Research Article An Efficient Ranking Technique for...

Research ArticleAn Efficient Ranking Technique forIntuitionistic Fuzzy Numbers with Its Application inChance Constrained Bilevel Programming

Animesh Biswas1 and Arnab Kumar De2

1Department of Mathematics, University of Kalyani, Kalyani 741235, India2Department of Mathematics, Government College of Engineering and Textile Technology, Serampore 712201, India

Correspondence should be addressed to Animesh Biswas; [email protected]

Received 22 November 2015; Revised 24 March 2016; Accepted 3 April 2016

Academic Editor: Kemal Kilic

Copyright © 2016 A. Biswas and A. K. De. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The aim of this paper is to develop a new ranking technique for intuitionistic fuzzy numbers using the method of defuzzificationbased on probability density function of the corresponding membership function, as well as the complement of nonmembershipfunction. Using the proposed ranking technique a methodology for solving linear bilevel fuzzy stochastic programming probleminvolving normal intuitionistic fuzzy numbers is developed. In the solution process each objective is solved independently to setthe individual goal value of the objectives of the decision makers and thereby constructing fuzzy membership goal of the objectivesof each decision maker. Finally, a fuzzy goal programming approach is considered to achieve the highest membership degree tothe extent possible of each of the membership goals of the decision makers in the decision making context. Illustrative numericalexamples are provided to demonstrate the applicability of the proposed methodology and the achieved results are compared withexisting techniques.

1. Introduction

The concept of bilevel programming problem (BLPP) wasfirst introduced by Candler and Townsley [1]. The BLPP isconsidered as a class of optimization problems where twodecision makers (DMs) locating at two different hierarchicallevels independently control a set of decision variables payingserious attention to the benefit of the others in a highlyconflicting decision making situation. The upper level DM istermed as leader and the lower level DM as follower. Thereare several applications of BLPP in many real life problemssuch as agriculture, biofuel production, economic systems,finance, engineering, banking, management sciences, andtransportation problem. Several methods were proposed tosolve BLPPs by different researchers [2, 3] in the past. Butthese traditional approaches are unable to provide a satisfac-tory solution if the parameter values involved with a BLPPinevitably contain some uncertain data or linguistic informa-tion. Stochastic programming (SP) and fuzzy programming

(FP) are two powerful techniques to handle such type ofproblems. Using probability theory, Dantzig [4] introducedSP. The SP was developed in various directions like chanceconstrained programming (CCP), recourse programming,multiobjective SP, and so forth. Charnes and Cooper [5]developed the concept of CCP.

Again, from the viewpoint of uncertainty or fuzzinessinvolved in human’s judgments, Zimmermann [6] firstapplied fuzzy set theory [7] in decisionmaking problemswithseveral conflicting objectives. The concept of membershipfunctions in BLPPs was introduced by Lai and Hwang [8].Lai’s solution concept was then extended by Shih et al. [9]and a supervised search procedure with the use of max-min operator of Bellman and Zadeh [10] was proposed. Thebasic concept of this procedure is that the follower optimizeshis/her objective function, taking into consideration leader’sgoal. Recently, Lodwick and Kacprizyk [11] developed amethodology for solving decision making problems underfuzziness. The main difficulty of FP approach is that the

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016, Article ID 6475403, 12 pageshttp://dx.doi.org/10.1155/2016/6475403

2 Advances in Fuzzy Systems

objectives of the DMs are conflicting. So there is possibilityof rejecting the solution again and again by the DMs and thesolution process is continued by redefining the membershipfunctions repeatedly until a satisfactory solution is obtained.This makes the solution process a very lengthy and tediousone. To remove these difficulties fuzzy goal programming(FGP) [12–14] is used as an efficient tool for making decisionin an imprecisely defined multiobjective decision making(MODM) arena. Baky [15] developed a FGP technique forsolving multiobjective multilevel programming problem.

Also it is observed that the fuzzy sets (FSs) are not alwayscapable of dealing with lack of knowledge with respect todegrees of membership. Realizing the fact Atanassov [16–18] introduced the concept of intuitionistic FSs (IFSs) byimplementing a nonmembership degree which can handlethe drawback of FSs and express more abundant and flexibleinformation than the FSs. In recent years, there is a growinginterest in the study of decision making problems withintuitionistic fuzzy numbers (IFNs) [19–21] andwith interval-valued intuitionistic fuzzy information [22–25].

The ranking of IFNs [26, 27] plays an important rolein dealing with IFNs as ranking of fuzzy numbers (FNs).Grzegoraewski [28] suggested some methods for measuringdistances between IFNs and interval-valued FNs, based onHausdorff metric.Themethodology for solving intuitionisticfuzzy linear programming problems with triangular IFNs(TIFNs) was developed byDubey andMehra [29] by convert-ing themodel into crisp linear programming problem. Li [30]developed a ratio ranking method for the TIFNs. Nehi [31]put forward a new ordering method of IFNs in which twocharacteristic values for IFNs are defined by the integral ofthe inverse fuzzymembership andnonmembership functionsmultiplied by the grade with powered parameters. Recently,numerous ranking methods for IFNs have been proposed inliterature to rank IFNs [32–34]. Although many defuzzifica-tion methods have already been proposed so far, no methodgives a right effective defuzzification output. Most of theexisting defuzzificationmethods tried tomake the estimationof IFNs in an objective way. This paper proposes a methodusing the concept probability density function of IFNs andMellin’s transform [35, 36] to find the ranking of normal IFNs.This ranking method removes the ambiguous outcomes anddistinguishes the alternatives clearly.

In fuzzy BLPP [37] it is sometimes realized that the con-cept of membership function does not provide satisfactorysolutions in a highly conflicting decisionmaking situation. Inthis context IFNs can be used to capture both themembershipand nonmembership degrees of uncertainties of both theDMs. Also there are some real world situations, where ran-domness and fuzziness occur simultaneously. The decisionmaking problem having such type of ambiguous informationis known as fuzzy stochastic programming problem. Manyresearchers [38–41] derived different methods to solve suchtype of decision making problems. But FGP approach forsolving fuzzy stochastic linear BLPPwith IFNs is yet to appearin the literature.

