Cloud model based approach to group decision making with uncertain pure linguistic information

Abstract

The cloud model, which mainly reflects the uncertainty in the real life and the concepts in human knowledge: fuzziness and randomness. Actually, many practical problems occur in uncertain environments, especially in the situations where the information about all the criteria weights, criteria values and expert weights are uncertain linguistic variables called uncertain pure linguistic problems. For this purpose, In this paper, we present an approach to multi-criteria group decision-making with uncertain pure linguistic information based on the cloud model. To do so, firstly, the uncertain linguistic values are converted into integrated cloud and interval integrated cloud, respectively, and the cloud decision-making information are converted into generating floating cloud and generating floating interval cloud by the cloud operational laws. Secondly, by means of the Hamming distance and closeness degree, the ranking of all alternatives is determined. Finally, a numerical example and comparative analysis with related decision-making methods are provided to illustrate the practicality and feasibility of the proposed method.

Keywords

Pure linguistic cloud model uncertain multi-criteria group decision-making

1 Introduction

Probability theory and fuzzy set theory are the most wide range of two effective tools in uncertainty knowledge representation. Probabilistic models mainly research random phenomenon and deal with random uncertainty. However, when the uncertainty is not probabilistic in nature other models have arisen such as fuzzy logic and the fuzzy linguistic approach. As to fuzzy set theory, proposed by Zadeh [54], have been the most important tools dealing with fuzzy uncertainty. And then, Zadeh [55] proposed the notion of a linguistic variable and computing with words (CWW), as a methodology for reasoning and computing with human-sourced decision-making information expressed in the form of natural language. So, the methods and applications to linguistic modelling in problems solving non-probabilistic uncertainty seem reasonable and have presented scientific results in many fields, for example, risk assessment [11, 37], data mining [14], decision support systems [25], engineering evaluation [24], group decision making [38 , 62], entropy and distance measure [2 , 60].

Recently, multi-criteria group decision making (MCGDM) problems under fuzzy environments arise from an increasing range of real life situations. There are several different kinds of discussion on fuzzy MCGDM, such as linguistic preference MCGDM [27 , 42], 2-tuple linguistic MCGDM [39 , 47], 4-tuple linguistic MCGDM [28], intuitionistic fuzzy MCGDM [58], hesitant fuzzy MCGDM [33, 34], pythagorean fuzzy MCGDM [52, 56]. However, due to the increasing complexity of the social environment and the vagueness of human thinking about the group decision making problems domain, in many situations, the decision information including the criteria weights, criteria values and expert weights are given in the form of fuzzy information. Xu [50, 51] developed methods of multiple attribute decision making with pure linguistic information, in which all the weights and preference values are linguistic variables. Peng et al. [29] presented an approach to group decision making based on the uncertain pure linguistic hybrid harmonic averaging operator and generalized interval aggregation operator. Peng and Ye [31, 32] introduced methods for aggregating induced uncertain pure linguistic and interval-valued intuitionistic pure linguistic information.

In these existing methods based on fuzzy information including sorts of above mentioned, the transformation between the linguistic preference information and exact numbers, which is implemented through the linguistic preference scale, is the most important aspect [4–10 , 16]. However, among all kinds of uncertainties involved in natural language, such as randomness, fuzziness, incompleteness and so on, randomness and fuzziness are the two key aspect and have attracted more and more concerns of people [36, 41]. Therefore, professor Li introduced the concept of cloud models as a new cognition model of uncertainty based on probability theory and fuzzy sets theory [17]. The prominent advantage of cloud model are the randomness of membership degree and the inherent relation between randomness and fuzziness taken into consideration. Also, the cloud model renders the transformation between the qualitative concepts and quantitative values more believable and interchangeable [18, 19].

Cloud model has been gaining ground during the last ten years in different theory and practice areas. The researches of theory areas in cloud model mainly reflect in the extension of concept was conducted by some scholars, such as floating clouds [20], integrated clouds [21], virtual clouds [3], expectation curve [22], and generic normal cloud [41]. The constructive studies have been widely applied into intelligent control [59], system evaluation [1], image segmentation [35, 46], data mining [23], multi-criterion group decision making [41, 42] and so on. From a large number of studies based on the cloud model, it is clear that it could be a strong instrument to solve uncertain linguistic MCGDM problems. Note that in the existing literature, however, few authors have done some research on the MCGDM problems under pure linguistic environment based on the cloud model.

