An ELECTRE TRI-based outranking approach for multi-attribute group decision making with picture fuzzy sets

Abstract

The aim of this paper is to develop a multi-attribute group decision making (MAGDM) with picture fuzzy sets based on ELECTRE TRI method, i.e., an ELECTRE TRI based group decision making with picture fuzzy information is given. The MAGDM with picture fuzzy information based on picture fuzzy ELECTRE TRI outranking method is divided into three stages, i.e., the group decision information aggregation stage, determination of parameters and ELECTRE TRI outranking based outranking stage. A novel comparison law for picture fuzzy sets is introduced. In the group decision information aggregation stage, the concept of picture fuzzy normalized weighted Bonferroni mean (PFNWBM) is developed. The developed decision procedure is further applied to the assessment of energy security. The numerical example shows that the developed group decision procedure is feasible and valid.

Keywords

Multi-attribute group decision making picture fuzzy sets ELECTRE TRI supplier selection picture fuzzy normalized Bonferroni mean

1 Introduction

Group decision making (or collaborative decision making) is a decision-making situation that individuals make a selection from the candidates before them. Since the decision is made by groups but not by individuals, the decision can reflect the wisdom of crowds. As a typical group decision situation, the multi-attribute group decision making (MAGDM) is a common decision environment with several decision makers (DMs), candidates and attributes, whose aim is to make a selection (or selections) from all candidates by comprehensively considering the evaluations given by the DMs with respect to different attributes. MAGDM has been widely studied in theory and applied to economic and management problems [1–6 , 67].

Among current studies on MAGDM, group decision making under different types of decision information environments and their applications is a hot topic. Especially, the case of uncertainty that the edge of decision information is unclear, or fuzzy in other words. The concept of fuzzy sets is introduced by Zadeh [7] in 1965 to express such uncertainty, which has been generalized as diverse kinds of fuzzy information [8–13 , 65]. Among those different types of fuzzy information, the concept of intuitionistic fuzzy sets (IFSs) extends the classical notion of fuzzy sets by detailing the degree of an element in a given set according to three dimensions, i.e., the membership degree (μ), the non-membership degree (ν) and the hesitancy degree (π). The mentioned three degrees satisfy the condition that μ + ν + π = 1. Recently, taking the American presidential election, electors in American are allowed to have 4 choices, i.e, vote for, abstain, refusal of voting and vote against, while not all Americans have the right to vote, in other words, when all Americans are seen as 1, then the sum of electors could be less than 1, i.e., μ + ν + π ⩽ 1 (The electors with deterministic attitudes: support, no support and abstention). To associate with such situations, Cuong and Kreinovich [13] introduce the concept of picture of fuzzy sets (PFSs). Since the definition of PFSs is developed, theoretical studies on PFSs and their applications have been reported. Singh [14] introduces the correlation coefficient of PFSs. Wei [15 –17] consider the similarity measures and the cross-entropy measure of PFSs. While Son [18] develops the notion of generalized picture distance measure of PFSs. Thong [19], Phuong and Thong [27] and Van and Van [20] propose the clustering analysis and inference models under the picture fuzzy environment, respectively. Son [21] puts forward some picture association measures to measure the analogousness in PFSs by using picture distance measures. Wei [22], Garg [23] and Thong [24 –26] consider the applications of PFSs to multicriteria decision making (MADM), optimization and forecasting.

It can be seen that decision making is the main application fields of PFSs. By using these fundamental and theoretical studies on PFSs, some typical decision technologies include TODIM (an acronym in Portuguese of interactive and multiple attribute decision making, a method for solving the multiple attribute decision making (MADM) problem considering DM’s behavior) [22], VIKOR method [20] and picture fuzzy information fusion based decision methods [23, 29] have been introduced. The developments of these decision technologies provide objective and selective stylized decision approaches for practical applications with PFSs.

As an another typical decision procedure, ELECTRE TRI outranking method is now a powerful tool to produce the ranking of candidates by considering multiple criteria, which has been generalized to diverse information environments [31 –33]. The advantages of ELECTRE TRI outranking method include the unnecessary of decision information’s algebra structure and the fusion of DM’s subjective attitude and objective ranking procedure. Actually, ELECTRE TRI assigns a candidate to one of the predefined categories according to compare the candidate with referenced profiles separating the categories. Herein, the determination of profiles and thresholds corresponding to those profiles could be subjective, while the outranking process is objective when all parameters are determined. However, by reviewing the literatures on ELECTRE TRI and PFSs, it can be found that the combination of ELECTRE TRI outranking method and PFSs has merely been reported. Besides, the application of ELECTRE TRI outranking method to group decision making with PFSs has also merely been reported.

By considering the issues highlighted before, the aim of this paper is to propose the combination of ELECTRE TRI and PFSs so that a novel decision technology named as picture fuzzy ELECTRE TRI outranking method can be put forward. Furthermore, the application of the developed picture fuzzy ELECTRE TRI outranking method to MAGDM problems with PFSs would also be discussed. In order to simplify the outranking procedure of alternatives in MAGDM with PFSs, we introduce the concept of picture fuzzy weighted Bonferroni mean to aggregate picture fuzzy group decision information. Herein, the weighted Bonferroni mean is selected to aggregate PFSs for the reason that such form of mean value contains possible links among aggregated information [35–38 , 46]. Then the picture fuzzy ELECTRE TRI is introduced and applied to MAGDM with PFSs.

To realize the aims mentioned above, the rest of this paper is structured as follows: Section 2 briefly presents the fundamental concepts of PFSs and ELECTRE TRI outranking method. Section 3 puts forward the picture fuzzy ELECTRE TRI outranking method and its application to MAGDM with PFSs, in which the concept of picture fuzzy normalized weighted Bonferroni mean and its properties are discussed. While a numerical example about supplier selection assessment is shown in Section 4 to illustrate the feasibility and validity of the developed model. Comparisons of the developed group decision method with other existed group decision approaches for MAGDM with PFSs are also presented. Section 5 gives some conclusions and remarks on the developed model.

