Hexagonal fuzzy approximation of fuzzy numbers and its ...

Complex & Intelligent Systems (2021) 7:1459–1487https://doi.org/10.1007/s40747-020-00242-4

ORIG INAL ART ICLE

Hexagonal fuzzy approximation of fuzzy numbers and its applicationsin MCDM

V. Lakshmana Gomathi Nayagam1 · Jagadeeswari Murugan1

Received: 31 August 2020 / Accepted: 25 November 2020 / Published online: 16 February 2021© The Author(s) 2021

AbstractNumerous research papers and several engineering applications have proved that the fuzzy set theory is an intelligent effectivetool to represent complex uncertain information. In fuzzy multi-criteria decision-making (fuzzyMCDM) methods, intelligentinformation systemand fuzzy control-theoreticmodels, complexqualitative information are extracted fromexpert’s knowledgeas linguistic variables and are modeled by linear/non-linear fuzzy numbers. In numerical computations and experiments, theinformation/data are fitted by nonlinear functions for better accuracy which may be little hard for further processing to applyin real-life problems. Hence, the study of non-linear fuzzy numbers through triangular and trapezoidal fuzzy numbers isvery natural and various researchers have attempted to transform non-linear fuzzy numbers into piecewise linear functions ofinterval/triangular/trapezoidal in nature by different methods in the past years. But it is noted that the triangular/trapezoidalapproximation of nonlinear fuzzy numbers has more loss of information. Therefore, there is a natural need for a betterpiecewise linear approximation of a given nonlinear fuzzy number without losing much information for better intelligentinformation modeling. On coincidence, a new notion of Generalized Hexagonal Fuzzy Number has been introduced and itsapplications on Multi-Criteria Decision-Making problem (MCDM) and Generalized Hexagonal Fully Fuzzy Linear System(GHXFFLS) of equations have been studied by Lakshmana et al. in 2020. Therefore, in this paper, approximation of nonlinearfuzzy numbers into the hexagonal fuzzy numbers which includes trapezoidal, triangular and interval fuzzy numbers as specialcases of Hexagonal fuzzy numbers with less loss/gain of information than other existing methods is attempted. Since anyfuzzy information is satisfied fully by its modal value/core of that concept, any approximation of that concept is expected tobe preserved with same modal value/core. Therefore, in this paper, a stepwise procedure for approximating a non-linear fuzzynumber into a new Hexagonal Fuzzy Number that preserves the core of the given fuzzy number is proposed using constrainednonlinear programming model and is illustrated numerically by considering a parabolic fuzzy number. Furthermore, theproposed method is compared for its efficiency on accuracy in terms of loss of information. Finally, some properties of thenew hexagonal fuzzy approximation are studied and the applicability of the proposed method is illustrated through the GroupMCDM problem using an index matrix (IM).

Keywords LR hexagonal fuzzy number · Parabolic fuzzy number · Hexagonal approximation · MCDM

Introduction

Most of the real-life problems are involved with complexforms of uncertain information which are continuous transi-tions. Zadeh introduced a fuzzy set (FS) in 1965 to deal with

B Jagadeeswari [email protected]

V. Lakshmana Gomathi [email protected]

1 Department of Mathematics, National Institute ofTechnology, Tiruchirappalli, India

such information which has a wide scope of applications inmany research fields such as expert systems, pattern recog-nition, data mining. In this queue, defuzzification methodshave been widely developed and are implemented in sev-eral research areas like MCDM, fuzzy control, data analysis,and clustering, etc. The approximation is a better kind ofdefuzzification with a reliable agreement balancing appro-priate form of approximation to enhance better computationand preventing loss of information as much as possible. Inthe past years, various researchers have investigated approx-imation of general non-linear fuzzy numbers by linear fuzzynumbers of simpler forms such as interval, triangular and

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http://orcid.org/0000-0002-8251-8529

1460 Complex & Intelligent Systems (2021) 7:1459–1487

trapezoidal to avoid complex calculations in the problemsinvolving nonlinear fuzzy numbers. All these reasons causea natural need for a better approximation of fuzzy numbersthat are easy to handle without losing much information. Inthis section, a detailed literature review and motivation aregiven as subsections.

Literature review

The interval approximation of fuzzy numbers (FNs) isinitially started using Hamming distance on intervals byStephen chanas [23] and later is developed using Euclideanmetric on intervals based on the α-cuts by Grzegorzewski[33]. Later, it is extended to another simpler form of triangu-lar approximation with symmetricity and non-symmetricityby Ma et al. [47] and Zeng and Li [68]. In that sequel,Abbasbandy and Asady [3] have proposed a new techniqueto defuzzify a fuzzy quantity by trapezoidal fuzzy num-bers (TPFNs). In 2005, Grzegorzewski and Mrowka [36]propounded trapezoidal approximation using the metric bypreserving the expected interval, and later, it was proved thatthe method does not lead always a FN. Hence they [37] revis-ited the trapezoidal approximation operator by preservingthe expected interval (EI). Abbasbandy and Amirfakhrian[1] have recommended an ideal method to enumerate theproximate approximation of a fuzzy number as a polyno-mial. They [2] also suggested a nearest trapezoidal form ofa FN using pseudo metric on the set of all FNs by consider-ing generalized LR type fuzzy number. In 2007, Zeng and Li[68] put forth the approximation procedure by a triangularfuzzy number (TRFN) followed by the approach of Grze-gorzewski’s trapezoidal approximation technique defined in[36].

To overcome the flaws that occurred in [36,37,68], Ban[8,9] redefined the approximation methods in 2008 usingKarush Kuhn Tucker (KKT) Theorem by upholding theexpected interval, value, ambiguity, width, and weightedexpected value. Continuing with that, Grzegorzewski [34]has expressed the algorithms and properties for computingthe proper trapezoidal approximation for a fuzzy numberby preserving the expected interval. Moreover in 2008, Yeh[63] has investigated the Zeng and Li approximation and heimproved it by extended triangular and trapezoidal approx-imation. He [64] improved Grzegorzewski’s approximationoperator defined in [34] and proposed a procedure for calcu-lating the improved approximation. Ban [10] has proposedthe parametric approximation using the average Euclideandistance of a given FN. He [11] also suggested the trian-gular and parametric approximations of FNs by correctingthe inadvertences of Zeng and Li’s [68] and Gregorzewski’s[36] nearest parametric approximation using the KKT theo-rem. In 2010, Grzegorzewski [35] has recommended a newapproach for finding trapezoidal approximation preserving

the expected interval. Ban [4,19] put forward a new approxi-mation method of trapezoidal type FN by perpetuating core,expected value, ambiguity, value, and discussed some ofits properties. Ban [12] has discussed the significance oftranslation invariance and scale invariance in approximationtechniques. He [13] also expressed the metric properties ofthe nearest extended parametric FN which generalizes theextended trapezoidal fuzzy number and parametric fuzzynumber. Ban [14,15] has proposed the nearest interval, tri-angular, trapezoidal, and weighted semi trapezoidal approxi-mation of FN preserving ambiguity and weighted ambiguityin 2012.

Coroianu [24] has introduced some properties of convexsubsets of topological spaces with Euclidean distance in thefield of fuzzy numbers. It is a very useful tool to compute theLipschitz constant of the trapezoidal operator preserving thevalue and ambiguity. In 2014, Ban [16] has recommendedthe conditions for existence, uniqueness, and continuity ofthe trapezoidal approximation of the FN. Coroianu [28]has considered different parameters named max productBernstein operators for the approximation of FNs. In 2014,Gregorzewski [38] has suggested a new trapezoidal operatorpreserving core, support, and expected interval which guar-antees the proper interpretation of the solution even for skewfuzzy numbers. Yeh in 2014, [67] introduced LR type fuzzynumbers to approximate FNs which generalize all recentapproximations without constraints in Euclidean class.

Furthermore, Ban [17] has investigated the existence,uniqueness, calculus, and properties of the triangular approx-imation with some simple general conditions. In 2015,Coroianu and Stefanni [26] have recommended a differentapproach, a monotonic F-transform approximation of thefuzzy distribution functionwhich produces an approximationof a FN. Ban and Coroianu [18] have suggested a symmet-ric triangular approximation of a FN that preserves the realparameter associated with a FN. Ban [20] has propoundedthe extended weighted LR approximation of a given FNby a method based on general results in Hilbert spaces, theweighted average euclidean distance. Coroianu and Stefanni[27] have recommended the extended inverse fuzzy trans-forms preserves the quasi concavity of a FN and hence itcan be used to generate FNs by approximating the restric-tion of the membership function to its support. Khastan andMoradi [42] have proposed width invariant trapezoidal andtriangular approximation of FNs which avoids the effortfulcomputation of KKT Theorem and preserves the expectedvalue. Huang [40] has designed the convolution method forconstructing approximations comprising FN sequences withuseful properties for a general FN.

Wang and Li [61] have proved that a fuzzy set of real num-ber field R is a simple FN if and only if it is a normal steptype fuzzy set whose result is dense in the FN space withrespect to some metric. Yeh [65] has studied the necessary

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Complex & Intelligent Systems (2021) 7:1459–1487 1461

and sufficient conditions of linear operators that are preservedby interval, triangular, symmetric triangular, trapezoidal orsymmetric trapezoidal approximations of FNs. The problemof triangular approximation of parabolic FNs is attemptedusing distance function in terms of α- cuts in 2017 [50]. In2018, Ban [66] has given the corrected version of symmet-ric triangular FN approximation of [18] by counterexample.Coroianu [29] has proposed the problem of the piecewise lin-ear approximation of FNs giving outputs nearest to the inputsfor the Euclidean metric which is a generalization of 1-knotFNs. Coroianu in 2020 [25] has proved that the nearest trape-zoidal approximation of FNwith respect toweighted L2-typemetrics with or without additional constraints via quadraticprograms.

Recently, a notion of Generalized Pentagonal FuzzyNumbers (GPNFNs), Generalized Hexagonal Fuzzy Num-bers (GHXFNs), Generalized Heptagonal Fuzzy Numbers(GHPFNs),GeneralizedOctagonal FuzzyNumbers (GOFNs),Generalized Nanogonal Fuzzy Numbers (GNFNs) and Gen-eralized Decagonal Fuzzy Numbers (GDFNs) have beenstudied widely by many researchers [21,22,41,48]. But theseare not properly defined in the literature. To overcome theseerrors, Lakshmana et al. [43] introduced a novel GHXFN byconsidering heights and slopes of an FN and is applied inMCDM problems and GHXFFLS of equations.

The concept of Intuitionistic Fuzzy Number (IFN) hasbeen introduced by Burillo et al and it is widely studied in[53–60]. In 2006, Ban [7] introduced an interval approxima-tion to the intuitionistic fuzzy numbers using Euclidean andTran-Duckstein distances defined on fuzzy numbers. Li andLi [44,45] have introduced a method of approximating IFNsby trapezoidal IFNswith respect to a standardEuclideanmet-ric on IFNs and approximated the output of aggregation ofIFNSwith the condition of preserving the width. Li andYuan[46] have investigated the representation of the weightedextended trapezoidal intuitionistic fuzzy approximation ofan IFN. Triangular approximation of the new type of IFNdefined by Lakshmana et al using Weighted Euclidean dis-tance on new IFNs by Karush Kuhn Tucker (KKT) Theoremhas been discussed in 2020 [51]. Likewise, the notion of aneutrosophic set has been introduced by Smarandache and itis applied in graph theory, algebra, signal and image process-ing, machine learning, etc., Furthermore, the approximationmethods are extended to neutrosophic numbers in [49].

From the literature survey, we can conclude that noform of better approximation of non-linear fuzzy numbersis available even in the trapezoidal, triangular, and intervalapproximations of fuzzy numbers. Therefore, it is requiredto define a better kind of approximation of a fuzzy num-ber that reduces the loss of information and fuzziness asmuch as possible but that should be easy in computation.Furthermore, since any fuzzy information is completelycharacterized by its modal value/core of that concept, any

approximation of that concept is expected to be preservedwith modal value/core. Hence, in this paper, it is proposedto define a better approximation of non-linear fuzzy num-bers by a Hexagonal fuzzy number (HXFN) which preservesthe core of the given fuzzy number and reduces the loss ofinformation as much as possible than existing methods.

The structure of the paper is as follows. In “Preludes”,the basic concepts of our task are introduced. In “Hexag-onal approximation of fuzzy number preserving the core”,we present a procedure of new hexagonal approximation offuzzy number and parabolic fuzzy number preserving thecore with algorithms and illustrations. “Results and discus-sions on the proposedmethod” deals with properties satisfiedby the new approximation for further implications and com-parison of proposed method with other existing methods.“Application of the hexagonal approximation in MCDMusing index matrix” consists of applications on group fuzzydecision-making using the proposed approximation methodin the index matrix with a suitable illustration based on [54].Finally, “Conclusion and future scope” concludes the paper.

Motivation

In real-time contexts, most of the problems are in the form ofimprecise numerical quantity with qualitative and linguistic(subjective) information which could not be modeled usingreal numbers. To model such cases, the theory of FNs hasbeen proposed as an alternative tool. Several researchershave been working in the area of defining new membershipfunctions such as interval, triangular, trapezoidal, hexagonal,octagonal, decagonal, etc., to represent different concepts.Some of them like Generalized Pentagonal Fuzzy Number,GeneralizedHexagonal FuzzyNumber,GeneralizedHeptag-onal Fuzzy Number, Generalized Octagonal Fuzzy Number,Generalized Nanogonal Fuzzy Number, and GeneralizedDecagonal Fuzzy Number are not properly defined in theexisting literature. But, many authors are working in thatarea with incomplete definitions and implemented in diverseapplication fields of engineering, multi-criteria decision-making problem, transportation problems, etc., because of itsusefulness and novelty. Therefore, it is necessary to rectifythose errors from the existing definitions for its wider scopeand usability. Hence in 2020, Lakshmana et al. introduceda novel generalized hexagonal fuzzy number by consider-ing heights and slopes of the left and right functions ofa GHXFN to overcome the inadvertences occurred in theexisting definition of GHXFN. In the literature, the approx-imation is available for nonlinear fuzzy numbers only byinterval, triangular and trapezoidal ones. The approximationof nonlinear fuzzy numbers by hexagonal fuzzy numberswillgive better approximation with minimum lack/abundance ofinformation than trapezoidal fuzzy number approximation.Furthermore, in the hexagonal fuzzy number approximation,

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the novel hexagonal fuzzy number approximation producesprecise approximation with minimum lack/abundance ofinformation than older hexagonal fuzzy number approxima-tion.

Preludes

In this paper, R, F(R), T F(R), HF(R) and PF(R) rep-resent the set of all real numbers, the class of all fuzzynumbers, the class of all trapezoidal fuzzy numbers, the classof all hexagonal fuzzy numbers and the class of all parabolicfuzzy numbers, respectively, andH represents the hexagonalapproximation operator.

