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Modified methods to solve interval-valued Fermatean fuzzy multi-criteria decision-making problems

  • In a recently published paper, two methods were proposed to solve interval-valued Fermatean fuzzy multi-criteria decision-making problems (those in which the rating value of each alternative over each criterion is represented by an interval-valued Fermatean fuzzy number). In this paper, some numerical examples are considered to show that these existing methods fail to find the correct ranking of the alternatives. Also, the reasons for the failure of these existing methods are pointed out. Furthermore, new methods are proposed to solve the interval-valued Fermatean fuzzy multi-criteria decision-making problems by modifying existing methods. Moreover, the proposed modified methods are illustrated with the help of numerical examples. Finally, the ranking of the alternatives of the two existing real-life interval-valued Fermatean fuzzy multi-criteria decision-making problems is obtained by the proposed methods.

    Citation: Raina Ahuja, Meraj Ali Khan, Parul Tomar, Amit Kumar, S. S. Appadoo, Ibrahim Al-Dayel. Modified methods to solve interval-valued Fermatean fuzzy multi-criteria decision-making problems[J]. AIMS Mathematics, 2025, 10(4): 9150-9170. doi: 10.3934/math.2025421

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  • In a recently published paper, two methods were proposed to solve interval-valued Fermatean fuzzy multi-criteria decision-making problems (those in which the rating value of each alternative over each criterion is represented by an interval-valued Fermatean fuzzy number). In this paper, some numerical examples are considered to show that these existing methods fail to find the correct ranking of the alternatives. Also, the reasons for the failure of these existing methods are pointed out. Furthermore, new methods are proposed to solve the interval-valued Fermatean fuzzy multi-criteria decision-making problems by modifying existing methods. Moreover, the proposed modified methods are illustrated with the help of numerical examples. Finally, the ranking of the alternatives of the two existing real-life interval-valued Fermatean fuzzy multi-criteria decision-making problems is obtained by the proposed methods.



    In the last few years, several extensions of the fuzzy set [1] have been proposed in the literature. In 2019, by generalizing the existing definitions [2] of a Pythagorean fuzzy set and a Pythagorean fuzzy number, Senapati and Yager [3] proposed the definitions of a Fermatean fuzzy set and a Fermatean fuzzy number. On the same direction, by generalizing the existing definitions [4] of an interval-valued Pythagorean fuzzy set and interval-valued Pythagorean fuzzy number, Jeevaraj [5] proposed the definitions of an interval-valued Fermatean fuzzy set (IVFFS) and an interval-valued Fermatean fuzzy number (IVFFN).

    Jeevaraj [5] also proposed a method for comparing two interval-valued Fermatean fuzzy numbers (IVFFNs). Before this definition of an IVFFN, there was no ranking method in the literature to compare IVFFNs. Also, interval-valued intuitionistic fuzzy numbers and interval-valued Pythagorean fuzzy numbers are special types of IVFFNs, considered by Jeevaraj [5] to show the advantage of the proposed ranking method. The author then showed that existing methods [6,7,8,9,10] for comparing interval-valued intuitionistic fuzzy numbers failed to distinguish the distinct interval-valued intuitionistic fuzzy numbers, and other existing methods [4,11] for comparing interval-valued Pythagorean fuzzy numbers failed to distinguish the considered distinct interval-valued Pythagorean fuzzy numbers. However, the proposed ranking method did not fail to distinguish the considered distinct interval-valued intuitionistic and interval-valued Pythagorean fuzzy numbers. The author also proved mathematically that their proposed ranking method will never fail to distinguish two distinct IVFFNs.

    Using the proposed ranking method, Jeevaraj [5] also proposed two methods to solve interval-valued Fermatean fuzzy multi-criteria decision-making (IVFFMCDM) problems. In the first method, an IVFFMCDM problem is solved by transforming it into its equivalent crisp multi-criteria decision-making (MCDM) problem. In the second method, an IVFFMCDM problem is solved without transforming it into its equivalent crisp MCDM problem.

    Afterward, several researchers used Jeevaraj [5]'s methods to solve IVFFMCDM problems. However, in this paper, it is pointed out that Jeevaraj [5]'s methods fail to find the correct ranking of the alternatives. Hence, it is not appropriate to use Jeevaraj [5]'s methods. To validate this claim, two IVFFMCDM problems are solved using Jeevaraj [5]'s methods, showing that the obtained alternative ranking is not correct. To resolve this, Jeevaraj [5]'s methods are modified. Furthermore, to illustrate the modified methods, the correct ranking of the alternatives of the considered IVFFMCDM problems is obtained by the modified methods. Finally, the ranking of the alternatives of the two real-life IVFFMCDM problems, considered by Jeevaraj [5] to illustrate his proposed approach, is obtained by the modified methods.

    This paper is organized as follows: In Section 2, some basic definitions are discussed. In Section 3, Jeevaraj [5]'s ranking method for comparing IVFFNs is discussed. In Section 4, Jeevaraj [5]'s methods for solving IVFFMCDM problems are discussed. In Section 5, two IVFFMCDM problems are solved using Jeevaraj [5]'s methods, and it is shown that the obtained alternative ranking is not correct. In Section 6, the reasons for the inappropriateness of Jeevaraj [5]'s methods are discussed. In Section 7, Jeevaraj [5]'s methods for solving IVFFMCDM problems are modified to resolve their inappropriateness. In Section 8, the appropriateness of the modified methods is discussed. In Section 9, the correct ranking of the alternatives of the considered IVFFMCDM problems is obtained by the modified methods. In Section 10, the real-life IVFFMCDM problems, considered by Jeevaraj [5] to illustrate his proposed method, are solved by the modified methods. Section 11 concludes the paper.

    In this section, some basic definitions are discussed.

    Definition 2.1. [3] A set F={x,μF(x),νF(x):xX}, defined on the universal set X, is said to be a Fermatean fuzzy set if the condition 0<(μF(x))3+(νF(x))31 xX is satisfied, where μF(x)[0,1] and νF(x)[0,1] represent the degree of membership and the degree of non-membership of the element x belonging to the set F, respectively. For any Fermatean fuzzy set F and xX,πF(x)=1(μF(x))3+(νF(x))3 is called the degree of hesitancy or the degree of indeterminacy of the element x belonging to the set F. Also, the number F=(μF,νF) is called a Fermatean fuzzy number.

    Definition 2.2. [5] A set F={x,[μFL(x),μFU(x)],[νFL(x),νFU(x)]:xX}, defined on the universal set X, is said to be an IVFFS if the condition 0<(μFU(x))3+(νFU(x))31 xX is satisfied, where [μFL(x),μFU(x)][0,1] and [νFL(x),νFU(x)][0,1] represent the intervals of the degree of membership and the degree of non-membership of the element x belonging to the set F, respectively. For any IVFFS F and xX,πF(x)=[πFL(x),πFU(x)]=[31(μFU(x))3(νFU(x))3,31(μFL(x))3(νFL(x))3] is called the interval of degree of hesitancy or the interval of degree of indeterminacy of the element x belonging to the set F. Also, the number F=([μFL(x),μFU(x)],[νFL(x),νFU(x)]) is called an IVFFN.

    Definition 2.3. [5] Let F=([μFL,μFU],[νFL,νFU]),F1=([μF1L,μF1U],[νF1L,νF1U]) and F2=([μF2L,μF2U],[νF2L,νF2U]) be three IVFFNs. Then

    (ⅰ) F1F2=([3μ3F1L+μ3F2Lμ3F1Lμ3F2L,3μ3F1U+μ3F2Uμ3F1Uμ3F2U],[νF1LνF2L,νF1UνF2U]).

    (ⅱ) F1F2=([μF1LμF2L,μF1UμF2U],[3ν3F1L+ν3F2Lν3F1Lν3F2L,3ν3F1U+ν3F2Uν3F1Uν3F2U]).

