A novel method to derive the intuitionistic multiplicative priority vector for the intuitionistic multiplicative preference relation

Abstract

Consistency is related with reasonableness of the priority vector derived from a preference relation. In this paper, it is pointed out by an example that the existing consistency for the intuitionistic multiplicative preference relations (IMPR) is weak that the ranking or the optimal alternative could not always be derived from the given consistent IMPR. We provide a novel consistency for the IMPRs by the score function and accuracy function and characterize it with the S-normalized and A-normalized intuitionistic multiplicative priority vectors (IMPV). Then, we propose methods to check and reach the S-normalization, the acceptable consistency of the IMPR by its local IMPVs. We also give some examples to show how the proposed methods work and make comparisons with the existing methods to demonstrate the advantages and disadvantages of the proposed methods.

Keywords

Intuitionistic multiplicative preference relation consistency intuitionistic multiplicative priority vector

Preference relation is a powerful tool to express the decision maker’s preferences over a set of feasible alternatives. Generally, preference relations are divided into multiplicative preference relations (MPR) [11] and fuzzy preference relations (FPR) [8, 12]. MPRs and FPRs were contacted by the transformation functions [1] which help to integrate different types of preference relations into a unified case. MPR is a kind of popular preference relations, and has been extensively investigated in Ref. [11]. However, due to the limitation of decision maker’s global perception, the decision maker may provide an inconsistent preference relation [2, 11]. Thus, an acceptable consistency is adopted to measure the consistency. Usually, if the consistency degree of a preference relation is smaller than a certain threshold parameter, it is of acceptable consistency.

From the concept of MPRs, one can find that they could only express the decision maker’s preferred information. However, in some situations, this is insufficient, and the decision maker might hope to give the positive and negative comparisons of an alternative over another simultaneously. Furthermore, considering the phenomenon that preferred information is usually asymmetrical distribution, Xia [14] introduced the concept of IMPRs by using intuitionistic multiplicative numbers (IMNs). The analyses of the existing literature indicate that works on the IMPRs focus on the following aspects:

Aggregation operators for intuitionistic multiplicative information. Xia [15] investigated group decision making based on intuitionistic multiplicative aggregation (IMA) operators. Yu [17] focused on the IMA operators for intuitionistic multiplicative preference information based on algebraic operational laws. Jiang [4] aggregated information and ranking alternatives in decision making with IMPRs by the ordered IMA operators. Qian [9] pointed out that the existing operations to define the IMA operators in Refs. [14, 15] are not closed, and then proposed some novel IMA operators. Xu [16] found that some existing IMA operators in Refs. [4 , 15] are not suitable to aggregate IMPRs in group decision making, and hence developed some symmetric IMA operators. Garg [3] developed several novel correlation coefficients under the intuitionistic multiplicative environment. Ren [10] developed the intuitionistic multiplicative analytic hierarchy process in group decision making.

Consistency for the IMPRs. Xu [13] defined the consistency for the IMPRs and derived the priority weight intervals. Jiang [4] discussed a more general consistency property of the IMPRs than that of Xu [13] by splitting an IMPR into two MPRs and then presented a novel consistency definition of the IMPRs. Jin [6] and Meng [7] made an approach to derive intuitionistic multiplicative weights based on intuitionistic multiplicative preference relations. Zhang [18] revealed that Jiang’s consistency extension is not robust to permutations of the decision maker’s pairwise judgments and not able to reflect the hesitancy in the decision maker’s preference. They proposed a novel consistency for the IMPRs based on which the intuitionistic multiplicative group analytic hierarchy process in group decision-making and group decision making with incomplete IMPRs were investigated in Refs. [19, 20].

However, the aforementioned studies in Refs. [18 –20] have several flaws, which are described as follows:

Although Zhang’s consistency in Ref. [18] is robust to permutations of the decision maker’s pairwise judgments, the ranking could fail to be derived from some IMPRs (See Example 1.3 for detail);

Zhang proposed two sufficient conditions to construct IMPRs by the IMPVs in Refs. [18, 19] where the derived IMPVs could not completely reflect the preference of the decision maker;

Method to check and reach the acceptable consistency in Ref. [19] is dependent on the preseted threshold and adjustment parameters, respectively, it could be time consuming or difficult for a decision maker to provide such suitable values to meet the needs.

