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International Journal of Computer Applications (0975 – 8887) Volume 32– No.8, October 2011

Multi-Objective Linear Plus Linear Fractional Programming Problem based on Taylor Series Approximation Surapati Pramanik

Partha Pratim Dey

Bibhas C. Giri

Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.Narayanpur, District – North 24 Parganas, Pin Code743126, West Bengal, India

Patipukur Pallisree Vidyapith, 1, Pallisree Colony, Patipukur, Kolkata-700048, West Bengal, India

Department of Mathematics, Jadavpur University, Kolkata – 700032, West Bengal, India

ABSTRACT This paper deals with fuzzy goal programming approach to multi-objective linear plus linear fractional programming problem based on Taylor series approximation. In the model formulation of the problem, we first construct the membership functions by determining individual optimal solutions of the objective functions subject to the system constraints. The membership functions are then transformed into equivalent linear membership functions by 1st order Taylor series approximation. Then fuzzy goal programming models are formulated in order to solve the problem by minimizing negative deviational variables. Euclidean distance function is then used to obtain compromise optimal solution. To demonstrate the efficiency and feasibility of the proposed approach, two numerical examples are solved and compared with existing methods in the literature.

General Terms Multi-objective linear plus linear fractional programming.

Keywords Fuzzy programming, Fuzzy goal programming, Linear Fractional programming, Multi-objective linear plus linear fractional programming, Taylor series.

1. INTRODUCTION Multi-objective linear fractional programming (MOLFP) technique is a very useful decision-making tool for modeling problems with more than one objective such as profit/cost, actual cost/standard cost, debt/equity, inventory/sales, etc. subject to the system constraints. MOLFP is applied to different real-life problems such as agricultural planning problem, production planning problem, inventory problem & other problems. Kornbluth and Steuer [1] introduced goalprogramming technique for solving multi-objective linear fractional programming problem (MOLFPP) based on variable change method. In 1984, Luhandjula [2] studied MOLFPP based on fuzzy set theory by using concept of linguistic variables due to Zadeh [3-5]. Dutta et al. [6] modified Luhandjula’s linguistic approach by using fuzzy set theoretic approach for solving MOLFPP. Minasian and Pop [7] slightly modified the theorems formulated by Dutta et al. [6] regarding MOLFPP and presented the correct proof for obtaining the efficient solution. Chakraborty and Gupta [8] studied MOLFPP by variable transformation method. With the help of transformation, they

transform the original MOLFPP into an equivalent multiobjective linear programming problem (MOLPP). They established equivalency of MOLFPP and MOLPP. Then they solve the resulting MOLPP by using min operator due to Zimmermann [9]. Sadjadi et al. [10] discussed inventory problems with multiple fractional objectives by using the concept of Chakraborty and Gupta [7]. Pal et al. [11] discussed fuzzy goal programming (FGP) procedure for solving fuzzy MOLFPP by using variable change method of Kornbluth and Steuer [1]. Guzel and Shivari [12] studied MOLFPP based on Taylor series approach in crisp environment. Toksarı [13] discussed the use of Taylor series approach for solving fuzzy MOLFPPs. Saad et al. [14] developed an algorithm for obtaining integer solution for MOLFPP involving fuzzy parameters. Recently, Pramanik and Dey [15] studied FGP approach to MOLFPP based on 1st order Taylor polynomial series. Teterev [16] derived optimality criteria for linear plus linear fractional programming problem (LPLFPP) at first. Chadda [17] and Hirche [18] have proposed different approaches for solving LPLFPP. Singh et al. [19] presented multi-parametric sensitivity analysis to LPLFPPs. Mangal and Sangeeta [20] studied alternative approach for solving LPLFPP based on branch and bound method. Kheirfan [21] investigated sensitivity analysis to LPLFPP when the right hand side vector and the coefficients of the objective function are allowed to vary. Sharma and Kumar [22] solved LPLFPP subject to two sided linear inequality constraints. They transformed the problem into fractional programming problem by parametric approach and obtained the solution of the problem by using programming theorems. Jain et al. [23] presented a method for solving multi-objective linear plus linear fractional programming problem (MOLPLFPP) in which non-differentiable term appeared in constraints. They transformed MOLPLFPP into multi-objective fractional programming problem by using suitable transformation and by using programming theorems. Jain and Lachhwani [24] studied MOLPLFPP under fuzzy rule constraints in which the fractional relationship between the decision variables and the objective functions is not completely known. Jain and Lachhwani [25] also developed an algorithm for solving MOLPLFPP when some constraints are homogeneous in nature. They formulated a transformation matrix that transformed the given MOLPLFPP into another MOLPLFPP with fewer constraints by using homogeneous constraints and they proved that the solution of the original 61

