Programming model-based method for ranking objects from group decision making with interval-valued hesitant fuzzy preference relations

Abstract

Interval-valued hesitant fuzzy preference relations (IVHFPRs) are useful that allow decision makers to apply several intervals in [0, 1] to denote the uncertain hesitation preference. To derive the reasonable ranking order from group decision making with preference relations, two topics must be considered: consistency and consensus. This paper focuses on group decision making with IVHFPRs. First, a multiplicative consistency concept for IVHFPRs is defined. Then, programming models for judging the consistency of IVHFPRs are constructed. Meanwhile, an approach for deriving the interval fuzzy priority weight vector is introduced that adopts the consistency probability distribution as basis. Subsequently, this paper builds several multiplicative consistency-based programming models for estimating the missing values in incomplete IVHFPRs. A consensus index is introduced to measure the agreement degree between individual IVHFPRs, and a method for increasing the consensus level is presented. Finally, a multiplicative consistency-and-consensus-based group decision-making method with IVHFPRs is offered, and a practical decision-making problem is selected to show the application of the new method.

1 Introduction

Various types of fuzzy sets are proposed [2, 10, 34] to cope with the increasing complexity of decision-making problems. Each of them has some advantages of denoting decision-making information in some aspects. For example, Atanassov’s intuitionistic fuzzy sets [1, 2] can denote the preferred and non-preferred information simultaneously; linguistic variables [34] can express decision makers (DMs)’ qualitative preferences, while hesitant fuzzy sets [22] can simply indicate the DMs’ hesitant recognitions. Based on these types of fuzzy sets, many decision-making methods are proposed, which are applied in many fields [6, 9, 11, 16, 17, 23, 24, 30].

In the current decision-making methods, preference relations or pairwise judgement matrices are good tools. Taking the advantages of preference relations and hesitant fuzzy sets, preference relations with hesitant fuzzy judgements are proposed. According to the structure of elements, they can be classified into three types: hesitant fuzzy preference relations (HFPRs) [20, 31], hesitant multiplicative preference relations (HMPRs) [40], and hesitant fuzzy linguistic preference relations (HFLPRs) [21, 41]. Following the principle of calculating the priority weight vector, there are two main types of methods: the $\alpha$-normalization methods [25, 26, 32, 35, 42] and the $\beta$-normalization methods [36,37,38,39]. The former derives the crisp priority weight vector by only considering a crisp preference relation, while the latter uses the normalized hesitant preference relations, and the hesitant fuzzy priority weight vector is obtained from the ordered preference relations. Because the former only considers one value in each hesitant fuzzy element, this type of methods loses information. On the other hand, the latter needs to add values to the shorter hesitant fuzzy judgments, which changes the original hesitant fuzzy preference relations. Furthermore, different ranking orders may be obtained with respect to different added values.

Although preference relations with hesitant fuzzy judgments have some advantages to denote the DMs’ hesitancy, they only permit the DMs to use the exact values [27, 31, 40, 41]. This may be still insufficient to express their preferences. To address this issue, interval-valued hesitant fuzzy sets (IVHFSs) introduced by Chen et al. [3] are good choices that permit the DMs to apply several intervals rather than exact ones to denote their inherent hesitancy and uncertainty. Then, the authors developed an approach to decision making with interval-valued hesitant fuzzy information. Considering the interactive characteristics between the weights of criteria and the DMs, Meng and Chen [14] applied the Shapley function to develop a method for interval-valued hesitant fuzzy decision making with incomplete weight information. Furthermore, Chen et al. [4] proposed several correlation coefficients of IVHFSs and applied them to clustering analysis. However, Chen et al.’s correlation coefficients require IVHFSs to have the same length. Otherwise, we need to add values to the shorter ones. In contrast to Chen et al.’s correlation coefficients, Meng et al. [15] defined several other correlation coefficients of IVHFSs that need not to add extra values and consider all information offered by the DMs. Jin et al. [8] introduced a cross entropy of IVHFSs and then developed a TOPSIS method for group decision making with interval-valued hesitant fuzzy information. Menwhile, Yuan et al. [33] used the defined confidence level and the Choquet integral to propose an approach for multi-attribute interval-valued hesitant fuzzy group decision making. More researches about decision making with interval-valued hesitant fuzzy information are available in the literature [5, 6, 18].

Similar to hesitant fuzzy preference relations, Chen et al. [3] introduced the concept of interval-valued hesitant fuzzy preference relations (IVHFPRs) by using interval-valued hesitant fuzzy elements (IVHFEs). Pérez-Fernándeza et al. [18] discussed the application of IVHFPRs in group decision making. However, both of these methods are based on the aggregations operators. Two vertical topics in preference relations, consistency and consensus, are not studied. This makes the ranking orders obtained from these methods be questionable because the illogical ranking orders may be derived from inconsistent IVHFPRs [7, 13, 24, 25, 30]. Furthermore, the final ranking orders cannot reflect the agreement degrees between individual opinions. To address these issues, this paper continues to study IVHFPRs. Considering the consistency of IVHFPRs, a natural and robust consistency concept is introduced. Then, inconsistent and incomplete IVHFPRs are studied. After that, a consensus index is defined, and an interactive method for improving the consensus level is proposed. On the basis of multiplicative consistency and consensus analysis, a group decision-making method with IVHFPRs is developed. To do these, the rest is organized as follows.

Section 2 reviews several related concepts, including hesitant fuzzy sets, interval-valued hesitant fuzzy sets, intervals, basic operations, and IVHFPRs. Section 3 first recalls a multiplicative consistency concept for interval fuzzy preference relations (IFPRs). Then, it defines a multiplicative consistency concept for IVHFPRs. Subsequently, several programming models for judging the multiplicative consistency of IVHFPRs are built. On the basis of the consistency probability distribution and the multiplicatively consistent IFPRs, a method for deriving the interval fuzzy priority weight vector is introduced. Section 4 focuses on incomplete IVHFPRs. Several multiplicative consistency-based programming models for determining missing IVHFEs are constructed. Then, a numerical example is offered to show the concrete application of the developed theoretical results. Section 5 introduces a consensus index and offers an improving consensus method. Then, a multiplicative consistency-and-consensus-based group decision-making method is presented. Section 6 offers a practical group decision-making problem about evaluating the air-conditioning manufacturers to show the concrete application of the new method. Conclusion and future remarks are performed in the last section.

2 Basic concepts

In some situations, decision makers (DMs) may be hesitant on several possible values for a pair of objects. Torra [22] introduced a new type of fuzzy sets called hesitant fuzzy sets (HFSs) to denote the DMs’ hesitancy. The facilitate application, Xu and Xia [29] offered the following formal definition of HFSs:

Definition 1

[29] A hesitant fuzzy set (HFS) E on $X =$ {x₁, $x_{2}$, … $x_{n}$} is a finite set, denoted by $E =$ {<$x_{i}, h_{E}(x_{i})$>| $x_{i}\in X$} where $h_{E}(x_{i})$ is a set of several values in [0,1] denoting the possible membership degrees of the element $x_{i}\in X$ to E.

From the concept of HFSs, we know that HFSs demand the DMs to offer the crisp membership degrees to express their judgments. This restricts the application because it may be still difficult due to the inherent vagueness of the DMs. Considering this issue, Chen et al. [3] further presented interval-valued hesitant fuzzy sets (IVHFSs).

Definition 2

[3] An IVHFS $\tilde {{A}}$ on $X =$ {x₁, $x_{2}$, … $x_{n}$} is a finite set, denoted by $\tilde {{A}}=\{<x_{i} ,\tilde {{h}}_{\tilde {{A}}} (x_{i} )>\vert x_{i} \in X\}$, where $\tilde {{h}}_{\tilde {{A}}} (x_{i} )$ is a set of all possible interval-valued membership degrees of the element x_i ∈ X to $\tilde {{A}}$ in [0,1]. For convenience, $\tilde {{h}}=\tilde {{h}}_{\tilde {{A}}} (x_{i} )$ is called an interval-valued hesitant fuzzy element (IVHFE).

Using IVHFEs, Chen et al. [3] introduced the following concept of interval-valued hesitant fuzzy preference relations (IVHFPRs):

Definition 3

[3] An IVHFPR $\tilde {{H}}$ on $X =$ {x₁, $x_{2}$, …, $x_{n}$} is defined as $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$, where $\tilde {{h}}_{ij}$ is an IVHFE denoting the possible interval-valued preferred degrees of the object $x_{i}$ over $x_{j}$. Elements in $\tilde {{H}}$ have the following characteristics:

$$ \left\{\! {\begin{array}{l} h_{l,ij}^{-} + h_{m_{ji} + 1-l,ji}^{+} = h_{l,ij}^{+} +h_{m_{ji} + 1-l,ji}^{-} = 1{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt},{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}l = 1,2,...,m_{ij} \\ h_{ii}^{-} = h_{ii}^{+} = 0.5 \end{array}} \right. $$

(1)

for all $i, j=$1, 2, …, n with $i\ne j$, and $m_{ij}$ is the number of elements in $\tilde {{h}}_{ij}$.

For example, let $X =$ {x₁, $x_{2}$, $x_{3}$} . Then, an IVHFPR $\tilde {{H}}$ on X may be defined as follows:

$$\tilde{{H}}=\left( {{\begin{array}{*{20}c} {\left\{ {\left[0.5,0.5\right]} \right\}} & {\left\{ {\left[0.2,0.3],[0.4,0.5\right]} \right\}} & {\left\{ {\left[0.3,0.4],[0.5,0.7],[0.8,0.9\right]} \right\}} \\ {\left\{ {\left[0.5,0.6],[0.7,0.8\right]} \right\}} & {\left\{ {\left[0.5,0.5\right]} \right\}} & {\left\{ {\left[0.4,0.6\right]} \right\}} \\ {\left\{ {\left[0.1,0.2],[0.3,0.5],[0.6,0.7\right]} \right\}} & {\left\{ {\left[0.4,0.6\right]} \right\}} & {\left\{ {\left[0.5,0.5\right]} \right\}} \end{array}} } \right). $$

From the concept of IVHFPRs, one can check that the IVHFPR $\tilde {{H}}$ reduces to a hesitant fuzzy preference relation [31] when the endpoints of all intervals in each IVHFE are equal. Furthermore, it reduces to an interval fuzzy preference relation (IFPR) [28] when there is only one interval in each IVHFE. Because IFPRs have a close relationship with our multiplicatively consistent concept for IVHFPRs, it needs to give the concept of IFPRs.

Definition 4

[28] An IFPR $\bar {{A}}$ on $X=$ {x₁, $x_{2}$, …, $x_{n}$} is defined as follows:

$$ \bar{{A}}=(\bar{{a}}_{ij} )_{n\times n} =\left( {{\begin{array}{*{20}c} {\left[0.5,0.5\right]} & {\left[a_{12}^{-} ,a_{12}^{+} \right]} & {...} &{\left[a_{1n}^{-} ,a_{1n}^{+} \right]} \\ \!{\left[a_{21}^{-} ,a_{21}^{+} \right]} & {\left[0.5,0.5\right]} & {...} &{\left[a_{2n}^{-} ,a_{2n}^{+} \right]} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\left[a_{n1}^{-} ,a_{n1}^{+} \right]} & {\left[a_{n2}^{-} ,a_{n2}^{+} \right]} &{...} & {\left[0.5,0.5\right]} \end{array}} } \!\right), $$

(2)

where $a_{ij}^{-} ,a_{ij}^{+} >0$ such that $a_{ij}^{-} \le a_{ij}^{+}$ and $a_{ij}^{-} +a_{ji}^{+} =a_{ij}^{+} +a_{ji}^{-} = 1$ for all $i, j=$1, 2, …, n, and $\bar {{a}}_{ij}$ is the interval intensity of the object $x_{i}$ over $x_{j}$

Let us review several Minkowski operations on intervals. Let $\bar {{a}}=[a^{-},a^{+}]$ and $\bar {{b}}=[b^{-},b^{+}]$ be any two positive intervals, then

• (i) $\bar {{a}}\oplus \bar {{b}}=[a^{-}+b^{-},a^{+}+b^{+}]$;

• (ii) $\bar {{a}}\otimes \bar {{b}}=[a^{-}b^{-},a^{+}b^{+}]$;

• (iii) ${\bar {{a}}} \left / {\bar {{b}}}\right .=[{a^{-}} \left / {b^{+}},{a^{+}}\right . \left / {b^{-}}\right .]$;

• (iv) $\lambda \bar {{a}}=[\lambda a^{-},\lambda a^{+}] \lambda \ge 0$;

• (v) $\log _{\lambda } \bar {{a}}=[\log _{\lambda } a^{-},\log _{\lambda } a^{+}] \lambda >0\wedge \lambda \ne 1$.