    (ⅲ) λF=([31(1μ3FL)λ,31(1μ3FU)λ],[νλFL,νλFU]),λ>0.

    (ⅳ) λF=(,[μλFL,μλFU][31(1ν3FL)λ,31(1ν3FU)λ]),λ>0.

    Definition 2.4. [5] Let F1=[μF1L,μF1U],[νF1L,νF1U] and F2=[μF2L,μF2U],[νF2L,νF2U] be any two IVFFNs. Then DGE(F1,F2) represents the generalized Euclidean distance between F1 and F2 and it can be evaluated by expression (1).

    DGE(F1,F2)=(μ3F1Lμ3F2L)2+(μ3F1Uμ3F2U)2+(ν3F1Lν3F2L)2+(ν3F1Uν3F2U)2+((1μ3F1Uν3F1U)(1μ3F2Uν3F2U))2+((1μ3F1Lν3F1L)(1μ3F2Lν3F2L))26. (1)

    Jeevaraj [5] proposed the following ranking method to compare two IVFFNs F1=([μF1L,μF1U],[νF1L,νF1U]) and F2=([μF2L,μF2U],[νF2L,νF2U]).

    Step 1: Evaluate JM(F1)=μ3F1L+μ3F1Uν3F1Lν3F1U2, JM(F2)=μ3F2L+μ3F2Uν3F2Lν3F2U2 and check that JM(F1)>JM(F2) or JM(F1)<JM(F2) or JM(F1)=JM(F2).

    Case (ⅰ): If JM(F1)<JM(F2) then F1<F2. Hence, maximum {F1,F2}=F2 and minimum {F1,F2}=F1.

    Case (ⅱ): If JM(F1)>JM(F2) then F1>F2. Hence, maximum {F1,F2}=F1 and minimum {F1,F2}=F2.

    Case (ⅲ): If JM(F1)=JM(F2) then go to Step 2.

    Step 2: Evaluate JH(F1)=μ3F1L+μ3F1U+ν3F1L+ν3F1U2, JH(F2)=μ3F2L+μ3F2U+ν3F2L+ν3F2U2 and check that JH(F1)>JH(F2) or JH(F1)<JH(F2) or JH(F1)=JH(F2).

    Case (ⅰ): If JH(F1)<JH(F2) then F1<F2. Hence, maximum {F1,F2}=F2 and minimum {F1,F2}=F1.

    Case (ⅱ): If JH(F1)>JH(F2) then F1>F2. Hence, maximum {F1,F2}=F1 and minimum {F1,F2}=F2.

    Case (ⅲ): If JH(F1)=JH(F2) then go to Step 3.

    Step 3: Evaluate JP(F1)=μ3F1L+μ3F1U+ν3F1Lν3F1U2, JP(F2)=μ3F2L+μ3F2U+ν3F2Lν3F2U2 and check that JP(F1)>JP(F2) or JP(F1)<JP(F2) or JP(F1)=JP(F2).

    Case (ⅰ): If JP(F1)>JP(F2) then F1<F2. Hence, maximum {F1,F2}=F2 and minimum {F1,F2}=F1.

    Case (ⅱ): If JP(F1)<JP(F2) then F1>F2. Hence, maximum {F1,F2}=F1 and minimum {F1,F2}=F2.

    Case (ⅲ): If JH(F1)=JH(F2) then go to Step 4.

    Step 4: Evaluate JC(F1)=μ3F1L+μ3F1Uν3F1L+ν3F1U2, JC(F2)=μ3F2L+μ3F2Uν3F2L+ν3F2U2 and check that JC(F1)<JC(F2) or JC(F1)>JC(F2) or JC(F1)=JC(F2).

    Case (ⅰ): If JC(F1)<JC(F2) then F1<F2. Hence, maximum {F1,F2}=F2 and minimum {F1,F2}=F1.

    Case (ⅱ): If JC(F1)>JC(F2) then F1>F2. Hence, maximum {F1,F2}=F1 and minimum {F1,F2}=F2.

    Case (ⅲ): If JC(F1)=JC(F2) then F1=F2. Hence, maximum {F1,F2}=minimum{F1,F2}=F1=F2.

    Jeevaraj [5] proposed two methods to solve IVFFMCDM problems. In this section, both methods are discussed.

    Jeevaraj [5] proposed the following method to solve IVFFMCDM problems having an interval-valued Fermatean fuzzy (IVFF) decision matrix M=(mij)r×s=([μLij,μUij],[νLij,νUij])r×s, where the IVFFN mij represents the rating value of the ith alternative Ai over the jth benefit criteria Crj.

    Step 1: Transform the IVFF decision matrix M into the weighted IVFF decision matrix M'=(m'ij)r×s, where m'ij=mij.wj,i=1tor,j=1tos, the non-negative real number wj is the weight of the jth benefit criteria, and sj=1wj=1.

    Step 2: Using the following steps, transform the weighted IVFF decision matrix M' into the weighted crisp decision matrix S=(sij)r×s.

    Step 2(a): Transform the weighted IVFF decision matrix M' into the weighted crisp decision matrix S=(JM(m'ij))r×s and check that all the elements JM(m'ij) of the weighted crisp decision matrix S are distinct or not.

    Case (ⅰ): If all the elements JM(m'ij) are distinct, then go to Step 3.

    Case (ⅱ): If some elements JM(m'ij) are equal, then replace such elements JM(m'ij) with JH(m'ij) and go to Step 2(b).

    Step 2(b): Check if all the elements JH(m'ij) are distinct or not.

    Case (ⅰ): If all the elements JH(m'ij) are distinct, then go to Step 3.

    Case (ⅱ): If some elements JH(m'ij) are equal, then replace such elements JH(m'ij) with JP(m'ij) and go to Step 2(c).

    Step 2(c): Check if all the elements JP(m'ij) are distinct or not.

    Case (ⅰ): If all the elements JP(m'ij) are distinct, then go to Step 3.

    Case (ⅱ): If some elements JP(m'ij) are equal, then replace such elements JP(m'ij) with JC(m'ij) and go to Step 3.

    Step 3: Evaluate the interval-valued positive ideal solution (IVFFPIS) PI=(pj)1×s and the interval-valued negative ideal solution (IVFFNIS) NI=(nj)1×s where

    pj=maximum1ir([μ'Lij,μ'Uij],[ν'Lij,ν'Uij])=([maximum1ir{μ'Lij},maximum1ir{μ'Uij}],[minimum1ir{ν'Lij},minimum1ir{ν'Uij}]),

    nj=minimum1ir([μ'Lij,μ'Uij],[ν'Lij,ν'Uij])=([minimum1ir{μ'Lij},minimum1ir{μ'Uij}],[maximum1ir{ν'Lij},maximum1ir{ν'Uij}]).

    Step 4: Using the following steps, transform the IVFFPIS PI=(pj)1×s into the crisp positive ideal solution CPI=(tj)1×s.

    Step 4(a): Transform the IVFFPIS PI=(pj)1×s into the crisp positive ideal solution CPI=(JM(pj))1×s and check if all the elements JM(pj) of the crisp positive ideal solution CPI are distinct or not.

    Case (ⅰ): If all the elements JM(pj) are distinct, then go to Step 5.

    Case (ⅱ): If some elements JM(pj) are equal, then replace such elements JM(pj) with JH(pj) and go to Step 4(b).

    Step 4(b): Check if all the elements JH(pj) are distinct or not.

    Case (ⅰ): If all the elements JH(pj) are distinct, then go to Step 5.

    Case (ⅱ): If some elements JH(pj) are equal, then replace such elements JH(pj) with JP(pj) and go to Step 4(c).

    Step 4(c): Check if all elements JP(pj) are distinct or not.