Inspired by the work in Refs. [14, 18], we investigate the consistency for the IMPRs to overcome the shortcomings in Refs. [18 –20]. To do this, the rest of this paper is structured as follows: In Section 2, a brief introduction to MPRs, IMPRs and their consistencies is made. In particular, some novel alternative definitions of consistencies for the MPRs and IMPRs are given by the corresponding priority vectors; example shows that Zhang’s consistency could fail to derive a ranking. Section 3 provides a novel definition of the consistency by the score function and the accuracy function, and equivalent conditions to derive the IMPVs are given. Furthermore, the S-normalization and A-normalization are given to guarantee the reasonability of the IMPRs. In Section 4, methods to check and repair these S-normalization and the acceptable consistency of an IMPR are investigated by their local IMPVs, respectively. Numerical examples are given to illustrate the effectiveness of our method. In Section 5, comparison with Zhang’s method are provided. Section 6 gives the conclusions.

1 Preliminary

To make the presentation self-contained, in what follows, we review some basic concepts on the MPRs and the IMPRs and their consistencies.

1.1 Multiplicative preference relations

Definition 1.1. [11] A multiplicative preference relation (MPR) R = (r_ij) _n×n on X is defined on the scale $[\frac{1}{9}, 9]$ , where r_ij indicates the preferred intensity of object x_i over x_j such that $r_{ij} \in [\frac{1}{9}, 9]$ and $r_{ij} = \frac{1}{r_{ji}}$ for all i, j = 1, 2, ⋯ , n.

Definition 1.2. [11] An MPR R is said to be consistent if it holds that r_ij = r_ikr_kj for all i, j, k = 1, 2, ⋯ , n.

In Ref. [11], a sufficient condition was provided to give the relationships between the MPRs and their corresponding priority vector. Here, an equivalent condition is given as follows:

Theorem 1.3. Let R be an MPR. Then R is consistent if and only if there exists a priority vector ω = (ω₁, ω₂, ⋯ , ω_n) ^T with $\sum_{k = 1}^{n} ω_{k} = 1, ω_{k} \in [0, 1], \frac{ω_{i}}{ω_{j}} \in [\frac{1}{9}, 9]$ for all i, j = 1, 2, ⋯ , n such that $r_{ij} = \frac{ω_{i}}{ω_{j}},$ (1) where $ω_{t} = \frac{\prod_{k = 1}^{n} r_{tk}^{\frac{1}{n}}}{\sum_{s = 1}^{n} \prod_{r = 1}^{n} r_{sr}^{\frac{1}{n}}}$ for t = 1, 2, ⋯ , n.

Proof. It is trivial, so we omit it.□

The constraint $\frac{ω_{i}}{ω_{j}} \in [\frac{1}{9}, 9]$ is necessary, without which the derived matrix by Eq. (1) could not be an MPR. This has been also pointed out in Ref. [18] for constructing the consistent IMPRs. In other words, this condition guarantees that we can always obtain an MPR from a priority vector.

Example 1.4. Let ω = (0.08, 0.12, 0.8). Then, a matrix R is derived by Eq. (1) as follows: $R = (\begin{matrix} 1 & \frac{2}{3} & \frac{1}{10} \\ \frac{3}{2} & 1 & \frac{3}{20} \\ 10 & \frac{20}{3} & 1 \end{matrix}),$ which is obviously not an MPR, because $\frac{1}{10}, 10 \notin [\frac{1}{9}, 9]$ .

1.2 Intuitionistic multiplicative sets

Here, let us review the concept of intuitionistic multiplicative sets (IMSs).

Definition 1.5. [14] An intuitionistic multiplicative set, IMS, for brevity, A on X is defined by two functions as: $A = 〈 x, (ρ_{A} (x), σ_{A} (x)) | x \in X 〉,$ where $ρ_{A} (x) : X \to [\frac{1}{9}, 9]$ indicates the degree of membership of x in A and $σ_{A} (x) : X \to [\frac{1}{9}, 9]$ indicates the degree of non-membership of x in A such that ρ_A (x) σ_A (x) ≤1. Specially, when ρ_A (x) σ_A (x) =1, the IMSs reduce to multiplicative sets.

For convenience, α = (ρ_α, σ_α) is called an intuitionistic multiplicative number, IMN for brevity in Ref. [14]. In order to compare the IMNs, a score function s and an accuracy function h are defined as follows: $s (α) = \frac{ρ_{α}}{σ_{α}}$ (2) $h (α) = ρ_{α} σ_{α}$ (3) Obviously, $s (\cdot) \in [\frac{1}{81}, 81]$ and $h (\cdot) \in [\frac{1}{81}, 1]$ . According to the score fucion and the accuracy function, Xia [14] presented the following ranking method for IMNs:

Definition 1.6. [14] Let α₁ and α₂ be two IMNs. The following comparison laws can be provided:

If s (α₁) > s (α₂), then α₁ ≻ α₂;

If s (α₁) = s (α₂),

if h (α₁) > h (α₂), then α₁ ≻ α₂;

if h (α₁) = h (α₂), then α₁ = α₂.

