Research on cloud-CRITIC-PDR method for hybrid multi-criteria decision making

Abstract

In order to solve the problems of insufficient uncertainty information measure, inaccuracy of weight calculation and incommensurability of indices in hybrid multi-criteria decision making, this paper introduces the Cloud-CRITIC weight calculation method and Cloud-CRITIC-PDR method, which combine cloud model, CRITIC method and Probabilistic Dominance Relation (PDR). In these two methods, the cloud model is used to characterize uncertainty, the Comprehensive information of CRITIC method has been modified in order to adapt to uncertain situation, the PDR method is used to compare schemes. A case study concerning supplier evaluation is given to demonstrate the merits of the cloud-CRITIC and cloud-CRITIC-PDR. The effectiveness and superiority of the developed methods are further illustrated through method comparison and sensitivity analysis. These combined methods can be used for dealing with decision-making problems with complex index types and strong data uncertainty, such as supplier evaluation and risk assessment. There are few papers about combining the cloud model, CRITIC method, and PDR method under hybrid indices decision-making situation at present, so this paper can provide a new perspective on hybrid MCDM.

Keywords

Hybrid multi-criteria decision making cloud model probabilistic dominance relationship CRITIC Gaussian criterion

1 Introduction

1.1 Hybrid multi-criteria decision-making

Multi-criteria Decision-Making (MCDM) is a decision-making model that aims to balance the conflict and immeasurability among criteria to obtain scientific ranking results. Its theory and methods are widely used in many fields such as economics, management, engineering, technology, military, etc. And it has become one of the research hotspots on decision theory. With the increasing complexity of the decision problems, some indices are difficult to quantify numerically and need to be portrayed by linguistic values in decision-making practice. For example, in the evaluation of suppliers, in order to obtain scientific and accurate rating results, decision makers often consider not only quantitative indices such as product market share and delivery time, but also qualitative indices such as fault handling satisfaction and performance described by linguistic values. Traditional MCDM methods have difficulty dealing with this problem. Therefore, it is of great practical significance to study the multi-criteria decision-making problem considering various data forms (hybrid MCDM), which has received extensive attention from scholars in various fields in recent years. Dinçer et al. [1] focused on the quality assessment of financial services, and used the hybrid multi-criteria decision-making method integrated by DEMATEL-ANP (DANP) and MOORA to make decisions; Gergin et al. [2] used the IFDEMATEL-IFTOPSIS method in the selection of automobile suppliers to improve the scientific nature of the decision results.

However, in hybrid multi-criteria decision-making, the data not only come from quantitative description but also from subjective evaluation, which makes the decision-making results more susceptible to the characteristics of subjective and objective indices. On the one hand, subjective cognitive bias and objective measurement error will cause random fluctuations in data and interfere with decision makers’ judgment; on the other hand, different metrics between indices will lead to data conflicts and incommensurability, affecting the integration of information. Without appropriate uncertainty measurement tools and scientific decision-making methods, the evaluation results will be inaccurate.

1.2 Related work

1.2.1 Uncertainty measurement tools

At present, most scholars prefer to use fuzzy sets (such as intuitionistic fuzzy numbers [3], interval intuitionistic fuzzy numbers [4], Pythagorean fuzzy numbers [5], Type-2 fuzzy numbers [6], etc.) to deal with the uncertainty in hybrid multi-criteria decision-making. However, the limitation of fuzzy sets is that they can only reflect the fuzziness of data and cannot reflect randomness. Therefore, academician Li et al. [7] proposed the concept of cloud model on the basis of probability theory and fuzzy mathematics in 1995, which can express uncertainty more comprehensively and has received extensive attention from the academic community.

Hassen et al. [8] used cloud models to construct fuzzy affiliation functions to effectively deal with the uncertainty and fuzziness of stock history data; Hu et al. [9] used cloud models to realize the transformation process from the actual value of indices to the ecological safety level in the ecological environmental safety evaluation; Wang et al. [10] proposed a decision-making method based on hierarchical structure and cloud model double integral (C-TI) to solve the MCDM problem with correlation between indices under uncertain information; Mandal et al. [11] used the cloud model and the joint compromise solution (CoCoSo) method to transform linguistic expressions into clear weighting matrices for supplier selection; Qiao et al. [12] constructed an evaluation index system and proposed a risk evaluation method for the anti-floating anchor system based on cloud model theory and system dynamics analysis. Although the cloud model theory has been widely used to deal with vagueness and uncertainty, there are relatively few studies in the field of hybrid MCDM. Fan et al. [13] designed a vector cloud weighting method based on cloud model and OWA method to solve the problem of weight uncertainty in hybrid MCDM. Zhang et al. [14] determined the grade threshold of the mixed index by cloud transformation, which reduced the influence of uncertainty on the evaluation results.

1.2.2 MCDM methods

Most scholars try to use the classical MCDM methods, such as TOPSIS [15], hierarchical analysis method [16] and integration operator [17], to solve the problem of conflict and incommensurability among evaluation criteria at present. However, these methods are easy to cause information loss and distortion in the standardization process. Therefore, a class of MCDM methods based on the comparison rules of the dominance relationship among schemes has gradually become a research hotspot in this field, such as interactive multi-criteria decision making method [18], PROMETHEE [19], probabilistic dominance relation method [20]. Relevant research has proved that such methods have higher prediction ability and lower information loss than classical MCDM methods. Among them, the Probabilistic Dominance Relation (PDR) method has received extensive attention from the academic community in recent years because of its good performance in dealing with the conflict of criteria.

Li et al. [21] introduced the α-probabilistic dominance relation with certain fault tolerance in the rough set model to solve the MCDM problem of incomplete systems; Wang [22] proposed a MCDM method based on the probabilistic dominance relation of linguistic values for the case where the characteristic attributes are linguistic values; Zhang et al. [23] explored the development level and spatial-temporal evolution of regional green logistics under the high-quality development of 30 provinces (cities) in China based on the improved PDR method. However, the PDR method does not consider the influence of the importance of the index on the decision-making results, and it is very important to solve the weight calculation problem of PDR under the mixed index.

1.2.3 Weight calculation methods

The existing weight calculation methods can be divided into two categories: subjective experience weighting method and objective weighting method. The subjective experience weighting method includes hierarchical analysis method, BWM [24], FUCOM [25] and so on. It is widely used and easy to operate, but it relies entirely on expert experience and lacks objective research and demonstration. Especially, when the problem involves hybrid indices, the process of obtaining information can be quite complicated [26]. The objective weighting method includes entropy weight method, standard deviation method, FANMA, CRITIC method, etc., which is based on the characteristics of the data and reduces the influence of decision makers’ subjective factors.Among them, the entropy weight method [27], FANMA [28] and the standard deviation method [29] have similar calculation principles, mainly relying on the index variability to determine the objective weight. In contrast, the CRITIC method determines the weight by evaluating the combination of correlation and information content. It not only considers the influence of the index variation on the weight, but also considers the conflict between the indices [30]. Therefore, compared with the above methods, the CRITIC method is more scientific and stable. With the deepening of research, some scholars began to improve the traditional CRITIC method. Malisa et al. [31] improved its data normalization process and comprehensive information calculation formula to make the weight calculation more accurate; Krishnan et al. [32] optimized the conflict parameters and proposed a D-CRITIC method based on distance correlation formula; Menekse et al. [33] combined the Pythagorean fuzzy set with the CRITIC method to expand the application field of the CRITIC method.

1.2.4 Overall assessment of the literature

It can be seen that decision-making results are easily affected by the characteristics of mixed indices. This problem limits the further development of hybrid MCDM research. Though most of the scholars have made explorations on uncertainty measurement and scientific decision-making, related research focusing on the field of hybrid MCDM is still in infancy. Specifically, in the current research, 1) fuzzy sets are difficult to fully describe uncertainty, 2) traditional decision-making methods are prone to cause information loss when dealing with index conflicts, 3) there is a lack of weight calculation method suitable for uncertain situation. In these regards, the existing studies believe that the cloud model, CRITIC method and PDR method are used as important tools for describe uncertainty, obtain objective weights and effectively deal with incommensurability between indices. However, few literatures explores the integration of these three methods for the solution of hybrid MCDM, and related research needs to be further developed.

1.3 Contributions and advantage of this new proposed method

Based on the key ideas and achievements of existing research, this paper combines cloud model theory and CRITIC method to improve the PDR method. The general view of the Cloud-CRITIC-PDR model is shown in Fig. 1. The main contributions and innovations are as follows.