International Journal of Computer Applications (0975 – 8887) Volume 32– No.8, October 2011 MOLPLFPP could be obtained from the transformed problem. Singh et al. [26] proposed fuzzy method for MOLPLFPP by transforming the problem into multi-objective linear programming problem using 1st order Taylor series. Then the problem is solved by reducing MOLPLFPP into single objective programming problem by assigning equal weight. Singh et al. [27] also studied MOLPLFPP by extending the concept of Pal et al. [11] and by using the method of approximation due to Frank and Wolfe [28]. In this paper, we have developed an alternative simple approach for solving MOLPLFPP. MOLPLFPP consisting of more than one objection functions that appear as a sum of linear function and linear fractional function and the system constraints are linear functions. In this study, we have transformed MOLPLFPP into an equivalent linear multi-objective programming problem by 1st order Taylor series approximation. Then FGP technique proposed by Pramanik and Dey [29-31] is used to solve the problem. To show the effectiveness of the proposed approach, we solve two MOLPLFPPs studied by P. Singh et al. [26, 27] and compare the solutions. Our main results in the paper are as follows: (i) alternative FGP models are formulated for solving MOLPLFPP. (ii) FGP approach due to Pramanik and Dey [31] is used to solve to the transformed MOLPLFPP by minimizing negative deviational variables. ii) Better optimal compromise solution is obtained by comparing distance function E2. Rest of the paper is organized as follows. Section 2 presents the formulation of MOLPLFPP. Section 3 provides fuzzy programming model of MOLPLFPP. Subsection 3.1 describes linearization of membership functions by 1st order Taylor series approximation. Subsection 3.2 presents FGP formulation to MOLPLFPP. In section 4, the selection of compromise optimal solution based on Euclidean distance function is presented. Section 5 provides FGP algorithm to MOLPLFPP. In section 6, we solve two numerical examples that help to understand the proposed FGP approach. Finally, section 7 provides the concluding remarks.

2. FORMULATION OF MOLPLFPP The general formulation of MOLPLFPP can be formulated as: T

T

maximize Zi ( x )  (c i x  d i ) 

pi x  i

(i = 1, 2, …, k)

T

qi x  βi

(2)

N

M N

that

T { qi x  βi

M

, b  R . S is assumed to be

non empty, convex and compact in R

Let,  iB = max imize Zi ( x ) and  iW = min imize Zi ( x ) (i = 1, 2, xS

xS

…, k). Then the fuzzy goals take the form: Zi ( x )  Z iB (i = 1, 2, …, k) ~

The membership function of i-th fuzzy objective goal can be constructed as: 1, if Zi ( x ) ≥ZiB ,  W  Z (x) - Z  i ( x ) =  i B Wi , if ZiW ≤Zi ( x ) ≤ZiB (i = 1, 2, …, k) (3)  Zi - Zi 0, if Zi ( x ) ≤ZiW B

W

Here, Zi and Zi (i = 1, 2, …, k) are respectively the lower limit and upper tolerance limit of i-th fuzzy objective goal. Then the problem (1) reduces to the following problem: maximize  i ( x ) (i = 1, 2, …, k) subject to    N   x  S = x  R | A x    b, x  0       

(4)

We assume that the objective function Zi ( x ) (i = 1, 2, …, k) and all of the partial derivatives of order less than or equal to N + 1 are continuous on the feasible region S. So the membership function  i ( x ) (i = 1, 2, …, k) corresponding to the objective function Zi ( x ) (i = 1, 2, …, k) has the same property in the feasible region S.

