Commentary on “On Intuitionistic Fuzzy Copula Aggregation Operators in Multiple-Attribute Decision Making”

Tao et al. [1, Definition 4, p. 613] proposed the operational law (1) and the operational law (2) respectively to evaluate the sum and the multiplication of two IFVs \( {\alpha}_1=\left\langle {\mu}_{\alpha_1},{\nu}_{\alpha_1}\right\rangle \) and \( {\alpha}_2=\left\langle {\mu}_{\alpha_2},{\nu}_{\alpha_2}\right\rangle \).

Hence, Tao et al. [1] proposed the operational law (3) to evaluate the sum of “n” IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n and the operational law (4) to evaluate the multiplication of n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n.

Also, using the operational law (3) and the operational law (4), Tao et al. [1, Theorem 7, p. 616] proposed the IFCAAO (5) to aggregate n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n by considering “w_i” as the normalized weight associated with the intuitionistic fuzzy value (IFV) “\( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \).”

where ϕ is a strictly decreasing function such that ϕ(1) = 0, ϕ(0) =  ∞ , ϕ⁻¹(0) = 1, and ϕ⁻¹(∞) = 0 [1, Proof of Theorem 2, p. 613].

The aim of this commentary is to make the researchers aware that

1. (i)

   The operational law (3), proposed by Tao et al. [1, Definition 4, p. 613] to evaluate the sum of n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈1, 0〉 for any i.

2. (ii)

   The operational law (4), proposed by Tao et al. [1, Definition 4, p. 613] to evaluate the multiplication n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈0, 1〉 for any i.

3. (iii)

   The IFCAAO (5), proposed by Tao et al. [1, Theorem 7, p. 616] to aggregate n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈1, 0〉 for any i.

In this section, it is shown that

1. (i)

   The operational law (3), proposed by Tao et al. [1] to evaluate the sum of n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈1, 0〉 for any i.

2. (ii)

   The operational law (4), proposed by Tao et al. [1] to evaluate the multiplication of n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈0, 1〉 for any i.

Limitation of Tao et al.’s Operational Law to Evaluate the Sum of Finite Number of IFVs

In this section, it is shown that if one of the n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n will be 〈1, 0〉. Then, the sum of all the n IFVs will also be 〈1, 0〉 i.e., if \( {\alpha}_p=\left\langle {\mu}_{\alpha_p},{\nu}_{\alpha_p}\right\rangle =\left\langle 1,0\right\rangle \) then the sum of the n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n will be independent from the remaining “(n − 1)” IFV values \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, p − 1, p + 1, …n. Hence, the operational law (3), proposed by Tao et al. [1] to evaluate the sum of n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈1, 0〉 for any i.

The operational law (6) represents an alternative form of the operational law (3).

Let \( {\alpha}_p=\left\langle {\mu}_{\alpha_p},{\nu}_{\alpha_p}\right\rangle =\left\langle 1,0\right\rangle \), i.e., \( {\mu}_{\alpha_p}=1 \) and \( {\nu}_{\alpha_p}=0 \). Then, using the operational law (6),

Using the existing relation ϕ(0) =  ∞ [1, Proof of Theorem 2, p. 613],

Using the existing relation ϕ⁻¹(∞) = 0 [1, Proof of Theorem 2, p. 613],

Limitation of Tao et al.’s Operational Law to Evaluate the Multiplication of IFVs

In this section, it is shown that if one of the n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n will be 〈0, 1〉, then the multiplication of all the n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n will also be 〈0, 1〉, i.e., if α_p = 〈0, 1〉, then the multiplication of the n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n will be independent from the remaining (n − 1) IFV values \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, p − 1, p + 1, …, n. Hence, the operational law (4), proposed by Tao et al. [1] to evaluate the multiplication of n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈0, 1〉 for any i.

The operational law (7) represents an alternative form of the operational law (4).

