Cloud model-based PROMETHEE method under 2D uncertain linguistic environment

Abstract

This paper proposes a cloud model-based Preference Ranking Organisation Method for Enrichment Evaluation (PROMETHEE) method with 2D uncertain linguistic variables (2DULVs). 2DULVs are adopted by decision makers (DMs) to evaluate each alternative under the criteria because they can provide extra evaluation information. Cloud model is adopted to depict randomness and fuzziness. The possibility degree and possibility degree index are defined to develop an improved PROMETHEE II method for sorting alternatives. Entropy weight method is used to calculate the weight of each criterion. A renewable energy performance sample is used to illustrate the applicability of the proposed method. Sensitivity analysis and four comparative experiments demonstrate the stability and accuracy of the proposed approach.

Keywords

2D uncertain linguistic variable cloud possibility degree possibility degree index improved PROMETHEE

1 Introduction

Conventional multicriteria decision-making (MCDM) methods represented by crisp number cannot deal with uncertainty and fuzziness because of the complexity of actual MCDM problems and the difficulty of obtaining complete information [1, 2]. Zadeh [3] firstly introduced the concept of fuzzy sets (FSs) to deal with fuzzy information. Various extension methods, such as interval valued intuitionistic FSs [4 –6], hesitant FSs (HFSs) [7], picture FSs [8, 9] and Z-numbers [10, 11], have been proposed on the basis of the high efficiency of FSs in handing fuzzy data. Group decision making is also widely investigated in MCDM [5 , 13]. Linguistic terms are intuitive for decision makers (DMs) to express the assessments because of the existence of vagueness in their judgments. For example, human beings are inclined to provide information in very fast, fast and slow manners whilst evaluating the performance of a car. Therefore, linguistic variables were introduced by Zadeh [14] to enhance the flexibility of decision process. Subsequently, many studies have investigated linguistic variables and their extended forms, including hesitant fuzzy linguistic term sets [15 –20], multigranular linguistic information [15, 16], unbalanced linguistic term sets [16], probabilistic linguistic term sets [18, 21], 2-tuple linguistic model [22 –25] and 2D linguistic variables [26 –28].

Liu et al. [29] proposed 2D uncertain linguistic variables (2DULVs) which are composed of two classes of linguistic terms. The first terms include I class linguistic variables describing the assessment results of the alternatives, and the second terms include II class linguistic variables showing the assessment of self-confidence. Compared with 2D linguistic information, 2DULVs allow experts to express their evaluation using more than one term. Evidently, 2DULVs can better describe uncertainty and are more reliable [26]. With the deepening of research on 2DULVs, several operators, including 2D uncertain linguistic power weighted aggregation (2DULPWA) and 2D uncertain linguistic power generalised weighted aggregation (2DULPGWA) [30], 2D uncertain linguistic weighted averaging (2DULWA), 2D uncertain linguistic weighted geometric (2DULWG) and 2D uncertain linguistic generalised weighted averaging (2DULGWA) [31], 2D uncertain linguistic generalised weighted average (2DULGWA), 2D uncertain linguistic generalised ordered weighted average operator (2DULGOWA) and 2D uncertain linguistic generalised hybrid weighted average (2DULGHWA) [32], 2D uncertain linguistic density geometric aggregation and 2D uncertain linguistic density generalised aggregation [33], 2D uncertain linguistic weighted Bonferroni mean (2DULWBM) and 2D uncertain linguistic improved weighted Bonferroni harmonic mean (2DULIWBHM) operators [34], have been introduced. Liu [29] acquired comprehensive weights of each DM on the basis of II class linguistic information. Most studies on 2DULVs have focused on operators. However, no study has investigated outranking methods. Attributes should be independent from each other whilst using aggregation operators, which cannot be ensure during complicated MCDM progress. Furthermore, a few studies have linked 2DULVs with other MCDM methods, such as decision making trial and evaluation laboratory method (DEMATEL) [35], TODIM (an acronym in Portuguese of interactive and multicriteria decision making) [36] and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [37]. Few studies have integrated 2DULVs with Preference Ranking Organisation Method for Enrichment Evaluation (PROMETHEE) methods.

Although 2DULVs can provide a comprehensive description of the performance of DMs on alternatives, they cannot reflect the randomness of the decision process. Uncertainty involves ambiguity and randomness, and they tend to appear simultaneously because of the complexity of objective objects and the limitations of individuals’ subjective cognition. Cloud model, which was proposed by Li et al. [38], is an effective tool to handle randomness. Conversion between quantitative and qualitative languages is achieved using cloud models described on the basis of three mathematical characteristics. With the maturity of cloud theory, an increasing number of studies on cloud models, such as integrated clouds [39, 40], interval integrated cloud [41, 42], linguistic intuitionistic normal clouds [43], and practical application of cloud models, have been conducted. These studies combine cloud model with linguistic variables, which agree with human expressing habits and effectively describe randomness and ambiguity. Inspired by them, it is reasonable and feasible to combine cloud model with 2DULVs. And some of them ignore the reliability of the evaluation results. Several studies have been conducted on cloud operators. Liu et al. [44] introduced cloud weighted averaging distance, cloud weighted geometric averaging distance operator and cloud generalised weighted averaging distance (CGWAD) operators, which are based on distance. Liu et al. [45] presented the cloud Maclaurin symmetric mean (MSM) and cloud weighted MSM operators. Aggregation operators are commonly used in MCDM methods, whereas cloud aggregation operators are generally complicated.

Many MCDM models and methods for 2DULVs have been successfully applied to various fields in real life [35, 42], but most of them cannot handle conditions where incomplete compensation exists between the criteria. Incomplete compensation is ubiquitous in real life. PROMETHEE can effectively overcome these shortcomings. PROMETHEE is a ranking method used to select the best alternative among various criteria that are frequently conflicting and can evaluate alternatives under quantitative and qualitative criteria. Studies [20 , 46–48] on PROMETHEE have provided many applications. Many studies have combined PROMETHEE with other forms of variables inclusive of cloud model [49], Pythagorean FSs [50], 2D linguistic variables [51], Z-numbers [10] and picture FSs [8]. Rare studies have been conducted on PROMETHEE methods with 2DULVs. PROMETHEE methods cannot directly handle 2DULVs, which are 2D extension of the linguistic term set. Although Zhao et al. [51] discussed the application about the PROMRTHEE methods with 2D linguistic variables through sign-area and PD, 2DULVs are rather complicated than 2D linguistic variables. Some new methods should be suggested.

This paper aims to provide a hybrid multiple criteria decision method through literature review. The motivations of this paper are provided as follows:

Although most studies on 2DULVs have focused on various aggregation operators, we cannot ensure that all attributes are independent from each other whilst using aggregation operators. To date, no literature has focused on the ranking method or probability degree of 2DULVs. The possibility degree ( PD) index can reveal a partial order of two 2DULVs. Outranking methods can effectively rank alternatives with different attributes, such as ‘price’ and ‘skin feeling’ of a cosmetic. They can use indifference and preference thresholds to model imperfect information [52].

Randomness is another important aspect of uncertainty, and cloud model is an effective solution to address it. However, the majority of the researches on cloud model neglect the reliability assessments. And the existing aggregation operators about cloud and 2DULVs are relatively complex. Converting 2DULVs into a cloud model and combining the probability cannot only effectively reduce the computational complexity during decision making but can also simultaneously describe the ambiguity and randomness.

PROMETHEE methods, including PROME THEE I and PROMETHEE II methods, are widely applied in MCDM through paired comparison between alternatives. They assume that an incomplete compensation is found between the criteria. Although PROMETHEE methods cannot directly deal with 2D linguistic variables, PD proposed by the paper can provide a new solution to solve them. Since 2DULVs are a more complicated extension of linguistic term set from one dimension to two dimensions, a new method should be suggested to handle PROMETHEE methods with 2DULVs.