Note that for a group of IMNs, the ranking derived by Definition 1.6 is totally ordered, that is, any two IMNs are comparable.

1.3 Intuitionistic multiplicative preference relations

Definition 1.7. [14] An IMPR R = (r_ij) _n×n on X, where r_ij = (ρ_ij, σ_ij) for each pair of (i, j), ρ_ij and σ_ij are the preferred and non-preferred intensity degrees of object x_i over x_j, respectively, such that

$ρ_{ij} = σ_{ji}, σ_{ij} = ρ_{ji},$ (4) where $ρ_{ij}, σ_{ij} \in [\frac{1}{9}, 9], ρ_{ij} σ_{ij} \leq 1$ , for all i, j = 1, 2, ⋯ , n.

When ρ_ijσ_ij = 1 for all i, j = 1, 2, ⋯ , n, the IMPR R reduces to an MPR. To derive a reasonable ranking, Xu introduced the following consistency concept for IMPRs.

Definition 1.8. [13] Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). If it holds that

$r_{ij} = r_{ik} \otimes r_{kj},$ (5) that is, ρ_ij = ρ_ikρ_kj and σ_ij = σ_ikσ_kj for each triple of (i, k, j) such that i ≤ k ≤ j, then R is said to be of Xu’s consistency.

Jiang et al. [5] defined their consistency by deleting the constraint i ≤ k ≤ j in Xu’s consistency.

Definition 1.9. [5] Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). If it holds that

$r_{ij} = r_{ik} \otimes r_{kj},$ (6) that is, ρ_ij = ρ_ikρ_kj and σ_ij = σ_ikσ_kj for each triple of (i, k, j), then R is said to be of Jiang’s consistency.

Zhang [18] presented the following consistency concept for IMPRs:

Definition 1.10. [18] Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). If it holds that $ρ_{ij} ρ_{jk} ρ_{ki} = ρ_{ik} ρ_{kj} ρ_{ji},$ (7) for all i, j, k = 1, 2, ⋯ , n, then R is said to be of Zhang’s consistency.

A vector whose entries are IMNs is called an intuitionistic multiplicative priority vector (IMPV). For an entry ω_i in an IMPV ω (resp. r_ij in an IMPR R), we write s (ω_i) = s_i (resp. s (r_ij) = s_ij) and h (ω_i) = h_i (resp. h (r_ij) = h_ij) for brevity in the following parts.

Theorem 1.11. [18] Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). Then R is of Zhang’s consistency if and only if it holds that $s_{ij} = s_{ik} s_{kj},$ (8)for all i, j, k = 1, 2, ⋯ , n.

Note that Eq. (9) was also used to define the consistency for IMPRs in Ref. [6] and some sufficient conditions to the consistency were provided. Next, we characterize this consistency for the IMPRs as follows:

Theorem 1.12. Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). Then R is of Zhang’s consistency if and only if there exists an IMPV ω = ((ρ₁, σ₁) , (ρ₂, σ₂) , ⋯ , (ρ_n, σ_n)) with $\frac{s_{i}}{s_{j}} \in [\frac{1}{81}, 81]$ such that $s_{ij} = \frac{s_{i}}{s_{j}},$ (9) where $s_{t} = \prod_{k = 1}^{n} s_{tk}^{\frac{1}{n}}$ for t = 1, 2, ⋯ , n.

Proof. Using Theorem 1.12 and Definition 1.7, we have $s_{ij} = s_{ik} s_{kj} = \frac{s_{ik}}{s_{jk}}$ , and hence $s_{ij} = \frac{\prod_{k = 1}^{n} s_{ik}^{\frac{1}{n}}}{\prod_{k = 1}^{n} s_{jk}^{\frac{1}{n}}}$ , i.e., $s_{ij} = \frac{s_{i}}{s_{j}}$ .

Conversely, $s_{ik} s_{kj} = \frac{s_{i}}{s_{k}} \frac{s_{k}}{s_{j}} = \frac{s_{i}}{s_{j}} = s_{ij}$ , and hence R is of Zhang’s consistency.□

Based on Theorem 1.12 and Definition 1.6, we find that the condition of Zhang’s consistency is weak that we could not always derive a ranking from the consistent IMPR.