A cloud-CRITIC weight calculation method is proposed. The hybrid indices are converted into cloud models according to relevant theory. On the basis of this, the cloud digital feature is used to improve the CRITIC method. Then, the cloud-CRITIC method suitable for the weight calculation of mixed indices is proposed, which solves the problem of uncertain information expression and inaccurate weighting of mixed indices;

A cloud-CRITIC-PDR hybrid multi-criteria decision making method is designed. Combined with the calculation results of the cloud-CRITIC method, we design and improve the comparison rules and calculation process of dominance degree in the PDR method. Then, by calculating the α-probabilistic dominance class, the dominance matrix and the comprehensive dominance, the ranking between schemes is finally obtained, which solves the problem of conflict and incommensurability between indices.

The novelty of the paper is reflected in the fact that the proposed cloud-CRITIC-PDR method has the capability to evaluate the index weight and capture the priority relationship among schemes in an uncertain mixed information environment, and has generality in mixed data processing. Compared with the traditional MCDM, this method describes the uncertainty more comprehensively, and increases the consideration of data credibility in the weight calculation, so it has a great improvement in practicability and accuracy.

Fig. 1

The general view of the Cloud-CRITIC-PDR hybrid MCDM model.

The advantage of this method is that it describes the uncertainty information more comprehensively. When calculating the index weight, it considers not only the conflict and variability, but also the reliability.

The remaining sections of this paper are structured as follows: Section 2 mainly introduces the basic knowledge of cloud model and PDR. Section 3 proposes a cloud-CRITIC weight calculation method. Section 4, a cloud-CRITIC-PDR hybrid MCDM method is established. In Section 5 applies the newly proposed method to supplier selection. The practicability and effectiveness of the method are illustrated by comparative analysis and sensitivity analysis. Finally, Section 6 concludes the paper and discusses potential avenues for future work.

2 Cloud model and probabilistic dominance relationship method

2.1 Cloud model theory

The cloud model is an uncertainty transformation model proposed by Professor Li Deyi to address the deficiencies of classical mathematical methods in describing information uncertainty. Usually, the cloud model consists of cloud drops distributed over the quantitative numerical domain and describes the uncertainty by three digital characteristics including Expectation Ex, Entropy En and Hyper entropy He, noted as X (Ex, En, He). Ex represents the gravity center of information and the ideal symbol for a qualitative concept; En and He represents the qualitative concept’s and the Entropy’s respective levels of uncertainty, the higher their values, the more uncertain the concept and the Entropy are. The concepts of cloud model in this study is defined as follows:

Definition 1 Cloud generator

Cloud generator is an algorithmic tool to realize the mutual transformation of qualitative and quantitative concepts. According to the transformation direction of qualitative and quantitative concepts, cloud generators can be divided into forward cloud generators and backward cloud generators. Forward cloud generators are algorithms to realize the transformation of qualitative concepts into quantitative values, and a large number of 2D cloud drops can be generated according to the given cloud digital characteristics; backward cloud generators are algorithms to realize the transformation of quantitative data into qualitative concepts.

Definition 2 Cloud comparison rules

Assume that X₁ (Ex₁, En₁, He₁) and X₂ (Ex₂, En₂, He₂) are cloud models on the quantitative numerical domain U. By calculating S_ab (X₁, X₂), the comparison rules of two clouds are as follows:

if S_ab (X₁, X₂) > 0, then X₁ > X₂.

If S_ab (X₁, X₂) = 0 and En₁ < En₂, then X₁ > X₂.

if S_ab (X₁, X₂) = 0, En₁ = En₂and He₁ < He₂, then X₁ > X₂.

if S_ab (X₁, X₂) = 0, En₁ = En₂ and He₁ = He₂, then X₁ = X₂.

where $S_{ab} (X_{1}, X_{2}) = 2 (\bar{a} - \underline{b}) - (\bar{a} - \underline{a} + \bar{b} - \underline{b})$ , $\underline{a} = {Ex}_{1} - 3 {En}_{1}$ , $\bar{a} = {Ex}_{1} + 3 {En}_{1}$ , $\underline{b} = {Ex}_{2} - 3 {En}_{2}$ , $\bar{b} = {Ex}_{2} + 3 {En}_{2}$ .

Definition 3 Cloud drop contribution degree

The contribution s of a cloud drop (x,y) to a qualitative concept can be calculated by s = xy. Its expectation $\bar{s}$ represents the certainty degree of cloud X to the qualitative concept and is calculated as Equation (1), where N cloud drops {(x₁, y₁) , (x₂, y₂) , …, (x_N, y_N)} are generated by using a forward cloud generator. Suppose there are two normal clouds, x₁ and x₂. If $\bar{s} (X_{1}) ⩾ \bar{s} (X_{2})$ , then X₁ ≻ X₂, i.e., x₁ is better than x₂. $\bar{s} (X) = \frac{1}{N} \sum_{i = 1}^{N} x_{i} y_{i}$ (1)

Definition 4 The cloud aggregation algorithm

Considering the difference of each cloud’s importance, the weighted average cloud aggregation algorithm is used for aggregating normal clouds. Let X₁ (Ex₁, En₁, He₁), X₂ (Ex₂, En₂, He₂),…, X_n (Ex_n, En_n, He_n) be n normal clouds in U, their weighted average aggregation cloud X (Ex, En, He) is given by Equation (2). $\begin{matrix} X = \sum_{i = 1}^{n} w_{i} X_{i} = \\ (\sum_{i = 1}^{n} w_{i} {Ex}_{i}, \sqrt{\sum_{i = 1}^{n} {(w_{i} {En}_{i})}^{2}}, \sqrt{\sum_{i = 1}^{n} {(w_{i} {He}_{i})}^{2}}) \end{matrix}$ (2) where w₁, w₂,…, w_n are the corresponding weights of each cloud.

2.2 Probabilistic dominance relationship method

The probabilistic dominance relationship (PDR) is a method to calculate the comprehensive dominance degree based on the comparison rules of dominance relationships under the ordered information system. The calculation involves the concepts of dominance degree, comparison rules, α-probabilistic dominance class, and dominance matrix. The relevant definitions are as follows.

Definition 1 Ordered information system

Supposing quadratic set Z = (U, C, V, f) is an information system, where U = (x₁, x₂, . . . , x_m) is a finite set of schemes, C = (c₁, c₂, . . . , c_n) is a finite set of indices, V = (v₁, v₂, . . . , v_l) is a finite set of index values, and f = {f_k : U × C → V, k ⩽ m} is the relationship set between U and C. When there is an ordinal relationship between index values, Z can be called as an ordinal information system.

Definition 2 Dominance degree and comparison rules

For schemes x_i, x_j ∈ U and index c ∈ C, supposing p_c (x_i, x_j) is the dominance degree of x_j compared to x_i under c, and the comparison rules are as follows. When x_j is better than x_i, p_c (x_i, x_j) is 1; when x_j is worse than x_i, p_c (x_i, x_j) is 0; when x_j is the same as x_i, p_c (x_i, x_j) is 0.5. Let p_C (x_i, x_j) be the dominance degree of x_j compared to x_i under the all-index set C, it means the average of p_c (x_i, x_j).

Definition 3 α-probabilistic dominance class

Supposing $R_{C}^{⩾ α}$ is the α-probabilistic dominance relation under the all-index set C, i.e., $R_{C}^{⩾ α} = {(x_{i}, x_{j}) | p_{C} (x_{i}, x_{j}) ⩾ α, 0.5 ⩽ α ⩽ 1}$ , where α is given by the decision maker. $[x_{i}]_{C}^{⩾ α}$ is defined as the α-probabilistic dominance class of x_i under C, and it denotes the set of elements that are inferior to the scheme x_i under α-probability conditions.

Definition 4 Dominance matrix and comprehensive dominance degree

The dominance matrix $D_{C}^{⩾ α} x_{i}, x_{j}$ is obtained by the set operation of the α-probabilistic dominance class. On this basis, the comprehensive dominance degree $d_{C}^{⩾ α} x_{i}$ can be obtained by calculating the arithmetic mean of matrix’s row.