The contributions of this paper constitute three aspects. Firstly, we define the PD and PD index of 2DULVs by converting them into integrated clouds and then transforming integrated clouds into interval values, thereby resulting in minimal computation and ensuring no correlations are found between attributes. Secondly, the partial order relationship between two 2DULVs can be obtained using the PD index. Subsequently, we can fully sort alternatives by combining PROMETHEE II method and PD index. Thirdly, we combine 2DULVs, cloud model and improved PROMETHEE II method to solve MCDM problems because PROMETHEE holds the hypothesis that criteria are not fully compensable [53]. This study is the first to combine 2DULVs, cloud model and improved PROMETHEE methods, which deserves to be mentioned.

The remainder of this paper is arranged as follows. Section 2 presents the basic concept of 2DULVs and cloud model. Section 3 defines the PD and PD index. Section 4 comprehensively explains the developed methodology and describes its steps. Section 5 presents a renewable energy project evaluation application of the proposed approach. Section 6 verifies the proposed method through sensitivity analysis and comparative experiments. Section 7 provides the conclusions.

2 Preliminaries

This section reviews the concept of linguistic term set (LTS), 2DULVs, cloud model and their transformation. These concepts are the basis of the entire paper.

2.1 LTS

The concept of LTS was proposed by Zadeh [14]. This paper introduces linguistic variables to mimic the habit of human expressions and decisions. A LTS is expressed by a series of ordered linguistic variables. A LTS S ={ s₀, s₁ , ⋯ , s_l-1 } can have odd l number of elements, where element s_i represents a possible value for a linguistic variable.

Example 1. S can be expressed as S = { s₀, s₁, s₂, s₃, s₄, s₅, s₆ } = { extremely poor, very poor, poor, fair, good, very good, extremely good} when t = 6.

To minimise the loss of linguistic information, Zhu et al. [27] extended the discrete LTS into continuous LTS $\bar{S} = \{s_{α}| α \in [0, q]\}$ , where q is an extremely large positive number. The extended LTS $\bar{S}$ only appears in actual computation. However, an operational law of $\bar{S}$ proposed by Xu [54], which is $S_{α} + S_{β} = S_{α + β}$ , may cause some problems. Set example 1 considered, then, we can calculate $s_{2} \oplus s_{3} = s_{5}$ . This example indicates that we aggregate ‘poor’ and ‘fair’ into ‘very good’ when we do not consider their weights, thereby opposing human intuition. To overcome this problem, a subscript-symmetric LTS $S = {s_{t}| t = - τ, \dots, - 1, 0, 1, \dots, τ}$ was proposed by Dai et al. [55].

2.2 2DULVs

DMs should not only evaluate the alternatives but also provide extra appraisal of the reliability for their own evaluation results when the actual decision-making environment becomes increasingly complicated. Thus, the concept of 2DULVs was presented by Liu [29].

Definition 1 [29]. Let $\hat{s} = ([s_{a}, s_{b}], [h_{c}, h_{d}])$ , where [ s_a, s_b] is I class uncertain linguistic information representing the DM’s assessment for each alternative under the corresponding criteria, and [ h_c, h_d] is the II class uncertain linguistic information, indicating the DMs’ self-assessment of the reliability of their own results. s_a, s_b are chosen from the predefined LTS $S = {s_{- l}, \dots, s_{0}, \dots, s_{l}}, h_{c}, h_{d}$ are the elements of predefined LTS $S_{=} {h_{- t}, \dots, h_{0}, \dots, h_{t}}$ , and $\hat{s}$ is the 2DULV.

A 2DULV degenerates to a 2D linguistic variable when a = b and c = d, degenerates to a ULV when a = c and b = d, and degenerates to a linguistic variable when a = b = c = d.

2.3 Basic concept of cloud model

Definition 2 [56]. Assume a universe of discourse U and T as a qualitative concept in U. The certainty degree of x belonging to T satisfies $y = e^{^{-} {(x - E x)}^{2} / 2 {(E n^{'})}^{2}}$ when $x (x \in U)$ is a random instantiation of concept T, which satisfies $x ~ N (E x, E {n^{'}}^{2})$ and En $~ N (E n, H e^{2})$ , where y belongs to [0, 1].

The distribution of X in universe U is named a normal cloud, and the cloud drop can be written as ( x, y). The cloud can effectively describe the fuzziness and randomness of a concept using three quantitative variables, namely, expectation Ex, entropy En and hyper entropy He.

Example 2. Assuming a cloud C (50, 10, 1), we can depict C with 1,000 clouds using a forward normal cloud generator, as illustrated inFig. 1.

Fig. 1

Number near 50 as represented by cloud C (50, 10, 1).

2.4 Description of the conversion between 2DULVs and integrated clouds

Definition 3 [39]. Let $[s_{i}, s_{j}]$ be an uncertain linguistic variable. Convert the left linguistic variable s_i into left cloud $y_{i} = (E x_{i}, E n_{i}, H e_{i})$ and the remaining variable s_j into right cloud $y_{j} = (E x_{j}, E n_{j}, H e_{j})$ . Then, aggregate the two clouds into an integrated cloud $\tilde{y} = (E x, E n, H e)$ .

$|E x_{j} - E x_{i}| > 3 | E n_{j} - E n_{i} |$ indicates that the distance between the two clouds is sufficiently large to consider them as two parted clouds. Then, the numerical characteristics of the integrated cloud obey the following formulas: $\begin{matrix} Ex = \frac{[({Ex}_{i} + 3 {En}_{i}) + ({Ex}_{j} - 3 {En}_{j})]}{2} \\ En = max {\frac{Ex - {Ex}_{i}}{3}, \frac{{Ex}_{j} - Ex}{3}} \\ He = \sqrt{{He}_{i}^{2} + {He}_{j}^{2}} \\ . \end{matrix}$ (1) $|E x_{j} - E x_{i}| \leq 3 | E n_{j} - E n_{i} |$ indicates that the two clouds are close and intersect. Then, the numerical characteristics of the integrated cloud can be calculated as follows: $\begin{matrix} Ex = \frac{[({Ex}_{i} {En}_{j}) + ({Ex}_{j} {En}_{i})]}{{En}_{i} + {En}_{j}} \\ En = max {\frac{{En}_{i} + (Ex - {Ex}_{i})}{3}, \frac{{En}_{j} + ({Ex}_{j} - Ex)}{3}} \\ He = \sqrt{{He}_{i}^{2} + {He}_{j}^{2}} \\ . \end{matrix}$ (2) The existing theory of generating adjacent normal clouds around $Y_{0} = (E x_{0}, E n_{0}, H e_{0})$ can only generate five clouds and is exclusively suitable for 2D linguistic variables. Therefore, an improved model for producing adjacent normal clouds based on golden ratio was proposed by [40].

Definition 4 [40]. Let ${\hat{s}}_{1} = ([s_{a}, s_{b}], [h_{c}, h_{d}])$ be a 2DULV, where s_a, s_b are chosen from the predefined linguistic term set $S_{=} {s_{- l}, \dots, s_{0}, \dots, s_{l}}$ , and h_c, h_d are the elements of predefined linguistic term set $S_{=} {h_{- t}, \dots, h_{0}, \dots, h_{t}}$ . Then, the numerical characteristics of the corresponding 2 l+1 clouds can be calculated as follows:

Ex_i calculation: $\begin{matrix} {Ex}_{0} = \frac{(X max + X min)}{2}, {Ex}_{- l} = X_{min}, \\ {Ex}_{l} = X max, \\ {Ex}_{i} = {Ex}_{0} + \frac{0.382 (\frac{X max + X min}{2})}{l - 1}, \\ {Ex}_{- i} = {Ex}_{0} - \frac{0.382 (\frac{X max + X min}{2})}{l - 1}, \\ (1 \leq i \leq l - 1) \end{matrix}$ (3)

En_i calculation: $\begin{matrix} {En}_{- 1} = {En}_{1} = \frac{0.382 (X max - X min)}{c + d + 2 (t + 1)}, \\ {En}_{0} = 0.618 {En}_{1}, {En}_{- i} = {En}_{i} = \frac{{En}_{i - 1}}{0.618}, \\ (2 \leq i \leq l) . \end{matrix}$ (4)

He_i calculation: ( k is previously provided) $\begin{matrix} {He}_{0} = \frac{6 k}{c + d + 2 (t + 1)}, \\ {He}_{- i} = {He}_{i} = \frac{{He}_{i - 1}}{0.618}, 1 \leq i \leq l \\ . \end{matrix}$ (5)

Remark 1. The adjacent clouds are symmetrical about Y₀. In this paper, we use subscript-symmetric LTSs to provide the assessment information of 2DULVs for maintaining consistency.