Example 1.13. For a decision making problem, there are four alternatives x_i (i = 1, 2, 3, 4). The decision maker provides his/her preferences over these four alternatives, and gives an IMPR as follows: $R = (\begin{matrix} (1, 1) & (\frac{2}{3}, \frac{2}{3}) & (2, \frac{1}{6}) & (\frac{1}{2}, \frac{1}{2}) \\ (\frac{2}{3}, \frac{2}{3}) & (1, 1) & (3, \frac{1}{4}) & (\frac{3}{4}, \frac{3}{4}) \\ (\frac{1}{6}, 2) & (\frac{1}{4}, 3) & (1, 1) & (\frac{1}{4}, 3) \\ (\frac{1}{2}, \frac{1}{2}) & (\frac{3}{4}, \frac{3}{4}) & (3, \frac{1}{4}) & (1, 1) \end{matrix})$ It is trivial to verify that R is of Zhang’s consistency. Solve the equations determined by Eq. (9), we have $\begin{matrix} s_{1} = & s_{2}, s_{1} = 12 s_{3}, s_{1} = s_{4}, \\ s_{2} = & 12 s_{3}, s_{2} = s_{4}, s_{4} = 12 s_{3} . \end{matrix}$ Thus, we get s₁ = s₂ = s₄ = 12s₃ and can not obtain the ranking or the optimal alternative by Definition 1.2.

2 Novel consistency for the IMPRs

Associated with Theorem 1.3 and Definition 1.2, we propose the following novel consistency for the IMPRs to overcome the shortcoming of Zhang’s consistency:

Definition 2.1. Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). If it holds that $s_{ij} = s_{ik} s_{kj},$ (10) $h_{ij} = h_{ik} h_{kj},$ (11) for all i, j, k = 1, 2, ⋯ , n, then R is said to be of consistency. It is routine to verify that the proposed consistency is equivalent to that of Jiang’s consistency [5].

Theorem 2.2. Let R = (r_ij) _n×n be an IMPR on X with r_ij = (ρ_ij, σ_ij) for each pair of (i, j). Then R is of consistency if and only if there exists an IMPV ω = ((ρ₁, σ₁) , (ρ₂, σ₂) , ⋯ , (ρ_n, σ_n)) with $\frac{ρ_{i}}{ρ_{j}}, \frac{σ_{i}}{σ_{j}} \in [\frac{1}{9}, 9]$ and $\frac{ρ_{i} σ_{i}}{ρ_{j} σ_{j}} \in [\frac{1}{81}, 1]$ such that $ρ_{ij} = \frac{ρ_{i}}{ρ_{j}},$ (12) $σ_{ij} = \frac{σ_{i}}{σ_{j}}$ (13) for i, j = 1, 2, ⋯ , n, where $ρ_{t} = \prod_{k = 1}^{n} ρ_{tk}^{\frac{1}{n}}$ and $σ_{t} = \prod_{k = 1}^{n} σ_{tk}^{\frac{1}{n}}$ (t = 1, 2, ⋯ , n).

Proof. The proof is similar to that of Theorem 1.12, so we omit it.□

Theorem 2.2 indicates that some necessary constraints should be given to well guarantee the correspondence between the IMPRs and the IMPVs, we state them as follows:

Definition 2.3. Let ω = ((ρ₁, σ₁) , (ρ₂, σ₂) , ⋯ , (ρ_n, σ_n)) be an IMPV.

If $\frac{ρ_{i}}{ρ_{j}}, \frac{σ_{i}}{σ_{j}} \in [\frac{1}{9}, 9]$ for i, j = 1, 2, ⋯ , n, then ω it is said to be S-normalized;

If $\frac{h_{i}}{h_{j}} \in [\frac{1}{81}, 1]$ for i ≤ j, i, j = 1, 2, ⋯ , n, then ω it is said to be A-normalized;

If ω is both S-normalized and A-normalized, then it is said to be normalized.

Based on the above, Theorem 2.2 could be rewritten as follows:

Theorem 2.4. Let R be an IMPR. Then R is consistent if and only if there exists a normalized IMPV such that Eqs. (12) and (13) hold.

Example 2.5. We continue Example 1.13. It is routine to verify that R is consistent in the proposed sense. By Eqs. (12) and (13), we have $\begin{matrix} \frac{ρ_{1}}{ρ_{2}} = \frac{2}{3}, \frac{ρ_{1}}{ρ_{3}} = 2, \frac{ρ_{1}}{ρ_{4}} = \frac{1}{2}, \frac{ρ_{2}}{ρ_{3}} = 3, \\ \frac{ρ_{2}}{ρ_{4}} = \frac{3}{4}, \frac{ρ_{3}}{ρ_{4}} = \frac{1}{4}, \frac{ρ_{1}}{ρ_{2}} = \frac{2}{3}, \frac{ρ_{1}}{ρ_{3}} = \frac{1}{6}, \\ \frac{ρ_{1}}{ρ_{4}} = \frac{1}{2}, \frac{ρ_{2}}{ρ_{3}} = \frac{1}{4}, \frac{ρ_{2}}{ρ_{4}} = \frac{3}{4}, \frac{ρ_{3}}{ρ_{4}} = 3, \end{matrix}$ and hence we obtain a normalized IMPV $ω = ((\frac{1}{2}, \frac{1}{2}), (\frac{3}{4}, \frac{3}{4}), (\frac{1}{4}, 3), (1, 1))$ . Using Definition 1.6, we have x₄ ≻ x₂ ≻ x₁ ≻ x₃. Thus, the optimal alternative is x₄.