3.1 Linearization of membership functions of MOLPLFPP by 1st order Taylor series approximation *

that x i  (x *i1 , x *i 2 ,..., x *iN ) be

the

individual

best

solution of the membership function  i ( x ) (i = 1, 2, …, k)

Here, ci T , pi T , q i  R (i = 1, 2, …, k) and i , βi , d i (i = 1, 2, …, k) are constants and A  R

In the proposed approach, to formulate the fuzzy programming model of MOLPLFPP, the objective function Zi ( x ) (i = 1, 2, …, k) would be transformed into fuzzy goal by means of assigning an aspiration level to each of them.

Suppose

subject to    N   x  S = x  R | A x    b, x  0        T

(1)

3. FUZZY PROGRAMMING FORMULATION OF MOLPLFPP

N

and further we assume

corresponding the objective function Zi ( x ) (i = 1, 2, …, k) subject to the system constraints, N is the total number of variables of the system. We transform the membership function

 i ( x ) (i = 1, 2, …, k) into equivalent linear membership function ˆ i ( x ) (i = 1, 2, …, k) by 1st order Taylor series approximation. The transformed linear membership function can be written as:

| x  S }> 0 (i = 1, 2, …, k).

62


 * *    i ( x )  + ( x 2  x *i 2 )  i ( x )   i ( x i ) + ( x1  x i1 )  *  x  1  at x  x i

The FGP model (II) for solving MOLPLFPP can be formulated as: minimize  =

         i ( x )  + … +(xN- x *iN )   x  i ( x )  *  x  2  at x  x i  N  at x  x 1* ˆ i ( x ) (i = 1, 2, …, k)

= (5)

3.2 FGP formulation of MOLPLFPP The problem (4) reduces to the following problem: maximize ˆ i ( x ) (i = 1, 2, …, k)

(6)

subject to    N   x  S = x  R | A x    b, x  0          The maximum value of a membership function is one, so for the defined membership function in (6), the flexible membership goal with the aspired level one can be formulated as:   ˆ i ( x ) + d i - d i = 1 (i = 1, 2, …, k)

(7) 

Here, d i ( 0) (i = 1, 2, …, k) and d i ( 0) (i = 1, 2, …, k) represent negative and positive deviational variables respectively. Then according to Pramanik and Dey [31], (7) can be written as: -

 ˆ i ( x ) + d i = 1 (i = 1, 2, …, k) The FGP model (I) can be formulated as: minimize  subject to

(8) (9)

 ˆ i ( x ) + d i = 1, (i = 1, 2, …, k)

   N   x  S= x  R | A x    b, x  0 ,          

 ≥ d i , (i = 1, 2, …, k)  0  d i  1,

d i ≥ 0 (i = 1, 2, …, k). Alternatively, the FGP model (I) can be explicitly presented as: minimize  (10)

 * *   *  i ( x )  + (x 2  x i 2 )  i ( x i ) + ( x1  x i1 )  *  x  1  at x  x i

      *    i ( x )  + … + (xN- x iN )  +  x  i ( x )  *  x  2  at x  x i  N  at x  x 1*

d i = 1, (i = 1, 2, …, k)    N   x  S = x  R | A x    b, x  0 ,        

 ≥ d i , (i = 1, 2, …, k)  0  d i  1,

d i ≥

0 (i = 1, 2, …, k).

k   w idi i 1

(11)

subject to  ˆ i ( x ) + d i = 1, (i = 1, 2, …, k)

   N   x  S = x  R | A x    b, x  0 ,       

0  d i  1,

d i ≥ 0 (i = 1, 2, k). Alternatively, the FGP model (II) can be explicitly formulated as: minimize  =

k   w idi i 1

(12)

subject to

 * *   *  i ( x )  + (x 2  x i 2 )  i ( x i ) + ( x1  x i1 )  *  x 1   at x  x i      + … + (xN- x *iN )  x  i ( x )  *  2  at x  x i = 1, (i = 1, 2, …, k)    N   x  S = x  R | A x    b, x  0 ,       

     + d i  x  i ( x )  *  N  at x  x 1

0  d i  1,

d i ≥ 0 (i = 1, 2, …, k). Here, the decision makers can take the normalized weight i.e. k  wi i 1

= 1 with wi = 1/k or any preference weight in the

decision-making situation.