After reviewing the previous consistency concepts for IFPRs, Meng et al. [13] found several limitations and introduced a new one using quasiIFPRs.

Definition 5

[13] Let $\bar {{A}}=(\bar {{a}}_{ij} )_{n\times n}$ be an IFPR $\bar {{B}}=(\bar {{b}}_{ij} )_{n\times n}$ is said to be a quasi IFPR (QIFPR) with respect to $\bar {{A}}$ if $\left \{ {{\begin {array}{*{20}c} {\bar {{b}}_{ij} =\bar {{a}}_{ij}} \\ {\bar {{b}}_{ji} =\bar {{a}}_{ji}^{\circ } } \end {array}} } \right .\textit {or}\left \{ {{\begin {array}{*{20}c} {\bar {{b}}_{ij} =\bar {{a}}_{ij}^{\circ } } \\ {\bar {{b}}_{ji} =\bar {{a}}_{ji}} \end {array}} } \right .$ for all $i, j =$ 1, 2, ..., n, where $\bar {{a}}_{ij}^{\circ }$ is called the quasi interval of $\bar {{a}}_{ij} =[a_{ij}^{-} ,a_{ij}^{+} ]$ such that $\bar {{a}}_{ij}^{\circ } =[a_{ij}^{+} ,a_{ij}^{-} ]$ for all $i, j =$ 1, 2, ..., n.

It is worth noting that the operational laws on quasi intervals are the same as those on intervals, where

• (i) $\bar {{a}}^{\circ } \oplus \bar {{b}}=[a^{+}+b^{-},a^{-}+b^{+}]$, $\bar {{a}}\oplus \bar {{b}}^{\circ } =[a^{-}+b^{+},a^{+}+b^{-}]$, $\bar {{a}}^{\circ } \oplus \bar {{b}}^{\circ } =[a^{+}+b^{+},a^{-}+b^{-}]$;

• (ii) $\bar {{a}}^{\circ } \otimes \bar {{b}}=[a^{+}b^{-},a^{-}b^{+}]$, $\bar {{a}}\otimes \bar {{b}}^{\circ } =[a^{-}b^{+},a^{+}b^{-}]$, $\bar {{a}}^{\circ } \otimes \bar {{b}}^{\circ } =[a^{+}b^{+},a^{-}b^{-}]$;

• (iii) ${\bar {{a}}^{\circ } } \left / {\bar {{b}}}\right .=[{a^{+}} \left / {b^{+}},{a^{-}}\right . \left / {b^{-}}\right .]$, ${\bar {{a}}} \left / {\bar {{b}}}^{\circ } \right .=[{a^{-}} \left / {b^{-}},{a^{+}}\right . \left / {b^{+}}\right .]$, ${\bar {{a}}^{\circ } } \left / {\bar {{b}}}^{\circ }\right . =[{a^{+}} \left / {b^{-}},{a^{-}}\right . \left / {b^{+}}\right .]$;

• (iv) $\lambda \bar {{a}}^{\circ } =[\lambda a^{+},\lambda a^{-}] \lambda \ge 0$;

• (v) $\log _{\lambda } \bar {{a}}^{\circ } =[\log _{\lambda } a^{+},\log _{\lambda } a^{-}] \lambda >0\wedge \lambda \ne 1$

with $\bar {{a}}=[a^{-},a^{+}]$ and $\bar {{b}}=[b^{-},b^{+}]$ being any two positive intervals. Because quasi intervals are only applied to judge the consistency of IFPRs, the operational results, intervals or quasi intervals, have no influence on the following discussions.

Definition 6

[13] Let $\bar {{A}}=(\bar {{a}}_{ij} )_{n\times n}$ be an IFPR. $\bar {{A}}$ is multiplicatively consistent if there is a multiplicatively consistent QIFPR $\bar {{B}}=(\bar {{b}}_{ij} )_{n\times n}$, namely,

$$ \bar{{b}}_{ij} \otimes \bar{{b}}_{jk} \otimes \bar{{b}}_{ki} =\bar{{b}}_{ji} \otimes \bar{{b}}_{ik} \otimes \bar{{b}}_{kj} $$

(3)

for all $i, k, j =$ 1, 2, ..., n.

One can check that Definition 6 degenerates to Tanino’s multiplicative consistency concept [19] when the IFPR $\bar {{A}}$ reduces to a reciprocal preference relation.

3 A multiplicative consistency concept for IVHFPRs

Because IFPRs can be seen as a special case of IVHFPRs, it makes us think of extending the multiplicative consistency concept for IFPRs to define the consistency of IVHFPRs. The question is that more than one interval in IVHFEs exists and the number of intervals in different IVHFEs is not equal. As we know, every element in crisp preference relations is used to judge the consistency, and their influence is same. According to this principle, we introduce the following multiplicative consistency concept:

Definition 7

Let $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$ be an IVHFPR on the object set $X =$ {x₁, x₂, …, $x_{n}$} , and let $\bar {{h}}_{l,i_{0} j_{0}}$ be an interval in $\tilde {{h}}_{i_{0} j_{0}}$. $\tilde {{H}}$ is $\bar {{h}}_{l,i_{0} j_{0}}$-multiplicatively consistent if there is a multiplicatively consistent IFPR $\bar {{A}}=(\bar {{a}}_{ij} )_{n\times n}$ , where $\bar {{a}}_{ij} \in \tilde {{h}}_{ij}$ for all i, j = 1, 2, …, $n$ with $\bar {{a}}_{i_{0} j_{0}} =\bar {{h}}_{l,i_{0} j_{0}}$ and $l\in \{1,2,...,m_{i_{0} j_{0}} \}.$

One can check that Definition 7 degenerates to Definition 6 when there is only one interval in every IVHFE. Similarly, we can further extend Definition 7 to give the following multiplicative consistency concept:

Definition 8

Let $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$ be an IVHFPR on the object set $X =$ {x₁, x₂, …, $x_{n}$} . $\tilde {{H}}$ is $\tilde {{h}}_{i_{0} j_{0}}$-multiplicatively consistent if there is a multiplicatively consistent IFPR for each interval in $\tilde {{h}}_{i_{0} j_{0}}$, namely, for any interva l $\bar {{h}}_{l,i_{0}j_{0}} \in \tilde {{h}}_{i_{0} j_{0}} l = 1,2,...,m_{i_{0} j_{0}}$, there is a multiplicatively consistent IFPR $\bar {{A}}=(\bar {{a}}_{ij} )_{n\times n}$ such that $\bar {{a}}_{ij} \in \tilde {{h}}_{ij}$ for all $i, j =$ 1, 2, …, $n$ with $\bar {{a}}_{i_{0} j_{0}} =\bar {{h}}_{l,i_{0} j_{0}}$.

Definition 6 shows that the IFPR $\bar {{A}}=(\bar {{a}}_{ij} )_{n\times n}$ is multiplicatively consistent if and only if it is $\bar {{a}}_{ij}$-multiplicatively consistent. However, this conclusion will not necessarily hold for IVHFPRs. For example, let $\tilde {{H}}$ be an IVHFPR on $X =$ {x₁, $x_{2}$, $x_{3}$} , where

$$\tilde{{H}} = \left( {{\begin{array}{*{20}c} {\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} &{\left\{ {\left[\textstyle{1 \over 5},\textstyle{3 \over {10}}\right]} \right\}} & {\left\{ {\left[\textstyle{1 \over 7},\textstyle{3 \over {10}}],[\textstyle{2 \over 5},\textstyle{1 \over 2}\right]} \right\}} \\ {\left\{ {\left[\textstyle{7 \over {10}},\textstyle{4 \over 5}\right]} \right\}} & {\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} & {\left\{ {\left[\textstyle{2 \over 5},\textstyle{1 \over 2}],[\textstyle{2 \over 3},\textstyle{3 \over 4}\right]} \right\}} \\ {\left\{ {\left[\textstyle{1 \over 2},\textstyle{3 \over 5}],[\textstyle{7 \over {10}},\textstyle{6 \over 7}\right]} \right\}} & {\left\{ {\left[\textstyle{1 \over 4},\textstyle{1 \over 3}],[\textstyle{1 \over 2},\textstyle{3 \over 5}\right]} \right\}} & {\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} \end{array}} } \right). $$

One can check that $\tilde {{H}}$ is $\tilde {{h}}_{12}$-multiplicatively consistent, where the multiplicatively consistent IFPR is $\bar {{A}}=\left ({{\begin {array}{*{20}c} {\left [\textstyle {1 \over 2},\textstyle {1 \over 2}\right ]} & {\left [\textstyle {1 \over 5},\textstyle {3 \over {10}}\right ]} & {\left [\textstyle {1 \over 7},\textstyle {3 \over {10}}\right ]} \\ {\left [\textstyle {7 \over {10}},\textstyle {4 \over 5}\right ]} & {\left [\textstyle {1 \over 2},\textstyle {1 \over 2}\right ]} & {\left [\textstyle {2 \over 5},\textstyle {1 \over 2}\right ]} \\ {\left [\textstyle {7 \over {10}},\textstyle {6 \over 7}\right ]} & {\left [\textstyle {1 \over 2},\textstyle {3 \over 5}\right ]} & {\left [\textstyle {1 \over 2},\textstyle {1 \over 2}\right ]} \end {array}} } \right )$, but it is inconsistent for $\tilde {{h}}_{13}$ and $\tilde {{h}}_{23}$ Thus, we need some extra conditions to define the multiplicative consistency of IVHFPRs.

Definition 9

Let $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$ be an IVHFPR on the object set $X =$ {x₁, $x_{2}$, …, $x_{n}$} . $\tilde {{H}}$is multiplicatively consistent if for each interval in $\tilde {{h}}_{ij}$, there is a multiplicatively consistent IFPR following Definition 6 for all i, j = 1, 2, …, n, namely, $\tilde {{H}}$is $\tilde {{h}}_{ij}$-multiplicatively consistent for all $i, j =$ 1, 2, …, n.

Example 3.1

Let $X =$ {x₁, $x_{2}$, $x_{3}$, $x_{4}$} be the object set, and let the IVHFPR $\tilde {{H}}$ on X be defined as follows:

$$\tilde{{H}}=\left( {{\begin{array}{*{20}c} {\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} &{\left\{ {\left[\textstyle{2 \over 3},\textstyle{3 \over 4}\right]} \right\}} &{\left\{ {\left[\textstyle{2 \over 5},\textstyle{3 \over 5}\right]} \right\}} &{\left\{ {\left[\textstyle{4 \over {13}},\textstyle{3 \over 5}],[\textstyle{3 \over 4},\textstyle{4 \over 5}\right]} \right\}} \\ {\left\{ {\left[\textstyle{1 \over 4},\textstyle{1 \over 3}\right]} \right\}} &{\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} &{\left\{ {\left[\textstyle{1 \over 4},\textstyle{1 \over 3}\right]} \right\}} &{\left\{ {\left[\textstyle{2 \over {11}},\textstyle{1 \over 3}],[\textstyle{1 \over 2},\textstyle{2 \over 3}\right]} \right\}} \\ {\left\{ {\left[\textstyle{2 \over 5},\textstyle{3 \over 5}\right]} \right\}} &{\left\{ {\left[\textstyle{2 \over 3},\textstyle{3 \over 4}\right]} \right\}} &{\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} &{\left\{ {\left[\textstyle{2 \over 5},\textstyle{1 \over 2}],[\textstyle{2 \over 3},\textstyle{6 \over 7}\right]} \right\}} \\ {\left\{ {\left[\textstyle{1 \over 5},\textstyle{1 \over 4}],[\textstyle{2 \over 5},\textstyle{9 \over {13}}\right]} \right\}} & {\left\{ {\left[\textstyle{1 \over 3},\textstyle{1 \over 2}],[\textstyle{2 \over 3},\textstyle{9 \over {11}}\right]} \right\}} & {\left\{ {\left[\textstyle{1 \over 7},\textstyle{1 \over 3}],[\textstyle{1 \over 2},\textstyle{3 \over 5}\right]} \right\}} &{\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} \end{array}} } \right). $$

One can check that the IVHFPR $\tilde {{H}}$ is multiplicatively consistent.

