Definition 1 ³⁴.

Let X = {x₁, x₂, … x_n} be a set of alternatives, then R = (r_ij)_n×n is called a FPR on X × X with the following conditions:

Definition 2 ³³.

Let R = (r_ij)_n×n be a FPR, then it is called a multiplicative consistent FPR if and only if

Remark 1.

In some case, additive transitivity property is in conflict with the [0,1] scale used for providing the preference values. Moreover, Chiclana et al.³⁵ have verified that multiplicative transitivity property is the most appropriate property to model and measure consistency of reciprocal preference relations. In this paper, multiplicative consistency is used.

Definition 3 ¹.

Let X = {x₁,x₂,…x_n} be a fixed set, a HFS on X is in terms of a function that when applied to x returns a subset of [0,1], which can be represented by the following:

where h_E(x) is a set of some values in [0,1], denoting the possible membership degree of the element x ∈ X. For convenience, Xia and Xu³ called h = h_E(x) a hesitant fuzzy element (HFE) and H the set of all the HFEs.

Given three HFEs h, h₁, h₂, Torra¹ defined some operations:

1. (1)

   hc=∪γ∈h{1−γ};

2. (2)

   h1∪h2=∪γ1∈h1,γ2∈h2{γ1∨γ2};

3. (3)

   h1∩h2=∪γ1∈h1,γ2∈h2{γ1∧γ2};

4. (4)

   h+=max{r|r∈h};

5. (5)

   h−=min{r|r∈h}.

Let #h denote the number of elements in the HFE h. In most cases, the numbers of possible values in different HFEs are generally different. In order to operate correctly when comparing them, one of the method is to make sure that they have the same number of elements³⁶. To solve this issue, Zhu and Xu³⁷ gave two opposite normalization principles: 1) α-normalization, remove some elements from h, which has more number of elements. 2) β-normalization, add some elements to h, which has fewer elements.

For the β-normalization, Zhu et al.² introduced the following method to add some elements to a HFE.

Definition 4 ².

Assume a HFE, h = {h^σ(s)|s = 1,2,…,#h}, let h⁺ and h⁻ be the maximum and minimum elements in h respectively. And ξ(0 ⩽ ξ ⩽ 1) be an optimized parameter, then we call h = ξh⁺ + (1 − ξ)h⁻ an added element.

ξ is used to reflect the DMs’ risk preference. Especially, Xu and Xia³⁶ introduced that ξ = 0 indicates the pessimists expect unfavorable outcomes; ξ = 1 indicates the optimized desirable outcomes.

Definition 5 ³.

For a HFE h, s(h)=1/#h∑r∈hr be the score function of h, where #h is the number of elements in h. For two HFEs, h₁ and h₂, if s(h₁) > s(h₂), then h₁ > h₂. If s(h₁) = s(h₂), then h₁ = h₂.

Definition 6 ³⁸.

Let X = {x₁,x₂,…x_n} be a fixed set, a HFPR H on X is denoted by a matrix H = (h_ij)_n×n ⊂ X × X, where hij={hijσ(s)|s=1,2,…,#hij} is a HFE, indicating hesitant degrees to which x_i is preferred to x_j. For all i, j ∈ N, h_ij (i < j) should satisfy the following conditions:

Where hijσ(s) and hjiσ(s) are the sth element in h_ij and h_ji, respectively, #h_{i j} is the number of the elements in h_ij.

Definition 7 ².

Let H = (h_ij)_n×n be a HFPR and an optimized parameter ξ(0 ⩽ ξ ⩽ 1), where ξ is used to add some elements to h_ij (i < j), and 1−ξ is used to add some elements to h_ji (i < j), then we obtain a HFPR H¯=(h¯ij)n×n . And for all i, j = 1,2,…,n, this preference relation should satisfy the following conditions:

Where h¯ijσ(s) and h¯jiσ(s) are the sth element in h¯ij and h¯ji, respectively. Then we call H¯=(h¯ij)n×n a normalized hesitant fuzzy preference relation (NHFPR) with the optimized parameter ξ, h¯ij is a normalized hesitant fuzzy element (NHFE).