Example 3. Let [0,100] be the universe of discourse and the predefined LTSs be S₌ ${s_{- 2}, s_{- 1}, s_{0}, s_{1}, s_{2}}$ and $S_{=} {h_{- 1}, h_{0}, h_{1}}$ . Assume a 2DULV ${\hat{s}}_{1} = ([s_{0}, s_{1}], [h_{- 1}, h_{0}])$ , where k = 0.1. The calculation of ${\hat{s}}_{1}$ converted into integrated cloud is expressed as follows: $\begin{matrix} s_{0} : y_{0} (50, 7.8692, 0.2), \\ s_{1} : y_{1} (69.1, 12.733, 0.3236), \\ \tilde{y} = (52.3, 5.62, 0.38) . \end{matrix}$

Definition 5 [57]. Let $\tilde{y}$ be an integrated could, and we can use the forward normal cloud generator to generate n cloud drops $((x_{1}, y_{1}), (x_{2}, y_{2}), \dots (x_{n}, y_{n}))$ . Then, the score function S of $\tilde{y}$ is expressed as follows: $S (\tilde{y}) = \frac{1}{n} \sum_{i = 1}^{n} x_{i} y_{i},$ (6) where x_i y_i indicates the expectation of cloud droplet $(x_{i}, y_{i})$ . Thus, the higher the score function value of ${\tilde{y}}_{ij}$ is, the better it will be.

2.5 Overview of PROMETHEE II method

Assuming a group of alternatives $A = {a_{1}, a_{2}, \dots a_{n}}$ and a set of criteria $C = {c_{1}, c_{2}, \dots, c_{m}}$ exist, where a_ij represents the performance of alternative a_i under criterion c_j. For convenience of description, all criteria belong to a maximising benefit criterion set.

PROMETHEE II method is based on a pair-wise type of relationship. The difference in performance between each pair of alternatives is expressed as $d (a_{i j}, a_{k j}) = P D I (a_{i j}) - P D I (a_{k j})$ , and the preference function of each criterion chosen by DMs is denoted as $P^{Ξ} (a_{i j}, a_{k j})$ = Function $(d (a_{i j}, a_{k j}))$ , where $d (a_{i j}, a_{k j})$ indicates the preference of alternative a_i over a_k for criterion c_j. $P^{Ξ} (Ξ = 1, 2, 3, 4, 5, 6)$ is a nondecreasing function and takes the value from 0 to 1. The difference between alternatives a_i and a_k increases with the increase in the function value. Alternative a_i is strictly superior to alternative a_k when the value reaches one.

After calculating the preference degree between two alternatives, we need to determine the multicriteria preference index II, that is, a global preference measure for alternative a_i over a_k. $Π (a_{i}, a_{k}) = \frac{\sum_{j = 1}^{m} w_{j} P^{Ξ} (a_{ij}, a_{kj})}{\sum_{j = 1}^{m} w_{j}},$ where w_j reflects the weight of criterion c_j

On the basis of the multicriteria preference index, we can derive outflow $Φ^{+}, inflow Φ^{-}$ and net flow Φ as follows: $\begin{matrix} Φ^{+} (a_{i}) = \sum_{j = 1}^{m} Π (a_{i}, a_{k}) \\ Φ^{-} (a_{i}) = \sum_{j = 1}^{m} Π (a_{k}, a_{i}) . \\ Φ (a_{i}) = Φ^{+} (a_{i}) - Φ^{-} (a_{i}) \end{matrix}$

Different from PROMETHEE I, PROMETHEE II is a complete ranking method based on net flow value. The higher the net flow value of the alternative is, the higher it ranks. Φ ( a_i) = Φ ( a_k) represents that the relationship of two alternatives is indifferent.

3 Improved PD for 2DULVs

3.1 PD

Definition 6 [58]. Let $a = [a^{-}, a^{+}], b = [b^{-}, b^{+}]$ be two interval numbers and $l_{a} = a^{+} - a^{-}, l_{b} = b^{+} - b^{-}$ . Then, the PD of a ≥ b is expressed as: $p (a \geq b) = \frac{min {l_{a} + l_{b}, max (a^{+} - b^{-}, 0)}}{l_{a} + l_{b}} .$ (7)

Definition 7 [59]. Supposed a normal cloud $y_{i} = (E x_{i}, E n_{i}, H e_{i})$ and a corresponding interval number ${y^{'}}_{i} = [y_{i}^{' -}, y_{i}^{' +}]$ , the conversion formula based on 3En principle between them is expressed as follows: $y_{i}^{' -} = {Ex}_{i} - 3 {En}_{i}, y_{i}^{' +} = {Ex}_{i} + 3 {En}_{i}$ (8)

Thus, we can derive the possibility definition of 2DULVs by referring to Definitions 3.1 and 3.2.

Definition 8. Let ${\tilde{y}}_{i} = (E x_{i}, E n_{i}, H e_{i})$ and ${\tilde{y}}_{j} = (E x_{j}, E n_{j}, H e_{j})$ be two clouds, and the PD of ${\tilde{y}}_{i}$ is equal or greater than ${\tilde{y}}_{j}$ , which is denoted as $P D ({\tilde{y}}_{i} \geq {\tilde{y}}_{j})$ , is defined as follows:

$\begin{matrix} PD ({\tilde{y}}_{i} \geq {\tilde{y}}_{j}) \\ = min {6 {En}_{i} + 6 {En}_{j}, max ({Ex}_{i} - {Ex}_{j} + 3 \\ \times ({En}_{i} + {En}_{j}), 0)} / (6 {En}_{i} + 6 {En}_{j}) . \end{matrix}$ (9)

Example 4. Let ${\tilde{y}}_{1} = (52.3, 5.62, 0.38), {\tilde{y}}_{2} = (54.1, 5.01, 0.29), {\tilde{y}}_{3} = (55.2, 4.64, 0.23),$ and ${\tilde{y}}_{4} = (55.9, 4.40, 0.19)$ be four clouds. The matrix of PD on the basis of Definition 3.3 is acquired as follows: $P = (\begin{matrix} 0.5 & 0.472 & 0.453 & 0.440 \\ 0.528 & 0.5 & 0.481 & 0.468 \\ 0.547 & 0.519 & 0.5 & 0.487 \\ 0.560 & 0.532 & 0.513 & 0.5 \end{matrix}) .$

Theorem 1. Let ${\tilde{y}}_{i}$ and ${\tilde{y}}_{j}$ be two clouds. The PD of ${\tilde{y}}_{i} \geq {\tilde{y}}_{j}$ is denoted as $P D ({\tilde{y}}_{i} \geq {\tilde{y}}_{j})$ . Then, the following properties hold:

Normative: 0 $\leq P ({\tilde{y}}_{i} \geq {\tilde{y}}_{j}) \leq 1$ ;

Complementary: $P ({\tilde{y}}_{i} \geq {\tilde{y}}_{j}) + P ({\tilde{y}}_{j} \geq {\tilde{y}}_{i}) = 1$ ;

Reflexivity: $P ({\tilde{y}}_{i} \geq {\tilde{y}}_{i}) = 0.5$ ;

3.2 PD outranking index (PDI) for 2DULV-cloud

We can measure the extent to which one alternative is superior to others under a certain criterion by proposing PDI.