In the present study FGP process is adopted for solvingbilevel intuitionistic FP problems where the parameters areexpressed in terms of normal IFNs. A ranking technique

1

0 a b c

Figure 1: TFN.

for normal IFNs is first proposed and then using the pro-posed technique the model is converted into a deterministicproblem. The individual optimal value of each objective isfound in isolation to construct the fuzzy membership goalsof each of the objectives. Finally, FGP model is developedfor the achievement of highest degree of each of the definedmembership goals to the extent possible byminimizing groupregrets in the decision making context. To explore the poten-tiality of the proposed approach, two illustrative examplesare considered and solved and the achieved solutions arecomparedwith the predefined technique developed byDubeyand Mehra [29].

2. Preliminaries

In this section some basic concepts on FNs, triangular FNs(TFNs), IFNs, and triangular IFNs (TIFNs) are discussed.

2.1. FN [42]. A fuzzy set 𝐴 defined on the set of real numbers,R, is said to be an FN if its membership function 𝜇

𝐴

(𝑥)

satisfies the following characteristics:

(i) 𝜇

𝐴

: R→ [0, 1] is continuous.(ii) 𝜇

𝐴

(𝑥) = 0 for all 𝑥 ∈ (−∞, 𝑎] ∪ [𝑑,∞).(iii) 𝜇

𝐴

(𝑥) is strictly increasing on [𝑎, 𝑏] and strictlydecreasing on [𝑐, 𝑑].

(iv) 𝜇

𝐴

(𝑥) = 1 for all 𝑥 ∈ [𝑏, 𝑐], where 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑.

2.2. TFN [43]. An FN

𝐴 = (𝑎, 𝑏, 𝑐) is said to be TFN if itsmembership function 𝜇

𝐴

(𝑥) is given by

𝜇

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

0 if 𝑥 < 𝑎 or 𝑥 > 𝑐𝑥 − 𝑎

𝑏 − 𝑎

if 𝑎 ≤ 𝑥 ≤ 𝑏𝑐 − 𝑥

𝑐 − 𝑏

if 𝑏 ≤ 𝑥 ≤ 𝑐.

(1)

The TFN can be expressed in the form of Figure 1.

2.3. IFN [44]. An IFN

𝐴 is

(i) an IFS defined on R;(ii) normal; that is, there exists 𝑥 ∈ R such that 𝜇

𝐴

(𝑥) = 1

and ]

𝐴

(𝑥) = 0;(iii) convex for the membership function 𝜇

𝐴

(𝑥); that is,𝜇

𝐴

(𝜆𝑥

1

+ (1 − 𝜆)𝑥

2

) ≥ min{𝜇

𝐴

(𝑥

1

), 𝜇

𝐴

(𝑥

2

)} for all𝑥

1

, 𝑥

2

∈ R, 0 ≤ 𝜆 ≤ 1;

Advances in Fuzzy Systems 3

1

0 ab c

ua

wa

Figure 2: TIFN.

(iv) concave for the nonmembership function ]

𝐴

(𝑥); thatis, ]

𝐴

(𝜆𝑥

1

+ (1 − 𝜆)𝑥

2

) ≤ max{]

𝐴

(𝑥

1

), ]

𝐴

(𝑥

2

)} for all𝑥

1

, 𝑥

2

∈ R, 0 ≤ 𝜆 ≤ 1.

2.4. TIFN [30]. Let 𝐴 = {(𝑏, 𝑎, 𝑐), 𝑤

𝑎

, 𝑢

𝑎

} be a TIFN. Thenthe membership and the nonmembership function of 𝐴 areexpressed as

𝜇

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

(𝑥 − 𝑏)

(𝑎 − 𝑏)

𝑤

𝑎

if 𝑏 ≤ 𝑥 ≤ 𝑎

(𝑐 − 𝑥)

(𝑐 − 𝑎)

𝑤

𝑎

if 𝑎 ≤ 𝑥 ≤ 𝑐

0 if 𝑥 < 𝑏 or 𝑥 > 𝑐,

]

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

(𝑎 − 𝑥)

(𝑎 − 𝑑)

𝑢

𝑎

if 𝑏 ≤ 𝑥 ≤ 𝑎

(𝑥 − 𝑎)

(𝑒 − 𝑎)

𝑢

𝑎

if 𝑎 ≤ 𝑥 ≤ 𝑐

1 if 𝑥 < 𝑏 or 𝑥 > 𝑐.

(2)

That TIFN is expressed by Figure 2.A TIFN is called normal if 𝑤

𝑎

= 1 and 𝑢𝑎

= 0 for at leastone 𝑥.

3. Proposed Ranking Technique forNormal TIFN

Let 𝐴 = {(𝑎

2

, 𝑎, 𝑎

3

), (𝑎

1

, 𝑎, 𝑎

4

)} be a normal TIFN. Thenthe membership and the nonmembership functions of 𝐴 areexpressed as

𝜇

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

(𝑥 − 𝑎

2

)

(𝑎 − 𝑎

2

)

if 𝑎2 ≤ 𝑥 ≤ 𝑎

(𝑎

3

− 𝑥)

(𝑎

3

− 𝑎)

if 𝑎 ≤ 𝑥 ≤ 𝑎3

0 if 𝑥 < 𝑎2 or 𝑥 > 𝑎3,

]

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

(𝑎 − 𝑥)

(𝑎 − 𝑎

1

)

if 𝑎1 ≤ 𝑥 ≤ 𝑎

(𝑥 − 𝑎)

(𝑎

4

− 𝑎)

if 𝑎 ≤ 𝑥 ≤ 𝑎4

1 if 𝑥 < 𝑎1 or 𝑥 > 𝑎4,

(3)

which is presented in Figure 3.

1

0 aa1

a2

a3

a4

Figure 3: Triangular normal IFN.

It is to be noted here that if the nonmembership valuedecreases, acceptance of the IFN increases. As a consequence,if the value of complement of the nonmembership functionincreases, the acceptance possibility of the IFN increases.Considering those concepts, membership functions as wellas the complement of the nonmembership functions aretaken into account to rank the IFNs. The complement of thenonmembership function of a normal TIFN

𝐴 is given by

1 − ]

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

(𝑥 − 𝑎

1

)

(𝑎 − 𝑎

1

)

if 𝑎1 ≤ 𝑥 ≤ 𝑎

(𝑎

4

− 𝑥)

(𝑎

4

− 𝑎)

if 𝑎 ≤ 𝑥 ≤ 𝑎4

0 if 𝑥 < 𝑎1 or 𝑥 > 𝑎4.

(4)

Let us define two probability density functions 𝑓1

(𝑥) and𝑓

2

(𝑥) corresponding to the membership function 𝜇

𝐴

(𝑥) andthe complement of the nonmembership function 1 − ]

𝐴

(𝑥),respectively.

