A multi-stage group decision making for strategic supplier selection based on prospect theory with interval-valued q-rung orthopair fuzzy linguistic sets

Abstract

With the aggravation of market competition, strategic supplier is becoming more and more critical for the success of manufacturing enterprises. Suppler selection, being the critical and foremost activity must ensure that selected suppliers are capable of supporting the long-term development of organizations. Hence, strategic supplier selection must be restructures considering the long-term relationships and prospects for sustainable cooperation. This paper proposes a novel multi-stage multi-attribute group decision making method under an interval-valued q-rung orthopair fuzzy linguistic set (IVq-ROFLS) environment considering the decision makers’ (DMs) psychological state in the group decision-making process. First, the initial comprehensive fuzzy evaluations of DMs are represented as IVq-ROFLS. Subsequently, two new operators are proposed for aggregating different stages and DMs’ preferences respectively by extending generalized weighted averaging (GWA) to IVq-ROFLS context. Later, a new hamming distance based linear programming method based on entropy measure and score function is introduced to evaluate the unknown criteria weights. Additionally, the Euclidean distance is employed to compute the gain and loss matrix, and objects are prioritized by extending the popular Prospect theory (PT) method to the IVq-ROFLS context. Finally, the practical use of the proposed decision framework is validated by using a strategic supplier selection problem, as well as the effectiveness and applicability of the framework are discussed by using comparative analysis with other methods.

Keywords

Strategic supplier selection multi-stage multi-attribute group decision making interval-valued q-rung orthopair fuzzy linguistic set hamming distance based linear programming prospect theory

1 Introduction

With the rapid development of Industry 4.0, the manufacturing industry is undergoing a breakthrough transformation [1]. Manufacturing enterprises are faced with urgently needed to be addressed problems which are embodied in huge inventory pressure, non-sharing of information with suppliers and low production profits. More and more enterprises seek long-term cooperation in supplier networks to form a stable manufacturing industry alliance, which can reduce cost through scale effect, reduce inventory pressure and realize information exchange. Thus, strategic supplier selection is becoming a critical aspect of enterprises’ competitiveness [2]. Considering the requirement of long-term sustainable development between strategic suppliers and enterprises, the process of strategic supplier selection often goes through the comprehensive consideration of the sales, technical, financial, purchasing and other relevant departments. It can be considered as a process of generating deterministic group preference by utilizing different preferences for a set of attributes and DMs [3]. Hence, strategic supplier selection can be also regarded as an extension of the multi-attribute group decision making (MAGDM) problem.

In recent years, many scholars have done a lot of research on the MAGDM problem for supplier selection [4 –8]. However, from the perspective of long-term cooperation of strategic suppliers, the comprehensive performances of suppliers in a multi-stage dimension are vital but rarely mentioned. Olanrewaju, Dong and Hu [9] studied a multi-stage stochastic programming model for supplier selection in disaster response. Xue, Fu, Feng, Lu, Chang and Yang [10] considered an evidential reasoning based multi-stage approach for evaluation of high-speed train supplier. Kaur and Prakash Singh [11] proposed a multi-stage hybrid model for supplier selection and order allocation problem. The above literature mainly considered the supplier selection problem with multi-stage single decision maker participation. Therefore, a novel multi-stage MAGDM structure for multi-stage evaluations and group decision making process respectively aggregated to select strategic supplier is proposed.

Since the judgments evaluated by DMs are subjective, vague and imprecise, uncertainty is an important challenging problem in strategic supplier selection problem [12]. To handle this concern, Zadeh [13] established the traditional fuzzy set (TFS) theory. Subsequently, various scholars extended the TFS into intuitionistic fuzzy set (IFS) [14], interval-valued intuitionistic fuzzy set (IVIFS) [15], Pythagorean fuzzy set (PFS) [16], interval-valued Pythagorean fuzzy set (IVPFS) [17] to describe the uncertain information in MAGDM problems. Recently, a novel fuzzy set called q-rung orthopair fuzzy set (q-ROFS) [18] was proposed which can provide a wider and more flexible space for preference elicitation than the above fuzzy sets [19]. The q-ROFS is redefined by Yager [18] by enlarging the scope which can be used to maximize the accuracy and integrity of fuzzy information, where the q ⩾ 1 in which qth power sum of membership degree (MD) and non-membership degree (NMD) is not bigger than one. Besides, IFS and PFS are the special cases of q-ROFS by setting q = 1 and q = 2 respectively. The difference of them can be seen in Fig. 1. Although q-ROFS is an effective tool for describing fuzzy and uncertain information, it cannot express evaluations qualitatively in strategic supplier selection. Considering this, linguistic term set (LTS) [20] can solve the problem of qualitative evaluations and it is regarded as one another effective way to express the ambiguity of DMs’ opinions. Hence, to better deal with qualitative evaluations and uncertainty, we combine LTS [21] and interval-valued q-rung orthopair fuzzy set (IVq-ROFS) [22] to a new fuzzy set called IVq-ROFLS. Based on IVq-ROFLS, the uncertainty of the preference information from DMs in strategic supplier selection process will be evaluated qualitatively well.

Fig. 1

Space range of IFS, PFS and q-ROFS.

In MAGDM, the determination of the optimal attribute weights is the crucial part of the rank of alternatives. In the existing approaches, the acquisition methods of attribute weights are mainly carried out in the following two ways: one is the objective weights using objective decision information [23 –25]; the other is the subjective weights considering the subjective preferences of DMs [26 –28]. But only considering the objective or subjective weights may ignore the effect of the other one. Given this situation, the method of obtaining attribute weights considering both objective information and subjective preferences is being studied. Liang, Goh and Wang [29] proposed a minimum relative entropy based method to determine the attribute weights considering both objective information and subjective preferences. Ding, Liu and Shi [30] employed a parameter to combine subjective and objective weights. However, they were somewhat arbitrary in their determination of final weights. Here, we apply the hamming distance based linear programming method to propose a novel method to determine the optimal attribute weights considering both subjective preferences and objective information on score function and entropy method.

Most of the existing MAGDM methods assume that DMs are totally rational [30], but in realistic decision making process, DMs exercise limited rationality under the risk which comes from the interaction of one person with others and is hard to avoid [31]. In consideration of this situation, the prospect theory (PT), proposed by Kahneman and Tversky [32] and developed by Tversky and Kahneman [33], is a decision theory that can accurately capture the risk characteristics, psychological behaviors and reference dependence of DMs. It is stated in PT that DMs select the alternatives with the highest prospect value under an assumed risky condition. The gain or loss of each prospect value is an outcome compared with a reference point. The DMs are risk aversion towards gains and risk seeking towards losses [34]. In addition, PT employs a value function and a probability weighting function to convert objective values to subjective ones and probabilities to decision weights respectively [35]. Xu, Huang and Li [36] proposed a PT based direct consensus framework to solve heterogeneous group decision making problem. Wu, Li and Dong [37] studied a PT based method for matching the technology suppliers and demanders. Cheng, Long and Chen [38] employed PT in the research of government policy making. Ding, Liu and Shi [30] applied PT to handle emergency decision making problem. Herrmann, Jong-A-Pin and Schoonbeek [39] incorporated PT in a game-theoretic model to predict voter turnout. Wan, Zou and Dong [40] conducted a research on hybrid fuzzy truth degrees of comparisons by PT. Surti, Celani and Gajpal [41] used PT to carry on newsvendor ordering behavior research. In general, strategic supplier selection entails risk, faced with risk the behavior of DMs conforms to prospect theory: the preferences rely on reference information and exhibit loss aversion, and probabilities are subjectively weighted. Thus, the ranking results can be more reasonable after applying the prospect theory to strategic supplier selection problem.

Therefore, this paper proposes a novel multi-stage multi-attribute group decision making method to select the strategic supplier with a long-term relationship. First, the IVq-ROFLS is used by DMs to qualitatively express the evaluation information of various attributes of potential strategic suppliers. Second, two aggregation operators called GIVq-ROFLWA and IVq-ROFLWA are used to aggregate the IVq-ROFLS evaluations of different periods and DMs respectively. Third, the optimal weights of attributes are calculated by a hamming distance based linear programming method incorporating objective information and subjective preferences of DMs on score function and entropy method. Finally, the Euclidean distance is employed to compute the gain and loss matrix and the prospect theory is used to find out the strategic supplier. Additionally, the case of an enterprise operating in machinery industry is adopted to verify the effectiveness and applicability of the proposed method.

The rest of this paper is structured as follows: In Section 2, the concepts of IVq-ROFLS, hamming distance based linear programming method (HDLP) and Prospect theory are reviewed briefly. In Section 3, the IVq-ROFLS-HDLP-PT approach is presented to solving strategic supplier selection problem. An illustrative case analysis and a comparison analysis are presented in Section 4. Finally, conclusions and future work are drawn in Section 5.

2 Preliminaries

This section presents some basic notions associated with q-ROFS, linear programming method and Prospect Theory.