    Case (ⅰ): If all the elements JP(pj) are distinct, then go to Step 5.

    Case (ⅱ): If some elements JP(pj) are equal, then replace such elements JP(pj) with JC(pj) and go to Step 5.

    Step 5: Using the following steps, transform the IVFFNIS NI=(nj)1×s into the crisp negative ideal solution CNI=(vj)1×s.

    Step 5(a): Transform the IVFFNIS NI=(nj)1×s into the crisp negative ideal solution CNI=(JM(nj))1×s and check if all the elements JM(nj) of the crisp negative ideal solution CNI are distinct or not.

    Case (ⅰ): If all the elements JM(nj) are distinct, then go to Step 6.

    Case (ⅱ): If some elements JM(nj) are equal, then replace such elements JM(nj) with JH(nj) and go to Step 5(b).

    Step 5(b): Check if all the elements JH(nj) are distinct or not.

    Case 1: If all the elements JH(nj) are distinct, then go to Step 6.

    Case 2: If some elements JH(nj) are equal, then replace such elements JH(nj) with JP(nj) and go to Step 5(c).

    Step 5(c): Check if all the elements JP(nj) are distinct or not.

    Case 1: If all the elements JP(nj) are distinct, then go to Step 6.

    Case 2: If some elements JP(nj) are equal, then replace such elements JP(nj) with JC(nj) and go to Step 6.

    Step 6: Using expressions (2) and (3), evaluate the distance between the ith row of weighted crisp decision matrix S with the crisp positive ideal solution CPI=(tj)1×s and the crisp negative ideal solution CNI=(vj)1×s.

    di(S,CPI)=sj=1(sijtj)2s,i=1,2,,r (2)
    di(S,CNI)=sj=1(sijvj)2s,i=1,2,,r. (3)

    Step 7: Using expression (4), evaluate the relative closeness CC(Ai) of the ith alternative Ai and check that CC(Ai)>CC(Aj) or CC(Ai)<CC(Aj) or CC(Ai)=CC(Aj).

    CC(Ai)=di(S,NI)di(S,PI)+di(S,NI),i=1,2,,r. (4)

    Case (ⅰ): If CC(Ai)>CC(Aj), then Ai>Aj.

    Case (ⅱ): If CC(Ai)<CC(Aj), then Ai<Aj.

    Case (ⅲ): If CC(Ai)=CC(Aj), then Ai=Aj.

    Jeevaraj [5] proposed the following method to solve IVFFMCDM problems having the IVFF decision matrix M=(mij)r×s=(([μLij,μUij],[νLij,νUij]))r×s, where IVFFN mij represents the ranking of the ith alternative Ai over the jth benefit criteria Crj.

    Step 1: Transform the IVFF decision matrix M into the weighted IVFF decision matrix M'=(m'ij)r×s, where m'ij=mij.wj,i=1tor,j=1tos,wj is the weight of the jth benefit criteria represented by a real number and sj=1wj=1.

    Step 2: Evaluate the IVFFPIS PI=(pj)1×s and the IVFFNIS NI=(nj)1×s where

    pj=maximum1ir([μ'Lij,μ'Uij],[ν'Lij,ν'Uij])=([maximum1ir{μ'Lij},maximum1ir{μ'Uij}],[minimum1ir{ν'Lij},minimum1ir{ν'Uij}]),

    nj=minimum1ir([μ'Lij,μ'Uij],[ν'Lij,ν'Uij])=([minimum1ir{μ'Lij},minimum1ir{μ'Uij}],[maximum1ir{ν'Lij},maximum1ir{ν'Uij}]).

    Step 3: Using expression (1), evaluate the distance DGE(Ai,PI) between the ith row of weighted IVFF decision matrix M' and the IVFFPIS PI and the distance DGE(Ai,NI) between the ith row of weighted IVFF decision matrix M' and the IVFFNIS NI.

    Step 4: Using expression (5), evaluate the relative closeness CC(Ai) of the ith alternative Ai and check that CC(Ai)>CC(Aj) or CC(Ai)<CC(Aj) or CC(Ai)=CC(Aj).

    CC(Ai)=DGE(Ai,NI)DGE(Ai,PI)+DGE(Ai,NI),i=1,2,,r. (5)

    Case (ⅰ): If CC(Ai)>CC(Aj), then Ai>Aj.

    Case (ⅱ): If CC(Ai)<CC(Aj), then Ai<Aj.

    Case (ⅲ): If CC(Ai)=CC(Aj), then Ai=Aj.

    In this section, two IVFFMCDM problems are solved using Jeevaraj [5]'s IVFFMCDM methods to show that the obtained results are not correct.

    Let M=(mij)2×2=[([0.2,0.3],[0,0])([0,0.4],[0.3,0.3])([0,30.035],[0,0])([0,0.4],[0.3,0.3])] represents the IVFF decision matrix of an IVFFMCDM problem having two alternatives (A1andA2) and two benefit criteria (Cr1andCr2). Since the rating value of both alternatives corresponding to the second criterion is the same, the ranking of the alternatives is independent from the second criterion. Also, as the rating values of both alternatives corresponding to the first criterion are distinct, the rank of both alternatives should be different. In this section, it is shown that by solving the considered IVFFMCDM problem by Jeevaraj [5]'s first IVFFMCDM method, the ranking of both alternatives is the same. Hence, it is inappropriate to use Jeevaraj [5]'s first IVFFMCDM method to solve IVFFMCDM problems.

    If it is assumed that w1=1 and w2=0, then by using Jeevaraj [5]'s first IVFFMCDM method, the ranking of both alternatives of the considered IVFFMCDM problem can be obtained as follows.

    Step 1: According to Step 1 of Jeevaraj [5]'s first IVFFMCDM method, the IVFF decision matrix M can be transformed into the weighted IVFF decision matrix

    M'=(m'ij)2×2=[JM([0.2,0.3],[0,0])JM([0,0],[1,1])JM([0,30.035],[0,0])JM([0,0],[1,1])].

    Step 2: According to Step 2(a) of Jeevaraj [5]'s first IVFFMCDM method, the weighted IVFF decision matrix M' can be transformed into the weighted crisp decision matrix

    S=(sij)2×2=[JM([0.2,0.3],[0,0])JM([0,0],[1,1])JM([0,30.035],[0,0])JM([0,0],[1,1])]=[0.01710.0171].

    Since JM(s11)=JM(s21) and JM(s12)=JM(s22). So, according to Case (ⅱ) of Step 2(a), there is a need to replace JM(s11),JM(s21),JM(s12) and JM(s22) with JH(s11),JH(s21),JH(s12), and JH(s22) respectively.

    S=(sij)2×2=[JH([0.2,0.3],[0,0])JH([0,0],[1,1])JH([0,30.035],[0,0])JH([0,0],[1,1])]=[0.01710.0171].

    Since JH(s11)=JH(s21) and JH(s12)=JH(s22). So, according to Case (ⅱ) of Step 2(b), there is a need to replace JH(s11),JH(s21),JH(s12) and JH(s22) with JP(s11),JP(s21),JP(s12) and JP(s22) respectively.

    S=(sij)2×2=[JP([0.2,0.3],[0,0])JP([0,0],[1,1])JP([0,30.035],[0,0])JP([0,0],[1,1])]=[0.00900.0170].

    Since JP(s12)=JP(s22). So, according to Case (ⅱ) of Step 2(c), there is a need to replace JP(s12) and JP(s22) with JC(s12) and JC(s22) respectively.

    S=(sij)2×2=[JP([0.2,0.3],[0,0])JC([0,0],[1,1])JP([0,30.035],[0,0])JC([0,0],[1,1])]=[0.00900.0170].