This paper presents an approach to MCGDM under uncertain pure linguistic information based on the cloud model. In the process of uncertain pure linguistic decision-making, the criteria weights, criteria values and expert weights are taken form of linguistic assessment scale. At first, we convert the uncertain linguistic values into integrated cloud. And then, by means of the cloud operational laws, we aggregate the cloud decision-making information into generating floating cloud and generating floating interval cloud. At last, utilizing the Hamming distance and closeness degree, we can obtain the ranking of all alternatives. For this reason, all procedures of group decision-making closely around cloud model will be expand discussed in the following text. The rest of this paper is structured as follows. In Section 2, we introduce the operational laws of uncertain linguistic label variables and the knowledge of some cloud aspect. In Section 3, we present the technology of conversion between linguistic variables and clouds, and the operational laws of some clouds. In Section 4, we develop an approach based on the cloud model to uncertain pure linguistic multiple criteria group decision making. Section 5 gives an illustrative example to show the feasibility and practicability of the developed approach. In Section 6, we provide a comparative analysis with other related group decision-making methods. And finally, the main conclusions of this paper are summarized in Section 7.

2 Preliminaries

2.1 Uncertain linguistic variables and their operational laws

Let S ={ s_α|α = - t, ⋯ , -1, 0, 1, ⋯ , t } be a finite and totally ordered discrete term set, where s_α represents a possible value for a linguistic variable. For example, a set of five terms S could be given as follows: $\begin{matrix} S & = & {s_{- 2} = very poor, s_{- 1} = poor, s_{0} = fair, \\ s_{1} = good, s_{2} = very good} . \end{matrix}$

Usually, it is required that there exist the following characteristics [12]:

The terms set is ordered: s_α > s_β if α > β;

There is the negative operator: neg (s_α) = s_-α. Especially, neg (s₀) = s₀;

Max operator: max(s_α, s_β) = s_α if s_α > s_β;

Min operator: min(s_α, s_β) = s_α if s_α < s_β.

To preserve all the given information, Xu [49] extended the discrete linguistic label set S to a continuous linguistic label set $\tilde{S} = {s_{α} | α \in [- q, q]}$ , where q (q > t) is a sufficiently large positive number.

Definition 1. [48] Let $\tilde{s} = [s_{α}, s_{β}]$ , where $s_{α}, s_{β} \in \tilde{S}$ , s_α and s_β are the lower and the upper limits, respectively, then we call $\tilde{s}$ the uncertain linguistic variable.

Consider any three uncertain linguistic variables $\tilde{s} = [s_{α}, s_{β}]$ , ${\tilde{s}}_{1} = [s_{α_{1}}, s_{β_{1}}]$ and ${\tilde{s}}_{2} = [s_{α_{2}}, s_{β_{2}}]$ , and let λ, λ₁, λ₂ > 0, then, the operational laws are defined as follows [49]:

${\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [s_{α_{1}}, s_{β_{1}}] \oplus [s_{α_{2}}, s_{β_{2}}] = [s_{α_{1}} \oplus s_{α_{2}}, s_{β_{1}} \oplus s_{β_{2}}] = [s_{α_{1} + α_{2}}, s_{β_{1} + β_{2}}];$

${\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = [s_{α_{1}}, s_{β_{1}}] \otimes [s_{α_{2}}, s_{β_{2}}] = [s_{α_{1}} \otimes s_{α_{2}}, s_{β_{1}} \otimes s_{β_{2}}] = [s_{α_{1} α_{2}}, s_{β_{1} β_{2}}];$

$λ \tilde{s} = λ [s_{α}, s_{β}] = [λ s_{α}, λ s_{β}] = [s_{λ α}, s_{λ β}];$

$λ ({\tilde{s}}_{1} \oplus {\tilde{s}}_{2}) = λ {\tilde{s}}_{1} \oplus λ {\tilde{s}}_{2};$