3 Cloud-CRITIC-based hybrid index conversion and weight calculation

Traditional CRITIC method is uesd to calculate index weights commonly. However, in the face of hybrid MCDM data complexity and uncertainty, it only considers how the relevance of index affects the weights, which may overestimate or underestimate the information value of part indices, leading to decision errors. Therefore, in this paper, we introduce the relevant parameters of the cloud model into the CRITIC method to improve: firstly design the hybrid index cloud conversion rules to obtain the digital characteristics (Ex, En, He); secondly, we use digital characteristics to optimize the calculation formula of the Comprehensive information; finally, we obtain the cloud-CRITIC weight calculation method.

3.1 Cloud conversion of hybrid indices

Based on the aforementioned definition, noting the ordered information system as Z = (U, C, V, f). The index set C is divided into quantitative index set C_A and qualitative index set C_B. The sets of index values are V_A ={ v_A1, v_A2, . . . , v_AK } and V_B ={ v_B1, v_B2, . . . , v_BS } respectively. Considering the difference of data types in hybrid indices, i.e., precise numbers, interval numbers, linguistic values, this paper adopts three methods to generate cloud.

3.1.1 Cloud conversion of quantitative indices

Quantitative index c_A in practical decision making is usually represented by a set of precise numbers or interval numbers, so the calculation of cloud model needs to be discussed in different situations. For indices expressed in precise numbers, modified backward cloud generator in literature [23] is used to calculate the digital characteristics; for indices expressed in interval numbers, firstly, according to the processing method of literature [34], the index approximation method is used to transform the upper and lower bounds [C_min, C_max] of each interval number into cloud. Then, the cloud is aggregated into a aggregation cloud by Equation (2).

The calculation processes above are shown in Equations (4) respectively, and the corresponding cloud is noted as X_cAiEx_cAi, En_cAi, He_cAi. $\begin{matrix} Ex = \frac{\sum_{k = 1}^{K} v_{Ak}}{K} En = \frac{1}{K} (\sum_{k = 1}^{K} | v_{Ak} - Ex |) \\ He = \frac{1}{K - 1} \sum_{k = 1}^{K} {(v_{Ak} - Ex)}^{2} \end{matrix}$ (3) $Ex = \frac{(C_{min} + C_{max})}{2} En = \frac{(C_{max} - C_{min})}{6} He = k_{0} En$ (4) where, k₀ is a constant that can be specifically adjusted by the decision maker according to the index uncertainty.

3.1.2 Cloud conversion of qualitative indices

Qualitative indices are usually represented by linguistic values in existing studies, the representation includes five-scale rubric sets and seven-scale rubric sets, such as very low, low, medium, high, very high and very poor, poor, slightly poor, fair, good, slightly good, very good . In this paper, qualitative indices are converted into cloud by using the improved Golden Segmentation method according to literature [35].

Firstly, the rubric sets for a single qualitative index c_B are defined as T = {T₁, T₂, …, T_t}, the effective numerical domain of experts is defined as [R_min, R_max], and He₀ is given; secondly, the rubric set T is converted into the standard cloud by Table 1. And r experts use standard clouds to evaluate all attributes under each scheme; finally, the expert weights W_R ={ w₁, w₂, . . . , w_r } given by the decision maker and the corresponding cloud X_cBiEx_cBi, En_cBi, He_cBi is calculated according to Equation (2).

Table 1
Improved golden segmentation method

Cloud model Expectation Ex Entropy En Hyper entropy He

$X_{+ \frac{(t - 1)}{2}} ({Ex}_{+ \frac{(t - 1)}{2}}, {En}_{+ \frac{(t - 1)}{2}}, {He}_{+ \frac{(t - 1)}{2}})$ R _max $\frac{{En}_{+ (n - 3) / 2}}{0.618}$ $\frac{{He}_{+ (n - 3) / 2}}{0.618}$

$X_{+ \frac{(t - 3)}{2}} ({Ex}_{+ \frac{(t - 3)}{2}}, {En}_{+ \frac{(t - 3)}{2}}, {He}_{+ \frac{(t - 3)}{2}})$ ${Ex}_{+ \frac{n - 5}{2}} + 0.382 (R_{max} - {Ex}_{+ \frac{n - 5}{2}})$ $\frac{{En}_{+ (n - 5) / 2}}{0.618}$ $\frac{{He}_{+ (n - 5) / 2}}{0.618}$

…… …… …… ……

X₊₂ (Ex₊₂, En₊₂, He₊₂) Ex₊₁ + 0.382 (R_max - Ex₊₁) $\frac{{En}_{+ 1}}{0.618}$ $\frac{{He}_{+ 1}}{0.618}$

X₊₁ (Ex₊₁, En₊₁, He₊₁) Ex₀ + 0.382 (R_max - Ex₀) $0.382 \frac{(R_{max} - R_{min})}{6}$ $\frac{{He}_{0}}{0.618}$

X₀ (Ex₀, En₀, He₀) $\frac{R_{max} + R_{min}}{2}$ 0.618En₁ He ₀

X_-1 (Ex_-1, En_-1, He_-1) Ex₀ - 0.382 (Ex₀ - R_min) $0.382 \frac{(R_{max} - R_{min})}{6}$ $\frac{{He}_{0}}{0.618}$

X_-2 (Ex_-2, En_-2, He_-2) Ex_-1 - 0.382 (Ex_-1 - R_min) $\frac{{En}_{- 1}}{0.618}$ $\frac{{He}_{- 1}}{0.618}$

…… …… …… ……

$X_{- \frac{(t - 3)}{2}} ({Ex}_{- \frac{(t - 3)}{2}}, {En}_{- \frac{(t - 3)}{2}}, {He}_{- \frac{(t - 3)}{2}})$ ${Ex}_{- \frac{n - 5}{2}} - 0.382 ({Ex}_{- \frac{n - 5}{2}} - R_{min})$ $\frac{{En}_{- (n - 5) / 2}}{0.618}$ $\frac{{He}_{- (n - 5) / 2}}{0.618}$

$X_{- \frac{(t - 1)}{2}} ({Ex}_{- \frac{(t - 1)}{2}}, {En}_{- \frac{(t - 1)}{2}}, {He}_{- \frac{(t - 1)}{2}})$ R _min $\frac{{En}_{- (n - 3) / 2}}{0.618}$ $\frac{{He}_{- (n - 3) / 2}}{0.618}$

Cloud model	Expectation Ex	Entropy En	Hyper entropy He
$X_{+ \frac{(t - 1)}{2}} ({Ex}_{+ \frac{(t - 1)}{2}}, {En}_{+ \frac{(t - 1)}{2}}, {He}_{+ \frac{(t - 1)}{2}})$	R _max	$\frac{{En}_{+ (n - 3) / 2}}{0.618}$	$\frac{{He}_{+ (n - 3) / 2}}{0.618}$
$X_{+ \frac{(t - 3)}{2}} ({Ex}_{+ \frac{(t - 3)}{2}}, {En}_{+ \frac{(t - 3)}{2}}, {He}_{+ \frac{(t - 3)}{2}})$	${Ex}_{+ \frac{n - 5}{2}} + 0.382 (R_{max} - {Ex}_{+ \frac{n - 5}{2}})$	$\frac{{En}_{+ (n - 5) / 2}}{0.618}$	$\frac{{He}_{+ (n - 5) / 2}}{0.618}$
……	……	……	……
X₊₂ (Ex₊₂, En₊₂, He₊₂)	Ex₊₁ + 0.382 (R_max - Ex₊₁)	$\frac{{En}_{+ 1}}{0.618}$	$\frac{{He}_{+ 1}}{0.618}$
X₊₁ (Ex₊₁, En₊₁, He₊₁)	Ex₀ + 0.382 (R_max - Ex₀)	$0.382 \frac{(R_{max} - R_{min})}{6}$	$\frac{{He}_{0}}{0.618}$
X₀ (Ex₀, En₀, He₀)	$\frac{R_{max} + R_{min}}{2}$	0.618En₁	He ₀
X_-1 (Ex_-1, En_-1, He_-1)	Ex₀ - 0.382 (Ex₀ - R_min)	$0.382 \frac{(R_{max} - R_{min})}{6}$	$\frac{{He}_{0}}{0.618}$
X_-2 (Ex_-2, En_-2, He_-2)	Ex_-1 - 0.382 (Ex_-1 - R_min)	$\frac{{En}_{- 1}}{0.618}$	$\frac{{He}_{- 1}}{0.618}$
……	……	……	……
$X_{- \frac{(t - 3)}{2}} ({Ex}_{- \frac{(t - 3)}{2}}, {En}_{- \frac{(t - 3)}{2}}, {He}_{- \frac{(t - 3)}{2}})$	${Ex}_{- \frac{n - 5}{2}} - 0.382 ({Ex}_{- \frac{n - 5}{2}} - R_{min})$	$\frac{{En}_{- (n - 5) / 2}}{0.618}$	$\frac{{He}_{- (n - 5) / 2}}{0.618}$
$X_{- \frac{(t - 1)}{2}} ({Ex}_{- \frac{(t - 1)}{2}}, {En}_{- \frac{(t - 1)}{2}}, {He}_{- \frac{(t - 1)}{2}})$	R _min	$\frac{{En}_{- (n - 3) / 2}}{0.618}$	$\frac{{He}_{- (n - 3) / 2}}{0.618}$

3.2 Calculation of hybrid index weights

Based on the above results, we integrate the digital characteristics of cloud into the Comprehensive information calculation process by improving the calculation equations of the Contrast intensity and the Conflict and adding the Data credibility. Through this processing, the weights’ inaccuracy caused by uncertainty of hybrid information can be solved.