    Step 3: On applying Step 3 of Jeevaraj [5]'s first IVFFMCDM method, the obtained IVFFPIS is

    PI=[p1p2]1×2

    where

    p1=([maximum(0.2,0),maximum(0.3,30.035)],[minimum(0,0),minimum(0,0)])=([0.2,0.3],[0,0]),

    p2=([maximum(0,0),maximum(0,0)],[minimum(1,1),minimum(1,1)])=([0,0],[1,1])

    and the obtained IVFFNIS is

    NI=[n1n2]1×2

    where

    n1=([minimum(0.2,0),minimum(0.3,30.035)],[maximum(0,0),maximum(0,0)])=([0,30.035],[0,0]),

    n2=([minimum(0,0),minimum(0,0)],[maximum(1,1),maximum(1,1)])=([0,0],[1,1]).

    Step 4: According to Step 4 of Jeevaraj [5]'s first IVFFMCDM method, the IVFFPIS PI can be transformed into the crisp positive ideal solution CPI.

    CPI=[JM([0.2,0.3],[0,0])JM([0,0],[1,1])]=[0.0171].

    Step 5: According to Step 5 of Jeevaraj [5]'s first IVFFMCDM method, the IVFFNIS NI can be transformed into the crisp negative ideal solution CNI.

    CNI=[JM([0,30.035],[0,0])JM([0,0],[1,1])]=[0.0171].

    Step 6: According to Step 6 of Jeevaraj [5]'s first IVFFMCDM method,

    d1(A1,PI)=(0.0090.017)2+(0+1)22=0.70713,

    d2(A2,PI)=(0.0170.017)2+(0+1)22=0.70711,

    d1(A1,NI)=(0.0090.017)2+(0+1)22=0.70713,

    d2(A2,NI)=(0.0170.017)2+(0+1)22=0.70711.

    Step 7: According to Step 7 of Jeevaraj [5]'s first IVFFMCDM method,

    CC(A1)=0.707130.70713+0.70713=0.5,CC(A2)=0.707110.70711+0.70711=0.5.

    Since CC(A1)=CC(A2)=0.5. So, according to Case (ⅲ) of Step 7 of Jeevaraj [5]'s first IVFFMCDM method, A1=A2.

    Let M=(mij)2×2=[([0.1,0.2],[0.1,0.2])([0.2,0.3],[0.2,0.3])([0.3,0.4],[0.3,0.4])([0.3,0.4],[0.3,0.4])] represent the IVFF decision matrix of an IVFFMCDM problem having two alternatives and two benefit criteria.

    If it is assumed that w1=1 and w2=0, then using Jeevaraj [5]'s second IVFFMCDM method, the ranking of the alternatives of the considered IVFFMCDM problem can be obtained as follows:

    Step 1: According to Step 1 of Jeevaraj [5]'s second IVFFMCDM method, the IVFF matrix M can be transformed into the weighted IVFF decision matrix M'=(m'ij)2×2=[([0.1,0.2],[0.1,0.2])([0.2,0.3],[0.2,0.3])([0,0],[1,1])([0,0],[1,1])].

    Step 2: On applying Step 2 of Jeevaraj [5]'s second IVFFMCDM method, the obtained IVFFPIS is

    PI=[p1p2]1×2

    where

    p1=maximum{([0.1,0.2],[0.1,0.2]),([0.2,0.3],[0.2,0.3])}=([maximum(0.1,0.2),maximum(0.2,0.3)],[minimum(0.1,0.2),minimum(0.2,0.3)])=([0.2,0.3],[0.1,0.2]),

    p2=maximum{([0,0],[1,1]),([0,0],[1,1])}=([maximum(0,0),maximum(0,0)],[minimum(1,1),minimum(1,1)])=([0,0],[1,1])

    and the obtained IVFFNIS is

    NI=[n1n2]1×2

    where

    n1=([minimum(0.1,0.2),minimum(0.2,0.3)],[maximum(0.1,0.2),maximum(0.2,0.3)])=([0.1,0.2],[0.2,0.3]),

    n2=([minimum(0,0),minimum(0,0)],[maximum(1,1),maximum(1,1)])=([0,0],[1,1]).

    Step 3: According to Step 3 of Jeevaraj [5]'s second IVFFMCDM method,

    DGE(A1,PI)=0.0116,DGE(A2,PI)=0.0116,DGE(A1,NI)=0.0116 and DGE(A2,NI)=0.0116.

    Step 4: According to Step 4 of Jeevaraj [5]'s second IVFFMCDM method,

    CC(A1)=0.01160.0116+0.0116=0.5,CC(A2)=0.01160.0116+0.0116=0.5.

    Since CC(A1)=CC(A2)=0.5.

    So, according to Case (ⅲ) of Step 4 of Jeevaraj [5]'s second IVFFMCDM method, A1=A2.

    In Section 5, it was shown that Jeevaraj [5]'s IVFFMCDM methods fail to find the correct ranking of the alternatives. Hence, it is inappropriate to use Jeevaraj [5]'s IVFFMCDM methods. In this section, reasons for this inappropriateness are discussed.

    Jeevaraj [5]'s first IVFFMCDM method fails to find the correct ranking of the alternatives due to the following reasons:

    In Step 2, the weighted IVFF decision matrix M is transformed into the weighted crisp decision matrix S; in Step 3, the IVFFPIS PI=(pj)1×s is transformed into the crisp positive ideal solution CPI; and in Step 4, IVFFNIS NI=(nj)1×s is transformed into the crisp negative ideal solution CNI. Then, the existing crisp TOPSIS [12] is used to find the ranking of the alternatives. However, the transformed weighted crisp decision matrix S will not necessarily be equivalent to the weighted IVFF decision matrix M, the transformed crisp positive ideal solution CPI will not necessarily be equivalent to IVFFPIS PI=(pj)1×s, and the transformed crisp negative ideal solution CNI will not necessarily be equivalent to IVFFNIS NI=(nj)1×s as it may happen that F1F2 but Ji(F1)=Ji(F2),i=M or H or P or C.

    The following validates this claim:

    (ⅰ) It is obvious from Step 1 of Section 5.1 that the first and second elements of the first column of the weighted IVFF decision matrix M are distinct. This indicates that the rating value of the first alternative and the second alternative, provided by a decision-maker, is distinct. However, it is obvious from Step 2 of Section 5.1 that the first and second elements of the first column of the transformed weighted crisp decision matrix S are equal. This indicates that the rating value of the first alternative and the second alternative, provided by a decision-maker, is the same.

    (ⅱ) It is obvious from Step 4 of Section 5.1 that the first and second elements of the crisp positive ideal solution are 0.017 and 1, respectively. However, according to the weighted crisp decision matrix S=[0.00900.0170], the first and second elements of the crisp positive ideal solution are maximum {0.009,0.017}=0.017 and maximum {0,0}=0, respectively.

    (ⅲ) It is obvious from Step 5 of Section 5.1 that the first and second elements of the crisp negative ideal solution are 0.017 and 1, respectively. However, according to the weighted crisp decision matrix S=[0.00900.0170], the first and second elements of the crisp positive ideal solution are minimum {0.009,0.017}=0.009 and minimum {0,0}=0, respectively.

    Jeevaraj [5]'s second IVFFMCDM method fails to find the correct ranking of the alternatives due to the following reasons:

    In Step 2 of Jeevaraj [5]'s second IVFFMCDM method, it is assumed that

    pj=maximum1ir([μ'Lij,μ'Uij],[ν'Lij,ν'Uij]) =([maximum1ir{μ'Lij},maximum1ir{μ'Uij}],[minimum1ir{ν'Lij},minimum1ir{ν'Uij}]) represents the jth element of the IVFFPIS PI=(pj)1×s and nj=minimum1ir([μ'Lij,μ'Uij],[ν'Lij,ν'Uij]) =([minimum1ir{μ'Lij},minimum1ir{μ'Uij}],[maximum1ir{ν'Lij},maximum1ir{ν'Uij}]) represents the jth element of the IVFFNIS NI=(nj)1×s.

