A novel cardinal-normalization method for Probabilistic Hesitant Fuzzy Elements with incomplete information

Abstract

Owing to the lack of information, it is more realistic that the sum of probabilities is less than or equal to one in the probabilistic hesitant fuzzy elements (P-HFEs). Probabilistic-normalization method and cardinal-normalization method are common processing methods for the P-HFEs with incomplete information. However, the existed probabilistic-normalization method of sharing the remaining probabilities will lose information and change the information integrity of the P-HFEs. The first existed cardinal-normalization method of adding maximum or minimum membership degree with probability zero are influenced by the subjectivity of the decision makers. And the second existed cardinal-normalization method named as reconciliation method only applicable to the P-HFEs with complete information. Aiming at solving those shortcomings, we propose a possibility degree method based on a novel cardinal-normalization method for the sake of comparing the P-HFEs in pairs. In the process of comparison, the information integrity remains unchanged. Then, we propose a multi-criteria decision making (MCDM) problem, where the attribute weight is determined by entropy measures of the integration results. Finally, an application case in green logistics area is given for the sake of illustrating the efficiency of the proposed method, where the evaluation values are given in the P-HFEs form with incomplete information. Numerical and theoretical results show that a MCDM problem based on the proposed cardinal-normalization method and possibility degree method have a wide range of application.

Keywords

Probabilistic hesitant fuzzy element possibility degree method entropy measures reconciliation method the identical membership method

1 Introduction

Owing to the existence of unknown factor and the depth and breadth of knowledge involved in the decision-making process are gradually improved, accurate digital model could not meet the needs of decision analysis. Therefore, in1965, Zadeh [1] proposed fuzzy sets (FSs), which allowed experts to use fuzzy numbers to describe the initial evaluation value. Many scholars extended and improved the FSs in practical decision-making processes widely, such as intuitionistic fuzzy set [2, 3], the dual FSs [4], type-n FSs [5], Pythagorean fuzzy set (PFS) [6], generalized orthopair fuzzy sets [7] etc.

The researchers found that experts might be hesitant between several evaluation values, that is, the inconsistency of experts’ preferences. However, the hesitant fuzzy set (HFS) which allows experts to take all the possible values as membership degrees in the evaluation proposed by Torra [8] solved this problem very well. Xia and Xu [9] used a set of values in [0,1] to define the hesitant fuzzy element (HFE). For instance, an investment expert is invited to evaluate the risk level of a project. Due to the lack of relevant information, the expert’s judgment on the risk level of the project is hesitant between 0.6 and 0.7, then the risk evaluation information of the project can be expressed by an HFE {0.6,0.7}. Since, many scholars have done a lot of works for the development of the HFS theory [10–15].

However, in most of the research on the HFSs, all the membership degrees in the evaluation provided by the decision-makers have equal importance or weight. It causes the serious loss of information and is inconsistent with reality. For the above-mentioned example, the experts think that 0.6 is more likely to happen than 0.7, but fuzzy sets cannot express this preference. For the sake of distinguishing the weights or preferences, some methods have been proposed [16, 17]. Based on these researches, Xu and Zhou [18] proposed probabilistic hesitant fuzzy set(P-HFS) and probabilistic hesitant fuzzy element (P-HFE), and construct the score function, deviation function, comparison laws and the basic operations of the P-HFEs. As the example of risk level evaluation mentioned above, we could get a P-HFE {0.6 (0.8), 0.7 (0.2)}, where 0.8 and 0.2 represent the occurrence probability of membership degree 0.6 and 0.7 respectively.

Compared with HFSs, the P-HFSs contain more uncertain information, which not only fully consider different membership degrees, but also gives the probability of occurrence of each membership degree. Thus, more and more scholars have been attracted to research it. Li and Wang [19] extended the preference ranking organization method and qualitative flexible multiple criteria method for enrichment evaluations II to the P-HFSs. Zhang et al. [20] proposed the concept of P-HFEs with the continuous form. Bashir et al. [21] defined the probabilistic hesitant fuzzy preference relation (PHFPR), and studied the consistency of PHFPR and consensus in the probabilistic hesitant fuzzy environment. Furthermore, in order to achieve consistency of PHFPRs, they developed many novel algorithms. Zhu and Xu [22] proposed the preference relation of the P-HFSs. Furthermore, a consensus index and a stochastic method were developed for the sake of measuring the consensus degrees of the PHFPRs. Li and Wang [23] discussed the shortcomings of the operations in P-HFSs which had been proposed in other literature. Then they developed the probabilistic hesitant fuzzy prioritized weighted average (PHFPWA) and probabilistic hesitant fuzzy prioritized weighted geometric (PHFPWG) operators based upon the idea of prioritized aggregation operator. Li et al. [24] proposed a dominance degree of the P-HFEs and the best worst method (BWM) in MCDM problem with probabilistic hesitant fuzzy complete information.

However, most of the existing researches on the P-HFEs are based on complete information [18, 24]. Obviously, due to the lack of information, the P-HFEs with complete information are not in accordance with reality [25]. Although we can realize the existence of incomplete information, it is difficult to study it deeply. Therefore, it is necessary to normalize the P-HFEs with incomplete information to those with complete information before performing research in depth. Probabilistic-normalization method and cardinal-normalization method are common processing methods to the P-HFEs. However, the existed probabilistic-normalization method proposed by Zhang et al. [20] of sharing the remaining probabilities will lose information and change the information integrity of the P-HFEs. The first existed cardinal-normalization method proposed by Zhang et al. [20] of adding maximum or minimum membership degree with probability zero are influenced by the subjectivity of the decision makers. Another cardinal-normalization method named as the reconciliation method proposed by Li et al. [26] and Fang et al. [27] is a probability splitting algorithm based on the P-HFES with identical information integrity. Aiming at solving those shortcomings about the existed normalization methods, we propose a possibility degree method based on a novel cardinal-normalization method. The main novelties of our study can be summarized as follows:

For the sake of comparing the P-HFEs in pairs, we propose a novel cardinal-normalization method and possibility degree method, where the normalization process keeps the sum of probabilities unchanged. That is, this novel cardinal-normalization method maintains the integrity of the information.

Based on the possibility degree method, we develop a MCDM problem, where the attribute weights are determined by entropy measures of the integration results.

We organize this paper in the following parts: Several basic concepts, entropy measures and aggregation operations of P-HFEs are reviewed in Section 2. In Section 3, the classical probabilistic- normalization method and cardinal-normalization method are reviews, and their drawbacks are described. In Section 4, a possibility degree method based on a novel cardinal-normalization method for the P-HFEs are proposed. In Section 5, we propose and give the steps of the MCDM problem based on entropy measures and the possibility degree method of the P-HEFs. Then in section 6, an application case in green logistics area is given for the sake of illustrating the efficiency of the proposed method, where the evaluation values are given in the P-HFEs form with incomplete information. At last, the paper ends with conclusions in Section 7.

2 Preliminaries

2.1 The concept of P-HFS and P-HFE

Definition 1 [18, 20]. A probabilistic hesitant fuzzy set (P-HFS) on a fixed set X can be expressed by a mathematical symbol: $H_{p} = {〈 x, h (p_{x}) 〉 | x \in X} .$

Where h (p_x) is the probabilistic hesitant fuzzy element (P-HFE), which can be described in simplified form: $h (p) = {γ_{k} (p_{k}) | k = 1, 2, \cdot \cdot \cdot, # h (p), \sum_{k = 1}^{# h (p)} p_{k} ⩽ 1}$

Where # h (p) is the number of all different membership degrees in h (p). γ_k (p_k) is a term of the P-HFE with the possible membership degree γ_k ∈ [0, 1], the corresponding probability p_k ∈ [0, 1]. If all probabilities are equal, then the P-HFSs reduce to HFSs.

Denote $I (h (p)) = \sum_{k = 1}^{# h (p)} p_{k}$ as the information integrity of the P-HFE. The essence of information integrity is to measure the integrity of information contained in the P-HFE. Note that if $I (h (p)) = \sum_{k = 1}^{# h (p)} p_{k} = 1$ , then we have the complete information of probabilistic distribution of all membership degrees in the P-HFE. If $I (h (p)) = \sum_{k = 1}^{# h (p)} p_{k} < 1$ , then we do not have the complete information of probabilistic distribution of all membership degrees owing to partial ignorance. Especially, if $I (h (p)) = \sum_{k = 1}^{# h (p)} p_{k} = 0$ it means completely ignorance. Generally, the closer the sum of probabilities is to one, the more complete the evaluation information is.

2.2 Entropy measures of the P-HFEs

Entropy weight method is a common method to determine the attribute weights. We will use the fuzzy entropy, hesitant entropy and total entropy of the P-HFEs to determine the attribute weight in this paper.

2.2.1 The fuzzy entropy of the P-HFEs

Definition 2 [28, 29]. Let $h (p) = {γ_{k} (p_{k}) | k = 1, 2, \cdot \cdot \cdot, # h (p), \sum_{k = 1}^{# h (p)} p_{k} ⩽ 1}$ be a P-HFE. The fuzzy entropy of h (p) is defined as $E_{FP} (h (p)) = \sum_{k = 1}^{# h (p)} p_{k} E_{F} (γ_{k})$ ,

where E_F is the entropy for the HFE h. E_FP (h (p)) satisfies several properties as follow:

E_FP (h (p)) =0 iff h (p) = {0 (p) , 1 (1 - p)};

E_FP (h (p)) =1 iff h (p) = {0.5 (1)};

Let $h_{i} (p) = {γ_{k}^{i} (p_{k}^{i}) | k = 1, 2, . . ., # h_{i} (p), \sum_{k = 1}^{# h_{i} (p)} p_{k}^{i} ⩽ 1}$ (i = 1, 2) be two ordered P-HFEs. If $γ_{k}^{1} ⩽ γ_{k}^{2} ⩽ 0.5$ or $γ_{k}^{1} ⩾ γ_{k}^{2} ⩾ 0.5$ and $p_{k}^{1} = p_{k}^{2}$ , # h₁ (p) = # h₂ (p), then E_FP (h₁ (p))⩽ E_FP (h₂ (p)) ;

E_FP (h (p)) = E_FP (h^c (p)).