2.1 IFS, PFS and q-ROFS

Definition 1 [14]. Suppose a universe set X, x is the element of X. Then, an IFS I of X is considered as follows: $I = {< x, μ_{I} (x), υ_{I} (x) > | x \in X}$ (1)

Her μ_I (x) ∈ [01] and υ_I (x) ∈ [01] denote a membership degree and a non-membership degree of x ∈ X to the set I, respectively, satisfying the condition 0 ⩽ μ_I (x) + υ_I (x) ⩽1 for all x ∈ X.

Definition 2 [16]. Suppose a universe set X, x is the element of X. Then, a PFS P of X is considered as follows: $P = {< x, μ_{P} (x), υ_{P} (x) > | x \in X}$ (2)

Her μ_P (x) ∈ [01] and υ_P (x) ∈ [01] denote a membership degree and a non-membership degree of x ∈ X to the set P, respectively, satisfying the condition $0 ⩽ μ_{P}^{2} (x) + υ_{P}^{2} (x) ⩽ 1$ for all x ∈ X.

But in some real-world decision-making process, the sum or the square sum of membership degree and non-membership degree may be bigger than 1. Based on this situation, the q-ROFS [18] is proposed to solve this problem.

Definition 3 [18]. Suppose a universe set X, x is the element of X. Then, a q-ROFS Q of X is considered as follows: $Q = {< x, μ_{Q} (x), υ_{Q} (x) > | x \in X}$ (3)

Here μ_Q (x) ∈ [01] and υ_Q (x) ∈ [01] indicate the membership degree and the non-membership degree of x ∈ X to the set Q, respectively, for all x ∈ X, it satisfies the condition $0 ⩽ μ_{Q}^{q} (x) + υ_{Q}^{q} (x) ⩽ 1$ where q ⩾ 1. The degree of indeterminacy of element x ∈ X is given as π_Q (x) = [(μ_Q (x)) ^q + (υ_Q (x)) ^q] ^1/q. For simplicity, Yager [18] called q = 〈μ, υ〉 a q-rung orthopair fuzzy number (q-ROFN).

Definition 4 [20]. Let S ={ s_i|i = 1, 2, . . . , t } be a linguistic term set (LTS) with finite and totally ordered elements, where t is an even number, then the s_i represents a possible value for a linguistic variable. Usually, the set S is required to satisfy the following characteristics [21].

The set is ordered, if i > j, then s_i > s_j.

neg (s_i) = s_j, where i + j = t + 1.

Besides, a continuous linguistic term set $\bar{S}$ [42] is proposed to extend extended the discrete term S, where $\bar{S} = {s_{1} ⩽ s_{θ} ⩽ s_{t} | λ \in [1, t]}$ , if s_θ ∈ S, then we call s_θ the original linguistic term, otherwise, we call s_θ the virtual linguistic term.

Definition 5 [43, 44]. Suppose that X is a finite and fixed set, x is the element of X, and $\bar{S}$ is a continuous linguistic term set. Then an IVq-ROFLS $\tilde{Q}$ defined on X is expressed as:

$\begin{matrix} \tilde{Q} = & {< x, [s_{θ (x)}, [μ_{\tilde{Q}}^{L} (x), μ_{\tilde{Q}}^{U} (x)], \\ [υ_{\tilde{Q}}^{L} (x), υ_{\tilde{Q}}^{U} (x)]] > | x \in X} \end{matrix}$ (4)

Where $s_{θ (x)} \in \bar{S}$ , $[μ_{\tilde{Q}}^{L} (x), μ_{\tilde{Q}}^{U} (x)] \subset [0, 1]$ and $[υ_{\tilde{Q}}^{L} (x), υ_{\tilde{Q}}^{U} (x)] \subset [0, 1]$ indicate the membership degree and non-membership degree of the element x ∈ X to $\tilde{Q}$ , $μ_{\tilde{Q}}^{L} (x), μ_{\tilde{Q}}^{U} (x), υ_{\tilde{Q}}^{L} (x), υ_{\tilde{Q}}^{U} (x)$ are the lower and upper bounds of the membership degree and non-membership degree, respectively, which meet the condition of $0 ⩽ {(μ_{\tilde{Q}}^{U} (x))}^{q} + {(υ_{\tilde{Q}}^{U} (x))}^{q} ⩽ 1, q ⩾ 1$ .

In addition, the indeterminacy degree of the element x ∈ X to $\tilde{Q}$ is ${\tilde{π}}_{\tilde{Q}} (x) = [π_{\tilde{Q}}^{L} (x), π_{\tilde{Q}}^{U} (x)] = [{(1 - {(μ_{\tilde{Q}}^{U} (x))}^{q} + {(υ_{\tilde{Q}}^{U} (x))}^{q})}^{1 / q}, (1 - {(μ_{\tilde{Q}}^{L} (x))}^{q} {+ {(υ_{\tilde{Q}}^{L} (x))}^{q})}^{1 / q}]$ . For the convenience, we call $\tilde{Q} = 〈 s_{θ}, [μ^{L}, μ^{U}], [υ^{L}, υ^{U}] 〉$ an interval-valued q-rung orthopair fuzzy linguistic number (IVq-ROFLN).

Definition 6 [44]. Let ${\tilde{Q}}_{1} = 〈 s_{θ_{1}}, [μ_{1}^{L}, μ_{1}^{U}], [υ_{1}^{L}, υ_{1}^{U}] 〉$ and ${\tilde{Q}}_{2} = 〈 s_{θ_{2}}, [μ_{2}^{L}, μ_{2}^{U}], [υ_{2}^{L}, υ_{2}^{U}] 〉$ be two IVq-ROFLNs, and then the operational laws of IVq-ROFLN are given as below:

${\tilde{Q}}_{1} \oplus {\tilde{Q}}_{2} = 〈 s_{θ_{1} + θ_{2}}, [\begin{matrix} {({(μ_{1}^{L})}^{q} + {(μ_{2}^{L})}^{q} - {(μ_{1}^{L})}^{q} {(μ_{2}^{L})}^{q})}^{1 / q}, \\ {({(μ_{1}^{U})}^{q} + {(μ_{2}^{U})}^{q} - {(μ_{1}^{U})}^{q} {(μ_{2}^{U})}^{q})}^{1 / q} \end{matrix}], [υ_{1}^{L} υ_{2}^{L}, υ_{1}^{U} υ_{2}^{U}] 〉$ (1) ${\tilde{Q}}_{1} \otimes {\tilde{Q}}_{2} = 〈 s_{θ_{1} \times θ_{2}}, [μ_{1}^{L} μ_{2}^{L}, μ_{1}^{U} μ_{2}^{U}], [\begin{matrix} {({(υ_{1}^{L})}^{q} + {(υ_{2}^{L})}^{q} - {(υ_{1}^{L})}^{q} {(υ_{2}^{L})}^{q})}^{1 / q}, \\ {({(υ_{1}^{U})}^{q} + {(υ_{2}^{U})}^{q} - {(υ_{1}^{U})}^{q} {(υ_{2}^{U})}^{q})}^{1 / q} \end{matrix}] 〉$ (2) $λ {\tilde{Q}}_{1} = 〈 s_{λ θ_{1}}, [\begin{matrix} {(1 - {(1 - {(μ_{1}^{L})}^{q})}^{λ})}^{1 / q}, \\ {(1 - {(1 - {(μ_{1}^{U})}^{q})}^{λ})}^{1 / q} \end{matrix}], [{(υ_{1}^{L})}^{λ}, {(υ_{1}^{U})}^{λ}] 〉, (λ > 0)$ (3)

${\tilde{Q}}_{1}^{λ} = 〈 s_{θ_{1}^{λ}}, [{(μ_{1}^{L})}^{λ}, {(μ_{1}^{U})}^{λ}], [\begin{matrix} {(1 - {(1 - {(υ_{1}^{L})}^{q})}^{λ})}^{1 / q}, \\ {(1 - {(1 - {(υ_{1}^{U})}^{q})}^{λ})}^{1 / q} \end{matrix}] 〉, (λ > 0)$ (4)

Definition 7 [45]. Suppose $\bar{S}$ is a continuous linguistic term set and $\tilde{Q} = 〈 s_{θ}, [μ^{L}, μ^{U}], [υ^{L}, υ^{U}] 〉$ is an IVq-ROFLN, then the score function $S (\tilde{Q})$ and the accuracy function $H (\tilde{Q})$ of $\tilde{Q}$ are defined as follows:

$S (\tilde{Q}) = \frac{θ}{4} [(1 + {(μ^{L})}^{q} - {(υ^{L})}^{q}) + (1 + {(μ^{U})}^{q} - {(υ^{U})}^{q})]$ (5)

$A (\tilde{Q}) = \frac{θ}{2} [{(μ^{L})}^{q} + {(μ^{U})}^{q} + {(υ^{L})}^{q} + {(υ^{L})}^{q}]$ (6)

Let $S ({\tilde{Q}}_{1})$ and $S ({\tilde{Q}}_{2})$ be the scores of ${\tilde{Q}}_{1}$ and ${\tilde{Q}}_{2}$ , $A (\tilde{Q})$ and $A (\tilde{Q})$ be the accuracy degrees of ${\tilde{Q}}_{1}$ and ${\tilde{Q}}_{2}$ , respectively, then when $S ({\tilde{Q}}_{1}) > S ({\tilde{Q}}_{2}), {\tilde{Q}}_{1} > {\tilde{Q}}_{2}$ . When $S ({\tilde{Q}}_{1}) = S ({\tilde{Q}}_{2})$ , if $A ({\tilde{Q}}_{1}) > A ({\tilde{Q}}_{2}), {\tilde{Q}}_{1} > {\tilde{Q}}_{2}$ .