3 Check and reach the consistency for the IMPR

In real decision making, it is impossible that the decision maker’s judgment accords with concrete mathematical formulae. Thus, the IMPVs either could not be normalized or provide various rankings. From Theorem 2.2, we find that for a given IMPR, it could be not consistent, if either the consistency formulae Eqs. (12) and (13) are not satisfied or the IMPV is not normalized. In this section, we investigate the consistency from these two aspects:

Apply the Logarithm log ₃ (·) to Theorem 2, we have ${log}_{3} ρ_{i} - {log}_{3} ρ_{j} \in [- 2, 2],$ (14) ${log}_{3} σ_{i} - {log}_{3} σ_{j} \in [- 2, 2],$ (15) ${log}_{3} h_{i} - {log}_{3} h_{j} \in [- 4, 0],$ (16) ${log}_{3} ρ_{ij} = {log}_{3} ρ_{i} - {log}_{3} ρ_{j},$ (17) ${log}_{3} σ_{ij} = {log}_{3} σ_{i} - {log}_{3} σ_{j} .$ (18) For brevity, we denote ${log}_{3} ρ = \tilde{ρ}$ and ${log}_{3} σ = \tilde{σ}$ , then we get

$\begin{matrix} \tilde{ρ} = & ({\tilde{ρ}}_{1}, {\tilde{ρ}}_{2}, \dots, {\tilde{ρ}}_{n})^{T}, {\tilde{ρ}}_{i} - {\tilde{ρ}}_{j} \in [- 2, 2], \\ \tilde{σ} = & ({\tilde{σ}}_{1}, {\tilde{σ}}_{2}, \dots, {\tilde{σ}}_{n})^{T}, {\tilde{σ}}_{i} - {\tilde{σ}}_{j} \in [- 2, 2], \\ {\tilde{b}}_{ρ} = & ({\tilde{ρ}}_{12}, \dots, {\tilde{ρ}}_{1 n}, {\tilde{ρ}}_{23}, \dots, {\tilde{ρ}}_{2 n}, \dots, {\tilde{ρ}}_{n - 1, n})^{T}, \\ {\tilde{b}}_{σ} = & ({\tilde{σ}}_{12}, \dots, {\tilde{σ}}_{1 n}, {\tilde{σ}}_{23}, \dots, {\tilde{σ}}_{2 n}, \dots, {\tilde{σ}}_{n - 1, n})^{T}, \\ A \tilde{ρ} = & {\tilde{b}}_{ρ} \\ A \tilde{σ} = & {\tilde{b}}_{σ}, \end{matrix}$ with

$A = {(\begin{matrix} \begin{matrix} 1 & - 1 & 0 & \dots & 0 & 0 \\ 1 & 0 & - 1 & \dots & 0 & 0 \\ 1 & 0 & 0 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 1 & 0 & 0 & \dots & 0 & - 1 \\ 0 & 1 & - 1 & \dots & 0 & 0 \\ 0 & 1 & 0 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 1 & 0 & \dots & 0 & - 1 \\ ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 1 & - 1 \end{matrix} \end{matrix})}_{\frac{n^{2} - n}{2} \times n} .$ (19) The following results are obvious but crucial.

Theorem 3.1. For the matrix A defined in Eq. (19), it holds that r (A) = n-1, where r (·) represents the rank of matrix ·.

Proof. It is trivial, so we omit it.□

The above theorem indicates that there exist n-1 independent membership degrees and n-1 independent non-membership degrees in R which always determine the same ranking, if the consistency is satisfied.

3.1 Check and reach the normality for the IMPR

In order to reach the normality for the IMPR, we develop the following concepts:

Definition 3.2. Let R be an IMPR. Then

an IMPV determined by both n-1 independent membership degrees and n-1 independent non-membership degrees is called a local IMPV of R;

if each local IMPV is normalized, then R is said to be normalized.

