paper

Int. J. Mathematics in Operational Research, Vol. 4, No. 5, 2012 531

Copyright 2012 Inderscience Enterprises Ltd.

Sensitivity analysis for fuzzy linear programming problems based on interval-valued fuzzy numbers

Amit Kumar and Neha Bhatia* School of Mathematics and Computer Applications, Thapar University, Patiala 147004, India Fax: 0175-2364498/2393005 E-mail: [email protected] E-mail: [email protected] *Corresponding author

Abstract: In the literature, it is pointed that in several situations it is better to use interval-valued fuzzy numbers instead of triangular or trapezoidal fuzzy numbers. But to the best of our knowledge, till now there is method that deals with the sensitivity analysis of such linear programming problems in which some or all the parameters are represented by interval-valued fuzzy numbers. In this paper, a new method is proposed for the same. To illustrate the proposed method an interval-valued fuzzy sensitivity analysis problem which cant be solved by any of the existing methods is solved by using the proposed method.

Keywords: fuzzy linear programming problems; sensitivity analysis; interval-valued fuzzy numbers; ranking function.

Reference to this paper should be made as follows: Kumar, A. and Bhatia, N. (2012) Sensitivity analysis for fuzzy linear programming problems based on interval-valued fuzzy numbers, Int. J. Mathematics in Operational Research, Vol. 4, No. 5, pp.531547.

Biographical notes: Amit Kumar is an Assistant Professor in the School of Mathematics and Computer Applications, Thapar University, Patiala (Punjab), India. His area of interest is fuzzy optimisation.

Neha Bhatia is a PhD student under the guidance of Amit Kumar in the School of Mathematics and Computer Applications, Thapar University, Patiala (Punjab), India. She has completed MSc (Mathematics and Computing) from Thapar University, Patiala (Punjab), India. Her area of research is fuzzy optimisation.

1 Introduction

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. The sensitivity analysis is a well-explored area in classical linear programming. Sensitivity analyses are basic tools for studying perturbations in optimisation problems. There is considerable research on sensitivity analyses for some operations research and management science models such as linear programming and investment analysis.

532 A. Kumar and N. Bhatia

In most practical applications of mathematical programming, the possible values of the parameters required in the modelling of the problem are provided either by a decision maker subjectively or by a statistical inference from past data, due to which there exists some uncertainty. To reflect this uncertainty, the model of the problem is often constructed with fuzzy data (Zadeh, 1965).

The concept of fuzzy mathematical programming on a general level was first proposed by Tanaka et al. (1974) in the framework of the fuzzy decision of Bellman and Zadeh (1970). The first formulation of fuzzy linear programming was proposed by Zimmermann (1978). Afterwards, many authors have considered various kinds of fuzzy linear programming problems, and have proposed several approaches for solving these problems. Maleki et al. (2000) proposed a new method for solving a fuzzy number linear programming problem and used its solution to obtain the fuzzy solution of the fuzzy variable linear programming problem.

Chiang (2001) used statistical data and a statistical confidence interval to derive (1 ) fuzzy numbers, (0 < < 1) and get a fuzzy linear programming problem. Chiang used two statistical confidence intervals to derive (1 , 1 ) interval-valued fuzzy numbers, 0 < < < 1 and get another fuzzy linear programming problem.

Mahdavi-Amiri and Nasseri (2006) extended the concepts of duality in fuzzy number linear programming problems. Ganesan and Veeramani (2006) introduced a new method for solving a kind of linear programming problem involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems.

Su (2007) proposed a method for fuzzy linear programming based on interval-valued fuzzy numbers and ranking. Mahdavi-Amiri and Nasseri (2007) used a certain linear ranking function to define the dual of fuzzy number linear programming problems and proposed the dual simplex algorithm for solving fuzzy variables linear programming problems. Moreover, Ebrahimnejad and Nasseri (2009) used the complementary slackness property to solve fuzzy number linear programming problems and fuzzy variables linear programming problems without the need for a simplex table.