Definition 2.1 [67] Let L, R : [0, 1] → [0, 1] be two fixedfunctions which are both upper semicontinuous and decreas-ing such that L(0) = R(0) = 1 and L(1) = R(1) = 0.A fuzzy number A is said to be LR type fuzzy number ifmembership function MA is given by

MA(x) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

L(CL−xη

), if CL − η ≤ x < CL ,

1, if CL ≤ x ≤ CU ,

R(x−CU

γ), if CU < x ≤ CU + γ,

0, otherwise

where η, γ are left and right spreads (widths) of MA and[CL ,CU ] is the core of MA. A LR type fuzzy number isdenoted as A = (CL ,CU ; η, γ ).

Definition 2.2 [6] A quadruple T = ((t1, t2, t3, t4); u, MT ),t1 ≤ t2 ≤ t3 ≤ t4, t1, t2, t3, t4 ∈ R, u ∈ [0, 1] is called gener-alized trapezoidal fuzzy numberwhosemembership functionis given by

MT (x) =

⎧⎪⎪⎨

⎪⎪⎩

MTL (x), for t1 ≤ x ≤ t2u, for t2 ≤ x ≤ t3MTR (x), for t3 ≤ x ≤ t40, otherwise

where MTL (x) = u( x−t1t2−t1

) and MTR (x) = u( t4−xt4−t3

).If t2 = t3, then generalized trapezoidal fuzzy number T

becomes generalized triangular fuzzy number (GTRFN) Tand is denoted by triplet T = ((t1, t2, t3); u, MT ).

Definition 2.3 [43] A fuzzy subset H of R of the form H =((h1, h2, h3, h4, h5, h6); u, uL , uR, MH ), h1 ≤ h2 ≤ h3 ≤h4 ≤ h5 ≤ h6, hi , i = 1, 2, . . . 6 ∈ R, u, uL , uR ∈ [0, 1] iscalled generalized hexagonal fuzzy number whose member-ship function is defined by

MH (x) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

M1L(x), if h1 ≤ x ≤ h2

M2L(x), if h2 ≤ x ≤ h3u, if h3 ≤ x ≤ h4

M2R(x), if h4 ≤ x ≤ h5

M1R(x), if h5 ≤ x ≤ h60, otherwise

where M1L(x) = uL

(x−h1h2−h1

), M2

L(x) = uL + (u −uL)

(x−h2h3−h2

), M2

R(x) = uR + (u−uR)(

h5−xh5−h4

), M1

R(x) =uR

(h6−xh6−h5

)are left lower, left upper, right upper and right

lower legs of H , respectively.A generalized hexagonal fuzzy number passes through

(h1, 0), (h2, uL), (h3, u), (h4, u), (h5, uR) and (h6, 0) andhence it is denoted by ((h1, h2, h3, h4, h5, h6); u, uL , uR,

MH ).In the GHXFN, H , uL , uR are heights of left lower leg

and right lower leg of H , respectively, and u is the height ofH .

Definition 2.4 [43] Let H = (h1, h2, h3, h4, h5, h6; u, uL ,

uR) be a GHXFN with ε-cut Hε = [LH (ε), RH (ε)]. Then,the score functions, The midpoint score of the GHXFN H isdefined as

M(H) = uLH (u) + RH (u)

2= u

(h3 + h4

2

)

The compass or span of the GHXFN H is defined as

S(H) = uRH (u) − LH (u)

2= u

(h4 − h3

2

)

The left dissimilitude of the slope score of the GHXFN H isdefined as

LD(H) = uL(h3 − h1) − u(h2 − h1)

4

The left aggregation of the slope score of the GHXFN H isdefined as

L A(H) = uL(h3 − h1) + u(h2 − h1)

4

The right dissimilitude of the slope score of the GHXFN His defined as

RD(H) = uR(h6 − h4) − u(h6 − h5)

4

The right aggregation of the slope score of the GHXFN H isdefined as

RA(H) = uR(h6 − h4) + u(h6 − h5)

4

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Definition 2.5 [50] A quadruple P = ((p1, p2, p3, p4); u,

MP ), p1 ≤ p2 ≤ p3 ≤ p4, p1, p2, p3, p4 ∈ R, u ∈ [0, 1] iscalled generalized parabolic fuzzy number whose member-ship function is given by

MP (x) =

⎧⎪⎪⎨

⎪⎪⎩

MPL (x), if p1 ≤ x ≤ p2u, if p2 ≤ x ≤ p3MPR (x), if p3 ≤ x ≤ p40, otherwise

where MPL (x) = u(x−p1p2−p1

)2 and MPR (x) = u(p4−xp4−p3

)2.If u = 1, then GPFN becomes PFN and is denoted by

P = (p1, p2, p3, p4; 1, MPL , MPR ).

Definition 2.6 [32] If P = (p1, p2, p3, p4; 1, MPL , MPR )

and Q = (q1, q2, q3, q4; 1, MQL , MQR ) are two PFNs, thenthe addition of P and Q is defined as

P ⊕ Q = (p1 + q1, p2 + q2, p3 + q3, p4

+q4, M(P⊕Q)L , M(P⊕Q)R ).

Definition 2.7 [68] Let A and B be arbitrary FNs with η-cuts ηA = [L A(η),UA(η)] and ηB = [LB(η),UB(η)]. Adistance d(A, B) between FNs A and B is given by

d(A, B) =(∫ 1

0d2(ηA, ηB)dη

) 12

(1)

where d2(ηA, ηB) = (L A(η)−LB(η))2+(UA(η)−UB(η))2

is the distance of η-cut sets of FNs A and B which reflectsthe nearness and overlap degree between ηA and ηB .

Definition 2.8 [5] The concept of index matrix (IM) wasintroduced by K. Atanassov in 1987 [5]. Let I be a fixed setof indices and R be the set of all real numbers. Let K ={k1, k2, . . . , km}, L = {l1, l2, . . . , ln} ⊂ I . The general formof IM with real numbers R − I M is given as

[K , L, {aki ,l j }] =

l1 · · · l j · · · lnk1 ak1,l1 · · · ak1,l j · · · ak1,ln...

......

...

ki aki ,l1 · · · aki ,l j · · · aki ,ln...

......

...

km akm ,l1 · · · akm ,l j · · · akm ,ln

where for (1 ≤ i ≤ m and 1 ≤ j ≤ n) : aki ,l j ∈ R.

In the above index matrix, if aki ,l j ∈ [0, 1], then IM iscalled (0, 1)− IM.

Now, we give the famous Karush Kuhn Tucker (KKT)Theorem on optimization theory.

Theorem 2.1 [8] Let F, g1, g2, . . . , gm : Rn → R be con-vex and differentiable functions. Then, x̄ solves the convexprogramming problem

Minimize F(x̄)

Subject to gi (x̄) ≤ ci , i ∈ {1, 2, . . . ,m}

if and only if there exists λi , i ∈ {1, 2, . . . ,m} such that

(i) ∇F(x̄) + ∑mi=1λi∇gi (x̄) = 0 where ∇ f denotes the

gradient of function f;(ii) gi (x̄) − ci ≤ 0;(iii) λi ≥ 0;(iv) λi (ci − gi (x̄)) = 0;

Novel definitions

The Definition 2.3 can also be represented as LR form in theforthcoming definition.

Definition 2.9 Let Li , Ri , : [0, 1] → [0, 1], i = 1, 2be linear functions which are both upper semicontinuousand decreasing such that Li (0) = Ri (0) = 1, Li (1) =Ri (1) = 0, i = 1, 2. A generalized hexagonal fuzzynumber H = ((CL ,CU ; h1, h2, h3, h4); u, uL , uR, MH ) =((CL ,CU ; h1, h2, h3, h4); u, uL , uR, L1, L2, R2, R1) is saidto be LR type generalized hexagonal fuzzy number if mem-bership function MH is given by

MH (x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

uL L1(CL−h2−xh1

), x ≤ CL − h2, h1 > 0

uL + (u − uL )L2(CL−xh2

), x ≤ CL , h2 > 0

u, CL ≤ x ≤ CU

uR + (u − uR)R2(x−CUh3

), CU ≤ x, h3 > 0

uR R1(x−CU−h3

h4), CU + h3 ≤ x, h4 > 0

0, otherwise

where hi ≥ 0, i = 1, 2, 3, 4 are left lower, left upper, rightupper and right lower spreads (widths) of H , respectively,and the interval [CL ,CU ] is core of H .

Definition 2.10 LetH = ((h1, h2, h3, h4, h5, h6); u, uL , uR)

be a GHXFN defined on R. Let η ∈ [0, 1]. The η-cut set ηH

of H is given by ηH = [LH (η), RH (η)]where

LH (η) =⎧⎨

⎩

L1H (η) for η ∈ [0, uL)

L2H (η) for η ∈ [uL , u]

RH (η) =⎧⎨

⎩

R2H (η) for η ∈ [uR, u]

R1H (η) for η ∈ [0, uR)

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and L1H (η) = h1 + η

uL(h2 − h1), L2

H (η) = h2 +(

η−uLu−uL

)(h3 − h2), R2

H (η) = h5 −(

η−uRu−uR

)(h5 − h4),

R1H (η) = h6 − η

uR(h6 − h5).

Hexagonal approximation of fuzzy numberpreserving the core

In this section, approximation of a fuzzy number by a hexag-onal fuzzy number that preserve the core of given fuzzynumber is proposed and hexagonal fuzzy approximationof parabolic fuzzy number is discussed in detail. Further-more, stepwise procedure for Hexagonal approximation of aParabolic fuzzy number is given and the proposed method isillustrated by numerical examples.

Hexagonal approximation of a fuzzy number

Consider a FN A = ((a1, a2, a3, a4); 1, MAL , MAU ) withη−cut set ηA = [MAL (η), MAU (η)]. In this sub-section, it isproposed to identify LR HXFNH(A) of A which preservesthe core of A = [a2, a3] with minimal loss of informationfrom the given A. Hence, the proximate LRHXFN of the FNis of the form H(A) = ((a2, a3; h1, h2, h3, h4); 1, u1, u2,L1, L2, R2, R1) in which heights of left lower leg u1 andright lower leg u2 are the functional values at (a2 − h2) and(a3+h3) in the given fuzzy number. Therefore, the left lower,left upper, right upper and right lower spreads (widths) hi ≥0, i = 1, · · · , 4 of H are determined by minimizing theloss of information measured by the distance d(A,H(A))

between A andH(A) by Eq. (1) with u1 = MAL (a2 −h2) =( a2−h2−a1

a2−a1)2 and u2 = MAU (a3 + h3) = ( a4−a3−h3

a4−a3)2.

Therefore, the aim of the subsection is reduced to findingof hi ≥ 0 by minimizing

d(A,H(A))

=

√√√√√√√√

∫ u1

0[MAL (η) − L1

H(A)(η)]2dη +∫ 1

u1[MAL (η) − L2

H(A)(η)]2dη

+∫ u2

0[MAU (η) − R1

H(A)(η)]2dη +∫ 1

u2[MAU (η) − R2

H(A)(η)]2dη

subject to constraints hi ≥ 0, where u1 = MAL (a2 − h2) =( a2−h2−a1

a2−a1)2, u2 = MAU (a3 + h3) = ( a4−a3−h3

a4−a3)2.

By definitions, L1H(A)

(η) = h1ηu1

+ a2 − h1 − h2,

L2H(A)

(η)= h2(η−1)+a2(1−u1)1−u1 , R2

H(A)(η) = −h3(η−1)+a3(1−u2)

1−u2,

R1H(A)

(η) = −h4ηu2

+ a3 + h3 + h4.

Finally, we have a problem of constrained minima, Mini-mize f (h1, h2, h3, h4) = [d(A,H(A))]2

=∫ u1

0

[

MAL (η) − h1η

u1− a2 + h1 + h2

]2

dη

+∫ 1

u1

[

MAL (η) − a2 − h2

(η − 1

1 − u1

)]2

dη

+∫ u2

0

[

MAU (η) − a3 − h3 − h4 + h4η

u2

]2

dη

+∫ 1

u2

[

MAU (η) − a3 + h3

(η − 1

1 − u2

)]2

dη (2)

subject to constraints −hi ≤ 0 with the integral limits u1 =( a2−h2−a1

a2−a1)2 and u2 = ( a4−a3−h3

a4−a3)2 which are functions of

variables h2 and h3 to be found, respectively.By KKT Theorem 2.1 and Leibnitz rule of general form

of differentiation under integral sign, we have

2I1 − 2

u1I2 + 2

3h1u1 + h2u1 − CLu1 − λ1 = 0 (3)

(2h1 I2u21

+ h213

+ h1h2 − CLh1

)

E

+2I1 + h1u1 + 2h2u1 − 2CLu1

+[6(I3 − I4)+h2(2u21 − 4u1 + 2)+3CL (−u21 + 2u1 − 1)

3

]

F−λ2 = 0

(4)(

−2h4 I6u22

+ h243

+ h3h4 + CUh4

)

G

−2I5 + h4u2 + 2h3u2 + 2CUu2

+[6(I7 − I8) + h3(−2u22 + 4u2 − 2) + 3CU (−u22 + 2u2 − 1)

3

]

H

−λ3 = 0 (5)

−2I5 + 2

u2I6 + 2

3h4u2 + h3u2 + CUu2 − λ4 = 0 (6)

−h1 ≤ 0 (7)−h2 ≤ 0 (8)−h3 ≤ 0 (9)−h4 ≤ 0 (10)λ1 ≥ 0 (11)λ2 ≥ 0 (12)λ3 ≥ 0 (13)λ4 ≥ 0 (14)−λ1h1 = 0 (15)−λ2h2 = 0 (16)−λ3h3 = 0 (17)−λ4h4 = 0 (18)

where λi , i = 1 to 4 are Lagrange’s multipliers,

123


I1 =∫ u1

0MAL (η)dη, I2 =

∫ u1

0ηMAL (η)dη,

I3 =∫ 1

u1MAL (η)dη,

I4 =∫ 1

u1ηMAL (η)dη, I5 =

∫ u2

0MAU (η)dη,

I6 =∫ u2

0ηMAU (η)dη, I7 =

∫ 1

u2MAU (η)dη,

I8 =∫ 1

u2ηMAU (η)dη,

du1dh2

= E,d

dh2

(h2

1 − u1

)

= F,

du2dh3

= G,d

dh3

( −h31 − u2

)

= H .

BysolvingEqs. (3)–(18),weobtain solutions forh1, h2, u1of left legs and h3, h4, u2 of right legs by which we get thehexagonal approximation H(A) = ((a2, a3; h1, h2, h3, h4);1, u1, u2, L1, L2, R2, R1) of given fuzzy number A.