    It is a well-known fact that if pj represents the jth element of the IVFFPIS PI=(pj)1×s and if nj represents the jth element of the IVFFNIS NI=(nj)1×s, then pj should be equal to ([μ'Lij,μ'Uij],[ν'Lij,ν'Uij]) for some i as well as nj should be equal to ([μ'Lij,μ'Uij],[ν'Lij,ν'Uij]) for some i. However, it is obvious from Step 2 of Section 5.2 that,

    p1=maximum{([0.1,0.2],[0.1,0.2]),([0.2,0.3],[0.2,0.3])} =([maximum(0.1,0.2),maximum(0.2,0.3)],[minimum(0.1,0.2),minimum(0.2,0.3)]) =([0.2,0.3],[0.1,0.2]) and

    n1=minimum{([0.1,0.2],[0.1,0.2]),([0.2,0.3],[0.2,0.3])} =([minimum(0.1,0.2),minimum(0.2,0.3)],[maximum(0.1,0.2),maximum(0.2,0.3)]) =([0.1,0.2],[0.2,0.3]).

    It is obvious that p1=([0.2,0.3],[0.1,0.2]) is neither ([0.1,0.2],[0.1,0.2]) nor ([0.2,0.3],[0.2,0.3]) and n1=([0.1,0.2],[0.2,0.3]) is neither ([0.1,0.2],[0.1,0.2]) nor ([0.2,0.3],[0.2,0.3]).

    In this section, Jeevaraj [5]'s methods are modified to resolve their inappropriateness.

    The steps of the first modified IVFFMCDM method are as follows:

    Step 1: Transform the IVFF decision matrix M=(mij)r×s into the weighted IVFF decision matrix M'=(m'ij)r×s, where m'ij=mij.wj,i=1tor,j=1tos, the non-negative real number wj is the weight of the jth benefit criteria and sj=1wj=1.

    Step 2: Using the existing method [5], discussed in Section 3, find the ranking of all the elements of the weighted IVFF decision matrix M' in increasing order.

    Step 3: Assign the natural number i to the IVFFN having ith position in the ranking.

    Step 4: Transform the weighted IVFF decision matrix M' into its equivalent weighted crisp decision matrix M''=(m''ij)r×s by replacing the IVFFN with the assigned natural number.

    Step 5: Apply the following steps of the existing crisp TOPSIS (Hwang and Yoon, 1981) to find the ranking of the alternatives.

    Step 5(a): Evaluate the crisp positive ideal solution CPI=(tj)1×s and the crisp negative ideal solution CNI=(vj)1×s where

    tj=maximum1ir{m''ij},j=1,2,,s.

    vj=minimum1ir{m''ij},j=1,2,,s.

    Step 5(b): Using expressions (6) and (7), evaluate the distance between the ith row of weighted crisp decision matrix M'' with the crisp positive ideal solution CPI=(tj)1×s and the crisp negative ideal solution CNI=(vj)1×s, respectively.

    di(M'',CPI)=sj=1(m''ijtj)2s,i=1,2,,r (6)
    di(M'',CNI)=sj=1(m''ijvj)2s,i=1,2,,r. (7)

    Step 5(c): Using expression (8), evaluate the relative closeness CC(Ai) of the ith alternative Ai and check that CC(Ai)>CC(Aj) or CC(Ai)<CC(Aj) or CC(Ai)=CC(Aj).

    CC(Ai)=di(M'',CNI)di(M'',CPI)+di(M'',CNI),i=1,2,,r. (8)

    Case (ⅰ): If CC(Ai)>CC(Aj) then Ai>Aj.

    Case (ⅱ): If CC(Ai)<CC(Aj) then Ai<Aj.

    Case (ⅲ): If CC(Ai)=CC(Aj) then Ai=Aj.

    The steps of the second modified IVFFMCDM method are as follows:

    Step 1: Transform the IVFF decision matrix M into the weighted IVFF decision matrix M'=(m'ij)r×s, where m'ij=mij.wj,i=1tor,j=1tos, the non-negative real number wj is the weight of the jth benefit criteria and sj=1wj=1.

    Step 2: Using the existing method [5], discussed in Section 3, evaluate pj=maximum1ir{m'ij},j=1,2,,s, nj=minimum1ir{m'ij},j=1,2,,s. Hence, evaluate the IVFFPIS PI=(pj)1×s and the IVFFNIS NI=(nj)1×s.

    Step 3: Using expression (1), evaluate the distance between the ith row of weighted IVFF decision matrix M' with the IVFFPIS PI=(pj)1×s and the IVFFNIS NI=(nj)1×s.

    Step 4: Using expression (5), evaluate the relative closeness CC(Ai) of the ith alternative Ai and check that CC(Ai)>CC(Aj) or CC(Ai)<CC(Aj) or CC(Ai)=CC(Aj).

    Case (ⅰ): If CC(Ai)>CC(Aj), then Ai>Aj.

    Case (ⅱ): If CC(Ai)<CC(Aj), then Ai<Aj.

    Case (ⅲ): If CC(Ai)=CC(Aj), then Ai=Aj.

    In this section, it is pointed out that the inappropriateness occurring in Jeevaraj [5]'s IVFFMCDM methods does not occur in the modified IVFFMCDM methods. Hence, it is appropriate to use these to solve IVFFMCDM problems.

    In Section 6.1, it is pointed out that in Jeevaraj [5]'s first IVFFMCDM method, the transformed weighted crisp decision matrix S is not equivalent to the weighted IVFF decision matrix M, the transformed crisp positive ideal solution CPI is not equivalent to IVFFPIS PI=(pj)1×s, and the transformed crisp negative ideal solution CNI is not equivalent to IVFFNIS NI=(nj)1×s.

    However, in the first modified IVFFMCDM method, the transformed weighted crisp decision matrix S is equivalent to the weighted IVFF decision matrix M, the transformed crisp positive ideal solution CPI is equivalent to IVFFPIS PI=(pj)1×s as two distinct natural numbers are assigned to two distinct IVFFNs, and the transformed crisp negative ideal solution CNI is equivalent to IVFFNIS NI=(nj)1×s as two distinct natural numbers are assigned to two distinct IVFFNs.

    In Section 6.2, it is pointed out that if pj and nj represent the jth element of the IVFFPIS PI=(pj)1×s and the IVFFNIS NI=(nj)1×s, respectively. Then, pj should be equal to ([μ'Lij,μ'Uij],[ν'Lij,ν'Uij]) for some i and nj should be equal to ([μ'Lij,μ'Uij],[ν'Lij,ν'Uij]) for some i. For the values of pj and nj obtained by Jeevaraj [5]'s second IVFFMCDM method, this condition is not satisfying, while it can be easily verified that for the values of pj and nj obtained by the second modified IVFFMCDM method, this condition is satisfying.

    In Section 5, two IVFFMCDM problems are considered to show that Jeevaraj [5]'s IVFFMCDM methods fail to find the correct ranking of the alternatives. In this section, the correct ranking of the alternatives of the same IVFFMCDM problems is obtained by the modified methods.

    Using the first modified IVFFMCDM method, the correct ranking of the alternatives of the first IVFFMCDM problem, considered in Section 5.1, can be obtained as follows:

    Step 1: According to Step 1 of the first modified IVFFMCDM method, the IVFF decision matrix M can be transformed into the weighted IVFF decision matrix M'.

    M'=(m'ij)2×2=[([0.2,0.3],[0,0])([0,0],[1,1])([0,30.035],[0,0])([0,0],[1,1])].