The specific steps are as follows.

(1) Cloud digital characteristic standardization. Based on cloud conversion of hybrid indices, the Expectation Ex_ij of positive and negative indices are standardized according to Equations (6) respectively. And the Entropy En_ij and Hyper entropy He_ij are standardized according to Equations (8). Then the standard cloud digital characteristics $({Ex}_{ij}^{'}, {En}_{ij}^{'}, {He}_{ij}^{'})$ can be obtained. Obviously, the higher the standard Expectation ${Ex}_{ij}^{'}$ , the higher the level of qualitative concept. And the greater the standard Entropy ${En}_{ij}^{'}$ and standard Hyper entropy ${He}_{ij}^{'}$ , the greater the uncertainty. ${Ex}_{ij}^{'} = \frac{{Ex}_{ij} - min_{i} {{Ex}_{ij}}}{max_{i} {{Ex}_{ij}} - min_{i} {{Ex}_{ij}}}$ (5) ${Ex}_{ij}^{'} = \frac{max_{i} {{Ex}_{ij}} - {Ex}_{ij}}{max_{i} {{Ex}_{ij}} - min_{i} {{Ex}_{ij}}}$ (6) ${En}_{ij}^{'} = \frac{{En}_{ij} - min_{i} {{En}_{ij}}}{max_{i} {{En}_{ij}} - min_{i} {{En}_{ij}}}$ (7) ${He}_{ij}^{'} = \frac{{He}_{ij} - min_{i} {{He}_{ij}}}{max_{i} {{He}_{ij}} - min_{i} {{He}_{ij}}}$ (8) where, $max_{i} {{Ex}_{ij}}$ , $min_{i} {{Ex}_{ij}}$ represent the maximum and minimum values of the Ex_ij respectively. En_ij and He_ij are calculated in the same way as above.

(2) Contrast intensity calculation. ${Ex}_{ij}^{'}$ replaces the index value of the original equation used to calculate the standard deviation of the jth index in the evaluation system. The calculation process of Contrast intensity σ_j is shown in Equation (9). $σ_{j} = \sqrt{\frac{1}{m} \sum_{i = 1}^{m} {({Ex}_{ij}^{'} - \bar{{Ex}_{j}^{'}})}^{2}} (j = 1, 2, . . ., n)$ (9)

Where, $\bar{{Ex}_{j}^{'}}$ is the arithmetic mean of ${Ex}_{ij}^{'}$ .

(3) Conflict calculation. More of the same, ${Ex}_{ij}^{'}$ is used to calculate the Conflict μ_j between the jth index and other indices in the evaluation system. The calculation process is shown in Equation (10). $μ_{j} = \sum_{l = 1}^{n} (1 - \frac{cov (j, l)}{σ_{j} σ_{l}}) (j = 1, 2, . . ., n)$ (10) where, cov (j, l) is the covariance of ${Ex}_{ij}^{'}$ between jth and lth index; σ_j is the standard deviation of ${Ex}_{ij}^{'}$ under jth index.

(4) Index weight calculation considering the Data credibility. As shown in Equation (11), we uses standard Entropy and Hyper entropy to calculate the Data credibility η_j, and the smaller η_j is, the more incredible the index is. $η_{j} = \frac{1}{\bar{{En}_{j}^{'}} + \bar{{He}_{j}^{'}}}$ (11)

On this basis, the Comprehensive information I_j and weight W_j are calculated by Equations (13). It can be seen that the higher values of the Contrast intensity, the Conflict and the Data credibility are, the greater the weight is. This is in line with the decision idea of “the extensive information volume and reliable information sources are the guarantee of high index reference value”. $I_{j} = σ_{j} \cdot μ_{j} \cdot η_{j} (j = 1, 2, . . ., n)$ (12) $W_{j} = \frac{I_{j}}{\sum_{j = 1}^{n} I_{j}} (j = 1, 2, . . ., n)$ (13) where, $\bar{{En}_{j}^{'}}$ and $\bar{{He}_{j}^{'}}$ are the arithmetic mean of ${En}_{ij}^{'}$ and ${He}_{ij}^{'}$ respectively.

4 Cloud-CRITIC-PDR hybrid MCDM method

As previously mentioned, PDR is a decision method that can effectively deal with conflict and incommensurability of data. However, the traditional PDR ignore the influence of data uncertainty and index weights on decision making. Therefore, we integrate the cloud-CRITIC method into the PDR and propose the hybrid cloud-CRITIC-PDR method:

On the one hand, based on the cloud conversion of hybrid index, the cloud dominance comparison rules are proposed for the characteristics of clouds; on the other hand, the all-index dominance degree is obtained by the weighted average method, where the index weights are calculated by cloud-CRITIC.

4.1 Cloud dominance comparison rules for hybrid indices

According to the characteristics of the data in the hybrid MCDM, the cloud dominance comparison rules of quantitative and qualitative indices are designed respectively.

(1) Cloud dominance comparison rules for quantitative indices

The cloud comparison rule in definition 2 is used to the quantitative index cloud comparison. Suppose x_i and x_j are any schemes in the U, their corresponding clouds under a single quantitative index c_A are X_{c
_A
i} (Ex_{c
_A
i}, En_{c
_A
i}, He_{c
_A
i}) and X_{c
_A
j} (Ex_{c
_A
j}, En_{c
_A
j}, He_{c
_A
j}). According to the comparison rule. If S_ab (X_{c
_A
i}, X_{c
_A
j}) > 0 then X_{c
_A
i} is better than X_{c
_A
j}; if S_ab (X_{c
_A
i}, X_{c
_A
j}) < 0 then X_{c
_A
i} is inferior to X_{c
_A
j}; if S_ab (X_{c
_A
i}, X_{c
_A
j}) = 0 and En_{c
_A
i} < En_{c
_A
j}, then X_{c
_A
i} is better than X_{c
_A
j}; if S_ab (X_{c
_A
i}, X_{c
_A
j}) = 0, En_{c
_A
i} = En_{c
_A
j} and He_{c
_A
i} = He_{c
_A
j}, then X_{c
_A
i} is equivalent to X_{c
_A
j}. Finally, the dominance degree p_{c
_A} (x_i, x_j) is obtained from Equation (14). $p_{c_{A}} (x_{i}, x_{j}) = {\begin{matrix} 1 \\ 0.5 \\ 0 \end{matrix} \begin{matrix} X_{c_{A} i} ≻ X_{c_{A} j} \\ X_{c_{A} i} \sim X_{c_{A} j} \\ X_{c_{A} i} ≺ X_{c_{A} j} \end{matrix}$ (14)

(2) Cloud dominance comparison rules for qualitative index

Qualitative index values are generated by subjective judgments of decision makers. Ex alone cannot adequately represent the will of decision makers, and the effects of En and He should be considered to avoid information deficiency. Therefore, we use contribution degree in definition 3 to compare each cloud under qualitative indices.

Meanwhile, considering the contribution of qualitative indices to the cloud dominance degree is nonlinear[36], we integrate a continuous priority function, Gaussian criterion, into the dominance comparison rule.

