Probability multi-valued neutrosophic ELECTRE method for multi-criteria group decision-making

Abstract

Probability multi-valued neutrosophic sets (PMVNSs) can better describe the incomplete and indeterminate evaluation information, and the ELECTRE method can rank the alternatives in the light of the outranking relations among criteria. To combine their advantages, this paper introduces an extended ELECTRE method to address multi-criteria group decision-making (MCGDM) problems with the information of PMVNSs. Firstly, we introduce the definitions of PMVNSs and the classical ELECTRE method, discuss the ELECTRE-based outranking relations for PMVNSs and analyze some properties of them. Furthermore, the probability multi-valued neutrosophic ELECTRE method is developed to address MCGDM problems based on the proposed distance measure and outranking relations for PMVNSs. Finally, a typical example for logistics outsourcing provider selection is devoted to demonstrate the feasibility of the proposed approach. Moreover, the same example-based comparisons with other existing methods are carried out, the results show our proposed approach outperforms the existing methods in solving the MCGDM problems with PMVNSs.

Keywords

ELECTRE outranking relations probability multi-valued neutrosophic sets MCGDM

1 Introduction

Multi-criteria decision making (MCDM) [14–16 , 18] and multi-criteria group decision-making (MCGDM) [11, 12] are an important tool, and are widely used in all aspects of life, such as education, ecology, energy and logistics, etc. In most cases, the decision information with respect to multiple criteria is obtained from multiple decision-makers (DMs), and it is difficult to be expressed by real numbers accurately due to the uncertainty and complexity of the MCGDM problems. In order to solve these problems, Zadeh [40] presented fuzzy sets (FSs). However, since FSs can only describe single membership degree, it cannot characterize the hesitancy of DMs. Then, Atanassov [1] initiated the idea of intuitionistic FSs (IFSs), which could depict membership (MS), and non-membership (NMS) degrees. Thus, IFSs are more practical and feasible in processing fuzziness than traditional FSs. However, sometimes, the MS and NMS degrees in IFSs may be fuzzy, thus, Atanassov and Gargovit [2] extended IFSs into interval-valued IFSs (IVIFSs) where the MS and NMS degrees are described by interval numbers. In addition, in the face of the situation that DMs are always hesitant to express their preferences to optional objects, Torra and Narukawa [31, 32] introduced a novel conception, named hesitant FSs (HFSs), and then, some investigations on HFSs are developed, such as operational laws, distance measure, entropy, correlation coefficient, and so on [5 , 37].

Although the theories of FSs have been generalized and extended, however, they cannot handle all forms of uncertain and incomplete information in real decision-making process, especially, for inconsistent and indeterminate information. For example, in order to judge the speed of cars about 120 Km/h on the highway, you may believe the true probability of given assertion is 0.5, the false probability is 0.6 and the indeterminate probability is 0.2. This situation is far beyond the scope of IFSs. To deal with this situation, Smarandache [30] presented neutrosophic sets (NSs) which are with the truth-membership function (TF), indeterminacy-membership function (IF) and falsity-membership function (FF). Additionally, IF is independent on TF and FF, which is obviously different from the hesitancy degree 1 - T (x) - F (x) in IFSs. As for aforementioned example, the evaluation result can be denoted by x〈 0.5, 0.2, 0.6 〉 in NSs. However, the value of each element lies in non-standard interval, so NSs are inadequate to tackle practical decision-making problems. Consequently, single-valued NSs (SVNSs) [33] were proposed, in which each value of TFs, IFs and FFs falls in standard intervals.

However, in most circumstance, DMs are hesitant to state their opinions on each membership of SVNSs. For aforementioned example, he /she may consider the TF is 0.7 or 0.6, the IF is 0.2 or 0.3, and the FF is 0.4 or 0.5. This instance cannot be described by SVNSs. Later, Wang and Li [34] introduced multi-valued NSs (MVNSs) and defined the operational laws for MVNSs. In the past few years, there have been a series of extended researches on MVNSs. For instance, Liu et al. [17] proposed the MVNWBM and MVNWGBM operators for MVNSs, Peng et al. [24] presented an extended ELECTRE approach to handle MVNSs. In MVNSs, there is a default that several values in each part of TF, IF and FF have the same importance, but it is not realistic. In real decision making, some DMs may prefer to some possible degrees, and distinct possible degrees in each part of MVNSs may have different belief degrees. To overcome this problem, Peng et al. [21] proposed a new concept probability MVNSs (PMVNSs) and applied the QUALIFLEX method to deal with this MCGDM issue. In a probability multi-valued neutrosophic number (PMVNN), each value has its probability which reflects the importance or weight of this evaluation value. Up to now, the researches about PMVNSs are few, there still exists much more research space, especially for the outranking relations for PMVNNs.

Recently, outranking relations have been widespread since the final ranking results are more accuracy and reliable in real decision-making problems, and they are established in term of pair-wise comparisons and can rank the alternatives on the basis of their priority according to the criteria. ELECTRE and PROMETHEE are two typical methods in this category. ELECTRE method is first proposed by Benayoun and Roy [3, 27], then some researches on ELECTRE were developed. For instance, Hatami-Marbini and Tavana [9] extended ELECTRE I to address fuzzy group decision-making problems. Chen [6] introduced an outranking method based on ELECTRE to handle interval type-2 fuzzy information. Peng et al. [22] extended ELECTRE to multi-HFSs to solve MCDM problems. Besides, Chen [7] proposed an extended ELECTRE method on the basis of Chebyshev distance measure to address Pythagorean fuzzy information. Liao et al. [10] provided the algorithm of probabilistic linguistic ELECTRE III methodology, which is applied to tackle nurse-patient relationship problem, Chen et al. [8] proposed a hybrid MCGDM approach based on QFD and ELECTRE III, and then was used to solve the sustainable building material selection problems.

In a word, the motivations of this paper can be given as follows.

PMVNSs can better describe the incomplete and indeterminate evaluation information by adding probability on the basis of MVNSs, however, the researches about PMVNSs are few, it is very necessary to study the outranking relations and other basic theories about MVNSs so that they can be used to solve the real MCGDM problems.

The weights are the important factors which can significantly influence the decision results. However, sometime, weight information for attributes and DMs is completely unknown. How to determine these weights is a vital problem in solving MCGDM problems.

The main strength of ELECTRE is that a very low score under some criterion cannot be compensated by a big score with respect to other criterion. Obviously, it is very consistent with the actual decision-making situations. However, to date, there have been no study on the extended ELECTRE method to deal with PMVNSs.

Based on above motivations, the contributions of this paper are summarized as follows:

To propose the outranking relations for any two PMVNNs, and verify all proper properties;

To establish two reliable models to obtain the attribute weights and DMs’ weights based on minimum deviation principle between expected value and subjective preference value.

To construct probability multi-valued neutrosophic ELECTRE method based on outranking relations;

To verify the usefulness and efficiency of the proposed MCGDM method.

To achieve these goals, the rest is organized as follows: in Sect. 2, we review basic notions of PMVNSs and traditional ELECTRE method. In Sect. 3, we propose some outranking relations for PMVNNs and discuss corresponding desirable properties. In Sect. 4, probability multi-valued neutrosophic ELECTRE method is developed to cope with MCGDM problems. In Sect. 5, based on the same example, we make comparable analysis between our proposed approach and other existing methods. Finally, conclusions are drawn reasonably at the end of this paper.

2 Preliminaries

2.1 PMVNSs

Definition 1 [21]. Let X be a space of points (objects) with generic elements in X denoted by x. A PMVNS A in X can be defined as $A = {〈 x, T_{A} (P_{t} (x)), I_{A} (P_{i} (x)), F_{A} (P_{f} (x)) 〉 | x \in X},$ (1) where $T_{A} (P_{t} (x)) = ⋃_{t_{A}^{j} \in T_{A}, p_{t}^{j} \in P_{t} (x)} {t_{A}^{j} (p_{t}^{j})}$ is a set including all possible TFs $t_{A}^{j} \in T_{A}$ associated with the probability $p_{t}^{j} \in P_{t} (x)$ , $I_{A} (P_{i} (x)) = ⋃_{i_{A}^{k} \in I_{A}, p_{i}^{k} \in P_{i} (x)} {i_{A}^{k} (p_{i}^{k})}$ is a set including all possible IFs $i_{A}^{k} \in I_{A}$ associated with the probability $p_{i}^{k} \in P_{i} (x)$ , and $F_{A} (P_{f} (x)) = ⋃_{f_{A}^{l} \in F_{A}, p_{f}^{l} \in P_{f} (x)} {f_{A}^{l} (p_{f}^{l})}$ is a set including all possible FFs $f_{A}^{l} \in F_{A}$ associated with the probability $p_{f}^{l} \in P_{f} (x)$ , satisfying $0 ⩽ t_{A}^{j}, i_{A}^{k}, f_{A}^{l} ⩽ 1$ , $0 ⩽ t_{A}^{j +} + i_{A}^{k +} + f_{A}^{l +} ⩽ 3$ , $0 ⩽ p_{t}^{j}, p_{i}^{k}, p_{f}^{l} ⩽ 1$ , $\sum_{j = 1}^{# T_{A}} p_{t}^{j} ⩽ 1$ , $\sum_{k = 1}^{# I_{A}} p_{i}^{k} ⩽ 1$ and $\sum_{l = 1}^{# F_{A}} p_{f}^{l} ⩽ 1$ , where $t_{A}^{j +} = sup (T_{A})$ , $i_{A}^{k +} = sup (I_{A})$ and $f_{A}^{l +} = sup (F_{A})$ , # T_A, # I_A and # F_A are the numbers of all elements in T_A (P_t (x)), I_A (P_i (x)) and F_A (P_f (x)), respectively.