Let 𝑓1

(𝑥) = 𝑘

1

𝜇

𝐴

(𝑥); then ∫∞−∞

𝑓

1

(𝑥)𝑑𝑥 = 1 implies𝑘

1

∫

∞

−∞

𝜇

𝐴

(𝑥)𝑑𝑥 = 1; that is,

𝑘

1

[∫

𝑎

𝑎

2

(𝑥 − 𝑎

2

)

(𝑎 − 𝑎

2

)

𝑑𝑥 + ∫

𝑎

3

𝑎

(𝑎

3

− 𝑥)

(𝑎

3

− 𝑎)

𝑑𝑥] = 1;(5)

that is,

𝑘

1

=

2

(𝑎

3

− 𝑎

2

)

. (6)

Thus

𝑓

1

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

2 (𝑥 − 𝑎

2

)

(𝑎

3

− 𝑎

2

) (𝑎 − 𝑎

2

)

if 𝑎2 ≤ 𝑥 ≤ 𝑎

2 (𝑎

3

− 𝑥)

(𝑎

3

− 𝑎

2

) (𝑎

3

− 𝑎)

if 𝑎 ≤ 𝑥 ≤ 𝑎3

0 if 𝑥 < 𝑎2 or 𝑥 > 𝑎3.

(7)


Similarly, defining 𝑓2

(𝑥) = 𝑘

2

(1 − ]

𝐴

(𝑥)), the constant 𝑘2

andthe function 𝑓

2

(𝑥) are obtained as

𝑘

2

=

2

(𝑎

4

− 𝑎

1

)

,

𝑓

2

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

2 (𝑥 − 𝑎

1

)

(𝑎

4

− 𝑎

1

) (𝑎 − 𝑎

1

)

if 𝑎1 ≤ 𝑥 ≤ 𝑎

2 (𝑎

4

− 𝑥)

(𝑎

4

− 𝑎

1

) (𝑎

4

− 𝑎)

if 𝑎 ≤ 𝑥 ≤ 𝑎4

0 if 𝑥 < 𝑎1 or 𝑥 > 𝑎4.

(8)

Let𝑓(𝑥) be the probability density function corresponding tothe normal TIFN

𝐴 which is defined as

𝑓 (𝑥) = 𝜆𝑓

1

(𝑥) + (1 − 𝜆) 𝑓

2

(𝑥) , (0 ≤ 𝜆 ≤ 1) . (9)

A technique for defuzzification of fuzzy number usingMellin’s transform was developed by R. Saneifard and R.Saneifard [36]. There are lots of successful applications [45–49] of this technique. From that viewpoint, Mellin’s transfor-mation has been applied to find the defuzzified value of IFNs.Mellin’s transform is given by

𝑀

𝑋

(𝑡) = ∫

∞

0

𝑥

𝑡−1

𝑓 (𝑥) 𝑑𝑥. (10)

Here 𝑋 denotes the random variable corresponding to thenormal TIFN

𝐴. Thus

𝑀

𝑋

(𝑡) = ∫

∞

0

𝑥

𝑡−1

(𝜆𝑓

1

(𝑥) + (1 − 𝜆) 𝑓

2

(𝑥)) 𝑑𝑥

= 2𝜆[∫

𝑎

𝑎

2

𝑥

𝑡−1

(𝑥 − 𝑎

2

)

(𝑎 − 𝑎

2

) (𝑎

3

− 𝑎

2

)

𝑑𝑥

+ ∫

𝑎

3

𝑎

𝑥

𝑡−1

(𝑎

3

− 𝑥)

(𝑎

3

− 𝑎) (𝑎

3

− 𝑎

2

)

𝑑𝑥] + 2 (1 − 𝜆)

⋅ [∫

𝑎

𝑎

1

𝑥

𝑡−1

(𝑥 − 𝑎

1

)

(𝑎 − 𝑎

1

) (𝑎

4

− 𝑎

1

)

𝑑𝑥

+ ∫

𝑎

4

𝑎

𝑥

𝑡−1

(𝑎

4

− 𝑥)

(𝑎

4

− 𝑎) (𝑎

4

− 𝑎

1

)

𝑑𝑥]

=

2𝜆

(𝑎 − 𝑎

2

) (𝑎

3

− 𝑎

2

)

[

[

[

(𝑎

𝑡+1

− (𝑎

2

)

𝑡+1

)

(𝑡 + 1)

−

𝑎

2

(𝑎

𝑡

− (𝑎

2

)

𝑡

)

𝑡

]

]

]

+

2𝜆

(𝑎

3

− 𝑎) (𝑎

3

− 𝑎

2

)

[

[

[

𝑎

3

((𝑎

3

)

𝑡

− 𝑎

𝑡

)

𝑡

−

((𝑎

3

)

𝑡+1

− 𝑎

𝑡+1

)

(𝑡 + 1)

]

]

]

+

2 (1 − 𝜆)

(𝑎 − 𝑎

1

) (𝑎

4

− 𝑎

1

)

[

[

[

(𝑎

𝑡+1

− (𝑎

1

)

𝑡+1

)

(𝑡 + 1)

−

𝑎

1

(𝑎

𝑡

− (𝑎

1

)

𝑡

)

𝑡

]

]

]

+

2 (1 − 𝜆)

(𝑎

4

− 𝑎) (𝑎

4

− 𝑎

1

)

[

[

[

𝑎

4

((𝑎

4

)

𝑡

− 𝑎

𝑡

)

𝑡

−

((𝑎

4

)

𝑡+1

− 𝑎

𝑡+1

)

(𝑡 + 1)

]

]

]

.

(11)

For 𝑡 = 2, Mellin’s transform converted to the definition ofexpectation of a random variable. Since the target is to findthe expected or defuzzified value of TIFNs, 𝑡 = 2 has beenconsidered.