2 Preliminaries

2.1 Picture fuzzy sets

In traditional fuzzy set given by Zadeh [7], the degree of an element in a given fuzzy set is represented by a real number between [0, 1], which is different from the characteristic function whose value is either 0 or 1. In order to enrich the expression of fuzzy sets, the following notion of intuitionistic fuzzy sets is given by Atanassov [8]:

Definition 1 ([4]). Let X be a universe of discourse, an Atanassov’s intuitionistic fuzzy set (AIFS) A in X is defined by $A = {〈 x, μ_{A} (x), v_{A} (x) 〉 | x \in X}$ where 0 ⩽ μ_A (x) + v_A (x) ⩽1, μ_A (x) ∈ [0, 1] and v_A (x) ∈ [0, 1] denoted the degrees of membership and non-membership of the element x ∈ X to the set A, respectively. π_A (x) =1 - μ_A (x) - v_A (x) is called the hesitancy degree of x to A.

It can be seen that an AIFS can provide more detailed information rather than classical fuzzy set, but there are still some situations in real life which cannot be represented by Atanassov’s IFSs. For instance, the example of voting situation in the section of Introduction. To illustrate such situations, the following concept of picture fuzzy sets is introduced:

Definition 2 ([13]). Given that X = {x₁, x₂, ⋯, x_n} is a finite universal set. A picture fuzzy set (PFS) A on X is defined as an object of the following form: $A = {〈 x, μ_{A} (x), η_{A} (x), v_{A} (x) 〉 | x \in X}$ where μ_A (x) ∈ [0, 1] is called the “degree of positive membership of A”, η_A (x) ∈ [0, 1] is called the “degree of neutral membership of A” and v_A (x) ∈ [0, 1] is called the “degree of negative membership of A”, the sum of μ_A (x) , η_A (x) and v_A (x) satisfies the condition that 0 ⩽ μ_A (x) + η_A (x) + v_A (x) ⩽1, ∀x ∈ X. Similarly, π_A (x) =1 - μ_A (x) - η_A (x) - v_A (x) is said to be the degree of refusal membership of the element x to the set A.

For convenience, the notation α = (μ_α, η_α, v_α) is called as a picture fuzzy number (PFN) [39], where μ_α ∈ [0, 1], η_α ∈ [0, 1], v_α ∈ [0, 1], 0 ⩽ μ_α + η_α + v_α ⩽ 1.

By the concepts and notations mentioned above, the following algebra structure is defined:

Definition 3 ([39]). Given that α = (μ_α, η_α, v_α) and β = (μ_β, η_β, v_β) are two PFNs, and λ ∈ [0, 1] is a parameter, the algebra operational laws of PFNs can be shown in the following:

α ⊕ β = (μ_α + μ_β - μ_αμ_β, η_αη_β, v_αv_β);

α ⊗ β = (μ_αμ_β, η_α + η_β - η_αη_β, v_α + v_β - v_αv_β);

$α^{λ} = (μ_{α}^{λ}, 1 - (1 - η_{α})^{λ}, 1 - (1 - v_{α})^{λ})$ .

$λ α = (1 - (1 - μ_{α})^{λ}, η_{α}^{λ}, v_{α}^{λ})$ .

Let X = { x₁, x₂, ⋯ , x_n } be a finite universal set, A = {(μ_A (x_i) , η_A (x_i) , v_A (x_i))} and B = {(μ_B (x_i) , η_B (x_i) , v_B (x_i))} are two sets of PFNs. To measure the difference between A and B, the following normalized Hamming distance d(A, B) can be utilized, $\begin{matrix} d (A, B) = & \frac{1}{3 n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | η_{A} (x_{i}) \\ - η_{B} (x_{i}) | + | ν_{A} (x_{i}) - v_{B} (x_{i}) |) . \end{matrix}$

In order to compare picture fuzzy numbers, the score function S and the accuracy function H are introduced.

Definition 4 ([39]). Let α = (μ_α, η_α, v_α) be a PFN, the score function S of α can be defined as S (α) = μ_α - v_α, S (α) ∈ [-1, 1].

Definition 5 ([39]). Let α = (μ_α, η_α, v_α) be a PFN, an accuracy function H of α can be defined according to H (α) = μ_α + η_α + v_α, H (α) ∈ [0, 1].

By the score function S and the accuracy function H, the comparison between α and β can be given according to:

Definition 6 ([39]). Let α = (μ_α, η_α, ν_α) and β = (μ_β, η_β, ν_β) be two PFNs, S (α) and S (β) are scores of α and β, H (α) and H (β) are accuracy degrees of α and β, then

if S (α) < S (β), then α is smaller than β, denoted by α < β;

if S (α) = S (β), then

if H (α) = H (β), then α is equal to β, denoted by α = β;

if H (α) < H (β), α is smaller than β, denoted by α < β.

Since there is a neutral membership in a PFN π_A (x), two extreme cases would be the ratio η_A (x) is assigned to μ_A (x) and ν_A (x), i.e., when assigning the part of refusal membership to the other three membership degrees, one can derive a AIFS located between 〈μ+ η, ν, π 〉 and 〈μ, ν+ η, π 〉. It can be easily seen that given a PFN, one can uniquely obtain the two AIFNs, while given two AIFNs with the forms, one can also determine the corresponding PFN.

Theorem 1. The score function and the accuracy function of a given PFN P =〈 μ, η, ν 〉 is equal to the average of score functions corresponding to the two derived AIFNs AN₁ =〈 μ + η, ν, π 〉 and AN₂ =〈 μ, ν + η, π 〉.

Proof. According to Xu [53], the score function of an AIFN AN =〈 μ, ν, η 〉 is defined as S (AN) = μ - ν and H (AN) = μ + ν. The proof can be easily obtained, thus it’s omitted.

