Interval probability hesitant fuzzy linguistic analytic hierarchy process and its application in talent selection

Abstract

Hesitant fuzzy linguistic term set (HFLTS) can handle the qualitative and hesitant information in multiple attribute decision making (MADM) problems which are widely used in various fields. However, the experts’ evaluation of information is not completely reliable in the situation where their own knowledge background is insufficient. In order to deal with deviations due to incomplete reliability of the evaluation, this paper first proposes the interval probability hesitant fuzzy linguistic variable (IPHFLV), which takes the HFLTS as the evaluation part and adds a novel element-reliability of evaluation, thus can describe the different credibility of information evaluation due to the familiarity of experts with schemes and the differences in knowledge cognition. The operation rules and comparison methods are also illustrated. Particularly, under the inspiration of probability theory, we propose the possibility degree of the IPHFLVs. Then we propose IPHFL-AHP based on the AHP and interval probability hesitant fuzzy linguistic variable. Especially, the general geometric consistency index (G-GCI) based on the unbiased estimator of the variance is presented to measure the consistency and the iterative algorithm is constructed to improve the consistency. We use the possibility degree to calculate the priority vector to acquire the total ranking and introduce the process of IPHFL-AHP. Finally, case study of talent selection is given to illustrate the effectiveness and feasibility of the proposed method.

Keywords

Interval probability hesitant fuzzy linguistic variable interval probability hesitant fuzzy linguistic analytic hierarchy processe general geometric consistency index,possibility degreee reliability

1 Introduction

Multiple attribute decision making (MADM) [7], with qualitative and hesitant information, can be commonly seen in various fields such as industry, agriculture, politics, economy and management. Effective evaluation of these uncertain information is necessary and attract many scholars attention. Zadeh [26] first proposed the fuzzy set (FS) which used the membership degree to express the information. Considering that sometimes the membership degree might be several possible values rather than a certain value, Torra [13] investigated the hesitant fuzzy set (HFS). HFS was more relevant to the actual situation and can reduce the loss of information effectively. In practical decision making process, linguistic variable is more flexible for experts to express their opinions. Zadeh [27] presented the concept of the linguistic term set (LTS). Since linguistic expressions are much closer than single or simple linguistic term to the way of human mind and cognition, Rodriguez [9] proposed the hesitant fuzzy linguistic term set (HFLTS), a powerful tool in representing and eliciting the comparative linguistic expressions. Liao [33] redefined the HFLTS and proposed the operator of the HFLTS. Also, a lot of other research on HFLTS have done [48]. HFLTS can not only deal with the difficulties to determine the membership degree of an element due to the hesitation among different possible values, but also can flexible express experts’ opinions with linguistic expressions. According to the above literature, it can be seen that HFLTS is useful and attracts many scholars’ attention. Due to the limited knowledge of experts, there is a phenomenon that experts’ evaluation of information is not completely reliable in the decision-making process. However, HFLTS believes that the probability of occurrence of each possible linguistic term is the same when evaluating information, while in the face of practical problems, the degree of different linguistic terms may be different. Thus Xu [20] investigated the probability linguistic term set(PLTS). Subsequently, under the environment of probabilistic linguistic term set, scholars studied the measure degrees, the aggregation operators, the preference relationships and the decision models [24 , 47]. Considering that sometimes when evaluating information, the evaluation given by the decision maker may not be completely reliable. Zadeh [28] proposed Z-number which is an ordered pair of fuzzy numbers (A, B), with the first component A, a restriction on the values, and the second component B, a measure of reliability of A. Despite the advantages of HFLTS, PLTS and Z-number, the information expression that considers the reliability of evaluation information given by decision makers is not perfect under the situation that conforms to the expression habits of decision makers.

Although the appropriate expression of uncertain information is important in MADM problems, effective decision making methods to help decision makers select the most suitable scheme(s) can not be ignored for MADM. There are many effective methods in the decision making process such as principal components analysis (PCA) [25], TOPSIS [8], the VIKOR method [8], the ELECTRE method [10] and the analytic hierarchy process (AHP) [11]. People often face with a complex system consisting of interrelated, mutual restraint of many factors on the social [3], economic [30] as well as in the field of management systems analysis [14]. The AHP provides a concise and practical decision making method for studying such complex systems. AHP first proposed by Saaty [11], makes it extremely convenient to deal with the decision making problem which is difficult to quantify and used in almost any scientific field. To help experts make effective decisions with less loss of information, Many extended forms of AHP have been studied by scholars under different forms of information expression, such as fuzzy numbers [4],triangular fuzzy numbers[15], linguistic terms [29], interval numbers [18], intuitionistic fuzzy numbers [21], interval-valued intuitionistic fuzzy set [16], pythagorean fuzzy set [41], hesitant fuzzy numbers [17, 19], hesitant fuzzy linguistic term set [6], simulated hesitant fuzzy linguistic term set [12], probabilistic linguistic term set [24], Z-number [2], intuitionistic 2-tuple linguistic set [5]. In the above AHP studies, most of them convert different expressions into real numbers and then deal with the problem according to the traditional AHP model. Part of them is to propose the consistency test method applicable to its own environment and the priority in its unique environment. However, few extended AHP models that can be applied to other complex linguistic environments have been studied.

Comparison matrix, as an important part of AHP model, is worth discussing. Scholars have discussed the comparison matrix and its consistency in different linguistic environments. Xu [18] discussed the interval reciprocal comparison matrix, Yang [42] and Zhang [26] investigated the linguistic comparison matrix. Zhang [31] studied the hesitant fuzzy linguistic comparison matrix. Due to the advantages of HFLTS, many scholars have explored the hesitant fuzzy linguistic preference relationship(HFLPR) from different aspects. Zhao [43] combined the shape similarity with the HFLPR. Liu [44] mainly discusses how to improve consistency for HFLPR. The automatic iterative algorithm to improve the consistency of HFLPR is investigated by Wu [45] and the multiplicative hesitant fuzzy linguistic preference relations is proposed by Tang [46]. However, the studies do not take the reliability of the information into consideration.

Focusing on those problems, this paper presents the IPHFLV and constructs a novel IPHFL-AHP model for decision making. The contributions of this paper are summarized as follows:

Considering reliability in the application of HFLTS, we propose the interval probability hesitant fuzzy linguistic variables (IPHFLVs), which use the probability interval to express the reliability restriction on HFLTS. For example, “Bill has 70% to 80% confidence that the quality of this umbrella is between general and good” can be expressed as: ({general, good} ; [0.7, 0.8]). Then the operation rules and comparison methods are introduced.

Inspired by probability theory, we present the possibility degree of IPHFLVs to obtain the priority vectors of schemes and attributes.

The general geometric consistency index (G-GCI) based on the unbiased estimator of the variance to deal with the consistency under the interval probability hesitant fuzzy linguistic environment is proposed and the iterative algorithm to improve the consistency of comparison matrix is constructed.

The rest of this paper is organized as follows: Section 2 reviews some basic definitions and the general process of AHP. Section 3 proposes the IPHFLV and its possible degree. Section 4 introduces the interval probability hesitant fuzzy linguistic averaging (IPHFLA) operator and the interval probability hesitant fuzzy linguistic geometric (IPHFLG) operator. The G-GCI, iterative algorithm and the process of IPHFL-AHP are presented in Section 5. A case study on talent selection is provided in Section 6 to illustrate the effectiveness and practicality of the IPHFL-AHP and some comparative analyses with previous researches are presented in Section 7. Finally, Section 8 ends with conclusions and future work.

2 Preliminaries

This section briefly introduces some basic concepts such as HFLTS, PLTS, the hesitant fuzzy weighted averaging (HFWA) operator and the hesitant fuzzy weighted geometric (HFWG) operator. Also, the process of AHP is mentioned.

2.1 Hesitant fuzzy linguistic term set

There are situations where people find it’s more straightforward and comfortable to use linguistic variables to express the information, therefore Zadeh [27] proposed the concept of linguistic term set (LTS).

Definition 1. [27] Let S = {s_α|α = - t, ⋯ , -1, 0, 1, ⋯ , t} be a set with finite and discrete linguistic terms, where s_i denotes the ith linguistic term of S. It can be seen that 2t is the cardinality of S. For example, a set S with five linguistic terms could be described as follows: $\begin{matrix} S = {s_{- 2} = very low, s_{- 1} = low, s_{0} = fair, \\ s_{1} = high, s_{2} = very high} \end{matrix}$ The linguistic term s_i has the following characteristics:

(1) The set is ordered: s_α ≻ s_β if α > β;

(2) The negation operator is defined: neg (s_α) = s_-α, especially neg (s₀) = s₀.

Subsequently, Xu [22] extended discrete linguistic term set to continuous linguistic term set: S ={ s_α|α ∈ [- t, t] }. However, there exits hesitation when people make decisions. To deal with this problem, Rodriguez [9] proposed the hesitant fuzzy linguistic term set(HFLTS).

Definition 2. [9] Let S be the LTS, S = {s_α|α = - t, ⋯, -1, 0, 1, ⋯ , t}, then the hesitant fuzzy linguistic term set (HFLTS) H_S is an ordered finite subset of consecutive linguistic terms of S. An example is used to explain it clearer.

Example 1. Let S be the LTS, S = {s_-3 = very low, s_-2 = low, s_-1 = lightly low, s₀ = fair, s₁ = slight high, s₂ = high, s₃ = very high}, then the two different HFLTSs can be obtained. $H_{S}^{1} = {s_{- 3}, s_{- 1}, s_{0}}$ , $H_{S}^{2} = {s_{0}, s_{1}, s_{2}, s_{3}}$ .

Definition 3. [9] Let S = {s_α|α = - t, ⋯ , -1, 0, 1, ⋯, t} be a LTS. The context-free grammar is defined as G_H =< V_N, V_T, I, P >, where V_N is the set of nonterminal symbol, V_T is the set of terminals’ symbols, I is the starting symbol, and P is the production rules that are defined in an extended Backus-Naur form.