Based on the above concepts, we develop the following method to S-normalize an arbitrary IMPR R as follows:

Algorithm I

Solve $A \tilde{ρ} = {\tilde{b}}_{ρ}$ with n-1 independent membership degrees in ${\tilde{b}}_{ρ}$ and $A \tilde{σ} = {\tilde{b}}_{σ}$ with n-1 independent non-membership degrees in ${\tilde{b}}_{σ}$ , the sets of the solutions are denoted as Ω and Σ;

Check the S-normalization of $\tilde{R}$ by Definition 2, if it is of S-normalization, denote $\tilde{R}$ as ${\tilde{R}}^{'}$ and go to Step 4; otherwise, go to next step;

Transform the IMPR $\tilde{R}$ to ${\tilde{R}}^{'} = ((\frac{4}{{\tilde{ρ}}_{m}} {\tilde{ρ}}_{ij}, \frac{4}{{\tilde{σ}}_{m}} {\tilde{σ}}_{ij}))_{n \times n}$ , where ${\tilde{ρ}}_{m} = max {max \tilde{ρ} - min \tilde{ρ} | \tilde{ρ} \in Ω}$ and ${\tilde{σ}}_{m} = max {max \tilde{σ} - min \tilde{σ} | \tilde{σ} \in Σ}$ , go to next step;

Output the IMPR ${\tilde{R}}^{'}$ .

Theorem 3.3. Let R be an IMPR. Then R′ provided in Algorithm I is an S-normalized IMPR.

Proof. It is trivial to verify that R′ is an IMPR. If R is S-normalized, it is natural that R′ is so. If R is not S-normalized, that is, ${\tilde{ρ}}_{m} > 4$ or ${\tilde{σ}}_{m} > 4$ . Without losing generality, we assume that ${\tilde{ρ}}_{m} > 4$ . Thus, $A {\tilde{ρ}}^{'} = A \frac{4}{m} \tilde{ρ} = \frac{4}{m} {\tilde{b}}_{ρ} = {\tilde{b}}_{ρ}^{'}$ , i.e., $ρ_{ij}^{'} = \frac{4}{m} ρ_{ij}$ and max ρ′-min ρ′ ≤ 4. Similarly, it holds that $σ_{ij}^{'} = \frac{4}{m} σ_{ij}$ and max σ′-min σ′ ≤ 4. Thus, R′ is S-normalized.□

Although the IMPR is modified by Algorithm I, the modification is based on a linear function in Step 3, and hence the local IMPVs of the original IMPR and that of the modified one always determine the same ranking. Thus, the proposed method can preserve the original preference information as much as possible in this sense.

Example 3.4. For a decision making problem, there are four alternatives x_i (i = 1, 2, 3, 4). The decision maker provides his/her preferences over these four alternatives, and gives an IMPR as follows: $R = (\begin{matrix} (1.0, 1.0) & (5, 1 / 7) & (4, 1 / 6) & (5 / 2, 1 / 5) \\ (1 / 7, 5) & (1.0, 1.0) & (5, 1 / 7) & (5, 1 / 7) \\ (1 / 6, 4) & (1 / 6, 4) & (1.0, 1.0) & (5 / 2, 1 / 5) \\ (1 / 5, 5 / 2) & (1 / 7, 5) & (1 / 5, 5 / 2) & (1.0, 1.0) \end{matrix})$ We compute Algorithm I with the mathematical software Sagemath, and find that ${\tilde{ρ}}_{m} = 3.764 < 4$ , and ${\tilde{σ}}_{m} = 4.867 > 4$ . Thus R is not S-normalized. Then, we modify R as R′:

$(\begin{matrix} (1.0, 1.0) & (3.75, 0.41) & (3.13, 0.44) & (2.12, 0.48) \\ (0.20, 2.08) & (1.0, 1.0) & (3.75, 0.41) & (3.75, 0.41) \\ (0.23, 1.88) & (0.23, 1.88) & (1.0, 1.0) & (2.12, 0.48) \\ (0.27, 1.52) & (0.20, 2.08) & (0.27, 1.52) & (1.0, 1.0) \end{matrix}) .$

3.2 Check and reach the consistency formulae for an IMPR

For an IMPR R, we construct an IMPR $\hat{R}$ , which is S-normalized and satisfies the consistency formulae, such that the distance between $\tilde{R}$ and $\hat{R}$ is the minimal, that is,

$min ({\hat{ρ}}_{i} - {\hat{ρ}}_{j} - {\tilde{ρ}}_{ij})^{2} + ({\hat{σ}}_{i} - {\hat{σ}}_{j} - {\tilde{σ}}_{ij})^{2} .$ (20) The optimal solution of Eq. (20) is equivalent to the solution of the equation $A^{T} A \hat{ρ} = A^{T} {\tilde{b}}_{ρ},$ (21) $A^{T} A \hat{σ} = A^{T} {\tilde{b}}_{σ},$ (22) which provides us with a novel method to investigate the consistency.