The proposedmethod is explained in the following sectionby considering the parabolic fuzzy number as fuzzy numberto have a Hexagonal approximation.

Hexagonal approximation of a parabolic fuzzynumber

Consider a parabolic fuzzynumber P = ((p1, p2, p3, p4); 1,MPL , MPU )withη-cut setηP = [MPL (η), MPU (η)] = [p1+√

η(p2−p1), p4−√η(p4−p3)]. In this sub-section, it is pro-

posed to identify LRHXFN of P which preserves the core ofP . Hence the proximate LR HXFN of the PFN is of the formH(P) = ((p2, p3; h1, h2, h3, h4); 1, u1, u2, L1, L2, R2, R1)

in which heights of left lower leg u1 and right lower leg u2are the functional values at (p2 − h2) and (p3 + h3) in thegiven parabolic fuzzy number.

From Eqs. (11)–(14), there are two possibilities λi = 0and λi > 0 for every multiplier λi , i = 1 to 4. The combi-nation of these multipliers becomes 16(24) possible cases.To make ease the process, the above 16 cases are partitionedinto 2 divisions by considering the Eqs. (3), (4), (7), (8), (11),(12), (15), (16) and (5), (6), (9), (10), (13), (14), (17), (18)which will not affect the solutions of minimization problem.i.e., we find the solutions for h1, h2 of left legs and h3, h4 ofright legs separately in the upcoming 2 lemmas.

Lemma 3.1 Let P = ((p1, p2, p3, p4); 1, MPL , MPU ) bea parabolic fuzzy number with η-cut set ηP = [MPL (η),

MPU (η)] = [p1 + √η(p2 − p1), p4 − √

η(p4 − p3)]. LetT 2 = (

p2−h2−p1p2−p1

)2 = u1. The solutions for h1, h2 which sat-isfy Eqs. (3), (4), (7), (8), (11), (12), (15), (16) are obtainedfrom h1 = 7T

10 (p2 − p1), h2 = p2 − p1 − T p2 + T p1 bysolving 4T 7 p31 −8T 9 p31 +4T 11 p31 −60T 4 p51 +320T 5 p51 −660T 6 p51 +600T 7 p51 −100T 8 p51 −240T 9 p51 +180T 10 p51 −40T 11 p51−12T 7 p21 p2+24T 9 p21 p2−12T 11 p21 p2+300T 4 p41 p2−

1600T 5 p41 p2+3300T 6 p41 p2−3000T 7 p41 p2+500T 8 p41 p2+1200T 9 p41 p2−900T 10 p41 p2+200T 11 p41 p2+12T 7 p1 p22 −24T 9 p1 p22 + 12T 11 p1 p22 − 600T 4 p31 p

22 + 3200T 5 p31 p

22 −

6600T 6 p31 p22+6000T 7 p31 p

22−1000T 8 p31 p

22−2400T 9 p31 p

22

+ 1800T 10 p31 p22 − 400T 11 p31 p

22 − 4T 7 p32 + 8T 9 p32 −

4T 11 p32 + 600T 4 p21 p32 − 3200T 5 p21 p

32 + 6600T 6 p21 p

32 −

6000T 7 p21 p32+1000T 8 p21 p

32+2400T 9 p21 p

32−1800T 10 p21 p

32

+400T 11 p21 p32−300T 4 p1 p42+1600T 5 p1 p42−3300T 6 p1 p42

+3000T 7 p1 p42−500T 8 p1 p42−1200T 9 p1 p42+900T 10 p1 p42−200T 11 p1 p42+60T 4 p52−320T 5 p52+660T 6 p52−600T 7 p52+ 100T 8 p52 + 240T 9 p52 − 180T 10 p52 + 40T 11 p52 = 0, 0 ≤T ≤ 1.

Proof Thehypothesis of convexity and differentiability in theKKT theorem are satisfied by the function and the conditions.

Now we have,

I1 =∫ u1

0MPL (η)dη =

∫ T 2

0(p1 + √

η(p2 − p1))dη

= 2p2T 3 − 2p1T 3 + 3p1T 2

3

I2 =∫ u1

0ηMPL (η)dη =

∫ T 2

0(p1η + η3/2(p2 − p1))dη

= 4p2T 5 − 4p1T 5 + 5p1T 4

10

I3 =∫ 1

u1ηMPL (η)dη =

∫ 1

T 2(p1 + √

η(p2 − p1))dη

= −(2p2 − 2p1)T 3 − 3p1T 2 + 2p2 + p13

I4 =∫ 1

u1ηMPL (η)dη =

∫ 1

T 2(p1η + η3/2(p2 − p1))dη

= −(4p2 − 4p1)T 5 − 5p1T 4 + 4p2 + p110

E = du1dh2

= 2(h2 − p2 + p1)

(p2 − p1)2

= 2T (p1 − p2)

(p2 − p1)2

F = d

dh2

(h2

1 − u1

)

= (p2 − p1)2(h22(2p2 − 2p1))

(h2 − 2p2 + 2p1)2(h2)2= 2(p2 − p1)3

(−1 − T )2

WhenEqs. (3), (4), (7), (8), (11), (12), (15), (16) are solvedforh1, h2,wehave following4 cases basedon themultipliers.

Case 1: λ1 = 0, λ2 = 0By substituting λ1 = λ2 = 0 and the values for I1, I2, I3, I4,E, F from Eqs. (19)–(24) in Eqs. (3), (4), we get h1 =7T10 (p2 − p1). Since h1 ≥ 0 and u1 ∈ [0, 1], T ∈ [0, 1].Since 0 ≤ h2 ≤ p2 − p1, T 2 = (

p2−h2−p1p2−p1

)2 = u1, we get

h2 = p2−p1−T p2+T p1. By substituting h1 = 7T10 (p2−p1)

123


and h2 = p2 − p1 − T p2 + T p1 in Eq. (4), we get equa-tion given in hypothesis. Hence, h1 = 7T

10 (p2 − p1) andh2 = p2 − p1 − T p2 + T p1 by solving hypothesis withT ∈ [0, 1].Case 2: λ1 = 0, λ2 = 0FromEq. (16), it is concluded that h2 = 0. Hence, T = 1 andu1 = 1. By substituting λ1 = 0, h2 = 0 and above requiredvalues in (3), (4), we get h1 = 7

10 (p2 − p1), h2 = 0, andλ2 = 1

75 (p2 − p1) ≥ 0. The solutions for h1 and h2 areparticular case of the solutions in case 1 in which T = 1.

Case 3 λ2 = 0, λ1 = 0From Eq. (15), it is concluded that h1 = 0. By substitutingλ2 = 0, h1 = 0 and required values in (3), (4),we get h1 = 0,h2 = p2 − p1 − T p2 + T p1 and λ1 = 7T 3

15 (p1 − p2) ≥ 0,where T ∈ [0, 1] which is arrived from equation in hypothe-sis. Hence, T = 0 and h2 = p2− p1. Therefore, the solutionsfor h1 and h2 are particular case of the solutions in case 1 inwhich T = 0.

Case 4: λ1 = 0, λ2 = 0From Eqs. (15), (16), it is concluded that h1 = 0, h2 = 0.Hence T = 1 and u1 = 1. By substituting h1 = 0, h2 = 0and other required values in (3), (4), we get λ1 = 7

15 (p1 −p2) < 0 which is a contradiction to λ1 ≥ 0. Therefore, inthis case, multipliers do not satisfy KKT. ��Lemma 3.2 Let P = ((p1, p2, p3, p4); 1, MPL , MPU ) bea parabolic fuzzy number with η-cut set ηP = [MPL (η),

MPU (η)] = [p1+√η(p2−p1), p4−√

η(p4−p3)]. Let S2 =(p4−p3−h3p4−p3

)2 = u2. The solutions for h3, h4 which satisfyEqs. (5), (6) (9), (10), (13), (14), (17), (18) are obtained fromh3 = −p3 + p4 − Sp4 + Sp3, h4 = 7S

10 (p4 − p3) by solving4S7 p33−8S9 p33+4S11 p33−60S4 p53+320S5 p53−660S6 p53+600S7 p53 − 100S8 p53 − 240S9 p53 + 180S10 p53 − 40S11 p53 −12S7 p23 p4 + 24S9 p23 p4 − 12S11 p23 p4 + 300S4 p43 p4 −1600S5 p43 p4 + 3300S6 p43 p4 − 300S7 p43 p4 + 500S8 p43 p4 +1200S9 p43 p4 − 900S10 p43 p4 + 200S11 p43 p4 + 12S7 p3 p24 −24S9 p3 p24 + 12S11 p3 p24 − 600S4 p33 p

24 + 3200S5 p33 p

24 −

6600S6 p33 p24+6000S7 p33 p

24−1000S8 p33 p

24−2400S9 p33 p

24+

1800S10 p33 p24 −400S11 p33 p

24 −4S7 p34 +8S9 p34 −4S11 p34 +

600S4 p23 p34−3200S5 p23 p

34+6600S6 p23 p

34−6000S7 p23 p

34+

1000S8 p23 p34+2400S9 p23 p

34−1800S10 p23 p

34+400S11 p23 p

34−

300S4 p3 p44+1600S5 p3 p44−3300S6 p3 p44+3000S7 p3 p44−500S8 p3 p44 −1200S9 p3 p44 +900S10 p3 p44 −200S11 p3 p44 +60S4 p54 − 320S5 p54 + 660S6 p54 − 600S7 p54 + 100S8 p54 +240S9 p54 − 180S10 p54 + 40S11 p54 = 0, 0 ≤ S ≤ 1.

Proof Thehypothesis of convexity and differentiability in theKKT theorem are satisfied by the function and the conditions.By solving the Eqs. (5), (6) (9), (10), (13), (14), (17), (18),we can obtain the solutions for h3, h4 in the form of fourcases based on the multipliers.

Case 1: λ3 = 0, λ4 = 0

By substituting λ3 = λ4 = 0 and the values of

I5 =∫ u2

0MPU (η)dη =

∫ S2

0(p4 − √

η(p4 − p3))dη

= −2p4S3 + 2p3S3 + 3p4S2

3,

I6 =∫ u2

0ηMPU (η)dη =

∫ S2

0(p4η − η3/2(p4 − p3))dη

= −4p4S5 + 4p3S5 + 5p4S4

10,

I7 =∫ 1

u2MPU (η)dη =

∫ 1

S2(p4 − √

η(p4 − p3))dη

= −(2p4 − 2p3)S3 − 3p4S2 + 2p3 + p43

,

I8 =∫ 1

u2ηMPU (η)dη =

∫ 1

S2(p4η − η3/2(p4 − p3))dη

= −(4p4 − 4p3)S5 − 5p4S4 + 4p3 + p410

,

G = du2dh3

= 2(h3 + p3 − p4)

(p4 − p3)2

= 2S(p3 − p4)

(p4 − p3)2,

H = ddh3

(−h31−u2

)

= −(p4−p3)2(h23(2p3−2p4))(h3−2p4+2p3)2(h3)2 = −2(p3−p4)3

(−1−S)2in

(5), (6) and solving these equations, we get h3 = −p3+ p4−Sp4 + Sp3 and h4 = 7S

10 (p4 − p3), where S ∈ [0, 1] whichis arrived from polynomial equation given in hypothesis.

Case 2: λ3 = 0, λ4 = 0From Eq. (18), it is concluded that h4 = 0. By substitutingλ3 = h4 = 0 and required values given above in (5), (6) andsolving these equations, we get h3 = −p3+ p4−Sp4+Sp3,h4 = 0 and λ4 = 7S3

15 (p3 − p4) ≥ 0, where S ∈ [0, 1] whichis arrived from equation in hypothesis. Hence, S = 0 andh3 = p4 − p3. Therefore, the solutions for h3 and h4 areparticular case of the solutions in case 1 in which S = 0.

Case 3: λ4 = 0, λ3 = 0From Eq. (17), it is concluded that h3 = 0. Hence S = 1and u2 = 1 By substituting λ4 = h3 = 0 and requiredvalues in (5), (6) and solving these equations, we get h3 = 0,h4 = 7

10 (p4 − p3) and λ3 = 175 (p4 − p3) ≥ 0. The solutions

for h3 and h4 are particular case of the solutions in case 1 inwhich S = 1.

Case 4: λ3 = λ4 = 0From Eqs. (17), (18), it is concluded that h3 = 0, h4 = 0.Hence S = 1 and u2 = 1. By substituting h3 = 0, h4 = 0and other required values in (5), (6), we get λ3 = 7

15 (p3 −p4) < 0 which is a contradiction to λ3 ≥ 0. So in this casemultipliers does not satisfy KKT. ��

123


Notes 1 The calculations have been done by Mathematicato arrive these equations and conditions given in the aboveLemmas 3.1 and 3.2.

The hexagonal approximationH(P) of a parabolic fuzzynumber P is obtained using Lemmas 3.1 and 3.2 in the fol-lowing stepwise procedure.

Stepwise procedure for hexagonal approximation ofa parabolic fuzzy number

In this sub-section, we discuss the stepwise procedure tofind the hexagonal fuzzy number approximation to a givenparabolic fuzzy number.

Consider a parabolic fuzzynumber P = ((p1, p2, p3, p4);1, MPL , MPU ) for which the hexagonal approximation is tobe found.

Step 1: Substitute the values of p1, p2, p3, p4 in polyno-mial equations given in Lemmas 3.1 and 3.2.

Step 2: Solve for T and S from the equations given in Lem-mas 3.1 and 3.2. Let them be T1, T2, . . . , Tm andS1, S2, . . . , Sn .

Step 3: Find h1, h2, h3 and h4 using Lemmas 3.1 and 3.2for each pair of Ti , S j , i = 1, 2, . . . ,m, j =1, 2, . . . , n and compute distances using Eq. 2.

Step 4: Fix h1, h2, h3, h4 and uL = T 2, uR = S2 for whichthe distance is minimum in the distances computedin step 3.

Step 5: The proximate LR HXFN for the PFN P is givenby H(P) = ((p2, p3; h1, h2, h3, h4); 1, uL , uR) =((p2−h1−h2, p2−h2, p2, p3, p3+h3, p3+h3+h4); 1, uL , uR).

Definition 3.1 Let P = ((p1, p2, p3, p4); 1, MPL , MPU ) bea parabolic fuzzy number with η-cut set ηP = [MPL (η),

MPU (η)] = [p1 + √η(p2 − p1), p4 − √

η(p4 − p3)].The hexagonal approximation of P is given by H(P) =((p2, p3; h1, h2, h3, h4); 1, u1, u2), where h1 = 7T

10 (p2 −p1), h2 = p2− p1−T p2+T p1, h3 = −p3+ p4−Sp4+Sp3,h4 = 7S

10 (p4 − p3), u1 = T 2, u2 = S2 where T and S are

obtained from the Lemmas 3.1 and 3.2 for which the distanceis minimum.

