Information Sciences 243 (2013) 95–99

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Information Sciences

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A note on ‘Taylor series approach to fuzzy multiple objectivelinear fractional programming’

0020-0255/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ins.2013.04.034

⇑ Tel.: +381 11 2630170; fax: +381 11 2186105.E-mail address: bgdnpop@mi.sanu.ac.rs

Bogdana Stanojevic ⇑Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001 Belgrade, Serbia

a r t i c l e i n f o

Article history:Received 2 March 2013Received in revised form 17 April 2013Accepted 29 April 2013Available online 4 May 2013

Keywords:Fuzzy mathematical programmingLinear fractional programmingMulti-objective optimization

a b s t r a c t

Toksari, in his paper ‘Taylor series approach to fuzzy multiobjective linear fractional pro-gramming’, published in Information Sciences 178 (2008), proposed a new method to solvemultiple objective linear fractional programming problems and declared that it is moreeffective when compare to previous methods. He constructed fuzzy goals and then usedfirst order Taylor-polynomials to approximate the corresponding linear fractional member-ship functions by linear functions. Aggregating the linear functions Toksari obtained a crisplinear programming problem and claimed that it is ‘equal’ to the fuzzy fractional one. Inthis paper, we indicate the fallacy that arises by using Taylor approximation and proposean improved method that guarantees the efficiency of the solution it provides. Both prac-tical applications from Toksari’s paper are recalled to show that the improvements we sug-gest are effective.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

In this work, the multiple objective linear fractional programming problem (MOLFPP) is considered. The state of the art inthe theory, methods and applications of fractional programming is presented in Stancu-Minasian [6]. When conflictingobjectives exist, the optimal solution for an objective is not necessarily optimal for the other objectives, and hence the notionof the ‘best compromise’ solution, also known as non-dominated or efficient solution was introduced at the end of the 19thcentury. Since then, the efficient solutions were aimed by any solving methodology. An overview of multiple objective prob-lems and solving methods can be found in [2].

Various approaches for dealing with fractional objectives exist. In the case of a single fractional objective optimization,the most common solving methods are based on linearization and/or similarities to linear optimization. When the MOLFPPis concerned, the linearization becomes quite delicate and the specific tools for handling multiple objectives (hierarchizationand/or aggregation of the objectives, goal programming, fuzzy programming) must be used before any linearization. See [1],[3] and [8] for details about significant theoretical and optimization issues that can appear in solving MOLFPPs.

Toksari [9] proposed a new method to solve the MOLFPP. He constructed fuzzy goals and then used first order Taylor-polynomials to approximate the corresponding linear fractional membership functions by linear functions. In Section 2we review Toksari’s approach, discuss some shortcomings and suggest some improvements. Our improved approach to solvethe MOLFPP is introduced in Section 3. Section 4 reconsiders the two practical applications and one numerical example firstused by Toksari to describe his method. Our new approach improves the solution values for both practical application. Thenumerical example is used to describe the new approach. Finally, conclusions are offered in Section 5.

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2. Review of Toksari’s approach

The general form of a multiple objective linear fractional programming problem discussed by Toksari [9] is represented asfollows:

maxx2X

ZðxÞ ¼ ½Z1ðxÞ; Z2ðxÞ; . . . ; ZpðxÞ�; ð1Þ

where X = {x 2 RnjAx 6 b,x P 0} with b 2 Rm, A 2 Rm�n and ZiðxÞ ¼ cixþaidixþbi

¼ NiðxÞDiðxÞ

with ci, di 2 Rn, ai, bi 2 R, for each i = 1, 2, . . . , p.