    Step 2: According to Step 2 of the first modified IVFFMCDM method, there is a need to apply the existing method [5], discussed in Section 3, to evaluate the ranking of all the elements of the weighted IVFF decision matrix M'.

    The following steps of the existing method clearly indicate that ([0,0],[1,1])<([0,30.035],[0,0])<([0.2,0.3],[0,0]).

    Step 2(a): It can be easily verified that JM([0.2,0.3],[0,0])=0.017,JM([0,30.035],[0,0])=0.017,JM([0,0],[1,1])=1.

    Since JM([0,0],[1,1])<(JM([0.2,0.3],[0,0])=JM([0,30.035],[0,0])). So, according to Step 1 of the existing method, discussed in Section 3, ([0,0],[1,1])<([0.2,0.3],[0,0]) and ([0,0],[1,1])<([0,30.035],[0,0]). But, as JM([0.2,0.3],[0,0])=JM([0,30.035],[0,0]), i.e., Case (ⅲ) of Step 1 of the existing method, discussed in Section 3, is satisfying. So, there is a need to go to Step 2 of the existing method to find the ranking of ([0.2,0.3],[0,0]) and ([0,30.035],[0,0]).

    Step 2(b): It can be easily verified that

    JH([0.2,0.3],[0,0])=0.017,JH([0,30.035],[0,0])=0.017.

    Since JH([0.2,0.3],[0,0])=JH([0,30.035],[0,0]), i.e., Case (ⅲ) of Step 2 of the existing method, discussed in Section 3, is satisfying. So, there is a need to go to Step 3 of the existing method to find the ranking of ([0.2,0.3],[0,0]) and ([0,30.035],[0,0]).

    Step 2(c): It can be easily verified that

    JP([0.2,0.3],[0,0])=0.009,JP([0,30.035],[0,0])=0.017.

    Since JP([0.2,0.3],[0,0])<JP([0,30.035],[0,0]). So, according to Case (ⅱ) of Step 3 of the existing method, discussed in Section 3, ([0.2,0.3],[0,0])>([0,30.035],[0,0]).

    Hence, ([0,0],[1,1])<([0,30.035],[0,0])<([0.2,0.3],[0,0]).

    Step 3: According to Step 3 of the first modified IVFFMCDM method, the natural numbers 1−3 can be assigned to ([0,0],[1,1]),([0,30.035],[0,0]) and ([0.2,0.3],[0,0]), respectively.

    Step 4: According to Step 4 of the first modified IVFFMCDM method, the weighted IVFF decision matrix M' is transformed into the weighted crisp decision matrix M''=(m''ij)3×2=[3121].

    Step 5: According to Step 5 of the first modified IVFFMCDM method, there is a need to apply the following steps of the existing crisp TOPSIS (Hwang and Yoon, 1981) to evaluate the ranking of the alternatives.

    Step 5(a): According to Step 5(a) of the first modified IVFFMCDM method,

    CPI=[maximum{3,2}maximum{1,1}]=[31].

    CNI=[minimum{3,2}minimum{1,1}]=[21].

    Step 5(b): According to Step 5(b) of the first modified IVFFMCDM method,

    d1(A1,CPI)=(33)2+(11)22=0,d2(A2,CPI)=(23)2+(11)22=0.707,

    d1(A1,CNI)=(32)2+(11)22=0.707,d2(A2,CNI)=(22)2+(11)22=0.

    Step 5(c): According to Step 5(c) of the first modified IVFFMCDM method,

    CC(A1)=0.7070+0.707=1,CC(A2)=00.707+0=0.

    Since CC(A1)>CC(A2). So, according to Case (ⅰ) of Step 5(c) of the first modified IVFFMCDM method, A1>A2.

    Using the second modified IVFFMCDM method, the correct ranking of the alternatives of the second IVFFMCDM problem, considered in Section 5.2, can be obtained as follows:

    Step 1: According to Step 1 of the second modified IVFFMCDM method, the IVFF decision matrix M can be transformed into the weighted IVFF decision matrix M'.

    M'=(m'ij)r×s=[([0.1,0.2],[0.1,0.2])([0.2,0.3],[0.2,0.3])([0,0],[1,1])([0,0],[1,1])].

    Step 2: According to Step 2 of the second modified IVFFMCDM method, there is a need to apply the existing method [5], discussed in Section 3, to evaluate p1=maximum{([0.1,0.2],[0.1,0.2]),([0.2,0.3],[0.2,0.3])},p2= maximum{([0,0],[1,1]),([0,0],[1,1])},n1=minimum{([0.1,0.2],[0.1,0.2]),([0.2,0.3],[0.2,0.3])},n2=minimum{([0,0],[1,1]),([0,0],[1,1])} and hence, PI=[p1p2]1×2 and NI=[n1n2]1×2.

    It can be easily verified that according to the existing method [5], discussed in Section 3, p1=([0.2,0.3],[0.2,0.3]),p2=([0,0],[1,1]),

    n1=([0.1,0.2],[0.1,0.2]),n2=([0,0],[1,1]) and hence,

    PI=[([0.2,0.3],[0.2,0.3])([0,0],[1,1])]1×2,

    NI=[([0.1,0.2],[0.1,0.2])([0,0],[1,1])]1×2.

    Step 3: According to Step 3 of the second modified IVFFMCDM method,

    DGE(A1,PI)=0.0202,DGE(A2,PI)=0,DGE(A1,NI)=0,DGE(A2,NI)=0.0202.

    Step 4: According to Step 4 of the second modified IVFFMCDM method,

    CC(A1)=00.0202+0=0,CC(A2)=0.02020+0.0202=1.

    Since CC(A2)>CC(A1). So, according to Case (ⅰ) of Step 4 of the second modified IVFFMCDM method, A2>A1.

    Jeevaraj [5] solved two real-life IVFFMCDM problems using his proposed methods.

    In this section, the ranking of the alternatives of the same real-life IVFFMCDM problems is obtained by the proposed methods.

    Jeevaraj [5] applied the first IVFFMCDM method to obtain the ranking of the alternatives of the following real-life IVFFMCDM problem.

    An investment company wants to invest in a software development project as the best option. The four possible alternatives are:

    (ⅰ) A1 is a mail development project.

    (ⅱ) A2 is a game development project.

    (ⅲ) A3 is a browser development project.

    (ⅳ) A4 is a music player.

    The development project is to be evaluated for potential investment based on the following four criteria:

    (ⅰ) Cr1 is economic feasibility.

    (ⅱ) Cr2 is technological feasibility.

    (ⅲ) Cr3 is staff feasibility.

    (ⅳ) Cr4 is period feasibility.

    The opinion of decision-makers, provided in terms of IVFFNs, about the performance of the alternatives with respect to criteria is represented by the IVFF decision matrix M. The weights assigned to the first, second, third, and fourth criteria, provided by the decision-makers, are w1=0.35, w2=0.20, w3=0.15, and w4=0.30, respectively.

    M=(mij)4×4=[([0.3,0.45],[0.1,0.2])([0.4,0.7],[0.15,0.35])([0.5,0.7],[0.6,0.8])([0.6,0.8],[0.3.0.3])([0.4,0.6],[0.3,0.5])([0.5,0.65],[0.6,0.8])([0.6,0.7],[0.2,0.4])([0.5,0.75],[0.1,0.3])([0.45,0.8],[0.3,0.7])([0.1,0.3],[0.3,0.3])([0.4,0.7],[0.35,0.4])([0.3,0.6],[0.3,0.35])([0.1,0.4],[0.4,0.5])([0.3,0.4],[0.3,0.7])([0.4,0.7],[0.5,0.75])([0.2,0.7],[0.2,0.4])]4×4.

    Using the first modified IVFFMCDM method, the ranking of the alternatives of the first real-life IVFFMCDM problem can be obtained as follows.