The fuzzy degree of h (p) is determined by the difference between h (p) and the P-HFE h^* (p) = {0.5 (1)}. By changing the corresponding rule of E_F, several fuzzy entropy formulas of h (p) can be obtained as follow: $E_{FP 1} (h (p)) = - \frac{1}{ln 2} \sum_{k = 1}^{# h (p)} p_{k} [γ_{k} ln (γ_{k}) + (1 - γ_{k}) ln (1 - γ_{k})];$ $E_{FP 2} (h (p)) = \frac{1}{\sqrt{e} - 1} \sum_{k = 1}^{# h (p)} p_{k} [γ_{k} e^{1 - γ_{k}} + (1 - γ_{k}) e^{γ_{k}} - 1];$ $\begin{matrix} E_{FP 3} (h (p)) = \frac{1}{(1 - q) ln 2} \sum_{k = 1}^{# h (p)} p_{k} ln [(γ_{k})^{q} + (1 - γ_{k})^{q}], \\ \begin{matrix} \end{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} q > 0, q \neq 1; \end{matrix}$ $E_{FP 4} (h (p)) = 4^{t} \sum_{k = 1}^{# h (p)} p_{k} (γ_{k})^{t} (1 - γ_{k})^{t}, 0 < t ⩽ 1 .$

For the sake of reducing the error caused by different formulas, the above four fuzzy entropy can be weighted into a generalized fuzzy entropy [29]: ${\bar{E}}_{FP} (h (p)) = \sum_{θ = 1}^{4} λ_{θ} E_{FP θ} (h (p)),$ (1) where λ_θ ∈ [0, 1] , θ = 1, . . . , 4 and $\sum_{θ = 1}^{4} λ_{θ} = 1$ .

2.2.2 The hesitant entropy of the P-HFEs

Definition 3 [28, 29]. Let $h (p) = {γ_{k} (p_{k}) | k = 1, 2, \cdot \cdot \cdot, # h (p), \sum_{k = 1}^{# h (p)} p_{k} ⩽ 1}$ be a P-HFE. The hesitant entropy of h (p) is defined as $E_{Hp} (h (p)) = {\begin{matrix} 0, & # h (p) = 1; \\ \sum_{i = 1}^{# h (p) - 1} \sum_{j = i + 1}^{# h (p)} 4 p_{i} p_{j} g (u_{ij}), & # h (p) > 1 . \end{matrix}$ where u_ij = |γ_i - γ_j|, i = 1, . . . , # h (p) -1 ; j = i + 1,. . . , # h (p), and g (x) : [0, 1] → [0, 1] is strictly monotone increasing, that is g (0) =0, and g (1) =1.

The essence of hesitation entropy is to describe the discrete degree of memberships contained in the P-HFEs. By changing the corresponding rule of g (x), several hesitant entropy formulas of h (p) can be obtained as follow: $E_{Hp 1} (h (p)) = \sum_{i = 1}^{# h (p) - 1} \sum_{j = i + 1}^{# h (p)} 4 p_{i} p_{j} \cdot (u_{ij})^{r}, r > 0;$ $E_{Hp 2} (h (p)) = \sum_{i = 1}^{# h (p) - 1} \sum_{j = i + 1}^{# h (p)} 4 p_{i} p_{j} \cdot sin (0.5 π u_{ij});$ $E_{Hp 3} (h (p)) = \sum_{i = 1}^{# h (p) - 1} \sum_{j = i + 1}^{# h (p)} 4 p_{i} p_{j} \cdot (1 - cos (0.5 π u_{ij}));$ $E_{Hp 4} (h (p)) = \sum_{i = 1}^{# h (p) - 1} \sum_{j = i + 1}^{# h (p)} 4 p_{i} p_{j} \cdot (\frac{2 u_{ij}}{u_{ij} + 1}) .$

In order to reduce the error caused by different formulas, the above four hesitant entropy can be weighted into a generalized hesitant entropy [29]: ${\bar{E}}_{HP} (h (p)) = \sum_{θ = 1}^{4} β_{θ} E_{HP θ} (h (p)),$ (2) where β_θ ∈ [0, 1] , θ = 1, . . . , 4 and $\sum_{θ = 1}^{4} β_{θ} = 1$ .

2.2.3 The total entropy of the P-HFEs

Definition 4 [28, 29]. Let $h (p) = {γ_{k} (p_{k}) | k = 1, 2, \cdot \cdot \cdot, # h (p), \sum_{k = 1}^{# h (p)} p_{k} ⩽ 1}$ be a P-HFE. The total entropy of h (p) is defined as $E_{Tp} (h (p)) = z ({\bar{E}}_{Fp} (h (p)), {\bar{E}}_{Hp} (h (p)))$ , where z (x, y) : [0, 1] × [0, 1] → [0, 1] satisfies four properties:

z (x, y) =0 if and only if x = 0, y = 0;

z (0, 1) = z (1, 0) =1;

z (x, y) = z (y, x);

z (x, y) is monotone increasing in respect of x and y.

Total entropy is more comprehensively to investigate the uncertainty of the P-HFEs. By changing the corresponding rule of z (x, y), several total entropy formulas of h (p) can be obtained:

$E_{Tp 1} (h (p)) = max ({\bar{E}}_{Fp} (h (p)), {\bar{E}}_{Hp} (h (p)))$ ;

$E_{Tp 2} (h (p)) = min ({\bar{E}}_{Fp} (h (p)) + {\bar{E}}_{Hp} (h (p)), 1)$ ; $\begin{matrix} E_{Tp 3} (h (p)) = {\bar{E}}_{Fp} (h (p)) + {\bar{E}}_{Hp} (h (p) \\ \begin{matrix} \end{matrix} - {\bar{E}}_{Fp} (h (p)) \times {\bar{E}}_{Hp} (h (p) . \end{matrix}$

For the sake of reducing the error caused by different formulas, the above three total entropy can be weighted into a generalized total entropy [29]: ${\bar{E}}_{TP} (h (p)) = \sum_{θ = 1}^{3} φ_{θ} E_{TP θ} (h (p)),$ (3) where φ_θ ∈ [0, 1] , θ = 1, 2, 3 and $\sum_{θ = 1}^{3} φ_{θ} = 1$ .

2.3 Aggregation operations for the P-HFEs

The probabilistic hesitant fuzzy weighted averaging (P-HFWA) operator and probabilistic hesitant fuzzy weighted geometric (P-HFWG) operator aggregation operators [20] for P-HFEs in decision making process are used commonly. However, with the increase of the number of elements involved in the P-HFEs, the computing processes shows a geometric progression by using the P-HFWA or P-HFWG operators [25]. The integration operator proposed by Li and Wang [19] could be adopted in this situation.

Definition 5 (Li and Wang [19]).

Let E_p ={ 〈 x, h (p_x) 〉 |x = 1, 2, . . . , n } be n P-HFEs, the integrated P-HFE h (p) = {γ_i (p_i) |i = 1, 2, . . . , m} spread out all possible values of the P-HFEs. The probability of value γ_i can be calculated according to the following formula: $P (x = γ_{i}) = \sum_{i = 1}^{n} P (x = h (p_{x})) P (x = γ_{i} | x = h (p_{x}))$ (4)

It states that the probability of hesitant fuzzy (x = γ_i) is a weighted average of the conditional probability of (x = γ_i) given that x = h (p_x) has occurred. According to the calculated structure, we name this aggregation operator as the probabilistic hesitant fuzzy weighted Total Probability (P-HFWTP) operator.

Example 1. Let h₁ (p) and h₂ (p) be two ordered P-HFEs, and h₁ (p) = {0.7 (0.6) , 0.6 (0.4)}, $h_{2} (p) = {0.8 (0.3), 0.7 (0.3), 0.5 (0.3)} .$

Suppose that h₁ (p) and h₂ (p) are integrated with the weight w = (0.4, 0.6) ^T based on the P-HFWTP operator. Then the probability of the membership degree 0.7 in the integrated P-HFE can be calculated as follow: $P (γ = 0.7) = 0.4 \times 0.6 + 0.6 \times 0.3 = 0.42 .$

Thus, the integrated P-HFE is $h (p) = {0.8 (0.18), 0.7 (0.42), 0.6 (0.16), 0.5 (0.18)} .$

3 The classical normalization processes and their drawbacks

When we deal with P-HFEs, two kinds of normalization processes may need to be considered: the probabilistic-normalization and the cardinal- normalization processes.

3.1 The probabilistic-normalization process and its drawbacks

Definition 6. Given an ordered P-HFE

$h (p) = {γ_{k} (p_{k}) | k = 1, 2, \cdot \cdot \cdot, # h (p), \sum_{k = 1}^{# h (p)} p_{k} ⩽ 1}$ , where the terms γ_k (p_k) are lined up based on the value of γ_k, k = 1, 2, ·· · , # h (p) in descending order. Then the generalized normalized P-HFE can be described as $h (\bar{p}) = {γ_{k} ({\bar{p}}_{k}) | k = 1, 2, . . ., # h (p), \sum_{k = 1}^{# h (p)} {\bar{p}}_{k} = 1},$

Where ${\bar{p}}_{k} = p_{k} + ɛ_{k} (1 - \sum_{k = 1}^{# h (p)} p_{k})$ , and parameter ɛ_k ∈ [0, 1] with conditions $\sum_{k = 1}^{# h (p)} ɛ_{k} = 1$ can be determined in advance according to the psychological preference of the decision maker.

Note:

If ɛ₁ = 1, ɛ₂ = ɛ₃ = . . . = ɛ_#h(p) = 0, it means that the probability of unknown information is assigned to the maximum hesitant fuzzy number, which means that the risk attitude of the decision makers is risk preference;

If ɛ₁ = ɛ₂ = . . . = ɛ_#h(p) - 1 = 0, ɛ_#h(p) = 1, it means that the probability of unknown information is assigned to the minimum hesitant fuzzy number, which means that the risk attitude of the decision makers is risk averse;

If $ɛ_{1} = ɛ_{2} = . . . = ɛ_{# h (p)} = p_{k} / \sum_{k = 1}^{# h (p)} p_{k}$ , it represents that the decision maker’s risk attitude is risk neutral proposed by Zhang et al. [20]. In this case, we can get

{\bar{p}}_{k} = p_{k} + ɛ_{k} (1 - \sum_{k = 1}^{# h (p)} p_{k})

\begin{matrix} = p_{k} + \frac{p_{k}}{\sum_{k = 1}^{# h (p)} p_{k}} (1 - \sum_{k = 1}^{# h (p)} p_{k}) \\ = \frac{p_{k}}{\sum_{k = 1}^{# h (p)} p_{k}} . \end{matrix}

Exampl e 2. Let h₁ (p) and h₂ (p) be two ordered P-HFEs, and $h_{1} (p) = {0.4 (0.4), 0.3 (0.2), 0.2 (0.2)},$ $h_{2} (p) = {0.4 (0.5), 0.2 (0.3)} .$

If the risk attitude of the decision makers is risk preference, then $h_{1} (\bar{p}) = {0.4 (0.6), 0.3 (0.2), 0.2 (0.2)},$ $h_{2} (\bar{p}) = {0.4 (0.7), 0.2 (0.3)} .$

If the risk attitude of the decision makers is risk averse, then $h_{1} (\bar{p}) = {0.4 (0.4), 0.3 (0.2), 0.2 (0.4)},$ $h_{2} (\bar{p}) = {0.4 (0.5), 0.2 (0.5)} .$

If the risk attitude of the decision makers is risk neutral, then $h_{1} (\bar{p}) = {0.4 (0.5), 0.3 (0.25), 0.2 (0.25)},$ $h_{2} (\bar{p}) = {0.4 (5 / 8), 0.2 (3 / 8)} .$

Obviously, I (h_i (p)) =0.8, $I ({\bar{h}}_{i} (p)) = 1, (i = 1, 2)$ .