Introducing an aspiration level gi to each objective of the MOLFPP, the following fuzzy multiple objective linear fractionalprogramming problem (FMOLFPP) is defined

Find x

so as to satisfy ZiðxÞ ~Pgi; i ¼ 1;2; . . . ;p

s:t: x 2 X;

where P�

is a fuzzy relation which indicates that the value of the left-hand-side expression is ‘essentially greater’ than thevalue of the right-hand side. Introducing a lower tolerance limit ti to each objective of MOLFPP, the following membershipfunctions

liðxÞ ¼1; ZiðxÞ P giZiðxÞ�ti

gi�ti; ti < ZiðxÞ < gi;

0; ZiðxÞ 6 ti

8><>: i ¼ 1;2; . . . ;p:

are usually used for describing the fuzzy goals. Detailed information about similarities between goal programming (GP) andfuzzy programming (FP) and a description of how each one can lead to the other can be found in [5]. According to [5], modelsFMOLFPP and MOLFPP are equivalent.

Toksari proposed a linearization of functions li, i = 1, 2, . . . , p by using first order Taylor-polynomials. He constructed thelinear functions

liðxÞ ¼ li x�i� �þXn

j¼1

xj � x�ij� � @li x�i

� �@xj

; i ¼ 1;2; . . . ; p;

where x�i ¼ x�i1; x�i2; . . . ; x�in

� �is such that Zi x�i

� �¼maxx2X ZiðxÞ, i = 1, 2, . . . , p, and used them to describe the fuzzy goals of

FMOLFPP. Since li; i ¼ 1;2; . . . ; p, are linear functions, Toksari called them linear membership functions of the fuzzy problem.Solving the crisp linear programming problem

maxx2X

Xp

i¼1

liðxÞ; ð2Þ

he found a ’satisfactory’ solution to FMOLFPP and presented it as solution to the original MOLFPP.In what follows, we point out some shortcomings and suggest some improvements in Toksari’s methodology [9] for solv-

ing the MOLFPP.

1. A kth order Taylor-polynomial generally gives a relatively good approximation to a k times differentiable function butonly around a given point, and not over the entire domain. Hence, for each i = 1, 2, . . . , p, function li approximates thefunction li around its optimal solution but cannot approximate properly li around the optimal solution to problem(2). In our opinion, there is no way to correct this fallacy without ruining Toksari’s method.

2. Functions li cannot be called membership functions since their range is not necessarily included in the interval [0,1]. Oneway to improve Toksari’s approach at this point is to normalize the functions li such that, over the feasible set X, theybecome positive and less then or equal to 1 whilst keeping linearity and the location of the maximum point.

3. Toksari claimed that FMOLFPP that uses membership functions li is ‘equal to’ the crisp problem (2) obtained by aggre-gating functions li. From the mathematical viewpoint, the two models cannot be equivalent. First, the FMOLFPP is notequivalent to the corresponding fuzzy multiple objective linear programming problem (FMOLPP) that uses functionsli as membership functions (see problem (3) and Fig. 1). Second, solving problem (2) instead of FMOLPP is simply a com-promise method and not a methodology that proposes an equivalent model to be solved instead of the original one. Anessential improvement at this stage is to test whether the optimal solution to problem (2) is efficient in MOLFPP and torestore the efficiency if it is not.

3. The improved approach

Based on Toksari’s method and being aware of the weakness of Taylor’s approximation we describe further an improvedapproach for solving the MOLFPP.

Fig. 1. The feasible sets and the sets of efficient solutions to Problems (3)–(5).

B. Stanojevic / Information Sciences 243 (2013) 95–99 97

Step 1. (see for instance [10]) Determine the marginal solutions to MOLFPP, i.e. x�i such that Zi x�i� �

¼maxx2X ZiðxÞ,i = 1, 2, . . . , p. Set the goals gi, i = 1, 2, . . . , p and the tolerance limits ti, i = 1, 2, . . . , p. Define the fuzzy goals throughtheir membership functions li,i = 1, 2, . . . , p.

Step 2. (see [9]) Approximate the linear fractional membership functions li, i = 1, 2, . . . , p by the linear functions li,i = 1, 2, . . . , p.

