Multiparametric sensitivity analysis in programming problem with linear-plus-linear fractional objective function

$Page 1: Multiparametric sensitivity analysis in programming problem with linear-plus-linear fractional objective function$
European Journal of Operational Research 160 (2005) 232–241

www.elsevier.com/locate/dsw

Continuous Optimization

Multiparametric sensitivity analysis in programmingproblem with linear-plus-linear fractional objective function

Sanjeet Singh a, Pankaj Gupta b,*, Davinder Bhatia a

a Department of Operational Research, University of Delhi, Delhi-110007, Indiab Department of Mathematics, Deen Dayal Upadhyaya College, Shivaji Marg, Karampura, New Delhi-110015, India

Received 27 June 2002; accepted 30 October 2002

Available online 14 November 2003

Abstract

In this paper, we study multiparametric sensitivity analysis for programming problems with linear-plus-linear

fractional objective function using the concept of maximum volume in the tolerance region. We construct critical re-

gions for simultaneous and independent perturbations in the objective function coefficients and in the right-hand-side

vector of the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as

�focal� and �non-focal�. Non-focal parameters can have unlimited variations, because of their low sensitivity in practice,

these parameters can be deleted from the analysis. For focal parameters, a maximum volume tolerance region is

characterized. Theoretical results are illustrated with the help of a numerical example.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Linear-plus-linear fractional programming; Fractional programming; Multiparametric sensitivity analysis; Tolerance

approach; Parametric analysis

1. Introduction

A general linear-plus-linear fractional programming problem has the following form:

* Co

E-m

0377-2

doi:10

ðLLFPÞ Maximize F ðxÞ ¼ ctxþ ptxqtxþ h

subject to Ax ¼ b;

xP 0;

where A is m� n coefficient matrix with m < n; ct, pt and qt are n-dimensional row vectors; x and b are n-dimensional and m-dimensional column vectors respectively and h is a scalar quantity. It is assumed that

the feasible region of the problem (LLFP) is bounded.

rresponding author. Address: Flat No-01, Kamayani Kunj, Plot No-69, I.P. Extension, Delhi-110092, India.

ail address: [email protected] (P. Gupta).

217/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

.1016/j.ejor.2002.10.001

mail to: [email protected]

S. Singh et al. / European Journal of Operational Research 160 (2005) 232–241 233

Teterev [13] pointed out that such problems arise when a compromise between absolute and relativeterms is to be maximized. Major applications of the problem (LLFP) can be found in; transportation,

problems of optimizing enterprise capital, the production development fund and the social, cultural and

construction fund. Teterev [13] derived an optimality criteria for (LLFP) using the simplex type algorithm.

Several authors [2,5,6,9,10,12,13] studied the problem (LLFP) and its variants and have discussed their

solution properties.

In practical applications the data collected may not be precise, we would like to know the effect of data

perturbation on the optimal solution. Hence, the study of sensitivity analysis is of great importance. In

general, the main focus of sensitivity analysis is on simultaneous and independent perturbation of theparameters. Besides this, all the parameters are required to be analyzed at their independent levels of

sensitivity. If one parameter is more sensitive than the others, the tolerance region characterized by treating

all the parameters at equal levels of sensitivity would be too small for the less sensitive parameters. If the

decision maker has prior knowledge that some parameters can be given unlimited variations without

affecting the original solution then we consider those parameters as non-focal and these non-focal

parameters can be deleted from the analysis. Wang and Huang [14] proposed the concept of maximum

volume in the tolerance region for the multiparametric sensitivity analysis of a single objective linear

programming problem. Their theory allows the more sensitive parameters called as �focal� to be investigatedat their independent levels of sensitivity, simultaneously and independently. This approach is a significant

improvement over the earlier approaches primarily because besides reducing the number of parameters in

the final analysis, it also handles the perturbation parameters with greater flexibility by allowing them to be

investigated at their independent levels of sensitivity.

In this paper, we extend the approach of Wang and Huang [14] to discuss multiparametric sensitivity

analysis for the problem (LLFP).

2. Problem formulation and sensitivity models

To address perturbations of the right-hand-side vector, and the coefficients of the objective function, in

problem (LLFP), we consider the following perturbed model:

ðPLLFPÞ Maximize F ðxÞ ¼ ðcþ DcÞtxþ ðp þ DpÞtxqtxþ h

subject to Ax ¼ bþ Db;

xP 0;

where

Dc ¼XHh¼1

cjhch

" #n�1

; Dp ¼XHh¼1

pjhch

" #n�1

; Db ¼XHh¼1

bjhch

" #m�1

are the multiparametric perturbations defined by the perturbation parameter c ¼ ðc1; c2; . . . ; cH Þt. Here H is

the total number of parameters.