Example 2. Let S = {s_-3 = nothing, s_-2 = very low, s_-1 = low, s₀ = medium, s₁ = high, s₂ = very high, s₃ = perfect} be a LTS, some linguistic expressions that are obtained by the context-free grammar might be:

$\begin{matrix} l l_{1} & = high; l l_{2} = lower than medium; l l_{3} \\ = between medium and perfect \end{matrix}$ Let E_{G
_H} be the function that transforms linguistic expression ll into HFLTS H_S: $E_{G_{H}} : ll \to H_{S}$

The specific transformation is presented as follows:

(1) E_{G
_H} (s_i) = {s_i|s_i ∈ S};

(2) E_{G
_H} (less than s_i) = {s_j|s_j ∈ S, s_j ≤ s_i};

(3) E_{G
_H} (greater than s_i) = {s_j|s_j ∈ S, s_j ≥ s_i};

(4) E_{G
_H} (between s_i and s_j) = {s_k|s_k ∈ S, s_i ≤ s_k ≤ s_j}.

Direct calculations between linguistic variables are difficult. In order to solve this problem, Wang [32] defined the equivalent transformation function.

Definition 4. [32] Let S = {s_i|i = - t, · , -1, 0, 1, · , t} be a LTS. H_S = {s_j|j ∈ [- t, t]} be a HFLTS and H = {γ|γ ∈ [0, 1]} be the hesitant fuzzy set(HFS). The equivalent transformation functions f which converts linguistic variable s_j into the membership degree γ and inverse function f^-1 which converts the membership degree γ into the linguistic variable s_j are defined as following form respectively:

$\begin{matrix} f : [- t, t] \to [0, 1] : f (H_{S}) \\ = {f (s_{j}) = \frac{j}{2 t} + \frac{1}{2} | j \in [- t, t]} = H \end{matrix}$ (1)

$\begin{matrix} f^{- 1} : [0, 1] \to [- t, t] : f^{- 1} (H) = {f^{- 1} (γ) \\ = s_{(2 γ - 1) t} | γ \in [0, 1]} = H_{S} \end{matrix}$ (2)

In order to facilitate the calculation of HFLTSs, we need to make the numbers of each HFLTSs are the same. So we use expectation to add linguistic terms into the HFLTS with a small number of linguistic terms.

Definition 5. Let H_S = {s_j|j ∈ [- t, t]} be a HFLTS, # H_S is the number of the linguistic terms in H_S, θ is a parameter and 0 ≤ θ ≤ 1. Then we add the linguistic term in H_S by (3). $\bar{s} = \sum_{i = 1}^{# H_{S}} θ_{i} s_{i} = s_{\sum_{i = 1}^{# H_{S}} i θ_{i}}$ (3) In this paper we set $θ_{i} = \frac{1}{# H_{S}}$ , # H_S is the number of linguistic terms in H_S.

To study the HFLTS effectively, Liao [33] redefined the HFLTS and proposed the operator of the HFLTS.

Definition 6. [33] Let S = {s_i| - t, ⋯ , -1, 0, 1, ⋯, t} be a LTS. H_S = {s_j|j ∈ [- t, t], $H_{S}^{1} =$ {s_j|j ∈ [- t, t]}, $H_{S}^{2} = {s_{j} | j \in [- t, t]}$ be three HFLTS and H = {γ|γ ∈ [0, 1]} be a HFS. And # H_S is the number of the linguistic terms in H_S, ${# H}_{S}^{1} {= # H}_{S}^{2}$ (we can use Equation (3) to extend the shorter one by adding the linguistic terms to ensure ${# H}_{S}^{1} {= # H}_{S}^{2}$ ). f and f^-1 be the equivalent transformation functions of hesitant fuzzy linguistic elements and hesitant fuzzy elements, and λ be a real number. Then

(1) $H_{S}^{1} \oplus H_{S}^{2} = f^{- 1} (\underset{γ_{1} \in f (H_{S}^{1}), γ_{2} \in f (H_{S}^{2})}{\cup} (γ_{1} + γ_{2} - γ_{1} γ_{2}))$ ;

(2) $H_{S}^{1} \otimes H_{S}^{2} = f^{- 1} (\underset{γ_{1} \in f (H_{S}^{1}), γ_{2} \in f (H_{S}^{2})}{\cup} (γ_{1} γ_{2}))$ ;

(3) $λ H_{S} = f^{- 1} (\underset{γ \in f (H_{S})}{\cup} (1 - (1 - γ)^{λ}))$ ;

(4) $(H_{S})^{λ} = f^{- 1} (\underset{γ \in f (H_{S})}{\cup} γ^{λ})$ .

2.2 Probabilistic linguistic term set

To perform the different important degrees of the linguistic terms, Xu [20] proposed the probabilistic linguistic term set(PLTS).

Definition 7. [20] Let S = {s_i| - t, ⋯ , -1, 0, 1, ⋯, t} be a LTS, then the PLTS is defined as:

$\begin{matrix} L (p) = {L^{(k)} (p^{(k)}) | L^{(k)} \in S, p^{(k)} \geq 0, \\ k = 1, 2, \dots, # L (p), \sum_{k = 1}^{# L (p)} p^{(k)} \leq 1} \end{matrix}$ (4) where L^(k) (p^(k)) is the linguistic term L^(k) associated with the probability p^(k), and # L (p) is the number of all different linguistic terms in L (p).

Bai [34] introduced the possibility degree of PLTS which makes the applications of PLTS more convenient.

Definition 8. [34] Let S = {s_i| - t, ⋯ , -1, 0, 1, ⋯, t} be a LTS, L₁ (p) and L₂ (p) be two PLSs. The possibility degree that L₁ (p) is not less than L₂ (p) is defined as: $\begin{matrix} p (L_{1} (p) \geq L_{2} (p)) = 0.5 \cdot (1 + \\ \frac{(a {(L_{1})}^{-} - a {(L_{2})}^{-}) + (a {(L_{1})}^{+} - a {(L_{2})}^{+})}{| a {(L_{1})}^{-} - a {(L_{2})}^{-} | + | a {(L_{1})}^{+} - a {(L_{2})}^{+} | + a (L_{1} \cap L_{2})}) \end{matrix}$ (5) where a (L₁) ^- and a (L₁) ⁺ are respectively the lower area and upper area of L₁ (p). a (L₁ ∩ L₂) represents the overlap between L₁ (p) and L₂ (p).

2.3 The HFWA operator and HFWG operator

To aggregate evaluations effectively, some authors have investigated several types of aggregation operators. Here, we reviews the hesitant fuzzy weighted averaging (HFWA) operator and hesitant fuzzy weighted geometric (HFWG) operator.

Definition 9. [23] Let h_i be the elements of hesitant fuzzy set H_i, then the hesitant fuzzy weighted averaging(HFWA) operator is defined as follows:

$\begin{matrix} HFW A_{w} (H_{1}, H_{2}, \dots, H_{n}) = w_{1} H_{1} \oplus w_{2} \\ H_{2} \oplus \dots \oplus w_{n} H_{n} \\ = \underset{h_{i} \in H_{i}}{\cup} {(1 - \prod_{i = 1}^{n} {(1 - h_{i})}^{w_{i}})} \end{matrix}$ (6) when w = (1/n, 1/n, ⋯ , 1/n) ^T, the HFWA operator degrades to hesitant fuzzy averaging(HFA) operator.

Definition 10. [23] Let h_i be the elements of hesitant fuzzy set H_i, then the hesitant fuzzy weighted geometric(HFWG) operator is defined as follows: $HFW G_{w} (H_{1}, H_{2}, \dots, H_{n}) = \underset{h_{i} \in H_{i}}{\cup} {\prod_{i = 1}^{n} {(1 - h_{i})}^{w_{i}}}$ (7) when w = (1/n, 1/n, ⋯ , 1/n) ^T, the HFWG operator degrades to hesitant fuzzy geometric(HFG) operator.

2.4 The process of AHP

When people make decisions about semi-qualitative and semi-quantitative problems, they need to quantify elements contained in the problems. To achieve this, AHP [10], an effective method is developed. AHP layers the complex decision making system and provides a quantitative basis for analysis. The steps of AHP are shown as follows:

Step 1. Establish a hierarchical structure.

As shown in Fig. 1, The hierarchy structure generally consists of the following three levels: the target level, the criterion level, and the measure level.

Fig. 1

The model of AHP.

Step 2. Construction comparison matrix. The comparison matrix is the core of the analytic hierarchy. Comparison matrix evaluates attributes qualitatively by comparing the importance of each two attributes. According to the approach proposed by Saaty, the comparison matrix A = (a_ij) _n×n can be expressed as: $A = (\begin{matrix} 1 & a_{12} & a_{13} & \dots & a_{1 n} \\ 1 / a_{12} & 1 & a_{23} & \dots & a_{2 n} \\ 1 / a_{13} & 1 / a_{23} & 1 & \dots & a_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & \dots \\ 1 / a_{1 n} & 1 / a_{2 n} & 1 / a_{3 n} & \dots & 1 \end{matrix})$

Step 3.Consistency test. The decision making results will be inaccurate if the comparison matrix fails the consistency. It is necessary to check whether the matrix satisfies the consistency. The consistency test is divided into the following three steps:

(1) Calculate the consistence index CI: $CI = \frac{λ_{max} - n}{n - 1}, CR = \frac{CI}{RI}$ (2) Determine the average random index RI. Table 1 shows the corresponding random index RI.

Table 1

The value of RI

n	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

(3) Calculate the consistence rate CR. The comparison matrix can be accepted when $CR = \frac{CI}{RI} \leq 0.1$ , otherwise it needs to be corrected.

Step 4. Calculate the priority of the attributes.

As AHP is no longer a versatile tool for decision makers struggled with complex decision situations featuring uncertainty and ambiguity, FAHP, and other AHP-based methodology for decision making in fuzzy environment has attracted much attention in recent years. Subsequently, Xu [24] extended AHP to the probabilistic linguistic analytic hierarchy process(PL-AHP), whose process is illustrated in Fig. 2.

Fig. 2

The process of PL-AHP.