Since the information degree of the P-HFEs will change after normalization, this classical probabilistic normalization method has its two sides:

Advantages: After the generalized normalization, it can avoid the phenomenon that the probability information of the integrated results will continuously decay until approaching zero as the number of elements involved in the calculation gradually increases;

Disadvantages: It is more realistic that the sum of probabilities is less than or equal to one, but the probabilistic-normalization method of sharing the remaining probabilities goes against the original intention. The information integrity of the P-HFEs will be changed from I (h (p)) ⩽1 to $I (\bar{h} (p)) = 1$ .

3.2 Two classical cardinal- normalization processes and their drawbacks

Given two P-HFEs

$h_{i} (p) = {γ_{k}^{i} (p_{k}^{i}) | k = 1, 2, \cdot \cdot \cdot, # h_{i} (p), \sum_{k = 1}^{# h_{i} (p)} p_{k}^{i} ⩽ 1}$ , (i = 1, 2), where the terms $γ_{k}^{i} (p_{k}^{i})$ are lined up based on the value of $γ_{k}^{i}$ in descending order, and # h₁ (p) < # h₂ (p). The P-HFEs with unequal cardinality need to be cardinal-normalized in advance before integrated calculation or distance calculation. There are two classical cardinal-normalization methods can be used.

Method 1. Zhang et al. [20] proposed a method to expand elements according to the subjectivity of decision makers.

If the risk attitude of the decision makers is risk averse, the minimum membership degree in h₁ (p) with the probability 0 will be added into $h_{1}^{*} (p)$ until $# h_{1}^{*} (p) = # h_{2} (p)$ ;

If the risk attitude of the decision makers is risk preference, the maximum membership degree in h₁ (p) with the probability 0 will be added into $h_{1}^{*} (p)$ until $# h_{1}^{*} (p) = # h_{2} (p)$ .

Example 3. Let h₁ (p) and h₂ (p) be two ordered P-HFEs, and $h_{1} (p) = {0.4 (0.7), 0.2 (0.3)}$ $h_{2} (p) = {0.4 (0.6), 0.3 (0.2), 0.2 (0.2)} .$

If the risk attitude of the decision makers is risk preference, then $h_{1}^{*} (p) = {0.4 (0.7), 0.4 (0), 0.2 (0.3)},$ $h_{2}^{*} (p) = h_{2} (p) = {0.4 (0.6), 0.3 (0.2), 0.2 (0.2)} .$

If the risk attitude of the decision makers is risk averse, then $h_{1}^{*} (p) = {0.4 (0.7), 0.2 (0.3), 0.2 (0)},$ $h_{2}^{*} (p) = h_{2} (p) = {0.4 (0.6), 0.3 (0.2), 0.2 (0.2)} .$

Generally speaking, the cardinal-normalization Method 1 can consider the risk attitude of decision- makers and apply it to the results. Conversely, if it is not brought into play properly, its advantages are also its disadvantages. That is, the normalized results are influenced by the subjectivity of the decision makers.

Method 2 (Reconciliation): Li et al. [26] and Fang et al. [27] proposed a probability splitting algorithm, named as the reconciliation method. Using this algorithm, the normalized P-HFEs not only have the same cardinality, but also have the same probability distribution. Let’s understand this algorithm through Example 4.

Example 4. Let H be a set made up by four ordered P-HFEs h_i (p) (i = 1, 2, 3, 4): $h_{1} (p) = {0.4 (0.6), 0.3 (0.2), 0.2 (0.2)},$ $h_{2} (p) = {0.6 (0.7), 0.5 (0.3)},$ $h_{3} (p) = {0.9 (0.4), 0.8 (0.4)},$ and h₄ (p) = {0.8 (0.2) , 0.6 (0.3) , 0.4 (0.2) , 0.2 (0.1)}.

Obviously, those four P-HFEs have different information integrity: $I (h_{1} (p)) = I (h_{2} (p)) = 1$ and $I (h_{3} (p)) = I (h_{4} (p)) = 0.8$ .

The reconciliation process of h₁ (p) and h₂ (p) can be shown in Fig. 1. Thus, the reconciled P-HFEs $h_{1}^{*} (p)$ and $h_{2}^{*} (p)$ can be expressed as:

Fig. 1

Reconciliation process of h₁ (p) and h₂ (p).

$h_{1}^{*} (p) = {0.4 (0.6), 0.3 (0.1), 0.3 (0.1), 0.2 (0.2)},$ $h_{2}^{*} (p) = {0.6 (0.6), 0.6 (0.1), 0.5 (0.1), 0.5 (0.2)} .$

Similarly, the reconciled P-HFEs about h₃ (p) and h₄ (p) can be expressed as: $\begin{matrix} h_{3}^{*} (p) = {0 . 9 (0 . 2), 0 . 9 (0 . 2), \\ 0 . 8 (0 . 1), 0 . 8 (0 . 2), 0 . 8 (0 . 1)}; \end{matrix}$ $\begin{matrix} h_{4}^{*} (p) = {0 . 8 (0 . 2), 0 . 6 (0 . 2), \\ 0 . 6 (0 . 1), 0 . 4 (0 . 2), 0 . 2 (0 . 1)} . \end{matrix}$

However, Owing to $I (h_{1} (p)) = I (h_{2} (p)) = 1$ $\neq I (h_{3} (p)) = I (h_{4} (p)) = 0.8$ , $h_{1} (p)$ or $h_{2} (p)$ cannot reconciled with $h_{3} (p)$ or h₄ (p).

It can be seen from Example 4 that the reconciliation method is not affected by the psychology of decision makers. However, it requires the P-HFEs to be reconciled have the same information integrity.

4 The possibility degree method based on the novel cardinal-normalization processes for the P-HFEs

4.1 The identical membership method

Considering the disadvantages of the above methods described in Section 3, a novel supplementary element method named as the identical membership method is proposed in this section. This novel method is not only not affected by the subjectivity of decision makers, but also can be used in situations with different information integrity.

Definition 7. (The identical membership method)

Given two ordered P-HFEs

$h_{i} (p) = {γ_{k_{i}}^{i} (p_{k_{i}}^{i}) | k_{i} = 1, 2, \cdot \cdot \cdot, # h_{i} (p), \sum_{k_{i} = 1}^{# h_{i} (p)} p_{k_{i}}^{i} ⩽ 1}$ , (i = 1, 2), where the terms $γ_{k_{i}}^{i} (p_{k_{i}}^{i})$ are lined up based on the value of $γ_{k_{i}}^{i}$ in descending order, and # h₁ (p) ≠ # h₂ (p). The transforming rules of the identical membership method are defined as follow:

If $γ_{k_{1}}^{1} \in h_{1} (p)$ and $γ_{k_{1}}^{1} \notin h_{2} (p)$ , then $γ_{k_{1}}^{1} (0)$ is added into $h_{2}^{*} (p)$ ;

If $γ_{k_{2}}^{2} \in h_{2} (p)$ and $γ_{r_{2}}^{1} \notin h_{1} (p)$ , then $γ_{k_{2}}^{2} (0)$ is added into $h_{1}^{*} (p)$ .

Then the corresponding transformed P-HFEs are $h_{1}^{*} (p) = {γ_{(k_{1})}^{1 *} (p_{(k_{1})}^{1 *}) | γ_{(k_{1})}^{1 *} \in [0, 1], p_{(k_{1})}^{1 *} \in [0, 1], k_{1} = 1, 2, ., # h_{1}^{*} (p), \sum_{k_{1} =1}^{# h_{1}^{*} (p)} p_{(k_{1})}^{1*} \leq 1}$

and $\begin{matrix} h_{2}^{*} (p) = {γ_{(k_{2})}^{2 *} (p_{(k_{2})}^{2 *}) | γ_{(k_{2})}^{2 *} \in [0, 1], p_{(k_{2})}^{2 *} \in [0, 1], \\ \begin{matrix} \end{matrix} k_{2} = 1, 2, . . ., # h_{2}^{*} (p), \sum_{k_{2} = 1}^{# h_{2}^{*} (p)} p_{(k_{2})}^{2 *} ⩽ 1} . \end{matrix}$

Where $γ_{(k_{i})}^{i *} (p_{(k_{i})}^{i *})$ is the membership degree $γ_{(k_{i})}^{i *}$ associated with the probability $p_{(k_{i})}^{i *}$ , and the terms $γ_{(k_{i})}^{i *} (p_{(k_{i})}^{i *})$ are lined up based on the value of $γ_{(k_{i})}^{i *}$ in descending order. $# h_{1}^{*} (p)$ and $# h_{2}^{*} (p)$ represent the number of the membership degree of $h_{1}^{*} (p)$ and $h_{2}^{*} (p)$ respectively, which are satisfied with the relationship of $# h_{1}^{*} (p) = # h_{2}^{*} (p)$ .

Example 5. Let H be a set made up by four ordered P-HFEs h_i (p) (i = 1, 2, 3, 4) described in example 4.

Thus, the P-HFEs transformed by the identical membership method about h₁ (p) and h₂ (p) can be expressed as: $h_{1}^{*} (p) = {0 . 6 (0), 0 . 5 (0), 0 . 4 (0 . 6), 0 . 3 (0 . 2), 0 . 2 (0 . 2)},$ $h_{2}^{*} (p) = {0 . 6 (0 . 7), 0 . 5 (0 . 3), 0 . 4 (0), 0 . 3 (0), 0 . 2 (0)} .$

The P-HFEs transformed by the identical membership method about h₃ (p) and h₄ (p) can be expressed as: $h_{3}^{*} (p) = {0 . 9 (0 . 4), 0 . 8 (0 . 4), 0 . 6 (0), 0 . 4 (0), 0 . 2 (0)},$ $h_{4}^{*} (p) = {0 . 9 (0), 0 . 8 (0 . 2), 0 . 6 (0 . 3), 0 . 4 (0 . 2), 0 . 2 (0 . 1)} .$

The P-HFEs transformed by the identical membership method about h₁ (p) and h₃ (p) can be expressed as:

$h_{1}^{*} (p) = {0 . 9 (0), 0 . 8 (0), 0 . 4 (0 . 6), 0 . 3 (0 . 2), 0 . 2 (0 . 2)},$ $h_{3}^{*} (p) = {0 . 9 (0 . 4), 0 . 8 (0 . 4), 0 . 4 (0), 0 . 3 (0), 0 . 2 (0)} .$

And the P-HFEs transformed by the identical membership method about h_i (p) (i = 1, 2, 3, 4) can be expressed as: $h_{1}^{*} (p) = {\begin{matrix} 0.9 (0), 0.8 (0), 0.6 (0), 0.5 (0), \\ 0.4 (0.6), 0.3 (0.2), 0.2 (0.2) \end{matrix}};$ $h_{2}^{*} (p) = {\begin{matrix} 0.9 (0), 0.8 (0), 0.6 (0.7), 0.5 (0.3), \\ 0.4 (0), 0.3 (0), 0.2 (0) \end{matrix}};$ $h_{3}^{*} (p) = {\begin{matrix} 0.9 (0.4), 0.8 (0.4), 0.6 (0), 0.5 (0), \\ 0.4 (0), 0.3 (0), 0.2 (0) \end{matrix}};$ $h_{4}^{*} (p) = {\begin{matrix} 0.9 (0), 0.8 (0.2), 0.6 (0.3), 0.5 (0), \\ 0.4 (0.2), 0.3 (0), 0.2 (0.1) \end{matrix}} .$

Thus, the advantages about the identical membership method can be listed as below:

It applies not only to the case of I (h₁ (p)) = I (h₂ (p)) =1, but also to the case of I (h₃ (p)) = I (h₄ (p)) =0.8. What’s more, h₁ (p) and h₃ (p) with different information integrity can also be normalized by this method;

Obviously, transformed by the identical membership method, the score function, the deviation function and the fuzzy entropy of the P-HFEs remain unchanged.