The complement set of A is defined as $A^{c} = {< x, F_{A} (P_{f} (x)), I_{A} (P_{i} (x)), T_{A} (P_{t} (x)) > | x \in X}$ (2)

Definition 2 [21]. Given a PMVNN a =< T_A (p_t) , I_A (p_i) , F_A (p_f) > with ∑p_α < 1 (α = t, i, f), then its normalized PMVNN is denoted as $\hat{a} = < T_{A} ({\hat{P}}_{t}), I_{A} ({\hat{P}}_{i}), F_{A} ({\hat{P}}_{f}) >,$ (3) where ${\hat{p}}_{t}^{j} = \frac{p_{t}^{j}}{\sum_{j = 1}^{# T_{A}} p_{t}^{j}} \in {\hat{P}}_{t}$ , ${\hat{p}}_{i}^{k} = \frac{p_{i}^{k}}{\sum_{k = 1}^{# I_{A}} p_{i}^{k}} \in {\hat{P}}_{i}$ and ${\hat{p}}_{f}^{l} = \frac{p_{f}^{l}}{\sum_{l = 1}^{# F_{A}} p_{f}^{l}} \in {\hat{P}}_{f}$ .

In addition, we give the operation laws and distance measure of PMVNNs.

Definition 3 [13]. Let A =< T_A (P_At) , I_A (P_Ai) , F_A (P_Af) > and B =< T_B (P_Bt) , I_B (P_Bi) , F_B (P_Bf) > be two arbitrary normalized PMVNNs, then the operations can be given by: $(1) A + B = 〈 ⋃_{\begin{matrix} t_{A} \in T_{A}, p_{At} \in P_{At}, \\ t_{B} \in T_{B}, p_{Bt} \in P_{Bt} \end{matrix}} {t_{A} + t_{B} - t_{A} t_{B} (p_{At} p_{Bt})}, ⋃_{\begin{matrix} i_{A} \in I_{A}, p_{Ai} \in P_{Ai}, \\ i_{B} \in I_{B}, p_{Bi} \in P_{Bi} \end{matrix}} {i_{A} i_{B} (p_{Ai} p_{Bi})} ⋃_{\begin{matrix} f_{A} \in F_{A}, p_{Af} \in P_{Af}, \\ f_{B} \in F_{B}, p_{Bf} \in P_{Bf} \end{matrix}} {f_{A} f_{B} (p_{Af} p_{Bf})} 〉;$ (4) $(2) A \times B = 〈 ⋃_{\begin{matrix} t_{A} \in T_{A}, p_{At} \in P_{At}, \\ t_{B} \in T_{B}, p_{Bt} \in P_{Bt} \end{matrix}} {t_{A} t_{B} (p_{At} p_{Bt})}, ⋃_{\begin{matrix} i_{A} \in I_{A}, p_{Ai} \in P_{Ai}, \\ i_{B} \in I_{B}, p_{Bi} \in P_{Bi} \end{matrix}} {i_{A} + i_{B} - i_{A} i_{B} (p_{Ai} p_{Bi})}, ⋃_{\begin{matrix} f_{A} \in F_{A}, p_{Af} \in P_{Af}, \\ f_{B} \in F_{B}, p_{Bf} \in P_{Bf} \end{matrix}} {f_{A} + f_{B} - f_{A} f_{B} (p_{Af} p_{Bf})} 〉;$ (5) $(3) λ A = 〈 ⋃_{t_{A} \in T_{A}, p_{At} \in P_{At}} {1 - (1 - t_{A})^{λ} (p_{At})}, ⋃_{i_{A} \in I_{A}, p_{Ai} \in P_{Ai}} {i_{A}^{λ} (p_{Ai})}, ⋃_{f_{A} \in F_{A}, p_{Af} \in P_{Af}} {f_{A}^{λ} (p_{Af})} 〉 (λ > 0);$ (6) $(4) A^{λ} = 〈 ⋃_{t_{A} \in T_{A}, p_{At} \in P_{At}} {t_{A}^{λ} (p_{At})}, ⋃_{i_{A} \in I_{A}, p_{Ai} \in P_{Ai}} {1 - (1 - i_{A})^{λ} (p_{Ai})}, ⋃_{f_{A} \in F_{A}, p_{Af} \in P_{Af}} {1 - (1 - f_{A})^{λ} (p_{Af})} 〉 (λ > 0) .$ (7)

Definition 4 [13]. Let A =< T_A (P_At) , I_A (P_Ai) , F_A (P_Af) > and B =< T_B (P_Bt) , I_B (P_Bi) , F_B (P_Bf) >be two arbitrary PMVNNs, then the generalized distance measure between A and B can be defined as follows: $\begin{matrix} d_{gd} (A, B) & = & \frac{1}{6} [\frac{1}{# T_{A}} \sum_{t_{A} \in T_{A}, p_{At} \in P_{At}} min_{t_{B} \in T_{B}, p_{Bt} \in P_{Bt}} | t_{A} p_{At} - t_{B} p_{Bt} | + \frac{1}{# T_{B}} \sum_{t_{B} \in T_{B}, p_{Bt} \in P_{Bt}} min_{t_{A} \in T_{A}, p_{At} \in P_{At}} | t_{B} p_{Bt} - t_{A} p_{At} | \\ + \frac{1}{# I_{A}} \sum_{i_{A} \in I_{A}, p_{Ai} \in P_{Ai}} min_{i_{B} \in I_{B}, p_{Bi} \in P_{Bi}} | i_{A} p_{Ai} - i_{B} p_{Bi} | + \frac{1}{# I_{B}} \sum_{i_{B} \in I_{B}, p_{Bi} \in P_{Bi}} min_{i_{A} \in I_{A}, p_{Ai} \in P_{Ai}} | i_{B} p_{Bi} - i_{A} p_{Ai} | \\ + \frac{1}{# F_{A}} \sum_{f_{A} \in F_{A}, p_{Af} \in P_{Af}} min_{f_{B} \in F_{B}, p_{Bf} \in P_{Bf}} | f_{A} p_{Af} - f_{B} p_{Bf} | + \frac{1}{# F_{B}} \sum_{f_{B} \in F_{B}, p_{Bf} \in P_{Bf}} min_{f_{A} \in F_{A}, p_{Af} \in P_{Af}} | f_{B} p_{Bf} - f_{A} p_{Af} |] \end{matrix}$ (8)

Where, # T_A, # T_B, # I_A, # I_B, # F_A, # F_B represent the numbers of all elements in T_A, T_B, I_A, I_B, F_A, F_B, respectively.

We also give the comparison method for PMVNNs.

Definition 5 [13]. Let A =< T_A (P_t) , I_A (P_i) , F_A (P_f) > be a normalized PMVNN, then its score functions (A) and accuracy function u (A) can be given by $(1) s (A) = \sum_{t_{A} \in T_{A}, p_{t} \in P_{t}, i_{A} \in I_{A}, p_{i} \in P_{i}, f_{A} \in F_{A}, p_{f} \in P_{f}} \frac{t_{A} - i_{A} - f_{A}}{3} p_{t} \cdot p \cdot_{i} p_{f},$ (9) $(2) u (A) = \sum_{t_{A} \in T_{A}, p_{t} \in P_{t}, i_{A} \in I_{A}, p_{i} \in P_{i}, f_{A} \in F_{A}, p_{f} \in P_{f}} \frac{t_{A} + i_{A} + f_{A}}{3} p_{t} \cdot p \cdot_{i} p_{f},$ (10) where # T_A, # I_A and # F_A denote the numbers of all the elements in T_A, I_A and F_A, respectively.

2.2 The classical ELECTRE method

The ELECTRE method is a kind of outranking methods to rank alternatives, which was first proposed by Benavoun, Roy and Sussman [3], then some extensions on it such as the ELECTRE-I [26], ELECTRE-II [28], ELECTRE-III [19], ELECTRE-IV [29], ELECTRE-IS [39] and ELECTRE-TRI [4] were put forward subsequently. The main steps to deal with MCDM problems using ELECTRE are provided as follows.

Step 1: Construct the prioritized relation based on the comparison of any two alternatives.

If the value of alternative a_iis better than that of alternative a_sunder the criterion j, then alternative a_i is prior to alternativea_s,denoted bya_ij ≻ a_sj; if the value of alternative a_i is equal to that of alternative a_s under the criterion j, then alternative a_i is equivalent to alternative a_s, denoted by a_ij ∼ a_sj; if the value of alternative a_i is worse than that of alternative a_s under the criterion j, then alternative a_i is inferior to alternative a_s, denoted by a_ij ≺ a_sj.

Step 2: Define concordance and discordance sets as follows. $U^{+} (a_{i}, a_{s}) = {j | 1 ⩽ j ⩽ n, \forall c_{j} : a_{i} ≻ a_{s} or a_{i} \sim a_{s}}, U^{-} (a_{i}, a_{s}) = {j | 1 ⩽ j ⩽ n, \forall c_{j} : a_{i} ≺ a_{s}}$

where, U⁺ (a_i, a_s) and U^- (a_i, a_s) represent the concordance set and discordance set, respectively.