Example 3.1 Consider aPFN P = (0, 1, 2, 144; 1, ML , MR).TheLRhexagonal approximation isH(P) = ((1, 2; h1, h2, h3,h4); 1, u1, u2, L1, L2, R2, R1).

Step 1: Substitute p1 = 0, p2 = 1, p3 = 2, p4 = 144 inpolynomial equations given in Lemmas 3.1 and 3.2, we have

60T 4 − 320T 5 + 660T 6 − 604T 7 + 100T 8

+248T 9 − 180T 10 + 36T 11 = 0 and

3464120353920S4 − 18475308554240S5

+38105323893120S6 − 34641214992352S7

+5773533923200S8 + 13856504321984S9

−10392361061760S10 + 2309402116128S11 = 0

Step 2: The roots T and S are given by

T = 1.50218, 0, 0, 0, 0, 0.598212, 0.725965

−0.478917i, 0.725965 + 0.478917i, 1, 1, 2.45204.

S = −1.5, 0, 0, 0, 0, 0.949347, 0.99747

−0.0529555i, 0.99747 + 0.0529555i, 1, 1, 1.05573.

The roots T and S which lies between [0, 1] are0, 0.598212, 1 and 0, 0.949347, 1, respectively, and the cor-responding distances (gain of information, loss of informa-tion and change of information due to left and right legs) aretabulated in the Table 1.

From the Table 1, the smaller change of information givesthe better approximation. Therefore, the approximated valuesare

p2 = 1, p3 = 2, h1 = 7T10 (p2 − p1) = 0.7,

h2 = p2 − p1 − T p2 + T p1 = 0, h3 = −p3 + p4 −Sp4 + Sp3 = 7.1928, h4 = 7S

10 (p4 − p3) = 94.3651,u1 = T 2 = 1, u2 = S2 = 0.9. Therefore, H(A) =((1, 2; 0.7, 0, 7.1928, 94.3651); 1, 1, 0.9).Example 3.2 Consider a PFN P = (1, 2, 3, 5; 1, ML , MR),The LR hexagonal approximation H(P) = ((2, 3; h1, h2,h3, h4); 1, u1, u2, L1, L2, R2, R1).

Table 1 Change of Informationin Example 3.1

T h1 h2 Gain of information Loss of information Change of information

0.0000 0.0000 1.0000 0.1667 0.0000 0.1667

0.5982 0.4187 0.4018 0.3213 0.0035 0.3248

1.0000 0.7000 0.0000 0.0310 0.0144 0.0454

S h3 h4 Gain of information Loss of information Change of information

0.0000 142.00 0.0000 23.667 0.0000 23.667

0.9494 7.1928 94.365 0.0000 1.7533 1.7533

1.0000 0.0000 99.400 4.4159 2.0493 6.4652

123




0.0000 0.0000 1.0000 0.1667 0.0000 0.1667

0.5982 0.4187 0.4018 0.3213 0.0035 0.3248

1.0000 0.7000 0.0000 0.0310 0.0144 0.0454


0.0000 0.0000 2.0000 0.0000 0.6667 0.6667

0.6817 0.9544 0.6366 0.2400 0.0084 0.2484

1.0000 1.4000 0.0000 0.0747 0.3413 0.4160

From the approximation procedure, the polynomial equa-tion of T and S are

60T 4 − 320T 5 + 660T 6 − 604T 7 + 100T 8

+248T 9 − 180T 10 + 36T 11 = 0 and

1920S4 − 10240S5 + 21120S6 − 19232S7

+3200S8 + 7744S9 − 5760S10 + 1248S11 = 0.

The obtained roots of T and S are

T = 1.50218, 0, 0, 0, 0, 0.598212, 0.725965

−0.478917i, 0.725965 + 0.478917i, 1, 1, 2.45204.

S = −1.50054, 0, 0, 0, 0, 0.68169, 0.844414

+0.385567i, 0.844414 − 0.385567i, 1, 1, 1.74541.

The roots T and S which lies between [0, 1] are0, 0.598212, 1 and 0, 0.68169, 1, respectively, and the corre-sponding distances (gain of information, loss of informationand change of information due to left and right legs) are tab-ulated in Table 2.From the Table 2, the smaller change of information givesthe better approximation. Therefore, the approximated valuesare

p2 = 2, p3 = 3, h1 = 7T10 (p2 − p1) = 0.7,

h2 = p2 − p1 − T p2 + T p1 = 0, h3 = −p3 + p4 −Sp4 + Sp3 = 0.9544, h4 = 7S

10 (p4 − p3) = 0.6366,u1 = T 2 = 1, u2 = S2 = 0.5. Therefore, H(P) =((2, 3; 0.7, 0, 0.9544, 0.6366); 1, 1, 0.5).Example 3.3 Consider a PFN P = (−2, 5, 7, 12; 1, 1, ML ,

MR). The LR hexagonal approximationH(P) = ((5, 7; h1,h2, h3, h4); 1, u1, u2, L1, L2, R2, R1).

From the approximation procedure, the polynomial equa-tion of T and S are

1008420T 4 − 5378240T 5 + 11092620T 6 − 10085572T 7

+1680700T 8+4036424T 9−3025260T 10

+670908T 11 = 0and 187500S4 − 1000000S5

+2062500S6 − 1875500S7 + 312500S8 + 751000S9

−562500S10 + 124500S11 = 0

The obtained roots of T and S are

T = −1.50004, 0, 0, 0, 0, 0.803393, 0.950779

+0.22862i, 0.950779 − 0.22862i, 1, 1, 1.3043

S = −1.50009, 0, 0, 0, 0, 0.774727, 0.932254

−0.265865i, 0.932254 + 0.265865i, 1, 1, 1.37893.

The rootsT and Swhich lies between [0, 1] are 0, 0.803393, 1and 0, 0.774727, 1, respectively, and the corresponding dis-tances (gain of information, loss of information and changeof information due to left and right legs) are tabulated inTable 3.

From Table 3, the smaller change of information gives thebetter approximation. Therefore, the approximated values are

p2 = 5, p3 = 7, h1 = 7T10 (p2 − p1) = 3.9366,

h2 = p2 − p1 − T p2 + T p1 = 1.3763, h3 = −p3 +p4 − Sp4 + Sp3 = 1.1264, h4 = 7S

10 (p4 − p3) = 2.7115,u1 = T 2 = 0.65, u2 = S2 = 0.6. Therefore, H(P) =((5, 7; 3.9366, 1.3763, 1.1264, 2.7115); 1, 0.65, 0.6).

Results and discussion on the proposedmethod

Approximation is a sort of defuzzification which metamor-phose nonlinear functions into piecewise linear functionswhich diminishes the fuzziness and impreciseness. In thelast 2 decades, many researchers have grappled with thevarious approximations of FNs into real scores, intervals,triangular, trapezoidal numbers (piecewise linear). But stillthere is loss/gain of information in the context. Therefore,it is required to define better kind of fuzzy number thatreduce the loss of information and fuzziness as much aspossible with ease in computation. The approximation ofnonlinear fuzzy numbers by hexagonal fuzzy numbers willgive better approximation with minimum lack/abundance ofinformation than trapezoidal fuzzy number approximation.To overcome the flaws in the existing definition of HXFN,Lakshmana et al. introduced a novel GHXFN by consid-ering heights and slopes of a GHXFN. The approximation

123




0.0000 0.0000 7.0000 1.1667 0.0000 1.1667

0.8034 3.9366 1.3763 0.1335 0.0521 0.1856

1.0000 4.9000 0.0000 0.2177 0.1010 0.3187


0.0000 5.0000 0.0000 0.8333 0.0000 0.8333

0.7747 1.1264 2.7115 0.0815 0.0336 0.1151

1.0000 0.0000 3.5000 0.1555 0.0722 0.2277

using this kind of GHXFN produces accurate and piecewiseapproximation than existing hexagonal, pentagonal, trape-zoidal, triangular, interval and real approximations. Theseapproximation methods can be applied in different researchfields such as pattern recognition, transportation problems,decision-making problems, TOPSIS and complex intelli-gent information systems. In this section, some propertiesof hexagonal approximation operator are studied and com-parison with existing methods are discussed in detail.

Properties of hexagonal approximation operator

In this section, some properties of hexagonal approximationoperator which are very much useful in the multi-criteriadecision making are proved.

Definition 4.1 (Value and Ambiguity of a FN) [30,31]The value, the ambiguity of a fuzzy number A = (L A(η),

RA(η)) ∈ F(R) are defined as

1. Val(A) = ∫ 10 η(L A(η) + RA(η))dη,

2. Amb(A) = ∫ 10 η(RA(η) − L A(η))dη.

Definition 4.2 (Expected interval, Expected value andWidthof a FN) [23,31,39,62]

The expected interval, the expected value and the width of afuzzy number A = (L A(η), RA(η)) ∈ F(R) are defined as

1. E I (A) = [ ∫ 10 L A(η)dη,

∫ 10 RA(η)dη

]

2. EV (A) = 12

∫ 10 (L A(η) + RA(η))dη

3. W (A) = ∫ 10 (RA(η) − LA(η))dη

Remark 4.1 Let P = ((p1, p2, p3, p4); 1, MPL , MPU ) ∈PF(R) be a parabolic fuzzy number. Then

Val(P) = p1 + 4p2 + 4p3 + p410

,

Amb(P) = −p1 − 4p2 + 4p3 + p410

,

E I (P) =[p1 + 2p2

3,2p3 + p4

3

]

,

EV (P) = p1 + 2p2 + 2p3 + p46

and

W (P) = −p1 − 2p2 + 2p3 + p43

.

Remark 4.2 Let H = ((h1, h2, h3, h4, h5, h6); 1, uL , uR) ∈HF(R) be a hexagonal fuzzy number. Then,

Val(H) = (h1 − h3)u2L + (h2 − h3)uL + h2 + 2h3 + 2h4 + h5 + uR(h5 − h4) + u2R(h6 − h4)

6,

Amb(H) = (h3 − h1)u2L + (h3 − h2)uL − h2 − 2h3 + 2h4 + h5 + uR(h5 − h4) + u2R(h6 − h4)

6,

E I (H) =[(h1 − h3)uL + h2 + h3

2,h4 + h5 + uR(h6 − h4)

2

]

,

EV (H) = (h1 − h3)uL + h2 + h3 + h4 + h5 + uR(h6 − h4)

4,

W (H) = (h3 − h1)uL − h2 − h3 + h4 + h5 + uR(h6 − h4)

2.

Theorem 4.1 If the LR hexagonal approximation of a PFN PisH(P) = ((p2, p3; h1, h2, h3, h4); 1, uL , uR), then hexag-onal operator H preserving core satisfies

123


1. Val(H(P)) = Val(P) − Z, with Z = 160

[(p1 −

p2)(3T 5 − 10T 4 + 10T 3 − 10T 2 + 10T − 4) + (p3 −p4)(−3S5 + 10S4 − 10S3 + 10S2 − 10S + 4)

],

2. Amb(H(P)) = Amb(P) − Z, with Z = 160

[(p1 −

p2)(−3T 5 + 10T 4 − 10T 3 + 10T 2 − 10T + 4) + (p3 −p4)(−3S5 + 10S4 − 10S3 + 10S2 − 10S + 4)

],

3. E I (H(P)) = E I (P) − Z with Z =[(p1−p2)(9T 3−30T 2+30T−10)

60 ,(p3−p4)(−9S3+30S2−30S+10)

60

]

,

4. EV (H(P)) = EV (P) − Z with Z = 1120

[(p1 −

p2)(9T 3 − 30T 2 + 30T − 10) + (p3 − p4)(−9S3 +30S2 − 30S + 10)

],

5. W (H(P)) = W (P)−Z with Z = 160

[(p1− p2)(−9T 3+

30T 2−30T+10)+(p3−p4)(−9S3+30S2−30S+10)],

where uL = T 2 and uR = S2.

Proof Let P = ((p1, p2, p3, p4); 1, MPL , MPU ) be a PFN.By Remark 1, Val(P) = p1+4p2+4p3+p4

10 . The LR hexagonalapproximation of P is H(P) = ((p1, p2; h1, h2, h3, h4); 1,uL , uR) with h1 = 7T (p2−p1)

10 , h2 = p2 − p1 − T (p2 − p1),

h3 = p4− p3−S(p4− p3), h4 = 7S(p4−p3)10 , uL = T 2, uR =

S2, where T and S are obtained from equations given inLemmas 3.1 and 3.2.

We can rewrite H(P) in the general representation asH(P) = ((p2 − h1 − h2, p2 − h2, p2, p3, p3 + h3, p3 +h3 + h4); 1, uL , uR) with p2 − h1 − h2 = 3T p2−3T p1+10p1

10 ,p2 − h2 = p1 + T p2 − T p1, p3 + h3 = p4 − Sp4 + Sp3,p3 + h3 + h4 = 10p4−3Sp4+3Sp3

10 .So, Val(H) =

(h1−h3)u2L+(h2−h3)uL+h2+2h3+2h4+h5+(h5−h4)uR+(h6−h4)u2R6 .By

substituting (h1 − h3)u2L = (p2 − h1 − h2 − p2)T 4 =3T 5 p2−3T 5 p1+10T 4 p1−10T 4 p2

10 , (h2 − h3)uL = (p2 − h2 −h3)T 2 = p1T 2 − p2T 2 + p2T 3 − p1T 3, h2 = p2 − h2 =p1 + T p2 − T p1, 2h3 = 2p2, 2h4 = 2p3, h5 = p3 + h3 =p4 − Sp4 + Sp3, (h5 − h4)uR = (p3 + h3 − p3)S2 =p4S2 − p3S2 − p4S3 + p3S3, (h6 − h4)u2R = (p3 +h3 + h4 − p3)S4 = 10p4S4−10p3S4−3p4S5+3p3S5

10 , we haveVal(H(P)) = 1

60

[(p1−p2)(−3T 5+10T 4−10T 3+10T 2−

10T ) + 10p1 + 20p2 + (p3 − p4)(3S5 − 10S4 + 10S3 −10S2 + 10S) + 20p3 + 10p4

].

Therefore, Val(H(P)) = Val(P) − Z , where Z =160

[(p1 − p2)(3T 5 − 10T 4 + 10T 3 − 10T 2 + 10T − 4) +

(p3 − p4)(−3S5 + 10S4 − 10S3 + 10S2 − 10S + 4)].