Similar to the ranking vector ω in Xu [60], we sort the interval number $y_{i}^{'}$ by establishing a probability matrix as $P = {(P_{i j})}_{n \times m}$ , where $P_{i j} = P ({\tilde{y}}_{i} \geq {\tilde{y}}_{j})$ . Thus, we have Definition 3.4.

Definition 9. Let $S = {s_{- l}, \dots, s_{0}, \dots, s_{l}}$ and $S_{=} {h_{- t}, \dots, h_{0}, \dots, h_{t}}$ be two LTSs, ${\hat{s}}_{i j} = ([s_{a i j}, s_{b i j}], [h_{c i j}, h_{d i j}])$ be the 2DULV of alternative a_i under attribute c_j, $i = {1, 2, \dots n}$ and ${\tilde{y}}_{ij}$ be the corresponding integrated cloud. Furthermore, $P D ({\tilde{y}}_{i j} \geq {\tilde{y}}_{k j})$ be the PD between ${\tilde{y}}_{ij}$ and ${\tilde{y}}_{kj}$ . Then, the PDI for ${\tilde{y}}_{ij}$ is expressed as follows:

$\begin{matrix} PDI ({\tilde{y}}_{ij}) \\ = \frac{1}{n (n - 1)} (\sum_{k = 1}^{n} PD ({\tilde{y}}_{ij} \geq {\tilde{y}}_{kj}) + \frac{n}{2} - 1) . \end{matrix}$ (10)

Example 5. As shown in Example 3.1, a set of clouds $({\tilde{y}}_{1}, {\tilde{y}}_{2}, {\tilde{y}}_{3}, {\tilde{y}}_{4})$ is obtained. In accordance with Definition 3.4, we have: $\begin{matrix} PDI ({\tilde{y}}_{1}) = 0.239, PDI ({\tilde{y}}_{2}) = 0.248, \\ PDI ({\tilde{y}}_{3}) = 0.254, PDI ({\tilde{y}}_{4}) = 0.259 . \end{matrix}$

Hence, the partial outranking order is ${\tilde{y}}_{4} ≻ {\tilde{y}}_{3} ≻ {\tilde{y}}_{2} ≻ {\tilde{y}}_{1}$ .

Subsequently, the characteristics of the PDI are given as follows:

$0 \leq P D I ({\tilde{y}}_{i j}) \leq 1$ ;

$\sum_{i = 1}^{n} P D I ({\tilde{y}}_{i j}) = 1$ .

4 Framework of the proposed method

Assume a MADM problem with a set of alternatives $A = {A_{1}, A_{2}, \dots A_{n}}$ and a set of criteria $C = {C_{1}, C_{2}, \dots, C_{m}}$ , where A_i represents the ith alternative, and C_j is the jth criterion. We have a set of experts $E = {E_{1}, E_{2}, \dots, E_{t}}$ , where E_k is on behalf of the kth expert, and each expert has a corresponding weight, which expressed as $ω = {ω_{1}, ω_{1}, \dots ω_{t}}, \sum_{k = 1}^{t} ω_{k} = 1$ .

Let $S_{I} = {s_{- l}, \dots, s_{0}, \dots, s_{l}}$ be the I class LTS, which describes the value of the performance of the alternative by DMs, and $S_{I I} = {h_{- t}, \dots, h_{0}, \dots, h_{t}}$ be the II class LTS, which denotes the self-assessment of DMs. We use ${\hat{s}}_{i j}^{k} = ([s_{a i j}^{k}, s_{b i j}^{k}], [h_{c i j}^{k}, h_{d i j}^{k}])$ to represent the performance of alternative A_i concerning the criterion C_j evaluated by E_k, where $[s_{a i j}^{k}, s_{b i j}^{k}] \in S_{I} and [h_{c i j}^{k}, h_{d i j}^{k}] \in S_{I I}$ .

Phase 1. Establish the normalised interval cloud decision matrix

Step 1. Evaluate alternative A_i under each attribute C_j using ${\hat{s}}_{i j}^{k} = ([s_{a i j}^{k}, s_{b i j}^{k}], [h_{c i j}^{k}, h_{d i j}^{k}])$ . Then, 2DULV decision making matrix $D^{k} = {[{\hat{s}}_{i j}^{k}]}_{n \times m}$ can be obtained.

Step 2. Normalise the decision-making matrix.

In actual decision making, the criteria are generally divided into the maximising benefit criterion set S^B and minimising cost benefit criterion set S^C. S^B aims to achieve a high assessment value. S^C indicates that low value achieves. To obtain evaluation values at the same scale, the following formula is used to obtain standardised decision matrix ${\tilde{D}}^{k} = {[{\tilde{\hat{s}}}_{i j}^{k}]}_{n \times m}$ . ${\tilde{\hat{s}}}_{ij} = {\begin{matrix} [s_{aij}, s_{bij}], [h_{cij}, h_{dij}], C_{j} \in S^{B} \\ [s_{- bij}, s_{- aij}], [h_{cij}, h_{dij}], C_{j} \in S^{C} \end{matrix}$ (11)

Step 3. Transform standardised decision matrix ${\tilde{D}}^{k} = {[{\tilde{\hat{s}}}_{i j}^{k}]}_{n \times m}$ into integrated cloud matrix ${\tilde{y}}^{k} = {[{\tilde{y}}_{i j}^{k}]}_{n \times m}$ .

Using the formulas in Definitions 2.3 and 2.4, standardised decision matrix ${\tilde{D}}^{k} = {[{\tilde{\hat{s}}}_{i j}^{k}]}_{n \times m}$ can be converted into integrated cloud matrix ${\tilde{y}}^{k} = {[{\tilde{y}}_{i j}^{l}]}_{n \times m}$ .

Step 4. Aggregate integrated clouds ${\tilde{y}}^{k} = {[{\tilde{y}}_{i j}^{k}]}_{n \times m}$ into group integrated clouds $Y = {[{\tilde{y}}_{i j}]}_{n \times m}$ .

Generally, disagreements exist between experts. Therefore, we should aggregate integrated cloud ${\tilde{y}}^{k}$ into Y using a cloud-weighted arithmetic averaging operator [61]:

$\begin{matrix} {CAA}_{ω} (Y_{1}, Y_{2}, \dots Y_{n}) \\ = \sum_{i = 1}^{n} ω_{i} Y_{i} \\ {CAA}_{ω} (Y_{1}, Y_{2}, \dots Y_{n}) \\ = (\sum_{i = 1}^{n} ω_{i} {Ex}_{i}, \sqrt{\sum_{i = 1}^{n} ω_{i} {En}_{i}^{2}}, \sqrt{\sum_{i = 1}^{n} ω_{i} {He}_{i}^{2}}) . \end{matrix}$ (12)

Phase 2. Determining the weights of criteria

The approaches to obtain the attribute weights can be divided into three major categories, namely, objective approaches, subjective methods and integrated means [62]. DMs may have strategic weight manipulation or uncooperative behaviour during decision making because of personal interests [62]. Therefore, we use entropy weight method (EWM), which belongs to objective approaches, to avoid this situation.

Entropy, which is a measure of the degree of system disorder, was firstly used in thermodynamics. Shannon applied it to the field of information theory for dealing with uncertainty. Entropy theory states that low entropy value contain many information. Therefore, a criterion with small entropy value should be assigned a large weight [63]. We use the EWM because it only depends on the data and is a widely used method. The EWM proceeds with the following calculation procedures.

Step 5. Calculate the score function matrix.

We use Equation (6) to calculate the score function of ${\tilde{y}}_{ij}$ .

Step 6. Calculate Entropy e_j for criterion j. $e_{j} = K \sum_{i = 1}^{n} \frac{S ({\tilde{y}}_{ij})}{\sum_{i = 1}^{n} S ({\tilde{y}}_{ij})} ln \frac{S ({\tilde{y}}_{ij})}{\sum_{i = 1}^{n} S ({\tilde{y}}_{ij})},$ (13) where constant K is generally considered $K = 1 / \ln (n)$ .