According to Theorem 1, the following dominance degree of a PFN can be derived:

Definition 7. Given that P =〈 μ, η, ν 〉 is a PFN and AN₁ =〈 μ + η, ν, π 〉 and AN₂ =〈 μ, ν + η, π 〉 are two derived AIFNs corresponding to P. Let PI =〈 1, 0, 0 〉 and NI =〈 0, 1, 0 〉 be the positive and the negative ideal points of AIFNs, then the dominance degree of a PFN can be defined according to

$\begin{matrix} {dd}_{P}^{C} = & \frac{1}{2} (\frac{d ({AN}_{1}, NI)}{d ({AN}_{1}, NI) + d ({AN}_{1}, PI)} \\ + \frac{d ({AN}_{2}, NI)}{d ({AN}_{2}, NI) + d ({AN}_{2}, PI)}) . \end{matrix}$ (1) where the notation d () represents the distance measure between any two AIFNs.

By Equation (1), the dominance degree of a PFN is an average of the dominance degrees corresponding to two AIFNs defined by the TOPSIS [54, 67]. it can be seen that the larger the dominance degree is, the larger the picture fuzzy number would be.

2.2 ELECTRE TRI

ELECTRE is a family of multi-criteria decision analysis approaches developed by Roy [40] in the mid-1960 s, which aims to choose the best action(s) by considering concrete and multiple criteria. There are two main stages to an ELECTRE application, i.e., the construction of one or several outranking relations and an exploitation procedure elaborating the recommendations.

ELECTRE TRI is a method pertaining to the ELECTRE family which are based on outranking methods. It assigns alternatives to predefined and ordered categories. The ordered categories are defined with lower and upper limits adjusted for each criteria being considered [27 , 41–43]. For each criteria C_j (j = 1, 2, ⋯ , n), the category CT_h+1 will be limited by a lower limit b_h and an upper limit b_h+1. The assignment of a candidate a into a category CT_h+1 results from the comparison of the alternative with the limits b_h (h = 1, 2, ⋯ , p) and comparisons are based on an outranking relation S [44, 45]. The credibility index σ (a, b_h) and σ (b_h, a) are proposed to verify the relationship between alternative a and b_h, which could be calculated as: $σ (a, b_{h}) = c (a, b_{h}) \times \prod_{j = 1}^{n} \frac{1 - d_{j} (a, b_{h})}{1 - c (a, b_{h})}$ (2) where c (a, b_h) and d_j (a, b_h) (j = 1, 2, ⋯ , n) are comprehensive concordance index and discordance index, respectively. The comprehensive concordance index c (a, b_h) is calculated as: $c (a, b_{h}) = \frac{\sum_{j = 1}^{n} W_{j} \times c_{j} (a, b_{h})}{\sum_{j = 1}^{n} W_{j}}$ (3) where W_j is the weight of criteria C_j and c_j (a, b_h) is partial concordance index computing by:

$c_{j} (a, b_{h}) = {\begin{matrix} 0, if g_{j} (b_{h}) - g_{j} (a) ⩾ p_{j} (b_{h}) \\ 1, if g_{j} (b_{h}) - g_{j} (a) ⩽ q_{j} (b_{h}) \\ [p_{j} (b_{h}) + g_{j} (a) - g_{j} (b_{h})] / [p_{j} (b_{h}) - q_{j} (b_{h})], otherwise \end{matrix}$ (4) where g_j (a) is the evaluation of candidate a on the attribute C_j, g_j (b_h) is profile between categories CT_h and CT_h+1 under criteria C_j. The parameters p_j (b_h) and q_j (b_h) are predefined preference and indifference thresholds corresponding to b_h.

The discordance indices d_j (a, b_h) , j = 1, 2, ⋯ , n, is calculated by:

$d_{j} (a, b_{h}) = {\begin{matrix} 0, if g_{j} (b_{h}) - g_{j} (a) ⩽ p_{j} (b_{h}) \\ 1, if g_{j} (b_{h}) - g_{j} (a) > v_{j} (b_{h}) \\ [g_{j} (b_{h}) - g_{j} (a) - p_{j} (b_{h})] / [v_{j} (b_{h}) - p_{j} (b_{h})], otherwise \end{matrix}$ (5) where v_j (b_h) is the veto threshold for the attribute C_j and limit b_h.

When the credibility indices σ (a, b_h) and σ (b_h, a) are calculated, the following rules are used to determine the preference relation between alternative a and the limit b_h:

If σ (a, b_h) ⩾ λ and σ (b_h, A) ⩾ λ, aSb_h and b_hSa, then a is indifferent in relation to b_h (aIb_h);

If σ (a, b_h) ⩾ λ and σ (b_h, a) < λ, aSb_h and not b_hSa, then a is preferred in relation to b_h (a ≻ b_h);

If σ (a, b_h) < λ and σ (b_h, a) ⩾ λ, b_hSa and not aSb_h, then b_h is preferred in relation to a (b_h ≻ a);

If σ (a, b_h) < λ and σ (b_h, a) < λ, neither aSb_h or b_hSa exists, then a is incomparable in relation to b_h (aRb_h);

Then, the following two assignment procedures can be used to derive the outranking of candidates [41 –43]:

The pessimistic procedure

Compare successively each alternative a with the limits b_h (h = 1, 2, ⋯ , p).

Assign the alternative a to the highest category CT_h+1 such that a ≻ b_h-1.

The optimistic procedure

Compare successively each alternative a with the limits b_h (h = 1, 2, ⋯ , p).

Assign the alternative a to the lowest category CT_h+1 such that b_h-1 ≻ a

3 MAGDM with PFSs via ELECTRE TRI

Let X = {X₁, X₂, ⋯ , X_m} be a set of candidates (or alternatives). Given that each alternative is evaluated by a set of decision makers Exp = {D₁, D₂, ⋯ , D_l} according to a set of criteria (or attributes) C = {C₁, C₂, ⋯ , C_n}. The decision matrices are denoted as $(p_{ij}^{k})_{m \times n}$ , k = 1, 2, ⋯ , l, where $p_{ij}^{k}$ represents the assessment of the i-th alternative with respect to the j-th attribute given by the k-th DM. Thus, a MAGDM problem with PFNs is to produce the outranking of alternatives according to the picture fuzzy decision information $p_{ij}^{k}$ .

In this section, the picture fuzzy ELECTRE TRI method will be developed and further be applied to derive the preferences of all alternatives.