Hosseinzadeh Lotfi et al. (2009) proposed a method to obtain the approximate solution of fully fuzzy linear programming problems. Ebrahimnejad and Nasseri (2010) introduced a new method called the bounded dual simplex method for bounded fuzzy number linear programming problems in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. Ebrahimnejad et al. (2010a) developed a method for solving such fuzzy linear programming problems in which some or all fuzzy decision variables are bounded. Ebrahimnejad et al. (2010b) proposed a method namely primal-dual simplex algorithm to obtain a fuzzy solution of fuzzy variables linear programming problems. Nasseri and Ebrahimnejad (2010) applied a fuzzy primal simplex method to solve flexible linear programming problems directly without solving any auxiliary problem. Kheirfam and Hasani (2010) studied the basis invariance sensitivity analysis for fuzzy linear programming problems.

Ebrahimnejad (2011) generalized the concept of sensitivity analysis in fuzzy number linear programming problems by applying fuzzy simplex algorithms and using the general linear ranking function on fuzzy numbers. Nasseri and Ebrahimnejad (2011) proposed a method for sensitivity analysis on linear programming problem with trapezoidal fuzzy variables. Hatami-Marbini and Tavana (2011) proposed a general and interactive method for solving linear programming problems with fuzzy parameters.

In the literature, it is pointed out that the values of the membership of an element, in a fuzzy set, are represented by a real number in [0, 1] but a specialist is always

Sensitivity analysis for fuzzy linear programming problems 533

uncertain about the values of the membership. Hence, it is better to represent the values of the membership of an element in a set by intervals of possible real numbers instead of real numbers. To the best of our knowledge, till now there is no method to deal with the sensitivity analysis of linear programming problems in which some or all the parameters are represented by interval-valued fuzzy numbers. In this paper, a new method is proposed for the same. To illustrate the proposed method, an interval-valued fuzzy sensitivity analysis problem which cant be solved by using any of the existing methods is solved by using the proposed method.

This paper is organised as follows. In Section 2, some basic definitions and arithmetic operations are presented. In Section 3, a new method, named Mehars method, is proposed to deal with the sensitivity analysis of interval-valued fuzzy linear programming problems. In Section 4, the proposed method is illustrated with the help of numerical examples. Finally, conclusions are discussed in Section 5.

2 Preliminaries

In this section, some basic definitions, arithmetic operations and comparison of interval-valued fuzzy numbers are presented.

2.1 Basic definitions

In this section, some basic definitions are presented.

Definition 2.1 (Gupta and Mehlawat, 2007): If the membership function of the fuzzy set ( , , )A a b c= on R = (,) is

1,( )

0,Ax a

xx a

==

then A is called a fuzzy point. Definition 2.2 (Gupta and Mehlawat, 2007): If the membership function of the fuzzy set A on R is

( ) ,( )( )( ) ,

( )0, otherwise

A

x a a x bb ac xx b x c

c b

<

=


Definition 2.3 (Gupta and Mehlawat, 2007): An interval-valued fuzzy set A on R is given by {( ,[ ( ), ( )]) : },L UA AA x x x x R = where ( ), ( ) [0,1]L UA Ax x and

( ) ( )L UA Ax x x R and is denoted as [ , ].L UA A A=

This means that the grade of membership of x belongs to the interval [ ( ), ( )],L UA Ax x the least grade of membership at x is ( )LA x and greatest grade of membership at x is )(~ xUA .

Interval-valued fuzzy sets were first used by Gorzafczany (1983) and have been applied to the fields of approximate inference, signal transmission and controller, etc.

Let

( ) ,( )( )( ) , .