Similarly, Amb(H(P)) = 160

[(p1 − p2)(3T 5 − 10T 4 +

10T 3 − 10T 2 + 10T ) − 10p1 − 20p2+ (p3 − p4)(3S5 −10S4 + 10S3 − 10S2 + 10S) + 20p3 + 10p4

]. Therefore,

Amb(H(P)) = Amb(P) − Z , where Z = 160

[(p1 −

p2)(−3T 5 + 10T 4 − 10T 3 + 10T 2 − 10T + 4) + (p3 −p4)(−3S5 + 10S4 − 10S3 + 10S2 − 10S + 4)

].

Similarly, 3), 4), and 5) are proved. ��

Theorem 4.2 The hexagonal approximation operator H istranslation invariant.

Proof Let k be a real number. Let P = ((p1, p2, p3, p4); 1,MPL , MPU ) be any PFN andH(P) = ((CL ,CU ; h1, h2, h3,h4); 1, u1, u2) be the LR hexagonal approximation of Pwhere CL = p2,CU = p3, h1 = 7T (p2−p1)

10 , h2 = p2 −p1 − T (p2 − p1), h3 = p4 − p3 − S(p4 − p3), h4 =7S(p4−p3)

10 , u1 = T 2, u2 = S2. Now, H(P) + k = ((CL +k,CU + k; h1, h2, h3, h4); 1, u1, u2), where CL + k =p2 + k,CU + k = p3 + k, h1 = 7T ((p2+k)−(p1+k))

10 , h2 =(p2 + k) − (p1 + k) − T ((p2 + k) − (p1 + k)), h3= (p4 + k) − (p3 + k) − S((p4 + k) − (p3 + k)), h4= 7S((p4+k)−(p3+k))

10 , u1 = T 2, u2 = S2.On the other hand, P+k = ((p1+k, p2+k, p3+k, p4+

k); 1, MPL (x−k), MPU (x−k)). The approximation of P+kis H(P + k) = ((DL , DU ; g1, g2, g3, g4); 1, v1, v2), whereDL = p2 + k, DU = p3 + k, g1 = 7Y ((p2+k)−(p1+k))

10 , g2 =(p2 + k) − (p1 + k) − Y ((p2 + k) − (p1 + k)), g3 =(p4 + k) − (p3 + k) − Z((p4 + k) − (p3 + k)), g4= 7Z((p4+k)−(p3+k))

10 , v1 = Y 2,Y = (p2+k)−g2−(p1+k)(p2+k)−(p1+k) , v2

= Z2, Z = (p4+k)−g3−(p3+k)(p4+k)−(p3+k) .

Substituting the values, p1 + k, p2 + k, p3 + k, p4 + kinstead of p1, p2, p3, p4 in the equations of Lemmas 3.1 and3.2, we obtain the same values for T and Y as well as for Sand Z . Hence, hi = gi , i = 1, 2, 3, 4 and ui = vi , i = 1, 2.Hence, H(P + k) = H(P) + k. ��

Thegeometrical representation of translation of above the-orem is given in Fig. 1.

Remark 4.3 The hexagonal approximation operator H satis-fies identity property. Equivalently, the hexagonal approx-imation H(A) of any hexagonal fuzzy number A is itselfsince the distance d(A,H(A)) = 0 if H(A) = A andd(A,H(A)) = 0 ifH(A) = A.

Remark 4.4 The hexagonal approximation operator need notbe a η-cut invariant operator. Equivalently, if H(P) is aHexagonal fuzzy number that approximates a fuzzy num-ber P , then H(P)η = Pη, η ∈ (0, 1] need not be true whichis shown in the following example.

Example 4.1 Consider P = (1, 2, 3, 5; 1, ML , MR) be aPFN. The η-cut set of P is Pη = [MPL , MPU ] = [1 +√

η, 5 − 2√

η]. The LR hexagonal approximation of P isH(P) = ((2, 3; 0.7, 0, 0.9544, 0.6366); 1, 1, 0.5). The η-cut set of H(P) isH(P)η = [MH(P)L , MH(P)U ], where

MH(P)L (η) = 0.7η + 1.3, 0 < η ≤ 1and

MH(P)U (η) ={ −0.9544(η−1)+1.5

0.5 i f 0 < η ≤ 0.5

−0.6366 η0.5 + 4.591 i f 0.5 < η ≤ 1

Clearly, [MPL , MPU ] = [MH(P)L , MH(P)U ]. Therefore,H(P)η = Pη.

123


Fig. 1 Translation of FNs

Fig. 2 Homogeneity of FNs

Remark 4.5 The hexagonal approximation operator does notsatisfy additivity. IfH(P) andH(Q) are two HXFNs whichapproximate fuzzy numbers P and Q, respectively, thenH(P + Q) need not imply H(P) + H(Q) which is shownin the following example.

Example 4.2 Consider P = (1, 2, 3, 5; 1, ML , MR) andQ = (−2, 5, 7, 12; 1, ML , MR) be two PFNs. The approxi-mation of P andQ areH(P) = (2, 3; 0.7, 0, 0.9544, 0.6366;1, 1, 0.5) andH(Q) = (5, 7; 3.9366, 1.3763, 1.1264, 2.7115;1, 0.65, 0.60). The addition of H(P) and H(Q) is

H(P) + H(Q) = ((7, 10; 4.6366, 1.3763,2.0808, 3.3481); 1, 1, 0.8) (19)

On the other hand, P + Q = (−1, 7, 10, 17; 1, ML , MR) bea PFN. The LR hexagonal approximation of P + Q is

H(P + Q) = ((7, 10; 4.5581, 1.4884,1.3763, 3.9366); 1, 0.66, 0.65) (20)

Hence, we obtain H(P) + H(Q) = H(P + Q).

Remark 4.6 The hexagonal approximation operator does notsatisfy homogeneity which supports our intuition since thecurvature of the parabolic fuzzy number is altered by themultiplication with k. If H(P) is the approximation of P ,then H(kP) need not imply kH(P) which is shown in thefollowing example.

Example 4.3 Consider P = (1, 2, 3, 5; 1, ML , MR) be anyPFN. The approximation of P is H(P) = ((2, 3; 0.7, 0,0.9544, 0.6366); 1, 1, 0.5). The approximation of scalarmultiplication of P isH(2P) is

H(2P) = ((4, 6; 1.4, 0, 0.9837, 2.1114); 1, 1, 0.5687) (21)

On the other hand, the scalar multiplication of approximationof P is 2H(P)

2H(P) = ((4, 6; 1.4, 0, 1.9088, 1.2732); 1, 1, 0.4647) (22)

Hence, we obtainH(2P) = 2H(P). The geometrical repre-sentation of homogeneity of FNs are given in Fig. 2.

Comparison with existingmethods

In this section, some existing methods of approximations offuzzy numbers in the literature and their flaws are discussedand they have been compared with the proposed approxima-tion with suitable examples.

Some of the methods of approximations of a given fuzzynumber available in the literature with their parameters ofapproximations and their flaws are given in the Table 4. Someapproximationmethods convert fuzzy numbers into intervalsand symmetric triangular fuzzy numbers without giving suit-able approximations which is mentioned in the column offlaw. Some of the methods may fails to produce triangular

123


Table4

Existingmethods

ofapproxim

ation

References

Type

oflin

earizatio

nApproximationparameters

Expressionof

parameters

Flaw

Maetal.[47

]Sy

mmetrictriang

ular

T(A)=

[t 1,σ]

=[t 1

−σ,t 1

,t 1

+σ]

t 1=

1 2

∫1 0[L

A(θ

)+U

A(θ

)]dθ

σ=

3 2

∫1 0[U

A(θ

)−

LA(θ

)](1

−θ)d

θThisapproxim

ationmethod

produces

only

symmetric

fuzzynumbers

Abbasbandyand

Asady

[3]

Trapezoidal

T(A)=

[t 1,t 2

,σ,β]

=[t 1

−σ,t 1

,t 2

,t 2

+β]

where

t 1=

LA(1

)−

h2 1,

t 2=

UA(1

)+

h2 2

i)h1

=h2

=0,

σ=

3∫1 0((LA(θ

)−

LA(1

))(θ

−1)dθ

,

β=

3∫1 0((U

A(θ

)−U

A(1

))(1

−θ)d

θ

ii)h1

=0,

h2 2

=−U

A(1

)−

β 2+

∫1 0U

A(θ

)dθ,

σ=

3∫1 0(L

A(θ

)−

LA(1

))(θ

−1)dθ

,β

=6

∫1 0U

A(θ

)(1

−2θ

)dθ

iii)

h2

=0,

h2 1

=LA(1

)−

σ 2−

∫2 0LA(θ

)dθ,

σ=

6∫1 0LA(θ

)(2θ

−1)dθ

,β

=3

∫1 0((U

A(θ

)−U

A(1

))(1

−θ)d

θ

iv)h2 1

=LA(1

)−

σ 2−

∫2 0LA(θ

)dθ,h2 2

=−U

A(1

)−

β 2+

∫1 0U

A(θ

)dθ,

σ=

6∫1 0LA(θ

)(2θ

−1)dθ

,β

=3

∫1 0((U

A(θ

)−U

A(1

))(1

−θ)d

θ

Thisapproxim

ationmethod

produces

both

symmetric

andnonsymmetricFN

s,butitisnotg

uaranteedto

have

thesamemodal

value

Abbasbandyand

Amirfakhrian

[1]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]

t 1=

Val

(A)−

Amb(A)−

2t2I

1−2I

,t 2

=p 2

,t 3

=p 3

,

t 4=

Val

(A)+

Amb(A)−

2t3I

1−2I

where

Val

(A)=

∫1 0r(

θ)[L

A(θ

)+U

A(θ

)]dθ,

Amb(A)=

∫1 0r(

θ)[U

A(θ

)−

LA(θ

)]dθ,I

=∫1 0r(

θ)θdθ

Thisapproxim

ationmethod

does

notalwaysproducea

FN

Grzegorzewskiand

Mrowka

[36]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]t 1

=∫1 0(4

−6θ

)LA(θ

)dθ,t 2

=∫1 0(6

θ−

2)LA(θ

)dθ

t 3=

∫1 0(6

θ−

2)U

A(θ

)dθ,t 4

=∫1 0(4

−6θ

)UA(θ

)dθ

Thisapproxim

ationmethod

does

notalwaysproducea

FN

Grzegorzewskiand

Mrowka

[37]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]

If∫1 0(U

A(θ

)−

LA(θ

))(1 3

−θ)d

θ>

0then

t 1=

∫1 0(3

−3θ

)LA(θ

)dθ

+∫1 0(1

−3θ

)UA(θ

)dθ

t 2=

t 3=

∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ

t 4=

∫1 0(1

−3θ

)LA(θ

)dθ

+∫1 0(3

−3θ

)UA(θ

)dθ

Other

wise

t 1=

∫1 0(4

−6θ

)LA(θ

)dθ,t 2

=∫1 0(6

θ−

2)LA(θ

)dθ

t 3=

∫1 0(6

θ−

2)U

A(θ

)dθ,t 4

=∫1 0(4

−6θ

)UA(θ

)dθ

Thisapproxim

ationmethod

does

notalwaysproducea

FN

Yeh

[65]

Trapezoidal,triangu

lar

T(A)=

[l e,ue;x e

,y e

]l e

=∫1 0LA(θ

)dθ,ue

=∫1 0U

A(θ

)dθ,

x e=

6∫1 0(2

θ−

1)LA(θ

)dθ,y e

=−6

∫1 0(2

θ−

1)U

A(θ

)dθ

Thisapproxim

ationmethod

fails

toprod

uceabette

rapproxim

ation

ZengandLi[68

]Weigh

tedtriang

ular

T(A)=

[t 1;t 2

,t 3

]=[t 1

−t 2

,t 1

,t 2

,t 2

+t 3

]

t 1=

∫1 0λ(θ

)(1−

θ)d

θ.∫

1 0λ(θ

)(1−

θ)[L

A(θ

)+U

A(θ

)]−∫1 0λ(θ

)(1−

θ)2dθ

.∫1 0λ(θ

)[LA(θ

)+U

A(θ

)]dθ

2(∫1 0λ(θ

)(1−

θ)d

θ)2

−∫1 0λ(θ

)(1−

θ)2dθ

t 2=

t 1∫1 0λ(θ

)(1−

θ)d

θ−∫

1 0λ(θ

)(1−

θ)L

A(θ

)dθ

∫1 0λ(θ

)(1−

θ)2dθ

t 3=

∫1 0λ(θ

)(1−

θ)U

A(θ

)dθ−t

1∫1 0λ(θ

)(1−

θ)d

θ∫1 0λ(θ

)(1−

θ)2dθ

Thisapproxim

ationmethod

does

notalwaysproducea

FN

123


Table4

continued

References

Type

oflin

earizatio


Expressionof

parameters

Flaw

Ban

[8,9]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]

If∫1 0(2

−3θ

)LA(θ

)dθ

+∫1 0(1

−3θ

)UA(θ

)dθ

>0then

t 1=

t 2=

t 3=

∫1 0LA(θ

)dθ,t 4

=2

∫1 0U

A(θ

)dθ

−∫1 0LA(θ

)dθ

If∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(3

θ−

2)U

A(θ

)dθ

>0then

t 1=

2∫1 0LA(θ

)dθ

−∫1 0U

A(θ

)dθ,t 2

=t 3

=t 4

=∫1 0U

A(θ

)dθ

If∫1 0(2

−3θ

)LA(θ

)dθ

+∫1 0(1

−3θ

)UA(θ

)dθ

≤0,

∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(3

θ−

2)U

A(θ

)dθ

≤0and

∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(1

−3θ

)UA(θ

)dθ

>0then

t 1=

∫1 0(3

−3θ

)LA(θ

)dθ

+∫1 0(1

−3θ

)UA(θ

)dθ

t 2=

t 3=

∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ

t 4=

∫1 0(1

−3θ

)LA(θ

)dθ

+∫1 0(3

−3θ

)UA(θ

)dθ

If∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(1

−3θ

)UA(θ

)dθ

≤0then

t 1=

∫1 0(4

−6θ

)LA(θ

)dθ,t 2

=∫1 0(6

θ−

2)LA(θ

)dθ

t 3=

∫1 0(6

θ−

2)U

A(θ

)dθ,t 4

=∫1 0(4

−6θ

)UA(θ

)dθ

There

ismuchloss

ofinform

ation

Ban

[10,11

]Triangu

lar

T(A)=

[t 1;t 2

,t 3

]