3.1 Picture fuzzy normalized weighted Bonferroni mean

The Bonferroni mean (BM) was originally introduced by Bonferroni [47, 48] and then be generalized by Yager [49]. By comparing with other aggregation operators, the desirable characteristic of the BM is the capability to capture the interrelationship between input information [47 –49]. Therefore, the normalized BM [56] is extended to aggregate picture fuzzy information of decision makers in this paper.

Definition 8. Given that α_j (j = 1, 2, ⋯ , n) is a collection of picture fuzzy numbers and ω_j (j = 1, 2, ⋯ , n) is the associated weight of α_j (j = 1, 2, ⋯ , n), and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ . Let

$\begin{array}{l} P F N W B M_{ω}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = {(\oplus_{\begin{array}{l} s, j = 1 \\ i \neq j \end{array}}^{n} (\frac{ω_{s} ω_{j}}{1 - ω_{s}} (α_{s}^{p} \otimes α_{j}^{q})))}^{\frac{1}{p + q}}, p, q > 0, \end{array}$ (6) then ${PFNWBM}_{ω}^{p, q}$ is called the picture fuzzy normalized weighted Bonferroni mean (PFNWBM).

According to Definition 8, the following conclusions can be derived:

Theorem 2. Let p, q > 0, and α_j (j = 1, 2, ⋯ , n) be a collection of PFNs, whose weight vector is ω = (ω₁, ω₂, ⋯ , ω_n) ^T, which satisfies ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ . Then the aggregated value, by using the PFNWBM, is also a picture fuzzy number, and $\begin{matrix} P F N W B M_{ω}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = (\begin{matrix} {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - μ_{α_{i}}^{p} μ_{α_{j}}^{q})}^{ω_{i} ω_{j} / (1 - ω_{i})})}^{1 / (p + q)}, \\ 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - η_{α_{i}})}^{p} {(1 - η_{α_{j}})}^{q})}^{ω_{i} ω_{j} / (1 - ω_{i})})}^{1 / (p + q)}, \\ 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - v_{α_{i}})}^{p} {(1 - v_{α_{j}})}^{q})}^{ω_{i} ω_{j} / (1 - ω_{i})})}^{1 / (p + q)} \end{matrix}) \end{matrix}$ (7)

Proof. The proof of this theorem is similar to Theorem 4.2 in Zhou and He [56], thus it’s omitted.

Besides, since the values of μ_{α
_i}, η_{α
_i} and v_{α
_i} are located in [0, 1], according to the operational rules defined in Definition 3, it’s easy to derive that the result obtained in Theorem 2 is also a PFN.

According to the conclusion mentioned above, Theorem 2 is proved. □

By using the PFNWBM, a set of PFNs can be aggregated to be a single PFN, which will be further used to simplify the group decision procedure.

3.2 MAGDM with PFNs via ELECTRE TRI

3.2.1 Determination of experts’ weighting vector

In order to measure the importance of experts, let ϖ = (ϖ₁, ϖ₂, ⋯ , ϖ_l) ^T be the weighting vector of experts. Given that ${PFM}^{(k)} = ({PFN}_{ij}^{k})_{m \times n}$ is the picture fuzzy decision matrix evaluated by the k-th expert. For the purpose of group decision making, it can be seen that the larger the difference of one expert with the other experts, the less consensus contribution to the group would be made. Correspondingly, the less expert weight should be given.

By using the normalized Hamming distance measure between two PFSs, the difference of one expert’s decision matrix with the other experts’ decision matrices can be given according to

$\begin{matrix} {dist}^{(k)} = & \frac{1}{l - 1} \sum_{h = 1}^{l} \bar{d} ({PFM}^{(k)}, {PFM}^{(h)}), \\ k = 1, 2, \dots, l, \end{matrix}$ (8) where $\bar{d} ({PFM}^{(k)}, {PFM}^{(h)}) = \frac{1}{mn} \sum_{i = 1}^{m} \sum_{j = 1}^{n} d ({PFN}_{ij}^{k}, {PFN}_{ij}^{h})$ is the distance between PFM^(k) and PFM^(h) and $d (p_{ij}^{k}, p_{ij}^{h}) = \frac{1}{3} (| μ_{p_{ij}^{k}} - μ_{p_{ij}^{h}} | + | η_{p_{ij}^{k}} - η_{p_{ij}^{h}} | + | v_{p_{ij}^{k}} - v_{p_{ij}^{h}} |)$ is the distance between two PFNs.

According to Equation (8), it can be seen that 0 ⩽ dist^(k) ⩽ 1. Then, by the differences given above, the weight of the k-th expert can be obtained according to $ϖ_{k} = \frac{1 - {dist}^{(k)}}{\sum_{h = 1}^{l} (1 - {dist}^{(h)})}, h = 1, 2, \dots, l .$ (9)

By Equation (9), as mentioned before, suggestions provided by different individuals would be aggregated in group decision making, so consensus among the group is encouraged.

3.2.2 Determination of attributes’ weighting vector

On the contrary, to measure the importance of attributes, the maximizing deviation method [50] is extended to assign the weights of attributes. It assigns a larger weight to the attribute which create larger difference among all alternatives. In other words, if all alternatives have similar performance value in term of a given attribute, then such attribute has less effect on ranking the alternatives. Given that ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weighting vector of attributes. Then the following optimization model can be used to obtain the weighting vector: $\begin{matrix} max Z (ω) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{s = 1}^{m} ω_{j} d (x_{ij}, x_{sj}) \\ s . t ., {\begin{matrix} \sum_{j = 1}^{n} ω_{j} = 1, \\ ω_{j} ⩾ 0, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (10) where x_ij represents the decision information of the i-th alternative under the j-th attribute.

By solving Equation (10), the solution can be represented according to

$\begin{matrix} ω_{j} = & \frac{\sum_{i = 1}^{m} \sum_{s = 1}^{m} d (x_{ij}, x_{sj})}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{s = 1}^{m} d (x_{ij}, x_{sj})}, \\ j = 1, 2, \dots, n . \end{matrix}$ (11)

According to Equation (11), let x_ij be the picture fuzzy decision information, then the weighting vector of attributes can be derived.