4. SELECTION OF OPTIMAL COMPROMISE SOLUTION The concept of ideal point (utopia point) and the use of distance functions for group decision problem were first studied by Yu [32]. The concept of Yu [32] has been widely used to several multi-objective decision making problem to arrive at a satisfactory decision [33, 34]. Since different FGP models offer different optimal solutions, we use Euclidean distance function to select which FGP model (FGP model (I), FGP model (II)) will provide optimal compromise solution. The Euclidean distance function is defined as follows: E2 =

k 2  [1   i ( x )] i 1

)]1/2

(13)

Here  i ( x ) represents the achieved membership value of i-th fuzzy objective goal. The solution for which E2 is minimal would be the optimal compromise solution.

5. FGP algorithm for MOLPLFPP The proposed FGP algorithm to MOLPLFPP is presented as follows:

63

International Journal of Computer Applications (0975 – 8887) Volume 32– No.8, October 2011 Step 1: Determine the individual best (maximal) and worst (minimal) solutions of each objective function Zi ( x ) (i = 1, 2, …, k) subject to the system constraints. Step 2: Construct the membership function μ i ( x ) (i = 1, 2, …, k) of i-th fuzzy objective goal as given by (3). Step 3: Find the individual best solution of the membership function μ i ( x ) (i = 1, 2, …, k) subject to the system constraints. Step 4: Transform the membership function μ i ( x ) (i = 1, 2, …, k) into equivalent linear membership function ˆ i ( x ) (i = 1, 2, *

…, k) at the best solution point x i  (x *i1 , x *i 2 ,..., x *iN ) by 1st order Taylor series approximation. Step 5: Construct the FGP model (I) and FGP model (II) as represented by (10) & (12). Step 6: Solve the FGP model (I) and FGP model (II).

(9x1  2x 2 )  2.595 (7 x1  3x 2  1) Z ( x )  2.595 = ,  2 (x) = 2 (3.205  2.595) (3.205 2.595) (3x1  8x 2 ) ( x1  1)   6.736 (4x1  5x 2  3) Z3 ( x )  6.736 = 3 (x) = (7.53  6.736) (7.53  6.736) ( x 2  1) 

The membership functions μ1( x ), μ 2 ( x ) and μ 3 ( x ) are maximal at the points (5, 1), (5, 1) and (5.806, 0.355) respectively subject to the system constraints. Then the membership functions are converted into equivalent linear membership functions at the best solution points by 1st order Taylor polynomial series as follows:    1 ( x )  + (x2 – 1) ˆ 1 ( x ) = µ1 (5, 1) + (x1 – 5)   x1  at x  (5,1)

     ,  x 1 ( x )   2  at x  (5,1)

 0.192,

Step 7: Determine E2 for solutions obtained from the FGP model (I) and FGP model (II).

ˆ 1 ( x ) = 1 + (x1 – 5)  (-1.024) + (x2 – 1)

Step 8: Select the solution for which E2 is minimal as compromise optimal solution.

    2 ( x )  + (xˆ 2 ( x ) = µ2 (5, 1) + (x1 – 5) (x2 – 1)   x  1  at x  (5, 1)

Step 9: End.