That is s (h (p)) = s (h^* (p)),

d (h (p)) = d (h^* (p)),

E_Fp (h (p)) = E_Fp (h^* (p)),

E_Hp (h (p)) = E_Hp (h^* (p))

and E_Tp (h (p)) = E_Tp (h^* (p));

This novel cardinal-normalization method is not affected by the psychology of decision makers;

The transformed result $h^{*} (p)$ is unique regardless of the order of membership degree $γ_{k}^{i}$ in h (p) are in ascending order, descending order or disorder;

Transformed by this novel method, the information degree of h (p) remains unchanged, that is I (h (p)) = I (h^* (p));

It is not only applicable to pairwise normalization between h₁ (p) and h₂ (p), but also applicable to the normalization between h_i (p) (i = 1, 2, 3, 4).

4.2 The possibility degree method for the P-HFEs

Yu et al. [30] proposed the possibility degree method for probabilistic linguistic term set (PLTS), which is a new research branch to express expert’s preferences using linguistic terms associated with probability. Motivated by Yu et al. [30], we propose the possibility degree method for P-HFEs.

4.2.1 The step of pairwise comparisons

Let H be a set made up by m P-HFEs h_i (p) (i = 1, 2, . . . , m). In order to rank them, we need to compare them in pairs to get the possibility degree p_ij = P (h_i (p) ⩾ h_j (p)), (i, j = 1, 2, . . . , m).

Taking the comparison of h₁ (p) and h₂ (p) as an example, the details for getting the possibility degree p₁₂ = P (h₁ (p) ⩾ h₂ (p)) are given as follow:

Step 1. Transform two P-HFEs h₁ (p) and h₂ (p) into $h_{1}^{*} (p)$ and $h_{2}^{*} (p)$ by the novel cardinal- normalization method shown as below: $h_{1}^{*} (p) = {γ_{(k_{1})}^{1 *} (p_{(k_{1})}^{1 *}) | γ_{(k_{1})}^{1 *} \in [0, 1], p_{(k_{1})}^{1 *} \in [0, 1], k_{1} = 1, 2, \dots, # h_{1}^{*} (p), \sum_{k_{1} =1}^{# h_{1}^{*} (p)} p_{(k_{1})}^{1*} \leq 1}$

Where $γ_{(k_{1})}^{1 *} (p_{(k_{1})}^{1 *})$ (or $γ_{(k_{2})}^{2 *} (p_{(k_{2})}^{2 *})$ ) is lined up in descending order based on the values of $γ_{(k_{1})}^{1 *}$ (or $γ_{(k_{2})}^{2 *}$ ).

Step 2. Let $γ_{(1)}^{1 *} = max_{k_{1}} {γ_{(k_{1})}^{1 *}} = max_{k_{2}} {γ_{(k_{2})}^{2 *}} = γ_{(1)}^{2 *}$ , then the comparison rule to compare all $p_{(k_{1})}^{1 *}$ with $p_{(k_{2})}^{2 *}$ is defined as below:

If $p_{(1)}^{1 *} > p_{(1)}^{2 *}$ , then numbers 1 and 1 are be added into the denominator and the numerator respectively. When comparing other probability values, use the following method, where $(k_{i}) = (2), (3) . . ., (# h_{i}^{*} (p)), (i = 1, 2) :$

If $p_{(k_{1})}^{1 *} > p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 0 to the numerator;

If $p_{(k_{1})}^{1 *} = p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 0.5 to the numerator;

If $p_{(k_{1})}^{1 *} < p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 1 to the numerator.

If $p_{(1)}^{1 *} = p_{(1)}^{2 *}$ , then numbers 1 and 0.5 are be added into the denominator and the numerator respectively. When comparing other probability values, use the following method, where $(k_{i}) = (2), (3) . . ., (# h_{i}^{*} (p)), (i = 1, 2)$ :

① If $p_{(k_{1})}^{1 *} > p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 0 to the numerator;

② If $p_{(k_{1})}^{1 *} = p_{(k_{2})}^{2 *}$ , then we add 0.5 to the denominator and 0.5 to the numerator;

③ If $p_{(k_{1})}^{1 *} < p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 1 to the numerator.

If $p_{(1)}^{1 *} < p_{(1)}^{2 *}$ , then numbers 1 and 0 are be added into the denominator and the numerator respectively. When comparing other probability values, use the following method, where $(k_{i}) = (2), (3) . . ., (# h_{i}^{*} (p)), (i = 1, 2)$ :

If $p_{(k_{1})}^{1 *} > p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 0 to the numerator;

If $p_{(k_{1})}^{1 *} = p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and 0.5 to the numerator;

If $p_{(k_{1})}^{1 *} < p_{(k_{2})}^{2 *}$ , then we add 1 to the denominator and1 to the numerator.

The comparison rule given above means that

As for the membership degrees $γ_{(1)}^{1 *}$ and $γ_{(1)}^{2 *}$ , the greater its probability value, the better;

As for the membership degrees $γ_{(k_{1})}^{1 *}$ and $γ_{(k_{2})}^{2 *}$ the smaller its probability value, the better, where $(k_{i}) = (2), (3) . . ., (# h_{i}^{*} (p)), (i = 1, 2)$ . □

Example 6. Let h₁ (p) and h₂ (p) be two ordered P-HFEs, and $h_{1} (p) = {0.7 (0.6), 06 (0.4)},$ $h_{2} (p) = {0.8 (0.3), 0.7 (0.3), 0.5 (0.3)} .$

Solution:

Step 1. h₁ (p) and h₂ (p) are transformed into $h_{1}^{*} (p)$ and $h_{2}^{*} (p)$ , respectively;

0.6 ∈ h₁ (p) and 0.6 ∉ h₂ (p), we add 0.6 (0) into $h_{2}^{*} (p)$ ;

0.8 ∈ h₂ (p) and 0.8 ∉ h₁ (p), we add 0.8 (0) into $h_{1}^{*} (p)$ ;

0.5 ∈ h₂ (p) and 0.5 ∉ h₁ (p), we add 0.5 (0) into $h_{1}^{*} (p)$ .

Then h₁ (p) and h₂ (p) can be transformed into $h_{1}^{*} (p) = {0.8 (0), 0.7 (0.6), 0.6 (0.4), 0.5 (0)},$ $h_{2}^{*} (p) = {0.8 (0.3), 0.7 (0.3), 0.6 (0), 0.5 (0.3)} .$

Step 2. According to $γ_{(1)}^{1 *} = max_{k_{1}} {γ_{(k_{1})}^{1 *}} = max_{k_{2}} {γ_{(k_{2})}^{2 *}} = γ_{(1)}^{2 *}$ , then $γ_{(1)}^{1 *} = max {0.8, 0.7, 0.6, 0.5} = 0.8 = γ_{(1)}^{2 *}$ , and $p_{(1)}^{1 *} = 0 < p_{(1)}^{2 *} = 0.3$ ,

Then we can get $p_{12} = P (h_{1} (p) ⩾ h_{2} (p)) = \frac{0 + 0 + 0 + 1}{1 + 1 + 1 + 1} = \frac{1}{4} .$

4.2.2 The definition and some basic properties of the possibility degree for the P-HFEs

Definition 8.

Let $# (p_{(k_{1})}^{1 *} > p_{(k_{2})}^{2 *}) = n_{1}$ represent the number of $p_{(k_{1})}^{1 *} > p_{(k_{2})}^{2 *}$ , $# (p_{(k_{1})}^{1 *} < p_{(k_{2})}^{2 *}) = n_{2}$ represent the number of $p_{(k_{1})}^{1 *} < p_{(k_{2})}^{2 *}$ , and $# (p_{(k_{1})}^{1 *} = p_{(k_{2})}^{2 *}) = n_{3}$ represent the number of $p_{(k_{1})}^{1 *} = p_{(k_{2})}^{2 *}$ , where $(k_{1}) = (2), . . ., (# h_{1}^{*} (p))$ , $(k_{2}) = (2), . . .$ , $(# h_{2}^{*} (p))$ , and $# h_{1}^{*} (p) = # h_{2}^{*} (p) = 1 + n_{1} + n_{2} + n_{3}$ . Then the possibility degree P (h₁ (p) ⩾ h₂ (p)) can be described as: $P (h_{1} (p) ⩾ h_{2} (p)) = {\begin{matrix} \frac{1 + n_{2} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}, & if p_{(1)}^{1 *} > p_{(1)}^{2 *}, \\ \frac{0.5 + n_{2} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}, & if p_{(1)}^{1 *} = p_{(1)}^{2 *}, \\ \frac{n_{2} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}, & if p_{(1)}^{1 *} < p_{(1)}^{2 *} . \end{matrix}$ (5)

Similarly, the possibility degree P (h₂ (p) ⩾ h₁ (p)) can be described as: $P (h_{2} (p) ⩾ h_{1} (p)) = {\begin{matrix} \frac{n_{1} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}, & if p_{(1)}^{1 *} > p_{(1)}^{2 *}, \\ \frac{0.5 + n_{1} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}, & if p_{(1)}^{1 *} = p_{(1)}^{2 *}, \\ \frac{1 + n_{1} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}, & if p_{(1)}^{1 *} < p_{(1)}^{2 *} . \end{matrix}$ (6)

Definition 9.