( )0, otherwise

LA

x a a x bb ac xx b x c

c b

<

=


2.3 Comparison of interval-valued fuzzy numbers (Gupta and Mehlawat, 2007)

Let 1 1 1 2 1 2[( , , ; ), ( , , ; )]A a b c a b c = and 1 1 1 2 1 2[( , , ; )], ( , , ; )]B p q r p q r = are two (, ) interval-valued fuzzy numbers, then

A B> if ( ) ( )A B >

A B= if ( ) ( )A B =

A B


3 Mehars method

There are few methods (Kheirfam and Hasani, 2010; Ebrahimnejad, 2011; Nasseri and Ebrahimnejad, 2011) to deal with the sensitivity analysis of such linear programming problems in which some or all the parameters are represented by triangular or trapezoidal fuzzy numbers. In the literature, it is pointed out that in several situations, it is better to use interval-valued fuzzy numbers instead of triangular or trapezoidal fuzzy numbers and for the same reason, several authors have used interval-valued fuzzy numbers (Chiang, 2001; Su, 2007; Gupta and Mehlawat, 2007) in different types of problems. But to the best of our knowledge, there is no method in the literature to deal with the sensitivity analysis of linear programming problems in which parameters are represented by interval-valued fuzzy numbers.

In this section, a new method, named Mehars method, is proposed to deal with the sensitivity analysis of fuzzy linear programming problems in which the decision variables are represented by real numbers and rest of the parameters are represented by interval-valued fuzzy numbers.

The steps of proposed method are as follows:

Step 1: In the existing method (Chiang, 2001; Su, 2007) the optimal solution of the interval-valued fuzzy linear programming problem (P1) is obtained by solving crisp linear programming problem (P2). So, solve the crisp linear programming problem (P2), to find the optimal solution X = [xj]n1.

Maximise (or Minimise)

( )TC X Subject to

( ) ( )AX or or b = (P2)

1[ ] where, 0j n jX x x=

where,

1 1[ ] , [ ] , [ ]T

k m kj m n j nb b A a C c = = = or

Maximise (or minimise)

3 2 4 11 1

1 [ (4 3 )( )]16

n n

j j j j j j jj j

c x x = =

+ +

Subject to

3 2 4 11 1

3 2 4 1

1 [ (4 3 )( )] or or16

1 [ (4 3 )( )], 1, 2,...,16

0, 1,2, , .

n n

kj j kj kj kj kj j kj j

k k k k

j

a x x b

k m

x j n

= =

+ + =

+ + =

=

(P2)


Step 2: Find the fuzzy optimal value of a chosen interval-valued fuzzy linear programming problem (P1) by putting the values of X in .TC X Step 3: The optimal solution of the interval-valued fuzzy linear programming problem (P1) is obtained by solving the crisp linear programming problem (P2). Thus, if any change is made in the interval-valued fuzzy linear programming problem (P1), then the optimal solution of the resulting interval-valued fuzzy linear programming problem can be obtained as follows:

Case 1: Change in the fuzzy cost vector

If the cost vector TC changes to 'TC in the original interval-valued fuzzy linear programming problem (P1) then replace ( )TC X by ( ' )TC X in crisp linear programming (P2) to obtain (P3)


( ' )TC X Subject to

( ) ( )AX or or b = )(P3

1[ ] where, 0.j n jX x x=

Case 2: Change in fuzzy requirement vector b If the change in fuzzy requirement vector is made i.e., b is changed to b in an original interval-valued fuzzy linear programming problem (P1) then, replace ( )b by ( )b in the crisp linear programming problem (P2) to obtain (P4):


( )TC X Subject to

( ) ( )AX or or b = (P4)


Case 3: Addition of a new fuzzy variable

Suppose a new non-negative variable, say xn+1, is added in the original interval-valued fuzzy linear programming problem (P1). Assume that if 1nc + is cost and 1nA + is the column associated with xn+1 then replace ( )AX by 1 1( )n nAX A x+ + and ( )TC X by

1 1( )T

n nC X c x+ + in (P2) to obtain the crisp linear programming problem (P5): Maximise (or Minimise)

1 1( )T

n nC X c x+ +


Subject to

1 1( ) ( )n nAX A x or or b+ + = (P5)


Case 4: Addition of a new fuzzy constraint

Suppose a new fuzzy constraint is added in the original interval-valued fuzzy linear programming problem (P1) then, replace ( ) ( )AX or or b = by