Next, we give some notations that are used throughout this study:

B � f1; 2; . . . ; ng denotes the index set of basic variables.

Without loss of generality, we suppose B ¼ f1; 2; . . . ;mg.N ¼ f1; 2; . . . ; ng n B denotes the index set of non-basic variables.

AB ¼The basis matrix with inverse b ¼ A1B ¼ ½bij�.

234 S. Singh et al. / European Journal of Operational Research 160 (2005) 232–241

AN ¼The submatrix of A corresponding to the non-basic variables.

cB ¼ ½c1; c2; . . . ; cm�t and cN ¼ ½cmþ1; cmþ2; . . . ; cn�t: The vectors of the objective function coefficients corre-

sponding to the basic and non-basic variables respectively.�b ¼ A1

B bP 0: The vector of the values corresponding to xB.A1B AN ¼ ½yij�.

uh ¼ ½b1h; b2h; . . . ; bmh�t: The vector of the coefficients of parameter ch in Db.vh ¼ ½c1h; c2h; . . . ; cmh�t: The vector of the partial coefficients of parameter ch in Dc.wh ¼ ½p1h; p2h; . . . ; pmh�t: The vector of the partial coefficients of parameter ch in Dp.�Dj ¼The vector of the reduced cost corresponding to non-basic variables.

P j ¼ jth column of the matrix P .

Under the assumptions ctxP 0 and qtxþ h > 0 over the feasible region, the optimality criteria for the

problem (LLFP) using the simplex type algorithm given by Teterev [13] is stated as follows:

Let AB denote the optimal basis matrix and let x ¼ xB0

� �¼ ½x1; x2; . . . ; xm; 0; . . . ; 0�t be the corresponding

basic feasible solution to the problem (LLFP). This solution will be an optimal solution if

Dj ¼ ðcj zcjÞ þZ 00ðpj zpj Þ Z 0ðqj zqj Þ

ðZ 00Þ26 0 for j ¼ 1; 2; . . . ; n;

where

Z 0 ¼ ptBxB; Z 00 ¼ qtBxB þ h;

zc ¼ ctBA1B AN zq ¼ qtBA

1B AN ;

zp ¼ ptBA1B AN :

Here cB, pB and qB are the subvectors of c, p and q respectively that correspond to the basis B.In general, the sensitivity analysis focuses on characterizing sets called as critical regions over which the

right-hand-side vector and objective function coefficients of the problem (LLFP) may vary while still

retaining the same optimal basis B [3]. Let S be a general notation for a critical region.

In the following proposition, we construct critical region for simultaneous and independent perturba-

tions with respect to b, p and c, respectively.

Proposition 1.When b, p and c are perturbed simultaneously, the critical region Rbpc of the problem (PLLFP)is given by

Rbpc ¼ ðc1;c2; . . . ;cH Þtj�bi

8>>><>>>:

þXHh¼1

ðbti uhÞchP0; i¼ 1;2; . . . ;m; �Djþ

XHh¼1

ðcmþj;h yt jvhÞch

þ

Pmi¼1

qixiþh

� �pj

PHh¼1

ðyt jwhpmþj;hÞch� �

qjPmi¼1

qiyij

� � PHh¼1

ðxtBwhÞch� ��

Pmi¼1

qixiþh

� �2 60; j¼ 1;2; . . . ;nm

9>>>=>>>;: ð1Þ


Proof. Let us assume that xB be the new basic solution, then

xB ¼ A1B ðbþ DbÞ ¼ A1

B bþ A1B Db ¼ �bþ A1

B

XHh¼1

b1hch;XHh¼1

b2hch; . . . ;XHh¼1

bmhch

" #t

¼ �bþXHh¼1

ðbt1 uhÞch;

XHh¼1

ðbt2 uhÞch; . . . ;

XHh¼1

ðbtm uhÞch

" #t:

Now ith component of xB is given by

xBi ¼ �bi þXHh¼1

ðbti uhÞch; i ¼ 1; 2 . . . ;m:

This new basic solution xB will be feasible if

�bi þXHh¼1

ðbti uhÞch P 0; i ¼ 1; 2; . . . ;m:

For the new solution xB, to satisfy optimality condition, the new values Dj�s of Dj are computed as follows:

Dj ¼ cj

þXHh¼1

cmþj;hch

! zcj þ

Z 00 pj þPHh¼1

pmþj;hch

� � zpj

� � Z 0 qj zqj

� �½Z 00�2

; j ¼ 1; 2; . . . ; n m:

Here,

Z 0 ¼Xmi¼1

pi

þXHh¼1

pihch

!xi; Z 00 ¼

Xmi¼1

qixi þ h;

zpj ¼Xmi¼1

pi

þXHh¼1

pihch

!yij; zqj ¼

Xmi¼1

qiyij;

zcj ¼Xmi¼1

ci

þXHh¼1

cihch

!yij:

Thus, Dj takes the form

Dj ¼ cj

þXHh¼1

cmþj;hch

!Xmi¼1

ci

þXHh¼1

cihch

!yij

þ

Pmi¼1

qixiþh

� �pjþ

PHh¼1

pmþj;hch

� �Pmi¼1

piþPHh¼1

pihch

� �yij

� �Pmi¼1

piþPHh¼1

pihch

� �xi

� �qj

Pmi¼1

qiyij

� �� Pmi¼1

qixiþh

� �2

¼ cj

Xmi¼1

ciyij

!þ

Pmi¼1

qixiþh

� �pj

Pmi¼1

piyij

� �Pmi¼1

pixi

� �qj

Pmi¼1

qiyij

� �Pmi¼1

qixiþh

� �2


þXHh¼1

cmþj;h�

yt jvh�ch þ

Pmi¼1

qixi þ h

� �pj

PHh¼1

ðyt jwh pmþj;hÞch� �

qj Pmi¼1

qiyij

� � PHh¼1

ðxtBwhÞch� �

Pmi¼1

qixi þ h

� �2

¼ �Dj þXHh¼1

cmþj;h�

yt jvh�ch þ

Pmi¼1

qixi þ h

� �pj

PHh¼1


qj Pmi¼1

qiyij

� � PHh¼1

ðxtBwhÞch� �

Pmi¼1

qixi þ h

� �2 :

Now solution xB will be optimal if

�Dj þXHh¼1

cmþj;h�

yt jvh�ch

þ

Pmi¼1

qixi þ h

� �pj

PHh¼1


qj Pmi¼1

qiyij

� � PHh¼1

ðxtBwhÞch� �

Pmi¼1

qixi þ h

� �2 6 0; j ¼ 1; 2; . . . ; n m:

Thus the critical region Rbpc is given by

Rbpc ¼ ðc1;c2; . . . ;cH Þtj�bi

8>>><>>>:

þXHh¼1

ðbti uhÞchP0; i¼ 1;2; . . . ;m; �Djþ

XHh¼1

ðcmþj;h yt jvhÞch

þ

Pmi¼1

qixiþh

� �pj

PHh¼1

ðyt jwhpmþj;hÞch� �

qjPmi¼1

qiyij

� � PHh¼1

ðxtBwhÞch� ��

Pmi¼1

qixiþh

� �2 60; j¼ 1;2; . . . ;nm

9>>>=>>>;: �

It can be seen that a parameter ch is absent in Rbpc if and only if

bti uh ¼ 0 for i ¼ 1; 2; . . . ;m; cmþj;h ¼ yt jvh;

yt jwh ¼ pmþj;h for j ¼ 1; 2; . . . ; n m and xtBwh ¼ 0:

Therefore, it is not necessary that all the parameters c1; c2; . . . ; cH appear simultaneously in Rbpc. Those

parameters that are not present in Rbpc are called the non-focal parameters.

Corollary 1. When perturbation is on b only (i.e. Dc ¼ 0, Dp ¼ 0), the critical region Rb of the problem(PLLFP) is given by

Rb ¼ c

(¼ ðc1; c2; . . . ; cH Þ

tj�bi þXHh¼1

ðbti uhÞch P 0; i ¼ 1; 2; . . . ;m

): ð2Þ

Parameter ch is non-focal in Rb if and only if bti uh ¼ 0 for i ¼ 1; 2; . . . ;m. That is, vector uh is orthogonal to

bi for i ¼ 1; 2; . . . ;m. Since A1B is invertible and bt

i is the ith row of A1B , the set of column vectors


fb1 ; b2 ; . . . ; bm g is a linearly independent set. Therefore, if bti uh ¼ 0 for i ¼ 1; 2; . . . ;m, then uh ¼ 0.