Step 3: Construct the relative concordance matrix.

According to the criteria weight vector w, the relative concordance matrix CM can be expressed by $CM = (\begin{matrix} {cm}_{11} & {cm}_{12} & \dots & {cm}_{1 m} \\ {cm}_{21} & {cm}_{22} & \dots & {cm}_{2 m} \\ \dots & \dots & \dots & \dots \\ {cm}_{m 1} & {cm}_{m 2} & \dots & {cm}_{mm} \end{matrix}),$ (11)

where, cm_is = ∑_{j∈U⁺(a_i,a_s)}w_j, i, s = 1, 2, ⋯ , m.

Step 4: Construct the relative discordance matrix.

The relative discordance matrix DM can be expressed by $DM = (\begin{matrix} {dm}_{11} & {dm}_{12} & \dots & {dm}_{1 m} \\ {dm}_{21} & {dm}_{22} & \dots & {dm}_{2 m} \\ \dots & \dots & \dots & \dots \\ {dm}_{m 1} & {dm}_{m 2} & \dots & {dm}_{mm} \end{matrix}),$ (12) where ${dm}_{is} = {\begin{matrix} 0 & i, s = 1, 2, \dots, m and i = s \\ \frac{max_{j \in U^{-} (a_{i}, a_{s})} | w_{j} (r_{ij} - r_{sj}) |}{max_{j \in U^{+} (a_{i}, a_{s}) + j \in U^{-} (a_{i}, a_{s})} | w_{j} (r_{ij} - r_{sj}) |} & i, s = 1, 2, \dots, m and i \neq s \end{matrix}$ . Here, r_ij and r_sj express the evaluation values of alternative a_i and a_s with respect to criterion c_j, respectively.

Step 5: Construct the corrected comprehensive weighted matrix.

The corrected comprehensive weighted matrix EM can be constructed by $EM = (\begin{matrix} {em}_{11} & {em}_{12} & \dots & {em}_{1 m} \\ {em}_{21} & {em}_{22} & \dots & {em}_{2 m} \\ \dots & \dots & \dots & \dots \\ {em}_{m 1} & {em}_{m 2} & \dots & {em}_{mm} \end{matrix}),$ (13)

where em_is = cm_is × (1 - dm_is), i, s = 1, 2, ⋯ , m.

Step 6: Calculate the net prioritized value.

The prioritized value of alternative a_s can be expressed by $γ_{s} = \sum_{i = 1, i \neq s}^{m} {em}_{si} - \sum_{i = 1, i \neq s}^{m} {em}_{is}, s = 1, 2, \dots, m$ (14)

Step 7: Rank the alternatives based on the prioritized values.

3 Outranking relations for PMVNSs

In this section, we define the outranking relations of PMVNSs.

Definition 6. Let A₁ =< T₁ (P_t1) , I₁ (P_i1) , F₁ (P_f1) > and A₂ =< T₂ (P_t2) , I₂ (P_i2) , F₂ (P_f2) > be any two PMVNNs, then the strong dominance relation, weak dominance relation and indifference relation of PMVNNs are proposed as follows.

(1) If $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ , or $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} = \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} = \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ , where # T₁, # T₂, # I₁, # I₂, # F₁, # F₂ denote the numbers of all elements in T₁, T₂, I₁, I₂, F₁, F₂, respectively, then A₁ strongly dominates A₂, denoted by A₁ > _SA₂.

(2) If $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} ⩾ \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ , or $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} ⩾ \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ , where# T₁, # T₂, # I₁, # I₂, # F₁, # F₂ denote the numbers of all elements in T₁, T₂, I₁, I₂, F₁, F₂, respectively, then A₁ weakly dominates A₂, denoted by A₁ > _WA₂.

(3) If $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} = \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} = \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} = \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ , where# T₁, # T₂, # I₁, # I₂, # F₁, # F₂ denote the numbers of all elements in T₁, T₂, I₁, I₂, F₁, F₂, respectively, then A₁ is indifferent to A₂, denoted by A₁ ∼ _IA₂.

(4) If any relations mentioned above don’t exist between A₁ and A₂, then A₁ and A₂ are incomparable, denoted by A₁ ⊥ A₂.

Property 1. Let A₁, A₂ and A₃be three normalized PMVNNs, if A₁ ≻ _SA₂and A₂ ≻ _SA₃, then A₁ ≻ _SA₃.

Proof.

Based on the strong dominance relation proposed in Definition 6, if A₁ ≻ _SA₂, then we can obtain $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ , or $\frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2}$ , $\frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} = \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2}$ and $\frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} = \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2}$ . If A₂ ≻ _SA₃, then we can obtain $\frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} ⩾ \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3}$ , $\frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} < \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3}$ and $\frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} < \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3}$ , or $\frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} > \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3}$ , $\frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} = \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3}$ and $\frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} = \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3}$ .

Thus, we can make following derivation.

Case 1. $\begin{matrix} \begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} \end{matrix} \\ \begin{matrix} \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} ⩾ \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} < \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} < \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix} \end{matrix}} \Rightarrow {\begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix},$

then by Definition 6, A₁ ≻ _SA₃ is achieved.

Case 2. $\begin{matrix} \begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} ⩾ \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} \end{matrix} \\ \begin{matrix} \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} > \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} = \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} = \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix} \end{matrix}} \Rightarrow {\begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix},$

then according to Definition 6, A₁ ≻ _SA₃ is obtained.

Case 3. $\begin{matrix} \begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} = \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} = \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} \end{matrix} \\ \begin{matrix} \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} ⩾ \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} < \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} < \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix} \end{matrix}} \Rightarrow {\begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} < \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} < \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix},$

then according to Definition 6, A₁ ≻ _SA₃is kept.

Case 4. $\begin{matrix} \begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} = \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} = \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} \end{matrix} \\ \begin{matrix} \frac{1}{# T_{2}} \sum_{t_{2} \in T_{2}, p_{t 2} \in P_{t 2}} t_{2} p_{t 2} ⩾ \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{2}} \sum_{i_{2} \in I_{2}, p_{i 2} \in P_{i 2}} i_{2} p_{i 2} = \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{2}} \sum_{f_{2} \in F_{2}, p_{f 2} \in P_{f 2}} f_{2} p_{f 2} = \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix} \end{matrix}} \Rightarrow {\begin{matrix} \frac{1}{# T_{1}} \sum_{t_{1} \in T_{1}, p_{t 1} \in P_{t 1}} t_{1} p_{t 1} > \frac{1}{# T_{3}} \sum_{t_{3} \in T_{3}, p_{t 3} \in P_{t 3}} t_{3} p_{t 3} \\ \frac{1}{# I_{1}} \sum_{i_{1} \in I_{1}, p_{i 1} \in P_{i 1}} i_{1} p_{i 1} = \frac{1}{# I_{3}} \sum_{i_{3} \in I_{3}, p_{i 3} \in P_{i 3}} i_{3} p_{i 3} \\ \frac{1}{# F_{1}} \sum_{f_{1} \in F_{1}, p_{f 1} \in P_{f 1}} f_{1} p_{f 1} = \frac{1}{# F_{3}} \sum_{f_{3} \in F_{3}, p_{f 3} \in P_{f 3}} f_{3} p_{f 3} \end{matrix},$

then by Definition 6, A₁ ≻ _SA₃ is obtained.

To sum up, if A₁ ≻ _SA₂ and A₂ ≻ _SA₃, then A₁ ≻ _SA₃.

Property 2. LetA₁,A₂and A₃ be three normalized PMVNNs, if A₁ ≻ _IA₂ andA₂ ≻ _IA₃, thenA₁ ≻ _IA₃.

Proof.

It is easy to prove this property, and the proof is omitted here.

Property 3. LetE,Fand Gbe three normalized PMVNNs, then the following properties hold.

(1) The strong dominance relations have:

(S1) irreflexivity: for any a PMVNN E, E ≻ _SEis false;

(S2) asymmetry: for any two PMVNNs E and F, E ≻ _SF ⇒ ¬ (F ≻ _SE);

(S3) transitivity: for any three PMVNNs E,F and G, E ≻ _SF, F ≻ _SG ⇒ E ≻ _SG.

(2) The weak dominance relations have:

(W1) irreflexivity: for any a PMVNN E, E ≻ _WE is false;

(W2) asymmetry: for any two PMVNNs E and F,E ≻ _WF ⇒ ¬ (F ≻ _WE);

(W3) non-transitivity: for any three PMVNNsE,F and G, if E ≻ _WF, F ≻ _WG, then E ≻ _WG can’t be derived.

(3) The indifference relations have:

(I1) reflexivity: for any a PMVNN E, E ≻ _IE;

(I2) symmetry: for any two PMVNNs E and F, E ≻ _IF ⇒ F ≻ _IE;

(I3) transitivity: for any three PMVNNs E, F and G, E ≻ _IF, F ≻ _IG ⇒ E ≻ _IG.

In the following, we give an example to prove the non-transitivity of weak dominance relations, other properties are easy to confirm and their proofs are omitted here.