Thus the crisp equivalent value of the normal TIFN isfound as

𝑉(

𝐴) = 𝑀

𝑋

(2) =

2𝜆

(𝑎

3

− 𝑎

2

)

[

[

[

((𝑎

2

)

2

+ 𝑎𝑎

2

+ (𝑎)

2

)

3

−

𝑎

2

(𝑎

2

+ 𝑎)

2

]

]

]

+

2𝜆

(𝑎

3

− 𝑎

2

)

[

[

[

𝑎

3

(𝑎

3

+ 𝑎)

2

−

((𝑎

3

)

2

+ 𝑎

3

𝑎 + (𝑎)

2

)

3

]

]

]


+

2 (1 − 𝜆)

(𝑎

4

− 𝑎

1

)

[

[

[

((𝑎)

2

+ 𝑎𝑎

1

+ (𝑎

1

)

2

)

3

−

𝑎

1

(𝑎 + 𝑎

1

)

2

]

]

]

+

2 (1 − 𝜆)

(𝑎

4

− 𝑎

1

)

[

[

[

𝑎

4

(𝑎

4

+ 𝑎)

2

−

((𝑎

4

)

2

+ 𝑎

4

𝑎 + (𝑎)

2

)

3

]

]

]

,

𝑉 (

𝐴) =

𝜆 (𝑎

2

+ 𝑎

3

− 𝑎

1

− 𝑎

4

) + (𝑎

1

+ 𝑎

4

+ 𝑎)

3

.

(12)

For any two normal TIFNs 𝐴 and

𝐵 if 𝑉(𝐴) and 𝑉(

𝐵)

represent their equivalent crisp values, then,

(1) 𝑉(𝐴) < 𝑉(𝐵) if and only if 𝐴 ≲ 𝐵;(2) 𝑉(𝐴) > 𝑉(𝐵) if and only if 𝐴 ≳ 𝐵;(3) 𝑉(𝐴) = 𝑉(𝐵) if and only if 𝐴 ≅ 𝐵.

It is worthy to mention here that Wang and Kerre [50] pro-posed seven axioms as the reasonable properties of orderingfuzzy quantities for an ordering approach. The properties ofIFNs depend on two FSs, namely, membership function andnonmembership function; and it can easily be shown that allthe seven axioms have been satisfied by the proposed rankingtechnique as described above.

The derived process of ranking of normal TIFNs is to besummarized through the following algorithm.

3.1. Solution Algorithm

Step 1. Write the membership function 𝜇

𝐴

(𝑥) and nonmem-bership function ]

𝐴

(𝑥) of the normal TIFN

𝐴 (refer to (3)).

Step 2. Calculate the complement 1 − ]

𝐴

(𝑥) of the nonmem-bership function ]

𝐴

(𝑥) of the normal TIFN


Step 3. Construct the probability density function 𝑓1

(𝑥) forthe membership function 𝜇

𝐴

(𝑥) and 𝑓2

(𝑥) for the comple-ment of the nonmembership function ]

𝐴

(𝑥) of the normalTIFN

𝐴 (refer to (7) and (8)).

Step 4. Take the convex combination of 𝑓1

(𝑥) and 𝑓2

(𝑥) toform the probability density function of the normal TIFN

𝐴

(refer to (9)).

Step 5. Use Mellin’s transform to calculate the crisp value ofthe normal TIFN


Step 6. Stop.

3.2. Illustrative Example. The following three TIFNs are con-sidered to find their equivalent crisp value by the proposedtechnique.

Let 𝐴 = {(1.5, 2, 2.5), (1, 2, 3)}, 𝐵 = {(0.6, 1, 1.4), (0.2, 1,

1.8)}, and 𝐶 = {(2.2, 3, 3.8), (1.8, 3, 4.2)} be three TIFNs.The membership function of the TIFN

𝐴 is given by

𝜇

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

{

{

𝑥 − 1.5

0.5

if 1.5 ≤ 𝑥 ≤ 2

2.5 − 𝑥

0.5

if 2 ≤ 𝑥 ≤ 2.5

0 otherwise.

(13)

The nonmembership function of 𝐴 is presented as

]

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

2 − 𝑥 if 1 ≤ 𝑥 ≤ 2

𝑥 − 2 if 2 ≤ 𝑥 ≤ 3

1 otherwise.

(14)

The complement of the non-membership function ]

𝐴

(𝑥) iscalculated as

1 − ]

𝐴

(𝑥) =

{

{

{

{

{

{

{

{

{

𝑥 − 1 if 1 ≤ 𝑥 ≤ 2

3 − 𝑥 if 2 ≤ 𝑥 ≤ 3

0 otherwise.

(15)

Let 𝑓1

(𝑥) and 𝑓2

(𝑥) be the respective density functions of themembership function and the complement of nonmember-ship function which are calculated as defined in the proposedmethodology. Then the density function of the TIFN

𝐴 isconsidered as (𝑥) = 𝜆𝑓

1

(𝑥) + (1 − 𝜆)𝑓

2

(𝑥), (0 ≤ 𝜆 ≤ 1).Using Mellin’s transform the crisp value of TIFN

𝐴 iscalculated as 𝑉(𝐴) = 2. Similarly the crisp value of TIFNs

𝐵 and 𝐶 is evaluated as 𝑉(𝐵) = 1, 𝑉(𝐶) = 3, (0 ≤ 𝜆 ≤ 1).Thus the ordering in the proposed ranking technique is

𝐵 ≲

𝐴 ≲

𝐶.

3.3. Comparison of the Proposed Ranking Method with thePredefined Methods. The proposed ranking technique iscompared with other predefined rankingmethods [29–34] toexplore the consistency of the proposed ranking methodol-ogy through Table 1.

FromTable 1 it is evident that the ordering obtained usingthe proposed methodology is identical in comparison withother techniques.This indicates that the proposedmethodol-ogy is consistent with the other predefined techniques whichcan be considered as an alternative technique for rankingIFNs. However, the superiority of the proposed approachwould be reflected in the context of solving BLPPs withTIFNs.

Using the proposed ranking method of normal TIFN afuzzy stochastic linear bilevel programming (FSLBLP) modelis developed and solved in the following section.


Table 1: Comparison between ranking techniques.