3.2.3 MAGDM with PFNs via ELECTRE TRI outranking method

With the notions and notations mentioned above, the picture fuzzy ELECTRE TRI method is put forward in this subsection, which will be further applied to MAGDM with PFNs. The decision procedure can be shown in the following:

Stage 1. Aggregation of group decision information

In order to simplify the group decision procedure, we first aggregate the group decision information of alternatives given by different experts.

By using the weighting method of experts ϖ = (ϖ₁, ϖ₂, ⋯ , ϖ_l) ^T provided by Equation (8), decision information of alternatives given by experts can be aggregated according to the developed PFNWBM, which can be shown as below:

$\begin{matrix} {\hat{p}}_{ij} = & {PFWNBM}_{ϖ} (p_{ij}^{1}, p_{ij}^{2}, \dots, p_{ij}^{l}), \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (12)

According to Equation (12), it can be seen that ${\hat{p}}_{ij}$ represents the synthesized decision information of the i-th alternative with respect to the j-th attribute. Consequently, the outranking of alternatives according to different experts and diverse attributes is simplified as an outranking of alternatives by considering different attributes.

Stage 2. Calculation of the credibility index

Let $x_{ij} = {\hat{p}}_{ij}$ , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, by Equation (11), the weighting vector associated to attributes with aggregated group evaluations can be obtained. According to Stage 1 and Definition 7, MAGDM with picture fuzzy information can be transformed to an outranking problem of MADM with crisp decision information.

For the purpose of constructing the comprehensive concordance index, the thresholds include the preference threshold (p), the indifference threshold (q) and the veto threshold (v) are firstly determined. Ramezanian [51] utilizes an optimization model to derive the profiles and thresholds of posteriori ELECTRE TRI. While Mousseau and Slowinski [52] build an optimization model to optimize the profiles and thresholds with given partitions of candidates and subjective assessments on the thresholds. Galo et al. [31] provide these parameters according to DMs’ subjective evaluations. Mousseau et al. [55] integrate specific functionalities supporting the decision maker (DM) in the preference elicitation process. However, there would be no deterministic partitions of candidates in most of practical cases. Besides, the profiles and thresholds given by subjective assessments would be changeable with respect to the attitude of DM. Thus, an objective way to determine the profiles and thresholds is needed.

Clustering analysis is a natural solution for the partition of candidates. Thus, the following procedure can be considered to produce the profiles and thresholds:

Step 1. Clustering analysis of the decision information under the j-th attribute provided by all experts.

Given that $℘_{j} = {{PFN}_{ij}^{k} | k = 1, 2, \dots, l; i = 1, 2, \dots, m .}$ is the set of decision information under the j-th attribute provided by all experts. Assume that all alternatives would be divided into h + 1 categories CT₁, CT₂, ⋯ , CT_h+1, then by using the Density-Based Spatial Clustering of Applications with Noise (DBSCAN), one can obtain classifications of alternatives.

Step 2. Determination of picture fuzzy profiles and thresholds.

Let ${pb}_{q}^{j}, q = 1, 2, \dots, h; j = 1, 2, \dots, n$ be the profiles between h + 1 categories under the j-th attribute. Then ${pb}_{q}^{j}$ can be set as the maximum point of the q-th category, i.e., $max {{\bar{PFN}}_{ij}^{k} | {\bar{PFN}}_{ij}^{k} \in {CT}_{p} \cap ℘_{j}}$ .

$\begin{matrix} {pb}_{q}^{j} = & max {{\bar{PFN}}_{ij}^{k} | {\bar{PFN}}_{ij}^{k} \in {CT}_{p} \cap ℘_{j}}, \\ q = 1, 2, \dots, h . \end{matrix}$ (13)

Herein, PFNs ${\bar{PFN}}_{ij}^{k}$ include the class-center and the class-centers would be compared by using their dominance degrees with increasing orders.

Next, the picture fuzzy thresholds associate to the profile $b_{p}^{j}$ can be given by: ${\begin{matrix} Q_{q}^{j} = max {{CT}_{p} \cap ℘_{j} - {{PFN}_{p}^{j}}, \\ q = 1, 2, \dots, h}, \\ P_{q}^{j} = max {{CT}_{p} \cap ℘_{j} - {{PFN}_{p}^{j}, Q_{p}^{j}}, \\ q = 1, 2, \dots, h}, \\ V_{q}^{j} = max {{CT}_{p} \cap ℘_{j} - {{PFN}_{p}^{j}, Q_{p}^{j}, P_{p}^{j}}, \\ q = 1, 2, \dots, h} . \end{matrix}$ (14)

Actually, the thresholds corresponding a profile reflect certain preferences of DM. The determination of these parameters is just a selective way, the parameters would be changeable with respect to different attributes.

Step 3. Transformation of picture fuzzy parameters to crisp values.

According to Theorem 1 and Definition 7, picture fuzzy profiles and thresholds determined by Equations (13, 14) would be transformed to crisp values. Since the larger the picture fuzzy number is, the larger the dominance degree would be, so the crisp values corresponding to these parameters would keep the order relations. Then, ${\begin{matrix} b_{q}^{j} = {dd}_{P}^{C} ({pb}_{q}^{j}) \\ q_{q}^{j} = {dd}_{P}^{C} ({pb}_{q}^{j}) - {dd}_{P}^{C} (P_{q}^{j}) \\ p_{q}^{j} = {dd}_{P}^{C} ({pb}_{q}^{j}) - {dd}_{P}^{C} (Q_{q}^{j}) \\ v_{q}^{j} = {dd}_{P}^{C} ({pb}_{q}^{j}) - {dd}_{P}^{C} (V_{q}^{j}) \end{matrix},$ (15) where $b_{q}^{j}, q_{q}^{j}, p_{q}^{j}$ and $v_{q}^{j}$ are crisp profile and thresholds under the j-th attribute.