If E = 〈 {0.6 (1)} , {0.4 (0.5) , 0.6 (0.5)} , {0.6 (0.5) , 0.8 (0.5)} 〉, F = 〈 {0.4 (0.5) , 0.5 (0.5)} , {0.5 (1)} , . . {0.5(0.5) , 0.7 (0.5)} 〉, G = 〈{0.3 (0.5) , 0.4 (0.5)} , {0 .4 (1)} , {0.7 (0.5) , 0.9 (0.5)} 〉 be three normalized PMVNNs, and E ≻ _WF, F ≻ _WG, then we can obtain E ≻ _SG in accordance with Definition 6.

Therefore, if E ≻ _WFandF ≻ _WG, then E ≻ _WGcan’t be deduced.

Property 4. Let m and n be any two alternatives, evaluation values of m and n are expressed by PMVNNs, and P = S ∪ W ∪ I mean that m is at least as good as n, then the following four conditions may hold.

mPn and not nPm, that means m ≻ _Sn or m ≻ _Wn;

nPm and not mPn, that means n ≻ _Sm or n ≻ _Wm;

mPn and nPm, that means m ≻ _In;

not mPn and not nPm, that means m ⊥ n.

4 Probability multi-valued neutrosophic ELECTRE method for the MCGDM problem

In this section, we give the description of decision-making problems with PMVNNs and then aggregate multiple decision information into group information. Furthermore, we extend the traditional ELECTRE method to process the PMVNSs and provide corresponding steps.

4.1 Description of decision-making problems

In the MCGDM circumstance, a group of alternatives are denoted by A ={ a₁, a₂, …, a_m }, a set of criteria are denoted by C ={ c₁, c₂, …, c_n }, whose weight vector is w ={ w₁, w₂, …, w_n }, satisfying w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ , and a group of DMs are denoted by D = {d₁, d₂, …, d_q}, whose weight vector isη ={ η₁, η₂, …, η_q }, satisfying η_l > 0 and $\sum_{l = 1}^{q} η_{l} = 1$ . Then, a DM d_l (l = 1, 2, …, q) gives the evaluation information of alternative a_i (i = 1, 2, …, m) with respect to criterion c_j (j = 1, 2, …, n), which is expressed by the PMVNN $z_{ij}^{l} = 〈 T_{ij}^{l} (P_{ijl}^{t}), I_{ij}^{l} (P_{ijl}^{i}), F_{ij}^{l} (P_{ijl}^{f}) 〉$ , where $T_{ij}^{l} (P_{ijl}^{t}) = ⋃_{t_{ij}^{l} \in T_{ij}^{l}, p_{ijl}^{t} \in P_{ijl}^{t}} {t_{ij}^{l} (p_{ijl}^{t})}$ , and $t_{ij}^{l}$ denotes the TF of alternative a_iunder the criterionc_j, and $p_{ijl}^{t}$ signifies corresponding probability; $I_{ij}^{l} (P_{ijl}^{i}) = ⋃_{i_{ij}^{l} \in I_{ij}^{l}, p_{ijl}^{i} \in P_{ijl}^{i}} {i_{ij}^{l} (p_{ijl}^{i})}$ , and $i_{ij}^{l}$ denotes the IF of alternative a_i under the criterion c_j, and $p_{ijl}^{i}$ signifies corresponding probability; $F_{ij}^{l} (P_{ijl}^{f}) = ⋃_{f_{ij}^{l} \in F_{ij}^{l}, p_{ijl}^{f} \in P_{ijl}^{f}} {f_{ij}^{l} (p_{ijl}^{f})}$ , and $f_{ij}^{l}$ denotes the FF of alternative a_i under the criterion c_j, and $p_{ijl}^{f}$ signifies corresponding probability. Finally, the decision matrix from the DM d_l (l = 1, 2, …, q)is constructed as follows. $R^{l} = {(z_{ij}^{l})}_{m \times n} = [\begin{matrix} z_{11}^{l} & z_{12}^{l} & \dots & z_{1 n}^{l} \\ z_{21}^{l} & z_{22}^{l} & \dots & z_{2 n}^{l} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ z_{m 1}^{l} & z_{m 2}^{l} & \dots & z_{mn}^{l} \end{matrix}]$ (15)

Suppose the weights of DMs and criteria are all completely unknown. Based on these decision information, this MCGDM problem aims to select the best alternatives.

4.2 Determination of the criteria weight vector

In practical MCGDM problems, sometimes, the weights of criteria are completely unknown. To determine the weights of criteria in each decision matrix from the DM d_l (l = 1, 2, …, q), we establish a reliable model based on deviations between expected value and subjective preference value. For each part of a PMVNN, there must be at least one preference value, we define the arithmetic mean value of these values as the expected value of this part in the PMVNN. Likewise, after calculating the expected values of other two parts, we can obtain the expected values of this PMVNN.

Thus, the deviation between expected value ${\bar{z}}_{ij}^{l}$ and subjective preference value $z_{ij}^{l}$ given by DMd_l can be denoted by $υ_{ijl} = d_{gd} ({\bar{z}}_{ij}^{l} - z_{ij}^{l})$ , then $υ_{ijl}^{2} = d_{gd} {({\bar{z}}_{ij}^{l} - z_{ij}^{l})}^{2}$ . All the deviations of alternative a_i given by DMd_l can be expressed as $υ_{il}^{2} = \sum_{j = 1}^{n} {(ω_{j}^{l} d_{gd} ({\bar{z}}_{ij}^{l} - z_{ij}^{l}))}^{2}$ . To make sure final results more reasonable and scientific, we construct the following single objective optimization model: $min υ^{l} (ω^{l}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {(ω_{j}^{l} d_{gd} ({\bar{z}}_{ij}^{l} - z_{ij}^{l}))}^{2}$ $s . t . ω_{j}^{l} ⩾ 0, \sum_{j = 1}^{n} ω_{j}^{l} = 1 (M - 1)$

Then Lagrange function is constructed to solve this model by

$\begin{matrix} υ^{l} (ω_{j}^{l}, λ) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {(ω_{j}^{l} d_{gd} ({\bar{z}}_{ij}^{l} - z_{ij}^{l}))}^{2} \\ + 2 λ (\sum_{j = 1}^{n} ω_{j}^{l} - 1), \end{matrix}$ (16) where λis the Lagrange multiplier. Then, the partial derivative can be computed as: ${\begin{matrix} \frac{\partial υ^{l}}{\partial ω_{j}^{l}} = 2 \sum_{i = 1}^{m} ω_{j}^{l} d_{gd} {({\bar{z}}_{ij}^{l} - z_{ij}^{l})}^{2} + 2 λ = 0, j = 1, 2, \dots, n \\ \frac{\partial υ^{l}}{\partial λ} = \sum_{j = 1}^{n} ω_{j}^{l} - 1 = 0 \end{matrix}$ (17)

By solving Equation (17), we can get $\begin{matrix} λ = - \frac{1}{\sum_{j = 1}^{n} 1 / \sum_{i = 1}^{m} d_{gd} {({\bar{z}}_{ij}^{l} - z_{ij}^{l})}^{2}} \\ ω_{j}^{l} = \frac{1}{\sum_{i = 1}^{m} d_{gd} {({\bar{z}}_{ij}^{l} - z_{ij}^{l})}^{2} \sum_{j = 1}^{n} 1 / \sum_{i = 1}^{m} d_{gd} {({\bar{z}}_{ij}^{l} - z_{ij}^{l})}^{2}} j = 11, 2, \dots, n \end{matrix}$ (18)

4.3 Determination of the weight vector for DMs

In practical MCGDM problems, because the DMs have different specialized knowledge and backgrounds, their roles are different in the decision process, so, their weights should be different. In general, the weight information is completely unknown. Inspired by the Xu [35] and Yager [38], an objective weight determination method for DMs is constructed in the following.

Compute the score function of each alternative in each decision matrix.

For each decision matrix, we can calculate the score function of each alternative using Equation (9), these scores are used to compute the deviations in the next step.

Construct the average deviation model.

Based on above scores, the average deviation model applied in PMVNSs is constructed as follows. $\begin{matrix} min AD = \sum_{l = 1}^{q} {(φ_{l} s (Δ h^{l}))}^{2} \\ s . t . \sum_{l = 1}^{q} φ_{l} = 1, φ_{l} ⩾ 0, l = 1, 2, \dots, q (M - 2) \end{matrix}$ where s (Δh^l) = s (h^l) - s (h^l-1), $h^{l} = \sum_{j = 1}^{n} ω_{j}^{l} \sum_{i = 1}^{m} s (a_{ij}^{l}), l = 1, 2, \dots, q$ , $s (a_{ij}^{l})$ denotes the score function of alternative a_i under the criterionc_jin the decision matrix given by DM d_l. For l = 1, let s (Δh¹) = s (Δh²).

Then the Lagrange function is constructed to solve this model by $L (φ_{l}, λ) = \sum_{l = 1}^{q} {(φ_{l} s (Δ h^{l}))}^{2} + 2 λ (\sum_{l = 1}^{q} φ_{l} - 1)$ (19) ,

where λ is the Lagrange multiplier.