As an extension of Definition 6, one can check that Definition 9 satisfies robustness, namely, the consistency conclusion is independent of the compared orders of objects. However, it is not an easy thing to judge the consistency of IVHFPRs because there may be several intervals in each IVHFE. To address this issue, we introduce a programming model based method for judging the multiplicative consistency of IVIFPRs.

Let $\alpha _{l,ij} =\left \{ {{\begin {array}{*{20}c} 1 & {\text {if}{\kern 1pt}{\kern 1pt}{\kern 1pt}\bar {{h}}_{l,ij} {\kern 1pt}{\kern 1pt}\text {is}{\kern 1pt}{\kern 1pt}\text {chosen}{\kern 1pt}{\kern 1pt}} \\ 0 & {\text {otherwise}} \end {array}} } \right .$ and $\alpha _{l,ij} =\alpha _{m_{ij} + 1-l,ji}$ for all i, $j=$1, 2, …, n and all $l =$ 1, 2, …, $m_{ij}$. Then, each interval in the IVHFE $\tilde {{h}}_{ij}$ can be denoted as: $\otimes _{l = 1}^{m_{ij}} \left ({\bar {{h}}_{l,ij}} \right )^{\alpha _{l,ij}}$ with $\sum \nolimits _{l = 1}^{m_{ij}} {\alpha _{l,ij}} = 1$. Thus, any IFPR $\bar {{A}}=(\bar {{a}}_{ij} )_{n\times n}$ obtained from the IVHFPR $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$ can be expressed as follows:

$$ \bar{{A}}=\left( {{\begin{array}{*{20}c} {\left[0.5,0.5\right]} & {\otimes_{l = 1}^{m_{12}} \left( {\bar{{h}}_{l,12}} \right)^{\alpha_{l,12}} } & {...} & {\otimes_{l = 1}^{m_{1n}} \left( {\bar{{h}}_{l,1n}} \right)^{\alpha_{l,1n}} } \\ {\otimes_{l = 1}^{m_{12}} \left( {\bar{{h}}_{l,21}} \right)^{\alpha_{l,21}} } & {\left[0.5,0.5\right]} & {\cdots} & {\otimes_{l = 1}^{m_{2n}} \left( {\bar{{h}}_{l,2n}} \right)^{\alpha_{l,2n}} } \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {\otimes_{l = 1}^{m_{1n}} \left( {\bar{{h}}_{l,n1}} \right)^{\alpha_{l,n1}} } & {\otimes_{l = 1}^{m_{2n}} \left( {\bar{{h}}_{l,n2}} \right)^{\alpha_{l,n2}} } & & {\left[0.5,0.5\right]} \end{array}} } \right) $$

(4)

with $\sum \nolimits _{l = 1}^{m_{ij}} {\alpha _{l,ij} = 1}$ and $\alpha _{l,ij} =\alpha _{m_{ij} + 1-l,ji}$ for all i, $j =$ 1, 2, …, n .

To judge the multiplicative consistency of the IVHFPR $\tilde {{H}}$, one can check whether the IFPR $\bar {{A}}$ is multiplicatively consistent, where $\alpha _{l,ij} = 1$ for all i, $j=$1, 2, …, n and all $l =$ 1, 2, …, $m_{ij}$.

When the IFPR $\bar {{A}}$ defined by formula (4) is multiplicatively consistent, there are the 0-1 indicator variables $\delta _{ij} =\left \{ \begin {array}{l} 1{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\text {if}{\kern 1pt}{\kern 1pt}{\kern 1pt}\bar {{a}}_{ij} \text {=}\otimes _{l = 1}^{m_{ij}} \left ({\bar {{h}}_{l,ij}} \right )^{\alpha _{l,ij}} \\ \text {0}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\text {if}{\kern 1pt}{\kern 1pt}{\kern 1pt}\bar {{a}}_{ij} \text {=}\left ({\otimes _{l = 1}^{m_{ij}} \left ({\bar {{h}}_{l,ij}} \right )^{\alpha _{l,ij}} } \right )^{\circ } \end {array}\right .$ with $\delta _{ij} \text {+}\delta _{ji} = 1$ for all i, $j=$1, 2, …, n satisfying formula (3), namely,

$$\begin{array}{@{}rcl@{}} &&\left( {\left( {\otimes_{l = 1}^{m_{ij}} \left( {\bar{{h}}_{l,ij}} \right)^{\alpha_{l,ij}} } \right)^{\delta_{ij}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ij}} \left( {\bar{{h}}_{l,ij}} \right)^{\alpha_{l,ij}} } \right)^{\circ} } \right)^{(1-\delta_{ij} )}} \right)~\otimes \left( {\left( {\otimes_{l = 1}^{m_{jk}} \left( {\bar{{h}}_{l,jk}} \right)^{\alpha_{l,jk}} } \right)^{\delta_{jk}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{jk}} \left( {\bar{{h}}_{l,jk}} \right)^{\alpha_{l,jk}} } \right)^{\circ} } \right)^{(1-\delta_{jk} )}} \right)~\\&&\quad~\!\!\otimes \left( {\left( {\otimes_{l = 1}^{m_{ki}} \left( {\bar{{h}}_{l,ki}} \right)^{\alpha_{l,ki}} } \right)^{\delta_{ki}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ki}} \left( {\bar{{h}}_{l,ki}} \right)^{\alpha_{l,ki}} } \right)^{\circ} } \right)^{(1-\delta_{ki} )}} \right)=\left( {\left( {\otimes_{l = 1}^{m_{ji}} \left( {\bar{{h}}_{l,ji}} \right)^{\alpha_{l,ji}} } \right)^{\delta_{ji}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ji}} \left( {\bar{{h}}_{l,ji}} \right)^{\alpha_{l,ji}} } \right)^{\circ} } \right)^{(1-\delta_{ji} )}} \right)\\&&~\quad\!\!\otimes \left( {\left( {\otimes_{l = 1}^{m_{ik}} \left( {\bar{{h}}_{l,ik}} \right)^{\alpha_{l,ik}} } \right)^{\delta_{ik}} \!\!\otimes\! \left( {\left( {\otimes_{l = 1}^{m_{ik}} \left( {\bar{{h}}_{l,ik}} \right)^{\alpha_{l,ik}} } \right)^{\circ} } \right)^{(1-\delta_{ik} )}} \!\right)~\!\!\otimes\! \left( {\left( {\otimes_{l = 1}^{m_{kj}} \left( {\bar{{h}}_{l,kj}} \right)^{\alpha_{l,kj}} } \right)^{\delta_{kj}} \!\!\otimes\! \left( {\left( {\otimes_{l = 1}^{m_{kj}} \left( {\bar{{h}}_{l,kj}} \right)^{\alpha_{l,kj}} } \right)^{\circ} } \right)^{(1-\delta_{kj} )}} \!\right), \end{array} $$

(5)

where i, k, $j=$1, 2, …, n.

We take the logarithm on both sides of formula (5) and get

$$\begin{array}{@{}rcl@{}} &&\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {\bar{{h}}_{l,ij}} \right)}} \right)\oplus (1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {\bar{{h}}_{l,ij}} \right)}} \right)^{\circ} ~\oplus \delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {\bar{{h}}_{l,jk}} \right)}} \right)\oplus (1-\delta_{jk})\\&&\quad\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {\bar{{h}}_{l,jk}} \right)}} \right)^{\circ} \oplus\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {\bar{{h}}_{l,ki}} \right)}} \right)\oplus (1-\delta_{ki} )\left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {\bar{{h}}_{l,ki}} \right)}} \right)^{\circ} \\&&~\quad=\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {\bar{{h}}_{l,ji}} \right)}} \right)\oplus (1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {\bar{{h}}_{l,ji}} \right)}} \right)^{\circ} \oplus\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {\bar{{h}}_{l,ik}} \right)}} \right)\\ &&\quad\oplus (1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {\bar{{h}}_{l,ik}} \right)}} \right)^{\circ} ~\oplus \delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {\bar{{h}}_{l,kj}} \right)}} \right)\oplus (1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {\bar{{h}}_{l,kj}} \right)}} \right)^{\circ} , \end{array} $$

(6)

where i, k, $j=$1, 2, …, n.

For per (6), we further derive

$$ \left\{ {\begin{array}{l} \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right) + (1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right) + \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right) + (1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)=\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right) + \\ \delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right) + (1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right) \\ \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right) + (1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right) + \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right) + (1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)=\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right) + \\ \delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right) + (1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right) \end{array}} \right.. $$

(7)

Because formula (7) will not necessarily hold for any given IVHFPR $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$, we relax it by introducing the positive and negative derivation values $\varepsilon _{ij}^{-} ,\varepsilon _{ij}^{+} ,\tau _{ij}^{-} ,\tau _{ij}^{+}$ such that $\varepsilon _{ij}^{-} ,\varepsilon _{ij}^{+} ,\tau _{ij}^{-} ,\tau _{ij}^{+} \ge 0$ for all i, $j=$1, 2, …, n, where

$$ \left\{\! {\begin{array}{l} \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+ \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+} \right. \\ \left. {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1 - \delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right) + (1 - \delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right)+\varepsilon_{ij}^{-} -\varepsilon_{ij}^{+} = 0 \\ \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+ \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+} \right. \\ \left. {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+(1 - \delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right) + (1 - \delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right)+\tau_{ij}^{-} -\tau_{ij}^{+} = 0 \end{array}} \right..\!\!\! $$

(8)

Let $\alpha _{l_{0} ,i_{0} j_{0}} = 1$ for all $i_{0}$, $j_{0} =$ 1, 2, …, n and all $l_{0} =$1, 2, …, $m_{i_{0} j_{0}}$. We construct the following programming model to judge the $\bar {{h}}_{l_{0} ,i_{0} j_{0}}$-multiplicative consistency of the IVHFPR $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$:

$$\begin{array}{@{}rcl@{}} &&J_{^{l_{0} ,i_{0} j_{0}} }^{\ast} =\min \sum\nolimits_{i,j = 1}^{n} {\left( {\varepsilon_{ij}^{-} +\varepsilon_{ij}^{+} +\tau_{ij}^{+} +\tau_{ij}^{+}} \right)}\\ &&s.t.\left\{ {\begin{array}{l} \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+ \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+} \right. \\ \left. {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right) \\ +\varepsilon_{ij}^{-} -\varepsilon_{ij}^{+} = 0,i,k,j = 1,2,...,n \\ \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+ \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+} \right. \\ \left. {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right) \\ +\tau_{ij}^{-} -\tau_{ij}^{+} = 0,i,k,j = 1,2,...,n \\ \alpha_{l,ij} = 0\vee 1,i,j = 1,2,...,n,l = 1,2,...,m_{ij} \\ \alpha_{l,ij} =\alpha_{m_{ij} + 1-l,ji} ,i,j = 1,2,...,n,l = 1,2,...,m_{ij} \\ \sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} = 1} ,i,j = 1,2,...,n \\ \delta_{ij} = 0\vee 1,i,j = 1,2,...,n \\ \delta_{ij} +\delta_{ji} = 1,i,j = 1,2,...,n \\ \alpha_{l_{0} ,i_{0} j_{0}} = 1 \end{array}} \right. . \end{array} $$

(9)

