4

� �

Subject to4m1

3þ 13n1

4þ 7b1

4� 2a1

3þ 2m2

3þ 15n2

4þ 9b2

4� 1a2

3þ 7m3

3þ 9n3

2þ 5b3

2� 4a3

3þ 4m4

3þ 15n4

2ðE6Þ

þ 7b4

2� 2a4

36

353

2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 13n2

4þ 7

4b2 þ

m4

3þ 7n4

2þ 2b4 6

476

m1 � a1 P 0; m2 � a2 P 0; m3 � a3 P 0; m4 � a4 P 0m1 6 n1; m2 6 n2; m3 6 n3; m4 6 n4

a1 P 0; a2 P 0; a3 P 0; a4 P 0b1 P 0; b2 P 0; b3 P 0; b4 P 0:

Now apply the existing sensitivity analysis techniques, the optimal solution of crisp linear programming problem (E6)is: m1 = 0, m2 = 0, m3 = 0, m4 = 0, n1 = 0, n2 = 0, n3 = 0, n4 = 0, a1 = 0, a2 = 0, a3 = 0, a4 = 0, b1 = 1.957, b2 = 0, b3 = 3.293,b4 = 0 and the optimal value is 21.005.Using Step 3 of the proposed method the fuzzy optimal solution is given by ~x1 ¼ ð0;0;0;1:957ÞLR; ~x2 ¼ ð0;0;0; 0ÞLR;~x3 ¼ ð0;0;0;3:293ÞLR; ~x4 ¼ ð0;0; 0;0ÞLR and the fuzzy optimal value is (0,0,0,42)LR.


(iv) Since, the column of the constraint matrix corresponding to the LR flat fuzzy variable ~x2 is changed to [(2,4,1,3)LR,(3,4,1,5)LR]T in the original fuzzy linear programming problem (E1) so replacing the crisp linear programming problem(E3) by E7):

Table 1Compa

Exam

3.11 [75.1

Maximize2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 7n2

2þ 5b2

2þm3 þ 6n3 þ 4b3

� �

Subject to4m1

3þ 13n1

4þ 7b1

4� 2a1

3þ 4m2

3þ 15n2

2þ 7b2

2� 2a2

3þ 7m3

3þ 9n3

2þ 5b3

2� 4a3

36

353

ðE7Þ

2m1

3þ 11n1

2þ 4b1 þ

7m2

3þ 13n2

2þ 9b2

2� 4a2

36

476


a1 P 0; a2 P 0; a3 P 0b1 P 0; b2 P 0; b3 P 0:

Now apply the existing sensitivity analysis techniques, the optimal solution of crisp linear programming problem (E7)is: m1 = 0, m2 = 0, m3 = 0, n1 = 0, n2 = 0, n3 = 0, a1 = 0, a2 = 0, a3 = 0, b1 = 1.957, b2 = 0, b3 = 3.293 and the optimal value is21.005.Using Step 3 of the proposed method the fuzzy optimal solution is given by ~x1 ¼ ð0;0;0;1:957ÞLR; ~x2 ¼ ð0;0;0;0ÞLR;~x3 ¼ ð0;0;0;3:293ÞLR and the fuzzy optimal value is (0,0,0,42)LR.

(v) Since, a new fuzzy constraint ð2;3;1;4ÞLR � ~x1 � ð1;2;1;3ÞLR � ~x26Rð2;4;1;1ÞLR is added to the original fuzzy linearprogramming problem (E1) so adding the constraint 4m1

3 þ13n1

4 þ7b1

4 �2a1

3 þm23 þ

7n22 þ

5b24 6

356 to (E3).