$(λ_{1} + λ_{2}) \tilde{s} = λ_{1} \tilde{s} \oplus λ_{2} \tilde{s} .$

2.2 Cloud related concepts

Definition 2. [3, 17] Given a qualitative concept T defined over a universe of discourse U. If x (x ∈ X) is a random instantiation of the concept T and G_T (x) ∈ [0, 1] : U → G_T (x) is the certainty degree of x belonging to T, which corresponds to a random number with a steady tendency. Then the distribution of the membership on the domain is called a membership cloud (a cloud). Generally, there are three numerical characters of a cloud, the first is the expected value Ex, which can be described by the center value of the qualitative concept domain. The second is the entropy En, which measures the ambiguity of the qualitative concept, and the third is the hyper entropy value He, also seen as the entropy of entropy, which reflects the degree of dispersion of a cloud’s droplets and the random changes of the membership. Moreover, the larger the He is, the larger is the degree of dispersion, the larger is the randomness of degree of membership, and the larger is the thickness of the cloud performance in graphics.

For different kinds of cloud including generic normal cloud [41], the trapezium-cloud [15] and so on, there are several approaches to describe the three numerical parameters, such as subjective method, objective method and interactive method. For a more intuitive representation, we provide an approximate shape of the cloud by means of an example, which is expressed by cloud C (0, 1, 0.1) with 3000 cloud drops shown in Fig. 1. As mentioned above, the cloud model reflects not only the fuzziness information but randomness information of the concepts in human knowledge. As shown in Fig. 1, the fuzziness information is about the extension span of independent variables, such as [-4, 4], and the randomness information is originated in the cognition for decision-maker. For example, a certain decision-maker may think the degree of membership of 1.5 belonging to the “number near 1” is 0.8, but the other decision-maker may consider it to be 0.76, or maybe the third decision-maker to be 0.82. Obviously, there exist non-uniform cognition or the lack of uniformity among these decision-makers, which may distort the aggregation process of uncertain information. Fortunately, the cloud model allows the degree of the independent variables’ certainty to follow a probability distribution, which allows the distortion induced by decision-makers in the uncertain information aggregation process to be neutralized to a very great extent.

3 The conversion between uncertain linguistic variables and interval integrated clouds

Definition 3. [12] Let n be the number of the decision-makers’ linguistic label variables, which can be selected randomly as appropriate. [X_min, X_max] is the effective domain designated by experts. n clouds can be generated corresponding to the expression of linguistic values. If we denote the middle cloud as Y₀ (Ex₀, En₀, He₀), then the left and right adjacent clouds are denoted as (suppose n be odd) Y_-1 (Ex_-1, En_-1, He_-1), Y₊₁ (Ex₊₁, En₊₁, He₊₁), Y_-2 (Ex_-2, En_-2, He_-2), $\begin{matrix} Y_{+ 2} ({Ex}_{+ 2}, {En}_{+ 2}, {He}_{+ 2}), \dots, \\ Y_{- (n - 1) / - 2} ({Ex}_{- (n - 1) / - 2}, {En}_{- (n - 1) / - 2}, {He}_{- (n - 1) / - 2}), \\ Y_{+ (n - 1) / - 2} ({Ex}_{+ (n - 1) / - 2}, {En}_{+ (n - 1) / - 2}, {He}_{+ (n - 1) / - 2}) . \end{matrix}$

There are five clouds generated by the Golden Section, and the corresponding numerical characters are shown as follows: $\begin{matrix} {Ex}_{0} = (X_{min} + X_{max}) / - 2, \\ {Ex}_{- 2} = X_{min}, {Ex}_{+ 2} = X_{max}, \\ {Ex}_{- 1} = {Ex}_{0} - 0.382 (X_{min} + X_{max}) / - 2, \\ {Ex}_{+ 1} = {Ex}_{0} + 0.382 (X_{min} + X_{max}) / - 2; \\ {En}_{- 1} = {En}_{+ 1} = 0.382 (X_{max} - X_{min}) / - 6, \\ {En}_{0} = 0.618 {En}_{+ 1}, {En}_{- 2} = {En}_{+ 2} = {En}_{+ 1} / - 0.618; \\ {He}_{- 1} = {He}_{+ 1} = {He}_{0} / - 0.618, \\ {He}_{- 2} = {He}_{+ 2} = {He}_{+ 1} / - 0.618 . \end{matrix}$

Note that He₀ is given in advance.