Step 3. Normalize the functions li, i = 1, 2, . . . , p and obtain

�liðxÞ ¼liðxÞ � ai

bi � ai; x 2 X; i ¼ 1;2; . . . ;p;

where ai and bi are the minimal and maximal values of liðxÞ over the feasible set X, "i = 1, 2, . . . , p.Step 4. (see for instance [8]) Aggregate the functions �li, i = 1, 2, . . . , p, using the weighted sum method and solve the single

objective crisp problem maxx2XPp

i¼1�liðxÞ.

Step 5. Test whether the optimal solution obtained in Step 4 is efficient solution to the MOLFPP and restore the efficiency ifit is not.

The normalization described in Step 3 is generally recommended in any weighted sum approach in order to adjustdifferent functions to have similar orders of magnitude. In this case it is done in order to turn functions li into actuallymembership functions whilst keeping their linearity and the location of the maximum point in the marginal solution x�i ,i = 1, 2, . . . , p.

Various algorithms that restore the efficiency of a feasible solution can be found in literature. For instance, Caballero andHernandez [1] introduced a simple and reliable test to establish whether a linear fractional goal programming problem hassolutions that verify all goals and, if so, how to find them by solving a linear programming problem. They also formulated anew technique, based on a mini-max philosophy, for restoring efficiency of any feasible solution to the MOLFPP. Stanojevicand Stanojevic [7] developed a procedure that starts from any feasible solution to the MOLFPP and generates an efficientsolution to the MOLFPP. The procedure is iterative and solves a linear programming problem in each step. Larabani and Aou-ni [4] proposed a general approach to determine an efficient solution to the multiple objective programming problem from anon-efficient solution obtained by goal programming, under the assumption that the functions are continuous and the fea-sible set is compact. Therefore, Step 5 can be completed by any of the above mentioned procedures. However, in order toapply the test proposed in [4], Theorem 3.1, to the particular case of the MOLFPP, a sum of linear ratios has to be optimizedover a set of linear constraints. This problem poses significant theoretical and optimization issues (see for instance [3]), dueto its multiple local optima that are not global optima.

For our computational experiments we used the procedure proposed in [7] to restore the efficiency in Step 5.

4. Practical applications and numerical examples

In this section we recall the practical applications and the numerical examples from Toksari’s paper in order to show thatthe improved approach we proposed in the previous section is effective.

In Practical application 1 ([9]), in a production planning context, Toksari formulated the following multiple objective lin-ear fractional programming problem

98 B. Stanojevic / Information Sciences 243 (2013) 95–99

maxx2X

z1ðxÞ ¼12x1 þ 13x2

40x1 þ 55x2 þ 500; z2ðxÞ ¼

12x1 þ 13x2

1:5y1 þ 1:6y2

� �;

subject to 2x1 þ x2 6 250;

5x1 þ 4x2 6 500;

45x1 þ 30x2 6 1500;

0:1x1 þ 0:1x2 6 y1 þ y2;

0:1x1 6 y1;

0:05x2 6 y2;

x1 P y1;

x2 P y2;

x1; x2; y1; y2 P 0:

The aspiration levels for the objective functions were 0.1 and 70, and the tolerance limits were �0.1 and 20.Toksari obtained the solution (0,50,0,5). Since, z1(0,50,0,5) = 0.2 and z2(0,50,0,5) = 81.25, Toksari’s solution satisfied

both goals. Despite that, solution (0,50,0,5) can be improved by running a procedure that restores the efficiency. Usingthe procedure proposed in [7] we obtained the efficient solution (3.504,44.744,2.588,2.2378). The corresponding valuesof the objective functions were 0.2011 and 83.598. Thus, the values of both criteria were improved.