If∫1 0(4

θ2−

3θ)L

A(θ

)dθ

+∫1 0(2

θ2−

θ)U

A(θ

)dθ

≥0and

∫1 0(2

θ2−

θ)L

A(θ

)dθ

+∫1 0(4

θ2−

3θ)U

A(θ

)dθ

≤0then

t 1=

3∫1 0(2

θ2−

θ)L

A(θ

)dθ

+3

∫1 0(2

θ2−

θ)U

A(θ

)dθ,

t 2=

6∫1 0(4

θ2−

3θ)L

A(θ

)dθ

+6

∫1 0(2

θ2−

θ)U

A(θ

)dθ

t 3=

−6∫1 0(2

θ2−

θ)L

A(θ

)dθ

−6

∫1 0(4

θ2−

3θ)U

A(θ

)dθ

If∫1 0(2

θ2−

θ)L

A(θ

)dθ

+∫1 0(4

θ2−

3θ)U

A(θ

)dθ

>0then

t 1=

∫1 0(3

θ2−

3 2θ)L

A(θ

)dθ

+3 2

∫1 0θU

A(θ

)dθ,

t 2=

3∫1 0(6

θ2−

5θ)L

A(θ

)dθ

+3

∫1 0θU

A(θ

)dθ,t 3

=0

If∫1 0(4

θ2−

3θ)L

A(θ

)dθ

+∫1 0(2

θ2−

θ)U

A(θ

)dθ

<0then

t 1=

3 2

∫1 0θLA(θ

)dθ

+∫1 0(3

θ2−

3 2θ)U

A(θ

)dθ,

t 2=

0,t 3

=−3

∫1 0θLA(θ

)dθ

−3

∫1 0(6

θ2−

5θ)U

A(θ

)dθ

There

ismuchloss

ofinform

ation

Ban

[14,15

]Trapezoidal,triangu

lar

T(A)=

[t 1,t 2

,t 3

,t 4

]

If∫1 0(1

−3θ

)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ

≥0then

t 1=

∫1 0(4

−6θ

)LA(θ

)dθ,t 2

=∫1 0(6

θ−

2)LA(θ

)dθ,

t 3=

∫1 0(6

θ−

2)U

A(θ

)dθ,t 4

=∫1 0(4

−6θ

)UA(θ

)dθ

If∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(θ

−1)U

A(θ

)dθ

>0then

t 1=

1 2

∫1 0(1

+9θ

)LA(θ

)dθ

+1 2

∫1 0(1

−9θ

)UA(θ

)dθ,

t 2=

t 3=

t 4=

1 2

∫1 0(1

−3θ

)LA(θ

)dθ

+1 2

∫1 0(1

+3θ

)UA(θ

)dθ

If∫1 0(θ

−1)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ

<0then

t 1=

t 2=

t 3=

1 2

∫1 0(1

+3θ

)LA(θ

)dθ

+1 2

∫1 0(1

−3θ

)UA(θ

)dθ,

t 4=

1 2

∫1 0(1

−9θ

)LA(θ

)dθ

+1 2

∫1 0(1

+9θ

)UA(θ

)dθ

If∫1 0(1

−3θ

)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ

<0,

∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(θ

−1)U

A(θ

)dθ

≤0and

∫1 0(θ

−1)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ

≥0then

t 1=

2∫1 0LA(θ

)dθ

−∫1 0(6

θ−

2)U

A(θ

)dθ,

t 2=

t 3=

∫1 0(3

θ−

1)LA(θ

)dθ

+∫1 0(3

θ−

1)U

A(θ

)dθ,

t 4=

∫1 0(2

−6θ

)LA(θ

)dθ

+2

∫1 0U

A(θ

)dθ

There

ismuchloss

ofinform

ation

123


Table5

Existingmethods

ofapproxim

ation

Reference

Type

oflin

-earizatio


Valuesof

parameters

Gainof

inform

ation

Lossof

inform

ation

Changeof

inform

ation

Maetal.[47

]Sy

mmetric

triang

ular

T(A)=

[t 1,σ]=

[t 1−

σ,t 1

,t 1

+σ]

T(A)=

[5.6667

,5.7]

=[−

0.033,5.667,11

.367

]0.2079

0.5080

0.7159

Abbasbandyand

Asady

[3]

Trapezoidal

T(A)=

[t 1,t 2

,σ,β]

=[t 1

−σ,t 1

,t 2

,t 2

+β]

t 1=

LA(1

)−

h2 1,

t 2=

UA(1

)+

h2 2

T(A)=

[5.4667

,6.667,5.6,4]

=[−

0.1333

,5.4667

,6.667,10

.667

]t 1

=1

−0,

t 2=

2+

0[0.]

0.2196

0.1861

0.4057

Abbasbandyand

Amirfakhrian

[1]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]T

(A)=

[0.8,5,7,10

]Val

(A)=

5.8,

Amb(A)=

2.2,

I=

0.3333

0.0741

0.4741

0.5482

Grzegorzewskiand

Mrowka

[36]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]T

(A)=

[−0.1333

,5.4667

,6.667,10

.667

]0.2196

0.1861

0.4057

Grzegorzewskiand

Mrowka

[37]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]T

(A)=

[−0.1333

,5.4667

,6.667,10

.667

]0.2196

0.1861

0.4057

Yeh

[65]

Trapezoidal,

Triangu

lar

T(A)=

[l e,ue;x e

,y e

]=

[l e−

x e,l e

,ue,ue+

y e]

T(A)=

[2.6667

,8.6667

;5.5998,4.0002

]=

[−2.9331

,2.6667

,8.6667

,12

.6669]

4.7907

0.0000

4.7907

ZengandLi[68

]Weigh

ted

Triangu

lar

T(A)=

[t 1;t 2

,t 3

]=[t 1

−t 2

,t 1

,t 2

,t 2

+t 3

]T

(A)=

[6.0375

,6.4796

,5.0678

]=

[−0.4421

,6.0375

,11

.1053]

0.2237

0.45

0.6737

Ban

[8,9]

Trapezoidal

T(A)=

[t 1,t 2

,t 3

,t 4

]T

(A)=

[−0.1333

,5.4667

,6.667,10

.667

]0.2196

0.1861

0.4057

Ban

[10,11

]Triangu

lar

T(A)=

[t 1;t 2

,t 3

]T

(A)=

[6.0288

,6.4572

,5.0856

]=

[−0.4284

,6.0288

,11

.1144]

0.2226

0.4512

0.6738

Ban

[14,15

]Trapezoidal,

Triangu

lar

T(A)=

[t 1,t 2

,t 3

,t 4

]T

(A)=

[−0.1333

,5.4667

,6.667,10

.667

]0.2196

0.1861

0.4057

Proposed

approxim

ation

Hexagonal

H(A)=

((CL,CU

;h1,h2,h3,h4);1

,uL,uR)=

((CL

−h1−

h2,CL

−h2,CL,CR,CR

+h3,

CR

+h3+

h4);1

,uL,uR)

H(A)=

((5,7;

3.9366

,1.3763

,1.1264

,2.7115

);1,0.65

,0.6)

=((

−0.3129,3.6237

,

5,7,8.1264

,10

.8379)

;1,0.65

,0.6)

0.2150

0.0857

0.3007

123


or trapezoidal FN/information for all FNs due to the lack ofspread constraints in the resultant FN.

The proposed hexagonal approximation is the generalisa-tion of all real, interval, triangular and trapezoidal approx-imations. Hence, it produces the approximations availablein the literature at different assumptions. Also our proposedapproximation gives any one of the approximation in the lit-erature for some fuzzy numberswhichmeans that ourmethodalso gives particular type of approximation like trapezoidal,triangular, interval and real numbers for some FNs becausewhich is the suitable approximation for given FN. Hence,we can conclude that our proposed hexagonal approxima-tion contains all sub type of approximations.

Some methods produces only intervals and symmetric tri-angular fuzzy numbers which are illogical and noted in theTable 4. Some more methods have not preserved even modalvalue / core which is also illogical since modal value / coreis completely a member of that fuzzy numbers. Furthermore,some methods will not produce triangular or trapezoidalfuzzy number for the positively skewed fuzzy number like(0, 1, 2, 144) given in Example 1. Due to the lack of the con-dition that the foots of the FN should be in ascending order(i.e., a1, a2, a3, a4) some of the methods like Zeng and Li[68], Gregorzewski’s (2005) approximation procedure mayfails to produce triangular or trapezoidal FN/information forthis FN. Hence, in our proposed method using KKT Theo-rem, we have included the condition. Our proposed approachgives linear approximation for the left leg and piecewise lin-ear approximation for the right leg by preserving the core.Therefore, trapezoidal approximation is the suitable approx-imation for the left leg and hexagonal approximation for theright leg, because it rectify all flaws in the other existingmethods which is mentioned in the table.

In Example 2, the given FN is asymmetric non linear FN.Some of the methods in the literature provides symmetrictriangular, interval FNs which are illogical. Based on theskewness in the given FN, our proposed approximation givesa suitable approximation, trapezoidal approximation for leftleg and hexagonal approximation for right leg. Hence, theloss/gain of information is less in our method which pro-duces better accurate approximation comparedwith the otherexisting methods.

The main objective of approximation is to de-fuzzify thefuzzy information/FNs into information of easy processingwithout loss/gain of much information. Geometrically, it isrepresented as the loss/gain of information of the resultantapproximated hexagonal fuzzy information/numbers is lessin area from thegiven fuzzy information/number. For the sakeof simplicity, Example 3 (−2, 5, 7, 12; 1, ML , MR) is shownin Table 5 as the illustration to compare the change of infor-mation with other existing methods. Based on the skewness,our proposed approximation gives a suitable approximationcalled hexagonal approximation for both left and right legsof a FN. The interval approximations to a FN are very easyfor processing but the change of information ismore compar-atively with the triangular and trapezoidal approximations.The superiority of the proposedmethod is shownandhence, itis concluded that the proposed method converts/de-fuzzifiesthe fuzzy numbers into hexagonal fuzzy numbers which arepiecewise linear that can be processed easier with minimalchange of information compared to triangular and trapezoidalinformation.

Application of the hexagonal approximationin MCDM using indexmatrix

In this section, (0, 1)-IMdefined inDefinition 2.8 is extendedto linguistic index matrix (LIM) and fuzzy index matrix(FIM) as follows.

The general form of LIM is given by [K , L, {LTki ,l j }] =

l1 · · · l j · · · lnk1 LTk1,l1 · · · LTk1,l j · · · LTk1,ln...

......

...

ki LTki ,l1 · · · LTki ,l j · · · LTki ,ln...

......

...

km LTkm ,l1 · · · LTkm ,l j · · · LTkm ,ln

where for every 1 ≤ i ≤ m and for 1 ≤ j ≤ n : LTki ,l j is alinguistic term that is taken from the set of linguistic terms S= {Very Low (VL), Fairly Low (FL), Moderately Low (ML),Low (L), Moderate (M), Good (G), Moderately Good (MG),Fairly Good (FG), Very Good (VG), High (H), ModeratelyHigh (MH), Fairly High (FH), Very High (VH)}.

123


The general form of FIM is given by [K , L, {( f1, f2, f3,f4; 1, ML , MU )ki ,l j }] =

l1 · · · lnk1 ( f1, f2, f3, f4; 1, ML , MU )k1,l1 · · · ( f1, f2, f3, f4; 1, ML , MU )k1,ln...

......

ki ( f1, f2, f3, f4; 1, ML , MU )ki ,l1 · · · ( f1, f2, f3, f4; 1, ML , MU )ki ,ln...

......

km ( f1, f2, f3, f4; 1, ML , MU )km ,l1 · · · ( f1, f2, f3, f4; 1, ML , MU )km ,ln

where for every 1 ≤ i ≤ m and for 1 ≤ j ≤ n :( f1, f2, f3, f4; 1, ML , MU )ki ,l j is a FNcorresponding to lin-guistic term.

Now, using LIM defined above, we introduce multi-criteria decision making based on LIM as follows.

Let I be a fixed set of indices and R be the setof all real numbers. Let G = {G1,G2, . . . ,Gm}, H ={H1, H2, . . . , Hn} ⊂ I . The general form of multi-criteriadecision making based on LIM ALIM is given as

H1 · · · Hj · · · Hn

G1 LTG1,H1 · · · LTG1,Hj · · · LTG1,Hn

......

......

Gi LTGi ,H1 · · · LTGi ,Hj · · · LTGi ,Hn

......

......

Gm LTGm ,H1 · · · LTGm ,Hj · · · LTGm ,Hn

where for every i, j (1 ≤ i ≤ m, 1 ≤ j ≤ n) : Gi is theobject being evaluated and Hj is the criterion taking part inthe evaluation and LTGi ,Hj is a linguistic term.

By converting each linguistic term into fuzzy number, weget the general form of multi-criteria decision making basedon FIM AF IM which is given as

H1 · · · Hn

G1 ( f1, f2, f3, f4; 1, ML , MU )G1,H1 · · · ( f1, f2, f3, f4; 1, ML , MU )G1,Hn...

......

Gi ( f1, f2, f3, f4; 1, ML , MU )Gi ,H1 · · · ( f1, f2, f3, f4; 1, ML , MU )Gi ,Hn...

......

Gm ( f1, f2, f3, f4; 1, ML , MU )Gm ,H1 · · · ( f1, f2, f3, f4; 1, ML , MU )Gm ,Hn

The general form of Hexagonal Fuzzy Index Matrix(HXFIM) [K , L, {((h1, h2, h3, h4, h5, h6); u, uL , uR)ki ,l j }],

the general form of multi-criteria decision making AHXF IM

based on HXFIM can be defined analogously.

Algorithm for groupmulti-criteria LIM

Let I be afixed set of indices.LetG = {G1,G2, . . . ,Gm}, H ={H1, H2, . . . , Hn} ⊂ I . Let LIM1,LIM2, . . . ,LIMk be lin-guistic index matrices given by multiple decision makersDM1,DM2, . . . ,DMk . The algorithmic procedure for thegroup multi-criteria LIM ALIM1 , ALIM2 , . . . , ALIMk can besummarized as follows:

1. Form the group multi-criteria linguistic index matrixALIM1 , ALIM2 , . . . , ALIMk given by multi-decision mak-ers DM1,DM2, . . . ,DMk .

2. Form the corresponding group multi-criteria ParabolicFuzzy Index Matrix (PFIM)APFIM1 , APFIM2 , . . . , APFIMk for the group multi-criterialinguistic index matrix ALIM1 , ALIM2 , . . . , ALIMk usingthe conversion Table.

3. The aggregated multi-criteria PFIM APF IM of cor-responding evaluations with respect to each decisionmakers of the given group multi-criteria PFIM by con-sidering equal weights to all decision makers is foundusing the row average aggregation for given PFIM[G, H , {(p1, p2, p3, p4; 1, ML , MU )Gi ,Hj }] defined by

123


H1 · · · Hn

G11

k

k∑

l=1(p1, p2, p3, p4; 1, ML , MU )G1,H1 · · · 1

k

k∑

l=1(p1, p2, p3, p4; 1, ML , MU )G1,Hn

......

...

Gi1

k

k∑

l=1(p1, p2, p3, p4; 1, ML , MU )Gi ,H1 · · · 1

k

k∑

l=1(p1, p2, p3, p4; 1, ML , MU )Gi ,Hn

......