Definition 8. Let $\tilde{Q} = 〈 s_{θ}, [μ^{L}, μ^{U}], [υ^{L}, υ^{U}] 〉$ be an IVq-ROLFS, then the transformation law of $\tilde{Q}$ into an interval-valued variable is defined as follows:

$\begin{matrix} Φ (\tilde{Q}) = 〈 [S (〈 s_{θ}, μ^{L}, υ^{U} 〉), S (〈 s_{θ}, μ^{U}, υ^{L} 〉)] 〉 \\ = 〈 [\frac{θ}{2} ({(μ^{L})}^{q} + 1 - {(υ^{U})}^{q}), \frac{θ}{2} ({(μ^{U})}^{q} + 1 - {(υ^{L})}^{q})] 〉 \end{matrix}$ (7)

Then, an interval-valued q-rung orthopair fuzzy linguistic weighted averaging (IVq-ROFLWA) operator [46] and a generalized interval-valued q-rung orthopair fuzzy linguistic weighted averaging (GIVq-ROFLWA) operator [46] are used to aggregate the linguistic evaluations expressed by IVq-ROLFS.

Definition 9 [46]. Suppose $\tilde{Q} = {{\tilde{Q}}_{j}} (j = 1, 2, . . ., n)$ is a collection of IVq-ROLFS, ${\tilde{Q}}_{j} = 〈 s_{θ_{j}}, [μ_{j}^{L}, μ_{j}^{U}], [υ_{j}^{L}, υ_{j}^{U}] 〉 (j = 1, 2, . . ., n)$ , w = (w₁, w₂, . . . , w_n) ^T is the aggregation-associated vector with w_j ∈ [0, 1] (j = 1, 2, . . . , n) and $\sum_{j = 1}^{n} w_{j} = 1$ . Then IVq-ROFLWA operator is a mapping IVq-ROFLWA: Ωⁿ → Ω, where:

$\begin{array}{l} I V q - R O F L W A ({\tilde{Q}}_{1}, {\tilde{Q}}_{2}, ..., {\tilde{Q}}_{n}) = \oplus_{j = 1}^{n} (w_{j} {\tilde{Q}}_{j}) \\ = 〈 S_{\sum_{j = 1}^{n} w_{j} θ_{j}}, [{(1 - \prod_{j = 1}^{n} {(1 - {(μ_{j}^{L})}^{q})}^{w_{j}})}^{1 / q}, (1 - \prod_{j = 1}^{n} {(1 - {(μ_{j}^{U})}^{q})}^{w_{j}}) 1 / q], [\prod_{j = 1}^{n} {(υ_{j}^{L})}^{w_{j}}, \prod_{j = 1}^{n} {(υ_{j}^{U})}^{w_{j}}] 〉 \end{array}$ (8)

Definition 10 [46]. Suppose $\tilde{Q} = {{\tilde{Q}}_{j}} (j = 1, 2, . . ., n)$ is a collection of IVq-ROLFS, ${\tilde{Q}}_{j} = 〈 s_{θ_{j}}, [μ_{j}^{L}, μ_{j}^{U}], [υ_{j}^{L}, υ_{j}^{U}] 〉 (j = 1, 2, . . ., n)$ , GIVq-ROFLWA operator can be defined as the following mapping GIVq-ROFLWA: Ωⁿ → Ω:

$\begin{matrix} GIVq - ROFLWA ({\tilde{Q}}_{1}, {\tilde{Q}}_{2}, . . ., {\tilde{Q}}_{n}) = {(\oplus_{j = 1}^{n} (w_{j} {({\tilde{Q}}_{j})}^{λ}))}^{1 / λ} \\ = 〈 \begin{matrix} {[S_{\sum_{j = 1}^{n} w_{j} {(θ_{j})}^{λ}}]}^{1 / λ}, [{({(1 - \prod_{j = 1}^{n} {(1 - {(μ_{j}^{L})}^{q λ})}^{w_{j}})}^{1 / q})}^{1 / λ}, {({(1 - \prod_{j = 1}^{n} {(1 - {(μ_{j}^{U})}^{q λ})}^{w_{j}})}^{1 / q})}^{1 / λ}], \\ [{(1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - {(υ_{j}^{L})}^{q})}^{λ})}^{w_{j}})}^{1 / λ})}^{1 / q}, {(1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - {(υ_{j}^{U})}^{q})}^{λ})}^{w_{j}})}^{1 / λ})}^{1 / q}] \end{matrix} 〉 \end{matrix}$ (9)

Where w = (w₁, w₂, . . . , w_n) ^T is the weight vector of ${\tilde{Q}}_{j} (j = 1, 2, . . ., n)$ , such that w_j∈ [0, 1] (j = 1, 2, . . . , n) and $\sum_{j = 1}^{n} w_{j} = 1$ , λ ∈ (0, + ∞) is a parameter and Ω is the set of all IVq-ROFLS.

2.2 Linear programming problem

Definition 11 [47]. A linear programming problem is formulated as follows: $\begin{matrix} Max ϖ = ξ_{1} x_{1} + ξ_{2} x_{2} + . . . + ξ_{n} x_{n} \\ Subject to {\begin{matrix} ζ_{11} x_{1} + ζ_{12} x_{2} + . . . + ζ_{1 n} x_{n} ⩽ ψ_{1} \\ ζ_{21} x_{1} + ζ_{22} x_{2} + . . . + ζ_{2 n} x_{n} ⩽ ψ_{2} \\ \begin{matrix} ⋮ \\ ζ_{m 1} x_{1} + ζ_{m 2} x_{2} + . . . + ζ_{mn} x_{n} ⩽ ψ_{m} \end{matrix} \\ x_{1}, x_{2}, . . . x_{n} ⩾ 0 \end{matrix} \end{matrix}$ (10)

Where n represents the number of decision variables and m represents the number of constraints. The linear programming technology aims at getting the optimal solution for the goal function ϖ.

Definition 12 [48]. Suppose that ${\tilde{E}}_{1} = [e_{1}^{L}, e_{1}^{U}]$ and ${\tilde{E}}_{2} = [e_{2}^{L}, e_{2}^{U}]$ are two interval-valued variables, then the normalized Hamming distance between ${\tilde{E}}_{1}$ and ${\tilde{E}}_{2}$ is defined as follows: $Hd ({\tilde{E}}_{1}, {\tilde{E}}_{2}) = \frac{1}{2} (| e_{1}^{L} - e_{2}^{L} | + | e_{1}^{U} - e_{2}^{U} |)$ (11)

2.3 Prospect theory

Prospect theory [32] is often used to reflect decision-makers’ subjective risk preference on the basis of limited rationality. Based on PT, a decision-making process is divided into two stages: the editing stage and the evaluation stage. In the editing stage, decision maker sets a reference point and then the outcomes of alternatives are coded as gains and losses relative to the reference point. In the evaluation stage, the edited prospects are evaluated by a value function and a weighting function, then the prospect of highest value is selected.

Suppose x₀ is the reference point and x_j is the existing prospect value, and then Δx_j = x_j - x₀ is the deviation degree between x₀ and x_j, Δx_j > 0 means existing prospect value can get gains, otherwise it means losses. Then the value function V (Δx_j) is computed as follows [33]: $V (Δ x_{j}) = {\begin{matrix} {(Δ x_{j})}^{α}, Δ x_{j} ⩾ 0 \\ - η {(- Δ x_{j})}^{β}, Δ x_{j} < 0 \end{matrix}$ (12)

Where α and β reflect the concave-convex degree parameter related to the gains and losses. 0 ⩽ α, β ⩽ 1, the greater the value α and β are, the more tendentious the decision maker is to the risk. η is loss-aware parameter, η > 1 means that decision maker is more sensitive to the losses than to the absolutely commensurate gains. Based on the previous experiments, the recommended values for α, β and η are 0.88, 0.88 and 2.25 [33].

The probability weighting function π (w_j) is expressed as follows: $π (w_{j}) = {\begin{matrix} \frac{w_{j}^{ɛ}}{{[w_{j}^{ɛ} + {(1 - w_{j})}^{ɛ}]}^{1 / ɛ}}, Δ x_{j} ⩾ 0 \\ \frac{w_{j}^{δ}}{{[w_{j}^{δ} + {(1 - w_{j})}^{δ}]}^{1 / δ}}, Δ x_{j} < 0 \end{matrix}$ (13)

Where ɛ and δ reflects the different attitude of DMs towards the risk of gains and losses. The values of ɛ and δ were 0.61 and 0.72 [32].