Step 7. Obtain the weight of each criterion. $w_{j} = \frac{1 - e_{j}}{\sum_{i = 1}^{m} (1 - e_{j})},$ (14) where w_j ≥ 0, and $\sum_{j = 1}^{m} w_{j} = 1$ . The higher the value of w_j, the more important the role of the criterion will be.

Phase 3. PROMETHEE II method

Step 8. Calculate the PD and PDI.

We can obtain the PD of ${\tilde{y}}_{ij}$ usingEquation (9) and the PDI using Equation (10).

Step 9. Select the preference functions.

On the basis of the classic PROMETHEE method [64, 65], we extend the six preference functions for 2DULV-cloud. Let $P^{Ξ} ({\tilde{y}}_{i}_{j}, {\tilde{y}}_{k}_{j}) (Ξ = 1, 2, 3, 4, 5, 6)$ be the different types of preference functions. The six improved criteria are displayed as follows:

improved usual criterion $\begin{matrix} P^{1} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj}) \\ = {\begin{matrix} 0, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq 0 \\ 1, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) > 0 \end{matrix}, \end{matrix}$

improved quasi-criterion $\begin{matrix} P^{2} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj}) \\ = {\begin{matrix} 0, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq q \\ 1, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) > q \end{matrix}, \end{matrix}$

improved linear preference criterion $\begin{matrix} P^{3} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj}) \\ = {\begin{matrix} 0, & if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq 0 \\ \frac{PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj})}{p}, & if 0 < PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq p \\ 1, & if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) > p \end{matrix}, \end{matrix}$

improved level criterion $\begin{matrix} P^{4} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj}) \\ = {\begin{matrix} 0, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq q \\ 0.5, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq p \\ 1, if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) > p \end{matrix}, \end{matrix}$

improved linear preference and indifference criterion $\begin{matrix} P^{5} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj}) \\ = {\begin{matrix} 0, & if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq q \\ \frac{PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) - q}{p}, & if 0 < PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq p \\ 1, & if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) > p \end{matrix}, \end{matrix}$

improved Gaussian criterion $\begin{matrix} P^{6} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj}) \\ = {\begin{matrix} 0, & if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) \leq 0 \\ 1 - e^{- \frac{{(PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}))}^{2}}{2 σ^{2}}}, & if PDI ({\tilde{y}}_{ij}) - PDI ({\tilde{y}}_{kj}) > 0 \end{matrix}, \end{matrix}$

where p, q and σ are the three variables decided by DMs, which denote indifference, preference and Gaussian threshold, respectively. DMs become conservative with the increase in the values of variables p, q and σ.

Remark 2. The superscript of the preference function represents its type rather than the power operation.

Step 10. Calculate multicriteria preference index Π. $Π ({\tilde{y}}_{i}, {\tilde{y}}_{k}) = \frac{\sum_{j = 1}^{m} w_{j} P^{Ξ} ({\tilde{y}}_{ij}, {\tilde{y}}_{kj})}{\sum_{j = 1}^{m} w_{j}} .$ (15)

Step 11. Calculate outflow $Φ^{+} and inflow Φ^{-}$ . $\begin{matrix} Φ^{+} ({\tilde{y}}_{i}) = \sum_{j = 1}^{m} Π ({\tilde{y}}_{i}, {\tilde{y}}_{k}) \\ Φ^{-} ({\tilde{y}}_{i}) = \sum_{j = 1}^{m} Π ({\tilde{y}}_{k}, {\tilde{y}}_{i}) \end{matrix} .$ (16)

Step 12. Calculate the net flow $Φ ({\tilde{y}}_{i}) = Φ^{+} ({\tilde{y}}_{i}) - Φ^{-} ({\tilde{y}}_{i}) .$ (17)

Step 13. Rank all the alternatives.

The ranking of alternatives can be achieved in accordance with each alternative’s net flow. The larger $Φ ({\tilde{y}}_{i})$ , the better ${\tilde{y}}_{i}$ will be.

5 Illustrative example

5.1 Background description

Traditional energy sources release large amounts of greenhouse gases during combustion, thereby leading to serious environment crisis, such as global warming. Therefore, renewable energy sources (RESs) have been widely investigated because they are environmentally friendly and conventional energy sources are limited. Choosing an appropriate renewable resource is crucial for a state-owned power generation company because many barriers should be overcome during this process [66]. Barriers are obtained from multiple aspects. For example, we should consider technical maturity, greenhouse gas emissions and investment cost during decision making. Careful MCDM strategies must be applied to overcome these obstacles.

A state-owned power generation company aims to choose a suitable RES for Fujian Province. The four potential RESs are solar PV ( A₁), biomass ( A₂), hydrogen ( A₃) and wind ( A₄), and three DMs ( D₁, D₂, D₃) have weights of $ω = (0.3, 0.3, 0.4)$ . The evaluation data of 2DULVs are obtained from Zhang et al. (2019).

The predefined LTSs are ${S_}_{I} {= { s_}_{2} {,s_}_{1} {,s_}_{0} {, s_}_{1} {, s_}_{2}}$ = { very poor, poor, fair,good, very good} and ${S_}_{II} {= { h_}_{1} {, h _}_{0} {, h _}_{1}}$ = { partial confident, confident, very confident}.

5.2 Description of decision-making process

Step 1. The evaluation matrices of four alternatives are shown inTable 1 –3.

Table 1
Decision matrix given by DM₁

C ₁ C ₂ C ₃ C ₄ C ₅ C ₆ C ₇ C ₈

A ₁ ([s₀, s₁], [h₀, h₁]) ([s₁, s₁], [h₀, h₂]) ([s₀, s₁], [h_-1, h₀]) ([s₁, s₂], [h₀, h₁]) ([s₁, s₂], [h₀, h₁]) ([s₀, s₂], [h₀, h₁]) ([s_-1, s₁], [h_-1, h₁]) ([s₀, s₂], [h_-1, h₀])

A ₂ ([s_-1, s₁], [h₀, h₁]) ([s₀, s₁], [h₀, h₁]) ([s_-2, s₀], [h_-1, h₀]) ([s₁, s₂], [h₁, h₁]) ([s₀, s₂], [h_-1, h₀]) ([s_-1, s₁], [h_-1, h₁]) ([s₀, s₂], [h₀, h₁]) ([s₀, s₁], [h₀, h₁])

A ₃ ([s₀, s₂], [h_-1, h₀]) ([s_-2, s_-1], [h_-1, h₁]) ([s₁, s₁], [h_-1, h₁]) ([s₀, s₁], [h₀, h₁]) ([s₀, s₂], [h₀, h₁]) ([s_-1, s₁], [h₀, h₁]) ([s₀, s₁], [h_-1, h₀]) ([s₀, s₁], [h₀, h₁])

A ₄ ([s₀, s₂], [h_-1, h₀]) ([s₁, s₂], [h₀, h₁]) ([s₁, s₂], [h₁, h₁]) ([s₀, s₁], [h_-1, h₁]) ([s₁, s₂], [h_-1, h₀]) ([s_-1, s₀], [h₀, h₁]) ([s₁, s₂], [h₀, h₁]) ([s₀, s₂], [h_-1, h₁])

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	([s₀, s₁], [h₀, h₁])	([s₁, s₁], [h₀, h₂])	([s₀, s₁], [h_-1, h₀])	([s₁, s₂], [h₀, h₁])	([s₁, s₂], [h₀, h₁])	([s₀, s₂], [h₀, h₁])	([s_-1, s₁], [h_-1, h₁])	([s₀, s₂], [h_-1, h₀])
A ₂	([s_-1, s₁], [h₀, h₁])	([s₀, s₁], [h₀, h₁])	([s_-2, s₀], [h_-1, h₀])	([s₁, s₂], [h₁, h₁])	([s₀, s₂], [h_-1, h₀])	([s_-1, s₁], [h_-1, h₁])	([s₀, s₂], [h₀, h₁])	([s₀, s₁], [h₀, h₁])
A ₃	([s₀, s₂], [h_-1, h₀])	([s_-2, s_-1], [h_-1, h₁])	([s₁, s₁], [h_-1, h₁])	([s₀, s₁], [h₀, h₁])	([s₀, s₂], [h₀, h₁])	([s_-1, s₁], [h₀, h₁])	([s₀, s₁], [h_-1, h₀])	([s₀, s₁], [h₀, h₁])
A ₄	([s₀, s₂], [h_-1, h₀])	([s₁, s₂], [h₀, h₁])	([s₁, s₂], [h₁, h₁])	([s₀, s₁], [h_-1, h₁])	([s₁, s₂], [h_-1, h₀])	([s_-1, s₀], [h₀, h₁])	([s₁, s₂], [h₀, h₁])	([s₀, s₂], [h_-1, h₁])