...

Gm1

k

k∑

l=1(p1, p2, p3, p4; 1, ML , MU )Gm ,H1 · · · 1

k

k∑

l=1(p1, p2, p3, p4; 1, ML , MU )Gm ,Hn

where∑

is found using Definition 2.6.4. The ranking of FNs/PFNs has not been uniquely defined

in literature. But, a totally ordering principle has beendefined on the entire class of generalized hexagonalfuzzy numbers in [43]. Hence, a ranking of two fuzzynumbers P and Q can be found by hexagonal approx-imation which is defined as P ≤ Q if H(P) ≤H(Q) where P = (p1, p2, p3, p4; 1, MPL , MPR ) andQ = (q1, q2, q3, q4; 1, MQL , MQR ). Therefore, com-pute hexagonal approximation is in the form of LRrepresentation and then convert it into general represen-tation corresponding to each parabolic fuzzy numbers.

5. Find the score M using Definition 2.4 and if requiredS, LD, L A, RD, RA accordingly, to conclude whetherGp > Gq orGp < Gq orGp = Gq for allGp,Gq ∈ U .

6. Enumerate EA(Gp,Gq) using EA(Gp,Gq) = {a ∈H |Gp > Gq} and FA(Gp,Gq) using FA(Gp,Gq) ={a ∈ H |Gp = Gq}. Compute the weighted fuzzy domi-nance relation using WDA(Gp,Gq) : U × U → [0, 1]is defined by WDA(Gp,Gq) = ∑

a∈EA(Gp,Gq ) wa +∑

a∈FA(Gp ,Gq ) wa

2 .7. Evaluate the entire dominance degree of each alternative

using DA(Gp) = 1|U |

∑|U |q=1 WDA(Gp,Gq). The alter-

natives are ordered using entire dominance degree. Thelarger value of DA(Gp) is the best alternative.

For the sake of simplicity, the algorithm for group multi-criteria LIM is given below as Fig. 3.

Numerical illustration

In the real-life environment, multi-criteria group decision-makingproblem includes imprecise, indefinite and subjectiveinformation from human judgement and preference. Fuzzyset theory provides the flexibility to deal such information. Inthis sub-section, a problem of supplier selection under groupdecision-making environment in which the evaluations of

alternatives against each criterion are considered as linguis-tic variables is considered to show the applicability of theproposed hexagonal approximation.

A company with two decision makers DM1, DM2 ofequal weightage has to select the best warehouse to exporttheir manufactured goods for the sale based on cost, localitypopulation, warehouses area, distance to warehouses, per-formance of alternatives. After some pre-screening, thesealternatives (warehouses) G1,G2,G3,G4,G5,G6,G7,G8

remain for further evaluations under those decision makersbased on the criteria H1: cost, H2 : locality population, H3:warehouses area, H4: distance to warehouses H5: perfor-mance and weights for each criteria wa is given by W ={wa |a ∈ H} = {0.28, 0.25, 0.12, 0.15, 0.20}. Let LIM bea combined linguistic index matrix given in Table 6 whichconsists of individual linguistic index matrices LIM1,LIM2

given by decision makers DM1,DM2.Step 1 & 2: The conversion table of linguistic opinions ofdecision makers into parabolic fuzzy numbers is given inTable 7 and the group multi-criteria PFIM (APFIM1 , APFIM2)

is formed using Table 6 and is depicted in Table 8.Step 3: The aggregated multi-criteria PFIM APF IM withrespect to each decision makers of the given group multi-criteria PFIM by considering equal weights to all decisionmakers is found and is tabulated in Table 9 using the Defini-tion 2.6.Step 4: The ranking of alternatives in the form of parabolicfuzzy numbers is not possible in the literature, hence it isrequired to approximate the parabolic fuzzy numbers with-out much loss of information. Therefore, compute hexagonalapproximation is in the form of LR representation and thenconvert it into general representation corresponding to eachparabolic fuzzy numbers (Tables 10, 11).Step 5: Using Definition 2.4, the midpoint score M is foundand is tabulated in Table 12. If M(Gi/Hj ) = M(G j/Hj )

for any alternatives Gi ,G j then S and other necessary scorefunctions LD, L A, RD and RA are foundwhenever required(Table 13).

123


Fig. 3 Flowchart of algorithm for group multi-criteria LIM

Step 6: The weighted fuzzy dominance relation is evaluatedand is depicted in Table 14. For instance EA(G2,G1) ={H2, H4, H5} and FA(G2,G1) = {H3} and henceWDA(G2,

G1) = 0.25 + 0.15 + 0.20 + 0.06 = 0.660.Step 7: Now, the entire dominance degree of each alterna-tive using DA(Gi ) = 1

|U |∑|U |

j=1 WDA(Gi ,G j ) is found. For

instance DA(G2) = 18

∑8j=1 WDA(G2,G j ) = 0.473750.

By step 7 G6 is selected as the best alternative from theabove table (Table 15).

Conclusion and future scope

Even several defuzzification methods for fuzzy numbersexists in the literature, approximation of fuzzy numbers isthe effective framework among them. We also have sug-gested a new approach of hexagonal approximation for fuzzynumbers. The propounded operator called the hexagonalapproximation operator preserving the core possess manydesired properties and less loss of information. Furthermore,

123


Table 6 Combined LIM(ALIM1 , ALIM2 )

DM1 Cost(H1)

Localitypopulation(H2)

Warehousesarea (H3)

Distance towarehouses(H4)

Performance(H5)

G1 FG G FL G M

G2 VL MG ML MH VH

G3 H FH FG L FL

G4 G MG VG MG FL

G5 FL H VG ML G

G6 FH VH M ML MH

G7 MG MH VL FL M

G8 MH H FL L ML

DM2 H1 H2 H3 H4 H5

G1 VG MG ML MG G

G2 FL FG FL FH MH

G3 FH VH VG ML VL

G4 MG G FG G ML

G5 VL MH FG L M

G6 H FH G FL VH

G7 M H FL ML G

G8 H MH VL ML FL

Table 7 The conversion table of linguistic opinions into PFNs

Linguistic variables PFN

Very low (0.25, 0.40, 0.45, 0.48; 1, MAL , MAU )

Fairly low (0.25, 0.40, 0.45, 0.50; 1, MAL , MAU )

Moderately low (0.30, 0.35, 0.50, 0.53; 1, MAL , MAU )

Low (0.32, 0.35, 0.50, 0.55; 1, MAL , MAU )

Moderate (0.50, 0.60, 0.80, 0.80; 1, MAL , MAU )

Good (0.55, 0.65, 0.75, 0.80; 1, MAL , MAU )

Moderately good (0.55, 0.65, 0.75, 0.83; 1, MAL , MAU )

Fairly good (0.58, 0.65, 0.75, 0.85; 1, MAL , MAU )

Very good (0.60, 0.65, 0.75, 0.88; 1, MAL , MAU )

High (0.70, 0.88, 0.92, 0.93; 1, MAL , MAU )

Moderately high (0.70, 0.88, 0.92, 0.95; 1, MAL , MAU )

Fairly high (0.80, 0.85, 0.95, 0.97; 1, MAL , MAU )

Very high (0.80, 0.85, 0.95, 1.00; 1, MAL , MAU )

the betterment from other approximation methods has alsobeen discussed. Finally, the applicability of the proposedapproximation method is described by MCDM problem

using Index Matrix. Since any numerical data involved intheresearch experiment of engineering problem can be fitted byquadratic fuzzy number which may be approximated intoHexagonal fuzzy number without affecting much informa-tion, this proposed method has a wide applications in allEngineering problems. In near future, this methodology canbe extended to intuitionistic fuzzy numbers, neutrosophicfuzzy numbers, hesitancy fuzzy numbers and dual hesitancyfuzzy numbers by whichMCDMmay be studied and appliedin many real-life applications where non-linear informationplays a vital role. Therefore, it opens a new area of researchof approximation of nonlinear fuzzy numbers by piecewiselinear fuzzy numbers of different shapes and applications ofdifferent engineering and management problems.

123


Table8

The

groupmulti-criteriaPF

IM(APF

IM1,PFIM

2)

PFIM

H1

H2

H3

H4

H5

Decisionmaker

1

G1

(0.58,0.65

,0.75

,0.85

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.84

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

(0.50,

0.60

,0.80

,0.80

;1,

MAL,M

AU)

G2

(0.25,0.40

,0.45

,0.48

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,1;

1,M

AL,M

AU)

G3

(0.70,

0.88

,0.92

,0.93

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,0.97

;1,

MAL,M

AU)

(0.58,0.65

,0.75

,0.85

;1,

MAL,M

AU)

(0.32,

0.35

,0.50

,0.55

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

G4

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.60,

0.65

,0.75

,0.88

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

G5

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.93

;1,

MAL,M

AU)

(0.60,

0.65

,0.75

,0.88

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

G6

(0.80,

0.85

,0.95

,0.97

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,1;

1,M

AL,M

AU)

(0.50,

0.60

,0.80

,0.80

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

G7

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.48

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.50,

0.60

,0.80

,0.80

;1,

MAL,M

AU)

G8

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.93

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.32,

0.35

,0.50

,0.55

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

Decisionmaker

2

G1

(0.60,

0.65

,0.75

,0.88

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

G2

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.58,0.65

,0.75

,0.85

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,0.97

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

G3

(0.80,

0.85

,0.95

,0.97

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,1;

1,M

AL,M

AU)

(0.60,

0.65

,0.75

,0.88

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.48

;1,

MAL,M

AU)

G4

(0.55,0.65

,0.75

,0.83

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

(0.58,0.65

,0.75

,0.85

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

G5

(0.25,0.40

,0.45

,0.48

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

(0.58,0.65

,0.75

,0.85

;1,

MAL,M

AU)

(0.32,

0.35

,0.50

,0.55

;1,

MAL,M

AU)

(0.50,

0.60

,0.80

,0.80

;1,

MAL,M

AU)

G6

(0.70,

0.88

,0.92

,0.93

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,0.97

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.80,

0.85

,0.95

,1;

1,M

AL,M

AU)

G7

(0.50,

0.60

,0.80

,0.80

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.93

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.55,0.65

,0.75

,0.80

;1,

MAL,M

AU)

G8

(0.70,

0.88

,0.92

,0.93

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.95

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.48

;1,

MAL,M

AU)

(0.30,

0.35

,0.50

,0.53

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.50

;1,

MAL,M

AU)

123


Table9

AggregatedPF

IM

APF

IMH1

H2

H3

H4

H5

G1

(0.59,0.65

,0.75

,0.865;

1,M

AL,M

AU)

(0.55,0.65

,0.75

,0.815;

1,M

AL,M

AU)

(0.275

,0.375,0.475,0.515;

1,M

AL,M

AU)

(0.55,0.65

,0.75

,0.815;

1,M

AL,M

AU)

(0.525

,0.625,0.775,0.8;

1,M

AL,M

AU)

G2

(0.25,0.40

,0.45

,0.49

;1,

MAL,M

AU)

(0.565

,0.65

,0.75

,0.84

;1,

MAL,M

AU)

(0.275

,0.375,0.475,0.575;

1,M

AL,M

AU)

(0.75,0.865,0.935,0.96

;1,

MAL,M

AU)

(0.75,0.865,0.935,0.975;

1,M

AL,M

AU)

G3

(0.75,0.865,0.935,0.95

;1,

MAL,M

AU)

(0.8

,0.85

,0.95

,0.985;

1,M

AL,M

AU)

(0.59,0.65

,0.75

,0.865;

1,M

AL,M

AU)

(0.31,0.35

,0.50

,0.54

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.49

;1,

MAL,M

AU)

G4

(0.55,0.65

,0.75

,0.815;

1,M

AL,M

AU)

(0.55,0.65

,0.75

,0.815;

1,M

AL,M

AU)

(0.59,0.65

,0.75

,0.865;

1,M

AL,M

AU)

(0.55,0.65

,0.75

,0.815;

1,M

AL,M

AU)

(0.275

,0.375,0.475,0.515;

1,M

AL,M

AU)

G5

(0.25,0.4,0.45

,0.49

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.94

;1,

MAL,M

AU)

(0.59,0.65

,0.75

,0.865;

1,M

AL,M

AU)

(0.31,0.35

,0.50

,0.54

;1,

MAL,M

AU)

(0.525

,0.625,0.775,0.80

;1,

MAL,M

AU)

G6

(0.75,0.865,0.935,0.95

;1,

MAL,M

AU)

(0.8

,0.85

,0.95

,0.985;

1,M

AL,M

AU)

(0.525

,0.625,0.775,0.8;

1,M

AL,M

AU)

(0.275

,0.375,0.475,0.515;

1,M

AL,M

AU)

(0.75,0.865,0.935,0.975;

1,M

AL,M

AU)

G7

(0.525

,0.625,0.775,0.815;

1,M

AL,M

AU)

(0.70,

0.88

,0.92

,0.94

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.49

;1,

MAL,M

AU)

(0.275

,0.375,0.475,0.515;

1,M

AL,M

AU)

(0.525

,0.625,0.775,0.80

;1,

MAL,M

AU)

G8

(0.70,

0.88

,0.92

,0.94

;1,

MAL,M

AU)

(0.70,

0.88

,0.92

,0.94

;1,

MAL,M

AU)

(0.25,0.40

,0.45

,0.49

;1,

MAL,M

AU)

(0.31,0.35

,0.50

,0.54

;1,

MAL,M

AU)

(0.275

,0.375,0.475,0.515;

1,M

AL,M

AU)

123


Table10

LRhexagonalapproximationof

PFIM

G/H

H1

H2

H3

H4

H5

G1

(0.650

,0.750,

0.01004210

,

0.04565414

,0.07893290

,0.02524700

;1,0.05716800

,0.09836190

)

(0.650

,0.750,

0.02077290

,

0.07032440

,0.04890060

,0.01126960

;1,0.08806400

,0.06134660

)

(0.375

,0.475,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

(0.650

,0.750,

0.02077290

,

0.09032440

,0.04890060

,0.01126960

;1,0.08806400

,0.06134660

)

(0.625

,0.775,0.02077290

,

0.07032440

,0.02106540

,0.00275420

;1,0.08806400

,0.02476930

)

G2

(0.400

,0.450,

0.03641310

,

0.09798120

,0.03206330

,0.00555569

;1,0.12026500

,0.03936960

)

(0.650

,0.750,

0.01652300

,

0.06139570

,0.06441040

,0.07791270

;1,0.07711590

,0.08084280

)

(0.375

,0.475,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

(0.865

,0.935,0.02524700

,

0.07893290

,0.02106540

,0.00275420

;1,0.09836190

,0.02476930

)

(0.865

,0.935,0.02524700

,

0.07893290

,0.03206330

,0.00555569

;1,0.09836190

,0.03936960

)