In Practical application 2 ([9]), in a financial planning context, Toksari formulated the following multiple objective linearfractional programming problem

minx2X

z1ðxÞ ¼�x11 � x12

x21 þ x22 þ x23 þ x24; z2ðxÞ ¼

�x21 � x22

x22 þ x24; z3ðxÞ ¼

�50x11 þ x12

� �;

subject to x11 þ x12 ¼ x21 þ x22 þ x23 þ x24;

x11 þ x12 P 300;100 6 x11 6 200;x12 6 250;x21 þ x22 P 200;x21 P 50;50 6 x23 6 100;75 6 x23 6 120;x11; x12; x21; x22; x23; x24 P 0:

The aspiration levels for the objective functions were �1, �1, and �0.5, and the tolerance limits were 1, 2 and 1.2.Toksari obtained the solution x⁄ = (200,250,175,150,50,75). Since, z1(x⁄) = �1, z2(x⁄) = �1.625, and z3(x⁄) = �0.111, Tok-

sari’s solution satisfied first two goals 100% and the third goal 77%. Solution (200,250,175,150,50,75) can be improvedby running a procedure that restores the efficiency. Using the procedure proposed in [7] we obtained the efficient solution(200,125,200,0,50,75). The corresponding values of the objective functions were �1, �4 and �0.1538. Thus, the values of allthree criteria were improved and the satisfaction of the third goal increased to 80%.

Further we recall the first numerical example from Troski’s paper and use it to describe the new approach.

maxx2X

z1ðxÞ ¼x1 � 4�x2 þ 3

; z2ðxÞ ¼�x1 þ 4x2 þ 1

� �; ð3Þ

where

X ¼ fx 2 R2j � x1 þ 3x2 6 0; x1 6 6; x1; x2 P 0g:

Toksari set the aspiration levels for the objective functions to 2 and 4 and the lower tolerance limits to �1 and �2. Thefeasible set and the set of the efficient solutions to Problem (3) are described in Fig. 1 by the triangle [OBD] and the line[OHGD], respectively. The rotational points of the objective functions are E and F, respectively. The corresponding FMOLFPPis

maxx2X

l1ðxÞ ¼x1 � x2 � 13ð�x2 þ 3Þ ;l2ðxÞ ¼

�x1 þ 2x2 þ 66ðx2 þ 1Þ

� �: ð4Þ

The set of efficient solutions to Problem (4) is the same as the set of efficient solutions to Problem (3). According toToksari’s approach, the following multiple objective linear programming problem should be constructed

maxx2X

l1ðxÞ ¼13

x1 þ23

x2 �73; l2ðxÞ ¼ �

16

x1 �23

x2 þ 1� �

: ð5Þ

B. Stanojevic / Information Sciences 243 (2013) 95–99 99

Due to a miscalculation, Toksari worked with l2ðxÞ ¼ �0:14x1 � 0:57x2 þ 1. Anyway, in Fig. 1, the set of efficient solutions toProblem (5) is the line [OBD] and Problems (4) and (5) cannot be considered equivalent.

Further, l1 and l2 are normalized and aggregated. Thus, �l1ðxÞ ¼ 110 ðx1 þ 2x2Þ; �l2ðxÞ ¼ 1

14 ð�x1 � 4x2 þ 14Þ, and using theirweighted sum aggregation with equal weights the following single objective linear programming problem

max PðxÞ ¼ 135ðx1 � 3x2 þ 35Þ

� �

subject to � x1 þ 3x2 6 0x1 6 6;x1; x2 P 0:

is constructed. The optimal solution of this problem is (6,0). This solution is not an efficient solution to the MOLFPP so thestep that restores the efficiency is needed. Applying procedure PsMOLF introduced in [7] we obtain the efficient solution (6,2)to the MOLFPP.

5. Conclusion

In this paper, we described critical flaws in Toksari’s approach to convert the multiple objective linear fractional program-ming problem into a crisp single objective linear programming problem. We provided an improved approach and showed itseffectiveness in solving the practical applications first used by Toksari’s to describe his approach.

Acknowledgments

This research was partially supported by the Ministry of Education and Science, Republic of Serbia, Project numberTR36006. The author wants to express her gratitude to the referees for their valuable suggestions and remarks.

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