Differentiating Equation (19) in accordance with φ_land λ, and setting these partial derivatives equal to zero, then we can get following equations easily. ${\begin{matrix} \frac{\partial L}{\partial φ_{l}} = s^{2} (Δ h^{l}) φ_{l} + λ = 0, l = 1, 2, \dots, q \\ \frac{\partial L}{\partial λ} = \sum_{l = 1}^{q} φ_{l} - 1 = 0 \end{matrix}$ (20)

So we can get $φ_{l} = \frac{- λ}{s^{2} (Δ h^{l})}, l = 1, 2, \dots, q$ (21)

According to $\sum_{l = 1}^{q} φ_{l} = 1$ , we have $\sum_{l = 1}^{q} \frac{- λ}{s^{2} (Δ h^{l})} = 1$ , then we get $λ = - \frac{1}{\sum_{l = 1}^{q} \frac{1}{s^{2} (Δ h^{l})}}$ (22)

Then according to Equation (21), we can deduce the following equation. $φ_{l} = \frac{1}{s^{2} (Δ h^{l}) \sum_{l = 1}^{q} \frac{1}{s^{2} (Δ h^{l})}}, l = 1, 2, \dots, q$ (23)

Based on Equation (23), the weights of DMs can be obtained effectively.

4.4 Probability multi-valued neutrosophic ELECTRE method

In the past few decades, the classical ELECTRE has been used to solve the decision-making problems. Nevertheless, the decision information is mostly real number. Because the PMVNSs can describe the uncertain and inconsistent probability multi-valued neutrosophic information and the hesitance of DMs [13], we try to extend the ELECTRE method to PMVNSs and propose probability multi-valued neutrosophic ELECTRE method for MCGDM problems.

Based on the above analysis, the main procedure of this ELECTRE method can be summarized as follows:

Step 1: Normalize the decision-making information.

It is of great importance to normalize all attributes to the same magnitude scale and type.

Motivation by the normalization method for the IFS proposed by Xu [34], we normalize the decision matrix by considering the criteria type and probability distribution by Equations (1) and (2).

The initial decision matrix is normalized as $R^{l^{'}} = (z_{ij}^{l^{'}})_{m \times n} (l = 1, 2, \dots, q)$ , where $z_{ij}^{l^{'}} = {\begin{matrix} < T_{A} ({\hat{p}}_{t} (x)), I_{A} ({\hat{p}}_{i} (x)), F_{A} ({\hat{p}}_{f} (x)) >, for benefit criteria \\ < F_{A} ({\hat{P}}_{f} (x)), I_{A} ({\hat{p}}_{i} (x)), T_{A} ({\hat{P}}_{t} (x)) >, for cost criteria \end{matrix}$ (24)

Step 2: Determine the weights of criteria in each decision matrix $R^{l^{'}} = (z_{ij}^{l^{'}})_{m \times n} (l = 1, 2, \dots, q)$ .

The criteria weight vector $w^{l} = (w_{1}^{l}, w_{2}^{l}, \dots, w_{n}^{l})$ in each decision matrix from each DM can be achieved by Equation (18).

Step 3: Determine the weights of DMs.

The weight vector φ = (φ₁, φ₂, ⋯ , φ_q)of DMs can be achieved by Equation (23).

Step 4: Compute the collective weight vector for criteria.

After calculating the weights of criteria and DMs, the collective weight vector for criteria w = (w₁, w₂, ⋯ , w_n) in group decision-making matrix can be obtained as $w_{j} = \sum_{l = 1}^{q} w_{j}^{l} φ_{l}, j = 1, 2, \dots, n$ (25)

Step 5: Aggregate evaluation information.

Based on the weights of DMs and criteria obtained in Step 3 and Step 4, we can aggregate all the individual decision-making matrices into a group decision matrix $R^{*} = {(z_{ij}^{*})}_{m \times n}$ (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n) by using basic operational laws in Definition 3.

$\begin{matrix} z_{ij}^{*} = η_{1} z_{ij}^{1} + η_{2} z_{ij}^{2} + \dots + η_{q} z_{ij}^{q} = \\ 〈 ⋃_{t_{k} \in T_{k}, p_{tk} \in P_{tk}} 1 - \prod_{k = 1}^{q} (1 - t_{k})^{η_{k}} (\prod_{k = 1}^{q} p_{tk}), \\ ⋃_{i_{k} \in I_{k}, p_{ik} \in P_{ik},} \prod_{k = 1}^{q} i_{k}^{η_{k}} (\prod_{k = 1}^{q} p_{ik}), \\ ⋃_{f_{k} \in F_{k}, p_{fk} \in P_{fk},} \prod_{k = 1}^{q} f_{k}^{η_{k}} (\prod_{k = 1}^{q} p_{fk}) 〉 \end{matrix}$ (26)

Step 6: Construct outranking relations of any two alternatives.

After comparing any two evaluation values described in PMVNNs, we can define the strong dominance relation, weak dominance relation and indifference relation of any two alternatives under some criterion.

Step 7: Define concordance and discordance sets.

The concordance set should satisfy the constraint $z_{ij}^{*} {Pz}_{sj}^{*}$ , denoted as

$\begin{matrix} U^{+} (z_{ij}^{*}, z_{sj}^{*}) = \\ {j | z_{ij}^{*} {Pz}_{sj}^{*}, 1 ⩽ j ⩽ n}, (i, s = 1, 2, \dots, m), \end{matrix}$ (27) where $z_{ij}^{*} {Pz}_{sj}^{*}$ includes $z_{ij}^{*} ≻^{S} z_{sj}^{*}$ or $z_{ij}^{*} ≻^{W} z_{sj}^{*}$ or $z_{ij}^{*} \sim z_{sj}^{*}$ .

The discordance set is the complementary set of concordance set, denoted as $U^{-} (z_{ij}^{*}, z_{sj}^{*}) = J - U^{+} (z_{ij}^{*}, z_{sj}^{*})$ (28)

Step 8: Construct relative concordance and discordance matrices.

According to the criteria weight vectorw = (w₁, w₂, ⋯ , w_n), the relative concordance matrix CM can be expressed by $CM = (\begin{matrix} {cm}_{11} & {cm}_{12} & \dots & {cm}_{1 m} \\ {cm}_{21} & {cm}_{22} & \dots & {cm}_{2 m} \\ \dots & \dots & \dots & \dots \\ {cm}_{m 1} & {cm}_{m 2} & \dots & {cm}_{mm} \end{matrix}),$ (29)

where ${cm}_{is} = \sum_{j \in U^{+} (z_{ij}^{*}, z_{sj}^{*})} ω_{j}$ , i, s = 1, 2, ⋯ , m.

Meanwhile, the relative discordance matrix DM can be expressed by $DM = (\begin{matrix} {dm}_{11} & {dm}_{12} & \dots & {dm}_{1 m} \\ {dm}_{21} & {dm}_{22} & \dots & {dm}_{2 m} \\ \dots & \dots & \dots & \dots \\ {dm}_{m 1} & {dm}_{m 2} & \dots & {dm}_{mm} \end{matrix}),$ (30) where ${dm}_{is} = {\begin{matrix} 0 & i, s = 1, 2, \dots, m and i = s \\ \frac{max_{j \in U_{is}^{-}} | ω_{j} d_{gd} (z_{ij}^{*} - z_{sj}^{*}) |}{max_{j \in U_{is}^{+} + j \in U_{is}^{-}} | ω_{j} d_{gd} (z_{ij}^{*} - z_{sj}^{*}) |} & i, s = 1, 2, \dots, m and i \neq s \end{matrix}$ .

Step 9: Construct corrected comprehensive weighted matrix.

The corrected comprehensive weighted matrix is expressed as $EM = (\begin{matrix} {em}_{11} & {em}_{12} & \dots & {em}_{1 m} \\ {em}_{21} & {em}_{22} & \dots & {em}_{2 m} \\ \dots & \dots & \dots & \dots \\ {em}_{m 1} & {em}_{m 2} & \dots & {em}_{mm} \end{matrix}),$ (31)

where em_is = cm_is × (1 - dm_is), i, s = 1, 2, ⋯ , m.

Step 10: Calculate the net prioritized value.

The prioritized value of alternativea_scan be expressed by $γ_{s} = \sum_{i = 1, i \neq s}^{m} {em}_{si} - \sum_{i = 1, i \neq s}^{m} {em}_{is}, s = 1, 2, \dots, m$ (32)

Step 11: Rank the alternatives in accordance with net prioritized values.

5 An illustrative Example

In this section, a real example [21] of logistics outsourcing problem is cited to illustrate the proposed approach. Furthermore, a comparison analysis is done to verify the superiority of the proposed method.

ABC Company plans to outsource logistics business to a reliable third-party provider. To achieve this goal, a professional team is constructed from three experts including a production manager, a general manager and a logistics manager, denoted by{d₁, d₂, d₃}. Three managers are all positive to the decision-making process. After several rounds of screening, four logistics providers stand out, denoted by{a₁, a₂, a₃, a₄}. The team chooses four criteria to evaluate these alternatives, that is, c₁: information and equipment systems; c₂: service; c₃: quality; c₄: relationship. The weight information of DMs and criteria is completely unknown. The expert gives the evaluation information on each alternative under each criterion by PMVNN, and all the evaluation information can be collected in three decision-making matrices $R^{l} = {(z_{ij}^{l})}_{4 \times 4} (l = 1, 2, 3)$ , shown in Tables 1 –3.