Theorem 3.5. The solutions of Eqs. (19)cd and (19)cd2 always exist with only one membership degree and one non-membership degree as free unknowns, respectively.

Proof. It is obvious, so we omit it.□

Definition 3.6. Let R be an S-normalized IMPR. If all the local IMPVs determine the same ranking, then R is said to be of acceptable consistency.

Next, a procedure to modify the preference values of a given IMPR R to reach the acceptable consistency can be established by the following steps:

Algorithm II

Assume $R^{(t)} = \tilde{R}$ (that is, transform R by log ₃ (·)) and the iteration t = 0;

Check and reach the S-normalization of ${\tilde{R}}^{(t)}$ by Algorithm I and denote the set of all the local IMPVs ${\hat{ω}}_{k}^{(t)}$ as S_n.

Check the acceptable consistency of ${\tilde{R}}^{(t)}$ . If ${\tilde{R}}^{(t)}$ is of acceptable consistency, then go to Step 5; otherwise, go to next step;

Reach the acceptable consistency of ${\tilde{R}}^{(t)}$ . Solve the optimal IMPV ${\hat{ω}}^{(t)}$ by Eqs. (19)cd and (19)cd2 and find k₀ such that $max_{k \in S_{n}} {dis ({\hat{ω}}^{(t)}, {\hat{ω}}_{k}^{(t)})} = dis ({\hat{ω}}^{(t)}, {\hat{ω}}_{k_{0}}^{(t)})$ (dis is a distance measure between IMNs), and then replace the preference values in ${\tilde{b}}^{(t)}$ with those in ${\hat{b}}_{k_{0}}^{(t)}$ determined by k₀. The modified ${\tilde{b}}^{(t)}$ is denoted as ${\tilde{b}}^{(t + 1)}$ and return to Step 2;

Output the ranking.

The proposed algorithm possesses the following advantages:

The modification is locally executed, not for all the entries of an IMPR, and hence can preserve the initial preference information as much as possible;

The acceptable consistency is checked by the local IMPVs, and is independent on the preseted threshold and adjustment parameters, respectively;

The procedure of reaching the acceptable consistency can be visualized by the local IMPVs.

Example 3.7. We continue Example 3.4. After computing all the local IMPVs by mathematical software Sagemath, and providing the rankings of them in Figure 1 with t = 0, we find that R is not of acceptable consistency. Then, we use Algorithm II to modify its consistency. The rankings of local IMPVs for t = 1 are also provided in Figure 1.

Fig. 1

The rankings of the local IMPVs in Example 3.7

When t = 1, the iteration stops, the modified IMPR R⁽¹⁾ is given as follows:

$(\begin{matrix} (0.5, 0.5) & (1.61, 0.48) & (3.13, 0.23) & (4.53, 0.12) \\ (0.48, 1.61) & (0.5, 0.5) & (2.13, 0.46) & (3.75, 0.20) \\ (0.23, 3.13) & (0.46, 2.13) & (0.5, 0.5) & (2.12, 0.27) \\ (0.12, 4.53) & (0.20, 3.75) & (0.27, 2.12) & (0.5, 0.5) \end{matrix}),$

which is of acceptable consistency. Thus, the ranking is x₁ ≻ x₂ ≻ x₃ ≻ x₄.

4 Comparison analysis

In this section, we make a detailed comparison with the method based on Zhang’s consistency for the IMPR as follows:

Although Zhang’s consistency in Ref. [18] is independent of alternative labels of the IMPR, it could not derive a ranking from a given IMPR in some cases; although the proposed consistency overcomes the above issue, it is equivalent to that in Ref. [5], and hence dependent of alternative labels of the IMPR. Furthermore, Zhang also proposed several consistency formulae to relate the IMPRs with the IMPVs in Refs. [18, 19] which is only a sufficient condition for their consistency. Thus, the derived IMPV could not completely reflect the preference of the decision maker. The present paper provided an equivalent condition between the IMPRs and IMPVs, and hence the preference of the decision maker can be well-described.