( ) ( )A X or or b = in (P2) to obtain a crisp linear programming problem (P6): Maximise (or Minimise)

( )TC X Subject to

( ) ( )A X or or b = )(P6


Case 5: Change in fuzzy constraint matrix

Suppose the column of the constraint matrix, corresponding to the variable xj, is changed from jA to jA in the original interval-valued fuzzy linear programming problem then, replace ( ) ( )AX or or b = by ( ) ( )A X or or b = in (P2) to obtain new crisp linear programming problem (P7):

Maximise (or Minimise)

( )TC X Subject to

( ) ( )A X or or b = )(P7

1[ ] where, 0.j n jX x x =

Step 4: Now apply the existing sensitivity analysis technique to find the optimal solutions for (P3), (P4), (P5), (P6) and (P7) with the help of the optimal solution for (P2) and use Step 3 of the proposed method to find the optimal solution and fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem.

Remark 3.1: The other cases i.e., deletion of variables, deletion of fuzzy constraints, simultaneous change in coefficients of the decision variables in objective function and requirement vectors, etc., can also be solved by using Mehars method.

4 Numerical example

In this section, the proposed method is illustrated with the help of a numerical example:


Example 4.1: A factory produces automobiles and trucks. Each requires three processes. The production condition is shown in Table 1.

Table 1 Production condition

Type Process 1 (hour) Process 2 (hour) Process 3 (hour) Profit (hundred $) Automobile 15 24 21 25 Truck 30 6 14 48 Total hour 450 240 280

Assume the quantity of automobiles and trucks produced to be x1 and x2. Then, the chosen problem can be formulated into the following crisp linear programming problem:

Maximise

1 2[25 48 ]x x+

Subject to

1 215 30 450x x+

1 224 6 240x x+ (P8)

1 221 14 280x x+

1 20, 0.x x

Assuming

11 12 13 14 21 22 23 24 111 112 113

114 121 122 123 124 211 212 213 214 221

222 223 224 311 312 313 314 321 322

7, 6, 8, 9, 5, 4, 6, 8, 5, 1, 2,3, 7, 5, 4, 8, 4, 3, 3, 9, 4,2, 2, 5, 5, 4, 1, 5, 6,

= = = = = = = = = = =

= = = = = = = = = =

= = = = = = = = = 323

324 11 12 13 14 21 22 23 24

31 32 33 34 411 412 413 414 421

422 423 424 41 42 43 44

2, 5,8, 30, 20, 30, 70, 60, 20, 50, 60,50, 10, 30, 40, 4, 3, 2, 9, 4,2, 2, 5, 60, 20, 50, 60, 0.9

=

= = = = = = = = =

= = = = = = = = =

= = = = = = = =

the crisp linear programming problem (P8) is converted into an interval-valued fuzzy linear programming problem (P9).

Maximise

1 2[[(19, 25, 33; 0.9), (18, 25, 34;1)] [(44, 48, 54; 0.9), (43, 48, 56;1)] ]x x (P9)

Subject to

1

2

[(14,15,17;0.9), (10,15,18;1)] [(25,30,34;0.9),(23,30,38;1)] [(430, 450, 480;0.9), (420, 450,520;1)]

xx

1

2

[(21, 24,26;0.9), (20, 24,33;1)] [(4,6,8;0.9),(2,6,11;1)] [(220,240,290;0.9), (180,240,300;1)]

xx


1

2 1 2

[(17,21, 22;0.9), (16, 21,26;1)] [(12,14,19;0.9),(8,14,22;1)] [(270, 280,310;0.9), (230,280,320;1)] 0, 0.

xx x x

Discuss the effect of changing the cost coefficients [(19, 25,33;0.9), (18,25,34;1)] and [(44, 48,54;0.9), (43, 48,56;1)] of the decision variables x1 and x2 to [(22, 28,36;0.9), (21, 28,37;1)] and [(46,50,56;0.9), (45,50,58;1)] respectively on the optimal solution and the fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem.