3 Interval probability hesitant fuzzy linguistic variable and its possibility degree

Under complex circumstances where alternatives hard to be exactly evaluated due to their fuzziness and uncertainty, people have to consider the reliability of the overall evaluation. Based on the fact that linguistic expression agrees with human habits, we propose the interval probability hesitant fuzzy linguistic variable (IPHFLV) which takes the reliability of the linguistic expression into account in this section. To apply it to decision making problems, we investigate its operation rules and comparison method. Particularly, we introduce the IPHFLVs’ possible degree which can be used to obtain the priority.

3.1 Interval probability hesitant fuzzy linguistic variable

Although the HFLTS can effectively represent the evaluation information, it believes that the decision maker is completely confident in their own evaluation. However, due to the limitations of the decision maker’s own cognition and the complexity of the environment, the evaluation information is not always reliable. Thus we proposed the interval probability hesitant fuzzy linguistic variable (IPHFLV) which evaluates information with HFLTS and express the reliability with probability interval to overcome this problem.

Definition 11. Let H_S be a HFLTS, [p_l, p_h] is the probability interval, the p_l is the lowest reliability and the p_h is the highest reliability. The interval probability hesitant fuzzy linguistic variable (IPHFLV) is defined as: $V = (H_{S}, [p_{l}, p_{h}])$ (8)

The probabilistic linguistic term set (PLTS) describes the importance of every single linguistic evaluation. Sometimes we need to consider the evaluation as a whole. For example, the student C₁ is 80% to 90% sure he will get A good grade at the end of the semester. Therefore the concept of the IPHFLV is necessary.

According to Definition 6 and the operation rules of probability, we propose the operation rules for IPHFLVs.

Definition 12. Let V = (H_S, [p_l, p_h]), $V_{1} = (H_{S}^{1},$ $[p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2}, [p_{l}^{2}, p_{h}^{2}])$ be three IPHFLVs, S = {s_i| - t, ⋯ , -1, 0, 1, ⋯ , t} be a LTS. H_s = {s_j|j ∈ [- t, t] be a HFLTS, And # H_S is the number of the linguistic terms in H_S, ${# H}_{S}^{1} {= # H}_{S}^{2}$ (we can use Equation (3) to extend the shorter one by adding the linguistic terms ensure ${# H}_{S}^{1} {= # H}_{S}^{2}$ ). Then:

(1) $V_{1} \oplus V_{2} = (f^{- 1} (\underset{γ_{1} \in f (H_{S}^{1}), γ_{2} \in f (H_{S}^{2})}{\cup} (γ_{1} + γ_{2} - γ_{1} γ_{2})); [\frac{p_{l}^{1} p_{l}^{1} + p_{l}^{2} p_{l}^{2}}{p_{l}^{1} + p_{l}^{2}}, \frac{p_{h}^{1} p_{h}^{1} + p_{h}^{2} p_{h}^{2}}{p_{h}^{1} + p_{h}^{2}}])$ ;

(2) $λ V = (f^{- 1} (\underset{γ \in f (H_{S})}{\cup} (1 - (1 - γ)^{λ})); [p_{l}, p_{h}])$ ;

(3) $V_{1} \otimes V_{2} = (f^{- 1} (\underset{γ_{1} \in f (H_{S}^{1}), γ_{2} \in f (H_{S}^{2})}{\cup} (γ_{1} γ_{2})); [p_{l}^{1}$ $p_{l}^{2}, p_{h}^{1} p_{h}^{2}])$ ;

(4) $V^{λ} = (f^{- 1} (\underset{γ \in f (H_{S})}{\cup} γ^{λ}); [p_{l}^{λ}, p_{h}^{λ}])$ .

When we add two IPHFLVs, the the evaluation part follows the operation rule of HFLTS. And for the reliability part, taking into account the weight assigned higher reliability should be higher, so we take the proportion of $p_{l}^{i}$ (or $p_{h}^{i}$ ) itself as the its weight. Using the idea of series and parallel circuits in physics, our operation of reliability is defined as above. We regard the evaluation and reliability as two dimensions with the evaluation being the main part. Then the scoring function is defined.

Definition 13. Let V = (H_S, [p_l, p_h]) be a IPHFLV, then: $E (V) = \frac{\sum_{i = 1}^{# H_{S}} f (s_{i})}{# H_{S}} + \frac{β (p_{l} + p_{h})}{2}$ (9) where the # H_S denotes the number of the linguistic variables in the HFLTS H_S, β represents the rate of reliability in the scoring function and β ∈ [0, 1]. Here we set β = 0.5.

When the reliability of the evaluation is the same, if evaluation is higher, the value of the scoring function is higher; when the evaluation is the same, if the reliability is higher, the value of the scoring function is higher. Considering that the evaluation is dominant of the information, the influence of the evaluation on the scoring function should be higher. Thus we give the parameter β to adjust the evaluation and reliability impact ratio.

However, there is a phenomenon that we can not determine which one is better when A₁’s evaluation is better than A₂ but the reliability is lower. In addition, using the average value to obtain the evaluation score and the reliability score will bring about a issue that different evaluation and reliability have the same scores. For instance, V₁ = ({s_-1, s₀, s₁} , [0.6, 0.8]} and V₂ = ({s₀} , [0.65, 0.75]}. We can obtain E (V₁) = E (V₂) =0.675. Inspired by the idea of variance, the degree of dispersion is proposed to address such issue.

Definition 14. Let $E (α) = (\sum_{i = 1}^{# H_{S}} f (s_{i})) / # H_{S}$ , then the degree of dispersion is defined as follows:

$\begin{matrix} σ (V) = \frac{β}{2} [(p_{h} - \frac{p_{h} + p_{l}}{2})^{2} + (p_{l} - \frac{p_{h} + p_{l}}{2})^{2}] \\ + \frac{1}{# H_{S}} (\sum_{i = 1}^{# H_{S}} {(f (s_{i}) - E (α))}^{2}) \end{matrix}$ (10) When the value of scoring function is equal, the distribution that is closer to the average value will be more stable. For example, V₁ = ({s_-1, s₀, s₁} , [0.6, 0.8]} and V₂ = ({s₀} , [0.65, 0.75]}. Although the expectations of the evaluation are both s₀ and the expectations of the reliability are both 0.7, we prefer V₂ because V₂ makes it less risky.

Then the comparison method is defined as follows:

If E (V_α) > E (V_β), then V_α > V_β;

If E (V_α) < E (V_β), then V_α < V_β;

If E (V_α) < E (V_β), then

If σ (V_α) > σ (V_β), then V_α < V_β;

If σ (V_α) < σ (V_β), then V_α > V_β;

If σ (V_α) = σ (V_β), then V_α = V_β;

Here we give an example to illustrate the method.

Example 3. Let S = {s_-3 : nothing, s_-2 : very low, s_-1 : low, s₀ : medium, s₁ = high, s₂ : very high, s₃ :perfect} be a LTS, V₁ = (({s_-2} , {s_-1}) , [0 . 6, 0 . 8]) and V₂ = ((s₀, s₁, s₂) , [0.7, 0.8]) be two IPHFLVs. We can deduce that: $E (V_{1}) = \frac{0.5 \times (0.6 + 0.8)}{2} + \frac{((\frac{- 2}{2} + \frac{1}{2}) + (\frac{- 1}{2} + \frac{1}{2}))}{2} = 0.60$ , $E (V_{2}) = \frac{0.5 \times (0.7 + 0.8)}{2} + \frac{(\frac{0}{2} + \frac{1}{2}) + (\frac{1}{2} + \frac{1}{2}) + (\frac{2}{2} + \frac{1}{2})}{3} = 1.375$ . Then we have E (V₁) < E (V₂), therefore V₁ < V₂.

3.2 Comparison method of IPHFLVs

Based on the definition of the possibility degree of PLTS, we extend it to IPHFLVs.

Definition 15. Let $V_{1} = (H_{S}^{1}, [p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2}, [p_{l}^{2}, p_{h}^{2}])$ be two IPHFLVs, the possibility degree that V₁ is not less than V₂ is defined as: $\begin{matrix} D (V_{1} \geq V_{2}) = 0.5 \cdot (1 + \\ \frac{(a {(V_{1})}^{-} - a {(V_{2})}^{-}) + (a {(V_{1})}^{+} - a {(V_{2})}^{+})}{| a {(V_{1})}^{-} - a {(V_{2})}^{-} | + | a {(V_{1})}^{+} - a {(V_{2})}^{+} | + | a (V_{1} \cap V_{2}) |}) \end{matrix}$ (11) -3ptwhere $a (V)^{-} = α \cdot \frac{p_{l} + p_{h}}{2}$ (α is the smallest linguistic subscript of H_S), $a (V)^{+} = β \cdot \frac{p_{l} + p_{h}}{2}$ (β is the largest linguistic subscript of H_S) and $a (V_{1} \cap V_{2}) = \frac{(max (p_{l}^{1}, p_{l}^{2}) + min (p_{h}^{1}, p_{h}^{2}))}{2} \cdot \sum | γ_{i} |$ (γ_i is the common element subscript of $H_{S}^{1}$ and $H_{S}^{2}$ ).

Theorem 1. Let $V_{1} = (H_{S}^{1}, [p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2}, [p_{l}^{2}, p_{h}^{2}])$ be two IPHFLVs, D (V₁ ≥ V₂) is the possibility degree that V₁ is not less than V₂. Then it satisfies:

(1) 0 ≤ D (V₁ ≥ V₂) ≤1;

(2) D (V₁ ≥ V₂) =0 . 5 if only if V₁ = V₂;

(3) D (V₁ ≥ V₂) + D (V₂ ≥ V₁) =1.

In Equation (11), a lot of evaluation information is missed and the example 4 is used here to illustrate. To overcome this shortcoming, we define a new possibility degree of IPHFLVs, which takes into account sufficient information and more appropriately to the comparison for IPHFLVs.