G3

(0.865

,0.935,0.02524700

,

0.07893290

,0.01319590

,0.00126288

;1,0.09836190

,0.01446580

)

(0.850

,0.950,

0.00770756

,

0.03898920

,0.02848710

,0.00455900

;1,0.04849510

,0.03462630

)

(0.650

,0.750,

0.01004210

,

0.04565410

,0.07893290

,0.02524700

;1,0.05716800

,0.09836190

)

(0.350

,0.500,

0.00555569

,

0.03206330

,0.03206330

,0.00555569

;1,0.03936960

,0.03936960

)

(0.400

,0.450,

0.03641310

,

0.09798120

,0.03206330

,0.00555569

;1,0.12026500

,0.03936960

)

G4

(0.650

,0.750,

0.02077290

,

0.07032440

,0.04890060

,0.01126960

;1,0.08806400

,0.06134660

)

(0.650

,0.750,

0.02077290

,

0.07032440

,0.04890060

,0.01126960

;1,0.08806400

,0.06134660

)

(0.650

,0.750,

0.01004210

,

0.045654100,

0.07893290

,0.02524700

;1,0.05716800

,0.09836190

)

(0.650

,0.750,

0.02077290

,

0.07032440

,0.04890060

,0.01126960

;1,0.08806400

,0.06134660

)

(0.375

,0.475,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

G5

(0.400

,0.450,

0.03641310

,

0.09798120

,0.03206330

,0.00555569

;1,0.12026500

,0.03936960

)

(0.880

,0.920,

0.04665210

,

0.11335400

,0.01719540

,0.00196320

;1,0.13708900

,0.01966410

)

(0.650

,0.750,

0.01004210

,

0.04565410

,0.07893290

,0.02524700

;1,0.05716800

,0.09836190

)

(0.350

,0.500,

0.00555569

,

0.03206330

,0.03206330

,0.00555569

;1,0.03936960

,0.03936960

)

(0.625

,0.775,0.02077290

,

0.07032440

,0.02106540

,0.00275420

;1,0.08806400

,0.02476930

)

G6

(0.865

,0.935,0.02524700

,

0.07893290

,0.01319590

,0.00126288

;1,0.09836190

,0.01446580

)

(0.850

,0.950,

0.00770756

,

0.07032440

,0.02106540

,0.00275420

;1,0.08806400

,0.02476930

)

(0.625

,0.775,0.02077290

,

0.07032440

,0.02106540

,0.00275420

;1,0.08806400

,0.02476930

)

(0.375

,0.475,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

(0.865

,0.935,0.02524700

,

0.07893290

,0.03206330

,0.00555569

;1,0.09836190

,0.03936960

)

G7

(0.625

,0.775,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

(0.880

,0.920,

0.04665210

,

0.11335400

,0.01719540

,0.00196320

;1,0.13708900

,0.01966400

)

(0.400

,0.450,

0.03641310

,

0.09798120

,0.03206330

,0.00555569

;1,0.12026500

,0.03936960

)

(0.375

,0.475,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

(0.625

,0.775,0.02077290

,

0.07032440

,0.02106540

,0.02754200

;1,0.08806400

,0.02476930

)

G8

(0.880

,0.920,

0.04665210

,

0.11335400

,0.01719540

,0.00196320

;1,0.13708900

,0.01966410

)

(0.880

,0.920,

0.04665210

,

0.11335400

,0.01719540

,0.00196320

;1,0.13708900

,0.01966410

)

(0.400

,0.450,

0.03641310

,

0.09798120

,0.03206330

,0.00555569

;1,0.12026500

,0.03936960

)

(0.350

,0.500,

0.00555569

,

0.03206330

,0.03206330

,0.00555569

;1,0.03936960

,0.03936960

)

(0.375

,0.475,0.02077290

,

0.07032440

,0.03206330

,0.00555569

;1,0.08806400

,0.03936960

)

123


Table11

Generalrepresentatio

nof

hexagonalapproximation

G/H

H1

H2

H3

H4

H5

G1

(0.59430380,

0.60434590

,0.650,

0.750,

0.82893290

,0.85417990

;1,0.00571680

,0.09836190

)

(0.55890270,

0.57967560

,0.650,

0.750,

0.79890060

,0.81017020

;1,0.08806400

,0.06134660

)

(0.28390270,

0.30467560

,0.375,

0.475,0.50706330

,0.51261899

;1,0.08806400

,0.03936960

)

(0.55890270,

0.57967560

,0.650,

0.750,

0.79890060

,0.81017020

;1,0.08806400

,0.06134660

)

(0.53390270,

0.55467560

,0.625,

0.775,0.79606540

,0.79881960

;1,0.08806400

,0.02476930

)

G2

(0.26560570,

0.30201880

,0.400,

0.450,

0.48206330

,0.48761899

;1,0.12026500

,0.03936960

)

(0.57208130,

0.58860430

,0.650,

0.750,

0.81441040

,0.89232310

;1,0.07711590

,0.08084280

)

(0.28390270,

0.30468560

,0.375,

0.475,0.50706330

,0.51261999

;1,0.08806400

,0.03936960

)

(0.76082010,

0.78606710

,0.865,

0.935,0.95606000

,0.95881960

;1,0.09836190

,0.02476930

)

(0.76082010,

0.78606710

,0.865,

0.935,0.96706330

,0.97261899

;1,0.09836190

,0.03936960

)

G3

(0.76082010,

0.78606710

,0.865,

0.935,0.94819590

,0.94945878

;1,0.09836190

,0.01446580

)

(0.80330324,0.81101080

,0.850,

0.950,

0.97848710

,0.98304610

;1,0.04849510

,0.03462630

)

(0.59430380,

0.60434590

,0.650,

0.750,

0.82893290

,0.85417990

;1,0.05716800

,0.09836190

)

(0.31238101,0.31793670

,0.350,

0.500,

0.53206330

,0.53761899

;1,0.03936960

,0.03936960

)

(0.26560570,

0.30201880

,0.400,

0.450,

0.48206330

,0.48761899

;1,0.12026500

,0.03936960

)

G4

(0.55890270,

0.57967560

,0.650,

0.750,

0.79890060

,0.81017020

;1,0.08806400

,0.06134660

)

(0.55890270,

0.57967560

,0.650,

0.750,

0.79890060

,0.81017020

;1,0.08806400

,0.06134660

)

(0.59430380,

0.60434590

,0.650,

0.750,

0.82893290

,0.85417990

;1,0.05716800

,0.09836190

)

(0.55890270,

0.57967560

,0.650,

0.750,

0.79890060

,0.81017020

;1,0.08806400

,0.06134660

)

(0.28390270,

0.30467560

,0.375,

0.475,0.50706330

,0.51261899

;1,0.08806400

,0.03936960

)

G5

(0.26560570,

0.30201880

,0.400,

0.450,

0.48206330

,0.48761899

;1,0.12026500

,0.03936960

)

(0.71999390,

0.76664600

,0.880,

0.920,

0.93719540

,0.93915860

;1,0.13708900

,0.01966410

)

(0.59430380,

0.60434590

,0.650,

0.750,

0.82893290

,0.85417990

;1,0.05716800

,0.09836190

)

(0.31238101,0.31793690

,0.350,

0.500,

0.53206330

,0.53761899

;1,0.03936960

,0.03936960

)

(0.53390270,

0.55467560

,0.625,

0.775,0.79606540

,0.79881960

;1,0.08806400

,0.02476930

)

G6

(0.76082010,

0.78606710

,0.865,

0.935,0.94819590

,0.94945878

;1,0.09836190

,0.01446580

)

(0.80330324,0.81101080

,0.850,

0.950,

0.97848710

,0.98304610

;1,0.08806400

,0.02476930

)

(0.53309270,

0.55467560

,0.625,

0.775,0.79606540

,0.79881960

;1,0.08806400

,0.02476930

)

(0.28390270,

0.30467560

,0.375,

0.475,0.50706330

,0.51261899

;1,0.08806400

,0.03936960

)

(0.76082010,

0.78606710

,0.865,

0.935,0.96706330

,0.97261899

;1,0.09836190

,0.03936960

)

G7

(0.53390270,

0.55467560

,0.625,

0.775,0.80706330

,0.81268990

;1,0.08806400

,0.03936960

)

(0.71999390,

0.76664600

,0.880,

0.920,

0.93719540

,0.93915860

;1,0.13708900

,0.01966400

)

(0.26560570,

0.30201880

,0.400,

0.450,

0.48206330

,0.48761899

;1,0.12026500

,0.03936960

)

(0.28390270,

0.30467560

,0.375,

0.475,0.50706330

,0.51261899

;1,0.08806400

,0.03936960

)

(0.53390270,

0.55467560

,0.625,

0.775,0.79606540

,0.79881960

;1,0.08806400

,0.02476930

)

G8

(0.71999390,

0.76664600

,0.880,

0.920,

0.93719540

,0.93915860

;1,0.13708900

,0.01966410

)

(0.71999390,

0.76664600

,0.880,

0.920,

0.93719540

,0.93915860

;1,0.13708900

,0.01966410

)

(0.26560570,

0.30201880

,0.400,

0.450,

0.48206330

,0.48761899

;1,0.12026500

,0.03936960

)

(0.31238101,0.31793670

,0.350,

0.500,

0.53206330

,0.53761890

;1,0.03936960

,0.03936960

)

(0.28390270,

0.30467560

,0.375,

0.475,0.50706330

,0.51261899

;1,0.08806400

,0.03936960

)

123


Table12

Scorefunctio

nsM,S

,LDof

HXFN

s

MS

LD

G/H

H1

H2

H3

H4

H5

H1

H2

H3

H4

H5

H1

H2

H3

H4

H5

G1

0.70000000

0.70000000

0.42500000

0.70000000

0.70000000

0.05000000

0.05000000

0.05000000

0.05000000

0.07500000

−0.00171450

−0.00318760

−0.00318760

−0.00318760

−0.00318760

G2

0.42500000

0.70000000

0.42500000

0.90000000

0.90000000

0.02500000

0.05000000

0.05000000

0.03500000

−0.00506250

−0.00262860

−0.00318760

−0.00374990

G3

0.90000000

0.90000000

0.70000000

0.42500000

0.42500000

0.03500000

0.05000000

0.05000000

0.07500000

0.02500000

−0.00374990

−0.00136070

−0.00171450

−0.00101870

G4

0.70000000

0.70000000

0.70000000

0.70000000

0.42500000

0.05000000

0.05000000

0.05000000

0.05000000

0.05000000

−0.00318760

−0.00318760

−0.00171450

−0.00318760

−0.00318760

G5

0.42500000

0.90000000

0.70000000

0.42500000

0.70000000

0.02500000

0.02000000

0.05000000

0.07500000

0.07500000

−0.00506250

−0.00617930

−0.00171450

−0.00101870

−0.00318760

G6

0.90000000

0.90000000

0.70000000

0.42500000

0.90000000

0.03500000

0.05000000

0.07500000

0.05000000

0.03500000

−0.00374990

−0.00136070

−0.00318760

−0.00374990

G7

0.70000000

0.90000000

0.42500000

0.42500000

0.70000000

0.07500000

0.02000000

0.02500000

0.05000000

0.07500000

−0.00617930

−0.00506250

−0.00318760

−0.00318760

G8

0.90000000

0.90000000

0.42500000

0.42500000

0.42500000

0.02000000

0.02000000

0.02500000

0.07500000

0.05000000

−0.00617930

−0.00506250

−0.00101870

−0.00318760

The

bold

lettersareused

todistinguishtheequalityandnonequalityin

scores

fordifferentalternatives

which

may

bemaintained

Table13

Scorefunctio

nsLA,R

D,R

Aof

HXFN

s

LA

RD

RA

G/H

H1

H2

H3

H4

H5

H1

H2

H3

H4

H5

H1

H2

H3

H4

H5

G1

0.00719882

0.00719882

0.00719882

0.00719882

−0.0018946

−0.0007365

−0.0018946

−0.0005411

0.00374021

0.00204133

0.00374021

0.00083605

G2

0.01314401

0.00719882

0.00887358

−0.0010187

−0.0007365

−0.0010187

0.00175918

0.00204133

0.00175918

G3

0.00887358

0.00249303

0.00330654

0.00175918

−0.0002634

−0.0008537

−0.0037499

−0.0010187

0.00036801

0.00142582

0.00887358

0.00175918

G4

0.00719882

0.00330654

0.00719882

0.00719882

−0.0018946

−0.0037499

−0.0018946

−0.0010187

0.00374021

0.00887358

0.00374021

0.00175918

G5

0.01314401

0.01714679

0.00330654

0.00175918

0.00719882

−0.0010187

−0.0003966

−0.0037499

−0.0010187

−0.0005411

0.00175918

0.00058498

0.00887358

0.00175918

0.00083605

G6

0.00887358

0.00249303

0.00719882

0.00887358

−0.0002634

−0.0008537

−0.0010187

−0.0010187

0.00036801

0.00142582

0.00175918

0.00175918

G7

0.01714679

0.01314401

0.00719882

0.00719882

−0.0003966

−0.0010187

−0.0010187

−0.0005411

0.00058498

0.00175918

0.00175918

0.00083605

G8

0.01714679

0.01314401

0.00175918

0.00719882

−0.0003966

−0.0010187

−0.0010187

−0.0010187

0.00058498

0.00175918

0.00175918

0.00175918

The

bold

lettersareused

todistinguishtheequalityandnonequalityin

scores

fordifferentalternatives

which

may

bemaintained

123


Table 14 Weighted dominance relation

WDA(Gi ,G j ) G1 G2 G3 G4 G5 G6 G7 G8

G1 0.500 0.340 0.350 0.680 0.530 0.150 0.370 0.470G2 0.660 0.500 0.350 0.600 0.490 0.250 0.470 0.470G3 0.650 0.650 0.500 0.590 0.665 0.415 0.800 0.725G4 0.320 0.400 0.410 0.500 0.490 0.150 0.270 0.370G5 0.470 0.510 0.335 0.510 0.500 0.150 0.495 0.520G6 0.850 0.750 0.585 0.850 0.850 0.500 0.925 0.850G7 0.630 0.530 0.200 0.730 0.505 0.075 0.500 0.385G8 0.530 0.530 0.275 0.630 0.480 0.150 0.615 0.500

Table 15 Dominance relation

Gi G1 G2 G3 G4 G5 G6 G7 G8

DA(Gi ) 0.423750 0.473750 0.624375 0.363750 0.436250 0.770000 0.444375 0.463750

Acknowledgements The Authors thank the anonymous referees andeditors for their valuable suggestions and queries which improved thequality of the paper. This work was supported by UGC-Rajiv GanhiNational Fellowship Scheme (RGNF-2014-15-SC-TAM-74350).

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