Table 1
Decision matrix from DM d₁

c ₁ c ₂

a ₁ 〈{ 0.5 (0.4) , 0.7 (0.4) } , { 0.5 (0.6) , 0.7 (0.4) } , { 0.6 (1) } 〉〈{ 0.4 (1) } , { 0.6 (0.4) , 0.7 (0.3) } , { 0.5 (0.3) , 0.7 (0.6) } 〉

a ₂ 〈{ 0.5 (0.4) , 0.6 (0.4) } , { 0.5 (1) } , { 0.4 (0.6) , 0.6 (0.4) } 〉〈{ 0.4 (0.5) , 0.5 (0.3) } , { 0.5 (0.5) , 0.6 (0.4) } , { 0.4 (1) } 〉

a ₃ 〈{ 0.3 (0.4) , 0.5 (0.4) } , { 0.4 (0.4) , 0.5 (0.6) } , { 0.3 (1) } 〉〈{ 0.6 (1) } , { 0.5 (1) } , { 0.4 (0.5) , 0.5 (0.4) } 〉

a ₄ 〈{ 0.4 (0.4) , 0.5 (0.6) } , { 0.4 (1) } , { 0.5 (0.4) , 0.6 (0.2) } 〉〈{ 0.5 (0.6) , 0.7 (0.4) } , { 0.4 (1) } , { 0.3 (0.4) , 0.5 (0.5) } 〉

c ₃ c ₄

a ₁ 〈{ 0.4 (1) } , { 0.3 (1) } , { 0.3 (0.2) , 0.5 (0.5) , 0.6 (0.3) } 〉〈{ 0.3 (0.4) , 0.5 (0.6) } , { 0.7 (1) } , { 0.5 (0.4) , 0.6 (0.6) } 〉

a ₂ 〈{ 0.7 (1) } , { 0.4 (0 .6) , 0.7 (0.3) } , { 0.4 (0 .5) , 0.5 (0.5) } 〉〈{ 0.4 (1) } , { 0.4 (1) } , { 0.4 (0.3) , 0.5 (0.5) , 0.6 (0.2) } 〉

a ₃ 〈{ 0.4 (0.5) , 0.6 (0.5) } , { 0.4 (1) } , { 0.6 (0.4) , 0.8 (0.5) } 〉〈{ 0.4 (1) , 0.5 (1) } , { 0.5 (0.6) } , { 0.3 (0.5) , 0.5 (0.5) } 〉

a ₄ 〈{ 0.6 (0.5) , 0.8 (0.5) } , { 0.4 (0.6) , 0.6 (0.4) } , { 0.5 (1) } 〉〈{ 0.8 (1) } , { 0.4 (0.5) , 0.6 (0.3) , 0.8 (0.2) } , { 0.3 (1) } 〉

	c ₁	c ₂
a ₁	〈{ 0.5 (0.4) , 0.7 (0.4) } , { 0.5 (0.6) , 0.7 (0.4) } , { 0.6 (1) } 〉	〈{ 0.4 (1) } , { 0.6 (0.4) , 0.7 (0.3) } , { 0.5 (0.3) , 0.7 (0.6) } 〉
a ₂	〈{ 0.5 (0.4) , 0.6 (0.4) } , { 0.5 (1) } , { 0.4 (0.6) , 0.6 (0.4) } 〉	〈{ 0.4 (0.5) , 0.5 (0.3) } , { 0.5 (0.5) , 0.6 (0.4) } , { 0.4 (1) } 〉
a ₃	〈{ 0.3 (0.4) , 0.5 (0.4) } , { 0.4 (0.4) , 0.5 (0.6) } , { 0.3 (1) } 〉	〈{ 0.6 (1) } , { 0.5 (1) } , { 0.4 (0.5) , 0.5 (0.4) } 〉
a ₄	〈{ 0.4 (0.4) , 0.5 (0.6) } , { 0.4 (1) } , { 0.5 (0.4) , 0.6 (0.2) } 〉	〈{ 0.5 (0.6) , 0.7 (0.4) } , { 0.4 (1) } , { 0.3 (0.4) , 0.5 (0.5) } 〉
	c ₃	c ₄
a ₁	〈{ 0.4 (1) } , { 0.3 (1) } , { 0.3 (0.2) , 0.5 (0.5) , 0.6 (0.3) } 〉	〈{ 0.3 (0.4) , 0.5 (0.6) } , { 0.7 (1) } , { 0.5 (0.4) , 0.6 (0.6) } 〉
a ₂	〈{ 0.7 (1) } , { 0.4 (0 .6) , 0.7 (0.3) } , { 0.4 (0 .5) , 0.5 (0.5) } 〉	〈{ 0.4 (1) } , { 0.4 (1) } , { 0.4 (0.3) , 0.5 (0.5) , 0.6 (0.2) } 〉
a ₃	〈{ 0.4 (0.5) , 0.6 (0.5) } , { 0.4 (1) } , { 0.6 (0.4) , 0.8 (0.5) } 〉	〈{ 0.4 (1) , 0.5 (1) } , { 0.5 (0.6) } , { 0.3 (0.5) , 0.5 (0.5) } 〉
a ₄	〈{ 0.6 (0.5) , 0.8 (0.5) } , { 0.4 (0.6) , 0.6 (0.4) } , { 0.5 (1) } 〉	〈{ 0.8 (1) } , { 0.4 (0.5) , 0.6 (0.3) , 0.8 (0.2) } , { 0.3 (1) } 〉

Table 2

Decision matrix from DM d₂

	c ₁	c ₂
a ₁	〈{ 0.4 (0.6) , 0.6 (0.2) } , { 0.4 (1) } , { 0.3 (0.4) , 0.4 (0.5) } 〉	〈{ 0.3 (0.4) , 0.6 (0.4) } , { 0.5 (0.5) , 0.6 (0.4) } , { 0.3 (1) } 〉
a ₂	〈{ 0.5 (0.4) , 0.6 (0.3) } , { 0.4 (0.2) , 0.6 (0.5) } , { 0.3 (1) } 〉	〈{ 0.6 (1) } , { 0.4 (0.3) , 0.6 (0.5) } , { 0.4 (0.6) , 0.6 (0.3) } 〉
a ₃	〈{ 0.5 (1) } , { 0.4 (0.3) , 0.5 (0.4) } , { 0.4 (0.3) , 0.6 (0.5) } 〉	〈{ 0.4 (0.5) , 0 . 6 (0 . 5) } , { 0.5 (1) } , { 0.4 (0.4) , 0.5 (0.4) } 〉
a ₄	〈{ 0.5 (1) } , { 0.2 (0.1) , 0.4 (0.5) , 0.6 (0.2) } , { 0.5 (1) } 〉	〈{ 0 . 6 (1) } , { 0 . 4 (0 . 5) , 0 . 6 (0.5) } , { 0.5 (0 .3) , 0.6 (0.5) } 〉
	c ₃	c ₄
a ₁	〈{ 0 . 7 (0 . 5) , 0 . 8 (0 . 5) } , { 0 . 3 (0.5) , 0 . 4 (0.4) } , { 0 . 5 (1) } 〉	〈{ 0 . 5 (0 . 4) , 0 . 7 (0 . 6) } , { 0 . 3 (0.5) , 0 . 5 (0.4) } , { 0 . 5 (1) } 〉
a ₂	〈{ 0 . 7 (0 . 3) , 0.8 (0.5) } , { 0.4 (1) } , { 0.4 (0 .5) , 0.6 (0 .4) } 〉	〈{ 0 . 6 (0 . 4) , 0.8 (0 .4) } , { 0.4 (0 .2) , 0.6 (0.5) } , { 0.5 (1) } 〉
a ₃	〈{ 0 . 6 (1) } , { 0 . 4 (0 . 5) , 0 . 5 (0 . 3) } , { 0 . 4 (0 . 5) , 0.6 (0 .4) } 〉	〈{ 0 . 6 (1) } , { 0 . 5 (0 . 4) , 0 . 6 (0 . 4) } , { 0 . 5 (0 . 6) , 0.6 (0 .4) } 〉
a ₄	〈{ 0 . 6 (0 . 3) , 0 . 8 (0 . 5) } , { 0 . 4 (1) } , { 0 . 5 (0 . 3) , 0 . 6 (0 . 5) } 〉	〈{ 0 . 6 (0 . 5) , 0 . 8 (0 . 4) } , { 0 . 4 (1) } , { 0 . 4 (0 . 5) , 0 . 5 (0 . 4) } 〉