    2 ( x )  , 1)   x  2  at x  (5,1)

6. NUMERICAL EXAMPLES Example 1: Consider the following MOLPLFPP with three objective functions studied by P. Sinha et al. [26, 27].  (5x1  4x 2 )   Z1 ( x )  ( x1  1)  , (2x1  x 2  5)   (9x1  2x 2 )   , maximize  Z 2 ( x )  ( x 2  1)  (14) ( 7 x1  3x 2  1)    Z3 ( x )  ( x1  1)  (3x1  8x 2 )   (4x1  5x 2  3)   subject to x1 – x2 ≥ 2, 4x1 + 5x2 ≤ 25, x1 + 9x2 ≥ 9, x1 ≥ 5, x1, x2 ≥ 0. B We determine the individual best solutions as Z1 = -7.312 at (5, B B 1); Z 2 = 3.205 at (5, 1); Z3 = 7.53 at (5.806, 0.355) and the

individual worst solutions as Z1W = -8.433 at (5.806, 0.355);

Z 2W = 2.595 at (5.806, 0.355); Z3W = 6.736 at (5, 0.444). Then the fuzzy goals appear in the following form:

Z1 ( x ) ≥ 7.312, Z 2 ( x ) ≥3.205, Z3 ( x ) ≥7.53. ~

~

~

Then the membership functions are formulated as: (5x1  4x 2 ) ( x1  1)   8.433 (2x1  x 2  5) Z1 ( x )  8.433 = , 1 ( x ) = (7.312  8.433) (7.312  8.433)

ˆ 2 ( x ) = 1 + (x1 – 5) ˆ 3 ( x ) =

µ3

 0.024 + (x2 – 1)  1.571,

(5.806,

0.355)

     +  x  3 ( x )   1  at x  (5.806, 0.355)

+ (x2

–

(x1 –

5.806) 0.355)

     ,  x  3 ( x )   2  at x  (5.806, 0.355) ˆ 3 ( x ) = 1 + (x1 – 5.806)

 1.115 + (x2 – 0.355)  0.197.

The FGP model (I) can be presented as: minimize  subject to 1 + (x1 – 5)  (-1.024) + (x2 – 1)

 0.192+

(15)

d1- = 1,

 0.024 + (x2 – 1)  1.571+ d -2 = 1, 1 + (x1 – 5.806)  1.115 + (x2 – 0.355)  0.197 + 1 + (x1 – 5)

d 3- = 1,

x1 – x2 ≥ 2, 4x1 + 5x2 ≤ 25, x1 + 9x2 ≥ 9, x1 ≥ 5, 

 ≥ d i , (i = 1, 2, 3)  0  d i  1,

d i ≥ 0, (i = 1, 2, 3)

64

International Journal of Computer Applications (0975 – 8887) Volume 32– No.8, October 2011 x1, x2 ≥ 0. Then following the above approach, FGP model (15) provides the solution Z1 = -7.807, Z2 = 2.939, Z3 = 7.131 at x1 = 5.352, x2 = 0.718. The resulting membership values are µ1 = 0.559, µ2 = 0.564 and µ3 = 0.497. The FGP model (II) with equal weight can be written as: minimize  = 1/3 ( d1  d 2  d 3 ) subject to 1 + (x1 – 5)  (-1.024) + (x2 – 1)

 0.192+

(16)

d1- = 1,

 0.024 + (x2 – 1)  1.571+ d -2 = 1, 1 + (x1 – 5.806)  1.115 + (x2 – 0.355)  0.197 + 1 + (x1 – 5)

d 3- = 1,

x1 – x2 ≥ 2, 4x1 + 5x2 ≤ 25, x1 + 9x2 ≥ 9, x1 ≥ 5,

( x1  2x 2  5)  5.792 (7 x1  3x 2  1) Z1 ( x )  5.792 = , 1 ( x ) = (0.688 5.792) (0.688  5.792) (2x1  3x 2  5) (2x 2  1)   31 ( x1  1) Z 2 ( x )  31 = ,  2 (x) = (0.272  31) (0.272  31) (5x1  2x 2  19) (3x1  1)   15.9 (5x1  20) Z3 ( x )  15.9 = 3 (x) = (1.45  15.9) (1.45  15.9) ( x1  1) 