In order to transform the uncertain linguistic variables into clouds, Wang et al. [42, 43] introduced integrated cloud and interval integrated cloud, which defined as follows:

Definition 4. Suppose [s_i, s_j] be the interval linguistic value. Convert s_i and s_j into two clouds Y_i = (Ex_i, En_i, He_i) and Y_j = (Ex_j, En_j, He_j), which called the left cloud and the right cloud, respectively. By aggregating the two clouds, we can get $\tilde{Y} = (Ex, En, He)$ , called an integrated cloud, and the corresponding numerical characters are shown as follows:

$\begin{matrix} Ex = [({Ex}_{i} + 3 {En}_{i}) + ({Ex}_{j} - 3 {En}_{j})] / - 2, \\ En = max {(Ex - {Ex}_{i}) / - 3, ({Ex}_{j} - Ex) / - 3}, \\ He = \sqrt{{He}_{i}^{2} + {He}_{j}^{2}} . \end{matrix}$ (1)

Definition 5. Suppose [s_i, s_j] be the interval linguistic value. Convert s_i and s_j into two clouds Y_i = (Ex_i, En_i, He_i) and Y_j = (Ex_j, En_j, He_j), which called the left interval cloud and the right interval cloud, respectively. By aggregating the two clouds, we can get $\tilde{Y} = ([E \underline{x}, E \bar{x}], En, He)$ , called an interval integrated cloud, and the corresponding numerical characters are shown as follows:

$\begin{matrix} E \underline{x} = min {{Ex}_{i}, {Ex}_{j}}, E \bar{x} = max {{Ex}_{i}, {Ex}_{j}}, \\ En = \sqrt{{En}_{i}^{2} + {En}_{j}^{2}}, \\ He = \sqrt{{He}_{i}^{2} + {He}_{j}^{2}} . \end{matrix}$ (2)

Let C₁ and C₂ be two clouds, and the corresponding numerical characters are C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂), Yang et al. [53] proposed the operational laws between two clouds: $\begin{matrix} 1) C_{1} + C_{2} = ({Ex}_{1} + {Ex}_{2}, \sqrt{{En}_{1}^{2} + {En}_{2}^{2}}, \\ \sqrt{{He}_{1}^{2} + {He}_{2}^{2}}) \end{matrix}$ (3) $\begin{matrix} 2) C_{1} - C_{2} = ({Ex}_{1} - {Ex}_{2}, \sqrt{{En}_{1}^{2} + {En}_{2}^{2}}, \\ \sqrt{{He}_{1}^{2} + {He}_{2}^{2}}) \end{matrix}$ (4) $\begin{matrix} 3) C_{1} \times C_{2} = ({Ex}_{1} {Ex}_{2}, \sqrt{({En}_{1} {Ex}_{2})^{2} + ({En}_{2} {Ex}_{1})^{2}}, \\ \sqrt{({He}_{1} {Ex}_{2})^{2} + ({He}_{2} {Ex}_{1})^{2}}) \end{matrix}$ (5)

However, the operational laws between cloud above can only be used in situations where the clouds are the integrated clouds. In the following we shall extend operational laws to accommodate the situations where the clouds are interval integrated clouds. Based on the cloud operational laws and the generalized interval aggregation (GIA) operator proposed by Peng et al. [29], the operational laws between interval integrated clouds can be defined asfollows:

Definition 6. Assuming there are two interval integrated clouds $C_{1} ([E {\underline{x}}_{1}, E {\bar{x}}_{1}], {En}_{1}, {He}_{1})$ and $C_{2} ([E {\underline{x}}_{2}, E {\bar{x}}_{2}], {En}_{2}, {He}_{2})$ , we can have some operations between interval integrated clouds C₁ and C₂: $\begin{matrix} 1) C_{1} + C_{2} = ([E {\underline{x}}_{1} + E {\underline{x}}_{2}, E {\bar{x}}_{1} + E {\bar{x}}_{2}], \\ \sqrt{{En}_{1}^{2} + {En}_{2}^{2}}, \sqrt{{He}_{1}^{2} + {He}_{2}^{2}}) \end{matrix}$ (6) $\begin{matrix} 2) C_{1} - C_{2} = ([E x_{-}^{'}, E {\bar{x}}^{'}], \sqrt{{En}_{1}^{2} + {En}_{2}^{2}}, \\ \sqrt{{He}_{1}^{2} + {He}_{2}^{2}}) \end{matrix}$ (7) $\begin{matrix} 3) C_{1} \times C_{2} = ([\hat{E} \underline{x}, \hat{E} \bar{x}], \sqrt{({En}_{1} {Ex}_{2})^{2} + ({En}_{2} {Ex}_{1})^{2}}, \\ \sqrt{({He}_{1} {Ex}_{2})^{2} + ({He}_{2} {Ex}_{1})^{2}}) \end{matrix}$ (8) in which, ${Ex}_{1} = (E {\underline{x}}_{1} + E {\bar{x}}_{1}) / - 2$ , ${Ex}_{2} = (E {\underline{x}}_{2} + E {\bar{x}}_{2}) / - 2$ ; $\begin{matrix} E x_{-}^{'} & = & min_{i, j = 1, 2} {(E {\bar{x}}_{i} - E {\underline{x}}_{j}), (E {\underline{x}}_{i} - E {\underline{x}}_{j}), \\ (E {\bar{x}}_{i} - E {\bar{x}}_{j})}, \\ E {\bar{x}}^{'} & = & max_{i, j = 1, 2} {(E {\bar{x}}_{i} - E {\underline{x}}_{j}), (E {\underline{x}}_{i} - E {\underline{x}}_{j}), \\ (E {\bar{x}}_{i} - E {\bar{x}}_{j})}; \\ \hat{E} \underline{x} & = & min {E {\underline{x}}_{1} E {\underline{x}}_{2}, E {\underline{x}}_{1} E {\bar{x}}_{2}, E {\bar{x}}_{1} E {\underline{x}}_{2}, \\ E {\bar{x}}_{1} E {\bar{x}}_{2}}, \\ \hat{E} \bar{x} & = & max {E {\underline{x}}_{1} E {\underline{x}}_{2}, E {\underline{x}}_{1} E {\bar{x}}_{2}, E {\bar{x}}_{1} E {\underline{x}}_{2}, \\ E {\bar{x}}_{1} E {\bar{x}}_{2}} . \end{matrix}$

Example 1. Assume C₁ ([1, 3] , 1.02, 0.28) and C₂ ([1.6, 2] , 1.48, 0.19) are two interval integrated clouds, then, we can get C₁ + C₂ = ([2.6, 5] , 1.8, 0.34), C₁ - C₂ = ([0.6, 2] , 1.8, 0.34), C₁ × C₂ = ([1.6, 6] , 3.48, 0.63).

The interval integrated cloud has some desirable properties including the law of commutative, associative and so on, which can be described as follows:

Theorem 1. Let C₁, C₂ and C₃ are three interval integrated clouds, then

C₁ + C₂ = C₂ + C₁

C₁ × C₂ = C₂ × C₁

(C₁ + C₂) + C₃ = C₁ + (C₂ + C₃)

Proof. It is straight based on Equations (6) and (8), thus omitted.

The distance between two interval integrated clouds C₁ and C₂ introduced by Wang et al. [42], which can be defined as follows:

Definition 7. Assuming there are two interval integrated clouds $C_{1} ([E {\underline{x}}_{1}, E {\bar{x}}_{1}], {En}_{1}, {He}_{1})$ and $C_{2} ([E {\underline{x}}_{2}, E {\bar{x}}_{2}], {En}_{2}, {He}_{2})$ , the Hamming distance between them denoted by d (C₁, C₂) is defined as