If P (h₁ (p) ⩾ h₂ (p)) ⩾ P (h₂ (p) ⩾ h₁ (p)), then h₁ (p) is superior to h₂ (p) with the degree of P (h₁ (p) ⩾ h₂ (p)), which can be denoted by h₁ (p) > ^{P (h₁ (p) ⩾h₂ (p))}h₂ (p);

If P (h₁ (p) ⩾ h₂ (p)) =1, then h₁ (p) is absolutely superior to h₂ (p);

If P (h₁ (p) ⩾ h₂ (p)) =0 . 5, then h₁ (p) is indifferent with h₂ (p), denoted by h₁ (p) ∼ h₂ (p).

Next, some basic properties of P (h₂ (p) ⩾ h₁ (p)) are given.

Property 1. (Boundness). $0 ⩽ P (h_{1} (p) ⩾ h_{2} (p)) ⩽ 1 .$

Proof: Obviously, it satisfies boundness.

Property 2. (Complementary). $P (h_{1} (p) ⩾ h_{2} (p)) + P (h_{2} (p) ⩾ h_{1} (p)) = 1 .$

Proof: Take $p_{(1)}^{1 *} > p_{(1)}^{2 *}$ for example, $P (h_{1} (p) ⩾ h_{2} (p)) + P (h_{2} (p) ⩾ h_{1} (p))$ $= \frac{1 + n_{2} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}} + \frac{n_{1} + 0.5 n_{3}}{1 + n_{1} + n_{2} + n_{3}}$ $= \frac{1 + n_{1} + n_{2} + n_{3}}{1 + n_{1} + n_{2} + n_{3}} = 1 .$

The other two cases can be proved in the same way. □

Property 3. If h₁ (p) = h₁ (p)), then P (h₁ (p) ⩾ h₁ (p)) =0.5.

Proof: If h₂ (p) = h₁ (p), then $p_{(1)}^{1 *} = p_{(1)}^{2 *}$ , and n₁ = 0, n₂ = 0, n₃ = # h₁ (p) -1,

We can get $P (h_{1} (p) ⩾ h_{1} (p)) = \frac{0.5 + 0 + 0.5 (# h_{1}^{*} (p) - 1)}{# h_{1}^{*} (p)} = 0.5 .$ □

Other properties of P (h₂ (p) ⩾ h₁ (p)) are not described detailed here, but can be referred to Yu et al. [30].

Now, p₁₂ = P (h₁ (p) ⩾ h₂ (p)) in Example 6 can be calculated using Definition 8 once more.

Example 7. Let h₁ (p) and h₂ (p) be two P-HFEs, and $h_{1} (p) = {0.7 (0.6), 0.6 (0.4)},$ $h_{2} (p) = {0.8 (0.3), 0.7 (0.3), 0.5 (0.3)} .$

After the transformation, $h_{1}^{*} (p) = {0.8 (0), 0.7 (0.6), 0.6 (0.4), 0.5 (0)},$ $h_{2}^{*} (p) = {0.8 (0.3), 0.7 (0.3), 0.6 (0), 0.5 (0.3)} .$

We can get $γ_{(1)}^{1 *} = max {0.8, 0.7, 0.6, 0.5} = 0.8 = γ_{(1)}^{2 *}$ , and $p_{(1)}^{1 *} = 0 < p_{(1)}^{2 *} = 0.3$ .

According to definition 8, we can get $n_{1} = 2, n_{2} = 1, n_{3} = 0,$

then p₁₂ = P (h₁ (p) ⩾ h₂ (p)) $= \frac{1 + 0.5 \times 0}{1 + 2 + 1 + 0} = \frac{1}{4}$ .

4.2.3 The drawbacks of Song et al.’s possibility method for the P-HFEs

Definition 10 [31]. Let h₁ (p) and h₂ (p) be two P-HFEs. The possibility degree of h₁ (p) being not less than h₂ (p) is defined as: $\begin{matrix} p_{12} = P (h_{1} (p) ⩾ h_{2} (p)) \\ = 0.5 \cdot (1 + \frac{(a (h_{1})^{+} - a (h_{2})^{+}) + (a (h_{1})^{-} - a (h_{2})^{-})}{| a (h_{1})^{+} - a (h_{2})^{+} | + | a (h_{1})^{-} - a (h_{2})^{-} | + a (h_{1} \cap h_{2})}) \end{matrix}$

Where a (h₁ ∩ h₂) represents the intersection area between h₁ (p) and h₂ (p).

Now, we use an example to clarify the drawback of this possibility degree method.

Example 8. Let h₁ (p) and h₂ (p) be two P-HFEs, and $h_{1} (p) = {0.3 (0.4), 0.25 (0.2), 0.2 (0.1)},$ $h_{2} (p) = {0.4 (0.3), 0.3 (0.2), 0.2 (0.1)} .$

Through the comparison method proposed by Song et al. [31], we can get $p_{12} = P (h_{1} (p) ⩾ h_{2} (p)) = 0.5 \cdot (1 + \frac{0 + 0}{0 + 0 + 0.9}) = 0.5$

That is h₁ (p) ∼ h₂ (p).

However, through the novel comparison method proposed in this paper, we can get $h_{1}^{*} (p) = {0.4 (0), 0.3 (0.4), 0.25 (0.2), 0.2 (0.1)},$ $h_{2}^{*} (p) = {0.4 (0.3), 0.3 (0.2), 0.25 (0), 0.2 (0.1)},$

and $p_{(1)}^{1 *} = 0 < p_{(1)}^{2 *} = 0.3$ . Then $p_{12} = P (h_{1} (p) ⩾ h_{2} (p)) = \frac{0.5 + 0 + 0.5 \times 1}{1 + 2 + 0 + 1} = 0.25 .$

Thus, we can get h₂ (p) ≻ h₁ (p).

Therefore, the possibility degree method proposed in this paper is better to solve the ranking problem in Example 8 than the possibility degree method proposed by Song et al. [31].

5 MCDM problem based on possibility degree and entropy measures of the P-HFEs

In this section, we consider a MCDM problem which consists of a finite set of alternatives A = (A₁, . . . , A_m), a set of criteria C = (C₁, . . . , C_n) with the weight w = {w₁, . . . , w_n} being completely unknown, and a set of experts D = (d₁, . . . , d_K) with the given weight η = (η₁, . . . , η_K). The experts provide the evaluation values in the form of P-HFEs: $\begin{matrix} h_{ij} (p) = {γ_{k}^{ij} (p_{k}^{ij}) | k = 1, 2, . . ., # h_{ij} (p), \sum_{k = 1}^{# h_{ij} (p)} p_{k}^{ij} ⩽ 1}, \\ (i = 1, . . ., m; j = 1, . . ., n) . \end{matrix}$

5.1 Calculate the attribute weights based on the entropy measures

Step1. Unify the data type.

Because there are two types of attribute values in MCDM problems, namely benefit type and cost type. It is necessary to normalize the evaluation values before integration in order to remove the impact of different attribute types.

For the benefit type attribute, its attribute evaluation value remains unchanged;

For the cost attribute, the evaluation value of the attribute is transformed into the benefit attribute according to the formula:

\begin{matrix} h_{ij} (p) \Rightarrow h_{ij}^{c} (p) \\ = {(1 - γ_{k}^{ij}) (p_{k}^{ij}) | k = 1, 2, . . ., # h_{ij} (p), \sum_{k = 1}^{# h_{ij} (p)} p_{k}^{ij} ⩽ 1}, \\ (i = 1, . . ., m; j = 1, . . ., n) . \end{matrix}

Step2. Given the weight η_k of expert k (k = 1, 2, . . . , K), the P-HFWTP operator is adopted to integrate K experts’ evaluation values.

Step3. Calculate the weight of n attributes.

Step3.1 Set parameters and calculate the fuzzy entropy ${\bar{E}}_{FP} (h_{ij} (p))$ , hesitant entropy ${\bar{E}}_{HP} (h_{ij} (p))$ and total entropy ${\bar{E}}_{TP} (h_{ij} (p))$ of the integrated P-HFEs h_ij (p) (i = 1, . . . , m ; j = 1, . . . , n) according to formula (1), (2) and (3).

Step3.2 The entropy of attribute C_j can be calculated by the formula (7): ${\bar{E}}_{P} (C_{j}) = \frac{1}{m} \sum_{i = 1}^{m} {\bar{E}}_{TP} (h_{ij} (p)) .$ (7)Step3.3 According to the information of entropy theory, the smaller the entropy value is, the more important the corresponding evaluation index is. Thus, we can calculate the weights of the attributes according to the formula (8): $w_{j} = \frac{1 - {\bar{E}}_{P} (C_{j})}{n - \sum_{j = 1}^{n} {\bar{E}}_{P} (C_{j})} .$ (8)

5.2 Ranking alternatives based on the novel cardinal-normalization and possibility degree method

Step 1. Calculate the overall evaluation values of the alternatives with the calculated attribute weight w = {w₁, . . . , w_n} based on the P-HFWTP operator once more.

Step 2. Construct the possibility degree matrix.

Fig. 2

The algorithm flowchart of the proposed MCDM problem.

By pairwise comparisons, the possibility degree p_ij = P (h_i (p) ⩾ h_j (p)), (i, j = 1, 2, . . . , m .) can be obtained by the Eq. (5). Thus, the fuzzy complementary judgment matrix P = (p_ij) _m×m could be described as: $P = (\begin{matrix} p_{11} & p_{12} & \dots & p_{1 m} \\ p_{21} & p_{22} & \dots & p_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ p_{m 1} & p_{m 2} & \dots & p_{mm} \end{matrix})$ $= (\begin{matrix} 0.5 & p_{12} & \dots & p_{1 m} \\ 1 - p_{12} & 0.5 & \dots & p_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1 - p_{1 m} & 1 - p_{2 m} & \dots & 0.5 \end{matrix})$ (9)

According to the properties, the elements on the main diagonal of the possibility degree matrix are all 0.5.

Step 3. Formula (10) given by Xu [32] can be adopted to derive priorities μ_i of alternatives A_i (i = 1, 2, . . . , m): $μ = (μ_{1}, μ_{2}, . . ., μ_{m})^{T} = {(\frac{1}{\sum_{i = 1}^{m} \frac{p_{i 1}}{p_{1 i}}}, \frac{1}{\sum_{i = 1}^{m} \frac{p_{i 2}}{p_{2 i}}} . . ., \frac{1}{\sum_{i = 1}^{m} \frac{p_{im}}{p_{mi}}})}^{T}$ (10) Step 4. Let μ′ = (μ₍₁₎, μ₍₂₎, . . . , μ_(m)) ^T be the rank of μ in descending order, then the ranking order of alternatives is $A_{(1)} ≻ A_{(2)} ≻ \cdot \cdot \cdot ≻ A_{(m)}$ (11)

Then the best alternative is A₍₁₎.

6 An application case in green logistics area

6.1 The description of the example

Green logistics is a form of logistics which is calculated to be environmentally and often socially friendly in addition to economically functional. The connotation of green logistics includes the following five aspects:

Intensive resource. By integrating existing resources and optimizing resource allocation, the resource utilization can be improved and the resource waste can be reduced.