Definition 16. Let $V_{1} = (H_{S}^{1}, [p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2},$ $[p_{l}^{2}, p_{h}^{2}])$ be two IPHFLVs, S = {s_i| - t, ⋯ , -1, 0, 1, ⋯ , t} be a LTS. H_s = {s_j|j ∈ [- t, t] be a HFLTS, And # H_S is the number of the linguistic terms in H_S, ${# H}_{S}^{1} {= # H}_{S}^{2}$ (we can use Equation (3) to extend the shorter one by adding the linguistic terms ensure ${# H}_{S}^{1} {= # H}_{S}^{2}$ ). The possibility degree that V₁ is not less than V₂ is defined as:

$\begin{matrix} P (V_{1} \geq V_{2}) = 0.5 \cdot (1 + \\ \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |}) \end{matrix}$ (12) where $a_{i} (V) = α_{i} \cdot \frac{p_{l} + p_{h}}{2}$ (α_i is the ith linguistic subscript of H_S) and $a (V_{1} \cap V_{2}) = \frac{(max (p_{l}^{1}, p_{l}^{2}) + min (p_{h}^{1}, p_{h}^{2}))}{2} \cdot \sum | γ_{i} |$ (γ_i is the common element subscript of $H_{S}^{1}$ and $H_{S}^{2}$ ).

Theorem 2. Let $V_{1} = (H_{S}^{1}, [p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2}, [p_{l}^{2}, p_{h}^{2}])$ be two IPHFLVs, P (V₁ > V₂) is the possibility degree that V₁ is not less than V₂. Then it satisfies:

(1) 0 ≤ P (V₁ ≥ V₂) ≤1;

(2) P (V₁ ≥ V₂) = 0 . 5 if only if V₁ = V₂;

(3) P (V₁ ≥ V₂) + P (V₂ ≥ V₁) =1.

The proof process is shown in Appendix A.

Lemma 1. Let $V_{1} = (H_{S}^{1}, [p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2},$ $[p_{l}^{2}, p_{h}^{2}])$ be two IPHFLVs, if $p_{l}^{1} = p_{l}^{2}, p_{h}^{1} = p_{h}^{2}$ , when $s_{i} \leq s_{j}, s_{i} \in H_{S}^{1}, s_{j} \in H_{S}^{2}$ , we could obtain that P (V₁ ≥ V₂) ∈ [0, 0.5].

Lemma 2. Let $V_{1} = (H_{S}^{1}, [p_{l}^{1}, p_{h}^{1}])$ , $V_{2} = (H_{S}^{2},$ $[p_{l}^{2}, p_{h}^{2}])$ be two IPHFLVs, if $s_{i} = s_{j}, s_{i} \in H_{S}^{1}, s_{j} \in H_{S}^{2}$ , when $p_{l}^{1} \leq p_{l}^{2}, p_{h}^{1} \leq p_{h}^{2}$ , we could obtain that P (V₁ ≤ V₂) ∈ (0, 0.5).

An example is given to explain the effectiveness.

Example 4. Suppose that V₁ = ({s_-3, s₁, s₂} , [0.6, 0.8]), V₂ = ({s_-3, s_-1, s₀, s₂} , [0.65, 0.75]) and V₃ = ({s_-3, s_-1, s₁, s₂}, [0.55, 0.85]) are three IPHFLVs, then:

From Equation (11), we can obtain that D (V₁≥ V₂) =0.5 × (1 + $\frac{0}{| a {(V_{1})}^{-} - a {(V_{2})}^{-} | + | a {(V_{1})}^{+} - a {(V_{2})}^{+} | + | a (V_{1} \cap V_{2}) |})$ = 0.5. Similarly, we can obtain that D (V₁ ≥ V₃) =0.5, D (V₂ ≥ V₃) =0.5.

From Equation (12), first we add the linguistic terms $\bar{s} = \sum_{i = 1}^{# H_{S}^{1}} \frac{1}{3} s_{i} = s_{(- 3) \times \frac{1}{3} + 1 \times \frac{1}{3} + 2 \times \frac{1}{3}} = s_{0}$ into V₁ by Equation (3). Then V₁ = ({s_-3, s₀, s₁, s₂} , [0.6, 0.8]). Thus $\begin{matrix} P (V_{1} \geq V_{2}) = 0.5 \times (1 + \\ \frac{[\frac{(0.6 + 0.8)}{2} \times (- 3) - \frac{(0.65 + 0.75)}{2} \times (- 3)] + [\frac{(0.6 + 0.8)}{2} \times 0 - \frac{(0.65 + 0.75)}{2} \times (- 1)] + [\frac{(0.6 + 0.8)}{2} \times 1 - \frac{(0.65 + 0.75)}{2} \times 0] + [\frac{(0.6 + 0.8)}{2} \times 2 - \frac{(0.65 + 0.75)}{2} \times 2]}{| \frac{(0.6 + 0.8)}{2} \times (- 3) - \frac{(0.65 + 0.75)}{2} \times (- 3) | + | \frac{(0.6 + 0.8)}{2} \times 0 - \frac{(0.65 + 0.75)}{2} \times (- 1) | + | \frac{(0.6 + 0.8)}{2} \times 1 - \frac{(0.65 + 0.75)}{2} \times 0 | + | \frac{(0.6 + 0.8)}{2} \times 2 - \frac{(0.65 + 0.75)}{2} \times 2 | + | \frac{0.65 + 0.8}{2} (3 + 0 + 2) |}) \\ = 0.5 \times (1 + \frac{0.7}{0.7 + 3.625}) = 0.581 \end{matrix}$ Similarly, we can obtain: P (V₁ ≥ V₃) =0.569, P (V₂ ≥ V₃) =0.433.

From the Fig. 3 we can see that P (V₁ ≥ V₂) = P (V₁ ≥ V₃) = P (V₂ ≥ V₃) =0.5 which is obtained by the possibility degree proposed in Equation (11) and we can not effectively compare the V₁, V₂ and V₃. But the improved possibility degree (Equation (12)) can effectively compare the three IPHFLVs.

Fig. 3

Comparison results of possibility degree.

4 The IPHFLA operator and the IPHFLG operator

In the MADM problem, it is necessary to aggregate the information under different alternatives to facilitate the expert to make decisions. And the aggregation operator is the important tool to aggregate information. This section we discuss the interval probability hesitant fuzzy linguistic averaging (IPHFLA) operator and the interval probability hesitant fuzzy linguistic geometric (IPHFLG) operator under the interval probability hesitant fuzzy linguistic environment.

Definition 17. The interval probability hesitant fuzzy linguistic averaging (IPHFLA) operator is a n dimensional function: Φ_IPHFLA : Vⁿ → V. Let $V_{i} = (H_{S}^{i}, [p_{l}^{i}, p_{h}^{i}])$ be a IPHFLV, then $Φ_{IPHFLA} (V_{1}, V_{2}, \dots, V_{n}) = \frac{1}{n} (V_{1} \oplus V_{2} \oplus \dots \oplus V_{n})$ (13) The IPHFLA operator satisfies the following properties:

Theorem 3. (Commutativity) Let ${V_{1}^{*}, {V_{2}}^{*}, . . ., {V_{n}}^{*}}$ be a set of IPHFLVs in any order of {V₁, V₂, ⋯ , V_n}, then $\begin{matrix} Φ_{IPHFLA} ({V_{1}}^{*}, {V_{2}}^{*}, \dots, {V_{n}}^{*}) \\ = Φ_{IPHFLA} (V_{1}, V_{2}, \dots, V_{n}) \end{matrix}$

Theorem 4. (Idempotency) Let {V₁, V₂, ⋯ , V_n} be a set of IPHFLVs, V is an IPHFLV, and V_i = V. then $Φ_{IPHFLA} (V_{1}, V_{2}, \dots, V_{n}) = V$

Theorem 5. (Monotonicity) Let {V₁, V₂, ⋯ , V_n}, ${V_{1}^{*}, {V_{2}}^{*}, \dots, {V_{n}}^{*}}$ be two group IPHFLVs, and V_i < V_i^*}, then $\begin{matrix} Φ_{IPHFLA} (V_{1}, V_{2}, \dots, V_{n}) < Φ_{IPHFLA} \\ ({V_{1}}^{*}, {V_{2}}^{*}, \dots, {V_{n}}^{*}) \end{matrix}$

Theorem 6. (Boundedness) Let {V₁, V₂, ⋯ , V_n} be a set of IPHFLVs, then $min {V_{i}} \leq Φ_{IPHFLA} (V_{1}, V_{2}, \dots, V_{n}) \leq max {V_{i}}$

Remark 1. When p_l = p_h = 1, the IPHFLV becomes the HFLTS. Thus the IPHFLA operator degenerates into the hesitant fuzzy linguistic averaging (HFLA) operator. $Φ_{HFLA} (H_{S}^{1}, H_{S}^{2}, \dots, H_{S}^{n}) = \frac{1}{n} (H_{S}^{1} \oplus H_{S}^{2} \oplus \dots \oplus H_{S}^{n})$

Definition 18. The interval probability hesitant fuzzy linguistic geometric (IPHFLG) operator is a n dimensional function: Φ_IPHFLG : Vⁿ → V. Let $V_{i} = (H_{S}^{i}, [p_{l}^{i}, p_{h}^{i}])$ be a IPHFLV, then $Φ_{IPHFLG} (V_{1}, V_{2}, \dots, V_{n}) = (V_{1} \otimes V_{2} \otimes \dots \otimes V_{n})^{\frac{1}{n}}$ (14) The IPHFLG operator satisfies the following properties:

Theorem 7. (Commutativity) Let ${V_{1}^{*}, {V_{2}}^{*}, . . ., {V_{n}}^{*}}$ be a set of IPHFLVs in any order of {V₁, V₂, ⋯ , V_n}, then $\begin{matrix} Φ_{IPHFLG} ({V_{1}}^{*}, {V_{2}}^{*}, \dots, {V_{n}}^{*}) \\ = Φ_{IPHFLA} (V_{1}, V_{2}, \dots, V_{n}) \end{matrix}$

Theorem 8. (Idempotency) Let {V₁, V₂, ⋯ , V_n} be a set of IPHFLVs, V is an IPHFLV, and V_i = V. then $Φ_{IPHFLG} (V_{1}, V_{2}, \dots, V_{n}) = V$