Discuss the effect of changing the requirement vector from [(430,450, 480;0.9), (420,450,520;1)], [(220,240,290;0.9), (180, 240,300;1)] and [(230,280,310;0.9), (230, 280,320;1] to [(480,500,530;0.9), (470,500,570;1)], [(210,230,280;0.9), (170, 230, 290;1)] and [(290,300,330;0.9), (250,300,340;1)] on the optimal solution and the fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem.

Find the effect of addition of a new non-negative variable x3 with cost [(46,50,56;0.9), (45,50,58;1)] and [[(14,15,17;0.9), (10,15,18;1)], [(21, 24,26;0.9), (20, 24,33;1)], [(14,15,17;0.9), (10,15,18;1)]]T as column vectors on the optimal solution and the fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem.

Discuss the effect of changing the column of the constraint matrix corresponding to the variable 2x by [[(30,35,39;0.9), (28,35,43;1)],[(3,5,7;0.9), (1,5,10;1)], [(13,15,20;0.9), (9,15, 23;1]]T on the optimal solution and the fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem.

Find the effect of addition of a new constraint 1[(18, 21, 23;0.9), (17, 21,30;1)]x

2[(18,20, 22;0.9), (16, 20,25;1)] [(480,500,550;0.9), (440,500,560;1)]x on the optimal solution and the fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem.

Solution: The solution of the interval-valued fuzzy sensitivity analysis problem chosen in Example 4.1, can be obtained by using the following steps of the proposed method:

Step 1: As discussed in Step 1 of Mehars method, the optimal solution of the interval-valued fuzzy linear programming problem (P9) can be obtained by the solving crisp linear programming problem (P11)

Maximise

1 2 1 2125 48 (4.6 5.9 )

16x x x x + + +

Subject to

1 2 1 21 115 30 ( 1.6 .3 ) 450 (62)

16 16x x x x+ + + + (P10)


1 2 1 21 124 6 (5.5 1.3 ) 240 (30)

16 16x x x x+ + + +

1 2 1 21 121 14 ( 3 5.6 ) 280 (7)

16 16x x x x+ + + +

1 20, 0x x

or

Maximise

1 2[25.287 48.368 ]x x+

Subject to

1 214.9 30.01875 453.875x x+

1 224.34375 6.08125 241.875x x+ (P11)

1 220.8125 14.35 280.4375x x+

1 20, 0.x x

Step 2: The optimal solution of the crisp linear programming problem (P11) is x1 = 4.64, x2 = 12.82.

Step 3: Putting x1 = 4.64, x2 = 12.82 in 1[[(19,25,33;0.9), (18,25,34;1)]x

2[(44, 48,54;0.9), (43, 48,56;1)] ]x the fuzzy optimal value is [(652.24,731.36,845.4;0.9), (634.78,731.36,875.68;1)] .

The cost coefficients corresponding to the variables x1 and x2 change to [(22, 28,36;0.9), (21, 28,37;1)] and [(46,50,56;0.9), (45,50,58;1)] respectively in the original interval-valued fuzzy linear programming problem (P9), so replacing the crisp linear programming problem (P11) by (P12)

Maximise

1 2[28.287 50.368 ]x x+

Subject to

1 214.9 30.01875 453.875x x+ (P12)

1 224.34375 6.08125 241.875x x+

1 220.8125 14.35 280.4375x x+

1 20, 0.x x

Now applying the existing sensitivity analysis technique, the optimal solution for crisp linear programming problem (P12) is x1 = 4.64, x2 = 12.82 and the optimal value is 776.81. Using Step 3 of the proposed method the fuzzy optimal value of resulting


interval-valued fuzzy linear programming problem is [(691.8, 770.92, 884.96; 0.9), (674.34, 770.92, 915.24; 1)].