Through the value function V (Δx_j) and the probability weighting function π (w_j), we can calculate the prospect value by the following formula: $V = \sum_{j = 1}^{m} π (w_{i}) V (Δ x_{i})$ (14)

3 The proposed approach for multi-stage multi-attribute group decision making problem

In this section, a novel multi-stage multi-attribute group decision making approach is developed for selecting strategic supplier. Let K ={ k₁, k₂, . . . , k_m } be a set of alternative strategic suppliers, C ={ c₁, c₂, . . . , c_n } be a predefined set of attributes. T ={ t₁, t₂, . . . , t_l } is a set of periods and the weight vector of these periods is denoted as w_T = (w_{t
₁}, w_{t
₂}, . . . , w_{t
_l}) ^T, where $\sum_{k = 1}^{l} w_{t_{k}} = 1$ and w_{t
_k} ⩾ 0 (k = 1, 2, . . . , l). D ={ d₁, d₂, . . . , d_r } is a set of DMs and the weight of these DMs is denoted ad w_D = (w_{d
₁}, w_{d
₂}, . . . , w_{d
_r}) ^T, where $\sum_{s = 1}^{r} w_{d_{s}} = 1$ and w_{d
_s} ⩾ 0 (s = 1, 2, . . . , r). The basic procedure of the proposed approach is shown in Fig. 2 and the detailed steps of the approach are described as follows.

Fig. 2

Flowchart of the proposed approach for strategic supplier selection problem.

3.1 Obtain and standardize the decision matrices

Step 1: Obtain the interval linguistic evaluations for strategic suppliers. ${\tilde{Q}}^{(k s)} = {[{\tilde{Q}}_{i j}^{(k s)}]}_{m \times n} (k = 1, 2, ..., l s = 1, 2, ..., r)$ is the decision matrix given by d_s on t_k, where ${\tilde{Q}}_{i j}^{(k s)} = 〈 s_{θ_{i j}^{(k s)}}, [μ_{i j}^{(k s) L}, μ_{i j}^{(k s) U}], [υ_{i j}^{(k s) L}, υ_{i j}^{(k s) U}] 〉 (k = 1, 2, ..., l s = 1, 2, ..., r)$ is an IVq-ROFLS representing the evaluations of k_i with respect to attribute c_j, s_{θ
_ijks} is the linguistic evaluation on the basis of LTS S ={ s_i|i = 1, 2, . . . , t }, $[μ_{ij}^{(ks) L}, μ_{ij}^{(ks) U}]$ and $[υ_{ij}^{(ks) L}, υ_{ij}^{(ks) U}]$ are the lower and upper bounds of MD and NMD, respectively.

Step 2: Standardize the obtained decision matrix. The original matrix ${\tilde{Q}}^{(ks)}$ can be standardized to decision matrix ${\tilde{R}}^{(ks)} = {[r_{ij}^{(ks)}]}_{m \times n} (k = 1, 2, . . ., l s = 1, 2, . . ., r)$ by the formula as follows: $r_{ij}^{(ks)} = {\begin{matrix} 〈 s_{θ_{ij}^{(ks)}}, [μ_{ij}^{(ks) L}, μ_{ij}^{(ks) U}], [υ_{ij}^{(ks) L}, υ_{ij}^{(ks) U}] 〉, (forbenefitcriteria) \\ 〈 s_{θ_{ij}^{(ks)}}, [υ_{ij}^{(ks) L}, υ_{ij}^{(ks) U}], [μ_{ij}^{(ks) L}, μ_{ij}^{(ks) U}] 〉, (forcostcriteria) \end{matrix}$ (15)

Where, if this criterion is a benefit criterion (such as profit, quality.etc.), we take the upper standardized formula; if this criterion is a cost criterion (such as delivery time.etc.), we take the below standardized formula.

3.2 Aggregate the linguistic evaluations of different periods and DMs

Step 3: Aggregate the decision matrix ${\tilde{R}}^{(ks)}$ of all the periods by same DMs into a novel decision matrix ${\tilde{R}}^{(s)} (s = 1, 2, . . ., r)$ . To highlight the timeliness of information, an aggregation operator called GIVq-ROFLWA is shown as follows:

$\begin{matrix} \begin{matrix} GIVq - ROFLWA ({\tilde{R}}^{(1 s)}, {\tilde{R}}^{(2 s)}, . . ., {\tilde{R}}^{(ls)}) = {(\oplus_{k = 1}^{l} (w_{t_{k}} {({\tilde{R}}^{(ks)})}^{λ}))}^{1 / λ} \\ = 〈 \begin{matrix} {[S_{\sum_{k = 1}^{l} w_{t_{k}} {(θ_{ij}^{(ks)})}^{λ}}]}^{1 / λ}, [{({(1 - \prod_{k = 1}^{l} {(1 - {({(μ_{ij}^{(ks)})}^{L})}^{q λ})}^{w_{t_{k}}})}^{1 / q})}^{1 / λ}, {({(1 - \prod_{k = 1}^{l} {(1 - {({(μ_{ij}^{(ks)})}^{U})}^{q λ})}^{w_{t_{k}}})}^{1 / q})}^{1 / λ}], \\ [{(1 - {(1 - \prod_{k = 1}^{l} {(1 - {(1 - {({(υ_{ij}^{(ks)})}^{L})}^{q})}^{λ})}^{w_{t_{k}}})}^{1 / λ})}^{1 / q}, {(1 - {(1 - \prod_{k = 1}^{l} {(1 - {(1 - {({(υ_{ij}^{(ks)})}^{U})}^{q})}^{λ})}^{w_{t_{k}}})}^{1 / λ})}^{1 / q}] \end{matrix} 〉 \end{matrix} \end{matrix}$ (16)

Step 4: Aggregate the decision matrix ${\tilde{R}}^{(s)}$ of all DMs into a novel decision matrix $\tilde{R}$ . An aggregation operator called IVq-ROFLWA is shown as follows:

$\begin{matrix} IVq - ROFLWA ({\tilde{R}}^{(1)}, {\tilde{R}}^{(2)}, . . ., {\tilde{R}}^{(r)}) = \oplus_{s = 1}^{r} (w_{d_{s}} {\tilde{R}}^{(s)}) \\ = 〈 S_{\sum_{s = 1}^{r} w_{d_{s}} θ_{ij}^{(s)}}, [\begin{matrix} {(1 - \prod_{s = 1}^{r} {(1 - {(μ_{ij}^{(s) L})}^{q})}^{w_{d_{s}}})}^{1 / q}, \\ {(1 - \prod_{s = 1}^{r} {(1 - {(μ_{ij}^{(s) U})}^{q})}^{w_{d_{s}}})}^{1 / q} \end{matrix}], [\prod_{s = 1}^{r} {(υ_{ij}^{(s) L})}^{w_{d_{s}}}, \prod_{s = 1}^{r} {(υ_{ij}^{(s) U})}^{w_{d_{s}}}] 〉 \end{matrix}$ (17)

Step 5: After step 2, 3, 4, all the initial interval linguistic decision matrices ${\tilde{Q}}^{(ks)}$ become one decision matrix $\tilde{R} = {[{\tilde{r}}_{ij}]}_{m \times n}$ . Then all the IVq-ROFLNs can be transformed into interval-valued q-rung orthopair fuzzy variables by formula as follows:

$\begin{matrix} Φ ({\tilde{r}}_{ij}) = 〈 [S (〈 s_{θ_{ij}}, μ_{ij}^{L}, υ_{ij}^{U} 〉), S (〈 s_{θ_{ij}}, μ_{ij}^{U}, υ_{ij}^{L} 〉)] 〉 \\ = 〈 [\frac{θ_{ij}}{2} ({(μ_{ij}^{L})}^{q} + 1 - {(υ_{ij}^{U})}^{q}), \frac{θ_{ij}}{2} ({(μ_{ij}^{U})}^{q} + 1 - {(υ_{ij}^{L})}^{q})] 〉 \end{matrix}$ (18)

Step 6: Identify PIS and NIS.

${\begin{matrix} PIS = {\tilde{r}}_{j}^{+} = [{max}_{i = 1}^{m} ({\tilde{r}}_{ij}^{L}), {max}_{i = 1}^{m} ({\tilde{r}}_{ij}^{U})] \\ NIS = {\tilde{r}}_{j}^{-} = [{min}_{i = 1}^{m} ({\tilde{r}}_{ij}^{L}), {min}_{i = 1}^{m} ({\tilde{r}}_{ij}^{U})] \end{matrix}, (j = 1, 2, . . ., n)$ (19)

Where ${\tilde{r}}_{j}^{+}$ and ${\tilde{r}}_{j}^{-}$ represent the PIS and NIS of attribute c_j.

3.3 Employ linear programming method to calculate optimal weights

Step 7: Calculate the objective weight.