r

[,

Maximize2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 7n2

2þ 5b2

2þm3 þ 6n3 þ 4b3

� �

Subject to4m1

3þ 13n1

4þ 7

4b1 �

2a1

3þ 2m2

3þ 15n2

4þ 9b2

4� 1a2

3þ 7m3

3þ 9n3

2þ 5b3

2� 4a3

36

353

ðE8Þ

2m1

3þ 11n1

2þ 4b1 þ

m2

3þ 13n2

4þ 7b2

46

476

4m1

3þ 5n1 þ

7b1

2� 2a1

3þm2

3þ 7n2

2þ 5b2

46

356


a1 P 0; a2 P 0; a3 P 0b1 P 0; b2 P 0; b3 P 0:

Now apply the existing sensitivity analysis techniques, the optimal solution of the crisp linear programming problem(E8) is:m1 = 0, m2 = 0, m3 = 0, n1 = 0, n2 = 0, n3 = 0, a1 = 0, a2 = 0, a3 = 0, b1 = 1.665, b2 = 0, b3 = 3.498 and the optimal value is20.654. Using Step 3 of the proposed method the fuzzy optimal solution is given by ~x1 ¼ ð0;0;0;1:665ÞLR;~x2 ¼ ð0;0;0;0ÞLR; ~x3 ¼ ð0;0;0;3:498ÞLR and the fuzzy optimal value is (0,0,0,41.304)LR.

6. Results and discussion

To compare the proposed method with the existing methods [16,15,17] the results of fuzzy sensitivity analysis problems,obtained by using existing methods as well as proposed method, are shown in Table 1.

From Table 1 it can be concluded that:

(a) Since in the existing fuzzy sensitivity analysis problem [3, Example 3.1, pp. 1882] only the decision variables are rep-resented by real numbers and rest of the parameters are represented by triangular or trapezoidal fuzzy numbers. So, asdiscussed in Sections 3 and 4, this problem can be solved by using the proposed method but cannot be solved by usingthe existing methods [15,17].

ison of proposed method with the existing methods.

ple Existing method [16] Existing methods [15,17] Proposed method

3, pp.1882] Applicable Not Applicable Applicablepp. 263] Not Applicable Applicable Applicable

Not applicable Not applicable Applicable


(b) Since in the existing fuzzy sensitivity analysis problem [7, Example 1, pp. 263] the decision variables and right-handside vectors are represented by fuzzy numbers and rest of the parameters are represented by real numbers. So, as dis-cussed in Sections 3 and 4, this problem can be solved by using the proposed method but cannot be solved by using theexisting method [16].

(c) Since in the chosen fuzzy sensitivity analysis problem [Example 5.1] all the parameters are represented by LR flat fuzzynumbers. So, as discussed in Sections 3 and 4, this problem can only be solved by using the proposed method but notby using any of the existing methods [16,15,17].

7. Conclusions and future work

In this paper, limitations of existing methods [16,15,17] for solving fuzzy sensitivity analysis problems are pointed outand to overcome these limitations a new method, named as Mehar’s method, is proposed for solving fuzzy sensitivity anal-ysis problems.

Since, for the fuzzy number ~A the properties ~A� ~A ¼ ~0 and ~A~A¼ ~1 are not satisfied so there is no other option to solve the

fully fuzzy linear programming problems without converting it into crisp linear programming problems. In future, it may betried to think some new arithmetic operations of fuzzy numbers so that the properties ~A� ~A ¼ ~0 and ~A

~A¼ ~1 are satisfied and

then fully fuzzy linear programming problems can be solved without converting it into crisp linear programming problems.

Acknowledgements

The authors thank the Editor-in-Chief ‘‘Prof. M. Cross’’ and anonymous referees for various suggestions which have led toan improvement in both the quality and clarity of the paper. I, Dr. Amit Kumar, want to acknowledge the adolescent innerblessings of Mehar. I believe that Mehar is an angel for me and without Mehar’s blessings it was not possible to think the ideaproposed in this paper. Mehar is a lovely daughter of Parmpreet Kaur (Research Scholar under my supervision).

References

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mehar’s method for solving fuzzy sensitivity analysis problems with lr flat fuzzy numbers

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