$\begin{matrix} d (C_{1}, C_{2}) & = & \frac{1}{2} (| (1 - \frac{{En}_{1} + {He}_{1}}{{Ex}_{1}}) E {\underline{x}}_{1} - (1 - \frac{{En}_{2} + {He}_{2}}{{Ex}_{2}}) E {\underline{x}}_{2} | \\ + | (1 - \frac{{En}_{1} + {He}_{1}}{{Ex}_{1}}) E {\bar{x}}_{1} - (1 - \frac{{En}_{2} + {He}_{2}}{{Ex}_{2}}) E {\bar{x}}_{2} |) \end{matrix}$ (9) in which, ${Ex}_{1} = (E {\underline{x}}_{1} + E {\bar{x}}_{1}) / - 2$ , ${Ex}_{2} = (E {\underline{x}}_{2} + E {\bar{x}}_{2}) / - 2$ . Moreover, the interval integrated cloud is reduced to an interval number if and only if En₁ = He₁ = En₂ = He₂ = 0.

Furthermore, the Hamming distance in Definition 7 satisfied some operation properties, such as non-negativity d (C₁, C₂) ≥0, commutativity d (C₁, C₂) = d (C₂, C₁), and triangle inequality d (C₁, C₃) ≤ d (C₁, C₂) + d (C₂, C₃).

4 An uncertain pure linguistic multi-criteria group decision making method based on the cloud model

Let d_k ∈ D (k = 1, 2, ⋯ , m) be the set of decision makers, and uncertain linguistic variables $s_{ν} = (s_{ν_{1}}, s_{ν_{2}}, \dots, s_{ν_{m}})^{T} \in \tilde{S}$ be the weight vector of decision makers, in which $s_{ν_{k}} = [s_{ν_{k}^{L}}, s_{ν_{k}^{U}}]$ . Let G ={ g₁, g₂, ⋯ , g_l } be the set of criteria, and uncertain linguistic variables $s_{ω} = (s_{ω_{1}}, s_{ω_{2}}, \dots, s_{ω_{l}})^{T} \in \tilde{S}$ be the weight vector of criteria. X ={ x₁, x₂, ⋯ , x_n } be a discrete set of alternatives. Suppose that $A^{(k)} = (a_{ij}^{(k)})_{l \times n}$ is the decision matrix, in which $a_{ij}^{(k)} \in \tilde{S}$ is a preference value taken form of uncertain linguistic variable $a_{ij} = [s_{a_{ij}^{L}}, s_{a_{ij}^{U}}]$ , given by the decision maker, for alternative x_j ∈ X(j = 1, 2, ⋯ , n) with respect to criteria g_i ∈ G (i = 1, 2, ⋯ , l). We shall propose a method for multi-criteria group decision making with uncertain pure linguistic information based on the interval integrated cloud as follows:

Step 1. We shall normalize the uncertain linguistic variable to unify all criteria, unless all the criteria have the same measurement unit or belong to the maximizing criteria. Suppose that the decision matrix $A^{(k)} = (a_{ij}^{(k)})_{l \times n}$ is normalized into the corresponding matrix ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{l \times n}$ . For the minimizing criteria, the normalization formula is ${\tilde{a}}_{ij} = [neg (s_{a_{ij}^{U}}), neg (s_{a_{ij}^{L}})]$ .

Step 2. Then, we shall convert the decision information taken form of the uncertain linguistic variable provided by decision makers, the weight vector of criteria $s_{ω} = (s_{ω_{1}}, s_{ω_{2}}, \dots, s_{ω_{l}})^{T} \in \tilde{S}$ and the weight vector of decision makers $s_{ν} = (s_{ν_{1}}, s_{ν_{2}}, \dots, s_{ν_{m}})^{T} \in \tilde{S}$ into the interval integrated cloud. Utilize Equation (2), we can get the corresponding interval integrated cloud $C ([E \underline{x}, E \bar{x}], En, He)$ .

Step 3. By using Equations (6) and (8) to aggregate the criteria values, and we can have the individual overall preference values associated to the decision makers. Moreover, we can calculate the collective overall preference value of alternative x_j given by the decision maker taken form of the converted interval integrated cloud, which is based on cloud theory.