Table 3

Decision matrix from DM d₃

	c ₁	c ₂
a ₁	〈{ 0.6 (1) } , { 0.4 (0.2) , 0.6 (0.6) } , { 0.4 (0.6) , 0.6 (0.2) } 〉	〈{ 0.5 (0.4) , 07 (0.4) } , { 0.6 (1) } , { 0.4 (0.6) , 0.5 (0.4) } 〉
a ₂	〈{ 0.3 (1) } , { 0.5 (1) } , { 0.2 (0.2) , 0.4 (0.5) , 0.6 (0.3) } 〉	〈{ 0.5 (1) } , { 0.6 (1) } , { 0.5 (0.3) , 0.6 (0.4) , 0.7 (0.2) } 〉
a ₃	〈{ 0.4 (0.6) , 0.6 (0.2) } , { 0.6 (1) } , { 0.5 (0.4) , 0.6 (0.5) } 〉	〈{ 0.6 (0.4) , 0 . 8 (0 . 4) } , { 0.5 (0.3) , 0.7 (0.5) } , { 0.5 (1) } 〉
a ₄	〈{ 0.5 (0.4) , 0.6 (0.4) } , { 0.5 (1) } , { 0.3 (0.4) , 0.6 (0.5) } 〉	〈{ 0.7 (1) } , { 0.5 (0.6) , 0 . 6 (0.3) } , { 0.5 (1) } 〉
	c ₃	c ₄
a ₁	〈{ 0.5 (0.3) , 0.6 (0 .5) } , { 0.4 (0.4) , 0.6 (0.6) } , { 0.3 (1) } 〉	〈{ 0.6 (1) } , { 0 . 3 (1) } , { 0.4 (0 .4) , 0.5 (0.3) , 0.6 (0.3) } 〉
a ₂	〈{ 0.5 (0.4) , 0.6 (0.3) } , { 0.5 (0 .6) , 0.6 (0.3) } , { 0.5 (1) } 〉	〈{ 0.5 (0.6) , 0.6 (0 .4) } , { 0.4 (0.5) , 0.6 (0.3) } , { 0.3 (1) } 〉
a ₃	〈{ 0.5 (0.4) , 0.6 (0.5) } , { 0.5 (0.4) , 0.7 (0.5) } , { 0.5 (1) } 〉	〈{ 0.4 (0.6) , 0.7 (0.4) } , { 0.3 (0 .4) , 0.4 (0.6) } , { 0 . 5 (1) } 〉
a ₄	〈{ 0.5 (1) } , { 0.5 (1) } , { 0.4 (0.2) , 0 . 6 (0 . 5) , 0.7 (0.3) } 〉	〈{ 0.5 (0 .5) , 0 . 7 (0 . 5) } , { 0.5 (1) } , { 0 . 4 (0.6) , 0.6 (0.3) } 〉

5.1 The steps of the proposed approach

The main steps for selecting the highest quality logistics outsourcing provider can be summarized as follows.

Step 1: Normalize the criteria values.

We use Equation (3) to normalize the decision matrix. Limited by the space, the normalized decision matrices are omitted here.

Step 2: Compute the criteria weights in each decision matrix.

The criteria weight vector $w^{l} = (w_{1}^{l}, w_{2}^{l}, w_{3}^{l}, w_{4}^{l})$ in each decision matrix can be obtained by Equation (18), which is shown in Table 4.

Table 4
Criteria weights in each decision matrix

$w_{1}^{l}$ $w_{2}^{l}$ $w_{3}^{l}$ $w_{4}^{l}$

R ^1′ 0.2239 0.2552 0.2117 0.3092

R ^2′ 0.3353 0.2573 0.1973 0.2101

R ^3′ 0.2659 0.2451 0.2154 0.2736

	$w_{1}^{l}$	$w_{2}^{l}$	$w_{3}^{l}$	$w_{4}^{l}$
R ^1′	0.2239	0.2552	0.2117	0.3092
R ^2′	0.3353	0.2573	0.1973	0.2101
R ^3′	0.2659	0.2451	0.2154	0.2736

Step 3: Calculate the weights of DMs.

The expert weight vector η ={ η₁, η₂, η₃ }can be obtained by Equation (23), which is η = (0.2710, 0.2710, 0.4580).

Step 4: Determine the collective weights of criteria.

Based on the weights of criteria in each matrix acquired in Step 2 and the DMs’ weights acquired in Step 3, the collective criteria weight vector can be obtained by using Equation (25) as ω = (0.2733, 0.2512, 0.2095, 0.2660)

Step 5: Aggregate evaluation information.

The evaluation information given by each DM can be aggregated into comprehensive evaluation information by utilizing Equation (26), the corresponding aggregated decision matrix is shown in Table 5.

Table 5

The aggregated decision-making information

	c ₁
a ₁	$\begin{matrix} 〈 {0.575 (0.125), 0.630 (0.125), 0.526 (0.375), 0.587 (0.375)}, \\ {0.466 (0.1), 0.425 (0.15), 0.560 (0.3), 0.549 (0.45)} \\ {0.413 (0.333), 0.425 (0.417), 0.497 (0.111), 0.512 (0.139)} 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 {0.417 (0.286), 0.451 (0.499), 0.483 (0.215)}, \\ {0.471 (0.286) 0.525 (0.714),}, \\ {0.269 (0.12), 0.301 (0.08), 0.370 (0.3), 0.413 (0.38), 0.497 (0.12)} 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 {0.452 (0.12 5), 0.549 (0.125), 0.405 (0.375), 0.457 (0.375)}, \\ {0.482 (0.172), 0.425 (0.228), 0.512 (0.257), 0.544 (0.343)}, \\ {0.410 (0.167), 0.446 (0.208), 0.457 (0.277), 0.497 (0.348)} 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 {0.475 (0.2), 0.526 (0.2), 0.500 (0.3), 0.549 (0.3)}, \\ {0.367 (0.1 25), 0.494 (0.25), 0.443 (0.6 25)}, \\ {0.396 (0.296), 0.416 (0.148), 0.544 (0.371), 0.656 (0.185)} 〉 \end{matrix}$
	c ₂
a ₁	$\begin{matrix} 〈 {0.425 (0.25), 0.545 (0.25), 0.506 (0.25), 0.609 (0.25)}, \\ {0.626 (0.190), 0.595 (0.2 39), 0.600 (0.2 54), 0.571 (0.317)}, \\ {0.393 (0.2), 0.435 (0.133), 0.431 (0.400), 0.477 (0.267)} 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 {0.506 (0.37 5), 0.529 (0.625)}, \\ {0.538 (0.166), 0.512 (0.209), 0.607 (0.278), 0.577 (0.34 7)}, \\ {0.443 (0.222), 0.482 (0.296), 0.517 (0.148), 0.495 (0.111), \\ 0.538 (0.148), 0.577 (0.074)} 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 {0.554 (0.25), 0.600 (0.25), 0.675 (0.25), 0.709 (0.25)}, \\ {0.500 (0.375), 0.583 (0.625)}, \\ {0.443 (0.278), 0.471 (0.5), 0.500 (0.222)} 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 {0.676 (0.4), 0.628 (0.6)}, \\ {0.482 (0.167), 0.538 (0.167), 0.443 (0.333), 0.494 (0.333)}, \\ {0.435 (0.166), 0.4 57 (0.278), 0.500 (0.209), 0.525 (0.347)} 〉 \end{matrix}$
	c ₃
a ₁	$\begin{matrix} 〈 {0.543 (0.187), 0.5 90 (0.188), 0.526 (0.313), 0.587 (0.312)}, \\ {0.370 (0.178), 0.342 (0.222), 0.446 (0.266), 0.412 (0.334)}, \\ {0.345 (0.2), 0.396 (0.5), 0.416 (0.3)} 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 {0.658 (0.161), 0.621 (0.214), 0.693 (0.268), 0.660 (0.357)}, \\ {0.560 (0.111), 0.482 (0.222), 0.516 (0.222), 0.443 (0.445)}, \\ {0.443 (0.278), 0.471 (0.278), 0.495 (0.222), 0.525 (0.222)} 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 {0.506 (0.222), 0.557 (0.222), 0.554 (0.278), 0.600 (0.2 78)}, \\ {0.471 (0.166), 0.549 (0.209), 0.443 (0.278), 0.583 (0.347)}, \\ {0.495 (0.246), 0.535 (0.310), 0.552 (0.197), 0.597 (0.247)} 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 {0.557 (0.188), 0.633 (0.499), 0.696 (0.313)}, \\ {0.443 (0.6), 0.494 (0.4)}, \\ {0.451 (0.075), 0.474 (0.125), 0.544 (0.188), 0.583 (0.112), \\ 0.571 (0.312), 0.613 (0.188)} 〉 \end{matrix}$
	c ₄
a ₁	$\begin{matrix} 〈 {0.506 (0.16), 0.549 (0.24), 0.569 (0.24), 0.607 (0.36)}, \\ {0.377 (0.5 56), 0.433 (0.444)}, \\ {0.451 (0.16), 0.474 (0.2 4), 0.500 (0.12), 0.5 25 (0.3), 0.571 (0.18)} 〉 \end{matrix}$
a ₂	$\begin{matrix} 〈 {0.526 (0.2), 0.630 (0.2), 0.506 (0.3), 0.590 (0.3)}, \\ {0.400 (0.179), 0.482 (0.107), 0.446 (0.446), 0.538 (0.268)}, \\ {0.373 (0.3), 0.396 (0.5), 0.416 (0.2)} 〉 \end{matrix}$
a ₃	$\begin{matrix} 〈 {0.609 (0.2), 0.628 (0.2), 0.462 (0.3), 0.488 (0.3)}, \\ {0.396 (0.2), 0.416 (0.2), 0.451 (0.3), 0.474 (0.3)}, \\ {0.435 (0.3), 0.457 (0.2), 0.500 (0.3), 0.525 (0.2)} 〉 \end{matrix}$
a ₄	$\begin{matrix} 〈 {0.696 (0.222), 0.759 (0.222), 0.633 (0.278), 0.709 (0.278)}, \\ {0.443 (0.5), 0.494 (0.3), 0.535 (0.2)}, \\ {0.370 (0.371), 0.393 (0.2 96), 0.446 (0.185), 0.473 (0.148)} 〉 \end{matrix}$

Step 6: Determine the concordance and discordance sets.