Methodology IFN Ranking value Result

Dubey and Mehra [29]

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} 𝑅 (

𝐴, 𝜆) = 1.33 + 0.34𝜆

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} 𝑅 (

𝐵, 𝜆) = 0.47 + 0.26𝜆

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} 𝑅 (

𝐶, 𝜆) = 2.2 + 0.27𝜆

Li [30]

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} 𝑅 (

𝐴, 𝜆) =

2

1.67 − 0.34𝜆

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} 𝑅 (

𝐵, 𝜆) =

1

1.53 − 0.26𝜆

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} 𝑅 (

𝐶, 𝜆) =

3

1.8 − 0.27𝜆

Nehi [31]

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} 𝐶

𝑘

𝜇

(

𝐴) = 2

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} 𝐶

𝑘

𝜇

(

𝐵) = 1

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} 𝐶

𝑘

𝜇

(

𝐶) = 3

Wan [32]

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} 𝑉𝐶 (

𝐴

𝛼

) = 0.10, 𝑉𝐶(𝐴𝛽

) = 0.35

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} 𝑉𝐶 (

𝐵

𝛼

) = 0.16

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} 𝑉𝐶 (

𝐶

𝛼

) = 0.10, 𝑉𝐶(𝐶𝛽

) = 0.28

Wan and Yi [34]

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} 𝑚

𝜇

(

𝐴) = 2

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} 𝑚

𝜇

(

𝐵) = 1

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} 𝑚

𝜇

(

𝐶) = 3

Wan et al. [33]

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} ℎ (

𝐴) = 2

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} ℎ (

𝐵) = 1

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} ℎ (

𝐶) = 3

Proposed method

𝐴 = {(1.5, 2, 2.5) , (1, 2, 3)} 𝑉 (

𝐴) = 2

𝐵 ≲

𝐴 ≲

𝐶

𝐵 = {(0.6, 1, 1.4) , (0.2, 1, 1.8)} 𝑉 (

𝐵) = 1

𝐶 = {(2.2, 3, 3.8) , (1.8, 3, 4.2)} 𝑉 (

𝐶) = 3

4. FSLBLP Model Formulation

An LFSBLP problem with normal TIFNs as coefficients ispresented as follows:

Find 𝑋(𝑥

1

, 𝑥

2

, . . . , 𝑥

𝑛

) so as to

max𝑋

1

𝑍

1

≅

𝑛

∑

𝑗=1

��

1𝑗

𝑥

𝑗

,

where for given 𝑋1

; 𝑋

2

solves

max𝑋

2

𝑍

2

≅

𝑛

∑

𝑗=1

��

2𝑗

𝑥

𝑗

Subject to 𝑟(

𝑛

∑

𝑗=1

��

𝑖𝑗

𝑥

𝑗

≲

𝑏

𝑖

) ≥ 1 − 𝛾

𝑖

;

𝑖 = 1, 2, . . . , 𝑚

𝑥

𝑗

≥ 0; 𝑗 = 1, 2, . . . , 𝑛,

(16)

where ��𝑘𝑗

= {(𝑐

2

𝑘𝑗

, 𝑐

𝑘𝑗

, 𝑐

3

𝑘𝑗

), (𝑐

1

𝑘𝑗

, 𝑐

𝑘𝑗

, 𝑐

4

𝑘𝑗

)} and ��𝑖𝑗

= {(𝑎

2

𝑖𝑗

, 𝑎

𝑖𝑗

, 𝑎

3

𝑖𝑗

),

(𝑎

1

𝑖𝑗

, 𝑎

𝑖𝑗

, 𝑎

4

𝑖𝑗

)} (𝑘 = 1, 2; 𝑗 = 1, 2, . . . , 𝑛; 𝑖 = 1, 2, . . . , 𝑚)are normal TIFNs and 𝛾

𝑖

is any real number that lies

within [0, 1]. Here 𝑏𝑖

(𝑖 = 1, 2, . . . , 𝑚) denotes fuzzy ran-dom variable following normal distribution whose mean𝑚

𝑏

𝑖

= {(𝑚

𝑏

𝑖

2

, 𝑚

𝑏

𝑖

, 𝑚

𝑏

𝑖

3

), (𝑚

𝑏

𝑖

1

, 𝑚

𝑏

𝑖

, 𝑚

𝑏

𝑖

4

)} and standard devi-ation 𝜎

𝑏

𝑖

= {(𝜎

𝑏

𝑖

2

, 𝜎

𝑏

𝑖

, 𝜎

𝑏

𝑖

3

), (𝜎

𝑏

𝑖

1

, 𝜎

𝑏

𝑖

, 𝜎

𝑏

𝑖

4

)} are normal TIFNsand ≅, ≲, and ≳ denote fuzzily equal, less than or equal, andgreater than or equal, respectively. The decision vector 𝑋

1

=

(𝑥

11

, 𝑥

12

, . . . , 𝑥

1𝑛

1

) is controlled by the upper level DM and𝑋

2

= (𝑥

21

, 𝑥

22

, . . . , 𝑥

2𝑛

2

) is controlled by the lower level DM;𝑋 = 𝑋

1

∪ 𝑋

2

and 𝑛1

+ 𝑛

2

= 𝑛.

4.1. FP Model Formulation [5]. Applying CCP technique, theprobabilistic constraints Pr(∑𝑛

𝑗=1

��

𝑖𝑗

𝑥

𝑗

≲

𝑏

𝑖

) ≥ 1 − 𝛾

𝑖

(𝑖 =1, 2, . . . , 𝑚) are modified as follows:

Pr(𝑛

∑

𝑗=1

��

𝑖𝑗

𝑥

𝑗

≲

𝑏

𝑖

) ≥ 1 − 𝛾

𝑖

; that is,

Pr(𝑏𝑖

≲

𝑛

∑

𝑗=1

��

𝑖𝑗

𝑥

𝑗

) ≤ 𝛾

𝑖

; that is,

Pr(

𝑏

𝑖

− 𝑚

𝑏

𝑖

𝜎

𝑏

𝑖

≲

∑

𝑛

𝑗=1

��

𝑖𝑗

𝑥

𝑗

− 𝑚

𝑏

𝑖

𝜎

𝑏

𝑖

) ≤ 𝛾

𝑖

; that is,


Φ(

∑

𝑛

𝑗=1

��

𝑖𝑗

𝑥

𝑗

− 𝑚

𝑏

𝑖

𝜎

𝑏

𝑖

) ≤ 𝛾

𝑖

or

𝑛

∑

𝑗=1

��

𝑖𝑗

𝑥

𝑗

≤ 𝑚

𝑏

𝑖

+ Φ

−1

(𝛾

𝑖

) 𝜎

𝑏

𝑖

,

(17)

where the function Φ(⋅) represents the cumulative distribu-tion function of the standard normal fuzzy random variate.

Hence the linear fuzzy BLPP is written in the followingform as

Find 𝑋(𝑥

1

, 𝑥

2

, . . . , 𝑥

𝑛

) so as to

max𝑋

1

𝑍

1

≅

𝑛

∑

𝑗=1

��

1𝑗

𝑥

𝑗

,


; 𝑋

2

solves

max𝑋

2

𝑍

2

≅

𝑛

∑

𝑗=1

��

2𝑗

𝑥

𝑗

Subject to𝑛

∑

𝑗=1

��

𝑖𝑗

𝑥

𝑗

≤ 𝑚

𝑏

𝑖

+ Φ

−1

(𝛾

𝑖

) 𝜎

𝑏

𝑖

;

(𝑖 = 1, 2, . . . , 𝑚)

𝑥

𝑗

≥ 0; 𝑗 = 1, 2, . . . , 𝑛.