Step 4. Calculate the Hamming distance by using Equation (9) between the collective overall preference value of alternative and the positive or negative ideal solutions: $\begin{matrix} {\tilde{Y}}^{+} & = & ([max E {\underline{x}}_{j}, max E {\bar{x}}_{j}], min {En}_{j}, min {He}_{j}), \\ {\tilde{Y}}^{-} & = & ([min E {\underline{x}}_{j}, min E {\bar{x}}_{j}], max {En}_{j}, max {He}_{j}) . \end{matrix}$

We have the Hamming distances between the j-th alternative x_j and the positive and negative ideal solution $d_{j}^{+} = d ({\tilde{Y}}_{j}, {\tilde{Y}}^{+})$ and $d_{j}^{-} = d ({\tilde{Y}}_{j}, {\tilde{Y}}^{-})$ , respectively.

Step 5. Rank all the alternatives x_j (j = 1, 2, ⋯ , n) and select the best one(s) in accordance with the closeness degree $d_{j}^{*} = \frac{d_{j}^{+}}{d_{j}^{+} + d_{j}^{-}}$ , the smaller $d_{j}^{*}$ is, the better the alternative x_j will be.

5 Illustrative example

In the supply chain management, the choices of suppliers is an important issue. It determines which suppliers you want to develop a strategic partnership, which you want to grow the business, which is to maintain the status you have, which is actively eliminated, and which is uncertainty. Recently, purchasing department of an overseas multi-national corporation intends to pick a suitable supplier to get better development. Therefore, the prominent element in process is the selection of excellent suppliers. Five suppliers (alternatives) are taken into consideration, which are denoted by x₁, x₂, x₃, x₄ and x₅. The five possible alternatives x_j (j = 1, 2, 3, 4, 5) are to be evaluated by means of the linguistic label set by three decision makers d_k (k = 1, 2, 3) (whose weightvector s_ν = ([s_-1, s₀] , [s₁, s₂] , [s₀, s₁]) ^T). And then, the department must make a decision according to the following three factors (criteria), g₁: risk analysis including risk identification, risk assessment, risk management strategy; g₂: development analysis including external management environment, enterprise inner quality and resource conditions; g₃: social-political impact analysis. Assume the weight of the three criteria is s_ω = (s_{ω
₁}, s_{ω
₂}, s_{ω
₃}) = ([s₁, s₂] , [s₀, s₁] , [s_-1, s₁]) ^T. After an adequate research on the suppliers and the intense discussion under the three criteria above, the three decision makers came to a consensus on the decision matrix $\tilde{A}$ as listed in Tables 1–3, respectively.

Here, we use the Golden Section to generate five clouds (Y_-2, Y_-1, Y₀, Y₊₁, Y₊₂) mentioned in Definition 3. Ordering the effective domain [X_min, X_max] = [0, 10] and He₀ = 0.1, we can have the numerical characters of the five clouds Y_-2 (0, 1.031, 0.262), Y_-1 (3.09, 0.637, 0.162), Y₀ (5, 0.393, 0.1), Y₊₁ (6.91, 0.637, 0.162), Y₊₂ (10, 1.031, 0.262).

Then, we convert all the criteria values, the weight of criteria and experts taken form of uncertain linguistic variables into the interval integrated clouds by Equation (2), respectively. The conversion result of the criteria values are listed as Tables 4–6.

Calculating the individual overall preference values C_kj associated to the decision maker d_k by using Equations (6) and (8) to aggregating the criteria values and the collective overall preference value C_j of alternative x_j given by the three decision maker taken form of the converted interval integrated cloud, we can have $\begin{matrix} C_{1} ([804.7, 3596.66], 231.41, 58.83), \\ C_{2} ([1127.68, 4373.46], 307.85, 177.92), \\ C_{3} ([1158.78, 4221.68], 279.14, 80), \\ C_{4} ([1231.79, 4203.41], 285.12, 81.21), \\ C_{5} ([908.8, 3594], 246.06, 62.45) . \end{matrix}$

By using Equation (9) calculate the Hamming distance between the collective overall preference value C_j of alternative x_j. The positive or negative ideal solutions: $\begin{matrix} {\tilde{Y}}^{+} & = & ([1231.79, 4373.46], 231.41, 58.83), \\ {\tilde{Y}}^{-} & = & ([804.7, 3594], 307.85, 177.92) . \end{matrix}$

Rank the five alternatives in accordance with the closeness degree $d_{j}^{*} = \frac{d_{j}^{+}}{d_{j}^{+} + d_{j}^{-}}$ , we can have the corresponding order is x₄ ≻ x₃ ≻ x₂ ≻ x₅ ≻ x₁, and the best supplier (alternative) is x₄.