From the robustness of Definition 9, we can only apply the upper triangular part to judge the multiplicative consistency. Thus, we further construct the following programming model to judge the $\bar {{h}}_{l_{0} ,i_{0} j_{0}}$-multiplicative consistency:

$$\begin{array}{@{}rcl@{}} &&K_{^{l_{0} ,i_{0} j_{0}} }^{\ast} =\min \sum\nolimits_{i,j = 1,i<j}^{n} {\left( {\varepsilon_{ij}^{-} +\varepsilon_{ij}^{+} +\tau_{ij}^{+} +\tau_{ij}^{+}} \right)}\\ &&s.t.\left\{ {\begin{array}{l} \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+ \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+} \right. \\ \left. {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right) \\ +\varepsilon_{ij}^{-} -\varepsilon_{ij}^{+} = 0,i,k,j = 1,2,...,n,i<k<j \\ \delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+ \\ \delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+} \right. \\ \left. {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right) \\ +\tau_{ij}^{-} -\tau_{ij}^{+} = 0,i,k,j = 1,2,...,n,i<k<j \\ \alpha_{l,ij} = 0\vee 1,i,j = 1,2,...,n,l = 1,2,...,m_{ij} ,i<j \\ \alpha_{l,ij} =\alpha_{m_{ij} + 1-l,ji} ,i,j = 1,2,...,n,l = 1,2,...,m_{ij} ,i<j \\ \sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} = 1} ,i,j = 1,2,...,n,i<j \\ \delta_{ij} = 0\vee 1,i,j = 1,2,...,n,i<j \\ \delta_{ij} +\delta_{ji} = 1,i,j = 1,2,...,n,i<j \\ \alpha_{l_{0} ,i_{0} j_{0}} = 1 \end{array}} \right.. \end{array} $$

(10)

Solving model (10), if $K_{^{l_{0} ,i_{0} j_{0}} }^{\ast } = 0$, the IVHFPR $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$ is $\bar {{h}}_{l_{0} ,i_{0} j_{0}}$ -multiplicatively consistent. Furthermore, if $K_{^{l_{0} ,i_{0} j_{0}} }^{\ast } = 0$ for all $l_{0} =$1, 2, …,$m_{i_{0} j_{0}}$ , it is $\bar {{h}}_{i_{0} j_{0}}$ -multiplicatively consistent. Moreover, if $K_{^{l_{0} ,i_{0} j_{0}} }^{\ast } = 0$ for all i₀, $j_{0} =$ 1, 2, …, n with i₀< $j_{0}$ and all l₀ =1, 2, …,$m_{i_{0} j_{0}}$, it is multiplicatively consistent.

On the other hand, if $K_{^{l_{0} ,i_{0} j_{0}} }^{\ast } \ne 0$, then the IVHFPR $\tilde {{H}}$ is inconsistent. According to the obtained 0-1 indicatorvariables $\alpha _{l,ij}^{\ast }$ and $\delta _{ij}^{\ast }$ for all i, $j =$ 1, 2, …, n with i <j and all l =1, 2, …, m_ij , we obtain the QIFPR $\bar {{B}}_{l_{0}} = ({\bar {{b}}_{l_{0} ,ij}} )_{n\times n}$ , where $\bar {{b}}_{l_{0} ,ij} =({\otimes _{l = 1}^{m_{ij}} ({\bar {{h}}_{l,ij}} )^{\alpha _{l,ij}^{\ast } } } )^{\delta _{ij}^{\ast } } \otimes (({\otimes _{l = 1}^{m_{ij}} ({\bar {{h}}_{l,ij}} )^{\alpha _{l,ij}^{\ast } } } )^{\circ } )^{(1-\delta _{ij}^{\ast } )}$ for all i, $j =$ 1, 2, …, n, which has the highest consistency level with respect to $\bar {{h}}_{l_{0},i_{0} j_{0}}$ Then, we can adopt the method in Property 6 [13] to derive the multiplicatively consistent QIFPR $\bar {{C}}_{l_{0}} =(\bar {{c}}_{l_{0},ij} )_{n\times n}$, where

$$ \bar{{c}}_{l_{0} ,ij} =\left[ {\frac{\sqrt[n]{{\Pi}_{k = 1}^{n} b_{l_{0} ,ik}^{-} b_{l_{0} ,kj}^{-}} }{\sqrt[n]{{\Pi}_{k = 1}^{n} b_{l_{0} ,ik}^{-} b_{l_{0} ,kj}^{-}} +\sqrt[n]{{\Pi}_{k = 1}^{n} b_{l_{0} ,jk}^{-} b_{l_{0} ,ki}^{-}} },\frac{\sqrt[n]{{\Pi}_{k = 1}^{n} b_{l_{0} ,ik}^{+} b_{l_{0} ,kj}^{+}} }{\sqrt[n]{{\Pi}_{k = 1}^{n} b_{l_{0} ,ik}^{+} b_{l_{0} ,kj}^{+}} +\sqrt[n]{{\Pi}_{k = 1}^{n} b_{l_{0} ,jk}^{+} b_{l_{0} ,ki}^{+}} }} \right] $$

(11)

for all $i j =$ 1, 2, ..., n

Next, we introduce a method for deriving the collectively multiplicatively consistent QIFPR $\bar {{C}}=(\bar {{c}}_{ij} )_{n\times n}$, by which we can obtain the interval fuzzy priority weight vector. The DM applies several intervals rather than one to denote his/her uncertain judgment for a pair of objects because all of these intervals are regarded as the uncertain preferred degrees. Thus, when the DM does not offer the importance of intervals in an IVHFE, it is reasonable to consider that there is a uniform probability distribution on these intervals, namely, intervals in each IVHFE have the same importance, and their chosen probabilities are equal. From this point of view, we determine the consistency probability distribution on the derived QIFPRs.

Let $\tilde {{H}}=(\tilde {{h}}_{ij} )_{n\times n}$ be an IVHFPR, and let $\bar {{C}}_{q} =(\bar {{c}}_{q,ij} )_{n\times n}$, $q =$ 1, 2, …, ${\Delta }$, be the multiplicatively consistent QIFPR obtained from formula (11), where ${\Delta }$ is the number of QIFPRs. Let $p_{q}$ be the consistency probability of the multiplicatively consistent QIFPR $\bar {{C}}_{q}$ such that $\sum \nolimits _{q = 1}^{\Delta } {p_{q}} = 1$ and $p_{q} \ge 0$.

To derive the collectively multiplicatively consistent QIFPR $\bar {{C}}=(\bar {{c}}_{ij} )_{n\times n}$, we apply the interval weighted geometric mean aggregation (IWGMA) operator to calculate the collective interval preference relation $\bar {{S}}=(\bar {{s}}_{ij} )_{n\times n}$, where $\bar {{s}}_{ij} =\text {IWGMA}(\bar {{c}}_{1,ij} ,\bar {{c}}_{2,ij} ,...,\bar {{c}}_{q,ij} )={\Pi }_{q = 1}^{{\Delta }} \left ({\bar {{c}}_{q,ij}} \right )^{p_{q}}$ for all i, j = 1, 2, …, n. Because we usually have $\left \{ {\begin {array}{l} s_{ij}^{-} +s_{ji}^{-} \ne 1 \\ s_{ij}^{+} +s_{ji}^{+} \ne 1 \end {array}} \right .$ for $i j =$ 1, 2, ..., n, this means that $\bar {{S}}$ is not a QIFPR. To address this issue, we apply formula (11) to obtain the collectively multiplicatively consistent QIFPR $\bar {{C}}=(\bar {{c}}_{ij})_{n\times n}$.

Following $\bar {{C}}=(\bar {{c}}_{ij} )_{n\times n}$ we obtain the multiplicatively consistent IFPR $\bar {{D}}=(\bar {{d}}_{ij} )_{n\times n}$, where $\bar {{d}}_{ij} =\left \{ {\begin {array}{l} \bar {{c}}_{ij} {\kern 1pt}{\kern 1pt}{\kern 1pt}\bar {{c}}_{ij} {\kern 1pt}{\kern 1pt}\text {is}{\kern 1pt}{\kern 1pt}\text {an}{\kern 1pt}{\kern 1pt}{\kern 1pt}\text {interval} \\ \bar {{c}}_{ij}^{\circ } {\kern 1pt}{\kern 1pt}{\kern 1pt}\bar {{c}}_{ij} {\kern 1pt}{\kern 1pt}\text {is}{\kern 1pt}{\kern 1pt}\mathrm {a}{\kern 1pt}{\kern 1pt}{\kern 1pt}\text {quasi-interval} \end {array}} \right .$ for all $i j =$ 1, 2, ..., n

To show the above principle clearly, we consider Example 1 again. Using model (10), the following two multiplicatively consistent QIFPRs are derived:

$$\bar{{B}}_{1} =\left( {{\begin{array}{*{20}c} {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} & {\left[\textstyle{2 \over 3},\textstyle{3 \over 4}\right]} & {\left[\textstyle{2 \over 5},\textstyle{3 \over 5}\right]} & {\left[\textstyle{4 \over {13}},\textstyle{3 \over 5}\right]} \\ {\left[\textstyle{1 \over 3},\textstyle{1 \over 4}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} & {\left[\textstyle{1 \over 4},\textstyle{1 \over 3}\right]} & {\left[\textstyle{2 \over {11}},\textstyle{1 \over 3}\right]} \\ {\left[\textstyle{3 \over 5},\textstyle{2 \over 5}\right]} & {\left[\textstyle{3 \over 4},\textstyle{2 \over 3}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} & {\left[\textstyle{2 \over 5},\textstyle{1 \over 2}\right]} \\ {\left[\textstyle{9 \over {13}},\textstyle{2 \over 5}\right]} & {\left[\textstyle{9 \over {11}},\textstyle{2 \over 3}\right]} & {\left[\textstyle{3 \over 5},\textstyle{1 \over 2}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \end{array}} } \right) $$

for intervals $\bar {{h}}_{1,14} ,\bar {{h}}_{1,24}$ and $\bar {{h}}_{1,34}$, and

$$\bar{{B}}_{2} =\left( {{\begin{array}{*{20}c} {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} & {\left[\textstyle{3 \over 4},\textstyle{2 \over 3}\right]} & {\left[\textstyle{3 \over 5},\textstyle{2 \over 5}\right]} & {\left[\textstyle{3 \over 4},\textstyle{4 \over 5}\right]} \\ {\left[\textstyle{1 \over 4},\textstyle{1 \over 3}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} & {\left[\textstyle{1 \over 3},\textstyle{1 \over 4}\right]} & {\left[\textstyle{1 \over 2},\textstyle{2 \over 3}\right]} \\ {\left[\textstyle{2 \over 5},\textstyle{3 \over 5}\right]} & {\left[\textstyle{2 \over 3},\textstyle{3 \over 4}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} & {\left[\textstyle{2 \over 3},\textstyle{6 \over 7}\right]} \\ {\left[\textstyle{1 \over 4},\textstyle{1 \over 5}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 3}\right]} & {\left[\textstyle{1 \over 3},\textstyle{1 \over 7}\right]} & {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \end{array}} } \right) $$

for intervals $\bar {{h}}_{2,14} ,\bar {{h}}_{2,24}$ and $\bar {{h}}_{2,34}$.

Because the QIFPRs $\bar {{B}}_{1}$ and $\bar {{B}}_{2}$ have the same chosen probability, the consistency probability distribution on them is {1/2, 1/2} . Using the IWGMA operator and formula (11), the collectively multiplicatively consistent QIFPR is

$$\bar{{C}}=\left( {\begin{array}{llll} \left[0.5000,0.5000\right]&\left[0.6778,0.7404\right]\text{} &\left[0.4246,0.5754\right]& \left[0.4979,0.7404\right] \\ \left[0.3222,0.2596\right]&\left[0.5000,0.5000\right]\text{} &\left[0.2596,0.3222\right]& \left[0.3204,0.5000\right] \\ \left[0.5754,0.4246\right]&\left[0.7404,0.6778\right]\text{} &\left[0.5000,0.5000\right]& \left[0.5734,0.6778\right] \\ \left[0.5021,0.2596\right]&\left[0.6796,0.5000\right]\text{} &\left[0.4266,0.3222\right]& \left[0.5000,0.5000\right] \end{array}} \right) $$

by which the multiplicatively consistent IFPR is

$$\bar{{D}}=\left( {\begin{array}{llll} \left[0.5000,0.5000\right]&\left[0.6778,0.7404\right]\text{} &\left[0.4246,0.5754\right]& \left[0.4979,0.7404\right] \\ \left[0.2596,0.3222\right]&\left[0.5000,0.5000\right]\text{} &\left[0.2596,0.3222\right] &\left[0.3204,0.5000\right] \\ \left[0.4246,0.5754\right]&\left[0.6778,0.7404\right]\text{} &\left[0.5000,0.5000\right] &\left[0.5734,0.6778\right] \\ \left[0.2596,0.5021\right]&\left[0.5000,0.6796\right]\text{} &\left[0.3222,0.4266\right] &\left[0.5000,0.5000\right] \end{array}} \right) $$

4 Programming models for determining missing values

In some cases, we can only obtain incomplete preference relations or judgment matrices for various reasons [11, 12, 39]. Considering this situation, this section focuses on incomplete IVHFPRs, namely, some values in an IVHFPR are missing. Several multiplicative consistency-based programming models for estimating missing values are constructed. First, we consider the multiplicative consistency concept for incomplete IVHFPRs.