Let \( {\alpha}_p=\left\langle {\mu}_{\alpha_p},{\nu}_{\alpha_p}\right\rangle =\left\langle 0,1\right\rangle \) i.e., \( {\mu}_{\alpha_p}=0 \) and \( {\nu}_{\alpha_p}=1 \). Then, using the operational law (7),

Using the existing relation ϕ(0) = ∞ [1, Proof of Theorem 2, p. 613],

Using the existing relation ϕ⁻¹(∞) = 0 [1, Proof of Theorem 2, p. 613],

In this section, it is shown that if one of the n IFVs \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n will be 〈1, 0〉, then the aggregated IFV, obtained by Tao et al.’s IFCAAO, will also be 〈1, 0〉, i.e., if α_p = 〈1, 0〉, then the aggregated IFV will be independent from the remaining (n − 1) IFV values \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, p − 1, p + 1, …, n. Hence, the IFCAAO (5), proposed by Tao et al. [1] to aggregate \( {\alpha}_i=\left\langle {\mu}_{\alpha_i},{\nu}_{\alpha_i}\right\rangle \), i = 1, 2, …, n, can be used only if α_i ≠ 〈1, 0〉 for any i.

The IFCAAO (8) represents an alternative form of the IFCAAO (5).

Let \( {\alpha}_p=\left\langle {\mu}_{\alpha_p},{\nu}_{\alpha_p}\right\rangle =\left\langle 1,0\right\rangle \), i.e., \( {\mu}_{\alpha_p}=1 \) and \( {\nu}_{\alpha_p}=0 \). Then, using the IFCAAO (8),

Using the existing relation ϕ(0) = �� [1, Proof of Theorem 2, p. 613],

Using the existing relation ϕ⁻¹(∞) = 0 [1, Proof of Theorem 2, p. 613],

Tao et al. [1] considered a multiple-attribute decision-making problem of the ranking of four projectors to illustrate their proposed IFCAAO.

In this section, firstly, the multiple-attribute decision-making problem, considered by the Tao et al. [1], is discussed in a brief manner. Then, the impact of the limitation of Tao et al.’s IFCAAO [1] on the ranking of alternatives of multiple-attribute decision-making problem, considered by Tao et al. [1], is discussed.

A Brief Review of Tao et al.’s Multiple-Attribute Decision-Making Problem

Tao et al. [1] used the following method to rank the four projectors by considering that the (i, j)th IFV of the intuitionistic fuzzy decision matrix \( \overset{\sim }{D}={\left({\alpha}_{ij}\right)}_{4\times 5}=\left(\begin{array}{c}\left\langle \mathrm{0.4,0.3}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.6,0.1}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.5,0.4}\right\rangle \\ {}\left\langle \mathrm{0.6,0.3}\right\rangle \end{array}\end{array}\end{array}\kern0.5em \begin{array}{c}\left\langle \mathrm{0.5,0.2}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.4,0.3}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.6,0.1}\right\rangle \\ {}\left\langle \mathrm{0.4,0.5}\right\rangle \end{array}\end{array}\end{array}\kern0.5em \begin{array}{c}\left\langle \mathrm{0.7,0.2}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.3,0.5}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.6,0.2}\right\rangle \\ {}\left\langle \mathrm{0.5,0.3}\right\rangle \end{array}\end{array}\end{array}\kern0.5em \begin{array}{c}\left\langle \mathrm{0.4,0.6}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.6,0.2}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.7,0.1}\right\rangle \\ {}\left\langle \mathrm{0.8,0.2}\right\rangle \end{array}\end{array}\end{array}\kern0.5em \begin{array}{c}\left\langle \mathrm{0.6,0.2}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.5,0.3}\right\rangle \\ {}\begin{array}{c}\left\langle \mathrm{0.3,0.6}\right\rangle \\ {}\left\langle \mathrm{0.5,0.2}\right\rangle \end{array}\end{array}\end{array}\right) \) represents the rating value of the ith projector (i = 1, 2, 3, 4) over the jth benefit attribute (j = 1, 2, 3, 4, 5), provided by a decision-maker.

Step 1: Tao et al. [1] applied an existing method to evaluate the normalized weights w₁, w₂, w₃, w₄, and w₅ corresponding to the first, second, third, fourth, and fifth attributes respectively.

Step 2: Tao et al. [1] applied the IFCAAO to evaluate

1. (i)

   The IFV α₁ = w₁⊗_c〈0.4,0.3〉⊕_cw₂⊗_c〈0.5,0.2〉⊕_cw₃⊗_c〈0.7,0.2〉 ⊕_cw₄⊗_c〈0.4,0.6〉⊕_cw₅⊗_c〈0.6,0.2〉 corresponding to the first projector.