The requirement vector is changed from [(430,450,480;0.9), (420,450,520;1)], [(220,240,290;0.9), (180, 240,300;1)] and [(270,280,310;0.9), (230, 280,320;1)] to [(480,500,530;0.9), (470,500,570;1)],[(210,230,280;0.9), (170, 230, 290;1)] and [(290,300,330;0.9), (250,300,340;1)] respectively in the original interval-valued fuzzy linear programming problem, (P9) so replacing the crisp linear programming problem (P11) by (P13)

Maximise

1 2[25.287 48.368 ]x x+

Subject to

1 214.9 30.01875 503.875x x+ (P13)

1 224.34375 6.08125 231.875x x+

1 220.8125 14.35 300.4375x x+

1 20, 0.x x

Now applying the existing sensitivity analysis technique, the optimal solution for crisp linear programming problem (P13) is x1 = 4.35, x2 = 14.62 and the optimal value is 817.46. Using Step 3 of the proposed method, the fuzzy optimal value of the resulting interval-valued fuzzy linear programming problem is [(725.93, 810.51, 933.03; 0.9), (706.96, 810.51, 966.52; 1)].

A new variable x3 0 with cost [(46, 50, 56; 0.9), (45, 50, 58; 1)] and column vectors [[(14,15,17;0.9), (10,15,18;1)],[(21, 24,26;0.9), (20, 24,33;1)], [(14,15,17;0.9), (10,15,18;1)]]T is added in the original interval-valued fuzzy linear programming problem (P9) so replacing the crisp linear programming problem (P11) by the crisp linear programming problem (P14).

Maximise

1 2 3[25.287 48.368 50.368 ]x x x+ +

Subject to

1 2 314.9 30.01875 14.9 453.875x x x+ + (P14)

1 2 324.34375 6.08125 24.34375 241.875x x x+ +

1 2 320.8125 14.35 14.9 280.4375x x x+ +

1 2 30, 0, 0.x x x

Now applying the existing sensitivity analysis technique, the optimal solution for crisp linear programming problem (P14) is x1 = 0, x2 = 11.63, x3 = 7.03 and the optimal value is


916.72. Using Step 3 of the proposed method the fuzzy optimal value is [(835.1, 909.74, 1021.7, 0.9), (816.44, 909.74, 1059.02; 1)].

The column of the constraint matrix corresponding to the variable x2 is changed to [[(30,35,39;0.9), (28,35,43;1)],[(3,5,7;0.9), (1,5,10;1)], [(13,15,20;0.9), (9,15, 23;1)]]T in the original interval-valued fuzzy linear programming problem (P9) so replacing the crisp linear programming problem (P11) by crisp linear programming problem (P15).

Maximise

1 2[25.287 48.368 ]x x+

Subject to

1 214.9 35.01875 453.875x x+

1 224.34375 5.08125 241.875x x+ (P15)

1 220.8125 15.35 280.4375x x+

1 20, 0.x x

Now apply the existing sensitivity analysis technique, the optimal solution for crisp linear programming problem (P15) is x1 = 5.71, x2 = 10.53 and the optimal value is 653. Using Step 3 of the proposed method, the fuzzy optimal value is [(571.81, 648.19, 757.05; 0.9), (555.57, 648.19, 783.82; 1)].

A new fuzzy constraint 1[(18, 21,23;.9), (17, 21,30;1)] [(18, 20, 22;.9), (16, 20, 25;1)]x

2 [(480,500,550;.9), (440,500,560;1)]x is added to the original interval-valued fuzzy linear programming problem (P9) so replacing crisp linear programming problem (P11) by crisp linear programming problem (P16).

Maximise

1 2[25.287 48.368 ]x x+

Subject to

1 214.9 30.01875 453.875x x+ (P16)

1 224.34375 6.08125 241.875x x+

1 220.8125 14.35 280.4375x x+

1 221.34375 20.08125 501.875x x+

1 20, 0.x x

Now applying the existing sensitivity analysis technique, the optimal solution for crisp linear programming problem (P16) is x1 = 4.64, x2 = 12.82 and the optimal value is 737.26. Using Step 3 of the proposed method the fuzzy optimal value is [(652.24, 731.36, 845.4; 0.9), (634.78, 731.36, 875.68; 1)].