For the acceptable consistency in Ref. [19], it was defined by the distance measures between the original IMPR and the constructed consistent IMPR, and then checked by a preseted threshold parameter. Thus, whether the acceptable consistency for an IMPR reaches is dependent on the threshold parameter. It could be time consuming or difficult for a decision maker to provide such suitable values to meet the needs. In this paper, a novel acceptable consistency was proposed with the local IMPVs of the original IMPR without the help of the preseted threshold parameter which can avoid some opposed results by using different threshold parameters.

for reaching the acceptable consistency, the original IMPR was replaced with the weighted geometric averaging of the original IMPR and the constructed consistent IMPR with the weight as an adjustment parameter which is dependent on the threshold parameter. Moreover, the modified IMPR could still be unreasonable, because they could inherit some ill properties such as non-S-normalization from the original IMPR which can not happen in the proposed method.

Using the consistency index CI as follows:

$CI (R) = 1 - \frac{\sum_{i \neq j, i, j = 1}^{n} | ln ρ_{ij} - ln {\hat{ρ}}_{ij} |}{4 n (n - 1) ln 3}$ (23) then Zhang’s consistency improvement approach is given as follows:

Establish the matrix $\hat{R}$ by Eq. (24) as follows: $\hat{R} = (\prod_{k = 1}^{n} (\frac{ρ_{ik}}{ρ_{jk}})^{\frac{1}{n}}, \prod_{k = 1}^{n} (\frac{ρ_{ki}}{ρ_{kj}})^{\frac{1}{n}});$ (24)

Convert $\hat{R}$ into a consistent IMPR ${\hat{R}}^{'}$ by Eqs. (14) and (15) in Ref. [18].

Calculate the adjustment parameter $λ_{0} = \frac{t - CI ({\hat{R}}^{'})}{1 - CI ({\hat{R}}^{'})}$ , where $t (t > CI ({\hat{R}}^{'}))$ is a consistency threshold parameter.

Obtain the rectified IMPR $\hat{R} (λ_{0}) = ((ρ_{ij} (λ_{0}), ρ_{ji} (λ_{0})))_{n \times n}$ with acceptable consistency pursuant to $ρ_{ij} (λ) = ρ_{ij}^{1 - λ} {ρ^{'}}_{ij}^{λ}$ .

Example 4.1. We continue Example 3.4 and take t = 0.6, 0.7, 0.8, 0.9, 0.93, respectively. The consistent IMPR ${\hat{R}}^{'}$ is given as follows:

$(\begin{matrix} (1.0, 1.0) & (1.89, 0.46) & (4.92, 0.2) & (9.0, 0.12) \\ (1.89, 0.46) & (1.0, 1.0) & (2.6, 0.43) & (4.75, 0.26) \\ (4.92, 0.2) & (2.6, 0.43) & (1.0, 1.0) & (1.83, 0.61) \\ (9.0, 0.12) & (4.75, 0.26) & (1.83, 0.61) & (1.0, 1.0) \end{matrix})$

Then, the adjustment parameters λ₀ = 0.029, 0.27, 0.51, 0.76, 0.83, respectively. All the rectified IMPRs $\hat{R} (λ_{0})$ are of acceptable consistency by the corresponding thresholds, and derive the same ranking x₁ ≻ x₂ ≻ x₃ ≻ x₄ with Zhang’s method in Ref. [18]. In spite of this, we find that $\hat{R} (0.6)$ and $\hat{R} (0.7)$ are not S-normalization, and the rectified IMPRs $\hat{R} (0.6)$ , $\hat{R} (0.7)$ , $\hat{R} (0.8)$ and $\hat{R} (0.9)$ are not of acceptable consistency in the proposed sense, $\hat{R} (0.93)$ is of acceptable consistency in the proposed sense, whose local IMPVs are shown in Figure 2.

Fig. 2

The rankings of the local IMPVs in Example 4.1

5 Conclusions

The present paper proposed a novel consistency for the IMPRs by the score function and accuracy function which can overcome the shortcomings of the existing consistency. For the consistent IMPRs, they were characterized by the S-normalized IMPVs; for the inconsistent IMPRs, methods to check and reach the proposed S-normalization, acceptable consistency of the IMPRs were investigated, respectively. Compared with the existing methods, the proposed methods processes the following advantages:

The proposed consistency is completely characterized by the S-normalized and A-normalized IMPV which can guarantee the reasonability of the given IMPR;

Method to check and reach the acceptable consistency is independent on the preseted threshold and adjustment parameters, respectively;

The procedure of reaching the acceptable consistency can be visualized by the local IMPVs.

However, group decision making with the IMPRs have been developed in Refs. [7 , 18–20] and A-normalization of the IMPRs was not considered in the present paper without which the accuracy values of some entries in the modified IMPRs could be still unreasonable, that is, it could be bigger than 1. Thus, how to develop a method to simultaneously reach both S-normalization and A-normalization in group decision making with the IMPRs is the future work.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. This research was supported by the NSF of Shandong Province (No. ZR2017MG027).

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