6 Comparative analysis and discussion

In the process of dealing with fuzzy information, lots of aggregation operators in information fusion for decision making methods emerged in the past decade [26 , 49]. Also, some novel methods to tackle group decision making problems with fuzzy information are developed in recent years, such as integrated cloud model [43], generic normal cloud model [41] and so on. We now consider the uncertain linguistic aggregation operators developed by Xu [49], the pure linguistic hybrid arithmetic averaging operator developed by Peng et al. [30], integrated cloud based method developed by Wang and Liu [43].

Utilize the generalized interval aggregation operator [29] to aggregate the uncertain linguistic variables of the decision matrix to derive the individual overall preference value of alternative and the PLHAA operator to aggregate the individual overall preference value to get the collective overall preference value. By constructing a complementary matrix and summing all elements in each line of matrix, the ordering of all the alternatives is x₄ ≻ x₃ ≻ x₂ ≻ x₁ ≻ x₅, and the best supplier (alternative) is x₄.

And then, utilizing the integrated cloud based method developed by Wang and Liu [43], we convert all the decision information into integrated cloud. The final ordering of all the alternatives is x₄ ≻ x₃ ≻ x₂ ≻ x₅ ≻ x₁, and the best supplier (alternative) still be x₄.

As we can see above, for all the cases the best alternative is x₄, and the ordering of all the alternatives is approximately consistent with the developed method. However, the aggregated numerical results in method developed by Peng et al. [30] shown are quite different to the form of interval integrated cloud. The uncertain linguistic variables in the results have the incomparable advantage of the cloud model based method. As a result of the interval integrated cloud in the final ordering, we can obtain the information not only the advantages and disadvantages of the alternatives, but the stability and discrete type of them. Actually, the more En is, the more the discrete level be, and the less He is, the greater the stability be. Moreover, the results of uncertain linguistic variable depending on the aggregation operator has gone far beyond the assume linguistic label range. Furthermore, the developed method consider the decision information including the criteria weights and expert weights are uncertain linguistic variables, which broaden the method based the integrated cloud [43]. Therefore, in the situations where all the decision information are uncertain linguistic variables, the developed method in this paper is a betterchoice.

7 Concluding remarks

The cloud model reflects both the fuzziness and randomness in the uncertainty. By converting the uncertain linguistic values into integrated cloud and interval integrated cloud, respectively, we tackled the cloud decision making information by means of the generating floating cloud and generating floating interval cloud utilizing the cloud operational laws. And then, in virtue of the concept of the Hamming distance and closeness degree, we obtained the ranking of all alternatives. The prominent advantages of the developed method are not only the ability to deal effectively with the pure linguistic preference information, but also the ability to obtain the stability and discrete type of the alternatives.

Note that in classical multi-criteria group decision making, decision makers evaluate predefined alternatives based on predefined criteria. However, many real-world MCGDM problems (e.g., the decision processes of the United Nations Security Council) frequently have the practical features such as decision makers use individual sets of criteria to evaluate the individual alternatives, both the individual sets of criteria and the individual sets of alternatives can change dynamically in the decision process and so on. Therefore, in future research, we expect to develop further extensions by considering the contents mentioned above in the linguistic MCGDM problem and its applications.

Footnotes

Acknowledgments

The authors are thankful to the editor and the anonymous referees for your valuable comments and constructive suggestions that have led to an improved version of this paper. This work was supported by the National Natural Science Funds of China (Nos. 61364065 and 71272191), the China Postdoctoral Science Foundation (Nos. 2015T80990 and 2014M550473), the Applied Basic Research Programs of Yunnan Province, China (No. 2014FB136), Zhejiang Province Natural Science Foundation (No. LQ15G010003) and Ningbo Natural Science Foundation (No. 2015A610172).

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