Green transportation. In order to achieve the goal of energy conservation and emission reduction, the transportation routes should be shortened and the vehicle loading rate should be improved scientifically.

Resource storage. On the one hand, for saving transportation costs, the warehouse should be located in a reasonable place; On the other hand, it should make full use of the warehouse scientifically in order to maximize the utilization of the storage area and reduce the storage cost.

Green package. Green packaging can improve the recycling rate of packaging materials, control resource consumption effectively and avoid environmental pollution.

Waste material logistics. Waste material, which have lost their original value in economic activities, can be collected, classified, processed, packaged, handled and then distributed to special treatment sites.

Example 9. There are five logistics companies A_i (i = 1, 2, . . . , 5) competing for the product transportation project of an enterprise. They claim that their logistics services meet the requirements of green logistics. The green logistics system can be divided into the following five attributes: intensive resource (C₁), green transportation (C₂), resource storage (C₃), green package (C₄) and waste material logistics(C₅). Criteria weights w = {w₁, . . . w₅} are completely unknown and independent with each other. Suppose that three experts (d₁, d₂, d₃) with the weight vector (0.4,0.4,0.2)^T are invited to participate in the decision-making analysis to evaluate their logistics systems. Three experts provide their preference evaluations for five alternatives in the form of P-HFEs, which are shown in Tables 1 –5. For necessary comparative analysis, the data in Tables 1-5 are adapted from Zhang et al. [20].

In the following, we will describe the steps of the MCDM problem in details in order to give the consumer the best advice.

6.2 Calculate the attribute weights based on the entropy measures

Step1. Unify the data type.

The attribute evaluation values remain unchanged for the evaluation values of logistics systems are all benefit type in this example.

Step2. Given that the weight vector of three experts is (0.4,0.4,0.2)^T, the P-HFWTP operator is selected to integrate three experts’ evaluation values according to Eq. (4). The integrated results are shown in Table 6.

Table 1
The evaluation information of attribute C₁

D₁ d₂ d₃

A ₁ {0.8(0.25), 0.7(0.5),0.6(0.25)} {0.7(0.5), 0.5(0.5)} {0.8(0.2),0.7(0.5), 0.6(0.3)}

A ₂ {0.6(0.5), 0.5(0.5)} {0.6(0.6),0.5(0.4)} {0.7(0.3), 0.6(0.5)}

A ₃ {0.8(0.3), 0.7(0.5)} {0.7(0.5),0.6(0.3), 0.5(0.2)} {0.6(0.7), 0.5(0.3)}

A ₄ {0.9(0.2),0.85(0.3), 0.8(0.5)} {0.8(0.4), 0.7(0.4)} {0.7(0.3), 0.6(0.4), 0.5(0.3)}

A ₅ {0.75(0.5),0.65(0.5)} {0.8(0.1),0.7(0.4), 0.6(0.5)} {0.8(0.5), 0.7(0.5)}

	D₁	d₂	d₃
A ₁	{0.8(0.25), 0.7(0.5),0.6(0.25)}	{0.7(0.5), 0.5(0.5)}	{0.8(0.2),0.7(0.5), 0.6(0.3)}
A ₂	{0.6(0.5), 0.5(0.5)}	{0.6(0.6),0.5(0.4)}	{0.7(0.3), 0.6(0.5)}
A ₃	{0.8(0.3), 0.7(0.5)}	{0.7(0.5),0.6(0.3), 0.5(0.2)}	{0.6(0.7), 0.5(0.3)}
A ₄	{0.9(0.2),0.85(0.3), 0.8(0.5)}	{0.8(0.4), 0.7(0.4)}	{0.7(0.3), 0.6(0.4), 0.5(0.3)}
A ₅	{0.75(0.5),0.65(0.5)}	{0.8(0.1),0.7(0.4), 0.6(0.5)}	{0.8(0.5), 0.7(0.5)}

Table 2

The evaluation information of attribute C₂

	d₁	d₂	d₃
A ₁	{0.6(0.4), 0.5(0.6)}	{0.9(0.1),0.8(0.2), 0.7(0.7)}	{0.7(0.4), 0.6(0.4)}
A ₂	{0.6(0.4), 0.5(0.3), 0.4(0.3)}	{0.7(0.3), 0.6(0.7)}	{0.7(0.3), 0.6(0.4),0.5(0.3)}
A ₃	{0.5(0.4), 0.4(0.4), 0.3(0.2)}	{0.8(0.2), 0.7(0.6)}	{0.7(0.5), 0.5(0.5)}
A ₄	{0.8(0.2), 0.7(0.6)}	{0.8(0.4), 0.7(0.1), 0.6(0.5)}	{0.7(0.4), 0.6(0.3), 0.5(0.3)}
A ₅	{0.6(0.5),0.5(0.5)}	{0.7(0.6), 0.6(0.4)}	{0.7(0.4), 0.6(0.6)}

Table 3

The evaluation information of attribute C₃

	d₁	d₂	d₃
A ₁	{0.7(0.2),0.6(0.8)}	{0.6(0.5),0.4(0.3)}	{0.8(0.4),0.7(0.6)}
A ₂	{0.7(0.5), 0.6(0.3), 0.5(0.2)}	{0.8(0.3), 0.7(0.4), 0.6(0.3)}	{0.6(0.7), 0.5(0.3)}
A ₃	{0.6(0.8), 0.5(0.2)}	{0.5(0.6), 0.3(0.4)}	{0.8(0.2), 0.7(0.3),0.6(0.5)}
A ₄	{0.8(0.3), 0.7(0.3), 0.6(0.4)}	{0.9(0.2), 0.8(0.3), 0.7(0.5)}	{0.8(0.6),0.7(0.2)}
A ₅	{0.8(0.7), 0.7(0.3)}	{0.5(0.5),0.4(0.5)}	{0.7(0.4), 0.6(0.4), 0.5(0.2)}

Table 4

The evaluation information of attribute C₄

	d₁	d₂	d₃
A ₁	{0.4(0.6), 0.3(0.4)}	{0.7(0.3), 0.6(0.4), 0.5(0.3)}	{0.75(0.4),0.7(0.2), 0.65(0.4)}
A ₂	{0.7(0.2), 0.6(0.6)}	{0.7(0.5), 0.6(0.5)}	{0.8(0.2), 0.6(0.8)}
A ₃	{0.6(0.6), 0.5(0.4)}	{0.8(0.3), 0.7(0.2), 0.6(0.5)}	{0.8(0.4), 0.7(0.4), 0.6(0.2)}
A ₄	{0.6(0.5), 0.5(0.5)}	{0.6(0.6), 0.5(0.4)}	{0.7(0.6), 0.6(0.4)}
A ₅	{0.7(0.4), 0.6(0.3), 0.5(0.3)}	{0.8(0.4), 0.7(0.4)}	{0.8(0.2), 0.7(0.8)}

Table 5

The evaluation information of attribute C₅

	d₁	d₂	d₃
A ₁	{0.8(0.4),0.7(0.3),0.6(0.3)}	{0.8(0.6),0.7(0.4)}	{0.7(0.8), 0.6(0.2)}
A ₂	{0.8(0.5), 0.7(0.5)}	{0.8(0.1), 0.6(0.2), 0.5(0.7)}	{0.65(0.3), 0.6(0.3),0.55(0.4)}
A ₃	{0.7(0.7), 0.6(0.3)}	{0.7(0.5),0.6(0.5)}	{0.8(0.4), 0.7(0.4)}
A ₄	{0.8(0.3), 0.7(0.7)}	{0.8(0.6),0.7(0.2)}	{0.8(0.4), 0.7(0.6)}
A ₅	{0.9(0.4), 0.8(0.2),0.7(0.2)}	{0.8(0.4), 0.7(0.4),0.6(0.2)}	{0.7(0.7),0.6(0.3)}

Table 6

Integrated evaluation information the P-HFWTP operator

	C₁	C₂	C₃	C₄	C₅
A ₁	{0.8(0.14), 0.7(0.5), 0.6(0.16), 0.5(0.2)}	{0.9(0.04), 0.8(0.08), 0.7(0.36), 0.6(0.24) 0.5(0.24)}	{0.8(0.08), 0.7(0.2), 0.6(0.52), 0.4(0.12)}	{0.75(0.08), 0.7(0.16), 0.65(0.08), 0.6(0.16), 0.5(0.12), 0.4(0.24), 0.3(0.16)}	{0.8(0.4), 0.7(0.44), 0.6(0.16)}
A ₂	{0.7(0.06), 0.6(0.54), 0.5(0.36)}	{0.7(0.18),0.6(0.52), 0.5(0.18),0.4(0.12)}	{0.8(0.12), 0.7(0.36), 0.6(0.38), 0.5(0.14)}	{0.8(0.04), 0.7(0.28), 0.6(0.6)}	{0.8(0.24), 0.7(0.2), 0.65(0.06), 0.6(0.14), 0.55(0.08), 0.5(0.28)}
A ₃	{0.8(0.12), 0.7(0.4), 0.6(0.26), 0.5(0.14)}	{0.8(0.08), 0.7(0.34), 0.5(0.26), 0.4(0.16), 0.3(0.08)}	{0.8(0.04), 0.7(0.06), 0.6(0.42),0.5(0.32), 0.3(0.16)}	{0.8(0.2), 0.7(0.16), 0.6(0.48), 0.5(0.16)}	{0.8(0.08), 0.7(0.56), 0.6(0.32)}
A ₄	{0.9(0.08), 0.85(0.12), 0.8(0.36), 0.7(0.22), 0.6(0.08),0.5(0.06)}	{0.8(0.24), 0.7(0.36), 0.6(0.26), 0.5(0.06)}	{0.9(0.08), 0.8(0.36), 0.7(0.36), 0.6(0.16)}	{0.7(0.12), 0.6(0.52), 0.5(0.36)}	{0.8(0.44), 0.7(0.48)}
A ₅	{0.8(0.14), 0.75(0.2), 0.7(0.26), 0.65(0.2), 0.6(0.2)}	{0.7(0.32), 0.6(0.48), 0.5(0.2)}	{0.8(0.28), 0.7(0.2), 0.6(0.08), 0.5(0.24), 0.4(0.2)}	{0.8(0.2), 0.7(0.48), 0.6(0.12), 0.5(0.12)}	{0.9(0.16), 0.8(0.24), 0.7(0.38), 0.6(0.14)}

Step3. Calculate the weight of five attributes.

Firstly, set parameters as follow: $λ_{1} = λ_{2} = λ_{3} = 0, λ_{4} = 1;$ $β_{1} = 1, β_{2} = β_{3} = β_{4} = 0;$ $φ_{1} = φ_{2} = φ_{3} = \frac{1}{3} .$

Then the fuzzy entropy ${\bar{E}}_{FP} (h_{ij} (p))$ , hesitant entropy ${\bar{E}}_{HP} (h_{ij} (p))$ and total entropy ${\bar{E}}_{TP} (h_{ij} (p))$ of h_ij (p) (i, j = 1, . . . , 5) can be calculated according to the formula (1), (2) and (3). Then according to formula (7) and (8), we can get the attribute weights w = (0.2165, 0.1555, 0.1754, 0.1422, 0.3104) ^T, which are shown in the last column of Table 7.