Table 2

Decision matrix given by DM₂

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	([s₀, s₂], [h₀, h₁])	([s₁, s₁], [h_-1, h₀])	([s₀, s₂], [h₁, h₁])	([s_-1, s₁], [h₀, h₁])	([s₀, s₁], [h_-1, h₀])	([s₀, s₁], [h₀, h₀])	([s₀, s₂], [h_-1, h₁])	([s₁, s₂], [h_-1, h₁])
A ₂	([s₁, s₁], [h₀, h₁])	([s₀, s₂], [h_-1, h₀])	([s₀, s₂], [h₀, h₁])	([s_-1, s₁], [h₀, h₁])	([s₀, s₂], [h₁, h₁])	([s_-1, s₁], [h₁, h₁])	([s₀, s₁], [h_-1, h₀])	([s₁, s₂], [h₀, h₀])
A ₃	([s_-1, s₁], [h₀, h₁])	([s₁, s₂], [h₀, h₁])	([s_-1, s₀], [h_-1, h₀])	([s₀, s₁], [h₁, h₁])	([s₀, s₂], [h_-1, h₁])	([s₁, s₂], [h₀, h₁])	([s_-2, s_-1], [h₀, h₁])	([s_-1, s₀], [h₀, h₁])
A ₄	([s₀, s₁], [h₀, h₁])	([s_-1, s₀], [h₀, h₁])	([s₀, s₁], [h₀, h₁])	([s₀, s₂], [h₁, h₁])	([s_-1, s₁], [h₀, h₁])	([s₀, s₂], [h_-1, h₀])	([s₁, s₂], [h_-1, h₀])	([s₀, s₁], [h₀, h₁])

Table 3

Decision matrix given by DM₃

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	([s₀, s₂], [h₀, h₁])	([s_-1, s₁], [h_-1, h₁])	([s₀, s₂], [h₀, h₁])	([s₀, s₁], [h_-1, h₀])	([s_-2, s₀], [h₀, h₁])	([s₀, s₁], [h₁, h₁])	([s_-1, s₁], [h₀, h₂])	([s₀, s₂], [h₁, h₁])
A ₂	([s_-2, s₀], [h₀, h₁])	([s₀, s₁], [h₀, h₁])	([s₀, s₁], [h₀, h₀])	([s_-1, s₁], [h₁, h₁])	([s₀, s₁], [h₀, h₁])	([s_-2, s_-1], [h₀, h₁])	([s₀, s₁], [h₁, h₁])	([s_-2, s₀], [h_-1, h₀])
A ₃	([s₀, s₂], [h_-1, h₁])	([s_-1, s₁], [h_-1, h₁])	([s₀, s₁], [h_-1, h₁])	([s₀, s₂], [h₁, h₁])	([s₀, s₁], [h_-1, h₁])	([s₀, s₁], [h₀, h₂])	([s₁, s₂], [h₁, h₁])	([s₀, s₁], [h_-1, h₁])
A ₄	([s₀, s₂], [h₀, h₁])	([s_-1, s₁], [h_-1, h₁])	([s₀, s₁], [h_-1, h₀])	([s₀, s₁], [h₀, h₁])	([s_-1, s₀], [h₀, h₁])	([s_-1, s₁], [h₁, h₁])	([s₀, s₁], [h₀, h₁])	([s₀, s₂], [h₀, h₀])

Step 2. C₄, C₅ and C₆ are the cost criteria, and the remaining are the benefit criteria. Therefore, the normalised decision matrix can be obtained using Equation (11).

Step 3. The corresponding integrated clouds, which are calculated using Eqs (3)–(5), are shown inTable 4 –6.

Table 4

Converted integrated cloud matrix given by DM₁

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	(55.17,4.64,0.23)	(69.1,6.37,0.16)	(52.25,5.62,0.38)	(28.4,5.54,0.37)	(28.43,5.54,0.37)	(42.36,10.19,0.34)	(50,6.37,0.34)	(60.56,16.08,0.56)
A ₂	(50,6.37,0.27)	(55.17,4.64,0.23)	(39.44,16.08,0.56)	(27.25,5.15,0.31)	(39.44,16.08,0.56)	(50,6.37,0.34)	(57.64,10.19,0.34)	(55.17,4.64,0.23)
A ₃	(60.56,16.08,0.56)	(30.2,6.13,0.46)	(69.1,9.55,0.24)	(44.83,4.64,0.23)	(42.36,10.19,0.34)	(50,6.37,0.27)	(52.25,5.62,0.38)	(55.17,4.64,0.23)
A ₄	(60.56,16.08,0.56)	(71.57,5.54,0.37)	(69.8,6.13,0.46)	(45.92,5.01,0.29)	(23.6,10.08,0.62)	(55.17,4.64,0.23)	(71.57,5.54,0.37)	(54.77,11.14,0.42)

Table 5

Converted integrated cloud matrix given by DM₂

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	(57.64,10.19,0.34)	(69.1,12.73,0.32)	(59.55,9.55,0.28)	(50,6.37,0.27)	(47.75,5.62,0.38)	(45.92,5.01,0.29)	(54.77,11.14,0.42)	(69.8,6.13,0.46)
A ₂	(69.1,7.64,0.19)	(60.56,16.08,0.56)	(57.64,10.19,0.34)	(50,6.37,0.27)	(40.45,9.55,0.28)	(50,6.37,0.23)	(52.25,5.62,0.38)	(69.8,6.13,0.46)
A ₃	(50,6.37,0.27)	(71.57,5.54,0.37)	(47.75,5.62,0.38)	(44.1,4.4,0.19)	(45.23,11.14,0.42)	(28.43,5.54,0.37)	(28.43,5.54,0.37)	(44.83,4.64,0.23)
A ₄	(55.17,4.64,0.23)	(44.83,4.64,0.23)	(55.17,4.64,0.23)	(40.45,9.55,0.28)	(50,6.37,0.27)	(39.44,16.08,0.56)	(76.4,10.8,0.62)	(55.17,4.64,0.23)

Table 6

Converted integrated cloud matrix given by DM₃

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	(57.64,10.19,0.34)	(50,6.37,0.34)	(57.64,10.19,0.34)	(47.75,5.62,0.38)	(57.64,10.19,0.34)	(44.1,4.4,0.19)	(50,6.37,0.27)	(59.55,9.55,0.28)
A ₂	(42.36,10.19,0.34)	(55.17,4.64,0.23)	(54.08,5.01,0.29)	(50,6.37,0.23)	(44.83,4.64,0.23)	(71.57,5.54,0.37)	(55.9,4.4,0.19)	(39.44,16.08,0.56)
A ₃	(54.77,11.14,0.42)	(50,6.37,0.34)	(54.08,5.01,0.29)	(40.45,9.55,0.28)	(45.92,5.01,0.29)	(44.83,4.64,0.23)	(72.75,5.15,0.31)	(54.08,5.01,0.29)
A ₄	(57.64,10.19,0.34)	(50,6.37,0.34)	(52.25,5.62,0.38)	(44.83,4.64,0.23)	(55.17,4.64,0.23)	(50,6.37,0.23)	(55.17,4.64,0.23)	(54.77,11.14,0.42)

Step 4. Result of aggregating the integrated clouds, as shown inTable 7.