Definition 8 ².

Assume a HFPR H = (h_ij)_n×n and its NHFPR H¯=(h¯ij)n×n with ξ, then H is multiplicative consistent if and only if:

Theorem 1. ¹⁶

Given a HFPR H = (h_ij)_n×n, and its NHFPR H¯=(h¯ij)n×n with ξ, the following statements are equivalent (for all i, j,k = 1,2,…,n, s=1,2,…#h¯ij ):

1. (i)

   H is multiplicative consistent;

2. (ii)

   (7)h¯ijσ(s)=h¯ikσ(s)h¯kjσ(s)h¯ikσ(s)h¯kjσ(s)+(1−h¯ikσ(s))(1−h¯kjσ(s));

3. (iii)

   (8)h¯ijσ(s)=∏k=1nh¯ikσ(s)h¯kjσ(s)n∏k=1nh¯ikσ(s)h¯kjσ(s)n+∏k=1n(1−h¯ikσ(s))(1−h¯kjσ(s))n

Theorem 2. ²

Assume a HFPR H = (h_ij)_n×n and its NHFPR H¯=(h¯ij)n×n with ξ, for i, j = 1,2,…,n, i ≠ j ≠ k, s = 1,2,…,#h_ij, let

Then, mH = (mh_ij)_n×n is called multiplicative consistent HFPR with ξ.

Remark 2.

Generally, for the HFPR H = (h_ij)_n×n, each preference degree in h_ij is a possible value, H can be directly separated into all possible FPRs. Thus, the judgement of a HFPR’s consistency is based on the consistency of the corresponding FPRs. In currently researches, it consists of three stages: (1) Normalizing a HFPR H; (2) Dividing a HFPR into several corresponding FPRs in accordance with the number of elements in HFE; (3) Checking the consistency of these corresponding FPRs by Eq. (2). If all of them are consistent, the HFPR is consistent; otherwise, it is inconsistent.

However, there are some problems for the current definitions of HFPRs, as the values in each HFE should be rearranged in ascending or descending order according to Definition 6. This operation may lead to inconsistent. In the following, we will give two examples to show the problems of the Definition 6.

Example 1 ¹⁶.

Let H be a HFPR, which is shown as follows:

In order to get a multiplicative consistent HFPR mH of H, Zhang¹⁶ first normalized it by Definition 7 (where ξ=1), and we have:

By Eq.(9), Zhang¹⁶ obtained the multiplicative consistent HFPR mH = (mh_ij)_n×n of H is:

In mH,mh₂₃ = {0.3132,0.3184,0.2949}, its elements do not arrange in ascending order, because 0.2949 is smaller than 0.3132 or 0.3184, which do not meet the requirement hijσ(s)<hijσ(s+1) in Definition 6. If we rearrange the elements in mh₂₃ according to Definition 6, the mH will be:

According to Eq.(8), we have

the left side of the equation is equal to 0.2949. The right side of the equation is equal to 0.3132. Therefore, according to Theorem 1, mH′ is not multiplicative consistent. Although Zhu et al.² provided the proof for Theorem 2, however, they did not notice that the values mhijσ(s) obtained by Eq.(9) may do not satisfy the condition hijσ(s)<hijσ(s+1) in Definition 6, and thus making Definition 6 and Theorem 2 contradictory.

Example 2.

Consider a decision making process of three alternatives X = {x₁,x₂,x₃}, two DMs E = {e₁,e₂} are invited to give their preference degrees over paired comparisons of alternatives. The result furnishes in Table 1.

Remark 3.