The entropy method [49] calculates weights based on the information of each attribute. Based on the Chen [50], the steps of entropy method are as follows:

Step 7.1: Calculate score and accuracy matrix. By employing the following equation and the decision matrix $\tilde{R} = {[{\tilde{r}}_{ij}]}_{m \times n}$ , we can get score matrix S = [s_ij] _m×n.

$\begin{matrix} s_{ij} = & S ({\tilde{r}}_{ij}) = \frac{θ_{ij}}{4} [(1 + {(μ_{ij}^{L})}^{q} - {(υ_{ij}^{L})}^{q}) \\ + (1 + {(μ_{ij}^{U})}^{q} - {(υ_{ij}^{U})}^{q})] \end{matrix}$ (20)

Step 7.2: Normalize the score matrix S = [s_ij] _m×n and the calculation equation is written as: $b_{ij}^{S} = s_{ij} / \sum_{i = 1}^{m} s_{ij}$ (21)

Step 7.3: Calculate the entropy of each attribute by the equation: $e_{j}^{S} = - K \sum_{i = 1}^{m} b_{ij}^{S} ln b_{ij}^{S}$ (22)

Where $e_{j}^{S}$ is the score entropy of the jth attribute. K is a constant and K = 1/ln m. Obviously, $0 ⩽ e_{j}^{S} ⩽ 1$ . When $b_{ij}^{S} = 0$ , then $ln b_{ij}^{S} = 0$ [51].

Step 7.4: Calculate the objective weight by the following formula: $w_{S_{j}} = (1 - e_{j}^{S}) / \sum_{j = 1}^{n} (1 - e_{j}^{S})$ (23)

Where w_{S
_j} is the objective weight of the jth attribute based on the score matrix.

Step 8: Calculate the subjective weights.

Suppose $H = {[{\tilde{h}}_{sj}]}_{r \times n}$ is an IVq-ROFLS evaluation matrix and represents the importance of the attribute. h_sj is the evaluation conducted by the sth decision maker on the jth attribute. Then, the steps of calculating the subjective weights are as follows:

Step 8.1: Aggregate the evaluations of all the DMs about the jth attribute by using the following formula: $I V q - R O F L W A ({\tilde{h}}_{1 j}, {\tilde{h}}_{2 j}, ..., {\tilde{h}}_{r j}) = \oplus_{s = 1}^{r} (w_{d_{s}} {\tilde{h}}_{s j}) = 〈 S_{\sum_{s = 1}^{r} w_{d_{s}} θ_{s j}}, [{(1 - \prod_{s = 1}^{r} {(1 - {(μ_{s j}^{L})}^{q})}^{w_{d_{s}}})}^{1 / q}, {(1 - \prod_{s = 1}^{r} {(1 - {(μ_{s j}^{U})}^{q})}^{w_{d_{s}}})}^{1 / q}], [\prod_{s = 1}^{r} {(υ_{s j}^{L})}^{w_{d_{s}}}, \prod_{s = 1}^{r} {(υ_{s j}^{U})}^{w_{d_{s}}}] 〉$ (24)

Step 8.2: Calculate the score value of each attribute by

$\begin{matrix} s_{j} = & S ({\tilde{h}}_{j}) = \frac{θ_{j}}{4} [(1 + {(μ_{j}^{L})}^{q} - {(υ_{j}^{L})}^{q}) \\ + (1 + {(μ_{j}^{U})}^{q} - {(υ_{j}^{U})}^{q})] \end{matrix}$ (25)

Step 8.3: Calculate the subjective weight of each attribute by the following equation: $w_{H_{j}} = s_{j} / \sum_{j = 1}^{n} s_{j}$ (26)

Step 8.4: Get the range of the weight of each attribute:

$\begin{matrix} min [w_{S_{j}}, w_{H_{j}}] ⩽ w_{j} ⩽ max [w_{S_{j}}, w_{H_{j}}], \\ (j = 1, 2, . . ., n) \end{matrix}$ (27)

Step 9: Calculate the attribute weight by linear programming method: $\begin{matrix} MAX D (w) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{h = 1}^{m} Hd ({\tilde{r}}_{ij}, {\tilde{r}}_{hj}) w_{j} \\ = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{h = 1}^{m} \frac{1}{2} (| r_{ij}^{L} - r_{hj}^{L} | + | r_{ij}^{U} - r_{hj}^{U} |) w_{j} \\ subject to {\begin{matrix} min [w_{S_{j}}, w_{H_{j}}] ⩽ w_{j} ⩽ max [w_{S_{j}}, w_{H_{j}}] \\ \sum_{j = 1}^{n} w_{j} = 1, w_{j} ⩾ 0, j = 1, 2, . . ., n \end{matrix} \end{matrix}$ (28)

Where, ${\tilde{e}}_{ij}$ is an interval-valued q-rung orthopair fuzzy variable. $Hd ({\tilde{e}}_{ij}, {\tilde{e}}_{hj})$ is the hamming distance between two I of the ith strategic supplier and the hth strategic supplier under the jth attribute.

3.4 Rank strategic suppliers using prospect theory

Step 10: Calculate the gain and loss matrices. By using the Euclidean distance operator [52], the gain and loss matrix can be calculated by the following equation: $\begin{matrix} G = [g_{ij}]_{m \times n}, g_{ij} \\ = {(\frac{1}{2} ({({\tilde{r}}_{ij}^{L} - {\tilde{r}}_{j}^{L -})}^{2} + {({\tilde{r}}_{ij}^{U} - {\tilde{r}}_{j}^{U -})}^{2}))}^{1 / 2} \\ L = [l_{ij}]_{m \times n}, l_{ij} \\ = {(\frac{1}{2} ({({\tilde{r}}_{ij}^{L} - {\tilde{r}}_{j}^{L +})}^{2} + {({\tilde{r}}_{ij}^{U} - {\tilde{r}}_{j}^{U +})}^{2}))}^{1 / 2} \end{matrix}$ (29)

Where ${\tilde{r}}_{j}^{L +}$ and ${\tilde{r}}_{j}^{U +}$ are the lower and upper of PIS ${\tilde{r}}_{j}^{+}$ , respectively. ${\tilde{r}}_{j}^{L -}$ and ${\tilde{r}}_{j}^{U -}$ are the lower and upper of NIS ${\tilde{r}}_{j}^{-}$ , respectively.

Step 11: Compute positive and negative prospect value matrix by the following equation: $\begin{matrix} V^{+} (x_{ij}) = x_{ij}^{α}, x_{ij} \in G \\ V^{-} (x_{ij}) = - η {(x_{ij})}^{β}, x_{ij} \in L \end{matrix}$ (30)

Where V⁺ (x_ij) and V^- (x_ij) are positive and negative prospect value matrices, respectively.

Step 12: Calculate the probability weighting function values. The formula shows as follows: $\begin{matrix} π^{+} (w_{j}) = \frac{w_{j}^{ɛ}}{{[w_{j}^{ɛ} + {(1 - w_{j})}^{ɛ}]}^{1 / ɛ}}, x_{ij} \in G \\ π^{-} (w_{j}) = \frac{w_{j}^{δ}}{{[w_{j}^{δ} + {(1 - w_{j})}^{δ}]}^{1 / δ}}, x_{ij} \in L \end{matrix}$ (31)

Where, w_j is the weight of the jth attribute obtained by linear programming method.

Step 13: Compute the integrated prospect values and rank strategic suppliers by the following equation: $V_{i} = \sum_{j = 1}^{n} π^{+} (w_{j}) V^{+} (x_{ij}) + \sum_{j = 1}^{n} π^{-} (w_{j}) V^{-} (x_{ij})$ (32)

4 Illustrative example

In this section, an example of strategic supplier selection for a mechanical enterprise is used to analyze the efficiency and applicability of the proposed method by comparing with other existing methods.

4.1 The preparation process

This section shows the application of the proposed framework on the strategic supplier selection of a machinery manufacturing enterprise in China. Due to the various accessories of the production of bucket elevator and the rising cost of labor and land, the enterprise is facing huge inventory and economic pressure. Therefore, the enterprise urgently needs a long-term strategic supplier to construct manufacturing strategic alliance to relieve inventory pressure and improve profits. Through the selection of the cooperated enterprises, there are four enterprises that have the potential to become the strategic supplier and the proposed method is employed to select the best one as a strategic supplier. In order to protect the confidential information of the enterprise and its potential strategic suppliers, some irrelevant information is not be presented. A set K ={ k₁, k₂, k₃, k₄ } is used to refer the four potential strategic suppliers. In addition, a decision committee consisting of five DMs from the enterprise and represented by a set D ={ d₁, d₂, d₃, d₄, d₅ }, they are a mechanical engineer, a sales manager, a financial executive, a human resources supervisor and an experienced front-line employee, respectively. Meanwhile, the weights of the DMs are shown below: w_{d
_s} ={ 0.25, 0.15, 0.20, 0.30, 0.10 }. Let T ={ t₁, t₂, t₃ } is set of periods, t₁ represents the last one year, t₂ represents the current year and t₃ represents the coming year. Meanwhile, the weights of the periods are shown below: w_{t
_k} ={ 0.30, 0.45, 0.25 }. According to the previous researches [53, 54] and the actual needs of the enterprise itself, six main attributes are listed in the Fig. 3 The integrated prospect values of different periods.