Theorem 9. (Monotonicity) Let {V₁, V₂, ⋯ , V_n}, ${V_{1}^{*}, {V_{2}}^{*}, \dots, {V_{n}}^{*}}$ be two group IPHFLVs, and V_i < V_i^*, then $\begin{matrix} Φ_{IPHFLG} (V_{1}, V_{2}, \dots, V_{n}) \\ < Φ_{IPHFLG} ({V_{1}}^{*}, {V_{2}}^{*}, \dots, {V_{n}}^{*}) \end{matrix}$

Theorem 10. (Boundedness) Let {V₁, V₂, ⋯ , V_n} be a set of IPHFLVs, then $min {V_{i}} \leq Φ_{IPHFLG} (V_{1}, V_{2}, \dots, V_{n}) \leq max {V_{i}}$

Remark 2. When p_l = p_h = 1, the IPHFLV becomes the HFLTS. Thus the IPHFLG operator degenerates into the hesitant fuzzy linguistic averaging (HFLG) operator. $Φ_{HFLG} (H_{S}^{1}, H_{S}^{2}, \dots, H_{S}^{n}) = (H_{S}^{1} \otimes H_{S}^{2} \otimes \dots \otimes H_{S}^{n})^{\frac{1}{n}} .$

5 Interval probability hesitant fuzzy linguistic AHP (IPHFL-AHP)

AHP provides a simple and practical decision-making method for studying MADM problems in

complex environments. However, most of the current complex linguistic AHP convert complex linguistic information into basic fuzzy numbers and use the basic fuzzy AHP to deal with problems. This section we introduce the interval probability hesitant fuzzy linguistic AHP(IPHFL-AHP) to directly deal with the decision problem in the complex linguistic environment. We adopt IPHFLVs to obtain the interval probability hesitant fuzzy linguistic comparison matrix (IPHFLCM). Especially, to deal with the consistency of comparison matrix under linguistic environment, we propose the general geometric consistency index (G-GCI) and construct the iterative algorithm to improve the consistency of the IPHFLCM. The priority vector can be obtained with the possibility degree (Equation (12)) proposed in Section 3.

5.1 The interval probability hesitant fuzzy linguistic comparison matrix

This section first introduces the definition of the hesitant fuzzy linguistic comparison matrix (HFLCM) introduced by Zhang [31]. Based on the HFLCM and the multiplicative linguistic scale, we define the IPHFLCM.

Definition 19. Let S = {s_α|α ∈ [- t, t]} be a LTS. A HFLCM $\tilde{R}$ is presented by a matrix $\tilde{R} = (H_{S_{ij}})_{n \times n}$ , where H_{S
_ij} is a HFLTS indicating that v_i is preferred to v_j, such that: $s_{α_{ij}^{l}} \oplus s_{α_{ji}^{l}} = s_{0}, H_{S_{ii}} = {s_{0}}$ for all l = 1, 2, ⋯ , # H_{S
_ij}, i, j = 1, 2, ⋯ , n and i < j.

Then we give the expression of IPHFLCM.

Definition 20. An IPHFLCM R is represented by a matrix R = (r_ij) _n×n, where r_ij = (H_{S
_ij}, [p_{l
_ij}, p_{h
_ij}]) and i, j = 1, 2, ⋯ , n. Then it satisfies the conditions: $\begin{matrix} s_{α_{ij}^{l}} \oplus s_{α_{ji}^{l}} = s_{0}, H_{S_{ii}} = {s_{0}}, p_{l_{ij}} = p_{l_{ji}}, p_{h_{ij}} \\ = p_{h_{ji}}, p_{l_{ii}} = p_{h_{ii}} = 1 \end{matrix}$

Example 5. Given the linguistic term set S = {s_α|α ∈ [-3, 3]}. Let H_{S
_ij} be a HFLS, Let V = (H_S, [p_l, p_h]) be a IPHFLV. Then IPHFLCM R is shown as follows: $R = (\begin{matrix} ((s_{0}); [1, 1]) & ((s_{- 1}, s_{0}); [0.6, 0.8]) & ((s_{2}, s_{3}); [0.5, 0.7]) & ((s_{- 1}, s_{0}, s_{1}); [0.6, 0.7]) \\ ((s_{0}, s_{1}); [0.6, 0.8]) & ((s_{0}); [1, 1]) & ((s_{- 2}, s_{- 1}); [0.3, 0.5]) & ((s_{- 2}); [0.5, 0.78]) \\ ((s_{- 3}, s_{- 2}); [0.5, 0.7]) & ((s_{1}, s_{2}); [0.3, 0.5]) & ((s_{0}); [1, 1]) & ((s_{- 3}, s_{- 2}, s_{- 1}); [0.6, 0.7]) \\ ((s_{- 1}, s_{0}, s_{1}); [0.6, 0.7]) & ((s_{2}); [0.5, 0.78]) & ((s_{1}, s_{2}, s_{3}); [0.6, 0.7]) & ((s_{0}); [1, 1]) \end{matrix})$

5.2 The consistency of IPHFLCM

In the process of AHP, we need to check whether the IPHFLCM is consistent before we make decisions. It is difficult and unnecessary to achieve complete consistency because of the complexity of the environment and the difference of people’s acknowledge. Therefore we propose the G-GCI based on the definition of acceptable consistency for fuzzy linguistic comparison matrix.

Aguarón [1] exploited the unbiased estimator of the variance of the perturbations as the measure of the consistency: $s^{2} = S / d . f . = \frac{2 \sum_{i < j} {(log a_{ij} - log (w_{i} / w_{j}))}^{2}}{(n - 1) (n - 2)}$ where the w_i and w_j were obtained by the prioritization procedure RGMM $w_{i} = (\prod_{j = 1}^{n} a_{ij})^{1 / n}$ .

The number a_ij is a real value so it can be easy to apply to calculations. However the linguistic variable r_ij could not do the direct calculation. Therefore we extend the GCI to general linguistic situation inspired by the GCI proposed in [1],

Definition 21. Given a comparison matrix B = (b_ij) _n×n, where b_ij can be expressed in any form of complex linguistic information, and w is the vector of priorities obtained by the RGMM, then G-GCI is defined as: $G - GCI = \frac{2}{(n - 1) (n - 2)} \sum_{i < j} {(log φ (b_{ij}) - log \frac{w_{i}}{w_{j}})}^{2}$ (15) where the $w_{i} = (\prod_{j = 1}^{n} φ (b_{ij}))^{1 / n}$ , φ (b_ij) is the function to transform complex linguistic variable to real number and φ (b_ii) =1.

In Equation (15), the $log φ (b_{ij}) - log \frac{w_{i}}{w_{j}}$ represents the difference between the value obtained by the priorities and the complex linguistic variable which has been transformed.

Lemma 3. Given a comparison matrix B = (b_ij) _n×n, where b_ij can be expressed in any form of complex linguistic information, and w is the vector of priorities obtained by the RGMM, it holds that $G - GCI = \frac{2 n}{n - 2} CI + o (ς)$ (16) The proof process is shown in Appendix B.

In this paper we discuss the consistency of the IPHFLCM so we set that φ (r_ij) = E (r_ij).

We can measure whether GCI is consistent with the established threshold $GC {\bar{I}}^{(n)}$ proposed by Aguarón and Moreno-Jimeńez in Table 2. The approximated threshold for the GCI has been proved in [1]. If $G - GCI \leq GC {\bar{I}}^{(n)}$ , then we agree that the matrix satisfies the consistency. Otherwise, we need to modify the matrix.

Table 2

The thresholds of G - GCI

n	1	2	3	4	5	6
$GC {\bar{I}}^{(n)}$	0	0	0.3147	0.3526	0.3717	0.3755

As comparison matrix does not always bring about consistency, adjustments needs to be made. This paper proposes the Algorithm 1 to improve the consistency.

Algorithm 1

Adjustment of the consistency

Require: A comparison judgement matrix R = (r_ij) _n×n and a parameter λ.

Ensure: comparison judgement matrix R = (r_ij) _n×n satisfy the consistency.

1: Calculate GCI;

2: If

G - GCI \leq GC {\bar{I}}^{(n)}

, go to Step 5. Otherwise, go to Step 3;

3: Let

({\hat{r}}_{ij})_{n \times n} = (\frac{w_{i}}{w_{j}})^{1 - ɛ} (E (r_{ij}))^{ɛ} \cdot (r_{ij})_{n \times n}, ɛ \in [0, 1]

;

4: Let

(r_{ij})_{n \times n} = ({\hat{r}}_{ij})_{n \times n}

, go back to Step 1;

5: Let

\underset{n \times n}{(r_{ij})^{*}} = ({\hat{r}}_{ij})_{n \times n}

, then end.

The smaller the s² is, the shorter distance between E (r_ij) and w_i/w_j is. Thus we need to make the s² smaller to ensure the acceptable consistency. We use the parameter $(\frac{w_{i}}{w_{j}})^{1 - ɛ} (E (r_{ij}))^{ɛ}$ which can make the s² smaller to adjust the consistency. In this paper, we set ɛ = 0.5.

Here, the Lemma 4 is presented to verify the iterative algorithm’s feasibility and effectiveness.

Lemma 4. Let B = (b_ij) _n×n be the IPHFLCM, b_ij is the IPHFLV. Then G - GCI can be reduced by the iterative algorithm 1, namely, G - GCI^* ≤ G - GCI.

The proof process is shown in Appendix C.

5.3 Priority for the IPHFLCM

In the Section 5.2, we can measure and improve the consistency of the matrix. Then we need to determine the priority to make decisions in the AHP. To get prioritization, we use the possibility degree.

For each row of the shcemes, C_i can be aggregated by Equation (14). $Y_{i} = IPHFLG (r_{i 1}, r_{i 2}, . . ., r_{in}) = (r_{i 1} \otimes r_{i 2} \otimes . . . \otimes r_{in})^{\frac{1}{n}}$ (17) According to the Equation (12), we can get the matrix of possibility degree as follows: $D = (\begin{matrix} d_{11} & d_{12} & \dots & d_{1 n} \\ d_{21} & d_{22} & \dots & d_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ d_{n 1} & d_{n 2} & \dots & d_{nn} \end{matrix})$

Inspired by the formula of priority proposed by [24], we can get the priority vector of schemes. ${\bar{w}}_{i} = \frac{1}{n (n - 1)} (\sum_{j = 1}^{n} d_{ij} + \frac{n}{2} - 1)$ (18)

The procedure of the interval probability hesitant fuzzy linguistic analytic hierarchy process (IPHFL-AHP) is introduced as follows:

Step 1. Construct the comparison matrix A which the values express with the linguistic expression and probability interval. Here we give an example to illustrate the matrix.