According to Equation (27), the subscripts of concordance set can be acquired as follows: $U_{12}^{+} = {4}; U_{13}^{+} = {1, 3, 4}; U_{14}^{+} = {1}; U_{21}^{+} = {2, 3}; U_{23}^{+} = {1, 2, 3}; U_{24}^{+} = {1};$ $U_{31}^{+} = Ø; U_{32}^{+} = Ø; U_{34}^{+} = Ø; U_{41}^{+} = {2, 3, 4}; U_{42}^{+} = {2, 3, 4}; U_{43}^{+} = {2, 3, 4}$

Then the subscripts of discordance set can also be obtained by Equation (28), which is depicted as follows: $U_{12}^{-} = {1, 2, 3}; U_{13}^{-} = {2}; U_{14}^{-} = {2, 3, 4}; U_{21}^{-} = {1, 4}; U_{23}^{-} = {4}; U_{24}^{-} = {2, 3, 4};$ $U_{31}^{-} = {1, 2, 3, 4}; U_{32}^{-} = {1, 2, 3, 4}; U_{34}^{-} = {1, 2, 3, 4}; U_{41}^{-} = {1}; U_{42}^{-} = {1}; U_{43}^{-} = {1}$

Step 7: Construct relative concordance and discordance matrices.

The relative concordance matrix can be constructed with respect to the weight vector of criteria, which is presented as follows: $CM = (\begin{matrix} 1 & 0.2660 & 0.7488 & 0.2733 \\ 0.4607 & 1 & 0.7340 & 0.2733 \\ 0 & 0 & 1 & 0 \\ 0.7267 & 0.7267 & 0.7267 & 1 \end{matrix})$ (33)

Similarly, the discordance matrix is constructed as follows: $U^{-} = (\begin{matrix} 0 & 1 & 1 & 0.8159 \\ 1 & 0 & 0.4456 & 0.8700 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 0.5795 & 0 \end{matrix})$ (34)

Step 8: Compute the corrected comprehensive weighted matrix.

The corrected comprehensive weighted matrix can be obtained by Equation (31), expressed as follows: $EM = (\begin{matrix} 1 & 0 & 0 & 0.0503 \\ 0 & 1 & 0.4069 & 0.0355 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0.3056 & 1 \end{matrix})$ (35)

Step 9: Calculate the net prioritized value of each alternative.

The net prioritized value of each alternative can be acquired by Equation (32), shown as follows.

Step 10: Select the best logistics provider.

According to Table 6, the final ranking of alternatives isa₂ ≻ a₄ ≻ a₁ ≻ a₃. Consequently, the optimal alternative is identified as a₄.

Table 6

The net prioritized value of each alternative

Alternative	$\sum_{i = 1, i \neq s}^{m} {em}_{si}$	$\sum_{i = 1, i \neq s}^{m} {em}_{is}$	Net prioritized value
a ₁	0.0503	0	0.0503
a ₂	0.4425	0	0.4425
a ₃	1	1.7125	–0.7125
a ₄	0.3056	0.0858	0.2198

5.2 Comparative analysis

To verify the feasibility of the proposed approach, we make a comparative analysis with some existing methods.

5.2.1 Comparison with Peng et al.’s method with GSNNWG operator [23]

It is an unquestioned fact that PMVNSs are extended from MVNSs, and MVNSs are extended from simplified neutrosophic sets (SNSs), so we can use the aggregation operators based on SNSs to address above example. First of all, PMVNNs should be transformed into simplified neutrosophic numbers (SNNs). To keep original information as much as possible, we can calculate the average values of all possible TFs, IFs and FFs with corresponding probability in a PMVNN, which can be viewed as a SNN. Let A =〈 ⋃ { t_A (p_t) } , ⋃ { i_A (p_i) } , ⋃ { f_A (p_f) } 〉be a PMVNN, then $A^{'} = 〈 \frac{\sum t_{A} p_{t}}{# T_{A}}, \frac{\sum i_{A} p_{i}}{# I_{A}}, \frac{\sum f_{A} p_{f}}{# F_{A}} 〉$ is a SNN.

The ranking result derived from Peng et al.’s method with GSNMWG operator [23] is a₂ ≻ a₃ ≻ a₁ ≻ a₄. Compared with the ranking result obtained by the proposed approach, there remains some difference, which is discussed in the following.

Firstly, PMVNSs not only consider the hesitancy of DMs but also take their preference into account, so the fuzzy information denoted by PMVNSs is more realistic and credible than that denoted by SNSs.

Secondly, based on some related research of intuitionistic fuzzy number, Peng et al. [23] defined a comparison method for SNNs. In this paper, we define the strong dominance, weak dominance and indifference relations for PMVNNs, which can provide more specific relations between binary PMVNNs. Obviously, the comparison method in this paper is more meticulous, accurate and comprehensive.

Thirdly, Peng et al. [23] applied the aggregation operators to address MCGDM problems, there remain a drawback that computational process may be complex. Usually, the aggregation operators may ignore evaluation information partially. To ensure accuracy and decline complexity, we provide an outranking approach for MCGDM problems based on ELECTRE, this method can take all evaluation information into account and output the ranking results more easily. As a result, this extended ELECTRE method is more effective and available.

5.2.2 Comparison with Peng et al.’s QUALIFLEX method [21]

We cited the example from Peng et al.’s QUALIFLEX method [21], so we can compare our proposed approach with it directly. The ranking result obtained from Peng et al.’s QUALIFLEX method [21] is a₄ ≻ a₂ ≻ a₁ ≻ a₃, and the result acquired from proposed approach isa₂ ≻ a₄ ≻ a₁ ≻ a₃. Next, we will discuss the aspects in which the proposed method outperforms than Peng et al.’s QUALIFLEX method [21].

Initially, Peng et al. [21] presented two cross-entropy measures for PMVNNs to denote the deviation of any two PMVNNs, but when two PMVNNs are the same totally, the value of their cross-entropy isn’t equal to zero. Obviously, this is unreasonable. To overcome this drawback, we introduce a new general distance measure to describe the deviation of any two PMVNNs with unequal length. We can apply this distance measure to calculate the deviations of pair-wise PMVNNs in the light of DMs’ attitudes. The computational process of this distance measure is more convenient than that of Peng et al.’s QUALIFLEX method [21].

Furthermore, Peng et al. [21] mainly tested how each possible ranking of alternatives is supported by different criteria, but it ignored the differences of pair-wise alternatives under each criterion, so the result isn’t enough adequate and satisfactory. Our proposed method extends the classical ELECTRE to PMVNSs and ranks the alternatives in accordance with the priority degree of alternatives. Moreover, based on outranking relations among non-compensated criteria, the ranking result obtained from our method is more complete and comprehensive.

Lastly, ELECTRE method ranks the alternatives in the light of the net prioritized values not just relative concordance degree or relative discordance degree of alternatives, thus this measure escapes the potential deviation caused by relative concordance degree or relative discordance degree effectively. What’s more, there is no complementarity between relative concordance degree and relative discordance degree, it means, there is sufficient reason to confirm that alternative a is at least as good as alternative b and no evidence is to reject such statement.

As discussed above, our proposed method is more feasible and efficient than existing methods [21, 23].

6 Conclusions

In this paper, we define the outranking relations for any two PMVNNs and present some properties for these relations. Then establish two reliable models to obtain the attribute weights and DMs’ weights based on minimum deviation principle between expected value and subjective preference value. Furthermore, based on the distance measure and outranking relations for PMVNNs, an extended MCGDM ELECTRE method with PMVNSs is designed, and a real example is used to compare the experimental results of the proposed approach with some existing methods. The experimental results show that the proposed approach precedes the existing methods in computational complexity and realistic feasibility. In the future, we will further research the subjective weight model and the combination weight model so that we can relieve the existing shortcoming in the proposed models in which only the objective weights are considered. In addition, we will endeavor to extend the application scopes of the proposed approach, such as in sustainable energy planning [25], economics [41] and education [20], and apply other classical methods to handle with MCGDM problems in PMVNSs environments.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (No. 71771140), Project of cultural masters and “the four kinds of a batch” talents, the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Major bidding projects of National Social Science Fund of China (19ZDA080).

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Probability multi-valued neutrosophic ELECTRE method for multi-criteria group decision-making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 PMVNSs

4 Probability multi-valued neutrosophic ELECTRE method for the MCGDM problem

4.1 Description of decision-making problems

Table 4 Criteria weights in each decision matrix w 1 l w 2 l w 3 l w 4 l R 1′ 0.2239 0.2552 0.2117 0.3092 R 2′ 0.3353 0.2573 0.1973 0.2101 R 3′ 0.2659 0.2451 0.2154 0.2736

5.2.1 Comparison with Peng et al.’s method with GSNNWG operator [23]

5.2.2 Comparison with Peng et al.’s QUALIFLEX method [21]

6 Conclusions

Footnotes

Acknowledgments

References

Table 4
Criteria weights in each decision matrix

$w_{1}^{l}$ $w_{2}^{l}$ $w_{3}^{l}$ $w_{4}^{l}$

R ^1′ 0.2239 0.2552 0.2117 0.3092

R ^2′ 0.3353 0.2573 0.1973 0.2101

R ^3′ 0.2659 0.2451 0.2154 0.2736