Fig. 3

The integrated prospect values of different periods.

Table 1. As the need of the enterprise is to select a long-time strategic supplier, the stability of the supplier C₄ is an absolutely necessary attribute. C ₂ is a qualitative attribute and depends on the supplier’s stability and continuity. Based on the Planning, the enterprise is inclined to select the more deeply cooperative suppliers. Hence, the degree of cooperation C₅ is an indispensable attribute.

Table 1

The attributes and definitions for strategic supplier selection

Code	Attribute	Definition
C ₁	Price of product	The price to purchase one product.
C ₂	Stability of supplier	The stability and continuity of the supplier.
C ₃	Time of delivery	Time used by supplier to complete delivery
C ₄	Quality of product	Performance of products to meet the requirements.
C ₅	Degree of cooperation	The degree of cooperation between enterprise and supplier.
C ₆	Environmental competencies	Efforts of the supplier to improve green production and reduce pollution effects

Where C₁ and C₃ are cost attributes, the others are benefit attributes.

The linguistic term set is given as S={s_1=extremely poor, s_2=very poor, s_3=poor, s_4=medium, s_5=good, s_6=very good, s_7=extremely good.}

4.2 The decision making process

Step 1: Obtain and standardize the decision matrices. All the DMs give their evaluations by the form ${\tilde{Q}}_{ij}^{(ks)} = 〈 s_{θ_{ij}^{(ks)}}, [μ_{ij}^{(ks) L}, μ_{ij}^{(ks) U}], [υ_{ij}^{(ks) L}, υ_{ij}^{(ks) U}] 〉$ . By Equation (15), the original matrices ${\tilde{Q}}^{(ks)}$ are standardized into decision matrices ${\tilde{R}}^{(ks)} = {[{\tilde{r}}_{ij}^{(ks)}]}_{4 \times 6} (k = 1, 2, 3 s = 1, 2, 3, 4, 5)$ . Due to the limited space of this paper, we only list the standardized IVq-ROFLS evaluation matrices of D₁ ${\tilde{R}}^{(k 1)} = {[{\tilde{r}}_{ij}^{(k 1)}]}_{4 \times 6} (k = 1, 2, 3)$ and show it in Appendix 1.

Step 2: Aggregate the linguistic evaluations

By employing Equation (16), the standardized IVq-ROFLS evaluations of three periods are aggregated into a new decision matrix ${\tilde{R}}^{(s)} = {[{\tilde{r}}_{ij}^{(s)}]}_{4 \times 6} (s = 1, 2, 3, 4, 5)$ related to five DMs. Due to the limited space of this paper, we only list the IVq-ROFLS evaluation matrices of D₁ ${\tilde{R}}^{(1)} = {[{\tilde{r}}_{ij}^{(1)}]}_{4 \times 6}$ which are shown in Appendix 2. By using Equation (17), the IVq-ROFLS evaluations of five DMs are aggregated into a group decision matrix $\tilde{R} = {[{\tilde{r}}_{ij}]}_{4 \times 6}$ . The final results are presented in Appendix 3. The interval-valued numbers decision matrix $\hat{R} = {[{\hat{r}}_{ij}]}_{4 \times 6}$ is computed by Equation (18) and shown in Table 2 Then, the PIS and NIS matrix is calculated by Equation (19) and shown in Table 3.

Table 2
The interval-valued numbers decision matrix

Alternatives Attributes

C ₁ C ₂ C ₃ C ₄ C ₅ C ₆

K ₁ 〈1.464, 2.572〉〈0.253, 0.736〉〈1.731, 2.699〉〈1.731, 2.488〉〈3.434, 4.830〉〈2.328, 3.314〉

K ₂ 〈1.003, 1.443〉〈2.159, 3.093〉〈0.942, 2.648〉〈1.248, 2.078〉〈2.682, 3.831〉〈3.007, 4.304〉

K ₃ 〈1.592, 2.306〉〈1.798, 2.558〉〈1.385, 1.916〉〈3.515, 4.951〉〈0.721, 1.328〉〈0.443, 1.020〉

K ₄ 〈1.211, 2.712〉〈3.729, 5.424〉〈1.651, 2.443〉〈1.756, 2.552〉〈1.322, 1.986〉〈2.592, 3.630〉

Alternatives			Attributes
K ₁	〈1.464, 2.572〉	〈0.253, 0.736〉	〈1.731, 2.699〉	〈1.731, 2.488〉	〈3.434, 4.830〉	〈2.328, 3.314〉
K ₂	〈1.003, 1.443〉	〈2.159, 3.093〉	〈0.942, 2.648〉	〈1.248, 2.078〉	〈2.682, 3.831〉	〈3.007, 4.304〉
K ₃	〈1.592, 2.306〉	〈1.798, 2.558〉	〈1.385, 1.916〉	〈3.515, 4.951〉	〈0.721, 1.328〉	〈0.443, 1.020〉
K ₄	〈1.211, 2.712〉	〈3.729, 5.424〉	〈1.651, 2.443〉	〈1.756, 2.552〉	〈1.322, 1.986〉	〈2.592, 3.630〉

Table 3

The PIS and NIS of each attribute

Attributes	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
PIS:	〈1.592, 2.712〉	〈3.729, 5.424〉	〈1.731, 2.699〉	〈3.515, 4.951〉	〈3.434, 4.830〉	〈3.007, 4.304〉
NIS:	〈1.003, 1.443〉	〈0.253, 0.736〉	〈0.942, 1.916〉	〈1.248, 2.078〉	〈0.721, 1.328〉	〈0.443, 1.020〉

Step 3: Employ linear programming method to calculate attribute weights.

All the DMs give their evaluations about the importance of attributes by the form ${\tilde{h}}_{rj} = 〈 s_{θ_{rj}}, [μ_{rj}^{L}, μ_{rj}^{U}], [υ_{rj}^{L}, υ_{rj}^{U}] 〉 (r = 1, 2, 3, 4, 5 . j = 1, 2, 3, 4, 5, 6)$ and the evaluation matrix is shown in Appendix 4.

Through employing Equations (20)–(23), the object weights w_S are determined, where w_S = (0.05, 0.35, 0.02, 0.13, 0.25, 0.21). By employing Equations (24)–(26), the subjective weights w_H are determined, where w_H = (0.16, 0.14, 0.09, 0.27, 0.18, 0.16) Then by Equation (27), we can get the range of the weight of each attribute that 0.05 ⩽ w₁ ⩽ 0.35, 0.14 ⩽ w₂ ⩽ 0.35, 0.14 ⩽ w₃ ⩽ 0.35, 0.13 ⩽ w₄ ⩽ 0.27, 0.18 ⩽ w₅ ⩽ 0.25 and 0.16 ⩽ w₆ ⩽ 0.21.

Then the linear programming model is built through Equation (28) and shown as follows:

$\begin{matrix} MAX D (w) = \sum_{j = 1}^{6} \sum_{i = 1}^{4} \sum_{h = 1}^{4} Hd ({\tilde{r}}_{ij}, {\tilde{r}}_{hj}) w_{j} = \sum_{j = 1}^{6} \sum_{i = 1}^{4} \sum_{h = 1}^{4} \frac{1}{2} (| r_{ij}^{L} - r_{hj}^{L} | + | r_{ij}^{U} - r_{hj}^{U} |) w_{j} \\ = 4.270 w_{1} + 14.425 w_{2} + 3.656 w_{3} + 5.703 w_{4} + 9.416 w_{5} + 6.871 w_{6} \\ subject to {\begin{matrix} \begin{matrix} 0.05 ⩽ w_{1} ⩽ 0.16, 0.14 ⩽ w_{2} ⩽ 0.35, 0.02 ⩽ w_{3} ⩽ 0.09, \\ 0.13 ⩽ w_{4} ⩽ 0.27, 0.18 ⩽ w_{5} ⩽ 0.25, 0.16 ⩽ w_{6} ⩽ 0.21, \end{matrix} \\ \sum_{j = 1}^{n} w_{j} = 1, w_{j} ⩾ 0,, j = 1, 2, . . ., 6 \end{matrix} \end{matrix}$

After solving this linear programming model, the final attribute weights are w = (0.05, 0.35, 0.02, 0.17, 0.25, 0.16).

Step 4: Rank strategic suppliers using prospect theory.

Firstly, gain matrix G = [g_ij] _4 ×6 and loss matrix L = [l_ij] _4 ×6 are calculated by Equation (29) and shown in Table 4, respectively. Then, compute positive and negative prospect value matrix by Equation (30), and the results are shown in Table 5. The values of probability weighting function for gains and losses are calculated by Equation (31) as π⁺ (w) =(0.132, 0.345, 0.081, 0.241, 0.291, 0.234) and π^- (w) = (0.104, 0.363, 0.056, 0.229, 0.292, 0.220). Then, through Equation (32), the integrated prospect values of the four strategic suppliers are calculated as: V_{K
₁} = -0.617, V_{K
₂} = -0.610, V_{K
₃} = -1.327 and V_{K
₄} = - 0 . 136. Therefore, the ranking of the potential strategic suppliers is K₄ ≻ K₂ ≻ K₁ ≻ K₃. The K₁ is the most appropriate strategic supplier.