From Examples 1 and 2, we can know that the reorder of HFEs have an impact on the consistency judgment of a HFPR. For Example 1, if we use Theorem 1 to get a new consistency HFPR, and reorder the elements according to Definition 6, then the new HFPR may be not multiplicative consistent again according to Theorem 1. If we do not reorder the elements, the new HFPR will not satisfy Definition 6. Therefore, they are contradictory in some cases. For Example 2, we know that the original judgments for e₁ and e₂ are multiplicative consistent respectively. But if we use Definition 6 to combine the two DMs’ judgments into one HFPR directly, the HFPR is not multiplicative consistent. The main reason is that the reordering of the elements in HFEs, which will distort the DMs’ original information. Therefore, the existing definition for HFPR is problematic. In order to overcome these drawbacks, we will propose revised definitions of the HFPR and NHFPR as follows:

Definition 9.

Let X = {x₁,x₂,…x_n} be a fixed set, a HFPR H on X is denoted by a matrix H = (h_{i j})_n×n ⊂ X × X, where hij={hijs|s=1,2,…,#hij} is a HFE, indicating hesitant degrees to which x_i is preferred to x_j. For all i, j ∈ N, h_ij should satisfy the following conditions:

Where hijs and hjis are the sth element values in h_ij and h_ji, respectively, #h_ij is denoted the number of the elements in h_ij.

Definition 10.

Let H = (h_ij)_n×n be a HFPR and an optimized parameter ξ (0 ⩽ ξ ⩽ 1), where ξ is used to add some elements to h_ij (i < j), and 1−ξ is used to add some elements to h_ji (i < j), then we obtain a HFPR H¯=(h¯ij)n×n . And for all i, j = 1,2,…,n, this preference relation should satisfy the following conditions:

Where h¯ijs and h¯jis are the sth element in h¯ij and h¯ji , respectively. Then, we call H¯=(h¯ij)n×n a NHFPR with the optimized parameter ξ.

Definition 11.

Let H = (h_ij)_n×n be a HFPR and its NHFPR H¯=(h¯ij)n×n with ξ, then H is multiplicative consistent if and only if:

Theorem 3.

Given a HFPR H = (h_ij)_n×n, and its NHFPR H¯=(h¯ij)n×n with ξ, the following statements are equivalent (∀i, j,k = 1,2,…,n, s=1,2,…,#h¯ij):

1. (i)

   H is multiplicative consistent;

2. (ii)

   (13)h¯ijs=h¯iksh¯kjsh¯iksh¯kjs+(1−h¯iks)(1−h¯kjs);

3. (iii)

   (14)h¯ijs=∏k=1nh¯iksh¯kjsn∏k=1nh¯iksh¯kjsn+∏k=1n(1−h¯iks)(1−h¯kjs)n.

Theorem 4.

Assume a HFPR H = (h_ij)_n×n and its NHFPRH¯=(h¯ij)n×n with ξ, for i, j = 1,2,…,n, i ≠ j ≠ k, s = 1,2,…,#h_ij, let

where Ω = {k|k ≠ i, j}, #Ω is the cardinality of Ω. Then, mH = (mh_ij)_n×n is multiplicative consistent HFPR with ξ.

Remark 4.

The difference between the new definitions of HFPR, multiplicative consistent HFPR, NHFPR and the current definitions are that the new definitions do not arrange the elements in ascending (or descending) order. If we use Theorem 3 to get a multiplicative consistent HFPR, we would not need to reorder the elements, and it still conforms to Definition 9. Moreover, the new definitions can retain the DMs’ original information as much as possible.

Definition 12 ¹⁷.

Let H = (h_ij)_n×n be a HFPR, where hij={hijs|s=1,2,…,#hij} (i, j = 1,2,…,n), then H is called an incomplete HFPR, if some of its elements cannot be given by the DM, we denote it by unknown variable x, and the others can be provided by the DM, which satisfy:

Where hijs is the sth element of h_ij, for all h_ij ∈ EV, where EV is the set of all the known HFEs in H.

Definition 13.