2. (ii)

   The IFV α₂ = w₁⊗_c〈0.6,0.1〉⊕_cw₂⊗_c〈0.4,0.3〉⊕_cw₃⊗_c〈0.3,0.5〉 ⊕_cw₄⊗_c〈0.6,0.2〉⊕_cw₅⊗_c〈0.5,0.3〉 corresponding to the second projector.

3. (iii)

   The IFV α₃ = w₁⊗_c〈0.5,0.4〉⊕_cw₂⊗_c〈0.6,0.1〉⊕_cw₃⊗_c〈0.6,0.2〉 ⊕_cw₄⊗_c〈0.7,0.1〉⊕_cw₅⊗_c〈0.3,0.6〉 corresponding to the third projector.

4. (iv)

   The IFV α₄ = w₁⊗_c〈0.6,0.3〉⊕_cw₂⊗_c〈0.4,0.5〉⊕_cw₃⊗_c〈0.5,0.3〉 ⊕_cw₄⊗_c〈0.8,0.2〉⊕_cw₅⊗_c〈0.5,0.2〉 corresponding to the fourth projector.

Step 3: Tao et al. [1] used the following method to conclude that that the pth alternative is better than the qth alternative or vice versa.

Step (3a): Check Score (α_p) > Score (α_q) or Score (α_p) < Score (α_q) or Score (α_p) = Score (α_q),

where,

\( \mathrm{Score}\ \left({\alpha}_p\right)=\mathrm{Score}\ \left(\left\langle {\mu}_{\alpha_p},{\nu}_{\alpha_p}\right\rangle \right)={\mu}_{\alpha_p}-{\nu}_{\alpha_p} \) and \( \mathrm{Score}\ \left({\alpha}_q\right)=\mathrm{Score}\ \left(\left\langle {\mu}_{\alpha_q},{\nu}_{\alpha_q}\right\rangle \right)={\mu}_{\alpha_q}-{\nu}_{\alpha_q} \).

Case (i): If Score (α_p) > Score (α_q), then the pth alternative is better than the qth alternative.

Case (ii): If Score (α_p) < Score (α_q), then the qth alternative is better than the pth alternative.

Case (iii): If Score (α_p) = Score (α_q), then go to step (3b).

Step (3b): Check Accuracy (α_p) > Accuracy (α_q) or Accuracy (α_p) < Accuracy (α_q), where,

\( \mathrm{Accuracy}\ \left({\alpha}_p\right)=\mathrm{Accuracy}\ \left(\left\langle {\mu}_{\alpha_p},{\nu}_{\alpha_p}\right\rangle \right)={\mu}_{\alpha_p}+{\nu}_{\alpha_p} \) and \( \mathrm{Accuracy}\ \left({\alpha}_q\right)=\mathrm{Accuracy}\ \left(\left\langle {\mu}_{\alpha_q},{\nu}_{\alpha_q}\right\rangle \right)={\mu}_{\alpha_q}+{\nu}_{\alpha_q}. \)

Case (i): If Accuracy (α_p) > Accuracy (α_q), then the pth alternative is better than the qth alternative.

Case (ii): If Accuracy (α_p) < Accuracy (α_q), then the qth alternative is better than the pth alternative.

Case (iii): If Accuracy (α_p) = Accuracy (α_q), then the pth alternative is equivalent to the qth alternative.

Impact of the Limitation of Tao et al.’s IFCAAO

It is obvious from “A Brief Review of Tao Et al.’s Multiple-Attribute Decision-Making Problem” that in step 2, the IFCAAO (5) has been used to obtain

1. (i)

   The IFV α₁ = w₁⊗_c〈0.4,0.3〉⊕_cw₂⊗_c〈0.5,0.2〉⊕_cw₃⊗_c〈0.7,0.2〉

   ⊕_cw₄⊗_c〈0.4,0.6〉⊕_cw₅⊗_c〈0.6,0.2〉 corresponding to the first projector.

   If the IFV α₁₁ = 〈0.4,0.3〉 or the IFV α₁₂ = 〈0.5,0.2〉 or the IFV α₁₃ = 〈0.7,0.2〉 or the IFV α₁₄ = 〈0.4,0.6〉 or the IFV α₁₅ = 〈0.6,0.2〉 is replaced by the IFV 〈1, 0〉. Then, according to “Limitation of Tao et al.’s IFCAAO,” the obtained IFV α₁ will be 〈1, 0〉.