4.2 Physical interpretation of the results

In this section, the effect of changing the cost coefficients on the results is physically interpreted. Similarly, the results of other cases, presented in Table 2, can also be physically interpreted.

Thus, from the results obtained by using the original interval-valued fuzzy linear programming problem and resulting interval-valued fuzzy linear programming problem, it can be easily seen that in Case 1:

For original interval-valued fuzzy linear programming problems, the optimal value will be greater than 634.78 and less than 875.68 units, while for the interval-valued fuzzy linear programming problem, obtained by changing the cost coefficients corresponding to the variables x1 and x2, the optimal value will be greater than 674.34 and less than 915.24 units.

For an original interval-valued fuzzy linear programming problem, the overall level of satisfaction of the decision maker about the statement that the optimal value will be 731.36 units lies in [0.9, 1] and for an interval-valued fuzzy linear programming problem, obtained by changing the cost coefficients corresponding to the variables x1 and, x2 the overall level of satisfaction of the decision maker about the statement that the optimal value will be 770.92 units lies in [0.9, 1].

For an original interval-valued fuzzy linear programming problem, the overall level of satisfaction of the decision maker for the remaining values of optimal value can be obtained as follows:

Let x represents the optimal value, then the least value and greatest value of overall level of satisfaction of the decision maker for x i.e. ( ) 100LA x and ( ) 100UA x can be obtained as follows:

0.9( 652.24) , 652.24 731.36(731.36 652.24)0.9(845.4 )( ) , 731.36 845.4

(845.4 731.36)0, otherwise

LA

x x

xx x

<

=


Table 2 Results of original and modified interval-valued fuzzy linear programming problem

Fuzzy optimal value obtained from original interval-valued fuzzy linear programming

problem

Fuzzy optimal value obtained from modified interval-valued fuzzy linear

programming problem Case 1 [(652.24,731.36,845.4;0.9),(634.78,731.

36,875.68;1)]

[(691.8,770.92,884.96;0.9),(674.34,770.92,915.24;1)]

Case 2 [(652.24,731.36,845.4;0.9), (634.78,731.36,875.68;1)]

[(725.93,810.51,933.03;0.9),(706.96,810.51,966.52;1)]

Case 3 [(652.24,731.36,845.4;0.9),(634.78,731.36,875.68;1)]

[(835.1,909.74,1021.7,0.9),(816.44,909.74,1059.02;1)]

Case 4 [(652.24,731.36,845.4;0.9),(634.78,731.36,875.68;1)]

[(571.81,648.19,757.05;0.9),(555.57,648.19,783.82;1)]

Case 5 [(652.24,731.36,845.4;0.9),(634.78,731.36,875.68;1)]

[(652.24,731.36,845.4;0.9),(634.78,731.36,875.68;1)]

Let x represents the optimal value, then the least value and greatest value of overall level of satisfaction of the decision maker for x i.e., ( ) 100LA x and ( ) 100UA x can be obtained as follows:

0.9( 691.8) , 691.8 770.92(770.92 691.8)0.9(915.24 )( ) , 770.92 915.24

(915.24 770.92)0, otherwise

LA

x x

xx x

<

=


variables are represented by real numbers and rest of the parameters are represented by interval-valued fuzzy numbers. To illustrate the proposed method an interval-valued fuzzy sensitivity analysis problem which cant be solved by using any of the existing methods is solved by using the proposed method. We emphasise that there is no existing rule for the product of two arbitrary interval-valued fuzzy numbers. So the proposed method cannot solve an interval-valued fuzzy sensitivity analysis problem when the variables are also interval-valued fuzzy numbers. That is an interesting research work for the future.

Acknowledgements

The authors would like to thank the editor and all the three anonymous referees for the various suggestions which have led to an improvement in both the quality and clarity of the paper. I, Dr. Amit Kumar, want to acknowledge the adolescent inner blessings of Mehar. I believe that Mehar is an angel for me and without Mehars blessings it was not possible to think of the idea proposed in this paper. Mehar is the lovely daughter of Parmpreet Kaur (research scholar under my supervision).

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