Table 4

The gain and loss matrix

Gain	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	Loss	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
K ₁	0.862	0.000	0.786	0.448	3.132	2.100	K ₁	0.134	4.127	0.000	2.150	0.000	0.849
K ₂	0.000	2.144	0.518	0.000	2.248	2.947	K ₂	0.989	1.987	0.559	2.587	0.884	0.000
K ₃	0.738	1.690	0.313	2.587	0.000	0.000	K ₃	0.288	2.444	0.605	0.000	3.132	2.947
K ₄	0.909	4.127	0.625	0.491	0.630	2.391	K ₄	0.269	0.000	0.189	2.103	2.505	0.560

Table 5

The positive and negative prospect value matrix

V ⁺	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	V^–	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
K ₁	0.878	0.000	0.809	0.493	2.731	1.921	K ₁	–0.384	–7.833	0.000	–4.413	0.000	–1.948
K ₂	0.000	1.956	0.560	0.000	2.040	2.588	K ₂	–2.229	–4.118	–1.349	–5.194	–2.019	0.000
K ₃	0.766	1.587	0.360	2.308	0.000	0.000	K ₃	–0.751	–4.939	–1.446	0.000	–6.145	–5.823
K ₄	0.920	3.482	0.661	0.535	0.666	2.153	K ₄	–0.709	0.000	–0.520	–4.328	–5.048	–1.351

4.3 Comparative analysis

4.3.1 Comparison of periods

A comparative analysis is conducted to investigate the necessity of considering the comprehensive information of all periods in the selection of strategic supplier. The integrated prospect values of T₁, T₂, T₃ and all of them are shown in Fig. 3.

It can be seen from Fig. 3 that the rank of suppliers is completely different. If we only take consider of T₁, the supplier K₁ is the best solution. If we only obtain the information of T₂, the supplier K₁ becomes the strategic supplier. If we only obtain the information of T₃, the supplier K₄ becomes the best supplier. However, if we take all periods into consideration, the supplier K₄ turns out to be the optimal solution. And the final result is also corresponding to the situation of each supplier. K₁ is one of the earliest suppliers to cooperate with the enterprise. But the enterprise comes to a standstill this year due to financial problems, and its integrated prospect value decline over time. Therefore, it would be a bad decision to choose K₁. K₄ is a new coadjutant supplier, but it has recently made a breakthrough in technology and its quoted price is quite attractive. Therefore, based on the joint analysis of all periods, K₄ is selected as the strategic supplier. In addition, when each period is considered separately, the integrated prospect value curve fluctuates greatly. On the contrary, the integrated prospect value becomes stable after the aggregation of each period. Therefore, aggregating all periods avoids the extreme option.

Based on the comparison of periods, the necessity of aggregating all periods is proved and the applicability and effectiveness of the proposed method are also verified at the same time.

4.3.2 Compare with other methods

In order to further verify the applicability and effectiveness of the proposed method, three comparison methods are implemented to handle the decision making problem, i.e., TOPSIS, the VIKOR and TODIM [55]. In addition, to measure the differences between the proposed method and the above three methods, Spearman’s rank-correlation test [56] is employed to ascertain whether there is significant rank-correlation between two sets of values. The comparative results and the Spearman’s rank-correlation tests are shown in Table 6. Then, the distinct ranking results are displayed in Fig. 4.

Table 6
Ranking results and differences of the four methods

Suppliers Ranking results Ranking differences

proposed method (I) TOPSIS method (II) VIKOR method (III) TODIM method (IV) I vs. II I vs. III I vs. IV

K ₁ 3 3 4 2 0 –1 1

K ₂ 2 2 2 3 0 0 –1

K ₃ 4 4 3 4 0 1 0

K ₄ 1 1 1 1 0 0 0

Spearman’s test result

Squared sum S_d2 0 2 2

Spearman’s rank-correlation coefficient r_s 1 0.8 0.8

Test value Z 1.73 1.38 1.38

Suppliers	Ranking results		Ranking differences
K ₁	3	3	4	2		0	–1	1
K ₂	2	2	2	3		0	0	–1
K ₃	4	4	3	4		0	1	0
K ₄	1	1	1	1		0	0	0
Spearman’s test result
Squared sum					S_d2	0	2	2
Spearman’s rank-correlation coefficient					r_s	1	0.8	0.8
Test value					Z	1.73	1.38	1.38

Fig. 4

Ranking results with different methods.

With reference to the research conducted by Parkan and Wu [56], the range of r_s is –1 to 1. The closer the value of r_s is to the boundary of the range, the stronger the correlation between two ranking orders. If the test value Z ⩾ 1.625, it means that there is evidence of a positive relation between two ranking orders. Otherwise, it is considered that the two ranking orders are dissimilar. The Spearman’s rank-correlation coefficient of the method I and the method II is the upper boundary of 1, and the test value is 1.73≥1.625. Therefore, there is a significant positive relation between two ranking orders. However, the test values of the method III and the method IV by the method are lower than 1.625. Therefore, these two ranking orders are considered to be dissimilar, respectively. It can be seen from Table 6 that the best solution determined by the four methods are the same, which is K₄. The results make a difference in the rest options. People do not like to take risks when it comes to profits, so K₃ is the worst option for risk aversion. K₂ is ought to be better than K₁, because people tend to get a steady rather than a risky profit. Therefore, the proposed method is more practical.

By comparing with other methods, the advantages of the IVq-ROFLS-HDLP-PT approach can be summarized as follows: (1) By IVq-ROFLS, the proposed approach can better represent the fuzziness and uncertainty of evaluation information given by DMs. This is particularly useful for the strategic supplier selection problems characterized by multi-stage, comprehensive information and limited expertise. (2) Based on HDLP method, the optimal criteria weights can be obtained by considering both subjective and objective weights of evaluation criteria, which makes the proposed method more practical. (3) By applying the Euclidean distance to the prospect theory, the process of select the strategic supplier is simplified. Besides, the proposed method considers the psychological behaviors of DMs. Hence, the proposed method can get a more accurate ranking in the strategic supplier selection.

5 Conclusion

Considering the current actual situation and previous research, a new supplier selection method for selecting strategic supplier has been developed in this paper. The proposed method makes the following contributions to strategic supplier selection problem:

We take the strategic supplier selection problem as a special multi-stage multi-attribute group decision making problem and express all the evaluation information in the form of IVq-ROFLS in order to adopt a flexible, generalized preference style to minimize subjective randomness and describe the uncertainty.

The GIVq-ROFLWA operator and the IVq-ROFLWA operator are used to aggregate the evaluation information of different periods and DMs separately.

To comprehensively and optimally consider the weights of attributes, the Hamming distance based linear programming method is employed to get the optimal weights under the range of the subjective and objective weights of attributes based on score function and entropy method.

The proposed method based on prospect theory can reflect risk attitude with consideration of the bounded rationality of DMs, because it combines different attitudes toward gain and loss. This makes the ranking order more persuasive than some other methods.

The multi-stage evaluations are given independently in this paper. Nevertheless, in the realistic situation, there is a certain degree of coupling between multi-stage evaluations. Accordingly, the one main future research direction is to further enhance the coupling between multi-stage evaluations.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China under Project (No. 51705386); China Scholarship Council (No. 201606955091); Fundamental Research Funds for the Central Universities, China (No. 2018-IVB-010).

Appendix

References

Peralta

M.E.

and Soltero

V.M.

, Analysis of fractal manufacturing systems framework towards industry 4.0, Journal of Manufacturing Systems 57 (2020), 46–60.

Gligor

D.D.

, Birds of a feather: The impact of race on the supplier selection and evaluation process, International Journal of Production Economics 230 (2020), 107802.

Tao

, Shao

, Liu

, Zhou

and Chen

, Basic uncertain information soft set and its application to multi-criteria group decision making, Engineering Applications of Artificial Intelligence 95 (2020), 103871.

Rani

, Mishra

A.R.

, Krishankumar

, Mardani

, Cavallaro

, Ravichandran

K.S.

and Balasubramanian

, Hesitant Fuzzy SWARA-Complex Proportional Assessment Approach for Sustainable Supplier Selection (HF-SWARA-COPRAS), Symmetry-Basel 12 (2020), 19.

Liu

L.M.

, Bin

Z.L.

, Shi

and Cao

W.Z.

, Sustainable Supplier Selection Based on Regret Theory and QUALIFLEX Method, Int J Comput Intell Syst 13 (2020), 1120–1133.

Gao

H.X.

, Ju

Y.B.

, Gonzalez

and Zhang

W.K.

, Green supplier selection in electronics manufacturing: An approach based on consensus decision making, Journal of Cleaner Production 245 (2020), 17.

Liang

, Qin

, Martínez

and Liu

, A heterogeneous QUALIFLEX method with criteria interaction for multi-criteria group decision making, Information Sciences 512 (2020), 1481–1502.

Liu

Z.M.

, Li

, Zhao

X.L.

, Sha

L.B.

, Wang

X.Y.

and Liu

P.D.

, Selecting the Optimal Green Agricultural Products Supplier: A Novel Approach Based on GBWM and PROMETHEE II, Sustainability 12 (2020), 23.

Olanrewaju

O.G.

, Dong

Z.S.

and Hu

, Supplier selection decision making in disaster response, Computers & Industrial Engineering 143 (2020), 106412.

10.

Xue

, Fu

, Feng

N.-P.

, Lu

G.-Y.

, Chang

W.-J.

and Yang

S.-L.

, Evaluation of supplier performance of high-speed train based on multi-stage multi-criteria decision-making method, Knowledge-Based Systems 162 (2018), 238–251.

11.

Kaur

and Prakash Singh

, Multi-stage hybrid model for supplier selection and order allocation considering disruption risks and disruptive technologies, International Journal of Production Economics 231 (2021), 107830.

12.

, Shao

, Wang

and Zhang

, A group decision making sustainable supplier selection approach using extended TOPSIS under interval-valued Pythagorean fuzzy environment, Expert Systems with Applications 121 (2019), 1–17.

13.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

14.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

15.

Atanassov

and Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349.

16.

Yager

R.R.

and Abbasov

A.M.

, Pythagorean Membership Grades, Complex Numbers, and Decision Making, International Journal of Intelligent Systems 28 (2013), 436–452.

17.

Peng

X.D.

and Yang

, Fundamental Properties of Interval-Valued Pythagorean Fuzzy Aggregation Operators, International Journal of Intelligent Systems 31 (2016), 444–487.

18.

Yager

R.R.

, Generalized Orthopair Fuzzy Sets, IEEE Transactions on Fuzzy Systems 25 (2017), 1222–1230.

19.

Krishankumar

, Gowtham

, Ahmed

, Ravichandran

K.S.

and Kar

, Solving green supplier selection problem using q-rung orthopair fuzzy-based decision framework with unknown weight information, Applied Soft Computing 94 (2020), 106431.

20.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning—I, Information Sciences 8 (1975), 199–249.

21.

Herrera-Viedma

, Martinez

, Mata

and Chiclana

, A Consensus Support System Model for Group Decision-Making Problems With Multigranular Linguistic Preference Relations, IEEE Transactions on Fuzzy Systems 13 (2005), 644–658.

22.

Joshi

B.P.

, Singh

, Bhatt

P.K.

and Vaisla

K.S.

, Interval valued q-rung orthopair fuzzy sets and their properties, Journal of Intelligent & Fuzzy Systems 35 (2018), 5225–5230.

23.

Wang

and Wan

, Possibility degree and divergence degree based method for interval-valued intuitionistic fuzzy multi-attribute group decision making, Expert Systems with Applications 141 (2020), 112929.

24.

Mao

X.-B.

, Wu

, Dong

J.-Y.

, Wan

S.-P.

and Jin

, A new method for probabilistic linguistic multi-attribute group decision making: Application to the selection of financial technologies, Applied Soft Computing 77 (2019), 155–175.

25.

, Chan

F.T.S.

and Jiang

, A quantum framework for modelling subjectivity in multi-attribute group decision making, Computers & Industrial Engineering 124 (2018), 560–572.

26.

Sun

, Qi

, Ma

, Wang

, Zhang

and Jiang

, Variable precision diversified attribute multigranulation fuzzy rough set-based multi-attribute group decision making problems, Computers & Industrial Engineering 142 (2020), 106331.

27.

Xian

, Yang

and Guo

, Double parameters TOPSIS for multi-attribute linguistic group decision making based on the intuitionistic Z-linguistic variables, Applied Soft Computing 85 (2019), 105835.

28.

Zhang

, Zhan

, Xu

and Alcantud

J.C.R.

, Covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decision-making, Information Sciences 494 (2019), 114–140.

29.

Liang

, Goh

and Wang

Y.-M.

, Multi-attribute group decision making method based on prospect theory under hesitant probabilistic fuzzy environment, Computers & Industrial Engineering 149 (2020), 106804.

30.

Ding

X.-F.

, Liu

H.-C.

and Shi

, A dynamic approach for emergency decision making based on prospect theory with interval-valued Pythagorean fuzzy linguistic variables, Computers & Industrial Engineering 131 (2019), 57–65.

31.

Brünner

, Reiner

, Natter

and Skiera

, Prospect theory in a dynamic game: Theory and evidence from online pay-per-bid auctions, Journal of Economic Behavior & Organization 164 (2019), 215–234.

32.

Kahneman

and Tversky

, Prospect Theory: An Analysis of Decision under Risk, Econometrica 47 (1979), 263–292.

33.

Tversky

and Kahneman

, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992), 297–323.

34.

Wang

, Li

, Zhang

, Zhou

and Huang

, A three-way decision model based on cumulative prospect theory, Information Sciences 519 (2020), 74–92.

35.

Ghader

, Darzi

and Zhang

, Modeling effects of travel time reliability on mode choice using cumulative prospect theory, Transportation Research Part C: Emerging Technologies 108 (2019), 245–254.

36.

, Huang

and Li

, A novel consensus reaching framework for heterogeneous group decision making based on cumulative prospect theory, Computers & Industrial Engineering 128 (2019), 325–335.

37.

, Li

and Dong

, A novel two-stage method for matching the technology suppliers and demanders based on prospect theory and evidence theory under intuitionistic fuzzy environment, Applied Soft Computing 95 (2020), 106553.

38.

Cheng

, Long

and Chen

, A policy utility dislocation model based on prospect theory: A case study of promoting policies with low-carbon lifestyle, Energy Policy 137 (2020), 111134.

39.

Herrmann

, Jong-A-Pin

and Schoonbeek

, A prospect-theory model of voter turnout, Journal of Economic Behavior & Organization 168 (2019), 362–373.

40.

Wan

S.-P.

, Zou

and Dong

J.-Y.

, Prospect theory based method for heterogeneous group decision making with hybrid truth degrees of alternative comparisons, Computers & Industrial Engineering 141 (2020), 106285.

41.

Surti

, Celani

and Gajpal

, The newsvendor problem: The role of prospect theory and feedback, European Journal of Operational Research 287 (2020), 251–261.

42.

, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Information Sciences 166 (2004), 19–30.

43.

Joshi

B.P.

, Singh

, Bhatt

P.K.

and Vaisla

K.S.

, Interval valued q-rung orthopair fuzzy sets and their properties, Journal of Intelligent & Fuzzy Systems 35 (2018), 5225–5230.

44.

Feng

, Shang

, Xu

and Wang

, A method to multi-attribute decision-making based on interval-valued q-rung dual hesitant linguistic Maclaurin symmetric mean operators, Complex & Intelligent Systems (2020).

45.

Wang

, Garg

and Li

, Interval-Valued $q$ -Rung Orthopair 2-Tuple Linguistic Aggregation Operators and Their Applications to Decision Making Process, IEEE Access 7 (2019), 131962–131977.

46.

, Luo

, Ma

, Gao

, Santibanez Gonzalez

E.D.R.

and Wang

, Some interval-valued q-rung orthopair weighted averaging operators and their applications to multiple-attribute decision making, International Journal of Intelligent Systems 34 (2019), 2584–2606.

47.

Scales

, Linear Programming: Foundations and Extensions, Journal of the Operational Research Society 60 (2009), 146–146.

48.

Zhang

, Zhang

and Mei

, Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure, Knowledge-Based Systems 22 (2009), 449–454.

49.

Shannon

C.E.

, A mathematical theory of communication, The Bell System Technical Journal 27 (1948), 379–423.

50.

Chen

, Effects of normalization on the entropy-based TOPSIS method, Expert Systems with Applications 136 (2019), 33–41.

51.

Huang

, Chang

, Leng

and Huang

, Integrated index for drought assessment based on variable fuzzy set theory: A case study in the Yellow River basin, China, Journal of Hydrology 527 (2015), 608–618.

52.

Merigó

J.M.

and Casanovas

, Induced aggregation operators in the Euclidean distance and its application in financial decision making, Expert Systems with Applications 38 (2011), 7603–7608.

53.

Stević

Ž.

, Pamučar

, Puška

and Chatterjee

, Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to Compromise solution (MSARCO), Computers & Industrial Engineering 140 (2020).

54.

Ecer

and Pamucar

, Sustainable supplier selection: A novel integrated fuzzy best worst method (F-BWM) and fuzzy CoCoSo with Bonferroni (CoCoSo’B) multi-criteria model, Journal of Cleaner Production 266 (2020).

55.

Chai

and Ngai

E.W.T.

, Decision-making techniques in supplier selection: Recent accomplishments and what lies ahead, Expert Systems with Applications 140 (2020).

56.

Parkan

and Wu

M.-L.

, Decision-making and performance measurement models with applications to robot selection, Computers & Industrial Engineering 36 (1999), 503–523.