As a result, the calculation of the credibility index can be computed according to Eqs. (2–5). Noting that the thresholds could be provided with the same value by the descending order if referenced values are not sufficient.

Stage 3. Determination of the (P, I, R) structure

With the credibility index, by giving the “cut level” λ, the preference relation, the indifference relation and the incomparability relation can be further derived.

The flow chart of the developed decision procedure is shown in Fig. 1. As shown in Fig. 1, both of the aggregation of picture fuzzy information by using the normalized weighted Bonferroni mean and the combination of picture fuzzy group decision making and ELECTRE TRI are provided.

Fig. 1

Flow chat of the developed decision procedure.

4 Numerical example

4.1 Case study

In a narrow sense, supplier selection refers to the process of selecting one or more suppliers after the proposal and quotation of the enterprise, which plays an important role for the production of enterprises [57]. The illustrative case is adapted from a practical supplier selection of a pump enterprise. The enterprise requires three departments include accounting department, quality inspection department and production department to make evaluations on candidate suppliers after first round filtering. To avoid the multi-modal information provided by different apartments, the concept of picture fuzzy sets is utilized. Five candidates (A₁, A₂, A₃, A₄ and A₅) in China are evaluated and divided into 4 categories. The attributes used to evaluate these candidates are: C₁ - economic cost, C₂ - product quality, C₃ - deliver ability and C₄ - after-sales service. Three decision makers are involved in this decision process.

Stage 1. Aggregation of group decision information

According to Equation (8), the distances of decision matrix given by each expert with other decision matrices can be obtained, we have dist⁽¹⁾ = 0.0216, dist⁽²⁾ = 0.0279 and dist⁽³⁾ = 0.0298. Then, by Equation (9), the weighting vector of experts is derived and listed in the following: $ϖ = {0.3350, 0.3328, 0.3322} .$

By using the weighting method of experts, the judgements given by the decision makers in Table 1 are aggregated using the picture fuzzy normalized weighted Bonferroni mean (PFNWBM, p = q = 1). Aggregated results are shown in Table 2.

Table 1
Judgement of individual alternative

C ₁ C ₂ C ₃ C ₄

DM₁ (0.3350) A ₁ (0.88,0.02,0.09) (0.69,0.16,0.11) (0.78,0.09,0.11) (0.77,0.08,0.13)

A ₂ (0.90,0.01,0.08) (0.80,0.08,0.09) (0.83,0.08,0.07) (0.76,0.14,0.09)

A ₃ (0.91,0.02,0.05) (0.78,0.06,0.11) (0.81,0.10,0.08) (0.72,0.10,0.12)

A ₄ (0.89,0.01,0.04) (0.74,0.14,0.09) (0.78,0.12,0.09) (0.69,0.14,0.11)

A ₅ (0.89,0.02,0.08) (0.71,0.18,0.09) (0.79,0.12,0.07) (0.75,0.08,0.14)

DM₂ (0.3328) A ₁ (0.86,0.06,0.05) (0.67,0.15,0.12) (0.80,0.09,0.10) (0.76,0.08,0.13)

A ₂ (0.87,0.05,0.07) (0.82,0.08,0.08) (0.82,0.06,0.08) (0.78,0.10,0.08)

A ₃ (0.90,0.03,0.05) (0.82,0.03,0.12) (0.80,0.05,0.09) (0.75,0.08,0.12)

A ₄ (0.87,0.05,0.04) (0.73,0.13,0.04) (0.75,0.14,0.08) (0.72,0.10,0.12)

A ₅ (0.90,0.01,0.08) (0.67,0.20,0.08) (0.82,0.05,0.08) (0.71,0.12,0.13)

DM₃ (0.3322) A ₁ (0.86,0.02,0.11) (0.70,0.14,0.11) (0.75,0.11,0.10) (0.79,0.08,0.11)

A ₂ (0.91,0.01,0.05) (0.78,0.06,0.09) (0.83,0.07,0.05) (0.69,0.19,0.04)

A ₃ (0.92,0.01,0.04) (0.83,0.02,0.11) (0.78,0.13,0.05) (0.70,0.11,0.13)

A ₄ (0.89,0.01,0.04) (0.72,0.13,0.09) (0.80,0.10,0.09) (0.65,0.24,0.08)

A ₅ (0.84,0.10,0.05) (0.74,0.13,0.10) (0.76,0.09,0.10) (0.80,0.02,0.16)

		C ₁	C ₂	C ₃	C ₄
DM₁ (0.3350)	A ₁	(0.88,0.02,0.09)	(0.69,0.16,0.11)	(0.78,0.09,0.11)	(0.77,0.08,0.13)
	A ₂	(0.90,0.01,0.08)	(0.80,0.08,0.09)	(0.83,0.08,0.07)	(0.76,0.14,0.09)
	A ₃	(0.91,0.02,0.05)	(0.78,0.06,0.11)	(0.81,0.10,0.08)	(0.72,0.10,0.12)
	A ₄	(0.89,0.01,0.04)	(0.74,0.14,0.09)	(0.78,0.12,0.09)	(0.69,0.14,0.11)
	A ₅	(0.89,0.02,0.08)	(0.71,0.18,0.09)	(0.79,0.12,0.07)	(0.75,0.08,0.14)
DM₂ (0.3328)	A ₁	(0.86,0.06,0.05)	(0.67,0.15,0.12)	(0.80,0.09,0.10)	(0.76,0.08,0.13)
	A ₂	(0.87,0.05,0.07)	(0.82,0.08,0.08)	(0.82,0.06,0.08)	(0.78,0.10,0.08)
	A ₃	(0.90,0.03,0.05)	(0.82,0.03,0.12)	(0.80,0.05,0.09)	(0.75,0.08,0.12)
	A ₄	(0.87,0.05,0.04)	(0.73,0.13,0.04)	(0.75,0.14,0.08)	(0.72,0.10,0.12)
	A ₅	(0.90,0.01,0.08)	(0.67,0.20,0.08)	(0.82,0.05,0.08)	(0.71,0.12,0.13)
DM₃ (0.3322)	A ₁	(0.86,0.02,0.11)	(0.70,0.14,0.11)	(0.75,0.11,0.10)	(0.79,0.08,0.11)
	A ₂	(0.91,0.01,0.05)	(0.78,0.06,0.09)	(0.83,0.07,0.05)	(0.69,0.19,0.04)
	A ₃	(0.92,0.01,0.04)	(0.83,0.02,0.11)	(0.78,0.13,0.05)	(0.70,0.11,0.13)
	A ₄	(0.89,0.01,0.04)	(0.72,0.13,0.09)	(0.80,0.10,0.09)	(0.65,0.24,0.08)
	A ₅	(0.84,0.10,0.05)	(0.74,0.13,0.10)	(0.76,0.09,0.10)	(0.80,0.02,0.16)