Table 7

Getting the attribute weights

	C₁	C₂	C₃	C₄	C₅
A₁	$\begin{matrix} {\bar{E}}_{FP} = 0.8632 \\ {\bar{E}}_{HP} = 0.2299 \\ {\bar{E}}_{TP} = 0.9193 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8384 \\ {\bar{E}}_{HP} = 0.2163 \\ {\bar{E}}_{TP} = 0.9039 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8336 \\ {\bar{E}}_{HP} = 0.1754 \\ {\bar{E}}_{TP} = 0.8988 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.9056 \\ {\bar{E}}_{HP} = 0.3443 \\ {\bar{E}}_{TP} = 0.9479 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.7792 \\ {\bar{E}}_{HP} = 0.1498 \\ {\bar{E}}_{TP} = 0.8401 \end{matrix}$
A₂	$\begin{matrix} {\bar{E}}_{FP} = 0.9288 \\ {\bar{E}}_{HP} = 0.1080 \\ {\bar{E}}_{TP} = 0.9746 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.9456 \\ {\bar{E}}_{HP} = 0.1853 \\ {\bar{E}}_{TP} = 0.9671 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8840 \\ {\bar{E}}_{HP} = 0.1902 \\ {\bar{E}}_{TP} = 0.9300 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8368 \\ {\bar{E}}_{HP} = 0.3040 \\ {\bar{E}}_{TP} = 0.9077 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8698 \\ {\bar{E}}_{HP} = 0.2650 \\ {\bar{E}}_{TP} = 0.9247 \end{matrix}$
A₃	$\begin{matrix} {\bar{E}}_{FP} = 0.8024 \\ {\bar{E}}_{HP} = 0.1653 \\ {\bar{E}}_{TP} = 0.8684 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8176 \\ {\bar{E}}_{HP} = 0.2870 \\ {\bar{E}}_{TP} = 0.8959 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.9336 \\ {\bar{E}}_{HP} = 0.2587 \\ {\bar{E}}_{TP} = 0.9615 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8832 \\ {\bar{E}}_{HP} = 0.2099 \\ {\bar{E}}_{TP} = 0.9303 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8288 \\ {\bar{E}}_{HP} = 0.4557 \\ {\bar{E}}_{TP} = 0.9119 \end{matrix}$
A₄	$\begin{matrix} {\bar{E}}_{FP} = 0.6420 \\ {\bar{E}}_{HP} = 0.1930 \\ {\bar{E}}_{TP} = 0.7293 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.7656 \\ {\bar{E}}_{HP} = 0.1627 \\ {\bar{E}}_{TP} = 0.8326 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.7152 \\ {\bar{E}}_{HP} = 0.1709 \\ {\bar{E}}_{TP} = 0.7884 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.9600 \\ {\bar{E}}_{HP} = 0.1344 \\ {\bar{E}}_{TP} = 0.9751 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.6848 \\ {\bar{E}}_{HP} = 0.0211 \\ {\bar{E}}_{TP} = 0.6941 \end{matrix}$
A₅	$\begin{matrix} {\bar{E}}_{FP} = 0.8320 \\ {\bar{E}}_{HP} = 0.1490 \\ {\bar{E}}_{TP} = 0.8900 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.9296 \\ {\bar{E}}_{HP} = 0.1510 \\ {\bar{E}}_{TP} = 0.9566 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.8560 \\ {\bar{E}}_{HP} = 0.3430 \\ {\bar{E}}_{TP} = 0.9205 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.7664 \\ {\bar{E}}_{HP} = 0.1613 \\ {\bar{E}}_{TP} = 0.8327 \end{matrix}$	$\begin{matrix} {\bar{E}}_{FP} = 0.6648 \\ {\bar{E}}_{HP} = 0.1755 \\ {\bar{E}}_{TP} = 0.7429 \end{matrix}$
E_p (c_j)	0.8763	0.9112	0.8998	0.9188	0.8227
w _j	0.2165	0.1555	0.1754	0.1422	0.3104

6.3 Ranking alternatives based on the possibility degree method

Step 1. Using the attribute weights w = (0.2165, 0.1555, 0.1754, 0.1422, 0.3104) ^T, the P-HFEs h_i (p) for five alternatives A_i (i = 1, 2, 3, 4, 5) can be integrated based on the P-HFWTP operator once more. $h_{1} (p) = {\begin{matrix} 0.9 (0.01), 0.8 (0.18), 0.75 (0.01), 0.7 (0.36), \\ 0.65 (0.01), 0.6 (0.24), 0.5 (0.1), 0.4 (0.06), \\ 0.3 (0.02) \end{matrix}},$ $h_{2} (p) = {\begin{matrix} 0.8 (0.1), 0.7 (0.21), 0.65 (0.02), 0.6 (0.39), \\ 0.55 (0.02), 0.5 (0.22), 0.4 (0.02) \end{matrix}},$ $h_{3} (p) = {\begin{matrix} 0.8 (0.1), 0.7 (0.35), 0.6 (0.3), 0.5 (0.15), \\ 0.4 (0.02), 0.3 (0.04) \end{matrix}}$ $h_{4} (p) = {\begin{matrix} 0.9 (0.03), 0.85 (0.03), 0.8 (0.31), \\ 0.7 (0.33), 0.6 (0.16), 0.5 (0.07) \end{matrix}}$ $h_{5} (p) = {\begin{matrix} 0.9 (0.05), 0.8 (0.18), 0.75 (0.04), 0.7 (0.33), \\ 0.65 (0.04), 0.6 (0.19), 0.5 (0.09), 0.4 (0.04) \end{matrix}} .$

Step2. Construct the possibility degree matrix.

According to the properties, the elements on the main diagonal of the possibility degree matrix are all 0.5, and p_ij + p_ji = 1, (i, j = 1, 2, . . . , 5). Thus, we just need to calculate p_ij (j > i, i = 1, 2, . . . , 5). The detailed process of solving p₁₂ is given below.

Firstly, we can transform h₁ (p) and h₂ (p) into $h_{1}^{*} (p)$ and $h_{2}^{*} (p)$ respectively: $h_{1}^{*} (p) = {\begin{matrix} 0.9 (0.01), 0.8 (0.18), 0.75 (0.01), \\ 0.7 (0.36), 0.65 (0.01), 0.6 (0.24), \\ 0.55 (0), 0.5 (0.1), 0.4 (0.06), 0.3 (0.02) \end{matrix}},$ $h_{2}^{*} (p) = {\begin{matrix} 0.9 (0), 0.8 (0.1), 0.75 (0), \\ 0.7 (0.21), 0.65 (0.02), 0.6 (0.39), \\ 0.55 (0.02), 0.5 (0.22), 0.4 (0.02), 0.3 (0) \end{matrix}} .$

According to $γ_{(1)}^{1 *} = max_{k_{1}} {γ_{(k_{1})}^{1 *}} = max_{k_{2}} {γ_{(k_{2})}^{2 *}} = γ_{(1)}^{2 *}$ , we obtain that $γ_{(1)}^{1 *} = γ_{(1)}^{2 *} = 0.9$ , and $p_{(1)}^{1 *} = 0.01 > p_{(1)}^{2 *} = 0$ .

Since n₁ = 5, n₂ = 4, n₃ = 0, according to Eq. (5) we can get $p_{12} = p (h_{1} (p) ⩾ h_{2} (p)) = \frac{1 + 4}{1 + 5 + 4 + 0} = \frac{1}{2} .$

Other elements of the matrix can be solved in the same way.

Then the priority matrix can be shown as: $P = (\begin{matrix} 0.5 & \frac{1}{2} & \frac{4}{9} & \frac{1}{5} & \frac{5}{18} \\ \frac{1}{2} & 0.5 & \frac{3}{8} & \frac{1}{3} & \frac{5}{9} \\ \frac{5}{9} & \frac{5}{8} & 0.5 & \frac{1}{4} & \frac{4}{9} \\ \frac{4}{5} & \frac{2}{3} & \frac{3}{4} & 0.5 & \frac{11}{18} \\ \frac{13}{18} & \frac{4}{9} & \frac{5}{9} & \frac{7}{18} & 0.5 \end{matrix}) .$

Step 3. According to the formula (10),we can get μ = (0.1015, 0.1546, 0.1504, 0.3677, 0.2258) ^T.

Step 4. Depending upon the value of μ, we can get the ranking of alternatives is $A_{4} ≻ A_{5} ≻ A_{2} ≻ A_{3} ≻ A_{1} .$

Then, logistics company A₄ is the best choice for the enterprise to transport their product.

6.4 Further comparative analysis

6.4.1 Sort based on TOPSIS method

TOPSIS method is a classical decision-making sorting method based on double base points. Fang et al. [27] solved a reconciled probabilistic hesitant fuzzy MCDM problem based on the TOPSIS method. Qi [33] applied TOPSIS method to performance evaluation of public charging service quality with probabilistic hesitant fuzzy information. Next, we will verify Example 9 based on the TOPSIS method.

Step1. Calculate the score function matrix for the integrated P-HFEs in Table 6: $\begin{matrix} S = (s (h_{ij} (p)))_{5 \times 5} = {(\sum_{k = 1}^{# h_{ij} (p)} γ_{k}^{ij} \cdot p_{k}^{ij})}_{5 \times 5} \\ = (\begin{matrix} 0.658 & 0.616 & 0.564 & 0.524 & 0.724 \\ 0.546 & 0.576 & 0.646 & 0.588 & 0.639 \\ 0.602 & 0.52 & 0.534 & 0.64 & 0.648 \\ 0.694 & 0.63 & 0.708 & 0.576 & 0.688 \\ 0.693 & 0.612 & 0.612 & 0.628 & 0.686 \end{matrix}) . \end{matrix}$

Step 2. Determine the negative and positive ideal solution according to the score function matrix S. The negative ideal solution is $R^{-} = {h_{j}^{-} (p) = min_{i = 1, . . ., 5} s (h_{ij} (p)), j = 1, . . ., 5}$ $= {h_{21} (p), h_{32} (p), h_{33} (p), h_{14} (p), h_{25} (p)} .$

And the positive ideal solution is $R^{+} = {h_{j}^{+} (p) = max_{i = 1, . . ., 5} s (h_{ij} (p)), j = 1, . . ., 5}$ $= {h_{41} (p), h_{42} (p), h_{43} (p), h_{34} (p), h_{15} (p)} .$

Step 3. Calculate the negative and positive distance matrix.