    Step 1: According to Step 1 of the first modified IVFFMCDM method, the IVFF decision matrix M can be transformed into the weighted IVFF decision matrix M'=(m'ij)4×4=

    [([0.2120,0.3204],[0.4467,0.5639])([0.2360,0.4319],[0.6843,0.8106])([0.3574,0.5152],[0.8383,0.9249])([0.3622,0.5113],[0.7860,0.7860])([0.2839,0.4338],[0.6561,0.7849])([0.2976,0.3692],[0.9029,0.9564])([0.4338,0.5152],[0.5693,0.7256])([0.2976,0.4700],[0.6310,0.7860])([0.2423,0.4673],[0.8348,0.9479])([0.0670,0.2015],[0.6968,0.6968])([0.2145,0.3938],[0.8543,0.8716])([0.2015,0.4129],[0.6968,0.7298])([0.0531,0.2415],[0.8716,0.9013])([0.2015,0.2698],[0.6968,0.8985])([0.2145,0.3938],[0.9013,0.9578])([0.1340,0.4910],[0.6170,0.7597])]4×4.

    Step 2: According to Step 2 of the first modified IVFFMCDM method, there is a need to apply the existing method [5], discussed in Section 3, to evaluate the ranking of all the elements of the weighted IVFF decision matrix M'.

    The following steps of the existing method clearly indicate that

    ([0.2145,0.3938],[0.9013,0.9578])<([0.2976,0.3692],[0.9029,0.9564])< ([0.0531,0.2415],[0.8716,0.9013])<([0.2423,0.4673],[0.8348,0.9479])< ([0.2145,0.3938],[0.8543,0.8716])<([0.3574,0.5152],[0.8383,0.9249])< ([0.2015,0.2698],[0.6968,0.8985])<([0.3622,0.5113],[0.7860,0.7860])< ([0.2360,0.4319],[0.6843,0.8106])<([0.0670,0.2015],[0.6968,0.6968])<([0.2839,0.4338],[0.6561,0.7849])< ([0.2015,0.4129],[0.6968,0.7298])<([0.2976,0.4700],[0.6310,0.7860])< ([0.1340,0.4910],[0.6170,0.7597])<([0.4338,0.5152],[0.5693,0.7256])< ([0.2120,0.3204],[0.4467,0.5639]).

    Step 2(a): It can be easily verified that

    JM([0.2120,0.3204],[0.4467,0.5639])=0.1156,

    JM([0.3574,0.5152],[0.8383,0.9249])=0.5968,

    JM([0.2839,0.4338],[0.6561,0.7849])=0.3305,

    JM([0.4338,0.5152],[0.5693,0.7256])=0.1741,

    JM([0.2360,0.4319],[0.6843,0.8106])=0.3796,

    JM([0.3622,0.5113],[0.7860,0.7860])=0.3950,

    JM([0.2976,0.3692],[0.9029,0.9564])=0.7611,

    JM([0.2976,0.4700],[0.6310,0.7860])=0.3033,

    JM([0.2423,0.4673],[0.8348,0.9479])=0.6586,

    JM([0.2145,0.3938],[0.8543,0.8716])=0.6073,

    JM([0.0531,0.2415],[0.8716,0.9013])=0.6921,

    JM([0.2145,0.3938],[0.9013,0.9578])=0.7698,

    JM([0.0670,0.2015],[0.6968,0.6968])=0.3341,

    JM([0.2015,0.4129],[0.6968,0.7298])=0.3243,

    JM([0.2015,0.2698],[0.6968,0.8985])=0.5180 and

    JM([0.1340,0.4910],[0.6170,0.7597])=0.2762.

    It is obvious that

    JM([0.2145,0.3938],[0.9013,0.9578])<JM([0.2976,0.3692],[0.9029,0.9564])< JM([0.0531,0.2415],[0.8716,0.9013])<JM([0.2423,0.4673],[0.8348,0.9479])<JM([0.2145,0.3938],[0.8543,0.8716])< JM([0.3574,0.5152],[0.8383,0.9249])<JM([0.2015,0.2698],[0.6968,0.8985])< JM([0.3622,0.5113],[0.7860,0.7860])<JM([0.2360,0.4319],[0.6843,0.8106])< JM([0.0670,0.2015],[0.6968,0.6968])<JM([0.2839,0.4338],[0.6561,0.7849])< JM([0.2015,0.4129],[0.6968,0.7298])<JM([0.2976,0.4700],[0.6310,0.7860])< JM([0.1340,0.4910],[0.6170,0.7597])<JM([0.4338,0.5152],[0.5693,0.7256])< JM([0.2120,0.3204],[0.4467,0.5639]).

    Hence,

    ([0.2145,0.3938],[0.9013,0.9578])<([0.2976,0.3692],[0.9029,0.9564])< ([0.0531,0.2415],[0.8716,0.9013])<([0.2423,0.4673],[0.8348,0.9479])< ([0.2145,0.3938],[0.8543,0.8716])<([0.3574,0.5152],[0.8383,0.9249])< ([0.2015,0.2698],[0.6968,0.8985])<([0.3622,0.5113],[0.7860,0.7860])< ([0.2360,0.4319],[0.6843,0.8106])<([0.0670,0.2015],[0.6968,0.6968])< ([0.2839,0.4338],[0.6561,0.7849])<([0.2015,0.4129],[0.6968,0.7298])< ([0.2976,0.4700],[0.6310,0.7860])<([0.1340,0.4910],[0.6170,0.7597])< ([0.4338,0.5152],[0.5693,0.7256])<([0.2120,0.3204],[0.4467,0.5639]).

    Step 3: According to Step 3 of the first modified IVFFMCDM method, the natural numbers 1,2,,16 can be assigned to ([0.2145,0.3938],[0.9013,0.9578]),([0.2976,0.3692],[0.9029,0.9564]), ([0.0531,0.2415],[0.8716,0.9013]),([0.2423,0.4673],[0.8348,0.9479]), ([0.2145,0.3938],[0.8543,0.8716]),([0.3574,0.5152],[0.8383,0.9249]), ([0.2015,0.2698],[0.6968,0.8985]),([0.3622,0.5113],[0.7860,0.7860]), ([0.2360,0.4319],[0.6843,0.8106]),([0.0670,0.2015],[0.6968,0.6968]), ([0.2839,0.4338],[0.6561,0.7849]),([0.2015,0.4129],[0.6968,0.7298]), ([0.2976,0.4700],[0.6310,0.7860]),([0.1340,0.4910],[0.6170,0.7597]), ([0.4338,0.5152],[0.5693,0.7256]) and ([0.2120,0.3204],[0.4467,0.5639]), respectively.

    Step 4: According to Step 4 of the first modified IVFFMCDM method, the weighted IVFF decision matrix M' is transformed into the weighted crisp decision matrix

    M''=(m''ij)4×4=[16941068512112371513114]4×4.

    Step 5: According to Step 5 of the first modified IVFFMCDM method, there is a need to apply the following steps of the existing crisp TOPSIS [12] to evaluate the ranking of the alternatives.

    Step 5(a): According to Step 5(a) of the first modified IVFFMCDM method, CPI=[maximum{16,6,11,15}maximum{9,8,2,13}maximum{4,5,3,1}maximum{10,12,7,14}]=[1613514].

    CNI=[minimum{16,6,11,15}minimum{9,8,2,13}minimum{4,5,3,1}minimum{10,12,7,14}]=[6217].

    Step 5(b): According to Step 5(b) of the first modified IVFFMCDM method, d1(A1,CPI)=2.872,d2(A2,CPI)=5.678,d3(A3,CPI)=7.053, d4(A4,CPI)=2.06,d1(A1,CNI)=6.461,d2(A2,CNI)=4.387, d3(A3,CNI)=2.692,d4(A4,CNI)=7.921.