   Furthermore, as 〈1, 0〉 is the only IFV for which Score will be 1 and the Score of any IFV can never be more than 1, in such a situation, the first projector will be one of the best projectors, i.e., if α_1j = 〈1, 0〉 for any j, then the result “the first projector is one of the best projectors” is independent from all the remaining 19 IFVs of the intuitionistic fuzzy decision matrix \( \overset{\sim }{D} \).

2. (ii)

   The IFV α₂ = w₁⊗_c〈0.6,0.1〉⊕_cw₂⊗_c〈0.4,0.3〉⊕_cw₃⊗_c〈0.3,0.5〉⊕_c

   w₄⊗_c〈0.6,0.2〉⊕_cw₅⊗_c〈0.5,0.3〉, corresponding to the second projector.

   If the IFV α₂₁ = 〈0.6,0.1〉 or the IFV α₂₂ = 〈0.4,0.3〉 or the IFV α₂₃ = 〈0.3,0.5〉 or the IFV α₂₄ = 〈0.6,0.2〉 or the IFV α₂₅ = 〈0.5,0.3〉 is replaced by the IFV 〈1, 0〉, then, according to “Limitation of Tao et al.’s IFCAAO,” the obtained IFV α₂ will be 〈1, 0〉.

   Furthermore, as discussed in (i), in such a situation, the second projector will be one of the best projectors, i.e., if α_2j = 〈1, 0〉 for any j, then the result “the second projector is one of the best projectors” is independent from all the remaining 19 IFVs of the intuitionistic fuzzy decision matrix \( \overset{\sim }{D} \).

3. (iii)

   The IFV α₃ = w₁⊗_c〈0.5,0.4〉⊕_cw₂⊗_c〈0.6,0.1〉⊕_cw₃⊗_c〈0.6,0.2〉 ⊕_cw₄⊗_c〈0.7,0.1〉⊕_cw₅⊗_c〈0.3,0.6〉 corresponding to the third projector.

   If the IFV α₃₁ = 〈0.5,0.4〉 or the IFV α₃₂ = 〈0.6,0.1〉 or the IFV α₃₃ = 〈0.6,0.2〉 or the IFV α₃₄ = 〈0.7,0.1〉 or the IFV α₃₅ = 〈0.3,0.6〉 is replaced by the IFV 〈1, 0〉, then, according to “Limitation of Tao et al.’s IFCAAO,” the obtained IFV α₃ will be 〈1, 0〉.

   Furthermore, as discussed in (i), in such a situation, the third projector will be one of the best projectors, i.e., if α_3j = 〈1, 0〉 for any j, then the result “the third projector is one of the best projectors” is independent from all the remaining 19 IFVs of the intuitionistic fuzzy decision matrix \( \overset{\sim }{D} \).

4. (iv)

   The IFV α₄ = w₁⊗_c〈0.6,0.3〉⊕_cw₂⊗_c〈0.4,0.5〉⊕_cw₃⊗_c〈0.5,0.3〉

   ⊕_cw₄⊗_c〈0.8,0.2〉⊕_cw₅⊗_c〈0.5,0.2〉, corresponding to the fourth projector.

   If the IFV α₄₁ = 〈0.6,0.3〉 or the IFV α₄₂ = 〈0.4,0.5〉 or the IFV α₄₃ = 〈0.5,0.3〉 or the IFV α₄₄ = 〈0.8,0.2〉 or the IFV α₄₅ = 〈0.5,0.2〉 is replaced by the IFV 〈1, 0〉, then, according to “Limitation of Tao et al.’s IFCAAO,” the obtained IFV α₄ will be 〈1, 0〉.

   Furthermore, as discussed in (i), in such a situation, the fourth projector will be one of the best projectors, i.e., if α_4j = 〈1, 0〉 for any j, then the result “the fourth projector is one of the best projectors” is independent from all the remaining 19 IFVs of the intuitionistic fuzzy decision matrix \( \overset{\sim }{D} \).

It is shown that the operational law (3) to evaluate sum of finite number of IFVs and the IFCAAO (5) to aggregate finite number of IFVs, proposed by Tao et al. [1], can be used only if α_i ≠ 〈1, 0〉 for any i. Also, it is shown that the operational law (4) to evaluate the multiplication of finite number of IFVs, proposed by Tao et al. [1], can be used only if α_i ≠ 〈0, 1〉 for any i.