Table 2

Aggregation of judgements of individual alternative

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.8667,0.0318,0.0826)	(0.6867,0.1500,0.1133)	(0.7768,0.0966,0.1033)	(0.7733,0.0800,0.1233)
A ₂	(0.8936,0.0209,0.0665)	(0.8001,0.0732,0.0866)	(0.8267,0.0699,0.0664)	(0.7436,0.1426,0.0694)
A ₃	(0.9101,0.0196,0.0466)	(0.8101,0.0357,0.1133)	(0.7967,0.0923,0.0730)	(0.7234,0.0965,01233)
A ₄	(0.8834,0.0209,0.0400)	(0.7300,0.1333,0.0727)	(0.7767,0.1198,0.0866)	(0.6867,0.1581,0.1031)
A ₅	(0.8772,0.0369,0.0698)	(0.7067,0.1697,0.0899)	(0.7901,0.0858,0.0831)	(0.7534,0.0711,0.1432)

Stage 2. Calculation of the credibility index

According to Equation (11), the weights associate to attributes with aggregated PFNs can be obtained, and we have $ω = {0 . 2129, 0 . 3219, 0 . 1952, 0.2701}^{T} .$

Let ℘_j, j = 1, 2, 3, 4 be the picture fuzzy decision information given by all expert, assume that all alternatives would be partitioned into 4 categories (A, B, C, D and A ≻ B ≻ C ≻ D). Then, by using the evolutionary clustering analysis and Eqs. (13–15), the parameters can be collected and summarized in Table 3 † .

Table 3

Parameters for the outranking of alternatives in ELECTRE TRI

	C1	C2	C3	C4
b3	0.9275	0.8655	0.8746	0.8387
b2	0.9088	0.8187	0.8643	0.8353
b1	0.8990	0.8074	0.8550	0.7988
Q	0.0036	0.0137	0.0053	0.0163
P	0.0069	0.0158	0.0133	0.0278
V	0.0097	0.0178	0.0163	0.0291
W	0.2129	0.3219	0.1952	0.2701
a1	0.8240	0.7774	0.8335	0.8240
a2	0.8240	0.8456	0.8679	0.8240
a3	0.7889	0.8395	0.8504	0.7889
a4	0.7807	0.8094	0.8426	0.7807
a5	0.8038	0.8014	0.8419	0.8038

As shown in Table 3, a1 to a5 (i.e., A₁ to A₅) are dominance degrees of aggregated PFNs listed in Table 2. All profiles and thresholds are computed according to the evolutionary clustering analysis and determination rules. Let λ = 0.8, by using the software J-Electre-v2.0, solutions are listed in Table 4 to Table 6.

Table 4

The concordance index c_j (a, b_h)

Concordance	g1	g2	g3	g4	C(a;b)
c(ai;bh):
c(a1;b3)	0	0	0	1	0.2701
c(a2;b3)	0	0	0.8295	1	0.432
c(a3;b3)	0	0	0	0	0
c(a4;b3)	0	0	0	0	0
c(a5;b3)	0	0	0	0	0
c(a1;b2)	0	0	0	1	0.2701
c(a2;b2)	0	1	1	1	0.7871
c(a3;b2)	0	1	0	0	0.3219
c(a4;b2)	0	1	0	0	0.3219
c(a5;b2)	0	0	0	0	0
c(a1;b1)	0	0	0	1	0.2701
c(a2;b1)	0	1	1	1	0.7871
c(a3;b1)	0	1	1	1	0.7871
c(a4;b1)	0	1	0.1142	0.8478	0.5731
c(a5;b1)	0	1	0.0291	1	0.5976

Table 5

The discordance index c (A_i, b_h)

Discordance	g1	g2	g3	g4
d(ai;bh):
d(a1;b3)	1	1	1	0
d(a2;b3)	1	1	0	0
d(a3;b3)	1	1	1	1
d(a4;b3)	1	1	1	1
d(a5;b3)	1	1	1	1
d(a1;b2)	1	1	1	0
d(a2;b2)	1	0	0	0
d(a3;b2)	1	0	0.2018	1
d(a4;b2)	1	0	1	1
d(a5;b2)	1	0.7645	1	1
d(a1;b1)	1	1	1	0
d(a2;b1)	1	0	0	0
d(a3;b1)	1	0	0	0
d(a4;b1)	1	0	0	0
d(a5;b1)	1	0	0	0

Table 6

The credibility

Credibility:	(ai;bh)	(bh;ai)
cr(a1;b3)	0	cr(b3;a1)	1
cr(a2;b3)	0	cr(b3;a2)	1
cr(a3;b3)	0	cr(b3;a3)	1
cr(a4;b3)	0	cr(b3;a4)	1
cr(a5;b3)	0	cr(b3;a5)	1
cr(a1;b2)	0	cr(b2;a1)	1
cr(a2;b2)	0	cr(b2;a2)	0
cr(a3;b2)	0	cr(b2;a3)	0
cr(a4;b2)	0	cr(b2;a4)	1
cr(a5;b2)	0	cr(b2;a5)	1
cr(a1;b1)	0	cr(b1;a1)	0.7912
cr(a2;b1)	0	cr(b1;a2)	0
cr(a3;b1)	0	cr(b1;a3)	0
cr(a4;b1)	0	cr(b1;a4)	1
cr(a5;b1)	0	cr(b1;a5)	1

Stage 3. Determination of the (P, I, R) structure

Based on the preference relations in Table 6 and the procedure in Subsection 3.2.3, the five areas are categorized as presented in Table 7.