(18)

4.2. Construction of Linear Bilevel Programming Model UsingProposed Ranking Method. In this subsection the proposedranking function is applied to convert the FP model into adeterministic model. Using the linearity of ranking function,the above model is written as

Find 𝑋(𝑥

1

, 𝑥

2

, . . . , 𝑥

𝑛

) so as to

max𝑋

1

𝑉(

𝑍

1

) =

𝑛

∑

𝑗=1

𝑉(��

1𝑗

) 𝑥

𝑗

,


; 𝑋

2

solves

max𝑋

2

𝑉(

𝑍

2

) =

𝑛

∑

𝑗=1

𝑉(��

2𝑗

) 𝑥

𝑗

Subject to𝑛

∑

𝑗=1

𝑉(��

𝑖𝑗

) 𝑥

𝑗

≤ 𝑉 (𝑚

𝑏

𝑖

) + Φ

−1

(𝛾

𝑖

) 𝑉 (𝜎

𝑏

𝑖

) ;

𝑖 = 1, 2, . . . , 𝑚

𝑥

𝑗

≥ 0; 𝑗 = 1, 2, . . . , 𝑛.

(19)

Using the parameters of TIFNs, model (19) is expressed as

Find 𝑋(𝑥

1

, 𝑥

2

, . . . , 𝑥

𝑛

) so as to

max𝑋

1

𝑉(

𝑍

1

) =

𝑛

∑

𝑗=1

(𝜆 (𝑐

1𝑗

2

+ 𝑐

1𝑗

3

− 𝑐

1𝑗

1

− 𝑐

1𝑗

4

) + (𝑐

1𝑗

1

+ 𝑐

1𝑗

4

+ 𝑐

1𝑗

))

3

𝑥

𝑗

,


; 𝑋

2

solves

max𝑋

2

𝑉(

𝑍

2

) =

𝑛

∑

𝑗=1

(𝜆 (𝑐

2𝑗

2

+ 𝑐

2𝑗

3

− 𝑐

2𝑗

1

− 𝑐

2𝑗

4

) + (𝑐

2𝑗

1

+ 𝑐

2𝑗

4

+ 𝑐

2𝑗

))

3

𝑥

𝑗

Subject to𝑛

∑

𝑗=1

(𝜆 (𝑎

𝑖𝑗

2

+ 𝑎

𝑖𝑗

3

− 𝑎

𝑖𝑗

1

− 𝑎

𝑖𝑗

4

) + (𝑎

𝑖𝑗

1

+ 𝑎

𝑖𝑗

4

+ 𝑎

𝑖𝑗

))

3

𝑥

𝑗

≤

(𝜆 (𝑚

𝑏

𝑖

2

+ 𝑚

𝑏

𝑖

3

− 𝑚

𝑏

𝑖

1

− 𝑚

𝑏

𝑖

4

) + (𝑚

𝑏

𝑖

1

+ 𝑚

𝑏

𝑖

+ 𝑚

𝑏

𝑖

4

))

3

+ Φ

−1

(𝛾

𝑖

)

(𝜆 (𝜎

𝑏

𝑖

2

+ 𝜎

𝑏

𝑖

3

− 𝜎

𝑏

𝑖

1

− 𝜎

𝑏

𝑖

4

) + (𝜎

𝑏

𝑖

1

+ 𝜎

𝑏

𝑖

+ 𝜎

𝑏

𝑖

4

))

3

; 𝑖 = 1, 2, . . . , 𝑚

𝑥

𝑗

≥ 0; 𝑗 = 1, 2, . . . , 𝑛; 0 ≤ 𝜆 ≤ 1.

(20)

Now, the DMs are trying to optimize their objective inde-pendently under the system constraints defined above. Let

𝑉(

𝑍

1

)

𝑏

and 𝑉(

𝑍

2

)

𝑏

be the best value of the objective ofthe upper and lower level DMs, respectively, obtained by


solving each objective independently. Since the DMs aretrying to maximize their objective and they also want thatthe maximum value of the objective of the other levelDM cannot exceed his minimum value of the objective,therefore, the worst values 𝑉(𝑍

1

)

𝑤

and 𝑉(𝑍2

)

𝑤

are calculatedby considering the best solution point of the other level DM.That is,

𝑉(

𝑍

1

)

𝑤

= 𝑉 (

𝑍

1

)

(at best solution point of 2nd level DM) ,

𝑉 (

𝑍

2

)

𝑤

= 𝑉 (

𝑍

2

)

(at best solution point of 1st level DM) .

(21)

Hence the fuzzy goal for each objective is expressed as

𝑉(

𝑍

𝑘

) ≳ 𝑉 (

𝑍

𝑘

)

𝑏

; 𝑘 = 1, 2.(22)

Thus the membership function of the upper and lower levelobjective is constructed as

𝜇

𝑉(

𝑍

𝑘)

=

{

{

{

{

{

{

{

{

{

{

{

{

{

{

{

0 if 𝑉(𝑍𝑘

) ≤ 𝑉 (

𝑍

𝑘

)

𝑤

𝑉((

𝑍

𝑘

) − 𝑉 (

𝑍

𝑘

)

𝑤

)

(𝑉 (

𝑍

𝑘

)

𝑏

− 𝑉 (

𝑍

𝑘

)

𝑤

)

if 𝑉(𝑍𝑘

)

𝑤

≤ 𝑉 (

𝑍

𝑘

) ≤ 𝑉 (

𝑍

𝑘

)

𝑏

1 if 𝑉(𝑍𝑘

) ≥ 𝑉 (

𝑍

𝑘

)

𝑏

𝑘 = 1, 2.

(23)

With the help of the above membership functions, the FGPmodel is defined in the next section.

4.3. Weighted FGP Model Formulation. In FGP model for-mulation, the membership functions are first converted intoflexiblemembership goals by introducing under- and overde-viational variables to each of them and thereby assigning thehighest membership value (unity) as the aspiration level toeach of them. Also it is evident that full achievement of all themembership goals is not possible in MODM context. So theunderdeviational variables are minimized to achieve the bestgoal values of objectives in the decisionmaking environment.