Taking X_{c
_B
i}Ex_{c
_B
i}, En_{c
_B
i}, He_{c
_B
i} and X_{c
_B
j}Ex_{c
_B
j}, En_{c
_B
j}, He_{c
_B
j} under a single qualitative index c_B as an example, the number of cloud drops denoted as N is set by the decision maker, and N cloud drops are obtained through the forward cloud generator. The contribution degree $\bar{s_{cB}} (i)$ and $\bar{s_{cB}} (j)$ of the two clouds are calculated separately according to Equation (1). If $\bar{s_{cB}} (i) > \bar{s_{cB}} (j)$ then X_{c
_B
i} is better than X_{c
_B
j}; if $\bar{s_{cB}} (i) < \bar{s_{cB}} (j)$ then X_{c
_B
i} is worse than X_{c
_B
j}; if $\bar{s_{cB}} (i) = \bar{s_{cB}} (j)$ then X_{c
_B
i} is equal to X_{c
_B
j}. Subsequently, the dominance degree p_{c
_B} (x_i, x_j) is calculated by Equation (15).

$p_{c_{B}} (x_{i}, x_{j}) = {\begin{matrix} 1 - 0.5 \times exp (- \frac{{(\bar{s_{cB}} (i) - \bar{s_{cB}} (j))}^{2}}{2 σ^{2}}) \\ 0.5 \\ 0.5 \times exp (- \frac{{(\bar{s_{cB}} (i) - \bar{s_{cB}} (j))}^{2}}{2 σ^{2}}) \end{matrix} \begin{matrix} X_{c_{B} i} ≻ X_{c_{B} j} \\ \begin{matrix} X_{c_{B} i} \sim X_{c_{B} j} \end{matrix} \\ \begin{matrix} X_{c_{B} i} ≺ X_{c_{B} j} \end{matrix} \end{matrix}$ (15) where σ reflects the intensity of the decision maker’s preference changing with the contribution degree difference and is given by the decision maker.

4.2 All-index dominance degree calculation

The traditional PDR obtains the all-index dominance degree by averaging the dominance under a single index, which simplifies the calculation process but does not take into account the the index-weight influence. We obtain the index weights based on the proposed cloud-CRITIC method and calculate the all-index dominance degree by the weighted average method. The weight set W_c is calculated by Eqs. (5)–(13) and are substituted into Equation (16) to obtain the all-index dominance degree p_C (x_i . x_j). $\begin{matrix} p_{C} (x_{i}, x_{j}) = \sum_{c_{A} \in C_{A}} W_{c_{A}} p_{c_{A}} (x_{i}, x_{j}) \\ + W_{c_{B}} \sum_{c_{B} \in C_{B}} p_{c_{B}} (x_{i}, x_{j}) \end{matrix}$ (16) where W_{c
_A} ∈ W_c, W_{c
_B} ∈ W_c separately denotes the weights of quantitative indices and qualitative indices.

4.3 Comprehensive dominance degree calculation

Based on the all-index dominance degree, we calculate the α-probabilistic dominance class and dominance matrix. Then the comprehensive dominance and ranking results are obtained.

(1) α-probability dominance class calculation.

Given a value α (generally set to 0.5), the α-probabilistic dominance class of each scheme can be calculated by Equation (17), noted as ${[x_{i}]}_{C}^{⩾ α}$ , i = 1, 2,…, n. $\begin{matrix} R_{C}^{⩾ α} = {(x_{i}, x_{j}) | p_{C} (x_{i}, x_{j}) ⩾ α, 0.5 ⩽ α ⩽ 1}, \\ {[x_{i}]}_{C}^{⩾ α} = {x_{j} | (x_{i}, x_{j}) \in R_{C}^{⩾ α}} \end{matrix}$ (17)

(2) The dominance matrix and comprehensive dominance degree calculation.

The dominance matrix $D_{C}^{⩾ α} (x_{i}, x_{j})$ is calculated by Equation (18). Where, $| {[x_{j}]}_{C}^{⩾ α} - {[x_{i}]}_{C}^{⩾ α} |$ denotes the element number of the difference set of ${[x_{j}]}_{C}^{⩾ α}$ and ${[x_{i}]}_{C}^{⩾ α}$ . The comprehensive dominance degree $d_{C}^{⩾ α} (x_{i})$ is calculated according to Equation (19) and then the schemes can be compared. $\begin{matrix} D_{C}^{⩾ α} (x_{i}, x_{j}) = \\ {\begin{matrix} \frac{| {[x_{i}]}_{C}^{⩾ α} - {[x_{j}]}_{C}^{⩾ α} |}{| {[x_{i}]}_{C}^{⩾ α} - {[x_{j}]}_{C}^{⩾ α} | + | {[x_{j}]}_{C}^{⩾ α} - {[x_{i}]}_{C}^{⩾ α} |} \\ 0.5, else \end{matrix}, \\ {[x_{j}]}_{C}^{⩾ α} \neq {[x_{i}]}_{C}^{⩾ α} \end{matrix}$ (18) $d_{C}^{⩾ α} (x_{i}) = \frac{\sum_{j = 1}^{m} D_{C}^{⩾ α} (x_{i}, x_{j})}{m}$ (19)

5 Example analysis

5.1 Example background and data description

A manufacturing company wants to seek more competitive suppliers as part of its supply chain in the market. Considering the large number of suppliers with adequate supply capability, the company has identified four potential suppliers (x₁, x₂, x₃, x₄) through pre-qualification, short-term visits, in-depth research and other links [37]. Subsequently, the company evaluated the potential suppliers in six aspects, including product market share (%) c₁, fixed assets (ten thousand yuan) c₂, solid waste recycling rate (%) c₃, Delivery lead time (days) c₄, product troubleshooting satisfaction c₅, and performance c₆, of which c₁, c₂, c₃, c₄ are quantitative indices, c₅, c₆ are qualitative indices. All of these six indices are positive except c₄. As for the data sources shown in Table 2, the quantitative indices are obtained by sampling, while the qualitative indices are evaluated using the seven-scaled linguistic evaluation set T = {VP, P, SP, F, SG, G, VG} = {Very poor, poor, slight poor, fair, good, slight good, verygood}. The index values are determined by the decision-making group consisting of 6 experts.

Table 2
Raw data of supplier evaluation

Suppliers Supplier evaluation indices

c₁ c₂ c₃ c₄ c₅ c₆

x ₁ 2.00 503.00 [54,55] [4.2,5.6] F VG

2.30 502.00 [54,56] [4.0,5.5] SP VG

1.90 499.00 [53,55] [4.1,5.5] F VG

2.00 501.00 [54,55] [4.0,5.4] F G

2.20 501.00 [52,55] [4.0,5.5] SP VG

2.20 500.00 [54,56] [4.1,5.4] SG G

x ₂ 2.40 541.00 [29,40] [4.2,5.3] P F

2.50 538.00 [30,36] [4.1,5.3] SP F

2.50 540.00 [28,41] [4.2,5.1] P F

2.60 540.00 [30,40] [4.2,5.2] P SG

2.40 539.00 [33,42] [4.3,5.0] P F

2.50 539.00 [31,39] [4.1,5.2] VP F

X₃ 1.80 481.00 [51,62] [5.2,6.0] SG VG

2.00 483.00 [47,56] [5.2,6.0] G F

1.90 485.00 [50,60] [5.0,5.9] G G

1.80 475.00 [51,60] [4.8,5.8] G VG

1.90 477.00 [47,57] [4.9,6.1] SG G

1.80 479.00 [48,60] [4.8,6.2] G G

X₄ 2.30 525.00 [33,45] [4.4,5.5] F SP

2.40 519.00 [35,43] [4.8,5.6] SP F

2.00 520.00 [33,47] [4.5,5.3] F SG

2.00 522.00 [35,45] [4.3,5.6] F F

2.20 523.00 [36,44] [4.5,5.5] F F

2.10 516.00 [36,46] [4.6,5.5] F SP

Suppliers	Supplier evaluation indices
x ₁	2.00	503.00	[54,55]	[4.2,5.6]	F	VG
	2.30	502.00	[54,56]	[4.0,5.5]	SP	VG
	1.90	499.00	[53,55]	[4.1,5.5]	F	VG
	2.00	501.00	[54,55]	[4.0,5.4]	F	G
	2.20	501.00	[52,55]	[4.0,5.5]	SP	VG
	2.20	500.00	[54,56]	[4.1,5.4]	SG	G
x ₂	2.40	541.00	[29,40]	[4.2,5.3]	P	F
	2.50	538.00	[30,36]	[4.1,5.3]	SP	F
	2.50	540.00	[28,41]	[4.2,5.1]	P	F
	2.60	540.00	[30,40]	[4.2,5.2]	P	SG
	2.40	539.00	[33,42]	[4.3,5.0]	P	F
	2.50	539.00	[31,39]	[4.1,5.2]	VP	F
X₃	1.80	481.00	[51,62]	[5.2,6.0]	SG	VG
	2.00	483.00	[47,56]	[5.2,6.0]	G	F
	1.90	485.00	[50,60]	[5.0,5.9]	G	G
	1.80	475.00	[51,60]	[4.8,5.8]	G	VG
	1.90	477.00	[47,57]	[4.9,6.1]	SG	G
	1.80	479.00	[48,60]	[4.8,6.2]	G	G
X₄	2.30	525.00	[33,45]	[4.4,5.5]	F	SP
	2.40	519.00	[35,43]	[4.8,5.6]	SP	F
	2.00	520.00	[33,47]	[4.5,5.3]	F	SG
	2.00	522.00	[35,45]	[4.3,5.6]	F	F
	2.20	523.00	[36,44]	[4.5,5.5]	F	F
	2.10	516.00	[36,46]	[4.6,5.5]	F	SP

5.2 Data calculation and evaluation results

The supplier evaluation is performed with the proposed cloud-CRITIC-PDR method. The specific data calculation steps and results are as follows.