Table 7

Aggregated integrated cloud decision matrix

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	(56.9,8.9,0.31)	(61.46,8.78,0.29)	(56.6,8.85,0.34)	(42.63,5.83,0.35)	(45.91,7.76,0.36)	(44.12,6.81,0.27)	(51.43,8.1,0.34)	62.93,11.19,0.43)
A ₂	(52.67,8.44,0.28)	(56.79,9.63,0.36)	(50.76,10.9,0.4)	(43.17,6.03,0.27)	(41.9,10.66,0.37)	(58.63,6.05,0.32)	(55.33,6.95,0.3)	(52.37,11.01,0.45)
A ₃	(55.08,11.81,0.43)	(50.53,6.06,0.39)	(56.69,6.85,0.31)	(42.86,6.98,0.24)	(44.65,8.86,0.35)	(41.46,5.48,0.29)	(53.3,5.41,0.35)	(51.63,4.79,0.26)
A ₄	(57.77,11.21,0.4)	(54.92,5.65,0.32)	(58.39,5.51,0.37)	(43.84,6.6,0.26)	(44.15,7.16,0.4)	(48.38,10.01,0.36)	(66.46,7.27,0.42)	(54.89,9.66,0.37)

Step 5. The score function matrix using Equation (6) is represented inTable 8.

Table 8

Score function matrix

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	40.25	43.28	40.10	30.26	32.87	31.17	36.34	44.69
A ₂	37.33	40.16	35.69	30.62	29.31	41.46	39.33	36.97
A ₃	38.87	35.72	40.30	30.07	31.53	29.41	37.85	36.37
A ₄	40.85	38.95	41.29	31.17	31.14	34.19	47.01	38.75

Step 6. We can obtain the weight vector using the EWM. $\begin{matrix} w = (0.026, 0.103, 0.067, 0.004, 0.037, 0.389, \\ 0.223, 0.151) . \end{matrix}$

Step 7. Compute the PD using Equation (9). Then, apply Equation (10) to compute the PDI for alternative A_i against others under criteria C_j. Then, we construct the PDI matrix, as illustrated inTable 9. Take element $P D I ({\tilde{y}}_{2}_{1})$ as an example. On the basis of Equation (10), we can calculate $P D I (y_{2}_{1}) = \frac{1}{4 \times (4 - 1)} \times ((0.4593, 0.5, 0.4802, 0.4567) + \frac{4}{2} - 1) = 0.2415$ .

Table 9

PDI matrix under each criterion

PDI	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
A ₁	0.2540	0.2701	0.2523	0.2478	0.2557	0.2338	0.2309	0.2718
A ₂	0.2414	0.2537	0.2348	0.2502	0.2433	0.2952	0.2450	0.2424
A ₃	0.2488	0.2290	0.2528	0.2489	0.2513	0.2201	0.2352	0.2364
A ₄	0.2558	0.2472	0.2601	0.2531	0.2497	0.2509	0.2889	0.2495

Step 8. We use $P^{6} = ({\tilde{y}}_{i}_{j}, {\tilde{y}}_{k}_{j})$ as the preference function for each pair $(A_{i}, A_{k})$ because Gaussian function has nonlinear variation, which is realistic. Let σ = 0.2, and take $P^{6} = ({\tilde{y}}_{11}, {\tilde{y}}_{k 1})$ as an example. Given that $P D I ({\tilde{y}}_{11}) - P D I ({\tilde{y}}_{21}) = 0$ for criteria C₄. Thus, $P^{6} = ({\tilde{y}}_{11}, {\tilde{y}}_{21}) = 0, P^{6} = ({\tilde{y}}_{11}, {\tilde{y}}_{31}) = 1 - e^{^{-} \frac{(P D I ({\tilde{y}}_{i j}) - P D I ({\tilde{y}}_{k j}))}{2 σ^{2}}} = 1 - e^{^{-} \frac{{(0.04)}^{2}}{2 \times {0.2}^{2}}} = 0.021$ .

Step 9. We can calculate the multicriteria preference index Π using Equation (15). The results are shown inTable 10.

Table 10

Multicriteria preference index Π for each pair (A_i, A_k)

(A_i, A_k)	A ₁	A ₂	A ₃	A ₄
A ₁	0	0.0023	0.0054	0.0016
A ₂	0.0185	0	0.0276	0.0095
A ₃	5.26 × 10^–5	0.0003	0	1.285 × 10^–6
A ₄	0.0106	0.0007	0.0132	0

Step 10. Outflow $Φ^{+}, inflow Φ^{-}$ and net flow Φ are obtained usingEquations (16) and (17), as shown inTable 11.

Table 11

Results of outflow, inflow and net flow for each alternative

	Φ⁺	Φ^–	Φ
A ₁	0.0093	0.0094	–0.0198
A ₂	0.0556	0.0034	0.0522
A ₃	0.0004	0.0463	–0.0459
A ₄	0.0246	0.0111	0.0135

Step 11. On the basis of the net flow on the right block ofTable 11, the final ranking is $A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$ .

6 Sensitivity and comparative analysis

6.1 Sensitivity analysis

The weights play a crucial role in the entire decision-making process because the final ranking is affected by the weight of each criterion. We performed sensitivity analysis to observe the change in the final ranking with the change in the weight of each criterion. The proposed method has a certain degree of stability when the final ranking of each alternative changes within a certain range.

The weight value is adjusted with 50% increase of each criterion. For the validity of the experiment, only one criterion weight value is changed each time. Other criteria should be proportionally reduced with the increase in one criterion. For example, when the weight of criterion C₁ rises to 0.039, the weight of criteria C₂ can be calculated using $\frac{1 - 0.026}{1 - 0.039} \times 0.103 = 0.104$ . Similarly, the new weight vector $w^{'} = (0.039, 0.10.2, 0.066, 0.004, 0.037 0.384, 0.220, 0.149)$ can be obtained.

The results of sensitivity analysis are presented inFigs. 2 –9. In the figures, red, green, blue and black lines represent the net flow values of alternatives A₁, A₂, A₃ and A₄, respectively. As depicted in Figs. 2, 5, the weight changes of attributes C₁, C₄ and C₅ have small or no effect on the final ranking. The net flow of each alternative approaches the same value with the intensive change in attribute weights. The comparison ofFigs. 3, 4, 9 andFig. 8 shows that the weight changes of attributes C₂, C₃, C₇ and C₈ significantly affect the ranking of alternatives. The difference is that the ranking of A₄ only rises with the increase in the weight of C₇ and shows a downward trend when the weights of other criteria increase. The ranking of A₁ is not affected by attributes C₂ and C₈. The weight change in C₆ only affects the ranking of alternative A₂ once. Alternative A₃ is the worst in most cases. The ranking remains unchanged when the weight changes with a certain interval, thereby confirming the stability of the proposed method.

Fig. 2

Ranking by changing the weight of C₁.

Fig. 3

Ranking by changing the weight of C₂.

Fig. 4

Ranking by changing the weight of C₃.

Fig. 5

Ranking by changing the weight of C₄.

Fig. 6

Ranking by changing the weight of C₅.

Fig. 7

Ranking by changing the weight of C₆.

Fig. 8

Ranking by changing the weight of C₇.

Fig. 9

Ranking by changing the weight of C₈.