Thus a FGP model is formulated as

Find 𝑋(𝑥

1

, 𝑥

2

, . . . , 𝑥

𝑛

) so as to

Minimize 𝐷 =

𝐾

∑

𝑘=1

𝑤

𝑘

𝑑

−

𝑘

Subject to 𝜇

𝑉(

𝑍

𝑘)

+ 𝑑

−

𝑘

− 𝑑

+

𝑘

= 1; 𝑘 = 1, 2,

𝑛

∑

𝑗=1

(𝜆 (𝑎

𝑖𝑗

2

+ 𝑎

𝑖𝑗

3

− 𝑎

𝑖𝑗

1

− 𝑎

𝑖𝑗

4

) + (𝑎

𝑖𝑗

1

+ 𝑎

𝑖𝑗

4

+ 𝑎

𝑖𝑗

))

3

𝑥

𝑗

≤

(𝜆 (𝑚

𝑏

𝑖

2

+ 𝑚

𝑏

𝑖

3

− 𝑚

𝑏

𝑖

1

− 𝑚

𝑏

𝑖

4

) + (𝑚

𝑏

𝑖

1

+ 𝑚

𝑏

𝑖

+ 𝑚

𝑏

𝑖

4

))

3

+ Φ

−1

(𝛾

𝑖

)

(𝜆 (𝜎

𝑏

𝑖

2

+ 𝜎

𝑏

𝑖

3

− 𝜎

𝑏

𝑖

1

− 𝜎

𝑏

𝑖

4

) + (𝜎

𝑏

𝑖

1

+ 𝜎

𝑏

𝑖

+ 𝜎

𝑏

𝑖

4

))

3

;

𝑖 = 1, 2, . . . , 𝑚

𝑥

𝑗

≥ 0; 𝑗 = 1, 2, . . . , 𝑛; 0 ≤ 𝜆 ≤ 1,

(24)

where 𝑤𝑘

≥ 0 represents the numerical weights of the goalswhich are determined as

𝑤

𝑘

=

𝑝

𝑘

(𝑉 (

𝑍

𝑘

)

𝑏

− 𝑉 (

𝑍

𝑘

)

𝑤

)

, 𝑘 = 1, 2; 𝑝

𝑘

> 0. (25)

The developed model (24) is solved to find the most satisfac-tory solution in the decision making environment.

4.4. Solution Algorithm. Themethodology for developing theintuitionistic FSLBLP model is presented in the form of analgorithm as follows.

Step 1. Apply CCP technique to all the fuzzy probabilisticconstraints, to form a fuzzy linear bilevel programmingmodel (refer to (18)).

Step 2. Using the proposed ranking technique of IFNs, thedeterministic linear bilevel model is developed (refer to (19)).

Step 3. The individual optimal value of the objective of eachDM is found in isolation.

Step 4. The worst value of the objective of the DMs isevaluated at the best solution point of the other level DMs(refer to (21)).


Step 5. Construct the fuzzy membership goals of the objec-tive of each DM (refer to (22)).

Step 6. FGP approach is used to achieve maximum degree ofeach of the membership goals (refer to (24)).

Step 7. Stop.

5. Numerical Example

To illustrate the proposed methodology the followingFSLBLP problem is considered with coefficients of objectivesand system constraints are taken as normal TIFNs:

Find 𝑋(𝑥

1

, 𝑥

2

) so as to

Max

𝑍

1

≅

8𝑥

1

+

17𝑥

2

Max

𝑍

2

≅

15𝑥

1

+

4𝑥

2

Subject to Pr (10𝑥1

+

6𝑥

2

≲

𝑏

1

) ≥ 0.15

Pr (7𝑥1

+

12𝑥

2

≲

𝑏

2

) ≥ 0.09

𝑥

1

, 𝑥

2

≥ 0,

(26)

where 𝑏1

and 𝑏2

denote fuzzy random variables followingnormal distribution. Here 8, 17, 15, 4, 10, 6, 7, and 12 are allnormal TIFNs. Then the TIFNs can be expressed as

8 = {(7.6, 8, 8.5) , (7, 8, 9)} ,

17 = {(16, 17, 18) , (15, 17, 18.6)} ,

15 = {(14.4, 15, 15.6) , (14, 15, 16.5)} ,

4 = {(3, 4, 4.8) , (2.6, 4, 5)} ,

10 = {(9.4, 10, 10.8) , (8.5, 10, 11.5)} ,

6 = {(5.6, 6, 6.5) , (5, 6, 7)} ,

7 = {(6, 7, 8) , (5.6, 7, 8.9)} ,

12 = {(11.2, 12, 13) , (10.5, 12, 13.5)} .

(27)

The mean 𝑚

𝑏

𝑖

and standard deviation 𝜎

𝑏

𝑖

(𝑖 = 1, 2) of thefuzzy random variables 𝑏

𝑖

following normal distribution aregiven as

𝑚

𝑏

1

=

20 = {(19.4, 20, 21) , (19, 20, 21.2)} ,

𝜎

𝑏

1

=

5 = {(4, 5, 5.9) , (3.8, 5, 6)} ,

𝑚

𝑏

2

=

21 = {(19.8, 21, 21.7) , (19.5, 21, 23)} ,

𝜎

𝑏

2

=

4 = {(3.6, 4, 5.5) , (3, 4, 5.84)} .

(28)

Applying CCPmethodology the above linear fuzzy stochasticBLPP is converted into linear fuzzy BLPP as

Find 𝑋(𝑥

1

, 𝑥

2

) so as to

Max

𝑍

1

≅

8𝑥

1

+

17𝑥

2

Max

𝑍

2

≅

15𝑥

1

+

4𝑥

2

Subject to

10𝑥

1

+

6𝑥

2

≲

20 + 1.04 ×

5

7𝑥

1

+

12𝑥

2

≲

21 + 1.37 ×

4

𝑥

1

, 𝑥

2

≥ 0.

(29)

By applying the proposed ranking technique of normalTIFNs, the above IFNs are converted into the equivalent crispvalues as

𝑉(

8) = (8 + 0.033𝜆) ,

𝑉 (

17) = (16.87 + 0.133𝜆) ,

𝑉 (

15) = (15.17 − 0.17𝜆) ,

𝑉 (

4) = (3.87 + 0.07𝜆) ,

𝑉 (

10) = (10 + 0.07𝜆) ,

𝑉 (

6) = (6 + 0.03𝜆) ,

𝑉 (

20) = (20.07 + 0.07𝜆) ,

𝑉 (

5) = (4.93 + 0.03𝜆) ,

𝑉 (

7) = (7.17 − 0.17𝜆) ,

𝑉 (

12) = (12 + 0.07𝜆) ,

𝑉 (

21) = (21.17 − 0.33𝜆) ,

𝑉 (

4) = (4.28 + 0.09𝜆) .