(1) Cloud conversion of hybrid indices

Supposing k₀ = 0.1, the data of suppliers under quantitative indices c₁, c₂, c₃, and c₄ are converted into cloud models according to Equations (4). Furthermore, assuming the experts’ quantitative numerical domain as [0, 100], He₀ as 0.1, and the expert weights as 1/6, the data of supplier under c₅ and c₆ is converted into cloud models according to Table 1 and Equation (2). Then, the cloud model of suppliers under each index is shown in Table 3.

Table 3
Cloud model for supplier evaluation

Indices Suppliers

x ₁ x ₂ x₃ x₄

c₁ X₁₁(2.10,0.13,0.02) X₁₂(2.48,0.06,0.01) X₁₃(1.87,0.07,0.01) X₁₄(2.17,0.13,0.03)

c₂ X₂₁(501.00,1.00,2.00) X₂₂(539.50,0.83,1.10) X₂₃(480.00,3.00,14.00) X₂₄(520.83,2.50,10.17)

c₃ X₃₁(54.42,0.13,0.01) X₃₂(34.92,0.66,0.07) X₃₃(54.08,0.70,0.07) X₃₄(39.83,0.72,0.07)

c₄ X₄₁(4.78,0.10,0.01) X₄₂(4.68,0.07,0.01) X₄₃(5.49,0.07,0.01) X₄₄(5.01,0.07,0.01)

c₅ X₅₁(46.82,2.16,0.06) X₅₂(17.88,4.55,0.12) X₅₃(76.97,3.75,0.10) X₅₄(46.82,1.81,0.05)

c₆ X₆₁(93.63,6.07,0.15) X₆₂(53.18,1.81,0.05) X₆₃(82.12,4.98,0.13) X₆₄(46.82,2.16,0.05)

Indices	Suppliers
c₁	X₁₁(2.10,0.13,0.02)	X₁₂(2.48,0.06,0.01)	X₁₃(1.87,0.07,0.01)	X₁₄(2.17,0.13,0.03)
c₂	X₂₁(501.00,1.00,2.00)	X₂₂(539.50,0.83,1.10)	X₂₃(480.00,3.00,14.00)	X₂₄(520.83,2.50,10.17)
c₃	X₃₁(54.42,0.13,0.01)	X₃₂(34.92,0.66,0.07)	X₃₃(54.08,0.70,0.07)	X₃₄(39.83,0.72,0.07)
c₄	X₄₁(4.78,0.10,0.01)	X₄₂(4.68,0.07,0.01)	X₄₃(5.49,0.07,0.01)	X₄₄(5.01,0.07,0.01)
c₅	X₅₁(46.82,2.16,0.06)	X₅₂(17.88,4.55,0.12)	X₅₃(76.97,3.75,0.10)	X₅₄(46.82,1.81,0.05)
c₆	X₆₁(93.63,6.07,0.15)	X₆₂(53.18,1.81,0.05)	X₆₃(82.12,4.98,0.13)	X₆₄(46.82,2.16,0.05)

Taking calculation process of the supplier x₁ under index c₁, c₃ and c₅ as an example:

① The modified backward cloud generator is used to calculate the digital characteristics of the supplier x₁ under the index c₁. ${Ex}_{11} = \frac{(2 + 2.3 + 1.9 + 2 + 2.2 + 2.2)}{6} = 2.100$

${En}_{11} = \frac{(| 2 - 2.1 | + | 2.3 - 2.1 | + | 1.9 - 2.1 | + | 2 - 2.1 | + | 2.2 - 2.1 | + | 2.2 - 2.1 |)}{6} \approx 0.133$

$\begin{matrix} {He}_{11} = \frac{({(2 - 2.1)}^{2} + {(2.3 - 2.1)}^{2} + {(1.9 - 2.1)}^{2} + {(2 - 2.1)}^{2} + {(2.2 - 2.1)}^{2} + {(2.2 - 2.1)}^{2})}{6 - 1} \\ \begin{matrix} \end{matrix} = 0.024 \end{matrix}$

① The index approximation method is used to calculate the numerical characteristics of the supplier x₁ under the index c₃. The interval number [54, 55] is converted to ( ${Ex}_{131} = \frac{(54 + 55)}{2} = 54.5$ , ${En}_{131} = \frac{(55 - 54)}{6} \approx 0.167$ , He₁₃₁ = 0.1 × 0.167 ≈ 0.017) by cloud transformation. Similarly, the cloud model of other interval numbers can be obtained. Through the comprehensive cloud calculation formula, ${Ex}_{13} = \frac{(54.5 + 55 + 54 + 54.5 + 53.5 + 55)}{6} \approx 54.417$ ${En}_{13} = \frac{\sqrt{0 . 167^{2} + 0 . 333^{2} + 0 . 333^{2} + 0 . 167^{2} + 0 . 500^{2} + 0 . 333^{2}}}{6} \approx 0.133$ ${He}_{13} = \frac{\sqrt{0 . 017^{2} + 0 . 033^{2} + 0 . 033^{2} + 0 . 017^{2} + 0 . 050^{2} + 0 . 033^{2}}}{6} \approx 0.013$

③ The improved Golden Segmentation method is used to calculate the digital characteristics of supplier x₁ under the index c₅. The evaluation clouds corresponding to the rubric sets T = {VP, P, SP, F, SG, G, VG} are X_-3 =(0.000, 16.700, 0.424), X_-2 = (19.100, 10.310, 0.262), X_-1 = (30.900, 6.370, 0.162), X₀ =(50.000, 3.930, 0.100), X₊₁ = (69.100, 6.370, 0.162), X₊₂ = (80.900, 10.310, 0.262) and

X₊₃ = (100.000, 16.700, 0.424), respectively. The evaluation results of 6 experts are transformed and aggregated by cloud theory, and the process is as follows. ${Ex}_{15} = \frac{(50 + 30.9 + 50 + 50 + 30.9 + 69.1)}{6} \approx 46.817$ ${En}_{15} = \frac{\sqrt{3 . 93^{2} + 6 . 37^{2} + 3 . 93^{2} + 3 . 93^{2} + 6 . 37^{2} + 6 . 37^{2}}}{6} \approx 2.161$ ${He}_{15} = \frac{\sqrt{0 . 1^{2} + 0 . 162^{2} + 0 . 1^{2} + 0 . 1^{2} + 0 . 162^{2} + 0 . 162^{2}}}{6} \approx 0.055$

(2) Calculation of hybrid index weight

According to the Eqs. (5)–(13), the index weights are calculated by the cloud-CRITIC method. Specifically, Table 4 shows the calculation results of the Contrast intensity, Conflict, Data credibility, Comprehensive information and weights of each index.