Step 3.1 Before calculating the distance, the evaluation information under attribute C_j (j = 1,..,5) need to be normalized by the novel method proposed in this paper. For example, under attribute C₁, we can get: $h_{11}^{*} (p) = {\begin{matrix} 0.9 (0), 0.85 (0), 0.8 (0.14), 0.75 (0), \\ 0.7 (0.5), 0.65 (0), 0.6 (0.16), 0.5 (0.2) \end{matrix}},$ $h_{21}^{*} (p) = h_{1}^{- *} (p) = {\begin{matrix} 0.9 (0), 0.85 (0), \\ 0.8 (0), 0.75 (0), \\ 0.7 (0.06), 0.65 (0), \\ 0.6 (0.54), 0.5 (0.36) \end{matrix}},$ $h_{31}^{*} (p) = {\begin{matrix} 0.9 (0), 0.85 (0), 0.8 (0.12), 0.75 (0), \\ 0.7 (0.4), 0.65 (0), 0.6 (0.26), 0.5 (0.14) \end{matrix}},$ $h_{41}^{*} (p) = h_{1}^{+ *} (p) = {\begin{matrix} 0.9 (0.08), 0.85 (0.12), \\ 0.8 (0.36), 0.75 (0), \\ 0.7 (0.22), 0.65 (0), \\ 0.6 (0.08), 0.5 (0.06) \end{matrix}},$ $h_{51}^{*} (p) = {\begin{matrix} 0.9 (0), 0.85 (0), 0.8 (0.14), 0.75 (0.2), \\ 0.7 (0.26), 0.65 (0.2), 0.6 (0.2), 0.5 (0) \end{matrix}} .$

Step 3.2 According to the Hamming distance formula, the distance between $h_{11}^{*} (p)$ and $h_{1}^{- *} (p)$ under attribute C₁ can be obtained as: $\begin{matrix} d (h_{11}^{*} (p), h_{1}^{- *} (p)) \\ = d (h_{11}^{*} (p), h_{21}^{*} (p)) \\ = | 0 - 0 | \times 0.9 + . . . + | 0.2 - 0.36 | \times 0.5 \\ = 0.728 . \end{matrix}$

By using the same method to calculate other distances, the negative and positive distance matrix under attribute C_j (j = 1,2,3,4) can be obtained as: $\begin{matrix} d^{-} = {(d (h_{ij}^{*} (p), h_{j}^{- *} (p)))}_{5 \times 5} \\ = (\begin{matrix} 0.728 & 0.292 & 0.446 & 0 & 0.531 \\ 0 & 0.568 & 0.436 & 0.696 & 0 \\ 0.612 & 0 & 0 & 0.628 & 0.711 \\ 1 & 0.486 & 0.902 & 0.62 & 0.663 \\ 0.916 & 0.484 & 0.662 & 0.664 & 0.493 \end{matrix}) . \end{matrix}$ $\begin{matrix} d^{+} = {(d (h_{ij}^{*} (p), h_{j}^{+ *} (p)))}_{5 \times 5} \\ = (\begin{matrix} 0.664 & 0.266 & 0.672 & 0.628 & 0 \\ 1 & 0.582 & 0.466 & 0.364 & 0.531 \\ 0.64 & 0.486 & 0.902 & 0 & 0.436 \\ 0 & 0 & 0 & 0.312 & 0.156 \\ 0.76 & 0.422 & 0.496 & 0.46 & 0.326 \end{matrix}) . \end{matrix}$

Step 4. The weighted distance between alternative $A_{i}$ and negative ideal solution R^- can be calculated as: $\begin{matrix} D^{-} = (D_{1}^{-}, D_{2}^{-}, D_{3}^{-}, D_{4}^{-}, D_{5}^{-})^{T} \\ = {(\sum_{j = 1}^{5} w_{j} d (h_{ij}^{*}, h_{j}^{- *}), i = 1, 2, . . ., 5 .)}^{T} \\ = (0.446, 0.264, 0.442, 0.744, 0.637)^{T} . \end{matrix}$

And the weighted distance between alternative $A_{i}$ and positive ideal solution R⁺ can be calculated as: $\begin{matrix} D^{+} = (D_{1}^{+}, D_{2}^{+}, D_{3}^{+}, D_{4}^{+}, D_{5}^{+})^{T} \\ = {(\sum_{j = 1}^{5} w_{j} d (h_{ij}^{*}, h_{j}^{+ *}), i = 1, 2, . . ., 5 .)}^{T} \\ = (0.392, 0.605, 0.508, 0.093, 0.484)^{T} . \end{matrix}$

Step 5. Compute out the relative closeness to the ideal solution according the formula: $CI (A_{i}) = \frac{D_{i}^{-}}{D_{i}^{-} + D_{i}^{+}}, i = 1, 2, . . ., 5 .$

Thus, we can get CI (A₁) =0.5321, CI (A₂) =0.3035, CI (A₃) =0.4657, CI (A₄) =0.8891, CI (A₅) =0.5684.

Step 6. The higher the value of CI (A_i) , i = 1, 2, . . . , 5, the better the alternative A_i. The ranking results of five alternatives is A₄ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₂. Thus, A₄ is the optimal alternative.

This verifies that the novel cardinal-normalization method and possibility degree method proposed in this paper has the same selection as the classical TOPSIS method.

6.4.2 Comparison with Zhang et al. [20]

The data in Example 9 are adapted from Zhang et al. [20], where the weight vector of three experts is given as (0.4, 0.4, 0.2)^T directly and the probabilistic hesitant fuzzy weighted averaging (P-HFWA) operator have been used to integrate three experts’ evaluation values. Then the score function matrix for the integrated P-HFEs by the P-HFWA operator has been shown as: $\begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} C_{1} \end{matrix} & C_{2} \end{matrix} \begin{matrix} \begin{matrix} C_{3} \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} C_{4} \end{matrix} & C_{5} \end{matrix} \\ S = \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} (\begin{matrix} 0.67 & 0.66 & 0.62 & 0.55 & 0.73 \\ 0.57 & 0.58 & 0.65 & 0.64 & 0.66 \\ 0.67 & 0.60 & 0.55 & 0.65 & 0.68 \\ 0.78 & 0.70 & 0.75 & 0.58 & 0.75 \\ 0.70 & 0.61 & 0.64 & 0.70 & 0.77 \end{matrix}) . \end{matrix}$

Then base on the average weights of attributes w = (0.2, 0.2, 0.2, 0.2, 0.2) ^T given subjectively, the scores of five alternatives have been calculated as: $\begin{matrix} s ({PHFWA}_{1}) = \\ 0.2 \times (0.67 + 0.66 + 0.62 + 0.55 + 0.73) \\ \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} = 0.646, \end{matrix}$ $s ({PHFWA}_{2}) = 0.62, s ({PHFWA}_{3}) = 0.63,$ $s ({PHFWA}_{4}) = 0.712, s ({PHFWA}_{5}) = 0.684 .$

Thus, the final ranking is A₄ ≻ A₅ ≻ A₁ ≻ A₃ ≻ A₂. Thus, A₄ is the optimal alternative. Therefore, the best alternative derived by Zhang et al. [20] is the same as that derived by the proposed method in this paper. Meanwhile, the ranking results are the same roughly. Even so, there are several reasons to believe that our proposed method has some advantages over the method by Zhang et al. [20] in this case:

In the process of normalization, the new possibility degree method proposed in this paper keeps the information integrity of the P-HFEs unchanged. For example, in step 2 of Section 6.3, $I (h_{1} (p)) = 0.99 = I (h_{1}^{*} (p))$ , and $I (h_{2} (p)) = 0.98 = I (h_{2}^{*} (p))$ . That is, this new method to compare the P-HFEs in pairs does not cause the loss of information.

The weights of attributes w = (0.2165, 0.1555, 0.1754, 0.1422, 0.3104) ^T are determined objectively by using the entropy measure in step 3 of the Section 6.2. Yet the weights of attributes w = (0.2, 0.2, 0.2, 0.2, 0.2) ^T were given subjectively by Zhang et al. [20]. This is also the main reason why the ranking results are slightly different.

As in Example 9, the number of elements involved in the operation is large. Thus, the P-HFWTP operator is used to calculate the evaluation information in our method, which can increase the identification degree of fuzzy elements and decrease the burden of computation. Take the integrated P-HFE in Table 6 for example, using Eq. (4) we can get h₁₁ (p) = { 0.8 (0.14) , 0.7 (0.5) , 0.6 (0.16), 0.5 (0.2)} for A₁C₁. However, when keep two decimal places of the membership degree, the integrated result of h₁₁ (p) based on the P-HFWA by Zhang et al. [20] can be showed as $h_{11} (p) = {\begin{matrix} 0.56 (0.0375), 0.59 (0.0625), 0.61 (0.075), \\ 0.62 (0.025), 0.63 (0.125), 0.64 (0.0375), \\ 0.66 (0.1125), 0.67 (0.0375), 0.68 (0.075), \\ 0.69 (0.0875), 0.7 (0.125), 0.71 (0.025) \end{matrix}} .$

Obviously, as the number of elements involved in the operation is large, using the P-HFWA operator or the P-HFWG operator to integrate the evaluation information will make the computing processes are too complex and tedious. Especially, as the number of elements involved in those two operations increase, the operation times will increase in geometric series correspondingly [25]. Furthermore, the integration results are affected by the number of reserved bits. And the discrimination of fuzzy elements is very small. For instance, membership degrees 0.61 and 0.62, which are not very distinct, have appeared in the integration result h₁₁ (p). But if only one digit after the decimal point is retained, 0.61 and 0.62 will be combined to 0.6.

7 Conclusions, limitations and prospect

In this paper, the main contribution is the proposition of a cardinal-normalization method named as the identical membership method and the possibility degree method for the P-HFEs. The advantages about them can be summarized as below:

The identical membership method is not affected by the psychology of the decision makers;

The identical membership method is not only applicable to the normalization of two P-HFEs, but also to the normalization of multiple P-HFEs;

This identical membership normalization method can be applied between P-HFEs with different information integrity;

Transformed by the identical membership method, the score function, the deviation function and the fuzzy entropy of the P-HFEs remains unchanged;

The normalized result of a P-HFE is unique, and it is not affected by the ranking of membership in the P-HFEs;

Normalized by the identical membership method, the information degree of the P-HFEs remains unchanged;

The possibility degree method has some basic properties, such as Boundness and Complementary.

However, there are some limitations to the model, including:

The occurrence probability in the P-HFEs must be provided explicitly, which could be difficult for some practical applications;

When constructing the probability formula, it is classified according to the probability, which makes the result rough relatively. As the number of alternatives increases, it is easy to have parallel sorting results.

Therefore, in the future research, the improved possibility degree method based on the identical membership method should be considered. Furthermore, granular computing has been applied with Linguistic information successfully [34]. Whether the granular computing method can be applied into the P-HFEs deserves further research.

Funding

This research was funded by the National Natural Science Foundation of China, grant number 12171278.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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