Example 6. Let S = {s_-2 = bad, s_-1 = little bad, s₀ = equal, s₁ = little good, s₂ = good} be a LTS. C₁, C₂, C₃ be the attributes. Then the matrix can be obtained as shown in Table 3:

Table 3

The quantitative description of attributes

	C ₁	C ₂	C ₃
C ₁	(s₀, [1, 1])	(between {{s₁}} and {{s₂}} , [0 . 75, 0 . 8])	(less than {s_-1} , [0 . 68, 0 . 76])
C ₂	(between {s_-2} and {s_-1} , [0 . 75, 0 . 8])	(s₀, [1, 1])	(great than s₀, [0.62, 0.75])
C ₃	(greater than {s₁} , [0 . 68, 0 . 76])	(less than s₀, [0.62, 0.75])	(s₀, [1, 1])

Step 2. According to the IPHFLV and the context-free grammar of HFLTS, we use the E_{G
_H} to transform the values in comparison matrix into IPHFLVs.

Example 7. We use the E_{G
_H} to transform the values in Table 3 into IPHFLVs. The results are shown in Table 4:

Table 4

The attributes expressed by IPHFLVs

	C ₁	C ₂	C ₃
C ₁	((s₀) ; [1, 1])	((s₁, s₂) ; [0 . 75, 0 . 8])	((s_-2, s_-1) ; [0 . 68, 0 . 76])
C ₂	((s_-2, s_-1) ; [0 . 75, 0 . 8])	((s₀) ; [1, 1])	((s₁, s₂) ; [0 . 62, 0 . 75])
C ₃	((s₁, s₂) ; [0 . 68, 0 . 76]	((s_-2, s_-1) ; [0 . 62, 0 . 75])	((s₀) ; [1, 1])

Step 3. Check the consistency of IPHFLCM by the G - GCI (Equation (15)). If all of the IPHFLCMs are of acceptable consistency, go to Step 5; otherwise, go to Step 4.

Step 4. Repair the IPHFLCM by the Algorithm 1 until the matrix satisfies the consistency.

Step 5. First we use the IPHFLG operator (Equation (14)) to aggregate the information of each scheme. Then the possibility degrees of matrix of schemes are obtained with the improved possibility degree (Equation (12)) and the priority vectors of attributes and schemes are acquired with the formula of priority (Equation (18)).

Step 6. According to the priority vectors of attributes and schemes obtained with the formula of priority (Equation (18)), we calculate the weight of each schemes and choose the one(s) with the largest weight.

Step 7. End.

The process is illustrated in Fig. 4.

Fig. 4

The process of IPHFL-AHP.

Regardless of the form of information representation, one of the most important elements in the AHP approach is the consistency checking. The consistency test method proposed in this paper (Equation (15)) is applicable to all expressions of complex linguistic information and proves the theoretical basis of this consistency test method through Lemma 3. Here, we illustrate the comparison between the proposed IPHFL-AHP and other AHP extension methods through the following Table 5.

Table 5

Comparison of AHP extension methods

	the expression of r_ij	φ (r_ij)
IPHFL - AHP	IPHFLVs	φ (r_ij) = E (r_ij)
PL - AHP	PLTS	φ (r_ij) = E (r_ij)
HFL - AHP	HFLTS	φ (r_ij) = E (r_ij) (p_l = p_h = 1)
FL - AHP	Linguistic variables	φ (r_ij) = E (r_ij) (p_l = p_h = 1, s₁ = s₂ = ⋯ = s_k)

When p_l = p_h = 1, the IPHFLVs converts into the HFLTS. At the same time, the IPHFL-AHP converts into the HFL-AHP. When the evaluation converts into linguistic variables, the IPHFL-AHP converts into FL-AHP. Although the IPHFLVs can not convert into the PLTS, the consistency test method can be used under the PLTS. Therefore the IPHFL-AHP can be used in more situations.

6 Numerical example

Talents and human resources are of vital importance for the stability and long term prosperity of the modem society. However, experts have certain hesitation and uncertainty in the candidates’ comparison. In the process of the talent selection, experts are more accustomed to using linguistic variables to compare candidates. In addition, the reliability of the comparison is not complete because of the different familiarity with different candidates or the degree of understanding of certain aspects. Therefore, using the IPHFL-AHP to deal with the talent selection is suitable. Suppose that a company wants to select one of the three candidates(A₁,A₂,A₃) to assume a middle-level leader. The performances of the candidates are measured with four attributes: C₁:Health status, C₂:Business knowledge, C₃:Ethical standards, C₄:Written expression. Let S = {s_-2, s_-1, s₀, s₁, s₂} be a LTS, then the leaders of the company give the comparison matrix about the four attributes as shown in the Table 6. (The data adapts from the book“Decision Theory and Method”.)

Table 6
Comparison of attributes

C ₁ C ₂ C ₃ C ₄

C ₁ (s₀, [1, 1]) (between {{s₁}} and {{s₂}} , [0 . 65, 0 . 72]) (less than {s_-2} , [0 . 6, 0 . 72]) ({s₂} , [0 . 66, 0 . 78])

C ₂ (between {s_-2} and {s_-1} , (s₀, [1, 1]) (less than {s_-1} , [0 . 75, 0 . 8]) (between {{s₁}} and {{s₂}} ,

[0.62,0.75]) [0.6,0.8])

C ₃ (greater than {s₂} , [0 . 6, 0 . 72]) (more than {s₁} , [0 . 75, 0 . 8]) (s₀, [1, 1]) ((s₂) ; [0.7, 0.86])

C ₄ ({s_-2} , [0 . 66, 0 . 78]) (between {{s_-2}} and {{s_-1}} , [0 . 6, 0 . 8]) ((s_-2) ; [0 . 7, 0 . 86]) (s₀, [1, 1])

	C ₁	C ₂	C ₃	C ₄
C ₁	(s₀, [1, 1])	(between {{s₁}} and {{s₂}} , [0 . 65, 0 . 72])	(less than {s_-2} , [0 . 6, 0 . 72])	({s₂} , [0 . 66, 0 . 78])
C ₂	(between {s_-2} and {s_-1} ,	(s₀, [1, 1])	(less than {s_-1} , [0 . 75, 0 . 8])	(between {{s₁}} and {{s₂}} ,
	[0.62,0.75])			[0.6,0.8])
C ₃	(greater than {s₂} , [0 . 6, 0 . 72])	(more than {s₁} , [0 . 75, 0 . 8])	(s₀, [1, 1])	((s₂) ; [0.7, 0.86])
C ₄	({s_-2} , [0 . 66, 0 . 78])	(between {{s_-2}} and {{s_-1}} , [0 . 6, 0 . 8])	((s_-2) ; [0 . 7, 0 . 86])	(s₀, [1, 1])

Next we use E_{G
_H} to transform the evaluation of the attributes into IPHFLVs shown in Table 7:

Table 7

Comparison of attributes with IPHFLVs

	C ₁	C ₂	C ₃	C ₄
C ₁	((s₀) ; [1, 1])	((s₁, s₂) ; [0 . 82, 0 . 89])	((s_-2) ; [0 . 62, 0 . 66])	((s₂) ; [0.86, 0.90])
C ₂	((s_-2, s_-1) ; [0 . 82, 0 . 89])	(s₀, [1, 1])	((s_-2, s_-1) ; [0 . 62, 0 . 65])	((s₁, s₂) ; [0 . 85, 0 . 92])
C ₃	((s₂) ; [0 . 62, 0 . 66])	((s₁, s₂) ; [0 . 62, 0 . 65])	((s₀) ; [1, 1])	((s₂) ; [0.6, 0.72])
C ₄	((s_-2) ; [0 . 86, 0 . 90])	((s_-2, s_-1) ; [0 . 85, 0 . 92])	((s_-2) ; [0 . 60, 0 . 72])	((s₀) ; [1, 1])

The result that G - GCI > 0.3526 means that the comparison matrix is not acceptable, thus needs to be modified. By using the Algorithm 1 to correct the matrix, we obtain that the modified matrix which is shown as R*. The value of G - GCI of R* is 0.02256, satisfying the consistency.