6.2 Comparative analysis

The EWM is compared with the maximising deviation method to verify its rationality [67]. The Hamming distance between two clouds is represented as follows:

$d_{H} (y_{1}, y_{2}) = | \frac{{En}_{2}^{2} + {He}_{2}^{2}}{{En}_{1}^{2} + {He}_{1}^{2} + {En}_{2}^{2} + {He}_{2}^{2}} {Ex}_{1} - \frac{{En}_{1}^{2} + {He}_{1}^{2}}{{En}_{1}^{2} + {He}_{1}^{2} + {En}_{2}^{2} + {He}_{2}^{2}} {Ex}_{2} | .$ (18)

The programming model can be constructed using Equation (18), which can be expressed as: $\begin{matrix} max D (w) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1, k \neq 1}^{m} d ({\hat{y}}_{ij}, {\hat{y}}_{kj}) w_{j} \\ s . t . {\begin{matrix} \sum_{j = 1}^{m} w_{j}^{2} = 1, \\ w_{j} \geq 0, \end{matrix} \end{matrix}$ (19)

Therefore, a linear programming model is presented as follows: $\begin{matrix} max D (w) = 125.91 w_{1} + 190.22 w_{2} + 259.71 w_{3} \\ + 52.292 w_{4} + 120.91 w_{5} + 180.52 w_{6} \\ + 140.88 w_{7} + 233.36 w_{8} \\ s . t . {\begin{matrix} \sum_{j = 1}^{m} w_{j}^{2} = 1, \\ w_{j} \geq 0, \end{matrix} \end{matrix}$

The optimal weight is w_j = (0.086,0.130, 0.287,0.036,0.083,0.123,0.096,0.159). The net flow obtained is $Φ (A_{1}) = 0.0040, Φ (A_{2}) = 0.0110, Φ (A_{3}) = - 0.0197, Φ (A_{4}) = 0.0083$ . The final ranking is $A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$ , which is exactly the same as that obtained using the EWM. There the rationality of the EWM applied in this study is confirmed. However, the least important criterion may have a better division between alternatives than that of other important criteria in practical application, making EWM reasonable.

The proposed approach is compared with three other methods proposed by Liu [29], Liu and Yu [68] and Zhang [40], respectively, to confirm the accuracy of the proposed method and the PD of 2DULVs. We use the data, criteria weights and experts’ weight along the same lines of Section 5 to clearly show the difference between the two approaches.

Using aggregation operators is a common means when dealing with group decision problems. Therefore, the first two comparison experiments are compared with the proposed methods of Liu [29] and Liu and Yu [68]. The formulas of these studies are based on the assumption that the subscripts of LTSs S_I and S_II are asymmetrical. Thus, we need to adjust the subscripts in the evaluation value with the appropriate forms before their application. The adjusted LTSs are expressed as $S_{I} = {s_{0}, s_{1}, s_{2}, s_{3}, s_{4}}$ and $S_{I I} = {h_{0}, h_{1}, h_{2}}$ . We performed Experiment 1 using the method proposed by Liu [29], and the final relative proximity of each alternatives was calculated as c₁ = 0.275, c₂ = 0.741, c₃ = 0.190, c₄ = 0.660. The ranking result is $A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$ . Experiment 2 was conducted to compare with the method proposed by Liu and Yu [68]. On the basis of aggregation operators 2DULPGWA proposed by Liu and Yu [68], we acquired a comprehensive evaluation value ${\tilde{Z}}_{i}$ for each alternative. The results are ${\hat{Z}}_{1} = ([s_{1}_{.61}, s_{3}_{.32}], [h_{0}, h_{1}]), {\hat{Z}}_{2} = ([s_{2}_{.01}, s_{3}_{.69}], [h_{0}, h_{1}]), {\hat{Z}}_{3} = ([s_{1}_{.43}, s_{2}_{.76}], [h_{0}, h_{1}])$ and ${\hat{Z}}_{4} = ([s_{1}_{.95}, s_{3}_{.57}], [h_{0}, h_{1}])$ . The larger the expected value of ${\hat{Z}}_{i}$ is, the higher the ranking of alternative A_i will be. The alternatives are ranked as $A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$ . The largest difference between Experiment 3 and Experiments 1 and 2 is that the former uses a cloud model rather than aggregation operators. Moreover, the method applied by Zhang et al. [40] considers the psychological behaviour of DMs. We can obtain the global value $π (A_{i})$ for each alternative on the basis of the decision process described by Zhang et al. [40]. The final ranking results are obtained as $A_{4} ≻ A_{1} ≻ A_{2} ≻ A_{3}$ depending on global values $π (A_{1}) = 0.939, π (A_{2}) = 0.155, π (A_{3}) = 0 a n d π (A_{4}) = 1.000$ . The ranking order acquired from the three methods is presented inTable 12.

Table 12

Ranking obtained from different methods

Method	Ranking
Liu’s method [29]	$A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$
Liu and Yu’s method [68]	$A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$
Zhang et al.’s method [40]	$A_{4} ≻ A_{1} ≻ A_{2} ≻ A_{3}$
The proposed method	$A_{2} ≻ A_{4} ≻ A_{1} ≻ A_{3}$

The final ranking results based on the theories proposed by Liu [29] and Liu and Yu [68] are consistent with our results. Therefore, the accuracy of the proposed method is confirmed. However, some nonnegligible distinctions are found although the results are the same. The models proposed by Liu [29] and Liu and Yu [68] are constructed on aggregation operators, whereas the method is based on PD and cloud model. The results produced by the proposed model are dissimilar from the theory of Zhang et al. [40]. The proposed model produces different optimal alternatives compared with the method of Zhang et al. [40] and is superior to the three other methods because of the following reasons. Firstly, the proposed model considers fuzziness and randomness because it is combined with the cloud model, whereas the theory proposed by Liu [29] and Liu and Yu [68] only considers fuzziness. Therefore, the proposed model corresponds to reality. Secondly, the proposed approach considers the non-compensability between the criteria, whereas the three other experiments ignore this aspect. Criteria, which are not fully compensable, generally exist, thereby making the proposed method suitable in various applications. Thirdly, the II class linguistic information is indirectly involved in the operations of the model in Liu [29], thereby leading to information distortion. 2DULPGWA in Liu and Yu [68] simply used the minimum operator when dealing with I class linguistic variables, causing information loss. And the calculation progress based on aggregation operators is much more complicated than the proposed method. The PD can intensively reduce calculation time work. Moreover, the information loss caused by aggregation operators can be relieved to some extent. Fourthly, despite that the ranking results obtained by the proposed method are inconsistent with the approach used by Zhang et al. [40], thereby indicating that the proposed method has certain advantages because regardless of how the loss attenuation factor changed, alternative A₃ is constantly the worst in the model described by Zhang et al. [40]. This finding is the same as the ranking of alternative A₃ in this paper. A₃ is the worst in most cases regardless of the change in the weights of eight criteria during the sensitivity analysis of Section 6.1. This finding certainly proves that the superiority of our model not only considers the incomplete compensability between criteria, but a certain degree of stability should be achieved when dealing with experts with different risk preferences. Besides, alternative A₄ is always superior to alternative A₁ among the four experiments, which demonstrate our method is scientific and credible.

7 Conclusion

This study proposed a hybrid method to resolve MCDM problems. Firstly, we defined the PD and PDI of 2DULVs to replace the PD in the PROMETHEE method. Secondly, we established a model combining PROMETHEE II and EWM to solve MCDM problems with unknown weight. Finally, we used a renewable energy performance sample, sensitivity analysis and comparative experiments to verify the effectiveness of the proposed approach. The results of sensitivity analysis demonstrate that the ranking of each alternative has its own stable interval, and the results of the comparison experiments manifest that the ranking obtained by this model is consistent with the ranking calculated by aggregation operators. Although the results differ from the ranking of Zhang et al. [40], the ranking of alternative A₃ reveals the superiority of the proposed approach. The comparative experiment of weight method proves the rationality of EWM. These experiments illustrate the stability, accuracy and rationality of the proposed.

This study has several limitations that can be discussed in future research. Firstly, the same words may indicate different concepts to different people during linguistic decision making, and the personalised individual semantics (PIS) [23 –25] should be investigated. In the future, we can integrate cloud with 2-tuple linguistic model to handle PIS problems. Secondly, the objective weight approach ignores the subjective intentions of DMs, thereby making the results contrary to reality. Assuming that the process of setting criteria weights in MCDM problems is not immune to strategic manipulations [62], we can develop a comprehensive weight programming model considering strategic manipulations and apply intelligent algorithms to solve the problem.

Footnotes

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper. This work was supported by the National Natural Science Foundation of China (No. 71871228).

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