Definition 10

Let $\tilde {{H}} = (\tilde {{h}}_{ij} )_{n\times n}$ be an incomplete IVHFPR on the object set X, namely, there are missing IVHFEs. It is multiplicatively consistent if there are IVHFEs for missing judgements that make $\tilde {{H}}$ be multiplicatively consistent.

When the incomplete IVHFPR $\tilde {{H}}$ is multiplicatively consistent, by formula (5) we obtain:

$$\begin{array}{@{}rcl@{}} &&\prod\nolimits_{k = 1}^{n} \left( \left( {\left( {\otimes_{l = 1}^{m_{ij}} \left( {\bar{{h}}_{l,ij}} \right)^{\alpha_{l,ij}} } \right)^{\delta_{ij}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ij}} \left( {\bar{{h}}_{l,ij}} \right)^{\alpha_{l,ij}} } \right)^{\circ} } \right)^{(1-\delta_{ij} )}} \right) \otimes \left( {\left( {\otimes_{l = 1}^{m_{jk}} \left( {\bar{{h}}_{l,jk}} \right)^{\alpha_{l,jk}} } \right)^{\delta_{jk}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{jk}} \left( {\bar{{h}}_{l,jk}} \right)^{\alpha_{l,jk}} } \right)^{\circ} } \right)^{(1-\delta_{jk} )}} \right)\right.\\&&\quad\left.\otimes \left( {\left( {\otimes_{l = 1}^{m_{ki}} \left( {\bar{{h}}_{l,ki}} \right)^{\alpha_{l,ki}} } \right)^{\delta_{ki}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ki}} \left( {\bar{{h}}_{l,ki}} \right)^{\alpha_{l,ki}} } \right)^{\circ} } \right)^{(1-\delta_{ki} )}} \right)\right) =\prod\nolimits_{k = 1}^{n} \left( \left( {\left( {\otimes_{l = 1}^{m_{ji}} \left( {\bar{{h}}_{l,ji}} \right)^{\alpha_{l,ji}} } \right)^{\delta_{ji}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ji}} \left( {\bar{{h}}_{l,ji}} \right)^{\alpha_{l,ji}} } \right)^{\circ} } \right)^{(1-\delta_{ji} )}} \right)\right.\\&&\left.\quad\otimes \left( {\left( {\otimes_{l = 1}^{m_{ik}} \left( {\bar{{h}}_{l,ik}} \right)^{\alpha_{l,ik}} } \right)^{\delta_{ik}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ik}} \left( {\bar{{h}}_{l,ik}} \right)^{\alpha_{l,ik}} } \right)^{\circ} } \right)^{(1-\delta_{ik} )}} \right)\otimes \left( {\left( {\otimes_{l = 1}^{m_{kj}} \left( {\bar{{h}}_{l,kj}} \right)^{\alpha_{l,kj}} } \right)^{\delta_{kj}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{kj}} \left( {\bar{{h}}_{l,kj}} \right)^{\alpha_{l,kj}} } \right)^{\circ} } \right)^{(1-\delta_{kj} )}} \right)\right) \end{array} $$

(12)

for all i, $j =$ 1, 2, …, n and all $l =$ 1, 2, …, $m_{ij}$, where the notations as shown in formulae (4) and (5).

From formula (12), we derive:

$$\begin{array}{@{}rcl@{}} &&\left( {\left( {\otimes_{l = 1}^{m_{ij}} \left( {\bar{{h}}_{l,ij}} \right)^{\alpha_{l,ij}} } \right)^{\delta_{ij}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ij}} \left( {\bar{{h}}_{l,ij}} \right)^{\alpha_{l,ij}}} \right)^{\circ} } \right)^{(1-\delta_{ij} )}} \right)^{n-2}\otimes \prod\nolimits_{k = 1,k\ne i,j}^{n} \left( \left( \left( {\otimes_{l = 1}^{m_{jk}} \left( {\bar{{h}}_{l,jk}} \right)^{\alpha_{l,jk}} } \right)^{\delta_{jk}} \right.\right.\\ &&\left.\left.\otimes \left( {\left( {\otimes_{l = 1}^{m_{jk}} \left( {\bar{{h}}_{l,jk}} \right)^{\alpha_{l,jk}} } \right)^{\circ} } \right)^{(1-\delta_{jk} )}\right)\otimes \left( {\left( {\otimes_{l = 1}^{m_{ki}} \left( {\bar{{h}}_{l,ki}} \right)^{\alpha_{l,ki}} } \right)^{\delta_{ki}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ki}} \left( {\bar{{h}}_{l,ki}} \right)^{\alpha_{l,ki}} } \right)^{\circ} } \right)^{(1-\delta_{ki} )}} \right)\right)\\ &&=\left( {\left( {\otimes_{l = 1}^{m_{ji}} \left( {\bar{{h}}_{l,ji}} \right)^{\alpha_{l,ji}} } \right)^{\delta_{ji}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{ji}} \left( {\bar{{h}}_{l,ji}} \right)^{\alpha_{l,ji}} } \right)^{\circ} } \right)^{(1-\delta_{ji} )}} \right)^{n-2}\otimes \prod\nolimits_{k = 1,k\ne i,j}^{n} \left( \left( \left( {\otimes_{l = 1}^{m_{ik}} \left( {\bar{{h}}_{l,ik}} \right)^{\alpha_{l,ik}} } \right)^{\delta_{ik}} \right.\right.\\ &&\left.\left.\otimes \left( {\left( {\otimes_{l = 1}^{m_{ik}} \left( {\bar{{h}}_{l,ik}} \right)^{\alpha_{l,ik}} } \right)^{\circ} } \right)^{(1-\delta_{ik} )}\right)\otimes \left( {\left( {\otimes_{l = 1}^{m_{kj}} \left( {\bar{{h}}_{l,kj}} \right)^{\alpha_{l,kj}} } \right)^{\delta_{kj}} \otimes \left( {\left( {\otimes_{l = 1}^{m_{kj}} \left( {\bar{{h}}_{l,kj}} \right)^{\alpha_{l,kj}} } \right)^{\circ} } \right)^{(1-\delta_{kj} )}} \right)\right) \end{array} $$

(13)

for all i, $j =$ 1, 2, …, n and all $l =$ 1, 2, …, m_ij.

We take the logarithm on both sides of formula (13) and get

$$\begin{array}{@{}rcl@{}} &&(n-2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {\bar{{h}}_{l,ij}} \right)}} \right)\oplus (1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {\bar{{h}}_{l,ij}} \right)}} \right)^{\circ} } \right)\oplus\\ &&\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {\bar{{h}}_{l,jk}} \right)}} \right)\oplus (1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {\bar{{h}}_{l,jk}} \right)}} \right)^{\circ} } \right)} \oplus\\ &&\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {\bar{{h}}_{l,ki}} \right)}} \right)\oplus (1-\delta_{ki} )\left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {\bar{{h}}_{l,ki}} \right)}} \right)^{\circ} } \right)}\\ &&=(n-2)\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {\bar{{h}}_{l,ji}} \right)}} \right)\oplus (1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {\bar{{h}}_{l,ji}} \right)}} \right)^{\circ} } \right)\oplus\\ &&\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {\bar{{h}}_{l,ik}} \right)}} \right)\oplus (1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {\bar{{h}}_{l,ik}} \right)}} \right)^{\circ} } \right)} \oplus\\ && \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {\bar{{h}}_{l,kj}} \right)}} \right)\oplus (1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {\bar{{h}}_{l,kj}} \right)}} \right)^{\circ} } \right)} \end{array} $$

(14)

for all i, $j=$1, 2, …, n, which can be equivalently expressed as follows:

$$ \left\{ {\begin{array}{l} (n - 2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right) + (1 - \delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)} \right)+\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+} \right.} \\ \left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)} \right)=(n-2)\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)} \right)+ \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right)} \\ \!\!(n - 2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \!\log \left( {h_{l,ij}^{+}} \right)}} \right) + (1 - \delta_{ij} )\left( {\!\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( \!{h_{l,ij}^{-}} \right)}} \right)} \right) + \!\sum\nolimits_{k = 1,k\ne i,j}^{n} {\!\left( {\delta_{jk} \left( {\!\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right) + (1 - \delta_{jk} )\left( {\!\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \!\log \left( {h_{l,jk}^{-}} \right)}} \right) + } \right.} \\ \left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)} \right)=(n-2)\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)} \right)+ \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right) + \delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right)} \end{array}} \right. $$

(15)

for each pair of (i, $j)$.

When the multiplicative consistency of the incomplete IVHFPR $\tilde {{H}}$ cannot be guaranteed, we add the positive and negative derivation values $\varepsilon _{ij}^{-} ,\varepsilon _{ij}^{+} ,\tau _{ij}^{-} ,\tau _{ij}^{+}$ into formula (15), where $\varepsilon _{ij}^{-},\varepsilon _{ij}^{+} ,\tau _{ij}^{-} ,\tau _{ij}^{+} \ge 0$ for all i, j =1, 2, …, n. Then,

$$ \left\{ {\begin{array}{l} (n-2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)} \right)} \right)+ \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)} \right)-} \right.} \\ \left( {\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right)}} \right)+ \\ \varepsilon_{ij}^{-} -\varepsilon_{ij}^{+} = 0 \\ (n-2)\left( {\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)} \right)} \right)+ \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} \!{\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\!\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right) + (1 - \delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right) + \left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right) + (1 - \delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \!\log \left( {h_{l,ki}^{-}} \right)}} \right)} \right) - } \right.} \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \!\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right)+} \\ \tau_{ij}^{-} -\tau_{ij}^{+} = 0 \end{array}} \right. $$

(16)

for each pair of (i, $j)$.