$R^{*} = (\begin{matrix} ((s_{0.1253}); [1, 1]) & ((s_{1.1291}, s_{2}); [0.82, 0.89]) & ((s_{- 2}); [0.62, 0.66]) & ((s_{2}); [0.86, 0.90]) \\ ((s_{- 1.2220}, s_{- 2}); [0.82, 0.89]) & ((s_{- 0.1563}); [1, 1]) & ((s_{- 1.1268}, s_{- 2}); [0.62, 0.65]) & ((s_{0.8468}, s_{2}); [0.85, 0.92]) \\ ((s_{2}); [0.62, 0.66]) & ((s_{1.2731}, s_{2}); [0.62, 0.65]) & ((s_{0.7042}); [1, 1]) & ((s_{2}); [0.60, 0.72]) \\ ((s_{- 2}); [0.86, 0.90]) & ((s_{- 1.3986}, s_{- 2}); [0.85, 0.92]) & ((s_{- 2}); [0.60, 0.72]) & ((s_{- 0.5330}); [1, 1]) \end{matrix})$

Next, we can get: $\begin{matrix} Y_{1} = ((s_{- 2}); [0.813, 0.853]) \\ Y_{2} = ((s_{- 2}, s_{- 0.503}); [0.810, 0.837]) \end{matrix}$ $\begin{matrix} Y_{3} = ((s_{1.130}, s_{1.364}); [0.693, 0.745]) \\ Y_{4} = ((s_{- 2}); [0.804, 0.879]) \end{matrix}$ We get the possibility degree matrix D: $D = (\begin{matrix} 0.5 & 0.198 & 0 & 0.51 \\ 0.802 & 0.5 & 0 & 0.718 \\ 1 & 1 & 0.5 & 1 \\ 0.49 & 0.282 & 0 & 0.5 \end{matrix})$

The priority vector is obtained as follows: $w = (0.184, 0.252, 0.375, 0.189)$

Next, we obtain the candidates’ comparison matrixes under each attribute. The comparison results are shown as Tables 8 –11:

Table 8

Comparison of C₁

	A ₁	A ₂	A ₃
A ₁	{{s₀} , [1, 1]}	{less than {s_-1} , [0.7, 0.8]}	{{s_-1} , [0.7, 0.78]}
A ₂	{great than {s₁} , [0.7, 0.8]}	{{s₀} , [1, 1]}	{{s₁} , [0.68, 0.76]}
A ₃	{{s₁} , [0.7, 0.78]}	{{s_-1} , [0.68, 0.76]}	{{s₀} , [1, 1]}

Table 9

Comparison of C₂

	A ₁	A ₂	A ₃
A ₁	{{s₀} , [1, 1]}	{between {s_-2} and {s_-1} , [0.6, 0.8]}	{less than {s_-1} , [0.7, 0.8]}
A ₂	{between {s₁} and {s₂} , [0.6, 0.8]}	{{s₀} , [1, 1]}	{{s_-1} , [0.7, 0.76]}
A ₃	{great than {s₁} , [0.7, 0.8]}	{{s₁} , [0.7, 0.76]}	{{s₀} , [1, 1]}

Table 10

Comparison of C₃

	A ₁	A ₂	A ₃
A ₁	{{s₀} , [1, 1]}	{{s₀} , [1, 1]}	{{s₂} , [0.66, 0.82]}
A ₂	{{s₀} , [1, 1]}	{{s₀} , [1, 1]}	{{s₂} , [0.72, 0.86]}
A ₃	{{s_-2} , [0.66, 0.82]}	{{s_-2} , [0.72, 0.86]}	{{s₀} , [1, 1]}

Table 11

Comparison of C₄

	A ₁	A ₂	A ₃
A ₁	{{s₀} , [1, 1]}	{{s₁} , [0.6, 0.8]}	{{s_-1} , [0.68, 0.78]}
A ₂	{{s_-1} , [0.6, 0.8]}	{{s₀} , [1, 1]}	{{s₀} , [1, 1]}
A ₃	{{s₁} , [0.68, 0.78]}	{{s₀} , [1, 1]}	{{s₀} , [1, 1]}

The results of the G - GCI of matrixes are G - GCI₁ = 0.035, G - GCI₂ = 0.075, G - GCI₃ = 0.036, G - GCI₄ = 0.021

Then we get the possibility degree matrixes:

$D_{1} = (\begin{matrix} 0.5 & 0 & 0 \\ 1 & 0.5 & 1 \\ 1 & 0 & 0.5 \end{matrix})$ $D_{2} = (\begin{matrix} 0.5 & 0 & 0 \\ 1 & 0.5 & 0 \\ 1 & 1 & 0.5 \end{matrix})$ $D_{3} = (\begin{matrix} 0.5 & 0.48 & 1 \\ 0.52 & 0.5 & 1 \\ 0 & 0 & 0.5 \end{matrix})$ $D_{4} = (\begin{matrix} 0.5 & 0 & 0 \\ 1 & 0.5 & 0 \\ 1 & 1 & 0.5 \end{matrix})$

The vectors is obtained: $w_{1} = (0.17, 0.50, 0.33) w_{2} = (0.17, 0.33, 0.50)$ $w_{3} = (0.41, 0.42, 0.17) w_{4} = (0.17, 0.33, 0.50)$ So we get the decision making matrix: $B = (\begin{matrix} 0.17 & 0.17 & 0.41 & 0.17 \\ 0.50 & 0.33 & 0.42 & 0.33 \\ 0.33 & 0.50 & 0.17 & 0.50 \end{matrix})$

Then we get the result: $Q = B w^{T} = (0.26, 0.395, 0.345)^{T}$ Therefore, A₂ is the candidate to be selected.

7 Comparative analysis

To illustrate the necessity to consider reliability and the advantage of using probability intervals to represent reliability, we compare IPHFLVs with HFLTS and PHFLVs (probability hesitant fuzzy linguistic variables) in this section.

When we do not consider the reliability of the evaluation, IPHFLVs degenerate into HFLTS. In this case, the comparison matrix about the four attributes is displayed as Table 12:

Table 12
Comparison of attributes of HFLTS

C ₁ C ₂ C ₃ C ₄

C ₁ {s₀} {s₁, s₂} {s_-2} {s₂}

C ₂ {s_-2, s_-1} {s₀} {s_-2, s_-1} {s₁, s₂}

C ₃ {s₂} {s₁, s₂} {s₀} {s₂}

C ₄ {s_-2} {s_-2, s_-1} {s_-2} {s₀}

	C ₁	C ₂	C ₃	C ₄
C ₁	{s₀}	{s₁, s₂}	{s_-2}	{s₂}
C ₂	{s_-2, s_-1}	{s₀}	{s_-2, s_-1}	{s₁, s₂}
C ₃	{s₂}	{s₁, s₂}	{s₀}	{s₂}
C ₄	{s_-2}	{s_-2, s_-1}	{s_-2}	{s₀}

Using the HFLTS to evaluate the information means that we believe that the experts’ evaluation is completely reliable, that is to say p_l = p_h = 1. Thus Table 12 can be rewritten as Table 13:

Table 13

Comparison of attributes of HFLTS with reliability

	C ₁	C ₂	C ₃	C ₄
C ₁	((s₀) ; [1, 1])	((s₁, s₂) ; [1, 1])	((s_-2) ; [1, 1])	((s₂) ; [1, 1])
C ₂	((s_-2, s_-1) ; [1, 1])	((s₀) ; [1, 1])	((s_-2, s_-1) ; [1, 1])	((s₁, s₂) ; [1, 1])
C ₃	((s₂) ; [1, 1])	((s₁, s₂) ; [1, 1])	((s₀) ; [1, 1])	((s₂) ; [1, 1])
C ₄	((s_-2) ; [1, 1])	((s_-2, s_-1) ; [1, 1])	((s_-2) ; [1, 1])	((s₀) ; [1, 1])

We can get the G - GCI = 0.2066, satisfying the consistency.

Then we calculate the weight vector and obtain the result:w = (0.189, 0.247, 0.375, 0.189). Similarly we get the weight vectors of A₁, A₂ and A₃: $w_{1} = (0.17, 0.5, 0.33) w_{2} = (0.17, 0.33, 0.5)$ $w_{3} = (0.42, 0.42, 0.16) w_{4} = (0.17, 0.33, 0.5)$ The result is (0.264, 0.396, 0.340) ^T. Therefore the final talent selected is A₂.

When the probability interval degenerates to a number, the IPHFLVs would turn to the PHFLVs(probability hesitant fuzzy linguistic variables). Here, we set the mean of the endpoints of the probability interval as the probability number. For example in Table 14:

Table 14

Comparison of attributes of PHFLVs

	C ₁	C ₂	C ₃	C ₄
C ₁	((s₀) ;1)	((s₁, s₂) ;0.855)	((s_-2) ;0.64)	((s₂) ;0.88)
C ₂	((s_-2, s_-1) ;0.855)	((s₀) ;1)	((s_-2, s_-1) ;0.635)	((s₁, s₂) ;0.885)
C ₃	((s₂) ;0.64)	((s₁, s₂) ;0.635)	((s₀) ;1)	((s₂) ;0.66)
C ₄	((s_-2) ;0.88)	((s_-2, s_-1) ;0.885)	((s_-2) ;0.66)	((s₀) ;1)

We can get the G - GCI > 0.3526, thus we modify the matrix to satisfy the consistency. Then we get the matrix: $R = (\begin{matrix} ((s_{0.1253}); 1) & ((s_{1.1291}, s_{2}); 0.855) & ((s_{- 2}); 0.64) & ((s_{2}); 0.88) \\ ((s_{- 1.2220}, s_{- 2}); 0.855) & ((s_{- 0.1563}); 1) & ((s_{- 1.1268}, s_{- 2}); 0.635) & ((s_{0.8468}, s_{2}); 0.885) \\ ((s_{2}); 0.64) & ((s_{1.2731}, s_{2}); 0.635) & ((s_{0.7042}); 1) & ((s_{2}); 0.66) \\ ((s_{- 2}); 0.88) & ((s_{- 1.3986}, s_{- 2}); 0.885) & ((s_{- 2}); 0.66) & ((s_{- 0.5330}); 1) \end{matrix})$

Then we calculate the weight vector and obtain the result: w = (0.191, 0.244, 0.375, 0.190)

Similarly we get the weight vectors of A₁, A₂ and A₃: $w_{1} = (0.17, 0.50, 0.33) w_{2} = (0.17, 0.33, 0.50)$ $w_{3} = (0.41, 0.42, 0.17) w_{4} = (0.17, 0.33, 0.50)$ The result is (0.26, 0.396, 0.344) ^T. Therefore the final talent selected is A₂.

It can be seen from Fig. 5 that A₂ is the candidate to be selected in three different environments. However the scoring function values in the three different environments are different. Therefore, the different ways of evaluation of information have a negligible effect on talent selection. IPHFLV, which considers the reliability as an interval, is the most effective tool for decision making in the actual talent selection process since IPHFLVS not only considers the reliability of the evaluation, but also uses the probability interval to contain more information than using a separate probability value.

Fig. 5

Comparison results of weights of candidates.

8 Conclusion

In this paper we propose the interval probability hesitant fuzzy linguistic variables (IPHFLVs) to express the phenomenon that experts’ evaluation of information is not completely reliable in the decision making process and introduce the operation rules and comparison method. Especially, the possibility degree which is used to obtain the priority vector based on the probability theory is presented. Aiming to deal with the consistency of comparison matrix in complex linguistic environment, we introduce the general geometric consistency index (G-GCI) and the iterative algorithm. Based on the above research, we extend AHP to the interval probability hesitant fuzzy linguistic environment and proposed the interval probability hesitant fuzzy linguistic AHP (IPHFL-AHP).