    Step 5(c): According to Step 5(c) of the first modified IVFFMCDM method,

    CC(A1)=0.69,CC(A2)=0.43,CC(A3)=0.27,CC(A4)=0.79

    Since CC(A4)>CC(A1)>CC(A2)>CC(A3). So, according to Case (ⅰ) of Step 5(c) of the first modified IVFFMCDM method, A4>A1>A2>A3.

    Jeevaraj [5] applied the second IVFFMCDM method to obtain the ranking of the alternatives of the following real-life IVFFMCDM problem:

    There is a panel with five possible alternatives to invest money:

    (ⅰ) A1 is a car company.

    (ⅱ) A2 is a food company.

    (ⅲ) A3 is a computer company.

    (ⅳ) A4 is a construction company.

    (v) A5 is a textile company.

    The investment company must decide for the best alternative based on their performance with respect to the following three criteria:

    (ⅰ) Cr1 is the risk analysis.

    (ⅱ) Cr2 is the growth analysis.

    (ⅲ) Cr3 is the environmental impact analysis.

    The opinion of decision-makers, provided in terms of IVFFNs, about the performance of the alternatives with respect to criteria is represented by the IVFF decision matrix M. The weights assigned to the first, second, and third criteria, provided by the decision-makers, are w1=0.35, w2=0.20, and w3=0.45, respectively.

    M=(mij)5×3=[([0.15,0.3],[0.3,0.6])([0.5,0.6],[0.35,0.55])([0.2,0.35],[0.15,0.6])([0.4,0.7],[0.1,0.35])([0.1,0.7],[0.2,0.4])([0.35,0.55],[0.35,0.45])([0.15,0.3],[0.45,0.6])([0,0.25],[0.2,0.5])([0.4,0.6],[0.1,0.3])([0.2,0.7],[0,0.2])([0.15,0.3],[0.6,0.6])([0.25,0.6],[0.4,0.7])([0.2,0.45],[0.3,0.6])([0.4,0.35],[0.3,0.55])([0.2,0.5],[0.15,0.6])]5×3.

    Using the second modified IVFFMCDM method, the ranking of the alternatives of the second real-life IVFFMCDM problem can be obtained as follows:

    Step 1: According to Step 1 of the second modified method, the IVFF decision matrix M can be transformed into the weighted IVFF decision matrix

    M'=(m'ij)5×3=[([0.1057,0.2120],[0.6561,0.8363])([0.3574,0.4338],[0.6925,0.8112])([0.1411,0.2478],[0.5148,0.8363])([0.2839,0.5152],[0.4467,0.6925])([0.0705,0.5152],[0.5693,0.7256])([0.2059,0.3294],[0.8106,0.8524])([0.0878,0.1761],[0.8524,0.9029])([0,0.1465],[0.7248,0.8706])([0.2360,0.3622],[0.6310,0.7860])([0.1171,0.4319],[0,0.7248])([0.1150,0.2305],[0.5817,0.7941])([0.1919,0.4698],[0.6621,0.8517])([0.1534,0.3478],[0.5817,0.7946])([0.0766,0.2693],[0.5817,0.7641])([0.1534,0.3878],[0.4258,0.7946])]5×3.

    Step 2: According to Step 2 of the second modified IVFFMCDM method, there is a need to apply the existing method [5], discussed in Section 3, to evaluate p1=maximum{([0.1057,0.2120],[0.6561,0.8363]),([0.3574,0.4338],[0.6925,0.8112]),([0.1411,0.2478],[0.5148,0.8363]),([0.2839,0.5152],[0.4467,0.6925]),([0.0705,0.5152],[0.5693,0.7256])},

    p2=maximum{([0.2059,0.3294],[0.8106,0.8524]),([0.0878,0.1761],[0.8524,0.9029]),([0,0.1465],[0.7248,0.8706]),([0.2360,0.3622],[0.6310,0.7860]),([0.1171,0.4319],[0,0.7248])},

    p3=maximum{([0.1150,0.2305],[0.5817,0.7941]),([0.1919,0.4698],[0.6621,0.8517]),([0.1534,0.3478],[0.5817,0.7946]),([0.0766,0.2693],[0.5817,0.7641]),([0.1534,0.3878],[0.4258,0.7946])},

    n1=minimum{([0.1057,0.2120],[0.6561,0.8363]),([0.3574,0.4338],[0.6925,0.8112]),([0.1411,0.2478],[0.5148,0.8363]),([0.2839,0.5152],[0.4467,0.6925]),([0.0705,0.5152],[0.5693,0.7256])},

    n2=minimum{([0.2059,0.3294],[0.8106,0.8524]),([0.0878,0.1761],[0.8524,0.9029]),([0,0.1465],[0.7248,0.8706]),([0.2360,0.3622],[0.6310,0.7860]),([0.1171,0.4319],[0,0.7248])},

    n3=minimum{([0.1150,0.2305],[0.5817,0.7941]),([0.1919,0.4698],[0.6621,0.8517]),([0.1534,0.3478],[0.5817,0.7946]),([0.0766,0.2693],[0.5817,0.7641]),([0.1534,0.3878],[0.4258,0.7946])}

    and hence, PI=[p1p2p3]1×3 and NI=[n1n2n3]1×3.

    It can be easily verified that according to the existing method [3, Section 4, Definition 4.13], discussed in Section 3, p1=([0.2839,0.5152],[0.4467,0.6925]),p2=([0.1171,0.4319],[0,0.7248]), p3=([0.1534,0.3878],[0.4258,0.7946]),n1=([0.1057,0.2120],[0.6561,0.8363]),n2=([0.0878,0.1761],[0.8524,0.9029]), n3=([0.1919,0.4698],[0.6621,0.8517]) and hence,

    PI=[([0.2839,0.5152],[0.4467,0.6925])([0.1171,0.4319],[0,0.7248])([0.1534,0.3878],[0.4258,0.7946])]1×3,

    NI=[([0.1057,0.2120],[0.6561,0.8363])([0.0878,0.1761],[0.8524,0.9029])([0.1919,0.4698],[0.6621,0.8517])]1×3.

    Step 3: According to Step 3 of the second modified IVFFMCDM method, DGE(A1,PI)=0.572,DGE(A2,PI)=0.733,DGE(A3,PI)=0.425,DGE(A4,PI)=0.240,DGE(A5,PI)=0.05, DGE(A1,NI)=0.195,DGE(A2,NI)=0.598,DGE(A3,NI)=0.344,DGE(A4,NI)=0.555 and DGE(A5,NI)=0.669.

    Step 4: According to Step 4 of the second modified IVFFMCDM method, CC(A1)=0.25,CC(A2)=0.45,CC(A3)=0.44,CC(A4)=0.69, and CC(A5)=0.92.

    Since CC(A5)>CC(A4)>CC(A2)>CC(A3)>CC(A1), So, according to Case (ⅰ) of Step 4 of the second modified IVFFMCDM method, A5>A4>A2>A3>A1.

    It is shown that Jeevaraj [5]'s IVFFMCDM methods fail to find the correct ranking of the alternatives. Hence, it is inappropriate to use Jeevaraj [5]'s methods to solve IVFFMCDM problems. Modified IVFFMCDM methods are proposed corresponding to Jeevaraj [5]'s methods. Furthermore, the ranking of the alternatives of the existing IVFFMCDM problems [5] is obtained using the proposed methods.

    Raina Ahuja: Conceptualization, methodology, writing – original draft; Meraj Ali Khan: Funding acquisition, visualization, writing – review & editing; Parul Tomar: Software, writing – review & editing; Amit Kumar: Visualization, supervision; S. S. Appadoo: Visualization, supervision; Ibrahim Al-Dayel: Funding acquisition, visualization, writing – review & editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502).

    The authors declare that they have no competing interest.



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