Table 4

Cloud-CRITIC index weights calculation results

	Supplier evaluation indices
	c₁	c₂	c₃	c₄	c₅	c₆
Contrast intensity σ_j	0.36	0.37	0.44	0.39	0.35	0.42
Conflict μ_j	5.53	5.84	5.45	4.70	6.73	4.88
Data credibility η_j	0.98	1.11	0.70	1.77	1.09	1.10
Comprehensive	1.94	2.41	1.68	3.22	2.59	2.23
information I_j
Weights W_c	0.14	0.17	0.12	0.23	0.18	0.16

The weight calculation process of index c₁ is taken as an example to illustrate,

① The calculation of the Contrast intensity σ₁, the Conflict μ₁ and the Data credibility η₁. $σ_{1} = \sqrt{\frac{1}{4} \times ({(0.378 - 0.466)}^{2} + {(1 - 0.466)}^{2} + {(0 - 0.466)}^{2} + {(0.486 - 0.466)}^{2})} \approx 0.357$ $\begin{matrix} μ_{1} = (1 - \frac{0.170}{0.357 \times 0.357}) + (1 - \frac{0.172}{0.357 \times 0.373}) + (1 - \frac{- 0.180}{0.357 \times 0.442}) \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \begin{matrix} \end{matrix} + (1 - \frac{0.156}{0.357 \times 0.387}) + (1 - \frac{- 0.166}{0.357 \times 0.354}) + (1 - \frac{- 0.122}{0.357 \times 0.417}) = 5.528 \end{matrix}$ $η_{1} = \frac{1}{0.536 + 0.480} \approx 0.984$

② The calculation of the Comprehensive information I1 and the weight W1. Firstly, based on the calculation results of the Contrast intensity, the Conflict and the Data credibility, the Comprehensive information is calculated as follows. $I_{1} = 0.357 \times 5.528 \times 0.984 \approx 1.944$

Similarly, the the Comprehensive information I₂–I₆ is calculated, and accordingly the index weight W₁ is calculated as follows. $W_{1} = \frac{1.944}{1.944 + 2.408 + 1.678 + 3.223 + 2.590 + 2.227} \approx 0.138$

(3) Calculation of cloud dominance comparison rules and all-index dominance degree

Firstly, supposing σ = 1. The dominance degree p_{c
_A} (x_i, x_j) and p_{c
_B} (x_i, x_j) of each supplier comparison pair are obtained according to Equations (15). Subsequently, based on the index weights W_c, the all-index dominance degree p_C (x_i, x_j) of each supplier comparison pair is calculated according to Equation (16), and the results are shown in Table 5.

Table 5

Results of all-index dominance degree for each supplier comparison pairs

Pairs for	All-index	Pairs for	All-index
comparison	dominance	comparison	dominance
	degree		degree
	p_C (x_i, x_j)		p_C (x_i, x_j)
(x₁,x₁)	0.50	(x₃,x₁)	0.18
(x₁,x₂)	0.46	(x₃,x₂)	0.46
(x₁,x₃)	0.82	(x₃,x₃)	0.50
(x₁,x₄)	0.61	(x₃,x₄)	0.46
(x₂,x₁)	0.54	(x₄,x₁)	0.39
(x₂,x₂)	0.50	(x₄,x₂)	0.30
(x₂,x₃)	0.54	(x₄,x₃)	0.54
(x₂,x₄)	0.70	(x₄,x₄)	0.50

Take the supplier x₁ and x₂ for example, according to the cloud dominance comparison rules, the two suppliers satisfy S_ab (X₁₁, X₁₂) = -0.76 < 0 under c₁. Obviously, the cloud X₁₁ is inferior to X₁₂, the dominance degree p₁ (x₁, x₂) =0. Similarly, the dominance degrees under the indices c₂, c₃, c₄ are 0, 1, 0.

For the qualitative index c₅, let the cloud drop number N be 1000, the contribution degree of the two suppliers satisfies $\bar{s_{5}} (1) = 33.20 > \bar{s_{5}} (2) = 12.98$ under c₅. The cloud X₅₁ is clearly better than X₅₂, the dominance degree $p_{5} (x_{1}, x_{2}) = 1 - 0.5 \times exp (- \frac{{(33.20 - 12.98)}^{2}}{2 \times 1^{2}}) = 0.99$ . In the same way, the dominance degree under index c₆ is 0.99.

In summary, the all-index dominance degree is p_C (x₁, x₂) =0.14 × 0 +0.17 × 0 +0.12 × 1 +0.23 × 0 +0.18 × 0.99 + 0.16 × 0.99 = 0.46.

(4) Comprehensive dominance degree calculation

By calculating the α-probability dominance class and dominance matrix in turn according to Eqs. (17)–(19), suppliers’ comprehensive advantage degree can be obtained, see Table 6. It can be seen that the supplier are sorted as x₂ ≻ x₁ ≻ x₄ ≻ x₃ [20].

Table 6

Overall supplier dominance

Suppliers	x ₁	x ₂	x₃	x₄
Comprehensive advantage degree $d_{C}^{⩾ α} (x_{i})$	0.625	0.875	0.125	0.375

5.3 Method validity test

(1) The validity test of cloud-CRITIC weight calculation method

To illustrate the method validity, we compare the suppliers’ ranking results obtained by different weight calculation methods, see Table 7.

Table 7
Comparison of the results of different weight calculation methods

Weight calculation method Index weights Supplier ranking results

c ₁ c ₂ c ₃ c ₄ c ₅ c ₆

Cloud-CRITIC 0.14 0.17 0.12 0.23 0.18 0.16 x₂ ≻ x₁ ≻ x₄ ≻ x₃

CRITIC-M 0.14 0.19 0.21 0.20 0.13 0.13 x₂ ≻ x₁ ≻ x₄ ≻ x₃

CRITIC 0.15 0.17 0.19 0.14 0.19 0.16 x₁ ≻ x₃ ≻ x₂ ≻ x₄

Entropy-weighted method 0.15 0.16 0.20 0.15 0.14 0.20 x₁ ≻ x₃ ≻ x₂ ≻ x₄

Weight calculation method	Index weights	Supplier ranking results
Cloud-CRITIC	0.14	0.17	0.12	0.23	0.18	0.16	x₂ ≻ x₁ ≻ x₄ ≻ x₃
CRITIC-M	0.14	0.19	0.21	0.20	0.13	0.13	x₂ ≻ x₁ ≻ x₄ ≻ x₃
CRITIC	0.15	0.17	0.19	0.14	0.19	0.16	x₁ ≻ x₃ ≻ x₂ ≻ x₄
Entropy-weighted method	0.15	0.16	0.20	0.15	0.14	0.20	x₁ ≻ x₃ ≻ x₂ ≻ x₄

The results show that The ranking results of cloud-CRITIC are different from those of traditional methods (entropy weight method and CRITIC method), but are consistent with those of CRITIC-M method [31]. Respectively, ① The difference between cloud-CRITIC and traditional method ranking comes from the indices c₃, c₄, c₅, and c₆. Comparing the entropy weight method and the CRITIC method, it can be seen that the weights of indices c₅ and c₆ are different but do not affect the ranking results. Therefore, it is necessary to analyze the sources of differences from indices c₃ and c₄: traditional methods often only consider the Contrast intensity and the Conflict, while cloud-CRITIC increases the consideration of the Data credibility on this basis. It can be seen from Table 4 that the Contrast intensity and the Conflict of index c₃ are higher, but the reliability is much lower than that of c₄, which results in the difference of sorting results before and after improvement. ②Both cloud-CRITIC and CRITIC-M are improved from classical methods, and their rankings are consistent. CRITIC-M improves the normalization and weight calculation formulas. Like the cloud-CRITIC method, it improves the accuracy of the results by reflecting the original data information more comprehensively. However, the advantage of the cloud-CRITIC method is that it not only pays attention to the richness of index information, but also pays attention to the credibility of data sources, and the decision results are more in line with the actual situation.

(2) The validity test of cloud-CRITIC-PDR method

In order to verify the validity of the method, HB-SIR hybrid MCDM method based on fuzzy sets is used for supplier evaluation [36], and the results are compared with cloud-CRITIC-PDR method. Firstly, on the basis of Table 2, the trapezoidal fuzzy number and triangular fuzzy number are used to represent the qualitative indices c₅ and c₆ respectively. The supplier comprehensive index value are replaced by the average value and are shown in Table 8. Subsequently, by calculating the distance between the positive and negative ideal solutions of each supplier, the corresponding dominant flow, inferior flow and net flow are obtained. The suppliers are ranked based on the net flow values, and the results are shown in Table 9.