Let H = (h_ij)_n×n be an incomplete HFPR and an optimized parameter ξ (0 ⩽ ξ ⩽ 1), where ξ is used to add some elements to h_ij ∈ EV (i < j), and 1 − ξ is used to add some elements to h_ji ∈ EV (i < j), then we obtain an incomplete HFPRH¯=(h¯ij)n×n. And for all i, j = 1,2,…,n, h_ij ∈ EV, this preference relation should satisfy the following conditions:

Then, we call H¯=(h¯ij)n×n an incomplete NHFPR with the optimized parameter ξ.

Definition 14.

Let H = (h_ij)_n×n be an incomplete HFPR, if the missing HFEs of H can be estimated by the known HFEs, then H is called an acceptable incomplete HFPR, otherwise, H is not an acceptable incomplete HFPR.

Theorem 5.

Let H = (h_ij)_n×n be an incomplete HFPR, the necessary condition of acceptable incomplete HFPR H is that there is at least one known HFE in each row or column of H except for the diagonal HFE.

Remark 5.

For Eq.(20) and Eq.(14), they are both the forms of multiplicative consistency. The difference of them is the range of parameter k. For Eq.(14), it ranges from 1 to n , it is suit to all the HFEs include the unknown elements. However, when some of the elements are missing as in this paper, Eq.(14) does not work. Therefore, we use Eq.(20) to estimate the missing values. In order to obtain a complete HFPR, we develop an algorithm as follows:

Example 3.

Let H = (h_ij)_4×4 be an incomplete HFP shown as follows:

We apply Algorithm 1 to estimate the missing elements:

• Step 1: Obviously, H is an acceptable incomplete HFPR.

• Step 2: Using β-normalization with ξ = 1, then the iomplete NHFPR is obtained as follows:

  H=[{0.5}{0.5,0.6}{0.2,0.2}{0.3,0.3}{0.5,0.4}{0.5}{0.7,0.9}x{0.8,0.8}{0.3,0.1}{0.5}x{0.7,0.7}xx{0.5}].

• Step 3: Estimating the missing elements x using Eqs.(18), (19) and (20), the calculation process of h₂₄ is as follows:

  h241=h211h141h211h141+(1−h211)(1−h141)=0.5×0.30.5×0.3+(1−0.5)×(1−0.3)=0.3.h242=h212h142h212h142+(1−h212)(1−h142)=0.4×0.30.4×0.3+(1−0.4)×(1−0.3)=0.2222.

  Therefore,

  h₂₄ = {0.3,0.2222}.

  Similarly, the other missing elements can be obtained:

  h₃₄ = {0.6316,0.6316}.

• Step 4: The complete HFPR of H is:

  H=[{0.5}{0.5,0.6}{0.5,0.4}{0.5}{0.8,0.8}{0.3,0.1}{0.7,0.7}{0.7,0.7778}                {0.2,0.2}{0.3,0.3}{0.7,0.9}{0.3,0.2222}{0.5}{0.6316,0.6316}{0.3684,0.3684}{0.5}].

Remark 6.

Since the numbers of elements in HFE are often different, two methods α-normalization and β-normalization are introduced in Ref. 37 to make all the HFE have the same number of values. However, there are some problems in normalization process. On the one hand, the principle aims to have same number elements in two HFEs, for β-normalization, the added values are based on the maximum and minimum values of a HFE, and the added values are not the DM’s original preferences. β-normalization is an artificial process. On the other hand, for optimized parameter , it ranges from 0 to 1. There is no rule how to choose the value for it. For example, assume a HFE h₁₂ = {0.5,0.6} and #h¯ij=3. Let ξ = 0.6, the added element is h123=0.6×0.6+0.5×.04=0.56, then h₁₂ = {0.5,0.6,0.56}. The main problem of β-normalization is too artificial. In the following, we propose a multiplicative consistency based method to add the elements. The advantage of this method is that the added elements are determined by the known elements, which makes it more reasonable and can use the DM’s information as much as possible. In the multiplicative consistency based method, we look the elements to be added as unknown values. In the following, we will give two examples to illustrate this method.

Example 4 ²⁷.