The membership functions 1 ( x ) ,  2 ( x ) and  3 ( x ) are maximal at the points (0, 5), (4.5, 0) and (0, 5) respectively subject to the constraints. Then the membership functions  i ( x ) (i = 1, 2, 3) are transformed into equivalent linear membership functions at the best solution points by 1st order Taylor polynomial series as follows:    1 ( x )  + (x2 – 5) ˆ 1 ( x ) = µ1 (0, 5) + (x1 – 0)   x  1  at x  (0, 5)

0  d i  1,

d i ≥ 0, (i = 1, 2, 3) x1, x2 ≥ 0. FGP model (16) provides the solution Z1 = -7.312, Z2 = 3.025, Z3 = 6.821 at x1 = 5, x2 = 1. The membership values are µ1 = 1, µ2 = 1 and µ3 = 0.108. Example 2: Consider the following MOLPLFPP with three objective functions studied by P. Sinha et al. [26, 27].  ( x1  2x 2  5)   Z1 ( x )  ( x1  1)  ,  (7 x1  3x 2  1)   (2x1  3x 2  5)   , maximize  Z 2 ( x )  (2x 2  1)  (17) ( x1  1)    Z3 ( x )  (3x1  1)  (5x1  2x 2  19)   (5x1  20)   subject to x1 ≤ 6, x2 ≤ 6, 2x1 + x2 ≤ 9, -2x1 + x2 ≤ 5, x1 – x2 ≤ 5, x1, x2 ≥ 0. The individual best solution of the objective functions subject to B the system constraints are obtained as Z1 = - 0.688 at (0, 5);

     ,  x 1 ( x )   2  at x  (0, 5) ˆ 1 ( x ) = 1 + (x1 – 0)  (- 0.235) + (x2 – 5)  0.013,

    2 ( x )  + (x2 – ˆ 2 ( x ) = µ2 (4.5, 0) + (x1 – 4.5)   x  1  at x  ( 4.5, 0)     2 ( x )  0)  ,  x  2  at x  ( 4.5, 0) ˆ 2 ( x ) = 1 + (x1 – 4.5)  0.667 + (x2 – 0)  (- 0.163),

    3 ( x )  + (x2 – 5) ˆ 3 ( x ) = µ1 (0, 5) + (x1 – 0)   x  1  at x  (0, 5)      ,  x  3 ( x )   2  at x  (0, 5) ˆ 3 ( x ) = 1 + (x1 – 0)  (- 0.198) + (x2 – 5)  0.007. The FGP model (I) for solving MOLPLFPP can be formulated as: minimize  (18) subject to

B ZB 2 = - 0.272 at (4.5, 0); Z3 = - 1.45 at (0, 5) and the individual

1 + (x1 – 0)  (- 0.235) + (x2 – 5)  0.013 + d1 = 1,

W W worst solutions are obtained as Z1 = - 5.792 at (4.5, 0); Z 2 =

1 + (x1 – 4.5)  0.667 + (x2 – 0)  (- 0.163) + d 2 = 1,

- 31 at (0, 5);

Z3W

= - 15.9 at (4.5, 0).

Then the fuzzy goals appear in the form: Z1 (x) ≥- 0.688, Z2 (x) ≥- 0.272 and Z3 (x) ≥- 1.45. ~

~

~

Then membership functions  i ( x ) (i = 1, 2, 3) corresponding to the objective functions Zi ( x ) are constructed as:

1 + (x1 – 0)  (- 0.198) + (x2 – 5)  0.007 + d 3 = 1, x1 ≤ 6, x2 ≤ 6, 2x1 + x2 ≤ 9, -2x1 + x2 ≤ 5, x1 – x2 ≤ 5,



 ≥ d i , (i = 1, 2, 3)  0  d i  1,

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d i ≥ 0, (i = 1, 2, 3)

x1, x2 ≥ 0.

x1, x2 ≥ 0.

Then following the procedure, FGP model (19) offers the solution Z1 = -6, Z2 = -6, Z3 = -1.95 at x1 = 0, x2 = 0. The corresponding membership values are obtained as µ1 = 0, µ2 = 0.814 and µ3 = 0.965.