Because the higher the consistency is, the better will be. We build the following programming model to determine the missing values:

$$\begin{array}{@{}rcl@{}} &&\varphi_{^{l_{0} ,i_{0} j_{0}} }^{\ast} =\min \sum\nolimits_{i,j = 1}^{n} {\left( {\varepsilon_{ij}^{-} +\varepsilon_{ij}^{+} +\tau_{ij}^{+} +\tau_{ij}^{+}} \right)}\\ &&\!\!\!\!\!s.t.\left\{\begin{array}{ll} (n-2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)} \right)} \right)+ \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right) + (1 - \delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right) + (1 - \delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)} \right)-} \right.} \\ \left( {\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right) + (1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right)}} \right)+ \\ \varepsilon_{ij}^{-} -\varepsilon_{ij}^{+} = 0,i,j = 1,2,...,n \\ (n-2)\left( {\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} ) \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)} \right)} \right)+ \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right) + (1 - \delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)} \right)-} \right.} \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} \!\!{\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right) + (1 - \delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right) + \delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right) + (1 - \delta_{kj} )\!\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log\! \left( {h_{l,kj}^{-}} \right)}} \right)} \right) + \!\!}\\ \tau_{ij}^{-} -\tau_{ij}^{+} = 0,i,j = 1,2,...,n \\ h_{l,ij}^{-} +h_{l,ji}^{+} =h_{l,ij}^{+} +h_{l,ji}^{-}= 1,[h_{l,ij}^{-},h_{l,ij}^{+}]\subseteq \left[0,1\right],\ \tilde{{h}}_{ij} \in U=\{\tilde{{h}}_{ij} \vert \tilde{{h}}_{ij}, \text{is}\ \text{missing} \ i,j = 1,2,...,n\} \\ \alpha_{l,ij} = 0\vee 1,\alpha_{l,ij} =\alpha_{m_{ij} + 1-l,ji},i,j = 1,2,...,n,l = 1,2,...,m_{ij} \\ \sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} = 1} ,i,j = 1,2,...,n \\ \delta_{ij} = 0\vee 1,i,j = 1,2,...,n \\ \delta_{ij} +\delta_{ji} = 1,i,j = 1,2,...,n \\ \alpha_{l_{0} ,i_{0} j_{0}} = 1,i_{0} ,j_{0} = 1,2,...,n,\ l_{0} = 1,2,...,m_{i_{0} j_{0}} \ \text{with} \ \tilde{{h}}_{i_{0} j_{0}} \notin U \end{array}\right.. \end{array} $$

(17)

The first two constraints are constants if $\tilde {{h}}_{ij} \notin U$ for all $i =$ 1, 2, …, n or all $j =$ 1, 2, …, n, which have no influence on missing values. Thus, we can disregard such constraints. To do this, we further introduce the 0-1 indictor variables $\beta _{ij} =\left \{ {\begin {array}{l} 0{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\tilde {{h}}_{ij} \notin U,i = 1,2,..,n\vee j = 1,2,..,n \\ 1{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\text {otherwise} \end {array}} \right .$ for each pair of (i, $j)$, and construct the following programming model:

$$\begin{array}{@{}rcl@{}} &&\psi_{^{l_{0} ,i_{0} j_{0}} }^{\ast} =\min \sum\nolimits_{i,j = 1}^{n} {\left( {\varepsilon_{ij}^{-} +\varepsilon_{ij}^{+} +\tau_{ij}^{+} +\tau_{ij}^{+}} \right)}\\ &&\!\!\!s.t.\!\!\left\{ {\begin{array}{l} x\beta_{ij} \left( {(n-2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)} \right)} \right)+} \right. \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)} \right)-} \right.} \\ \left( {\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right)}} \right)+ \\ \left. {\varepsilon_{ij}^{-} -\varepsilon_{ij}^{+}} \right)= 0,i,j = 1,2,...,n \\ \beta_{ij} \left( {(n-2)\left( {\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)} \right)} \right)+} \right. \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)} \right)-} \right.} \\ \!\!\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\!\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \!\log \left( {h_{l,ik}^{+}} \right)}} \right) + (1 - \delta_{ik} )\left( {\!\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \!\log \left( {h_{l,ik}^{-}} \right)}} \right) + \delta_{kj} \left( {\!\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \!\log \left( {h_{l,kj}^{+}} \right)}} \right) + (1 - \delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} \!{\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right) + \!\!} \\ \left. {\tau_{ij}^{-} -\tau_{ij}^{+}} \right)= 0,i,j = 1,2,...,n \\ h_{l,ij}^{-} +h_{l,ji}^{+} =h_{l,ij}^{+} +h_{l,ji}^{-} = 1,[h_{l,ij}^{-} ,h_{l,ij}^{+} ]\subseteq \left[0,1\right],{\kern 1pt}{\kern 1pt}\tilde{{h}}_{ij} \in U=\{\tilde{{h}}_{ij} \vert \tilde{{h}}_{ij} {\kern 1pt}{\kern 1pt}\text{is}{\kern 1pt}{\kern 1pt}\text{missing}{\kern 1pt},{\kern 1pt}{\kern 1pt}i,j = 1,2,...,n\} \\ \alpha_{l,ij} = 0\vee 1,\alpha_{l,ij} =\alpha_{m_{ij} + 1-l,ji} ,i,j = 1,2,...,n,l = 1,2,...,m_{ij} \\ \sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} = 1} ,i,j = 1,2,...,n \\ \delta_{ij} = 0\vee 1,i,j = 1,2,...,n \\ \delta_{ij} +\delta_{ji} = 1,i,j = 1,2,...,n \\ \alpha_{l_{0} ,i_{0} j_{0}} = 1,i_{0} ,j_{0} = 1,2,...,n,{\kern 1pt}{\kern 1pt}{\kern 1pt}l_{0} = 1,2,...,m_{i_{0} j_{0}} {\kern 1pt}\text{with}{\kern 1pt}\tilde{{h}}_{i_{0} j_{0}} \notin U \end{array}} \right.. \end{array} $$

(18)

According to the construction of the elements in IVHFPRs, we can only determine missing IVHFEs in the upper triangular part. Thus, model (18) can be further transformed into the following model:

$$\begin{array}{@{}rcl@{}} &&\phi_{^{l_{0} ,i_{0} j_{0}} }^{\ast} =\min \sum\nolimits_{i,j = 1}^{n} {\left( {\varepsilon_{ij}^{-} +\varepsilon_{ij}^{+} +\tau_{ij}^{+} +\tau_{ij}^{+}} \right)}\\ &&s.t.\left\{ {\begin{array}{l} \beta_{ij} \left( {(n-2)\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)} \right)} \right)+} \right. \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)} \right)-} \right.} \\ \left( {\sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right)+(1-\delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{+}} \right)}} \right)+\delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)+(1-\delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right)} \right)}} \right)+ \\ \left. {\varepsilon_{ij}^{-} -\varepsilon_{ij}^{+}} \right)= 0,i,j = 1,2,...,n,i<j \\ \beta_{ij} \left( {(n-2)\left( {\left( {\delta_{ij} \left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{+}} \right)}} \right)+(1-\delta_{ij} )\left( {\sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} \log \left( {h_{l,ij}^{-}} \right)}} \right)} \right)-\left( {\delta_{ji} \left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{+}} \right)}} \right)+(1-\delta_{ji} )\left( {\sum\nolimits_{l = 1}^{m_{ji}} {\alpha_{l,ji} \log \left( {h_{l,ji}^{-}} \right)}} \right)} \right)} \right)+} \right. \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{jk} \left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{+}} \right)}} \right)+(1-\delta_{jk} )\left( {\sum\nolimits_{l = 1}^{m_{jk}} {\alpha_{l,jk} \log \left( {h_{l,jk}^{-}} \right)}} \right)+\left. {\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{+}} \right)}} \right)+(1-\delta_{ki} )\delta_{ki} \left( {\sum\nolimits_{l = 1}^{m_{ki}} {\alpha_{l,ki} \log \left( {h_{l,ki}^{-}} \right)}} \right)} \right)-} \right.} \\ \sum\nolimits_{k = 1,k\ne i,j}^{n} {\left( {\delta_{ik} \left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \!\log \left( {h_{l,ik}^{+}} \right)}} \right) + (1 - \delta_{ik} )\left( {\sum\nolimits_{l = 1}^{m_{ik}} {\alpha_{l,ik} \log \left( {h_{l,ik}^{-}} \right)}} \right) + \delta_{kj} \left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{+}} \right)}} \right) + (1 - \delta_{kj} )\left( {\sum\nolimits_{l = 1}^{m_{kj}} {\alpha_{l,kj} \log \left( {h_{l,kj}^{-}} \right)}} \right)} \right) + } \\ \left. {\tau_{ij}^{-} -\tau_{ij}^{+}} \right)= 0,i,j = 1,2,...,n,i<j \\ h_{l,ij}^{-} +h_{l,ji}^{+} =h_{l,ij}^{+} +h_{l,ji}^{-} = 1,[h_{l,ij}^{-} ,h_{l,ij}^{+} ]\subseteq \left[0,1\right],{\kern 1pt}{\kern 1pt}\tilde{{h}}_{ij} \in U'=\{\tilde{{h}}_{ij} \vert \tilde{{h}}_{ij} {\kern 1pt}{\kern 1pt}\text{is}{\kern 1pt}{\kern 1pt}\text{missing}{\kern 1pt},{\kern 1pt}{\kern 1pt}i,j = 1,2,...,n,i<j\} \\ \alpha_{l,ij} = 0\vee 1,\alpha_{l,ij} =\alpha_{m_{ij} + 1-l,ji} ,i,j = 1,2,...,n,l = 1,2,...,m_{ij} ,i<j \\ \sum\nolimits_{l = 1}^{m_{ij}} {\alpha_{l,ij} = 1} ,i,j = 1,2,...,n,i<j \\ \delta_{ij} = 0\vee 1,i,j = 1,2,...,n,i<j \\ \delta_{ij} +\delta_{ji} = 1,i,j = 1,2,...,n,i<j \\ \alpha_{l_{0} ,i_{0} j_{0}} = 1,i_{0} ,j_{0} = 1,2,...,n,{\kern 1pt}{\kern 1pt}{\kern 1pt}l_{0} = 1,2,...,m_{i_{0} j_{0}} {\kern 1pt}\text{with}{\kern 1pt}\tilde{{h}}_{i_{0} j_{0}} \notin U' \end{array}} \right.. \end{array} $$

(19)

Solving model (19) with respect to each interval in every IVHFE $\tilde {{h}}_{i_{0} j_{0}} \notin U^{\prime }$, we derive the missing IVHFEs in the upper triangular part, and the missing IVHFEs in the lower triangular part can be obtained following the additive reciprocity. On the basis of the above analysis, we introduce the following decision-making method with IVHFPRs:

Remark 1

Following previous theoretical research results, we offer algorithm I for ranking objects from IVHFPRs. Its main principle includes: (i) ascertaining missing judgments, (ii) judging the multiplicative consistency of IVHFPRs using QIFPRs, (iii) deriving multiplicatively consistent QIFPRs, (iv) calculating the collectively multiplicatively consistent QIFPR based on the consistency probability distribution and the IWGMA operator, (v) obtaining the multiplicatively consistent IFPR, (vi) calculating the interval fuzzy priority weight vector using formula (20), and (vii) ranking objects. There are several features of Algorithm I, such as it is based on the multiplicative consistency analysis, it neither adds extra value nor disregards any information, and it can address incomplete and inconsistent IVHFPRs.

Example 4.1

Let $X=${x₁, $x_{2}$, $x_{3}$, $x_{4}$} be the set of the compared objects, and let the incomplete IVHFPR $\tilde {{H}}$ be defined as follows:

$$\begin{array}{@{}rcl@{}} \tilde{{H}}\!&=&\!\!\left( \! {{\begin{array}{*{20}c} {\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} & x & & \\ x & {\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} & & \\ {\left\{ {\left[\textstyle{1 \over 4},\textstyle{2 \over 5}\right]} \right\}} & x & & \\ {\left\{ {\left[\textstyle{1 \over 2},\textstyle{{11} \over {20}}\right],\left[\textstyle{3 \over 5},\textstyle{2 \over 3}\right],\left[\textstyle{5 \over 7},\textstyle{6 \over 7}\right]} \right\}} & x & & \end{array}} } \right.\\&& \left.\begin{array}{cc}{\left\{ {\left[\textstyle{3 \over 5},\textstyle{3 \over 4}\right]} \right\}} &{\left\{ {\left[\textstyle{1 \over 7},\textstyle{2 \over 7}\right],\left[\textstyle{1 \over 3},\textstyle{2 \over 5}\right],\left[\textstyle{9 \over {20}},\textstyle{1 \over 2}\right]} \right\}}\\x&x\\{\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}}&{\left\{ {\left[\textstyle{1 \over 4},\textstyle{1 \over 3}\right],\left[\textstyle{2 \over 5},\textstyle{1 \over 2}\right]} \right\}} \\ {\left\{ {\left[\textstyle{1 \over 2},\textstyle{3 \over 5}\right],\left[\textstyle{2 \over 3},\textstyle{3 \over 4}\right]} \right\}}&{\left\{ {\left[\textstyle{1 \over 2},\textstyle{1 \over 2}\right]} \right\}} \end{array}\right). \end{array} $$

To rank the objects $x_{1}$, $x_{2}$, $x_{3}$, and $x_{4}$, the following steps are needed:

• Step 1: With respect to the incomplete IVHFPR $\tilde {{H}}$, missing IVHFEs based on model (19) are determined as shown in Table 1.