Table 8

Supplier comprehensive index values

Suppliers	Supplier evaluation indices
	c₁	c₂	c₃	c₄	c₅	c₆
x ₁	2.10	501.00	[53.50,55.33]	[4.07,5.48]	(0.36,0.44,0.51,0.59)	(0.78,0.94,1.00)
x ₂	2.48	539.50	[30.17,39.67]	[4.18,5.18]	(0.09,0.15,0.23,0.31)	(0.36,0.53,0.70)
x₃	1.87	480.00	[49.00,59.17]	[4.98,6.00]	(0.64,0.72,0.81,0.87)	(0.67,0.83,0.95)
x₄	2.17	520.83	[30.17,39.67]	[4.52,5.50]	(0.36,0.44,0.51,0.59)	(0.31,0.47,0.64)

Table 9

Evaluation results of HB-SIR method

Suppliers	Dominant flow γ_> (x_i)	Inferior flow γ_< (x_i)	Net flow γ_n (x_i)	Ranking results
x ₁	0.23	0.08	0.15	1
x ₂	0.21	0.17	0.04	2
x₃	0.16	0.22	–0.06	4
x₄	0.11	0.16	–0.04	3

It can be seen that there is a difference between the HB-SIR method and the cloud-CRITIC-PDR method in the ranking results of the x₁ and x₂, which may be caused by the processing of the qualitative indices. Furthermore, by comparing cloud model, there are always three indices under Expectation where one supplier is superior than another, but in terms of Entropy and Hyper entropy values, the indices of supplier x₁ are generally higher than those of the supplier x₂. The result indicates that the subjective cognitive bias and objective measurement error of the supplier x₁ are larger, and the fact that the supplier x₂ is superior can be confirmed. This result also echoes the view in previous studies [7] that “cloud model is more adequate for uncertainty consideration than fuzzy number”.

5.4 Method sensitivity analysis

Using the original evaluation information to weight each index objectively, when the weight changes, will the supplier priority order change? In response to this problem, this paper uses sensitivity analysis to explore the potential impact of index weight changes on decision-making results, as follows.

The perturbation method is used to analyze the sensitivity of the index weight, that is, to observe the corresponding changes of the potential suppliers’ priority order after the index weight is slightly disturbed. The initial weight of c_j is W_j. After disturbance, W_j is converted to , satisfying , and the change interval of parameter λ is 0 ⩽ λ ⩽ 1/W_j, where . Because of the weigh normalization, the remaining weights are changed due to the , which is recorded as and satisfies . Thus, φ = (1 - λW_j)/(1 - W_j).

Based on the above principles, this paper perturbs the weights of the six indices, and the parameters λ are 3, 2, 1/2, 1/3 in turn. A total of 24 experiments are carried out and the experimental results are shown in Table 10 and Fig. 2.

Table 10
Sensitivity analysis results

Experimental Perturbed λ Supplier ranking Optimal

times weight W_j supplier

1 W ₁ 3 x₂ ≻ x₄ ≻ x₁ ≻ x₃ x ₂

2 2 x₂ ≻ x₄ ≻ x₁ ≻ x₃ x ₂

3 1/2 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

4 1/3 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

5 W₂ 3 x₂ ≻ x₄ ≻ x₁ ≻ x₃ x ₂

6 2 x₂ ≻ x₄ ≻ x₁ ≻ x₃ x ₂

7 1/2 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

8 1/3 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

9 W₃ 3 x₁ ≻ x₃ ≻ x₄ ≻ x₂ x ₁

10 2 x₁ ≻ x₃ ≻ x₄ ≻ x₂ x ₁

11 1/2 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

12 1/3 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

13 W₄ 3 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

14 2 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

15 1/2 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

16 1/3 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

17 W₅ 3 x₃ ≻ x₁ ≻ x₄ ≻ x₂ x₃

18 2 x₁ ≻ x₃ ≻ x₄ ≻ x₂ x ₁

19 1/2 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

20 1/3 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

21 W₆ 3 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

22 2 x₁ ≻ x₃ ≻ x₂ ≻ x₄ x ₁

23 1/2 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

24 1/3 x₂ ≻ x₁ ≻ x₄ ≻ x₃ x ₂

Experimental	Perturbed	λ	Supplier ranking	Optimal
1	W ₁	3	x₂ ≻ x₄ ≻ x₁ ≻ x₃	x ₂
2		2	x₂ ≻ x₄ ≻ x₁ ≻ x₃	x ₂
3		1/2	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
4		1/3	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
5	W₂	3	x₂ ≻ x₄ ≻ x₁ ≻ x₃	x ₂
6		2	x₂ ≻ x₄ ≻ x₁ ≻ x₃	x ₂
7		1/2	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
8		1/3	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
9	W₃	3	x₁ ≻ x₃ ≻ x₄ ≻ x₂	x ₁
10		2	x₁ ≻ x₃ ≻ x₄ ≻ x₂	x ₁
11		1/2	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
12		1/3	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
13	W₄	3	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
14		2	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
15		1/2	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
16		1/3	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
17	W₅	3	x₃ ≻ x₁ ≻ x₄ ≻ x₂	x₃
18		2	x₁ ≻ x₃ ≻ x₄ ≻ x₂	x ₁
19		1/2	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
20		1/3	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
21	W₆	3	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
22		2	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
23		1/2	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂
24		1/3	x₂ ≻ x₁ ≻ x₄ ≻ x₃	x ₂

Fig. 2

Sensitivity analysis radar chart.

It can be seen that with the change of parameter λ, the supplier ranking and the optimal supplier also changed. Suppliers x₁ and x₂ are the optimal choice for decision makers in most cases, and can maintain stability within a certain range; supplier x₃ ranks first only once, and is sensitive to evaluation values of c₅. Supplier x₄ is always ranked lower, and is sensitive to evaluation values of c₁ and c₂. In summary, the weight is a very important parameter in the proposed cloud-CRITIC-PDR method. To a certain extent, the change of weight will lead to the change of the comprehensive information evaluation value, and the selection of the optimal supplier will also change accordingly.

6 Conclusion

Aiming at the problems of insufficient uncertainty expression, inaccurate index weight calculation and incommensurability of indices in existing hybrid MCDM, this paper designs a cloud-CRITIC weight calculation method based on cloud model, CRITIC and PDR related theoretical knowledge, and further proposes a cloud-CRITIC-PDR hybrid MCDM method. In the decision-making process, the fuzziness and randomness of hybrid data are fully considered, and the credibility parameters are added to the index weight calculation, which improves the accuracy of decision-making. Then, taking supplier evaluation as an example for empirical analysis, by comparing with entropy weight method, CRITIC method, CRITIC-M weight calculation method and HB-SIR hybrid MCDM method, the scientificity and feasibility of cloud-CRITIC and cloud-CRITIC-PDR methods are verified. Finally, the sensitivity analysis shows that the cloud-CRITIC-PDR decision result is relatively stable, but it will be affected by the index weight.

This paper proposes a cloud conversion and cloud dominance comparison method for different types of data, which is suitable for dealing with decision-making problems with complex index types and strong data uncertainty, such as supplier evaluation and risk assessment. Its advantages are mainly reflected in two aspects: in theory, compared with fuzzy sets, this method is more accurate in depicting uncertainty, and can avoid the problems of data loss and distortion caused by traditional MCDM methods, and the decision-making accuracy has been greatly improved; in terms of practical value, this method is applicable to both qualitative and quantitative index in decision-making. It can effectively solve the problem that the traditional evaluation method is only applicable to a single data type, meet the decision-making needs of the vast majority of practical problems, and provide decision-makers with more comprehensive decision-making references.

There are some limitations to the study as well. First, the evaluation results of the cloud-CRITIC-PDR method are easily affected by the change of weight, and the requirements for the accuracy of index weights are relatively high. Second, the objective weighting method ignores the experience and subjective intention of the decision maker, which may make the decision result contrary to the reality.

In the future, we will try to select a representative subjective and objective weighting method, and use a combining method to calculate the index weight to reduce the impact of calculation errors on the decision results. At the same time, the combination of cloud-CRITIC-PDR and more data types will be explored to expand its application scope.

Declaration of competing interest

The authors declare that they have no conflict of ispesothispaper has not been submitted elsewhere in identical or similarform, and they agree to the Author Copyright Agreement, the IOS Press Ethics Policy, and the IOS Press Privacy Policy.

CRediT authorship contribution statement

Xu Zhang: Methodology, Writing –originaldraft. Mingrui Lv: Conceptualization, Formal analysis, Writing –review & editing, Supervision. Xumei Yuan: Software, Visualization, Writing –review & editing.

Footnotes

Acknowledgments

The authors would like to thank the editors and anonymous reviewers for their helpful comments and suggestions. Support for this project was provided by National Social Science Fund Youth Program of China (21CJY051).

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