Despite the contributions mentioned above, there are several limitations and future work to do. Other decision making method such as TOPSIS and VIKOR can be used to deal with different practical problems under interval probability hesitant fuzzy linguistic environment. Besides, the best-worst method has a good effect in decision making. Associating IPHFLV-AHP with the best-worst method might have great results in the decision making process.

Footnotes

Appendix A

The proof of the Theorem 2.

Proof:

(1) Since $- 1 \leq \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) |} \leq 1$ , we can easily get: $- 1 \leq \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |} \leq 1$ Therefore, $0 \leq 0.5 (1 + \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |}) \leq 1$ . Then 0 ≤ P (V₁ ≥ V₂) ≤1.

(2) If V₁ = V₂, then α_i¹ = α_i² (α_i¹ and α_i² are respectively the ith linguistic subscript of $H_{S}^{1}$ and $H_{S}^{2}$ ), $p_{l}^{1} = p_{l}^{2}$ and $p_{h}^{1} = p_{h}^{2}$ . Then α_i (V₁) = α_i (V₂). Thus $\begin{matrix} P (V_{1} \geq V_{2}) = 0.5 \cdot (1 + \\ \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |}) = 0.5 \end{matrix}$ (3) $\begin{matrix} P (V_{1} \geq V_{2}) + P (V_{2} \geq V_{1}) = 0.5 \end{matrix}$ $\begin{matrix} (1 + \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |}) \\ + 0.5 (1 + \frac{\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{2}) - a_{i} (V_{1}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{2}) - a_{i} (V_{1}) | + | a (V_{1} \cap V_{2}) |}) \end{matrix}$ $\begin{matrix} = 0.5 + \frac{0.5 \sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |} + 0.5 \end{matrix}$ $\begin{matrix} + \frac{0.5 \sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{2}) - a_{i} (V_{1}))}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |} = 1 + \end{matrix}$ $\begin{matrix} \frac{0.5 [(\sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))) + \sum_{i = 1}^{# H_{S}^{1}} (a_{i} (V_{1}) - a_{i} (V_{2}))]}{\sum_{i = 1}^{# H_{S}^{1}} | a_{i} (V_{1}) - a_{i} (V_{2}) | + | a (V_{1} \cap V_{2}) |} \\ = 1 \end{matrix}$

Appendix B

The proof of the Lemma 3.

Proof: We set $e_{ij} = log φ (b_{ij}) - log \frac{w_{i}}{w_{j}}$ , then $\prod_{j = 1}^{n} e_{ij} = (\prod_{i, j = 1}^{n} φ (b_{ij}))^{1 / n}$ , ς_ij = log e_ij². The matrix E = (e_ij).

The vectors v will be approached to what is obtained by geometric mean if the inconsistency is low. Let us take v_i = 1 + x_i with ∑_ix_i = 0. Then we can verify Ev = λ_maxv. $(\begin{matrix} 1 & e_{12} & \dots & e_{1 n} \\ e_{21} & 1 & \dots & e_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ e_{n 1} & e_{n 2} & \dots & 1 \end{matrix}) (\begin{matrix} 1 + x_{1} \\ 1 + x_{2} \\ ⋮ \\ 1 + x_{n} \end{matrix}) = λ_{max} (\begin{matrix} 1 + x_{1} \\ 1 + x_{2} \\ ⋮ \\ 1 + x_{n} \end{matrix})$ therefore $\sum_{j = 1}^{n} e_{ij} (1 + x_{j}) = λ_{max} (1 + x_{i})$ .

Sum on i, then $\sum_{i, j = 1}^{n} e_{ij} (1 + x_{j}) = λ_{max} \sum_{i = 1}^{n} (1 + x_{i}) = n λ_{max}$ Since ς_ij = log e_ij², then e^ς_ij/2 = e_ij. So we can obtain that $\begin{matrix} n λ_{max} = \sum_{i, j} (1 + \frac{1}{2} ς_{ij} + o (ς_{ij})) (1 + x_{j}) \\ = \sum_{i, j} (1 + \frac{1}{2} ς_{ij} + x_{j} + \frac{1}{2} ς_{ij} x_{j}) + o (ς_{ij}) \\ = n^{2} + \frac{1}{2} \sum_{i, j} ς_{ij} + o (ς_{ij}) \end{matrix}$ then $λ_{max} = n + \frac{1}{2 n} \sum_{i, j} ς_{ij} + o (ς_{ij})$ $CI = \frac{λ_{max} - n}{n - 1} = \frac{1}{2 n (n - 1)} \sum_{i, j} ς_{ij} + o (ς_{ij})$ therefore we have $G - GCI = \frac{2 n}{n - 2} CI + o (ς)$

Appendix C

The proof of the Lemma 4.

Proof:

Suppose λ is a parameter. Then $\begin{matrix} λ E (b_{ij}) - E (λ b_{ij}) \\ = \frac{λ \sum_{- t \leq i \leq t} f (s_{i})}{# H_{S}} + \frac{λ β (p_{l} + p_{h})}{2} - \frac{\sum_{- t \leq i \leq t} (1 - {(1 - f (s_{i}))}^{λ})}{# H_{S}} - \frac{β (p_{l} + p_{h})}{2} \\ = \frac{λ \sum_{- t \leq i \leq t} (\frac{1}{2} + \frac{i}{2 t}) - # H_{S} + \sum_{- t \leq i \leq t} {(1 - \frac{1}{2} - \frac{i}{2 t})}^{λ}}{# H_{S}} + (λ - 1) \frac{β (p_{l} + p_{h})}{2} \\ = \frac{\frac{λ}{2} # H_{S} + \frac{λ}{2 t} \sum_{- t \leq i \leq t} i + \sum_{- t \leq i \leq t} \frac{1}{2^{λ}} {(1 - \frac{i}{t})}^{λ}}{# H_{S}} + (λ - 1) \frac{β (p_{l} + p_{h})}{2} - 1 \\ > \frac{λ}{2} + \frac{\frac{λ}{2 t} (- t # H_{S})}{# H_{S}} + \frac{\sum_{- t \leq i \leq t} \frac{1}{2^{λ}} \cdot 2^{λ}}{# H_{S}} + (λ - 1) \frac{β (p_{l} + p_{h})}{2} - 1 \\ = \frac{λ}{2} - \frac{λ}{2} + 1 + (λ - 1) \frac{β (p_{l} + p_{h})}{2} - 1 \\ = (λ - 1) \frac{β (p_{l} + p_{h})}{2} \\ > 0 \end{matrix}$ here $g (i) = \frac{λ}{2 t} i + (\frac{1}{2} - \frac{i}{2 t})^{λ}$ , we can obtain: $g^{'} (i) = \frac{λ}{2 t} - \frac{λ}{2 t} (\frac{1}{2} - \frac{i}{2 t})^{λ} = \frac{λ}{2 t} (1 - (\frac{1}{2} - \frac{i}{2 t})^{λ})$ since λ > 0, - t ≤ i ≤ t, we can acquire g′ (i) >0. Then g (i) > g (- t).

Thus λE (b_ij) > E (λb_ij). Let $(\frac{w_{i}}{w_{j}})^{1 - ɛ} (E (b_{ij}))^{ɛ} = λ$ , then $\begin{matrix} E ((\frac{w_{i}}{w_{j}})^{1 - ɛ} (E (b_{ij}))^{ɛ} \cdot b_{ij}) < (\frac{w_{i}}{w_{j}})^{1 - ɛ} \\ (E (b_{ij}))^{ɛ} \cdot E (b_{ij}) \end{matrix}$ $\begin{matrix} E ({b_{ij}}^{*}) \cdot \frac{{w_{j}}^{*}}{{w_{i}}^{*}} < (\frac{w_{i}}{w_{j}})^{1 - ɛ} (E (b_{ij}))^{ɛ} \\ \cdot E (b_{ij}) \frac{w_{j}}{w_{i}} = (\frac{w_{j}}{w_{i}})^{ɛ} (E (b_{ij}))^{ɛ} \end{matrix}$ $\begin{matrix} log (E ({b_{ij}}^{*}) \cdot \frac{{w_{j}}^{*}}{{w_{i}}^{*}}) < log (\frac{w_{j}}{w_{i}})^{ɛ} (E (b_{ij}))^{ɛ} \\ = ɛ log (E (b_{ij}) \cdot \frac{w_{j}}{w_{i}}) < log (E (b_{ij}) \cdot \frac{w_{j}}{w_{i}}) \end{matrix}$ namely, G - GCI^* ≤ G - GCI.

Acknowledgments

This work was supported by the Chongqing Social Science Planning Project (No.2018YB SH085), Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (No.YJG183074) and the Chongqing Research and Innovation Project of Graduate Students (No.CYS2019254).

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Interval probability hesitant fuzzy linguistic analytic hierarchy process and its application in talent selection

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy linguistic term set

3.1 Interval probability hesitant fuzzy linguistic variable

5.1 The interval probability hesitant fuzzy linguistic comparison matrix

5.2 The consistency of IPHFLCM

Table 12 Comparison of attributes of HFLTS C 1 C 2 C 3 C 4 C 1 {s0} {s1, s2} {s-2} {s2} C 2 {s-2, s-1} {s0} {s-2, s-1} {s1, s2} C 3 {s2} {s1, s2} {s0} {s2} C 4 {s-2} {s-2, s-1} {s-2} {s0}

Footnotes

Appendix A

Appendix B

Appendix C

Acknowledgments

References

Table 12
Comparison of attributes of HFLTS

C ₁ C ₂ C ₃ C ₄

C ₁ {s₀} {s₁, s₂} {s_-2} {s₂}

C ₂ {s_-2, s_-1} {s₀} {s_-2, s_-1} {s₁, s₂}

C ₃ {s₂} {s₁, s₂} {s₀} {s₂}

C ₄ {s_-2} {s_-2, s_-1} {s_-2} {s₀}