Let H = (h_ij)_4×4 be a HFPR shown as follows:

In order to obtain a NHFPR of H, we also use x denote the added elements. First, H is transformed into the following two FPRs:

Obviously, H⁽²⁾ is an incomplete and acceptable FPR. These added elements x can be estimated by using intermediate alternative x₃, the computation is given below:

After the estimation process is applied, corresponding NHFPR of H is obtained as follows:

Example 5.

Let H = (h_ij)_4×4 be a HFPR, which is shown as follows:

First, H is transformed into the following two FPRs:

For this HFPR, it is different from Example 4. It is a complete and acceptable HFPR, but the corresponding FPR H⁽²⁾ is unacceptable. The added elements x cannot be calculated immediately. In order to estimate it, we first use β-normalization with ξ = 1 to get an acceptable FPR H⁽²⁾. Let h132=0.3, H becomes:

Then the other added elements x can be estimated through the known elements.

We can know that MV = {(2,3,2),(3,2,2),(3,4,2), (4,3,2)}, using Eqs.(18), (19) and (20), we elaborate the computation process of the estimated value for h232 as follows:

Then

In a similar way, we can calculate the rest of x through the intermediate alternative x₁, after the computation is applied, the NHFPR is following:

Remark 7.

In this example, H⁽²⁾ is an unacceptable incomplete fuzzy preference relation. Xu et al.³⁹ have proposed some methods to deal with unacceptable situations. In this paper, we choose h132 as an added element randomly.

For Examples 4 and 5, in the process of normalization, we also use x to denote the added element, x is estimated based on multiplicative consistency with the known elements. The calculation process is the same as incomplete HFPR. In the following, we summarize the above process in the following Algorithm 2.

Definition 15.

Let h1={h1s|s=1,2,…,#h1} and h2={h2s|s=1,2,…,#h2} (#h₁ = #h₂ = #h) be two HFEs; then the distance between them is defined as:

Where h1s and h2s are the sth elements in h₁ and h₂, respectively.

Definition 16.

Let H₁ = (h_ij,1)_n×n and H₂ = (h_ij,2)_n×n be two HFPRs, their NHFPRs are H¯1=(h¯ij,1)n×n and H¯2=(h¯ij,2)n×n, then the distance between H₁ and H₁ is defined as:

Definition 17 ⁴².

An IOWA operator is defined as:

Where w = (w₁, w₂, …, w_m)^T is a weighting vector, such that w_t ∈ [0,1], ∑t=1mwt=1, σ is a permutation of {1,2,…,m} such that μ_σ(t) ⩾ μ_σ(t+1), ∀_t = 1,2,…,m−1, i.e.,< μ_σ(t), p_σ(t) > is the 2-tuple with μ_σ(t) the tth highest value in the set {μ₁, μ₂,…, μ_m}.

Definition 18.

Let E = {e₁, e₂,…,e_m} be a set of DMs, and {H₁,H₂,…,H_m} be the HFPRs provided by the DMs on a set of alternatives X = {x₁,x₂,…,x_n}. A MC-IOWA operator of dimension m, ϕwc is an IOWA operator whose set of order inducing values is the set of consistency index values, {CI₁,CI₂,…,CI_m}, associated with the set of DMs.

Then, the collective HFPR H_c = (h_ij,c)_n×n is computed as follows:

The weights of the MC-IOWA operator are obtained by the following expression:

Where s(σ(i))=∑k=1iCIk, and CI_k is the kth largest value in {CI₁,CI₂,…,CI_m}.

The linguistic quantifier is a Basic Unit-interval Monotone (BUM) function f: [0,1] → [0,1], such that f(0) = 0, f (1) = 1 and if x > y, then f(x) ⩾ f(y).

Then, the proximity measure can be obtained in the following three levels:

Level 1: Proximity index on pairs of alternatives (x_i, x_j), which is the similarity degree between one value of individual’s preference and the collective one:

Level 2: Proximity index on alternatives x_i, which is average degree of PP_ij,t on alternative x_i:

Level 3: Proximity index on the relation H_t, which is average degree of PA_i,t of expert e_t: