Application of catastrophe progression method on hybrid multiple attribute decision-making problems based on regret theory

Abstract

In order to effectively solve the decision-making problems with the diversity of evaluation information, the dynamics of research objects, the limitations of subjective authorization, and the irrational behavior of decision-makers, this paper extends catastrophe progression method to solve hybrid multiple attribute decision-making problems based on regret theory. Firstly, some basic theories are introduced. Secondly, the original catastrophe progression method is extended by using the regret theory, which is employed to solve the multiple attribute decision-making problems with hybrid evaluation information. Finally, a real-life case study of selecting a fresh cold chain logistics service provider is used to verify the practicality and effectiveness of the proposed method, and a comparative analysis with the TOPSIS method and the sensitivity of the regret avoidance coefficient is analyzed in this article.

Keywords

Regret theory catastrophe progression hybrid multi-attribute decision-making

1 Introduction

The multi attribute decision-making is widely used in the production, operation and management of the economy, society, technology and culture [1 –4]. People often may make mistakes in decision-making. When such mistakes occur, they usually feel very remorse. Therefore, decision-makers often exhibit an indecisive personality trait in order to avoid regretful attitudes, and rational decision theory cannot better explain regretful decision behavior [5], and a limited rational decision-making approach based on decision-makers’ behavior has attracted the attention of scholars [6 –8]. Bell et al., proposed the regret theory [9 –11], where the calculation function only includes risk aversion coefficient and regret aversion coefficient, because there is no reference point in regret theory and it is easy to operate and has been widely used in practice [12 –14]. Zhang and Tu proposed an interval intuition ladder fuzzy cluster decision method based on regret theory to describe a facility location decision for a cold chain logistics service provider [15]. Li et al., proposed a stochastic equilibrium model based on regret theory for traveler path selection and found that regret avoidance is an important factor influencing traveler path selection [16]. Based on regret theory, Liu et al., proposed a group decision method to evaluate the project financing risk with hesitant fuzzy elements [17]. Yuan et al., studied the path selection problem in the risk environment, a traffic network model based on the risk avoidance of regret theory is proposed, and the analysis shows that under low environmental risk, the influence of regret psychology on path selection is not obvious [18]. Wang et al., considered the psychological behavior of decision-makers who regret avoidance characteristics and a proposed intuitive linguistic multi-criteria decision method with expectation levels based on regret theory [19]. Wang et al developed new utility and regret-rejoice functions by integrating the proposed projection model with regret theory to solve MADM problems [20]. In summary, regret theory provides a new method to study the multi-criteria decision-making problem of expected utility in uncertain environments.

Decision-makers usually use different types of mixed data to describe decision attributes, because there are often problems of information asymmetry and data source diversity in real decision environment [31, 32]. In real life, it is very common for quantitative evaluation and qualitative evaluation to coexist, in which real number, interval number and triangular fuzzy number are generally used for quantitative description [33 –35], while linguistic number and intuitive fuzzy number are used for qualitative description [17 , 37]. Therefore, it is necessary to further study the problem of mixed multi-attribute decision-making in which decision-makers give multiple forms of attribute values according the dynamics, uncertainty and diversity of real problems.

The catastrophe theory is the leap of a dynamic system from one equilibrium state to another in the process of continuous change, and effectively describes and explains the internal and external influences and characteristics of mutating systems [21]. Based on catastrophe theory and combined with fuzzy mathematics, the catastrophe progression method fully considers the importance of each evaluation index and avoids the determination of index weight in the traditional evaluation model [22]. Catastrophe progression method is also widely used in decision-making problems [23 –25]. Jia and Dong proposed a risk evaluation method based on traditional catastrophe progression method to solve the problem of fire risk assessment in shopping malls, which solved the difficulty of determining the index weight of traditional evaluation methods [26]. Zhao et al., applied the evaluation model of catastrophe progression method based on rough set to the domestic green growth evaluation problem [27]. Based on the mutation progression method, Guo et al., proposed an improved decision model for the selection of science and technology park projects [28]. Similarly, Zhang et al., improved a new evaluation model of index ranking by using the maximizing deviation method for regional agricultural logistics performance evaluation problems [22]. Zha and Song used the catastrophe progression method to evaluate and select green building suppliers [29]. Chen et al optimized the rough set theory and the catastrophe progression method by entropy method and developed an evaluation model to scientifically measure the regional development of green building [30]. Considering that the catastrophe theory can well avoid the disadvantages of subjective weighting, which is computationally simple and has excellent dynamics, a hybrid multi-attribute decision-making method based on catastrophe theory and regret theory is proposed in this study. Based on the above analysis, this paper mainly extends the traditional catastrophe progression method considering the decision makers’ regret behavior to solve the decision-making problem with hybrid evaluating information, where we perform multilevel contradiction decomposition on the evaluation target of the system, and then use the catastrophe fuzzy membership function generated by the combination of catastrophe theory and regret theory to obtain the elementary catastrophe model and its bifurcation point set equation.

In this paper, our motivation is that the traditional catastrophe progression method is extended to solve the decision-making problems with the hybrid information, in which the decision-maker’s regret avoidance behavior is considered. The importance of the study is that we consider the hybrid information, different types of mutations and regret well-being in the process of decision-making. The rest of this paper is organized as follows: Section 2 mainly introduces the related theories and methods used in the proposed method. Section 3 proposes the catastrophe progression method based on regret theory to solve hybrid multiple attribute decision-making problem. Section 4 provides case studies and comparisons with other methods, and related discussions are presented. The conclusions are presented in Section 5.

2 Basic concepts and definitions

This section briefly introduces the concepts related to catastrophe progression and regret theory used in this paper.

2.1 Catastrophe theory

The catastrophe theory was put forward by Rene Tom on the basis of topology, structural stability and other mathematics to solve the problems of discontinuous change and abrupt change. In the catastrophe progression method, each alternative is regarded as a system, and the variables that change within the system are called state variables. The sub-level variables that can control the changes of the state variables are called control variables. The common types of primary catastrophe systems include fold catastrophe, cusp catastrophe, swallowtail catastrophe, butterfly catastrophe, and shack catastrophe, which has from 1 to 5 state variables. First of all, the first-order derivative of each catastrophe potential function is obtained. Let f′ (x) = 0, all the set of zero-bound points of each catastrophe potential function f (x) is get. Let f″ (x) = 0, the equilibrium surface singular point set equation is obtained by deriving the equilibrium surface equation. Combine the equilibrium surface equation and the equilibrium surface singular point set equation to eliminate the state variables and obtain the bifurcation set equation. Further derivation of the bifurcation set equation, in order to make the value of each control variable and state variable in the same range, based on the idea of fuzzy mathematical membership function, the catastrophe model is normalized. Through the recursive calculation of each index in the system, the catastrophe level value of each control variable is obtained, and the value of the total catastrophe membership function of the system state characteristics is also obtained, as shown in Table 1.

Table 1
Catastrophe progression method commonly used normalization model

Catastrophe type Potential function Bifurcation set equation Control dimension Normalization formula

Fold catastrophe f (x) = x³ + ax a = -3x² 1 $x_{a} = \sqrt{| a |}$

Cusp catastrophe f (x) = x⁴ + ax² + bx a = -6x², b = 8x³ 2 $x_{a} = \sqrt{| a |}, x_{b} = \sqrt{| b | 3}$

Swallowtail catastrophe f (x) = x⁵ + ax³ + bx² + cx a = -6x², b = 8x³, c = -3x⁴ 3 $x_{a} = \sqrt{| a |}, x_{b} = \sqrt{| b | 3}, x_{c} = \sqrt{| c | 4}$

Butterfly catastrophe f (x) = x⁶ + ax⁴ + bx³ + cx² + dx a = -10x², b = 20x³, c = -15x⁴, d = 4x⁵ 4 $x_{a} = \sqrt{| a |}, x_{b} = \sqrt{| b | 3}, x_{c} = \sqrt{| c | 4}, x_{d} = \sqrt{| d | 5}$

Shack catastrophe f (x) = x⁷ + ax⁶ + bx⁴ + cx³ + dx² + ex a = - x², b = 2x³, c = -2x⁴, d = 4x⁵, e = -5x⁶ 5 $x_{a} = \sqrt{| a |}, x_{b} = \sqrt{| b | 3}, x_{c} = \sqrt{| c | 4}, x_{d} = \sqrt{| d | 5}, x_{e} = \sqrt{| e | 6}$

Catastrophe type	Potential function	Bifurcation set equation	Control dimension	Normalization formula
Fold catastrophe	f (x) = x³ + ax	a = -3x²	1	$x_{a} = \sqrt{\| a \|}$
Cusp catastrophe	f (x) = x⁴ + ax² + bx	a = -6x², b = 8x³	2	$x_{a} = \sqrt{\| a \|}, x_{b} = \sqrt{\| b \| 3}$
Swallowtail catastrophe	f (x) = x⁵ + ax³ + bx² + cx	a = -6x², b = 8x³, c = -3x⁴	3	$x_{a} = \sqrt{\| a \|}, x_{b} = \sqrt{\| b \| 3}, x_{c} = \sqrt{\| c \| 4}$
Butterfly catastrophe	f (x) = x⁶ + ax⁴ + bx³ + cx² + dx	a = -10x², b = 20x³, c = -15x⁴, d = 4x⁵	4	$x_{a} = \sqrt{\| a \|}, x_{b} = \sqrt{\| b \| 3}, x_{c} = \sqrt{\| c \| 4}, x_{d} = \sqrt{\| d \| 5}$
Shack catastrophe	f (x) = x⁷ + ax⁶ + bx⁴ + cx³ + dx² + ex	a = - x², b = 2x³, c = -2x⁴, d = 4x⁵, e = -5x⁶	5	$x_{a} = \sqrt{\| a \|}, x_{b} = \sqrt{\| b \| 3}, x_{c} = \sqrt{\| c \| 4}, x_{d} = \sqrt{\| d \| 5}, x_{e} = \sqrt{\| e \| 6}$

2.2 Regret theory

When making a decision, the decision-maker not only focuses on the results of the chosen solution, but also compares the results of the chosen solution with the results of other possible solutions. This kind of comparison would make the decision-maker produce corresponding psychological behaviors of regret-rejoicing, and this psychological behavior of regret-rejoicing would also affect the final decision-making result of the decision-maker. The perceived utility function consists of the “regret-rejoicing” utility function.

The utility that the decision-maker obtains from alternatives a_x and a_y is v (a_x) and v (a_y), respectively. The perceived utility of alternative a_x relative to alternative a_y is: $u (a_{x}, a_{y}) = v (a_{x}) + R (v (a_{x}) - v (a_{y}))$ (1)

The function $R (v (a_{x}) - v (a_{y}))$ is a monotonically increasing concave function, which means the regret-rejoicing value produced by the decision-maker who chooses the alternative $a_{x}$ instead of the alternative $a_{y}$ . $R (v (a_{x}) - v (a_{y})) > 0$ means the rejoicing value produced by the decision-maker choosing the alternative $a_{x}$ without choosing the alternative $a_{y}$ , $R (v (a_{x}) - v (a_{y})) < 0$ means the regret value produced by the decision-maker choosing the alternative $a_{x}$ without choosing the alternative $a_{y}$ . The original regret theory is only applicable to the comparison between the two alternatives. To overcome this limitation, the scholars later modified the original regret theory model appropriately, making the regret theory more widely used. There are m alternatives (a₁, a₂, ⋯ , a_m), where a_i represents the i alternative (i = 1, 2, ⋯ m). The perceived utility of the decision-maker on a_i is: $u (a_{i}) = v (a_{i}) + R (v (a_{i}) - v (a^{+}))$ (2)

Among them $a^{+} = max {a_{i}}$ , R (v (a) - v (a⁺)) > 0 represents the rejoicing value produced relative to the positive ideal result, and R (v (a) - v (a⁺)) < 0 represents the regret value produced relative to the positive ideal result.

3 Decision-making method

In the hybrid multi-attribute decision-making problem, A ={ a₁, a₂, ⋯ , a_m } represents the set of m alternatives, where a_i represents the i alternative. C ={ c₁, c₂, ⋯ , c_n } represents a set of n attributes, where c_j represents the j attribute and is additive independent. W = (w₁, w₂, ⋯ , w_n) represents the weight vector of the attribute, where w_j is the weight of attribute c_j, satisfy w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ . The attributes include benefit type and cost type, where N_b and N_c denote the set of benefit-type attributes and cost-type attributes respectively, N_b ∪ N_c = N and N_b∩ N_c = ∅. Let the decision matrix be $X^{'} = {[x_{ij}^{'}]}_{m \times n}$ , where $x_{ij}^{'}$ represents the evaluation value of attribute c_j on alternative a_i, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. Five data types are considered: real numbers d_ij, interval numbers $[{\underline{e}}_{ij}, {\bar{e}}_{ij}]$ , triangular fuzzy numbers (a_ij, b_ij, c_ij), linguistic numbers s_ij ∈ S_h, and intuitionistic fuzzy numbers 〈u_ij, v_ij〉, where $S_{h} = {s_{0}, s_{1}, \dots, s_{T}}$ is the language set and T + 1 represents the number of element S_h. The linguistic numbers would be converted into triangular fuzzy numbers in order to simplify the processing and calculation [9]. The data normalization process is showed in Table 2.

Table 2
Normalized hybrid information

Type of information Benefit-type normalization x_ij Cost-type normalization x_ij

Real numbers $x_{ij} = d_{ij} / \sqrt{\sum_{i = 1}^{m} {(d_{ij})}^{2}}$ $x_{ij} = \sqrt{\sum_{i = 1}^{m} {(d_{ij})}^{2}} / d_{ij}$

Interval numbers $x_{ij} = [\begin{matrix} {\underline{e}}_{ij} / \sqrt{\sum_{i = 1}^{m} ({({\underline{e}}_{ij})}^{2} + {({\bar{e}}_{ij})}^{2})}, \\ {\bar{e}}_{ij} / \sqrt{\sum_{i = 1}^{m} ({({\underline{e}}_{ij})}^{2} + {({\bar{e}}_{ij})}^{2})} \end{matrix}]$ $x_{ij} = [\begin{matrix} (1 / {\bar{e}}_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / {\underline{e}}_{ij})}^{2} + {(1 / {\bar{e}}_{ij})}^{2})}, \\ (1 / {\underline{e}}_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / {\underline{e}}_{ij})}^{2} + {(1 / {\bar{e}}_{ij})}^{2})} \end{matrix}]$

Triangular fuzzy numbers $x_{ij} = [\begin{matrix} a_{ij} / \sqrt{\sum_{i = 1}^{m} ({(a_{ij})}^{2} + {(b_{ij})}^{2} + {(c_{ij})}^{2})}, \\ b_{ij} / \sqrt{\sum_{i = 1}^{m} ({(a_{ij})}^{2} + {(b_{ij})}^{2} + {(c_{ij})}^{2})}, \\ c_{ij} / \sqrt{\sum_{i = 1}^{m} ({(a_{ij})}^{2} + {(b_{ij})}^{2} + {(c_{ij})}^{2})} \end{matrix}]$ $x_{ij} = [\begin{matrix} (1 / c_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / a_{ij})}^{2} + {(1 / b_{ij})}^{2} + {(1 / c_{ij})}^{2})}, \\ (1 / b_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / a_{ij})}^{2} + {(1 / b_{ij})}^{2} + {(1 / c_{ij})}^{2})}, \\ (1 / a_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / a_{ij})}^{2} + {(1 / b_{ij})}^{2} + {(1 / c_{ij})}^{2})} \end{matrix}]$

Linguistic numbers $x_{ij} = [\begin{matrix} s_{f}^{1} / \sqrt{\sum_{i = 1}^{m} ({(s_{f}^{1})}^{2} + {(s_{f}^{2})}^{2} + {(s_{f}^{3})}^{2})}, \\ s_{f}^{2} / \sqrt{\sum_{i = 1}^{m} ({(s_{f}^{1})}^{2} + {(s_{f}^{2})}^{2} + {(s_{f}^{3})}^{2})}, \\ s_{f}^{3} / \sqrt{\sum_{i = 1}^{m} ({(s_{f}^{1})}^{2} + {(s_{f}^{2})}^{2} + {(s_{f}^{3})}^{2})} \end{matrix}]$ $x_{ij} = [\begin{matrix} (1 / s_{f}^{3}) / \sqrt{\sum_{i = 1}^{m} ({(1 / s_{f}^{1})}^{2} + {(1 / s_{f}^{2})}^{2} + {(1 / s_{f}^{3})}^{2})}, \\ (1 / s_{f}^{2}) / \sqrt{\sum_{i = 1}^{m} ({(1 / s_{f}^{1})}^{2} + {(1 / s_{f}^{2})}^{2} + {(1 / s_{f}^{3})}^{2})}, \\ (1 / s_{f}^{1}) / \sqrt{\sum_{i = 1}^{m} ({(1 / s_{f}^{1})}^{2} + {(1 / s_{f}^{2})}^{2} + {(1 / s_{f}^{3})}^{2})} \end{matrix}]$

Intuitive fuzzy numbers $x_{ij} = 〈 \begin{matrix} u_{ij} / \sqrt{\sum_{i = 1}^{m} ({(u_{ij})}^{2} + {(v_{ij})}^{2})}, \\ v_{ij} / \sqrt{\sum_{i = 1}^{m} ({(u_{ij})}^{2} + {(v_{ij})}^{2})} \end{matrix} 〉$ $x_{ij} = 〈 \begin{matrix} (1 / v_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / u_{ij})}^{2} + {(1 / v_{ij})}^{2})}, \\ (1 / u_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / u_{ij})}^{2} + {(1 / v_{ij})}^{2})} \end{matrix} 〉$

Type of information	Benefit-type normalization x_ij	Cost-type normalization x_ij
Real numbers	$x_{ij} = d_{ij} / \sqrt{\sum_{i = 1}^{m} {(d_{ij})}^{2}}$	$x_{ij} = \sqrt{\sum_{i = 1}^{m} {(d_{ij})}^{2}} / d_{ij}$
Interval numbers	$x_{ij} = [\begin{matrix} {\underline{e}}_{ij} / \sqrt{\sum_{i = 1}^{m} ({({\underline{e}}_{ij})}^{2} + {({\bar{e}}_{ij})}^{2})}, \\ {\bar{e}}_{ij} / \sqrt{\sum_{i = 1}^{m} ({({\underline{e}}_{ij})}^{2} + {({\bar{e}}_{ij})}^{2})} \end{matrix}]$	$x_{ij} = [\begin{matrix} (1 / {\bar{e}}_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / {\underline{e}}_{ij})}^{2} + {(1 / {\bar{e}}_{ij})}^{2})}, \\ (1 / {\underline{e}}_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / {\underline{e}}_{ij})}^{2} + {(1 / {\bar{e}}_{ij})}^{2})} \end{matrix}]$
Triangular fuzzy numbers	$x_{ij} = [\begin{matrix} a_{ij} / \sqrt{\sum_{i = 1}^{m} ({(a_{ij})}^{2} + {(b_{ij})}^{2} + {(c_{ij})}^{2})}, \\ b_{ij} / \sqrt{\sum_{i = 1}^{m} ({(a_{ij})}^{2} + {(b_{ij})}^{2} + {(c_{ij})}^{2})}, \\ c_{ij} / \sqrt{\sum_{i = 1}^{m} ({(a_{ij})}^{2} + {(b_{ij})}^{2} + {(c_{ij})}^{2})} \end{matrix}]$	$x_{ij} = [\begin{matrix} (1 / c_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / a_{ij})}^{2} + {(1 / b_{ij})}^{2} + {(1 / c_{ij})}^{2})}, \\ (1 / b_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / a_{ij})}^{2} + {(1 / b_{ij})}^{2} + {(1 / c_{ij})}^{2})}, \\ (1 / a_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / a_{ij})}^{2} + {(1 / b_{ij})}^{2} + {(1 / c_{ij})}^{2})} \end{matrix}]$
Linguistic numbers	$x_{ij} = [\begin{matrix} s_{f}^{1} / \sqrt{\sum_{i = 1}^{m} ({(s_{f}^{1})}^{2} + {(s_{f}^{2})}^{2} + {(s_{f}^{3})}^{2})}, \\ s_{f}^{2} / \sqrt{\sum_{i = 1}^{m} ({(s_{f}^{1})}^{2} + {(s_{f}^{2})}^{2} + {(s_{f}^{3})}^{2})}, \\ s_{f}^{3} / \sqrt{\sum_{i = 1}^{m} ({(s_{f}^{1})}^{2} + {(s_{f}^{2})}^{2} + {(s_{f}^{3})}^{2})} \end{matrix}]$	$x_{ij} = [\begin{matrix} (1 / s_{f}^{3}) / \sqrt{\sum_{i = 1}^{m} ({(1 / s_{f}^{1})}^{2} + {(1 / s_{f}^{2})}^{2} + {(1 / s_{f}^{3})}^{2})}, \\ (1 / s_{f}^{2}) / \sqrt{\sum_{i = 1}^{m} ({(1 / s_{f}^{1})}^{2} + {(1 / s_{f}^{2})}^{2} + {(1 / s_{f}^{3})}^{2})}, \\ (1 / s_{f}^{1}) / \sqrt{\sum_{i = 1}^{m} ({(1 / s_{f}^{1})}^{2} + {(1 / s_{f}^{2})}^{2} + {(1 / s_{f}^{3})}^{2})} \end{matrix}]$
Intuitive fuzzy numbers	$x_{ij} = 〈 \begin{matrix} u_{ij} / \sqrt{\sum_{i = 1}^{m} ({(u_{ij})}^{2} + {(v_{ij})}^{2})}, \\ v_{ij} / \sqrt{\sum_{i = 1}^{m} ({(u_{ij})}^{2} + {(v_{ij})}^{2})} \end{matrix} 〉$	$x_{ij} = 〈 \begin{matrix} (1 / v_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / u_{ij})}^{2} + {(1 / v_{ij})}^{2})}, \\ (1 / u_{ij}) / \sqrt{\sum_{i = 1}^{m} ({(1 / u_{ij})}^{2} + {(1 / v_{ij})}^{2})} \end{matrix} 〉$

Considering regret avoidance behavior of the decision-maker, the regret theory and the catastrophe progression method is employed to sequence the alternatives and achieve the maximum satisfaction of the decision-maker. The steps are as follows:

Step 1: Normalize the original decision matrix $X^{'} = {[x_{ij}^{'}]}_{m \times n}$ and get the normalized matrix X = [x_ij] _m×n.

Step 2: Calculate the distance between the evaluation value and the positive and negative ideal points and describe the degree of regret-rejoicing of the decision-maker to construct a matrix of regret and rejoicing, as shown in Table 3. Therefore, the regret value and rejoicing value of the alternative under each attribute are as follows:

Table 3

Distance from positive and negative ideal points

Type of information	Positive ideal point x⁺	Positive ideal point distance d (x_ij, x⁺)	Negative ideal point x^-	Negative ideal point distance d (x_ij, x^-)
Real numbers	$max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (d_{ij})$	$\sqrt{{(d_{ij} - max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (d_{ij}))}^{2}}$	$min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (d_{ij})$	$\sqrt{{(d_{ij} - d_{i}^{-})}^{2}}$
Interval numbers	$[max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} ({\bar{e}}_{ij}), max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} ({\bar{e}}_{ij})]$	$\sqrt{\frac{1}{2} [{({\underline{e}}_{ij} - {\underline{e}}_{ij}^{+})}^{2} + {({\bar{e}}_{ij} - {\bar{e}}_{ij}^{+})}^{2}]}$	$[min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} ({\underline{e}}_{ij}), min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} ({\underline{e}}_{ij})]$	$\sqrt{\frac{1}{2} [{({\underline{e}}_{ij} - {\underline{e}}_{ij}^{-})}^{2} + {({\bar{e}}_{ij} - {\bar{e}}_{ij}^{-})}^{2}]}$
Triangular fuzzy numbers	$[\begin{matrix} max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (a_{ij}), max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (b_{ij}), \\ max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (c_{ij}) \end{matrix}]$	$\sqrt{\frac{1}{3} [{(a_{ij} - a_{ij}^{+})}^{2} + {(b_{ij} - b_{ij}^{+})}^{2} + {(c_{ij} - c_{ij}^{+})}^{2}]}$	$[\begin{matrix} min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (a_{ij}), min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (b_{ij}), \\ min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (c_{ij}) \end{matrix}]$	$\sqrt{\frac{1}{3} [{(a_{ij} - a_{ij}^{-})}^{2} + {(b_{ij} - b_{ij}^{-})}^{2} + {(c_{ij} - c_{ij}^{-})}^{2}]}$
Linguistic numbers	$[\begin{matrix} max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (s_{f}^{3}), max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (s_{f}^{3}), \\ max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (s_{f}^{3}) \end{matrix}]$	$\sqrt{\frac{1}{3} [{(s_{f}^{1} - s_{f}^{1^{+}})}^{2} + {(s_{f}^{2} - s_{f}^{2^{+}})}^{2} + {(s_{f}^{3} - s_{f}^{3^{+}})}^{2}]}$	$[\begin{matrix} min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (s_{f}^{1}), min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (s_{f}^{1}), \\ min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (s_{f}^{1}) \end{matrix}]$	$\sqrt{\frac{1}{3} [{(s_{f}^{1} - s_{f}^{1^{-}})}^{2} + {(s_{f}^{2} - s_{f}^{2^{-}})}^{2} + {(s_{f}^{3} - s_{f}^{3^{-}})}^{2}]}$
Intuitive fuzzy numbers	$[max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (u_{ij}), min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (v_{ij})]$	$\sqrt{\frac{1}{2} [{(u_{ij} - u_{ij}^{+})}^{2} + {(v_{ij} - v_{ij}^{+})}^{2}]}$	$[min_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (u_{ij}), max_{1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n} (v_{ij})]$	$\sqrt{\frac{1}{2} [{(u_{ij} - u_{ij}^{-})}^{2} + {(v_{ij} - v_{ij}^{-})}^{2}]}$

$R_{ij}^{'} = 1 - exp (γ d (x_{ij}, x_{j}^{+}))$ (3) $G_{ij}^{'} = 1 - exp (- γ d (x_{ij}, x_{j}^{-}))$ (4)

Where $x_{j}^{+} = max (x_{ij})$ , $x_{j}^{-} = min (x_{ij})$ , γ represents the decision-makers’ regret avoidance coefficient, and γ > 0. Regret value $R_{ij}^{'}$ and rejoicing value $G_{ij}^{'}$ are normalized: $R_{ij} = R_{ij}^{'} / \sqrt{\sum_{i = 1}^{m} {(R_{ij}^{'})}^{2}}$ (5) $G_{ij} = G_{ij}^{'} / \sqrt{\sum_{i = 1}^{m} {(G_{ij}^{'})}^{2}}$ (6)

Step 3: Obtain the ranking of attributes’ relative importance degree by the maximizing deviation method. Currently, most methods are subjective judgments on the ranking of attributes, and the evaluation results lack objectivity. The maximizing deviation method effectively avoids this deficiency. $w_{j}^{*} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} | x_{ij} - x_{kj} |}{\sqrt{\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} \sum_{k = 1}^{m} | x_{ij} - x_{kj} |)}^{2}}}$ (7)

Step 4: Calculate total catastrophe membership value by the PROMETHEE Π method [38] and the normalized regret catastrophe membership value and the rejoicing catastrophe membership value of different attributes, as shown in Table 4. For example, if the control dimension is 5, the calculation function is divided into complementary state $δ_{1}$ and non-complementary state $δ_{2}$ .

Table 4

Regretted catastrophe progression normalization model

Catastrophe type	Status dimension	Control dimension	Normalization formula
			Regret normalization	Rejoicing normalization
Fold catastrophe	1	1	$x_{a} = \sqrt{\| R_{a} \|}$	$x_{a} = \sqrt{\| G_{a} \|}$
Cusp catastrophe	1	2	$x_{a} = \sqrt{\| R_{a} \|}, x_{b} = \sqrt{\| R_{b} \| 3}$	$x_{a} = \sqrt{\| G_{a} \|}, x_{b} = \sqrt{\| G_{b} \| 3}$
Swallowtail catastrophe	1	3	$x_{a} = \sqrt{\| R_{a} \|}, x_{b} = \sqrt{\| R_{b} \| 3}, x_{c} = \sqrt{\| R_{c} \| 4}$	$x_{a} = \sqrt{\| G_{a} \|}, x_{b} = \sqrt{\| G_{b} \| 3}, x_{c} = \sqrt{\| G_{c} \| 4}$
Butterfly catastrophe	1	4	$x_{a} = \sqrt{\| R_{a} \|}, x_{b} = \sqrt{\| R_{b} \| 3}, x_{c} = \sqrt{\| R_{c} \| 4}, x_{d} = \sqrt{\| R_{d} \| 5}$	$x_{a} = \sqrt{\| G_{a} \|}, x_{b} = \sqrt{\| G_{b} \| 3}, x_{c} = \sqrt{\| G_{c} \| 4}, x_{d} = \sqrt{\| G_{d} \| 5}$
Shack catastrophe	1	5	$x_{a} = \sqrt{\| R_{a} \|}, x_{b} = \sqrt{\| R_{b} \| 3}, x_{c} = \sqrt{\| R_{c} \| 4}, x_{d} = \sqrt{\| R_{d} \| 5}, x_{e} = \sqrt{\| R_{e} \| 6}$	$x_{a} = \sqrt{\| G_{a} \|}, x_{b} = \sqrt{\| G_{b} \| 3}, x_{c} = \sqrt{\| G_{c} \| 4}, x_{d} = \sqrt{\| G_{d} \| 5}, x_{e} = \sqrt{\| G_{e} \| 6}$

Complementary:

$\begin{matrix} δ_{1} = a v e r a g e (\sqrt{| G_{a} |}, \sqrt{| G_{b} | 3}, \sqrt{| G_{c} | 4}, \sqrt{| G_{d} | 5}, \sqrt{| G_{e} | 6}) \\ - a v e r a g e (\sqrt{| R_{a} |}, \sqrt{| R_{b} | 3}, \sqrt{| R_{c} | 4}, \sqrt{| R_{d} | 5}, \sqrt{| R_{e} | 6}) \end{matrix}$ (8)

Non-complementary:

$\begin{matrix} δ_{2} = min (\sqrt{| G_{a} |}, \sqrt{G_{b} 3}, \sqrt{| G_{c} | 4}, \sqrt{G_{d} 5}, \sqrt{| G_{e} | 6}) \\ - min (\sqrt{| R_{a} |}, \sqrt{R_{b} 3}, \sqrt{| R_{c} | 4}, \sqrt{R_{d} 5}, \sqrt{| R_{e} | 6}) \end{matrix}$ (9)

Step 5: Rank the alternatives.

4 Illustrative example

In order to show the effectiveness of the decision-making method, the method is applied to the selection of cold chain logistics service provider. For example, YONGHUI SUPERSTORES, as one of fresh produce management enterprises in China, is planning to find a cold chain logistics service provider as its strategic partner in order to better meet the demand of the distribution business in Shanghai. Through interviewing and scoring the cold chain service provider’s official website, e-commerce research center, and industry practitioners, a total of 5 local cold chain logistics service providers (a₁, a₂, a₃, a₄, a₅) in Shanghai were collected. In interviews with industry practitioners, the index data of the number of refrigerated trucks (c₆), cold storage ratio (c₇), and cold storage turnover ratio (c₁₆) can be calculated through the online survey and related formulas. Customer satisfaction (c₂₃) is expressed by the number of membership and non-membership, < $\frac{m}{m + n}$ , $\frac{n}{m + n}$ >, where m represents the number of approval in the survey, and n represents the number of disapproval in the survey. The data of remaining 19 attributes are assessment by 10-point scale (c₁₀, c₁₁, c₁₂, c₁₃, c₁₄, c₁₅, c₁₈, c₁₉, c₂₀, c₂₁) and language scoring (c₁, c₂, c₃, c₄, c₅, c₈, c₉, c₁₇, c₂₂), where $S = {s_{0}, s_{1}, \dots, s_{6}} = {\begin{matrix} VP (very poor), \\ P (poor), \\ MP (medium poor), \\ M (medium), \\ MG (medium good), \\ G (good), \\ VG (very good) . \end{matrix}}$ .

Step 1: Obtain the normalized matrix, and then normalize the original decision matrix by the normalized formula, as shown in in Table 5 and Table 6.

Table 5
Decision-making matrix

c ₁ c ₂ c ₃ c ₄ c ₅ c ₆ c ₇ c ₈ c ₉ c ₁₀ c ₁₁ c ₁₂ c ₁₃ c ₁₄ c ₁₅ c ₁₆ c ₁₇ c ₁₈ c ₁₉ c ₂₀ c ₂₁ c ₂₂ c ₂₃

a ₁ M M MG M MP 50 16% M M 6 7 6 7 5 6 70% G 9 3 8 8 M <0.7,0.3>

a ₂ G MG MG MG G 300 25% M MG 7 7 6 6 6 7 90% G 8 1 9 9 VG <0.8,0.2>

a ₃ G VG VG VG G 476 16.7% MP G 9 9 9 9 9 8 90% VG 13 1 9 10 VG <0.9,0.1>

a ₄ VG VG G MG G 209 20% MG MG 8 9 9 7 6 9 80% M 14.1 2 9 10 G <0.9,0.1>

a ₅ MG G MG MG MG 30 12.5% MP MG 8 8 7 7 6 8 80% MG 8 2 7 9 MG <0.7,0.3>

Table 6

Normalized decision-making matrix D

	c ₁	c ₂	c ₃	c ₄	c ₅	c ₆	c ₇	c ₈	c ₉	c ₁₀	c ₁₁	c ₁₂
a ₁	(0.1101, 0.1651, 0.2202)	(0.1069, 0.1604, 0.2138)	(0.1667, 0.2222, 0.2778)	(0.1204, 0.1806, 0.2408)	(0.0582, 0.1164, 0.1747)	0.0829	0.3862	(0.1715, 0.2572, 0.343)	(0.125, 0.1875, 0.25)	0.3499	0.3889	0.3567
a ₂	(0.2202, 0.2752, 0.3303)	(0.1604, 0.2138, 0.2673)	(0.1667, 0.2222, 0.2778)	(0.1806, 0.2408, 0.301)	(0.2329, 0.2911, 0.3493)	0.4975	0.6035	(0.1715, 0.2572, 0.343)	(0.1875, 0.25, 0.3125)	0.4082	0.3889	0.3567
a ₃	(0.2202, 0.2752, 0.3303)	(0.2673, 0.3207, 0.3207)	(0.2778, 0.3333, 0.3333)	(0.301, 0.3612, 0.3612)	(0.2329, 0.2911, 0.3493)	0.7893	0.4031	(0.0857, 0.1715, 0.2572)	(0.25, 0.3125, 0.375)	0.5249	0.5	0.535
a ₄	(0.2752, 0.3303, 0.3303)	(0.2673, 0.3207, 0.3207)	(0.2222, 0.2778, 0.3333)	(0.1806, 0.2408, 0.301)	(0.2329, 0.2911, 0.3493)	0.3466	0.4828	(0.2572, 0.343, 0.4287)	(0.1875, 0.25, 0.3125)	0.4666	0.5	0.535
a ₅	(0.1651, 0.2202, 0.2752)	(0.2138, 0.2673, 0.3207)	(0.1667, 0.2222, 0.2778)	(0.1806, 0.2408, 0.301)	(0.1747, 0.2329, 0.2911)	0.0497	0.3017	(0.0857, 0.1715, 0.2572)	(0.1875, 0.25, 0.3125)	0.4666	0.4444	0.4161
	c ₁₃	c ₁₄	c ₁₅	c ₁₆	c ₁₇	c ₁₈	c ₁₉	c ₂₀	c ₂₁	c ₂₂	c ₂₃
a ₁	0.4308	0.3418	0.3499	0.3802	(0.2202, 0.2752, 0.3303)	2.6685	1.453	0.424	0.3876	(0.1069, 0.1604, 0.2138)	<0.3752, 0.1608>
a ₂	0.3693	0.4102	0.4082	0.4888	(0.2202, 0.2752, 0.3303)	3.0021	4.3589	0.477	0.4361	(0.2673, 0.3207, 0.3207)	<0.4288, 0.1072>
a ₃	0.5539	0.6152	0.4666	0.4888	(0.2752, 0.3303, 0.3303)	1.8475	4.3589	0.477	0.4845	(0.2673, 0.3207, 0.3207)	<0.4825, 0.0536>
a ₄	0.4308	0.4102	0.5249	0.4345	(0.1101, 0.1651, 0.2202)	1.7033	2.1794	0.477	0.4845	(0.2138, 0.2673, 0.3207)	<0.4825, 0.0536>
a ₅	0.4308	0.4102	0.4666	0.4345	(0.1651, 0.2202, 0.2752)	3.0021	2.1794	0.371	0.4361	(0.1604, 0.2138, 0.2673)	<0.3752, 0.1608>

Step 2: Calculate the distance between the positive and negative ideal points is shown in Table 7 and Table 8. And then, the regret matrix R and the rejoicing matrix G are established by equations (5) and (6), the regret avoidance coefficient γ = 0.3 [9].

Table 7

The distance of positive ideal point

	c ₁	c ₂	c ₃	c ₄	c ₅	c ₆	c ₇	c ₈	c ₉	c ₁₀	c ₁₁	c ₁₂
a ₁	0.1933	0.1877	0.1283	0.2114	0.2772	0.7064	0.2173	0.198	0.2195	0.175	0.1111	0.1783
a ₂	0.0711	0.1234	0.1283	0.139	0.0752	0.2919	0	0.198	0.1443	0.1166	0.1111	0.1783
a ₃	0.0711	0.0309	0.0321	0.0348	0.0752	0	0.2004	0.3011	0.0807	0	0	0
a ₄	0.0318	0.0309	0.0717	0.139	0.0752	0.4428	0.1207	0.1107	0.1443	0.0583	0	0
a ₅	0.1271	0.069	0.1283	0.139	0.1345	0.7396	0.3017	0.3011	0.1443	0.0583	0.0556	0.1189
	c ₁₃	c ₁₄	c ₁₅	c ₁₆	c ₁₇	c ₁₈	c ₁₉	c ₂₀	c ₂₁	c ₂₂	c ₂₃
a ₁	0.1231	0.2734	0.175	0.1086	0.0711	0.3336	2.9059	0.0047	0.0969	0.1877	0.1072
a ₂	0.1846	0.2051	0.1166	0	0.0711	0	0	0.0483	0.0485	0.0309	0.0536
a ₃	0	0	0.0583	0	0.0318	1.1547	0	0.0483	0	0.0309	0
a ₄	0.1231	0.2051	0	0.0543	0.1933	1.2988	2.1794	0.0483	0	0.069	0
a ₅	0.1231	0.2051	0.0583	0.0543	0.1271	0	2.1794	0.0577	0.0485	0.1234	0.1072

Table 8

The distance of the negative ideal point

	c ₁	c ₂	c ₃	c ₄	c ₅	c ₆	c ₇	c ₈	c ₉	c ₁₀	c ₁₁	c ₁₂
a ₁	0.1146	0.1113	0.1156	0.1253	0.1212	0.0332	0.0845	0.2801	0.1301	0	0	0
a ₂	0.2482	0.1746	0.1156	0.1966	0.3361	0.4477	0.3017	0.2801	0.2041	0.0583	0	0
a ₃	0.2482	0.2637	0.2029	0.2969	0.3361	0.7396	0.1014	0.1785	0.2818	0.175	0.1111	0.1783
a ₄	0.2715	0.2637	0.1814	0.1966	0.3361	0.2968	0.181	0.3867	0.2041	0.1166	0.1111	0.1783
a ₅	0.1798	0.241	0.1156	0.1966	0.2625	0	0	0.1785	0.2041	0.1166	0.0556	0.0594
	c ₁₃	c ₁₄	c ₁₅	c ₁₆	c ₁₇	c ₁₈	c ₁₉	c ₂₀	c ₂₁	c ₂₂	c ₂₃
a ₁	0.0615	0	0	0	0.2482	0.9652	0	0.1668	0	0.1113	0
a ₂	0	0.0684	0.0583	0.1086	0.2482	1.2988	2.9059	0.2198	0.0485	0.2637	0.0536
a ₃	0.1846	0.2734	0.1166	0.1086	0.2715	0.1441	2.9059	0.2198	0.0969	0.2637	0.1072
a ₄	0.0615	0.0684	0.175	0.0543	0.1146	0	0.7265	0.2198	0.0969	0.241	0.1072
a ₅	0.0615	0.0684	0.1166	0.0543	0.1798	1.2988	0.7265	0.1138	0.0485	0.1746	0

$\begin{matrix} R = [\begin{matrix} - 0.0597 & - 0.0579 & - 0.0392 & - 0.0655 & - 0.0867 & - 0.2361 & - 0.0673 & - 0.0612 & - 0.0681 & - 0.0539 & - 0.0339 & - 0.055 & - 0.0376 \\ - 0.0215 & - 0.0377 & - 0.0392 & - 0.0426 & - 0.0228 & - 0.0915 & 0 & - 0.0612 & - 0.0443 & - 0.0356 & - 0.0339 & - 0.055 & - 0.057 \\ - 0.0215 & - 0.0093 & - 0.0097 & - 0.0105 & - 0.0228 & 0 & - 0.0619 & - 0.0945 & - 0.0245 & 0 & 0 & 0 & 0 \\ - 0.0096 & - 0.0093 & - 0.0217 & - 0.0426 & - 0.0228 & - 0.1421 & - 0.0369 & - 0.0338 & - 0.0443 & - 0.0177 & 0 & 0 & - 0.0376 \\ - 0.0389 & - 0.0209 & - 0.0392 & - 0.0426 & - 0.0412 & - 0.2484 & - 0.0947 & - 0.0945 & - 0.0443 & - 0.0177 & - 0.0168 & - 0.0363 & - 0.0376 \end{matrix} \\ \begin{matrix} - 0.0855 & - 0.0539 & - 0.0331 & - 0.0215 & - 0.1052 & - 0.7293 & - 0.0014 & - 0.0295 & - 0.0579 & - 0.0327 \\ - 0.0635 & - 0.0356 & 0 & - 0.0215 & 0 & 0 & - 0.0146 & - 0.0146 & - 0.0093 & - 0.0162 \\ 0 & - 0.0177 & 0 & - 0.0096 & - 0.414 & 0 & - 0.0146 & 0 & - 0.0093 & 0 \\ - 0.0635 & 0 & - 0.0164 & - 0.0597 & - 0.4764 & - 0.9229 & - 0.0146 & 0 & - 0.0209 & 0 \\ - 0.0635 & - 0.0177 & - 0.0164 & - 0.0389 & 0 & - 0.9229 & - 0.0175 & - 0.0146 & - 0.0377 & - 0.0327 \end{matrix}] \end{matrix}$ $\begin{matrix} G = [\begin{matrix} 0.0 338 & 0.0 328 & 0.0 341 & 0.0 369 & 0.0 357 & 0.00 99 & 0.0 25 & 0.0 806 & 0.0 383 & 0 & 0 & 0 & 0.0 183 \\ 0.0 718 & 0.0 51 & 0.0 341 & 0.0 573 & 0.0 959 & 0.1257 & 0.0 865 & 0.0 806 & 0.0 594 & 0.0 173 & 0 & 0 & 0 \\ 0.0 718 & 0.0 761 & 0.0 59 & 0.0 852 & 0.0 959 & 0.199 & 0.0 3 & 0.0 521 & 0.0 811 & 0.0 511 & 0.0 328 & 0.0 521 & 0.0 539 \\ 0.0 782 & 0.0 761 & 0.0 53 & 0.0 573 & 0.0 959 & 0.0 852 & 0.0 529 & 0.1095 & 0.0 594 & 0.0 344 & 0.0 328 & 0.0 521 & 0.0 183 \\ 0.0 525 & 0.0 698 & 0.0 341 & 0.0 573 & 0.0 757 & 0 & 0 & 0.0 521 & 0.0 594 & 0.0 344 & 0.0 165 & 0.0 177 & 0.0 183 \end{matrix} \\ \begin{matrix} 0 & 0 & 0 & 0.0 718 & 0.2514 & 0 & 0.0 488 & 0 & 0.0 328 & 0 \\ 0.0 203 & 0.0 173 & 0.0 321 & 0.0 718 & 0.3227 & 0.5818 & 0.0 638 & 0.0 144 & 0.0 761 & 0.0 16 \\ 0.0 788 & 0.0 344 & 0.0 321 & 0.0 782 & 0.0 423 & 0.5818 & 0.0 638 & 0.0 287 & 0.0 761 & 0.0 317 \\ 0.0 203 & 0.0 511 & 0.0 162 & 0.0 338 & 0 & 0.1958 & 0.0 638 & 0.0 287 & 0.0 698 & 0.0 317 \\ 0.0 203 & 0.0 344 & 0.0 162 & 0.0 525 & 0.3227 & 0.1958 & 0.0 335 & 0.0 144 & 0.0 51 & 0 \end{matrix}] \end{matrix}$

Step 3: Get the importance degree of attribute according to the weight value of each attribute by the equation (7), as shown in Table 9.

Table 9

Weight values of attributes

I indexes	weight of I indexes	weight of II indexes	weight of II indexes
Enterprise Management capacity (b₁)	W₁ = 0 . 1270	Enterprise qualification (c₁)	w₁ = 0.0264
		Enterprise reputation (c₂)	w₂ = 0.0284
		Business Scope (c₃)	w₃ = 0.0183
		Enterprise regulations (c₄)	w₄ = 0.024
		Enterprise anti-risk ability (c₅)	w₅ = 0.0299
Logistics infrastructure (b₂)	W₂ = 0 . 1594	Number of refrigerated trucks (c₆)	w₆ = 0.0801
		Cold storage ratio (c₇)	w₇ = 0.0296
		Employee literacy (c₈)	w₈ = 0.0314
		Other auxiliary hardware facilities (c₉)	w₉ = 0.0183
Logistics information technology (b₃)	W₃ = 0 . 0927	Technical level of temperature control (c₁₀)	w₁₀ = 0.0173
		Real-time monitoring technology level (c₁₁)	w₁₁ = 0.0141
		Timeliness of information transmission (c₁₂)	w₁₂ = 0.0226
		Information Traceability (c₁₃)	w₁₃ = 0.0156
		Information sharing rate (c₁₄)	w₁₄ = 0.0231
Logistics operations capability (b₄)	W₄ = 0 . 2138	Refrigerated truck utilization (c₁₅)	w₁₅ = 0.0173
		Cold storage turnover ratio (c₁₆)	w₁₆ = 0.0115
		Distribution diversity (c₁₇)	w₁₇ = 0.0264
		Distribution cost (c₁₈)	w₁₈ = 0 .1587
Logistics service quality (b₅)	W₅ = 0 . 4071	Freshness loss rate (c₁₉)	w₁₉ = 0 .338
		Delivery on time rate (c₂₀)	w₂₀ = 0.0112
		Delivery accuracy (c₂₁)	w₂₁ = 0.0102
		Product safety (c₂₂)	w₂₂ = 0.0284
		Customer satisfaction rate (c₂₃)	w₂₃ = 0.0192

(1) According to the weight values, the importance order of II indexes is sorted:

(2) Enterprise Management capacity: c₃ < c₄ < c₁ < c₂ < c₅.

(3) Logistics infrastructure: c₉ < c₇ < c₈ < c₆.

(4) Logistics information technology: c₁₁ < c₁₃ < c₁₀ < c₁₂ < c₁₄.

(5) Logistics operations capability: c₁₆ < c₁₅ < c₁₇ < c₁₈.

Logistics service quality: c₂₁ < c₂₀ < c₂₃ < c₂₂ < c₁₉.

And, I index: B₅ (Logistics service quality)> B₄ (Logistics operations capability)> B₂ (Logistics infrastructure)> B₁ (Enterprise Management capacity)> B₃ (Logistics information technology).

Step 4: Calculate total catastrophe membership value by equations (8) and (9). Through indicator definition analysis, it can be considered that the indexes constructed under each dimension are complementary.

For example, the regret catastrophe membership function value of the I index of the alternative cold chain logistics service provider a₁ is as follows: $\begin{matrix} R_{11} = \frac{{(| c_{5} |)}^{1 / 2} + {(| c_{2} |)}^{1 / 3} + {(| c_{1} |)}^{1 / 4} + {(| c_{4} |)}^{1 / 5} + {(| c_{3} |)}^{1 / 6}}{5} = 0.4677 \\ R_{12} = \frac{{(| c_{6} |)}^{1 / 2} + {(| c_{8} |)}^{1 / 3} + {(| c_{7} |)}^{1 / 4} + {(| c_{9} |)}^{1 / 5}}{4} = 0 . 4934 \\ R_{13} = \frac{{(| c_{14} |)}^{1 / 2} + {(| c_{12} |)}^{1 / 3} + {(| c_{10} |)}^{1 / 4} + {(| c_{13} |)}^{1 / 5} + {(| c_{11} |)}^{1 / 6}}{5} = 0.4485 \\ R_{14} = \frac{{(| c_{18} |)}^{1 / 2} + {(| c_{17} |)}^{1 / 3} + {(| c_{15} |)}^{1 / 4} + {(| c_{16} |)}^{1 / 5}}{4} = 0.3975 \\ R_{15} = \frac{{(| c_{19} |)}^{1 / 2} + {(| c_{22} |)}^{1 / 3} + {(| c_{23} |)}^{1 / 4} + {(| c_{20} |)}^{1 / 5} + {(| c_{21} |)}^{1 / 6}}{5} = 0 . 4981 \end{matrix}$

The rejoicing catastrophe membership function value of the I index of the alternative cold chain logistics service provider a₁ is as follows:

$G_{11} = \frac{{(| c_{5} |)}^{1 / 2} + {(| c_{2} |)}^{1 / 3} + {(| c_{1} |)}^{1 / 4} + {(| c_{4} |)}^{1 / 5} + {(| c_{3} |)}^{1 / 6}}{5} = 0.4048$ $G_{12} = \frac{{(| c_{6} |)}^{1 / 2} + {(| c_{8} |)}^{1 / 3} + {(| c_{7} |)}^{1 / 4} + {(| c_{9} |)}^{1 / 5}}{4} = 0.3625$ $G_{13} = \frac{{(| c_{14} |)}^{1 / 2} + {(| c_{12} |)}^{1 / 3} + {(| c_{10} |)}^{1 / 4} + {(| c_{13} |)}^{1 / 5} + {(| c_{11} |)}^{1 / 6}}{5} = 0 . 0899$ $G_{14} = \frac{{(| c_{18} |)}^{1 / 2} + {(| c_{17} |)}^{1 / 3} + {(| c_{15} |)}^{1 / 4} + {(| c_{16} |)}^{1 / 5}}{4} = 0 . 2293$ $G_{15} = \frac{{(| c_{19} |)}^{1 / 2} + {(c_{22})}^{1 / 3} + {(| c_{23} |)}^{1 / 4} + {(c_{20})}^{1 / 5} + {(| c_{21} |)}^{1 / 6}}{5} = 0 . 1733$

According to the PROMETHEE Π method and the I index ranking results, it can be seen that the total catastrophe membership value of the alternative cold chain logistics service providers a₁ is:

$\begin{matrix} δ (a_{1}) = average (\begin{matrix} {(| G_{15} |)}^{1 / 2} + {(G_{14})}^{1 / 3} + \\ {(| G_{12} |)}^{1 / 4} + {(G_{11})}^{1 / 5} + {(| G_{13} |)}^{1 / 6} \end{matrix}) - \\ average (\begin{matrix} {(| R_{15} |)}^{1 / 2} + {(R_{14})}^{1 / 3} + \\ {(| R_{12} |)}^{1 / 4} + {(R_{11})}^{1 / 5} + {(| R_{13} |)}^{1 / 6} \end{matrix}) = - 0 . 1499 \end{matrix}$

Similarly, the total catastrophe membership function values of other fresh cold chain logistics service providers are $δ (a_{2}) = 0.0620;$ $δ (a_{3}) = 0 . 2878;$ $δ (a_{4}) = 0.0433;$ $δ (a_{5}) = - 0.0321 .$

Step 5: The final alternative ranking is a₁ < a₅ < a₄ < a₂ < a₃.

In addition, this paper uses the traditional catastrophe progression method to verify and calculate this example, and the ranking is {0.7857, 0.9430, 0.9747, 0.9356, 0.8973}. The alternative is sorted as a₁ < a₅ < a₄ < a₂ < a₃, which consistent with the method in this paper. TOPSIS method calculates this example and the result is {0.2707, 0.4596, 0.7841, 0.7658, 0.4518}. The alternative is sorted as a₁ < a₅ < a₂ < a₄ < a₃. The order is slightly different from this paper, but the best results are all cold chain logistics service provider a₃.

Since the decision-maker’s regret avoidance behavior was not considered in the traditional decision-making method, the sensitivity of regret avoidance coefficient of the evaluation attribute is compared and analyzed. Due to γ > 0 and γ ∈(0, 1] , the sensitivity analysis of the second-level index and the I index is shown in Fig. 1 and Fig. 2, and the results of the final alternative rankings are shown in Fig. 3. Figure 1 and Fig. 2 show that with the continuous change of the regret avoidance coefficient in the decision-making process, the number of refrigerated trucks (c₆), technical level of temperature control (c₁₀), and freshness loss rate (c₁₉) in the second-level index fluctuate in a large range and are very sensitive. The sensitivity of the remaining second-level indexes is relatively weak, which should be paid attention to. The logistics service quality (b₅) of I indexes fluctuates the most, because the freshness loss rate is the second-level index of logistics service quality. Therefore, logistics service quality is the most sensitive among the I indexes, while logistics information technology (b₄) is the least sensitive.

Fig. 1

The range of II indexes with the changes γ.

Fig. 2

The range of I indexes with the changes γ.

Fig. 3

Sorting changes with the regret coefficient γ.

From the Fig. 3, when the regret avoidance coefficient is 0.1, the total value of the catastrophe membership function of a₄ is 0.0566, and the total value of the catastrophe membership function of a₂ is 0.0573. The sorting result is a₁ < a₅ < a₂ < a₄ < a₃, which is exactly the same as the result calculated by TOPSIS, but the ranking result fluctuates due to constant change of the regret avoidance coefficient. Therefore, in real life, the final sorting result may be different depending on the degree of regret in different situations because of the influence of external factors such as market environment. In this example, the best result is always the cold chain logistics service provider a₃, so Shanghai YH enterprises should choose a₃ as strategic partner. The example study results show that the cold chain logistics service provider ranking results obtained by the three decision-making methods are the same. However, compared with the traditional catastrophe progression method and TOPSIS method, the regret catastrophe progression method is closer to the actual decision-makers’ decision-making scenarios, and incorporates a variety of mixed information, which combines regret theory and catastrophe theory and help decision-makers make more scientific and reasonable decisions.

5 Conclusion

Considering the realistic decision-making situation, the degree of decision makers’ regret avoidance will affect the decision-making process and result. In order to solve the hybrid multi-attribute decision-making problem with bounded rationality, a multi-attribute decision-making method based on regret theory and catastrophe theory is proposed, where evaluation information contains real numbers, interval numbers, triangular fuzzy numbers, linguistic numbers, and intuitive fuzzy numbers. Hybrid evaluation information also improves the flexibility and richness of language information expression. The cognitive limitations of decision-makers and the influence of psychological and behavioral factors are fully considered, where the regret avoidance coefficient is introduced to deeply simulate bounded rational decision-making. And then, the maximizing deviation method is used to determine the importance of attributes, and the catastrophe progression is used to sort alternatives. In addition, the method given in this paper has the characteristics of simple calculation process, practical operability and strong practicability. Furthermore, this new approach can be used to select new product development and venture capital.

Footnotes

Acknowledgments

The authors would like to thank the anonymous reviewers and editors for their insightful and constructive comments on our paper. This work was supported in part by the Doctoral Project of Chongqing Federation of Social Science Circles under Grant 2016bs085 and the Humanities, Social Sciences Research General Project of Chongqing Education Commission under Grant 18SKGH045, Research Center for Cyber Society Development Problems under Chongqing Municipal Key Research (No. 2018skjd06), National Social Science Fund of Chongqing University of Posts and Telecommunications (No. 2017KZD10).

Conflict of interest

The authors declare no conflict of interest regarding the publication of this article.

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