(30)

Hence the deterministic model of (29) is represented as

Max 𝑉(

𝑍

1

)

= (8 + 0.033𝜆) 𝑥

1

+ (16.87 + 0.133𝜆) 𝑥

2

Max 𝑉(

𝑍

2

)

= (15.17 − 0.17𝜆) 𝑥

1

+ (3.87 + 0.07𝜆) 𝑥

2

Subject to (10 + 0.07𝜆) 𝑥

1

+ (6 + 0.03𝜆) 𝑥

2

≤ (25.2 + 0.1𝜆)

(7.17 − 0.17𝜆) 𝑥

1

+ (12 + 0.07𝜆) 𝑥

2

≤ (27.03 − 0.21𝜆)

𝑥

1

, 𝑥

2

≥ 0.

(31)


Table 2: Compromise solution obtained by proposed methodology.

Objective Solution point Ranking value of the objective Membership value

𝑍

1

𝑥

1

= 1.81, 𝑥2

= 1.17

𝑉(

𝑍

1

) = 34.48 𝜇

𝑉(

𝑍1)= 0.80

𝑍

2

𝑉(

𝑍

2

) = 31.78 𝜇

𝑉(

𝑍2)= 0.78

Table 3: Comparison of solutions between existing and proposed techniques.

Ranking technique Solution point Objective values Membership value

Dubey and Mehra [29] 𝑥

1

= 1.81 𝑉 (

𝑍

1

) = 33.98 𝜇

𝑉(

𝑍1)= 0.79

𝑥

2

= 1.29 𝑉 (

𝑍

2

) = 30.86 𝜇

𝑉(

𝑍2)= 0.76

Proposed technique 𝑥

1

= 1.81 𝑉(

𝑍

1

) = 34.48 𝜇

𝑉(

𝑍1)= 0.80

𝑥

2

= 1.17 𝑉 (

𝑍

2

) = 31.78 𝜇

𝑉(

𝑍2)= 0.78

Now, each objective is considered independently and is solvedwith respect to the system of constraints in (31) to find theindividual optimal values of the objectives. The results areobtained as

(𝑉 (

𝑍

1

))

𝑏

= 38.00, at the point (0, 2.25) ,

(𝑉 (

𝑍

2

))

𝑏

= 38.23, at the point (2.52, 0) .(32)

The worst values of 𝑉(𝑍1

) and 𝑉(

𝑍

2

) are obtained byputting the best solution point of the 2nd and 1st objective,respectively. Thus (𝑉(𝑍

1

))

𝑤

= 20.16 and (𝑉(𝑍2

))

𝑤

= 8.71.Then the fuzzy goals of the objectives are found as (𝑍

1

) ≳

38.00, 𝑉(

𝑍

2

) ≳ 38.23.The membership function of the objectives is written as

𝜇

𝑉(

𝑍

1)

= 0.056 ((8 + 0.033𝜆) 𝑥

1

+ (16.87 + 0.133𝜆) 𝑥

2

− 20.16) ,

𝜇

𝑉(

𝑍

2)

= 0.034 ((15.17 − 0.17𝜆) 𝑥

1

+ (3.87 + 0.07𝜆) 𝑥

2

− 8.71) .

(33)

Hence the FGP model can be presented by eliciting themembership goals as

Find 𝑋(𝑥

1

, 𝑥

2

)

So as to Min 𝐷 = 0.040𝑑

−

1

+ 0.034𝑑

−

2

Subject to 𝜇

𝑉(

𝑍

1)

+ 𝑑

−

1

− 𝑑

+

1

= 1

𝜇

𝑉(

𝑍

2)

+ 𝑑

−

2

− 𝑑

+

2

= 1

(10 + 0.07𝜆) 𝑥

1

+ (6 + 0.03𝜆) 𝑥

2

≤ (25.2 + 0.1𝜆)

(7.17 − 0.17𝜆) 𝑥

1

+ (12 + 0.07𝜆) 𝑥

2

≤ (27.03 − 0.21𝜆)

𝑥

1

, 𝑥

2

≥ 0,

(34)

where 𝑑−1

, 𝑑+1

, 𝑑−2

, and 𝑑+2

≥ 0 with 𝑑−1

⋅ 𝑑

+

1

, 𝑑

−

2

⋅ 𝑑

+

2

= 0.Now solving the abovemodel using software LINGO (ver.

11) the solutions are found as 𝑥1

= 1.81, 𝑥2

= 1.17 withobjective values 𝑉(𝑍

1

) = 34.48 and 𝑉(𝑍2

) = 31.78.The solutions achieved through FGP technique are sum-

marized in Table 2.If the numerical example presented in this section is

solved by applying the ranking technique of IFNdeveloped byDubey andMehra [29], then the objective values are obtainedas 𝑉(𝑍

1

) = 33.98 and 𝑉(𝑍2

) = 30.86. The solutions achievedby two ranking techniques are summarized in Table 3.

This comparison can also be shown graphically as inFigure 4.

From the comparison table and fromFigure 4 it is realizedthat better objective values are obtained if the example issolved by the proposed ranking technique. So the proposedranking technique is more acceptable to the DMs in real lifedecision making context.

6. Conclusions

This paper presents a new ranking technique for normalTIFNs. The proposed methodology can be used to find thedefuzzified value of intuitionistic trapezoidal FNs. Based onthe proposed ranking technique an intuitionistic LFSBLPmodel is developed. This methodology covers a wider range


Proposed method Dubey and Mehra method33.6

33.8

34

34.2

34.4

34.6

30

30.5

31

31.5

32

Proposed method Dubey and Mehra method

V(Z1) V(Z2)

Figure 4: Comparison of solutions.

of applications with enriched solutions than earlier tech-niques. The proposed model can also be applied to solvemany real life decisionmaking problems inwhich parametersare expressed in the form of IFNs. Further the proposedmethodology can be used efficiently to solve multiobjectivefractional programming problems, multilevel optimizationproblems, and other associated problems in an intuitionisticfuzzy stochastic decision making arena. However, it is hopedthat the proposed methodology may open up new vistas intotheway ofmaking decision in an imprecisely defined decisionmaking arena.

Competing Interests

The authors declare that they have no competing interests.

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