  According to the additive reciprocity of intervals in IVHFEs, the missing IVHFEs $\tilde {{h}}_{21} ,\tilde {{h}}_{32}$, and $\tilde {{h}}_{42}$ are obtained as shown in Table 2.

• Step 2: With respect to the complete IVHFPR $\tilde {{H}}$, we apply model (10) to judge its multiplicative consistency, where the objective function values, 0-1 indicator variables, QIFPRs, and their forming times are derived as shown in Table 3.

• Step 3: Table 3 shows that all of QIFPRs are inconsistent. Thus, formula (11) is used to obtain the multiplicatively consistent QIFPRs, which are shown in Table 4.

  Step 4: According to the forming times of QIFPRs, the consistency probability distribution is derived as follows:

  $$P=\left\{ {\frac{1}{20},\frac{1}{20},\frac{3}{20},\frac{5}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20},\frac{1}{20}} \right\}, $$

  by which the collectively multiplicatively consistent QIFPR is

  $$\bar{{C}}=\left( {\begin{array}{llll} \left[0.5000,0.5000\right]&\left[0.5152,0.5122\right]&\left[0.6917,0.6533\right]& \left[0.4565,0.4594\right] \\ \left[0.4848,0.4878\right]&\left[0.5000,0.5000\right]&\left[0.6786,0.6421\right]& \left[0.4415,0.4472\right] \\ \left[0.3083,0.3467\right]&\left[0.3214,0.3579\right]&\left[0.5000,0.5000\right]& \left[0.2724,0.3108\right] \\ \left[0.5435,0.5406\right]&\left[0.5585,0.5528\right]&\left[0.7276,0.6892\right]& \left[0.5000,0.5000\right] \end{array}} \right), $$

  and the collectively multiplicatively consistent IFPR is

  $$\bar{{D}}=\left( {\begin{array}{llll} \left[0.5000,0.5000\right]&\left[0.5122,0.5152\right]&\left[0.6533,0.6917\right]& \left[0.4565,0.4594\right] \\ \left[0.4848,0.4878\right]&\left[0.5000,0.5000\right]&\left[0.6421,0.6786\right]& \left[0.4415,0.4472\right] \\ \left[0.3083,0.3467\right]&\left[0.3214,0.3579\right]&\left[0.5000,0.5000\right]& \left[0.2724,0.3108\right] \\ \left[0.5406,0.5435\right]&\left[0.5528,0.5585\right]&\left[0.6892,0.7276\right]& \left[0.5000,0.5000\right] \end{array}} \right). $$

• Step 5: Using formula (20), the interval fuzzy priority weight vector is

  $$\begin{array}{@{}rcl@{}} \bar{{\omega} }&=&\left( \left[0.2633,0.2737\right],\left[0.2565,0.2668\right],\right.\\ &&\left.\left[0.1727,0.1885\right],\left[0.2839,0.2949\right]\right). \end{array} $$

• Step 6: Adopting the formula for comparing intervals in [13], we obtain $\rho (\bar {{\omega } }_{4} >\bar {{\omega } }_{1} )= 1,\rho (\bar {{\omega } }_{1} >\bar {{\omega } }_{2} )= 0.9424,\rho (\bar {{\omega } }_{2} >\bar {{\omega } }_{3} )= 1$. Thus, the ranking order is $x_{4} \succ x_{1} \succ x_{2} \succ x_{3}$.

Table 1 Determined missing IVHFEs

Table 2 Determined missing IVHFEs

Table 3 Objective function values, 0-1 indicator variables, QIFPRs, and forming times

Table 4 Multiplicatively consistent QIFPRs

5 A group decision-making method with IVHFPRs

In Section 4, we offer a method for ranking objects from incomplete and inconsistent IVHFPRs. However, it is insufficient to address group decision making. Therefore, this section further researches group decision making with IVHFPRs.

In general, for a group decision-making problem, there are n objects $X =$ {x₁, $x_{2}$, …, $x_{n}$} , which are evaluated by m DMs $E =$ {e₁, $e_{2}$, …, $e_{m}$} . Let $\tilde {{H}}^{k}=({\tilde {{h}}_{ij}^{k}} )_{n\times n}$ be the IVHFPR offered by the DM $e_{k}$, $k =$ 1, 2, …, m.

To measure the agreement degree between individual IVHFPRs, the consensus analysis is necessary [11, 13, 29]. Next, we apply the collectively multiplicatively consistent QIFPRs to define a distance measure between IVHFPRs, by which a consensus index is derived.

Definition 11

Let $\tilde {{H}}^{k}=({\tilde {{h}}_{ij}^{k}} )_{n\times n}$ and $\tilde {{H}}^{g}=({\tilde {{h}}_{ij}^{g}})_{n\times n}$ be any two IVHFPRs, and let $\bar {{C}}^{k} = ({\bar {{c}}_{ij}^{k}} )_{n\times n}$ and $\bar {{C}}^{g} = ({\bar {{c}}_{ij}^{g}} )_{n\times n}$ be their collectively multiplicatively consistent QIFPRs obtained from Algorithm I. Then, the distance measure between the QIFPRs $\bar {{C}}^{k}$ and $\bar {{C}}^{g}$ is defined as follows:

$$\begin{array}{@{}rcl@{}} V\left( {\bar{{C}}^{k},\bar{{C}}^{g}} \right)&=&\frac{1}{n(n-1)}\sum\nolimits_{i,j = 1,i<j}^{n}\\&&\times {\left( {\left| {c_{ij}^{k-} -c_{ij}^{g-}} \right|+\left| {c_{ij}^{k+} -c_{ij}^{g+}} \right|} \right)} . \end{array} $$

(21)

Property 1

Let $\tilde {{H}}^{k}=({\tilde {{h}}_{ij}^{k}})_{n\times n}$ and $\tilde {{H}}^{g}=({\tilde {{h}}_{ij}^{g}} )_{n\times n}$ be any two IVHFPRs, and let $\bar {{C}}^{k}=({\bar {{c}}_{ij}^{k}})_{n\times n}$ and $\bar {{C}}^{g}=({\bar {{c}}_{ij}^{g}} )_{n\times n}$ be their collectively multiplicatively consistent QIFPRs. Then, their distance measure defined by formula (21) has the following characteristics:

• (i) $V\left ({\bar {{C}}^{k},\bar {{C}}^{g}} \right )=V\left ({\bar {{C}}^{g},\bar {{C}}^{k}} \right );$

• (ii) $0\le V\left ({\bar {{C}}^{k},\bar {{C}}^{g}} \right )\le 1;$

• (iii) $V\left ({\bar {{C}}^{k},\bar {{C}}^{g}} \right )= 0$ if and only if $\bar {{C}}^{k}=\bar {{C}}^{g};$

• (iv) Let $\tilde {{H}}^{t}=({\tilde {{h}}_{ij}^{t}} )_{n\times n}$ be any another IVHFPR, and let $\bar {{C}}^{t}=({\bar {{c}}_{ij}^{t}} )_{n\times n}$ be its collectively multiplicatively consistent QIFPR. Then, $V\left ({\bar {{C}}^{k},\bar {{C}}^{g}} \right )\le V\left ({\bar {{C}}^{g},\bar {{C}}^{t}} \right )+V\left ({\bar {{C}}^{t},\bar {{C}}^{k}} \right )$.

Proof

From formula (21), one can easily derive the conclusions. □

Definition 12

Let $\tilde {{H}}^{k}=({\tilde {{h}}_{ij}^{k}} )_{n\times n}$, $k =$ 1, 2, …, m, be any m IVHFPRs, and let $\bar {{C}}^{k}=({\bar {{c}}_{ij}^{k}} )_{n\times n}$ be their collectively multiplicatively consistent QIFPR. Furthermore, let $\bar {{C}}=\left ({\bar {{c}}_{ij}} \right )_{n\times n}$ be the comprehensively multiplicatively consistent QIFPR. Then, the consensus measure of $\bar {{C}}^{k}$ is defined as follows:

$$\begin{array}{@{}rcl@{}} COI\left( {\bar{{C}}^{k}} \right)&=&1-\frac{1}{n(n-1)}\sum\nolimits_{i,j = 1,i<j}^{n}\\&&\times {\left( {\left| {c_{ij}^{k-} -c_{ij}^{-}} \right|+\left| {c_{ij}^{k+} -c_{ij}^{+}} \right|} \right)} , \end{array} $$

(22)

where $k =$ 1, 2, …, m.

When we calculate the comprehensively multiplicatively consistent QIFPR, the weights of the DMs are used. To determine the weighting vector on the DM set, we establish the following maximum consensus-based programming model:

$$\begin{array}{@{}rcl@{}} &&{\Lambda}^{\ast} =\min \sum\nolimits_{k = 1}^{m} {\varepsilon_{k}^{-} +\varepsilon_{k}^{+}}\\ &&s.t.\left\{\begin{array}{ll} \sum\nolimits_{i,j = 1,i<j}^{n} \left( \left( c_{ij}^{k-} -\frac{\sqrt[n]{{\Pi}_{k = 1}^{n} \left( \left( {\Pi}_{g = 1}^{m} \left( {c_{ik}^{g-}} \right)^{w_{e_{g}}} \right)\left( {\Pi}_{g = 1}^{m} \left( {c_{kj}^{g-}} \right)^{w_{e_{g}}} \right) \right)}}{\sqrt[n]{{\Pi}_{k = 1}^{n} \left( {\left( {{\Pi}_{g = 1}^{m} \left( {c_{ik}^{g-}} \right)^{w_{e_{g}} } } \right)\left( {{\Pi}_{g = 1}^{m} \left( {c_{kj}^{g-}} \right)^{w_{e_{g}} } } \right)} \right)}+\sqrt[n]{{\Pi}_{k = 1}^{n} \left( {\left( {{\Pi}_{g = 1}^{m} \left( {c_{jk}^{g-}} \right)^{w_{e_{g}} } } \right)\left( {{\Pi}_{g = 1}^{m} \left( {c_{ki}^{g-}} \right)^{w_{e_{g}} } } \right)} \right)}}\right)^{2}+\right.\\ \left. {\left( {c_{ij}^{k+} -\frac{\sqrt[n]{{\Pi}_{k = 1}^{n} \left( {\left( {{\Pi}_{g = 1}^{m} \left( {c_{ik}^{g+}} \right)^{w_{e_{g}} } } \right)\left( {{\Pi}_{g = 1}^{m} \left( {c_{kj}^{g+}} \right)^{w_{e_{g}} } } \right)} \right)}}{\sqrt[n]{{\Pi}_{k = 1}^{n} \left( {\left( {{\Pi}_{g = 1}^{m} \left( {c_{ik}^{g+}} \right)^{w_{e_{g}} } } \right)\left( {{\Pi}_{g = 1}^{m} \left( {c_{kj}^{g+}} \right)^{w_{e_{g}} } } \right)} \right)}+\sqrt[n]{{\Pi}_{k = 1}^{n} \left( {\left( {{\Pi}_{g = 1}^{m} \left( {c_{jk}^{g+}} \right)^{w_{e_{g}} } } \right)\left( {{\Pi}_{g = 1}^{m} \left( {c_{ki}^{g+}} \right)^{w_{e_{g}} } } \right)} \right)}}} \right)^{2}} \right)-\varepsilon_{k}^{-} +\varepsilon_{k}^{+} = 0,k = 1,2,...,m \\ \sum\nolimits_{k = 1}^{m} {w_{e_{k}} } = 1, \\ w_{e_{k}} \in \left[0,1\right],k = 1,2,...,m \end{array}\right.. \end{array} $$

(23)

One can check that ${\Lambda }^{\ast } = 0$ if and only if $\bar {{C}}^{k}=\bar {{C}}^{g}$ for all $k, g =$ 1, 2, …, m with $k\ne g$.

Following multiplicative consistency and consensus analysis, we further present the following algorithm for group decision making with IVHFPRs that can address incomplete and inconsistent cases.

To see the procedure of Algorithm II intuitively, please see Fig. 1.