Then, based on proposed approach, FGP model (18) provides the solution Z1 = -3.306, Z2 = -7.51, Z3 = -1.92 at x1 = 0, x2 = 0.302.The resulting membership values are obtained as µ1 = 0.487, µ2 = 0.764 and µ3 = 0.967. The FGP model (II) with equal weight can be written as: minimize  = 1/3 ( d1  d 2  d 3 ) subject to

(19)

1 + (x1 – 0)  (- 0.235) + (x2 – 5)  0.013 + d1- = 1,

Note 1: From table1, we observe that the FGP model (I) offers better optimal solution than FGP model (II) based on Euclidean distance function E2. It is to be noted that the proposed model (I) offers better optimal solution than the solution of P. Singh et al. [26, 27] based on E2. Note 2: On comparing the Euclidean distance function from table 2, we see that the proposed FGP model (I) provides better optimal solution than FGP model (II) for numerical example 2. It may also be noted that the proposed model (I) provides better optimal solution than the solution of P. Singh et al. [26, 27] based on E2.

1 + (x1 – 4.5)  0.667 + (x2 – 0)  (- 0.163) + d -2 = 1, 1 + (x1 – 0)  (- 0.198) + (x2 – 5)  0.007 + d 3- = 1, x1 ≤ 6, x2 ≤ 6, 2x1 + x2 ≤ 9, -2x1 + x2 ≤ 5, x1 – x2 ≤ 5,

Note 3: In general, we cannot say which FGP model (FGP model (I), FGP model (II)) provides better optimal solution for a particular problem. Therefore, it is better to solve the problem by both FGP models and then Euclidean distance function E2 is used to identify the better compromise optimal solution.

0  d i  1,

d i ≥ 0, (i = 1, 2, 3)

Note 4: All the solutions of the problem are determined by the software, Lingo 6.0.

Table1. Comparison of optimal solutions of the numerical example 1 based on Euclidean distance function Approach

Solution point

Objective values

Membership values

Euclidean distance

(x1, x2 )

(Z1, Z2, Z3)

(µ1, µ2, µ3)

(E2)

Proposed model (I)

5.352, 0.718

-7.807, 2.939, 7.131

0.559, 0.564, 0.497

0.798

Proposed model (II)

5, 1

-7.312, 3.025, 6.821

1, 1, 0.108

0.892

P. Singh et al. [26]

5, 1

-7.312, 3.025, 6.821

1, 1, 0.108

0.892


5, 1

-7.312, 3.025, 6.821

1, 1, 0.108

0.892

Table2. Comparison of optimal solutions of the numerical example 2 based on Euclidean distance function Approach

Solution point

Objective values

Membership values

Euclidean distance

(x1, x2 )

(Z1, Z2, Z3)

(µ1, µ2, µ3)

(E2)

Proposed model (I)

0, 0.302

-3.306, -7.51, -1.92

0.487, 0.764, 0.967

0.565

Proposed model (II)

0, 0

-6, -6, -1.95

0, 0.814, 0.965

1.058


0, 0

-6, -6, -1.95

0, 0.814, 0.965

1.058


0, 0

-6, -6, -1.95

0, 0.814, 0.965

1.058

7. CONCLUSION Alternative two FGP models are presented in this paper for solving multi-objective linear plus linear fractional programming problem. In the proposed approach, we transform the membership functions into linear membership functions by 1st order Taylor polynomial series and then fuzzy goal programming technique is used to solve MOLPLFPP by minimizing only negative deviational variables. The proposed

approach can be applied to solve bi-level multi-objective as well as multi-level multi-objective linear plus linear fractional programming problem. We further hope that the proposed concept will be helpful in solving real-life problems involving linear plus linear fractional programming problem in agriculture, planning, inventory problems, transportation problem etc.

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8. ACKNWLEDGEMENTS The authors are thankful to the anonymous referees for their valuable comments and helpful suggestions. The third author gratefully acknowledges the financial support provided by Jadavpur University under JU Research Grant (2009-2011).

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