Therefore a parameter ch is non-focal in Rb if and only if uh ¼ 0.

Corollary 2.When perturbation is in c and p, (i.e. Db ¼ 0), the critical region Rpc of the problem (PLLFP) isgiven by 8

Rpc ¼ c

>>><>>>:

¼ ðc1;c2; . . . ;cH Þtj�Dj þ

XHh¼1

ðcmþj;h yt:jvhÞch

þ

Pmi¼1

qixi þ h

� �pj

PHh¼1


qj Pmi¼1

qiyij

� � PHh¼1

ðxtBwhÞch� �

Pmi¼1

qixi þ h

� �2 60 for j¼ 1;2; . . . ;nm

9>>>=>>>;:

ð3Þt t
Parameter ch is non-focal in Rpc if and only if cmþj;h ¼ y jvh; y jwh ¼ pmþj;h for j ¼ 1; 2; . . . ; n m and
xtBwh ¼ 0.

Remark 1. For the problem (LLFP), a local optimal solution need not be global solution [4]. However, if

the objective function of the problem is pseudoconcave over the feasible region then the local optima is alsoa global optima.

Remark 2. Moreover, if the objective function of (LLFP) is pseudoconcave and hence quasiconcave, the

optimal solution is attained at an extreme point of the feasible region, which is a compact set [11].

These features of an optimal solution are very valuable from the computational point of view. The next

result provides the conditions which ensure the pseudoconcavity of the linear-plus-linear fractional

objective function.

Lemma [1]. The function F in the problem (LLFP) is pseudoconcave over the feasible region of the problem ifand only if one of the following conditions holds:

(i) c ¼ kq, kP 0;

(ii) there is s 2 R such that p ¼ sq and shP 0.

However, in this paper, sensitivity analysis has been carried out for the local/global optimal solution of the

problem (LLFP).

Definition 1 [14]. The maximum volume region (MVR) BS in a critical region S is given by

BS ¼ rn

¼ ðr1; r2; . . . ; rHÞtjjrjj6 k�j ; j ¼ 1; 2; . . . ;Ho;

where k� ¼ ðk�1 ; k�2 ; . . . ; k�H Þ is the optimal solution of the following maximization problem:

ðP1Þ Maximizek2KðSÞ

V ðkÞ ¼ k1 k2 k3 kH ;

where

KðSÞ ¼ fk ¼ ðk1; k2; . . . ; kH Þtjjrjj6 kj; j ¼ 1; 2; . . . ;H implies r ¼ ðr1; r2; . . . ; rH Þt 2 Sg:
The volume of BS is VolðBSÞ ¼ 2Hk�1 k�2 k�H .


Let us suppose that out of H parameters only J parameters are obtained as focal parameters. Since thecritical region is a polyhedral set, there exists L ¼ ½‘ij� 2 RI�J , d ¼ fdig 2 RI , I ; J 2 N , where I and J are the

number of constraints and variables of S, respectively, such that S ¼ fr ¼ ðr1; r2; . . . ; rJ ÞtjLr6 dg. For focalparameters, it is assumed that ‘ j 6¼ 0 for j ¼ 1; 2; . . . ; J .

Remark 3. It follows from Proposition 1 and Corollaries 1 and 2 that r ¼ 0 belongs to Rpc, Rb and Rbpc, and

thus we have d P 0.

Consider the case when d > 0. The set KðSÞ can be rewritten as

KðSÞ ¼ k�

¼ ðk1; k2; . . . ; kJ Þtjjrjj6 kj; j ¼ 1; 2; . . . ; J implies jLjr6 d�;

where jLj is the matrix obtained by changing the negative elements of L to positive. Since jLjP 0, followingWang and Huang [14], the equivalent form of KðSÞ is

KðSÞ ¼ k�

¼ ðk1; k2; . . . ; kJ ÞtjjLjk6 d; kP 0�:

The MVR BS of a polyhedral set S ¼ frjLr6 dg with d > 0 and ‘ j 6¼ 0 for all j is bounded and is defined by

BS ¼ rn

¼ ðr1; r2; . . . ; rJ Þtjjrjj6 k�j ; j ¼ 1; 2; . . . ; Jo;

where k� ¼ ðk�1 ; k�2 ; . . . ; k�J Þ is the unique optimal solution of the following problem:

ðP2Þ Maximize V ðkÞ ¼ k1 k2 k3 kJsubject to jLjk6 d;

kP 0:

Multiparametric sensitivity analysis of the problem (PLLFP) can now be performed as follows:

Obtain the critical regions as given in (1)–(3) by considering perturbations in the objective function

coefficients and in the right-hand-side vector. Delete all the non-focal parameters from the analysis. The

MVR of the critical regions is obtained by solving the problem (P2). The problem (P2) can be solved byexisting techniques such as Dynamic Programming. The detailed algorithm can be found in Wang and

Huang [15]. Software GINO [7] can also be used to solve the non-linear programming problem (P2).

3. Numerical example

We present the following numerical example to illustrate the theoretical results presented in this paper.

ðLLFPÞ Maximize F ðxÞ ¼ ð2x1 þ 6x2 þ 2x3Þ þ3x1 þ 5x2 þ 6x3x1 þ 3x2 þ x3 þ 2

subject to 3x1 x2 þ 2x3 þ x4 ¼ 7

2x1 þ 4x2 þ x5 ¼ 12

4x1 þ 3x2 þ 8x3 þ x6 ¼ 10

xj P 0; j ¼ 1; 2; . . . ; 6:

In view of Lemma, it can easily be seen that the objective function F is pseudoconcave. Hence, local

maxima of (LLFP) is also global.Using simplex type procedure of Teterev [13], the initial and the final simplex tables are given in Tables 1

and 2.

Table 2

Optimal simplex table

qj ! 1 3 1 0 0 0

pj ! 3 5 6 0 0 0

cj ! 2 6 2 0 0 0

qB pB cB Basic variables x1 ðz1Þ x2 ðz2Þ x3 ðz3Þ x4 ðz4Þ x5 ðz5Þ x6 ðz6Þ b

1 3 2 x1ðz1Þ 1 0 45

25

110

0 4

3 5 6 x2ðz2Þ 0 1 25

15

310

0 5

0 0 0 x6ðz3Þ 0 0 10 1 12

1 11

ðcj zjÞ ! 0 0 )2 )2 )2 0

ðpj zpj Þ ! 0 0 1.6 )2.2 )1.8 0 Z 0 ¼ 37

ðqj zqj Þ ! 0 0 )1 )1 )1 0 Z 00 ¼ 21

Dj ! 0 0 )1.84 )2.02 )2.00 0

Table 1

Initial simplex table

Basic variables x1 x2 x3 x4 x5 x6 b

x4 3 )1 2 1 0 0 7

x5 )2 4 0 0 1 0 12

x6 )4 3 8 0 0 1 10


The optimal solution is

x� ¼ ½x�1; x�2; . . . ; x�6�t ¼ ½4; 5; 0; 0; 0; 11�t ¼ ½4; 5; 11; 0; 0; 0�t ¼ ½Z1; Z2; Z3; . . . ; Z6�t ½say�:

Here B ¼ f1; 2; 3g. The matrix of the optimal basis is

AB ¼3 1 0

2 4 0

4 3 1

24

35 and its inverse b ¼ A1

B ¼25

110

015

310

01 1

21

24

35:

Also

�b ¼ ½4; 5; 11�t; A1B AN ¼

45

25

110

25

15

310

10 1 12

24

35:

Consider perturbations in b; c and p simultaneously

Db ¼ ½2c1 þ 5c2 þ 0c3; 0c1 þ 0c2 þ c3; c1 c2 þ c3�t;

Dc ¼ ½5c1 5c2 þ 5c3 5c4; 10c1 þ 5c2 10c3 þ 5c4; c1 c2 þ 0:1c3 0:2c4�t;

Dp ¼ ½10c1 þ 5c2 5c3 þ 5c4; 5c1 5c2 þ 10c3 5c4; c1 þ c2 þ 0:1c3 þ 5=11c4�t;

u1 ¼ ½2; 0; 1�t; u2 ¼ ½5; 0;1�t; u3 ¼ ½0; 1; 1�t; u4 ¼ ½0; 0; 0�t;

v1 ¼ ½5; 10; 1�t; v2 ¼ ½5; 5;1�t; v3 ¼ ½5;10; 0:1�t; v4 ¼ ½5; 5;0:2�t;


w1 ¼ ½10; 5; 1�t; w2 ¼ ½5;5; 1�t; w3 ¼ ½5; 10; 0:1�t; w4 ¼ 5;

� 5;

5

11

�t;

�Dj ¼ ½1:84;2:02;2:00�t;

c41 ¼ 5; c42 ¼ 1; c43 ¼ 1; c44 ¼ 4;

c51 ¼ 2; c52 ¼ 3; c53 ¼ 3; c54 ¼ 1:2;

c61 ¼ 3; c62 ¼ 1; c63 ¼ 3; c64 ¼ 1:1;

p41 ¼ 19; p42 ¼ 10; p43 ¼ 1; p44 ¼ 6:60;

p51 ¼ 5; p52 ¼ 2; p53 ¼ 1; p54 ¼ 1:46;

p61 ¼ 1; p62 ¼ 1; p63 ¼ 3; p64 ¼ 0:77:

Here n ¼ 6, m ¼ 3.

Because yt jv4 c3þj;4 ¼ 0, yt jw4 p3þj;4 ¼ 0, for j ¼ 1,2,3, xtBw4 ¼ 0 and u4 ¼ 0, therefore, c4 is a non-

focal parameter and can be varied unlimitedly. The critical region Rbpc defined by three focal parameters c1,c2 and c3 is given by

Rbpc ¼ c�

¼ ðc1; c2; c3Þtj4þ 0:8c1 þ 2c2 þ 0:1c3 P 0; 5þ 0:4c1 þ c2 þ 0:3c3 P 0; 11þ 3c1 þ 4c2 þ 0:5c3 P 0;

1:697 14:84c1 þ 10:92c2 2:68c3 6 0; 1:78 2:78c1 0:99c2 2:98c3 6 0;

1:72 5:88c1 2:52c2 0:44c3 6 0�:

The MVR of Rbpc is obtained by solving the following maximization problem

Max V ðkÞ ¼ k1 k2 k3subject to 0:8k1 þ 2k2 þ 0:1k3 6 4;

0:4k1 þ k2 þ 0:3k3 6 5;

3k1 þ 4k2 þ 0:5k3 6 11;

14:84k1 þ 10:92k2 þ 2:68k3 6 1:697;

2:78k1 þ 0:99k2 þ 2:98k3 6 1:78;

5:88k1 þ 2:52k2 þ 0:44k3 6 1:72;

k1; k2; k3 P 0:

The optimal solution of the above problem is k� ¼ ð0:04; 0:05; 0:21Þ
MVR BRbpc ¼ c
�¼ ðc1; c2; c3Þ

tjjc1j6 0:04; jc2j6 0:05; jc3j6 0:21�:

VolðBRbpcÞ ¼ 23ð0:04Þ ð0:05Þ ð0:21Þ ¼ 0:0034:

Comparison. For the above perturbations, the following tolerance region is obtained using the tolerance

approach [8,16] treating all four parameters at equal levels of sensitivity:

S ¼ c�

¼ ðc1; c2; c3; c4Þtjjc1j6 0:06; jc2j6 0:06; jc3j6 0:06; jc4j6 0:06

�:

The volume of the above region in the reduced three dimensional space is 23Æ(0.06)(0.06)(0.06)¼ 0.0017

which is less than Vol(Rbpc). Also all the four parameters are treated at equal levels of sensitivity using the


tolerance approach while the approach adopted in the present paper enables the less sensitive parameter c4to vary in R. Thus the present approach to sensitivity analysis not only gives the maximum volume tol-

erance region but also handles the simultaneous and independent perturbations of the parameters with

greater flexibility.

4. Conclusion

In the present paper, we discuss multiparametric sensitivity analysis for the problem (LLFP) by clas-

sifying the perturbation parameters as �focal� and �non-focal�. This approach not only reduces the number of

parameters in the final analysis but also allow the different parameters to be investigated at their inde-

pendent levels of sensitivity. The study of perturbations in the constraint matrix of the problem (LLFP)

would be investigated by the authors in their forthcoming research paper.

Acknowledgements

The authors are thankful to the unknown referees for their critical evaluation of the paper and sug-

gestions for the improvement in presentation of the paper. The authors also wish to express their deep

gratitude to Professor R.N. Kaul (Retd.), Department of Mathematics, University of Delhi, Delhi and

Professor M.C. Puri, Department of Mathematics, I.I.T., Delhi for their encouragement and inspiration to

complete the work.

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Multiparametric sensitivity analysis in programming problem with linear-plus-linear fractional objective function