Table 7

Result of ELECTRE TRI categorization

Alternative	Pessimist	Optimist
a1	D	C
a2	D	B
a3	D	B
a4	D	D
a5	D	D

The results listed in Table 7 show that both A₂ and A₃ are better than the other three candidates according to the optimist rule, A₃ locates in the middle, while the rankings of A₄ and A₅ are lower.

4.2 Comparative study

In order to show the efficiency and validity of our developed model, the picture fuzzy weighted averaging operator (PFWA), the picture fuzzy geometric averaging operator (PFWGA) and TOPSIS method (positive ideal point 〈1 0 0〉 and negative ideal point 〈0 0 1〉) are also considered. With the weighting vector of attributes, the calculated results are listed in the following Table 8.

Table 8
Comparative results given by different methods

PFWA PFWGA TOPSIS

Aggregated results dd Aggregated results dd Closeness degree

A ₁ (0.7492,0.1016,0.1159) 0.8118 (0.7449,0.1063,0.1162) 0.8099 0.7856

A ₂ (0.7808,0.1001,0.0709) 0.8397 (0.7772,0.1068,0.0744) 0.8390 0.8189

A ₃ (0.7692,0.0694,0.1083) 0.8177 (0.7645,0.0765,0.1105) 0.8161 0.8052

A ₄ (0.7205,0.1418,0.0890) 0.8035 (0.7174,0.1428,0.0902) 0.8012 0.7752

A ₅ (0.7474,0.0976,0.1109) 0.8092 (0.7449,0.1069,0.1148) 0.8101 0.7860

PFWA	PFWGA		TOPSIS
A ₁	(0.7492,0.1016,0.1159)	0.8118	(0.7449,0.1063,0.1162)	0.8099	0.7856
A ₂	(0.7808,0.1001,0.0709)	0.8397	(0.7772,0.1068,0.0744)	0.8390	0.8189
A ₃	(0.7692,0.0694,0.1083)	0.8177	(0.7645,0.0765,0.1105)	0.8161	0.8052
A ₄	(0.7205,0.1418,0.0890)	0.8035	(0.7174,0.1428,0.0902)	0.8012	0.7752
A ₅	(0.7474,0.0976,0.1109)	0.8092	(0.7449,0.1069,0.1148)	0.8101	0.7860

By Table 8, noting that the larger the dominance degree or closeness degree is, the better the candidate would be. The rankings of all candidates can be derived, we have:

PFWA: A₂ ≻ A₃ ≻ A₁ ≻ A₅ ≻ A₄;

PWGA: A₂ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₄;

TOPSIS: A₂ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₄.

As a result, it can be seen that the categories given by our method is the same as three selective approaches. Although efficient, there would be advantages and disadvantages among these decision technologies. Both of PFWA and PFWGA can provide detailed aggregation results of given decision matrices, in which the algebra structure and comparison laws of PFNs are needed. Besides, possible information loss would be existed when making aggregation. The TOPSIS method compares all candidates with respect to ideal points, but PFNs locate on the perpendicular line cannot be distinguished. While the developed group decision procedure utilizes algebra structure and operational rules of PFNs, and considers multiple attributes simultaneously with certain subjective preferences of DMs. In practical decision problems, the developed group decision model is suitable to the situation that both of objective algebra structure and subjective attitude need to be integrated.

It can be seen that the developed combination of ELECTRE TRI and picture fuzzy group decision making provides a novel method for the ranking of alternatives when considering multiple attributes (or criteria). The advantage of the developed method is that it could compare alternatives in an objective way with more feasibility in real decision problems.

4.3 Discussion on the developed outranking model

Group decision making is a common decision situation in practical economic and management problems. Decision information is the base to derive scientific decision, while the attitude of decision makers is also important in group decision process. In the developed ELECTRE TRI based outranking method for MAGDM with picture fuzzy information, the strengths of such outranking approach can be summarized according to:

To avoid multi-modal decision information provided by different apartments and to describe the uncertainties in real-world decision problems, picture fuzzy information is utilized. A novel comparison law and a novel aggregation method is given.

In group decision process, objective information aggregation is considered. With the information aggregation process, individual assessments can be integrated so that the MAGDM decision problem can be simplified.

In group decision process, the attitude of decision maker(s) can also be included by using the ELECTRE TRI method. Besides, an objective parameter determination method is discussed.

5 Conclusions and remarks

The multi-attribute group decision model with regard to categorization that combines picture fuzzy sets and ELECTRE TRI as an approach to deal with the evaluation of alternatives is proposed and tested in the previous part. The weight of specialist is calculated based on ranking vectors, the weight of attributes is computed based on the maximum deviation principle. The scores of the alternatives and the weights of the criteria have been aggregated. And then they were use as input to the ELECTRE TRI computation, as well as the parameters of ELECTRE TRI. ELECTRE TRI is used to categorize alternatives and output the results of the decision making problems.

In the future, the developed outranking can be extended in two ways. The first would be the application of other types of ELECTRE method include ELECTRE II, III, and ELECTRE TRI–B/C et al. Next, the outranking method can be further applied to other kinds of picture fuzzy information. For example, other types of picture fuzzy information [58, 63], complex intuitionistic fuzzy sets [59], Pythagorean fuzzy [60, 62] and interval-valued q-rung orthopair 2-tuple linguistic [61, 64].

Footnotes

Acknowledgments

The work was supported by National Natural Science Foundation of China (Nos. 71701001, 71771001, 71871001, 71901001, and 71901088), the Social Science Innovation and Development Research Project in Anhui Province (2019CX094), the Key Research Project of Humanities and Social Sciences in Colleges and Universities of Anhui Province (SK2018A0025, SK2019A0013). The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which would have helped immensely in improving the quality of this paper.

The calculation is realized by using the software J-Electre-v2.0, three thresholds are the averages of corresponding thresholds associate to profiles.

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