Fig. 1

The procedure of Algorithm II

Remark 2

Following algorithm II and Fig. 1, one can find that algorithms I and II have the similar procedure for ranking objects, and there are two more steps of algorithm II than Algorithm I: determining the weights of the DMs, and measuring and improving the consensus level.

Example 5.1

The technology of air conditioning has made extraordinary progress since Dr. Carrier invented the world’s first air-conditioning in 1902. According to the development of air conditioning, it can be summarized as four development phases: centrifugal air conditioning, solar air conditioning, inverter air conditioning, and gas air conditioning. At present, air conditioning has become one of the most important office equipment and appliances. The global air conditioning market is huge. According to the report of NIKKEI in 2015, the shipments of China’s domestic air conditioning to the world are 7398.8 million units. To gain more market share, the competition among air conditioning manufacturers is fierce.

In the Chinese market, there are about 38 brands of air conditioning. According to the data of internet consumer research center in 2014, there are four main brands of air conditioning: Gree, Haier, Hisense, and Midea. There is an air conditioning industry assessment panel that is composed by three experts. They were invited to compare these four brands of air conditioning in the next five years of developments. Because there are many factors, such as enterprise management, enterprise R D capability, corporate reputation, and brand utility, it is not an easy thing to give their comparisons using one exact or fuzzy variable. In this case, the experts can apply IVHFEs to express their uncertain hesitancy preferences. Furthermore, they are allowed to only give partial judgments, namely, the missing judgements are permitted. According to their expertise and known information, the individual IVHFPRs are listed in Tables 5, 6 and 7.

Table 5 IVHFPR $\tilde {{H}}^{1}$ offered by the DM $e_{1}$

Table 6 IVHFPR $\tilde {{H}}^{2}$offered by the DM $e_{2}$

Table 7 IVHFPR $\tilde {{H}}^{3}$offered by the DM $e_{3}$

To rank these four air conditioning brands, the following procedure is needed:

• Step 1: Because all of these three individual IVHFPRs are incomplete, model (19) is applied to determine the missing IVHFEs, which are obtained as shown in Table 8.

  With respect to each complete IVHFPR, the multiplicatively consistent QIFPRs, and their consistency probability distributions are derived as shown in Tables 9, 10 and 11.

  Using the IWGMA operator and formula (11), the multiplicatively consistent QIFPRs are:

  $$\bar{{C}}^{1}=\left( {\begin{array}{llll} \left[0.5000,0.5000\right]&\left[0.7271,0.7973\right]&\left[0.7844\text{,}0.7003\right]& \left[0.8484\text{,}0.9060\right]\\ \left[0.2729,0.2027\right]&\left[0.5000,0.5000\right]&\left[0.5772,0.3726\right]&\left[0.6775\text{,}0.7102\right]\\ \left[0.2156,0.2997\right]&\left[0.4228,0.6274\right]&\left[0.5000,0.5000\right]&\left[0.6060\text{,}0.8049\right]\\ \left[0.1516,0.0940\right]&\left[0.3225,0.2898\right]&\left[0.3940,0.1951\right]& \left[0.5000\text{,}0.5000\right] \end{array}} \right), $$
  $$\bar{{C}}^{2}=\left( {\begin{array}{llll} \left[0.5000\text{,}0.5000\right]&\left[0.8287\text{,}0.7206\right]&\left[0.8433\text{,}0.7336\right]& \left[0.8892\text{,}0.7101\right] \\ \left[0.1713\text{,}0.2794\right]&\left[0.5000\text{,}0.5000\right]&\left[0.5267\text{,}0.5163\right]& \left[0.6239\text{,}0.4871\right] \\ \left[0.1567\text{,}0.2664\right]&\left[0.4733\text{,}0.4837\right]&\left[0.5000\text{,}0.5000\right]& \left[0.5985\text{,}0.4708\right] \\ \left[0.1108\text{,}0.2899\right]&\left[0.3761\text{,}0.5129\right]&\left[0.4015\text{,}0.5292\right]& \left[0.5000\text{,}0.5000\right] \end{array}} \right), $$
  $$\bar{{C}}^{3}=\left( {\begin{array}{llll} \left[0.5000\text{,}0.5000\right]&\left[0.7984\text{,}0.7990\right]&\left[0.8099\text{,}0.7746\right]& \left[0.8041\text{,}0.8488\right] \\ \left[0.2016\text{,}0.2010\right]&\left[0.5000\text{,}0.5000\right]&\left[0.5182\text{,}0.4637\right]& \left[0.5089\text{,}0.5854\right] \\ \left[0.1901\text{,}0.2254\right]&\left[0.4818\text{,}0.5363\right]&\left[0.5000\text{,}0.5000\right]& \left[0.4907\text{,}0.6202\right] \\ \left[0.1959\text{,}0.1512\right]&\left[0.4911\text{,}0.4146\right]&\left[0.5093\text{,}0.3798\right]& \left[0.5000\text{,}0.5000\right] \end{array}} \right). $$

• Step 2: Following model (23), the weights of the DMs are $w_{e_{1}}= 0.3063,w_{e_{2}} = 0.3632,w_{e_{3}} = 0.3305$. Furthermore, the comprehensively multiplicatively consistent QIFPR based on the IWGMA operator and formula (11) is

  $$\bar{{C}}=\left( \begin{array}{llll} \left[0.5000,0.5000\right] &\left[0.7905,0.7720\right]&\left[0.8155,0.7380\right]&\left[0.8520,0.8306\right] \\ \left[0.2095,0.2280\right]&\left[0.5000,0.5000\right]&\left[0.5395,0.4542\right]&\left[0.6042,0.5914\right] \\ \left[0.1845,0.2620\right]&\left[0.4605,0.5458\right]&\left[0.5000,0.5000\right]&\left[0.5658,0.6350\right] \\ \left[0.1480,0.1694\right]&\left[0.3958,0.4086\right]&\left[0.4342,0.3650\right]&\left[0.5000,0.5000\right] \end{array}\right). $$

• Step 3: Letπ = 0.9. From formula (22), we have $\left \{ {\begin {array}{l} COI({\tilde {{H}}^{1}} )= 0.9468 \\ COI({\tilde {{H}}^{1}} )= 0.9500 \\ COI({\tilde {{H}}^{1}} )= 0.9722 \end {array}} \right .$.

• Step 4: With respect to the multiplicatively consistent QIFPR $\bar {{C}}$, the comprehensively multiplicatively consistent IFPR is

  $$\bar{{D}}=\left( {\begin{array}{llll} \left[0.5000,0.5000\right]&\left[0.7720,0.7905\right]&\left[0.7380,0.8155\right]&\left[0.8306,0.8520\right] \\ \left[0.2095,0.2280\right]&\left[0.5000,0.5000\right]&\left[0.4542,0.5395\right]&\left[0.5914,0.6042\right] \\ \left[0.1845,0.2620\right]&\left[0.4605,0.5458\right]&\left[0.5000,0.5000\right]&\left[0.5658,0.6350\right] \\ \left[0.1480,0.1694\right]&\left[0.3958,0.4086\right]&\left[0.3650,0.4342\right]&\left[0.5000,0.5000\right] \end{array}} \right). $$

Table 8 Determined missing IVHFEs

Table 9 Multiplicatively consistent QIFPRs obtained from the complete IVHFPR $\tilde {{H}}^{1}$ and the consistency probability distribution

Table 10 Multiplicatively consistent QIFPRs obtained from the complete IVHFPR $\tilde {{H}}^{2}$ and the consistency probability distribution

Table 11 Multiplicatively consistent QIFPRs obtained from the complete IVHFPR $\tilde {{H}}^{3}$ and the consistency probability distribution

Furthermore, the interval fuzzy priority weight vector is

$$\begin{array}{@{}rcl@{}} \bar{{\omega} }&=&\left( \left[0.3580,0.3916\right],\left[0.2106\text{,}0.2371\right],\left[0.2065\text{,}0.2442\right]\right.\\ &&\left.,\left[0.1652\text{,}0.1892\right] \right). \end{array} $$

Using the formula for comparing intervals in [13], the ranking is $\text {Gree}\succ \text {Hisense}\succ \text {Haier}\succ \text {Midea}$.

Note that the previous group decision-making methods with IVHFPRs [3, 18] only considered the complete case. Thus, none of them can be applied in this example directly.

Following the complete IVHFPRs obtained from our method, when Pérez-Fernándeza et al.’s method [18] is used in Example 5.1, the ranking is $\text {Gree}\succ \text {Hisense}\succ \text {Haier}\succ \text {Midea}$. It is the same as the above ranking. Notably, the aggregation operator uses the arithmetic mean and $\alpha$ equals 0.25, which are adopted by Pérez-Fernandez et al. [18].

Furthermore, when Chen et al.’s method [3] is adopted in Example 5.1, the weights of the DMs are $w_{e_{1}} = 0.\text {2787},w_{e_{2}} = 0.\text {3895}$, and $w_{e_{3}} = 0.\text {3318}$. Moreover, the score values of objects are

$$\begin{array}{@{}rcl@{}} s(\tilde{{h}}_{1} )&=&\left[0.\text{73,}0.89\right],s(\tilde{{h}}_{2} )=\left[0.\text{47,}0.59\right],s(\tilde{{h}}_{3} )\\ &&=\left[0.\text{42,}0.59\right],s(\tilde{{h}}_{4} )=\left[0.\text{32,}0.46\right], \end{array} $$

by which the ranking is $\text {Gree}\succ \text {Haier}\succ \text {Hisense}\succ \text {Midea}$, which is different from the above ranking. Notably, the used aggregation operators are the same as those adopted by Chen et al. [3].

Remark 3

There are several limitations of methods in [3, 18]: (i) they cannot address decision making with incomplete IVHFPRs, (ii) they did not study the consistency of IVHFPRs, which may lead to illogical ranking, (iii) they did not consider the consensus of individual IVHFPRs. Therefore, the final ranking cannot reflect the agreement degree of individual opinions, (iv) method in [18] is based on the assumption that all of individual IVHFPRs have the same important, while method in [3] needs to add extra values into shorter IVHFEs to determine the weights of the DMs.

Notably, new method is based on multiplicative consistency and consensus analysis that can guarantee the logical ranking and reflect the agreement level of individual opinions. Furthermore, new method determines the weighting information without any extra subjective information. To show the advantages of the new method clearly, it can be summarized as follows:

• (i) It is based on the consistency analysis that neither causes information loss nor disregards any information;

• (ii) It offers a method to obtain consistent IVHFPRs from inconsistent ones, which only applies the judgements offered by the DMs;

• (iii) It can address incomplete IVHFPRs by only using the provided preferences;

• (iv) It provides a consensus-based method to determine the weights of the DMs;

• (v) It defines a distance measure-based consensus formula to complete the consensus levels among DMs and proposes an interactive method to improve the consensus degree.

6 Conclusions

Interval-valued hesitant fuzzy preference relations are powerful to denote the decision makers’ uncertain and hesitant information, which is an extension of hesitant fuzzy preference relations. After reviewing previous researches, we find that there is no research about interval-valued hesitant fuzzy preference relations based on the consistency and consensus. To ensure their reasonable application, this paper continues to study decision making with interval-valued hesitant fuzzy preference relations. To do this, a multiplicative consistency concept based on interval fuzzy preference relations is presented. Unlike the previous consistency concepts for hesitant fuzzy preference relations, the new concept does not need to add values to interval-valued hesitant fuzzy elements or ignore any information. Then, several multiplicative consistency- based programming models are constructed to address inconsistent and incomplete interval-valued hesitant fuzzy preference relations. After that, group decision making is further researched and a group decision-making algorithm is developed. Finally, a practical decision-making problem is offered to show the application of the new method.

This paper focuses on the multiplicative consistency of interval-valued hesitant fuzzy preference relations, and we can similarly study the additive consistency. Furthermore, we can extend the new theoretical results to other types of preference relations, such as interval-valued multiplicative hesitant fuzzy preference relations and interval-valued linguistic hesitant fuzzy preference relations. Besides the theoretical aspect, we shall continue to research the application of the new method in some other fields, such as medical recommendation, evaluating double first-class universities in China, assessing online shopping platforms, and evaluating Chinese airlines.