Three-way linguistic group decisions model based on cloud for medical care product investment

Abstract

The long payback period for medical care products prevents investors from immediately recognizing risks arising throughout the entire period. To avoid risk loss, delay decision should be introduced to investment decision. In this study, we illustrate investment decision from the view of three-way group decisions. Linguistic scale is widely used during assessment, but the randomness and fuzziness of linguistic information are ignored. To cover these defects, this study introduces cloud to three-way group decisions and further extends cloud to medical care product investment decision in which the weights of experts are unknown. In our proposed model, the loss functions and conditional probability described by linguistic values in decision theoretic rough sets are converted to clouds, which can handle the fuzziness and randomness of linguistic information. The corresponding three-way decision rules are also derived from a cloud perspective. In addition, we define a new derivation degree based on the score function of cloud to determine the weights of experts in three-way group decisions. To validate the feasibility of our model, comparisons with the existing model are presented.

Keywords

Three-way group decisions linguistic assessment cloud decision theoretic rough sets

1 Introduction

The enlarged health industry includes provision of health service, production of health care products and equipment, and appropriation of other services closely related to human health. This emerging industry experiences good market and policy environment and thus possesses considerable potential in investment value. As a potential market, medical care products attract much attention from investors. Before making an investment decision, investors invite a group of experts from different fields to comprehensively evaluate market. Traditional investment decision results in either investment or not investment. Given the long payback period of medical care products, experts cannot know all the risk information of these products in detail. Delay decision is essential to further investigate these products and relatively decrease the misclassification cost (loss). Introducing delay decision to traditional binary decision forms three-way decisions. Researchers [1, 2] illustrated that the classification based on three-way decisions generates a precise classification in investment decision. In their research on recommendation, Zhang et al. [3, 4] demonstrated that the average recommendation cost of item in three-way decisions decreases significantly.

Three-way decisions theory proposed by Yao is derived from decision theoretic rough sets (DTRSs) [5, 6]. The system theory and applications of three-way decisions are then researched extensively. In theory research, three-way decisions theory mainly includes the modified and extended models. For example, Yao [7, 8] extended DTRS model to probabilistic rough sets and further utilized Bayes decision theory to derive three-way decision rules. Liu et al. [9] and Zhou [10] extended three-way decisions to multiple categories. Liang and Liu [11] studied three-way decisions with DTRSs in the context of multiple periods with intuitionistic fuzzy assessment information. In application research, three-way decision presents wide application prospect, including email spam filtering [12], medical treatment decision [13, 14] and three-way recommendation systems [3, 4].

In DTRSs model, conditional probability and loss functions are two important parameters. In extant research on three-way decisions, the format of conditional probability format and loss functions can be described by crisp numbers [15], fuzzy numbers [14 , 17], and linguistic values [18]. Linguistic decision problems [19 –21] have attracted much attention from numerous researchers. For example, Yu et al. [22] identified the current research status on linguistic decision making through a visualization method. Zhou and Chen proposed certain linguistic operators and applied them into linguistic group decision-making problems [23 –27]. Excluding studies on operators in linguistic decision problems, the quantization methods of linguistic information have also drawn attention from several researchers. The quantization methods in existing studies can be classified into four types: (I) the linguistic computational model based on membership functions [28, 29], (II) linguistic symbolic model [30, 31], (III) linguistic model based on 2-tuple representation [32], and (IV) cloud model [33, 34]. The linguistic computational model based on membership functions can describe fuzziness but not randomness. The linguistic symbolic model and 2-tuple linguistic model can neither describe fuzziness nor randomness.

By contrast, cloud model is an effective tool realizing the conversion between qualitative concept and quantitative description. This model considers fuzziness, randomness, and their association relationship. The main advantage of cloud model is that it transforms qualitative concepts to quantitative ones with three parameters, which can describe the randomness and fuzziness of the qualitative concepts [35]. Therefore, information loss is less when qualitative concepts are converted to quantitative ones through cloud model than with other methods. In recent years, cloud model has received much attention and brought new solutions for uncertain linguistic assessment in many fields, such as data mining [36], water quality assessment [37], and e-commerce [34].

Fuzzy decision-making problems [38 –41] have been widely studied by researchers. As a branch of fuzzy decision making, linguistic assessment [42] is extensively used by experts in group decision. However, before solving linguistic decision problems, quantifying qualitative assessment is crucial. Considering the advantage of cloud model in converting qualitative concepts into quantitative ones, utilizing this model for conversion is reasonable. An assessment model is also required to quantify the market risk of medical care product investment. DTRSs model can realize risk quantization through loss functions and conditional probability. DTRS model can be utilized to evaluate market risk and make a final decision for medical care product investment based on three-way decisions. In this study, we build a novel three-way group-decisions model with cloud-based linguistic DTRSs, and we apply this model in medical care product investment decision.

The rest of this paper is organized as follows. In Section 2, the basic concepts of DTRSs, linguistic variables, and cloud model are briefly reviewed. In Section 3, the three types of three-way decisions model and corresponding three-way decision rules based on cloud are discussed and proposed. In Section 4, a three-way group decisions model based on cloud and a new derivation degree calculation method between clouds are proposed. In Section 5, an example based on the background of enlarged health industry is provided to show the validity and feasibility of the proposed model. Finally, in Section 6, conclusions are drawn.

2 Preliminaries

In this section, the concepts of DTRSs, linguistic variables and cloud model are briefly reviewed.

2.1 DTRSs

In light of Bayesian decision and rough set theory [43], Yao proposed three-way decision theory. In his theory, DTRSs are composed of two states and three actions. The two states represented by a set Ω ={ C, ¬ C } indicate that an object belongs to C or not. In different contexts of decision making, the semantic interpretations of C and ¬C vary. The three actions are represented by a set A ={ a_P, a_B, a_N }, where a_P, a_B, and a_N represent the three actions for classifying object x. Positive action a_P should be taken when object x is classified into POS (C), non-commitment action a_B should be taken when object x is classified into BND (C), and negative action a_N should be taken when object x is classified into NEG (C). Table 1 lists the loss functions related to the cost of actions in different states.

Table 1
Loss functions of actions in different states

C (P) ¬C (N)

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

	C (P)	¬C (N)
a _P	λ _PP	λ _PN
a _B	λ _BP	λ _BN
a _N	λ _NP	λ _NN

In Table 1, λ_PP, λ_BP, and λ_NP denote the losses incurred for taking actions a_P, a_B, and a_N respectively, when object x belongs to C. Similarly, λ_PN, λ_BN, and λ_NN denote the losses incurred for taking the same actions when the object belongs to ¬C. Pr(C| [x]) is the conditional probability of an object x belonging to C given that the object is described by its equivalence class [x]. For object x, the expected loss R(a_i| [x]) associated with taking the individual actions can be expressed as $\begin{matrix} R (a_{P} | [x]) = λ_{pp} Pr (C | [x]) + λ_{NP} Pr (\neg C | [x]), \\ R (a_{B} | [x]) = λ_{BP} Pr (C | [x]) + λ_{BN} Pr (\neg C | [x]), \\ R (a_{N} | [x]) = λ_{NP} Pr (C | [x]) + λ_{NN} Pr (\neg C | [x]) . \end{matrix}$

Since that Pr(C| [x]) + Pr(¬ C| [x]) = 1, we simplify the rules based only on Pr(C| [x]) and the losses. Considering that losses λ_PP ≤ λ_BP < λ_NP and λ_NN ≤ λ_BN < λ_PN, the decision rules can be expressed as

(P) If Pr(C| [x]) ≥ α and Pr(C| [x]) ≥ γ, decide x ∈ POS (C).

(B) If Pr(C| [x]) ≤ α and Pr(C| [x]) ≥ β, decide x ∈ BND (C).

(N) If Pr(C| [x]) ≤ β and Pr(C| [x]) ≤ γ, decide x ∈ NEG (C).

The thresholds α, β, and γ are given by $α = \frac{λ_{PN} - λ_{BN}}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})},$ (1) $β = \frac{λ_{BN} - λ_{NN}}{(λ_{BN} - λ_{NN}) + (λ_{NP} - λ_{BP})},$ (2) $γ = \frac{λ_{PN} - λ_{NN}}{(λ_{PN} - λ_{NN}) + (λ_{NP} - λ_{PP})} .$ (3)

2.2 Linguistic variables

Definition 1. Let H = {h_α|α = - g, …, g, g ∈ N} be a finite and totally ordered discrete term set and N be the set of non-negative integers. Each term h_α represents a possible value of a linguistic variable. H must satisfy the following properties:

H is ordered, i.e., if α > β, h_α > h_β.

A negation operator exists on H, i.e., neg (h_α) = h_-α for ∀α.

2.3 Conversion between linguistic variables and cloud

Definition 2. [45] Let U be the universe of discourse and T be a qualitative concept in U. x ⊆ U. If x (x ∈ X) is a random instantiation of concept T that satisfies the conditions $x ~ N (E x, E n^{' 2})$ and $E n^{' 2} ~ N (E n, H e^{' 2})$ . y ∈ [0, 1] is the certainty degree of x belonging to concept T, calculated as follows: $y = e^{- \frac{{(x - E x)}^{2}}{2 {(E n^{'})}^{2}}}$ (4)

Definition 3. [46] Suppose that two clouds A (Ex_A, En_A, He_A) and B (Ex_B, En_B, He_B) are present, certain operations between cloud A and cloud B are defined as follows: $\begin{matrix} (1) A + B = ({Ex}_{A} + {Ex}_{B}, \sqrt{{En}_{A}^{2} + {En}_{B}^{2}}, \\ \sqrt{{He}_{A}^{2} + {He}_{B}^{2}}), \\ (2) A - B = ({Ex}_{A} - {Ex}_{B}, \sqrt{{En}_{A}^{2} + {En}_{B}^{2}}, \\ \sqrt{{He}_{A}^{2} + {He}_{B}^{2}}), \\ (3) λ A = (λ {Ex}_{A}, \sqrt{λ} {En}_{A}, \sqrt{λ} {He}_{A}) . \end{matrix}$

Definition 4. [46] Assuming linguistic assessment scale for decision makers 2g + 1 (g ∈ N) in natural linguistic assessment information group decision making, experts define effective domain universe [X_min, X_max]. Let h_α ∈ H correspond to Y_α = (Ex_α, En_α, He_α), the steps generating 2g + 1 (g ∈ N) level clouds are listed as follows:

Calculate θ_α and map h_α to θ_α using Equation (5). $θ_{α} = {\begin{matrix} \frac{a^{g} + a^{α}}{2 a^{g} - 2} & (- g \leq α \leq 0) \\ \frac{a^{g} + a^{α} - 2}{2 a^{g} - 2} & (0 < a < g) \end{matrix}$ (5)

Here, θ_α ∈ [0, 1] is a numerical value. The value of a can be obtained through experiments. In [47], Chang and Wang conducted experiment analysis on the value of a. In their final analysis results, when g equals 3 the best choice for the value of a is 1.36, when g equals 4, the best choice for the value of a is 1.4.

Calculate the value of Ex_α. ${Ex}_{α} = X_{min} + θ_{α} (X_{max} - X_{min})$ (6)

Calculate the value of En_α.

$\begin{matrix} {En}_{- α} = {En}_{α} \\ = {\begin{matrix} \frac{(θ_{| α - 1 |} + θ_{| α |} + θ_{| g |}) (X_{max} - X_{min})}{9} & (0 < | α | \leq g - 1) \\ \frac{(θ_{| α - 1 |} + θ_{| α |}) (X_{max} - X_{min})}{6} & (| α | = g) \\ \frac{(θ_{| α |} + 2 θ_{| α + 1 |}) (X_{max} - X_{min})}{9} & (α = 0) \end{matrix} \end{matrix}$ (7)

Calculate the value of He_α. $H e_{- α} = H e_{α} = \frac{(E n^{' +} - E n_{α})}{3}$ (8) where $E n^{' +} = \max_{k} {E n^{'}_{k}}$ , and $E n_{α}^{'} = {\begin{cases} \frac{(1 - θ_{α}) (X_{\max} - X_{\min})}{3} (- g \leq α \leq 0) \\ \frac{θ_{α} (X_{\max} - X_{\min})}{3} (0 < α \leq g) \end{cases}$

Definition 5. [48] Given a cloud drop (x, y), its contribution to concept T can be measured by the score function s = xy. Given the numerical characteristics and applied forward normal cloud generator, n cloud drops 〈 (x₁, y₁) , (x₂, y₂) , … , (x_n, y_n) 〉 can be generated. The estimated $\hat{s}$ of cloud A is as follows: $\hat{s} (A) = \frac{1}{n} \sum_{α = 1}^{n} x_{α} y_{α} .$ (9)

Given the numerical characteristics (Ex, En, He) of a cloud, the forward generator of this cloud can be described as the following algorithm [45]:

Steps:

Generate a normally distributed random number $E n^{'}$ with expectation En and variance He².

Generate a normally distributed random number x_α with expectation Ex and variance $H e^{2}$ .

Calculate $y_{α} = e^{- \frac{{(x_{α} - E x)}^{2}}{2 {(E n_{α}^{'})}^{2}}}$ .

x_α is a cloud drop in the universe and y_α is the certainty degree of x_α belonging to the concept T.

Repeat steps 1–4 until n cloud drops are generated.

3 Novel three-way decision model based on DTRSs and cloud

In three-way decisions, conditional probability and loss functions are two important parameters. Loss functions are evaluated by decision makers. The types of evaluation values vary, including numerical and non-numerical formats. Conditional probability is usually calculated through the rough membership implied by the nature of information systems or based on available historical data. However, we may lack historical data at our disposal; thus, drawing support from experts’ evaluation is a good choice. In this study, we use the linguistic term set H = {h_-4 = Absolutelylow, h_-3 = Verylow, h_-2 = Low, h_-1 = Fairlylow h₀ = Medium, h₁ = Fairlyhigh, h₂ = High, h₃ = Veryhigh, h₄ = Abso - lutelyhigh} to evaluate the loss functions and conditional probability. After evaluating the loss functions and conditional probability, we utilize the cloud model to convert qualitative concepts to quantitative ones.

Using the conversion results between linguistic variables and clouds as basis, the three-way decision based on DTRSs and cloud can be divided into three situations: (1) only the value of the conditional probability is a cloud variable, (2) only the values of the loss functions are cloud variables, and (3) the values of the loss functions and conditional probability are both cloud variables. According to the three different situations in three-way decisions based on DTRSs and cloud, Models 1, 2, and 3 are constructed.

3.1 Three-way decisions with the cloud interpretation of the conditional probability

Model 1 is constructed when the value of conditional probability in DTRSs is expressed as cloud variable and the loss functions are numeric. In this situation, the conditional probability is denoted as $\hat{Pr (C | [x])}$ . In light of Definition 1, $\hat{Pr (\neg C | [x])}$ is the negation of $\hat{Pr (C | [x])}$ . Combined with the conversion results between linguistic variables and clouds, we can derive that $(X_{min} + X_{max}, 0, 0) - Y (\hat{Pr (C | [x])}) = Y (\hat{Pr (\neg C | [x])})$ . Here, Y^* = (X_min + X_max, 0, 0) is an ideal cloud in our discourse. According to the relationship between $Y (\hat{Pr (\neg C | [x])})$ and $Y (\hat{Pr (C | [x])})$ , we can translate the three-way decision rules (P-N) to (P₁-N₁) based on cloud operations laws and Equation (9). The decision rules (P₁-N₁) are expresses as follows:

(P₁) If $\hat{S} (Y (\hat{Pr (C | [x])})) \geq α_{1}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \geq γ_{1}$ , decide x ∈ POS (C).

(B₁) If $\hat{S} (Y (\hat{Pr (C | [x])})) \leq α_{1}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \geq β_{1}$ , decide x ∈ BND (C).

(N₁) If $\hat{S} (Y (\hat{Pr (C | [x])})) \leq β_{1}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \leq γ_{1}$ , decide x ∈ NEG (C).

The thresholds values α₁, β₁, and γ₁ are givenby $α_{1} = \hat{s} (\frac{(λ_{PN} - λ_{BN}) Y^{*}}{(λ_{PN} - λ_{BN}) + (λ_{BP} - λ_{PP})})$ (10) $β_{1} = \hat{s} (\frac{(λ_{BN} - λ_{NN}) Y^{*}}{(λ_{BN} - λ_{NN}) + (λ_{NP} - λ_{BP})})$ (11) $γ_{1} = \hat{s} (\frac{(λ_{PN} - λ_{NN}) Y^{*}}{(λ_{PN} - λ_{NN}) + (λ_{NP} - λ_{PP})})$ (12)

Where 0 ≤ α₁ ≤ Y^*, 0 ≤ β₁ ≤ Y^*, and 0 ≤ γ₁ ≤ Y^*. As a valid boundary region in (B₁), it must satisfy that β₁ < α₁, indicating that 0 ≤ β₁ < γ₁ < α₁ ≤ Y^*. In this case, after tie breaking, the decision rules (P₁-N₁) are simplified as follows:

( $P_{1}^{'}$ ) If $Y (\hat{Pr (C | [x])}) \geq α_{1}$ , decide x ∈ POS (C).

( $B_{1}^{'}$ ) If $β_{1} \leq Y (\hat{Pr (C | [x])}) < α_{1}$ , decide x ∈ BND (C).

( $N_{1}^{'}$ ) If $Y (\hat{Pr (C | [x])}) \leq β_{1}$ , decide xinNEG (C).

However, if β₁ ≥ α₁, we can derive the following decision rules:

( $P_{1}^{″}$ ) If $Y (\hat{Pr (C | [x])}) \geq γ_{1}$ , decide x ∈ POS (C).

( $N_{1}^{″}$ ) If $Y (\hat{Pr (C | [x])}) < γ_{1}$ , decide xinNEG (C).

From the above decision rules (P₁^″ - N₁^″), the three-way decision rules ( $P_{1}^{'} {- N}_{1}^{'}$ ) are reduced to two-way decisions.

3.2 Three-way decisions with the cloud interpretation of the loss functions

In this subsection, we discuss Model 2 in which the values of the loss functions in three-way decision based on DTRSs are cloud variables, whereas the conditional probability is numerical. Table 2 presents the loss functions.

Table 2
Cost loss functions of actions with cloud variables

C (P) ¬C (N)

a _P $Y (\hat{λ_{PP}})$ $Y (\hat{λ_{PN}})$

a _B $Y (\hat{λ_{BP}})$ $Y (\hat{λ_{BN}})$

a _N $Y (\hat{λ_{NP}})$ $Y (\hat{λ_{NN}})$

	C (P)	¬C (N)
a _P	$Y (\hat{λ_{PP}})$	$Y (\hat{λ_{PN}})$
a _B	$Y (\hat{λ_{BP}})$	$Y (\hat{λ_{BN}})$
a _N	$Y (\hat{λ_{NP}})$	$Y (\hat{λ_{NN}})$

Since that Pr(C| [x]) + Pr(¬ C| [x]) = 1, we can simplify the three-way decision rules (P-N) into (P₂-N₂) based on Pr(C| [x]), cloud operations laws, and Equation (10). Considering the loss functions with $Y (\hat{λ_{PP}}) \leq Y (\hat{λ_{BP}}) < Y (\hat{λ_{NP}})$ and $Y (\hat{λ_{NN}}) \leq Y (\hat{λ_{BN}}) < Y (\hat{λ_{PN}})$ , the decision rules (P₂-N₂) are expressed as follows:

(P₂) If Pr(C| [x]) ≥ α₂ and Pr(C| [x]) ≥ γ₂, decide x ∈ POS (C).

(B₂) If Pr(C| [x]) ≤ α₂ and Pr(C| [x]) ≥ β₂, decide x ∈ BND (C).

(N₂) If Pr(C| [x]) ≤ β₂ and Pr(C| [x]) ≤ γ₂, decide x ∈ NEG (C).

The thresholds values α₂, β₂, and γ₂ are given as $α_{2} = \frac{\hat{s} (Y (\hat{λ_{PN}}) - Y (\hat{λ_{BN}}))}{\hat{s} ((Y (\hat{λ_{PN}}) - Y (\hat{λ_{BN}})) + (Y (\hat{λ_{BP}}) - Y (\hat{λ_{PP}})))}$ (13) $β_{2} = \frac{\hat{s} (Y (\hat{λ_{BN}}) - Y (\hat{λ_{NN}}))}{\hat{s} ((Y (\hat{λ_{BN}}) - Y (\hat{λ_{NN}})) + (Y (\hat{λ_{NP}}) - Y (\hat{λ_{BP}})))}$ (14)

$γ_{2} = \frac{\hat{s} (Y (\hat{λ_{PN}}) - Y (\hat{λ_{NN}}))}{\hat{s} ((Y (\hat{λ_{PN}}) - Y (\hat{λ_{NN}})) + (Y (\hat{λ_{NP}}) - Y (\hat{λ_{PP}})))}$ (15)

As a valid boundary region in (B₂), it must satisfy that β₂ < α₂, indicating that 0 ≤ β₂ < γ₂ < α₂ ≤ 1. In this case, after tie breaking, the decision rules (P₂-N₂) are simplified asfollows:

(P2^′) If Pr(C| [x])≥ α₂, decide x ∈ POS (C).

(B2^′) If β₂ ≤ Pr(C| [x])< α₂, decide xinBND (C).

(N2^′) If Pr(C| [x])≤ β₂, decide x ∈ NEG (C).

However, if β₂ ≥ α₂, 0 ≤ α₂ ≤ γ₂ ≤ β₂ ≤ 1 is implied, and we can derive the following decision rules:

(P₂^″) If Pr(C| [x])≥ γ₂, decide x ∈ POS (C).

(B₂^″) If Pr(C| [x])< γ₂, decide x ∈ NEG (C).

From the above decision rules (P ₂^″-N₂^″), the three-way decision rules (P ₂^′-N₂^′) are reduced to two-way decisions.

3.3 Three-way decisions with the cloud interpretation of both the conditional probability and loss functions

In this section, we discuss Model 3 in which the values of both the loss functions and conditional probability in three-way decisions based on DTRSs are cloud variables. Given that $(X_{min} + X_{max}, 0, 0) - Y (\hat{Pr (C | [x])}) = Y (\hat{Pr (\neg C | [x])})$ , we can simplify the three-way decision rules (P-N) to (P₃-N₃) based on conditional probability, cloud operations laws, and Equation (10). Considering the loss functions with $Y (\hat{λ_{PP}}) \leq Y (\hat{λ_{BP}}) < Y (\hat{λ_{NP}})$ and $Y (\hat{λ_{NN}}) \leq Y (\hat{λ_{BN}}) < Y (\hat{λ_{PN}})$ , the decision rules (P₃-N₃) are expressed as follows:

(P₃) If $\hat{S} (Y (\hat{Pr (C | [x])})) \geq α_{3}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \geq γ_{3}$ , decide x ∈ POS (C).

(B₃) If $\hat{S} (Y (\hat{Pr (C | [x])})) \leq α_{3}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \geq β_{3}$ , decide x ∈ BND (C).

(N₃) If $\hat{S} (Y (\hat{Pr (C | [x])})) \leq β_{3}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \leq γ_{3}$ , decide x ∈ NEG (C).

The thresholds values α₃, β₃, and γ₃ are givenby $α_{3} = \frac{\hat{s} (Y^{*} (Y (\hat{λ_{PN}}) - Y (\hat{λ_{BN}})))}{\hat{s} ((Y (\hat{λ_{PN}}) - Y (\hat{λ_{BN}})) + (Y (\hat{λ_{BP}}) - Y (\hat{λ_{PP}})))}$ (16) $β_{3} = \frac{\hat{s} (Y^{*} (Y (\hat{λ_{BN}}) - Y (\hat{λ_{NN}})))}{\hat{s} ((Y (\hat{λ_{BN}}) - Y (\hat{λ_{NN}})) + (Y (\hat{λ_{NP}}) - Y (\hat{λ_{BP}})))}$ (17) $γ_{3} = \frac{\hat{s} (Y^{*} (Y (\hat{λ_{PN}}) - Y (\hat{λ_{NN}})))}{\hat{s} ((Y (\hat{λ_{PN}}) - Y (\hat{λ_{NN}})) + (Y (\hat{λ_{NP}}) - Y (\hat{λ_{PP}})))}$ (18)

As a valid boundary region in (B₃), it must satisfy that β₃ < α₃, indicating that 0 ≤ β₃ < γ₃ < α₃ ≤ Y^*. In this case, after tie breaking, the decision rules (P₃-N₃) are simplified asfollows:

(P₃^′) If $\hat{S} (Y (\hat{Pr (C | [x])})) \geq α_{3}$ , decide x ∈ POS (C).

(B₃^′) If $\hat{S} (Y (\hat{Pr (C | [x])})) \leq α_{3}$ and $\hat{S} (Y (\hat{Pr (C | [x])})) \geq β_{3}$ , decide x ∈ BND (C).

(N₃^′) If $\hat{S} (Y (\hat{Pr (C | [x])})) \leq β_{3}$ , decide x ∈ NEG (C).

Considering the condition that β₃ ≥ α₃, 0 ≤ α₃ ≤ γ₃ ≤ β₃ ≤ Y^* is implied. In this case, we can derive the following decision rules (P₃^″-N₃^″):

(P₃^″) If $\hat{S} (Y (\hat{Pr (C | [x])})) \geq γ_{3}$ , decide x ∈ POS (C).

(N₃^″) If $\hat{S} (Y (\hat{Pr (C | [x])})) < γ_{3}$ , decide x ∈ NEG (C).

4 Three-way linguistic group decisions based on cloud

Group-decision problems have received much attention from researchers. For example, Merigoet al. [49] proposed generalized probabilistic weigh-ted aggregation operators to solve group-decision problem in strategic management. Using different fuzzy assessment environments as basis, Yu et al. extended group-decision problems to information systems security assessment [50], personal evaluation [51], and talent introduction [52]. Yang et al. [53] utilized triangular hesitant fuzzy preference relations and cooperative games method to solve group-decision problem in doctor assessment.

By contrast, three-way group decision problems [18, 54] are not extensively researched. In this section, we study three-way group decisions. In Section 3, we construct three models of three-way decision with linguistic DTRSs based on cloud. As Model 3 is the combination of Models 1 and 2, in this section, we take Model 3 as an example to extend our model to three-way group decisions in which the weights of experts are unknown.

4.1 Determination of experts’ weights

Assume m alternatives R ={ r₁, r₂, …, r_m } and seven criteria exist, denoted as C ={ c₁, c₂, …, c₇ }. If k experts E ={ e₁, e₂, …, e_k } exist whose corresponding weights are unknown, the evaluation matrix of the k-th expert e_k is denoted by $M^{k} = {(a_{ij}^{k})}_{m \times 7} (i = 1, \dots, m, j = 1, \dots, 7,)$ and $a_{ij}^{k} = h_{α} (α = - 4, \dots, 4)$ as alternative r_i under criterion c_j. The steps for calculating the weight of each expert e_k are as follows:

Step 1. Utilize the concerned method in Definition 4 to translate the linguistic variables to clouds.

Step 2. Apply the forward generator to generate n cloud drops $(x_{α}^{d}, y_{α}^{d}) (d = 1, 2, \dots, n)$ for each cloud Y_α.

Step 3. Calculate the comprehensive distance between each cloud Y_α based on the n cloud drops generated by the forward generator. The comprehensive distance is defined as

$\begin{matrix} d (Y_{α}, Y_{β}) & = & \frac{1}{n} \sum_{d = 1}^{n} | x_{α}^{d} y_{α}^{d} - x_{β}^{d} y_{β}^{d} |, \\ α, β = - 4, \dots, 4 \end{matrix}$ (19)

Evidently, the distance defined in Equation (20) satisfies the following properties:

d (Y_α, Y_β) ≥ 0.

d (Y_α, Y_β) = 0 if and only if Y_α = Y_β.

d (Y_α, Y_β) = d (Y_β, Y_α).

If cloud Y_γ ≥ Y_β ≥ Y_α in defined discourse, d (Y_α, Y_γ) ≥ d (Y_α, Y_β) and d (Y_α, Y_γ) ≥ d (Y_β, Y_γ).

Proof. Clearly, d (Y_α, Y_β) satisfies Properties (1–3) above. Now, it can be proved that d (Y_α, Y_β) also satisfies Property (4).

Y_γ ≥ Y_β ≥ Y_α, we can derive $\hat{s} (γ) \geq \hat{s} (β) \geq \hat{s} (α)$ , $\sum_{d = 1}^{n} x_{γ}^{d} y_{γ}^{d} \geq \sum_{d = 1}^{n} x_{β}^{d} y_{β}^{d} \geq \sum_{d = 1}^{n} x_{α}^{d} y_{α}^{d}$ .

Further, we can get $\sum_{d = 1}^{n} x_{γ}^{d} y_{γ}^{d} - \sum_{d = 1}^{n} x_{α}^{d} y_{α}^{d} \geq \sum_{d = 1}^{n} x_{γ}^{d} y_{γ}^{d} - \sum_{d = 1}^{n} x_{β}^{d} y_{β}^{d}$ and $\sum_{d = 1}^{n} x_{γ}^{d} y_{γ}^{d} - \sum_{d = 1}^{n} x_{α}^{d} y_{α}^{d} \geq \sum_{d = 1}^{n} x_{β}^{d} y_{β}^{d} - \sum_{d = 1}^{n} x_{α}^{d} y_{α}^{d}$ ;

Then, $\begin{matrix} d (Y_{α}, Y_{γ}) & = & \frac{1}{n} \sum_{d = 1}^{n} | x_{α}^{d} y_{α}^{d} - x_{γ}^{d} y_{γ}^{d} | \geq d (Y_{β}, Y_{γ}) \\ = & \frac{1}{n} \sum_{d = 1}^{n} | x_{β}^{d} y_{β}^{d} - x_{γ}^{d} y_{γ}^{d} |, \\ d (Y_{α}, Y_{γ}) & = & \frac{1}{n} \sum_{d = 1}^{n} | x_{α}^{d} y_{α}^{d} - x_{γ}^{d} y_{γ}^{d} | \geq d (Y_{α}, Y_{β}) \\ = & \frac{1}{n} \sum_{d = 1}^{n} | x_{α}^{d} y_{α}^{d} - x_{β}^{d} y_{β}^{d} | . \end{matrix}$

Therefore, the proof is completed.

Step 4. Calculate the weight of each expert e^k. The deviation degree between the evaluation matrix given by the expert e^k and the evaluation matrix given by the other expert e^t is determined by $g_{k} = \sum_{t \neq k} \sum_{i = 1}^{m} \sum_{j = 1}^{7} d (Y_{ij}^{k} - Y_{ij}^{t}) .$ (20)

The smaller the deviation degree between one expert and that of all other experts, the more accurate the evaluations of this expert, indicating that this expert should be given a higher weight. The weight of expert e^k can be defined as $w_{k} = (1 / - g_{k}) / - (\sum_{t = 1}^{k} (1 / - g_{k}))$ (21)

4.2 Aggregation of experts’ assessment information

After the weight of each expert is determined, certain tools are needed for aggregating the experts’ assessment. Operators are effective tools for aggregating assessment information. In recent years, many operators have been proposed under different fuzzy environments. For example, Harish proposed new confidence level-based Pythagorean fuzzy operators [55] and t-norm- and t-conorm-based generalized intuitionistic fuzzy interactive operators[56].

Weighted average (WA) operator is frequently used in aggregating the assessment of experts. We use WA operator to aggregate the evaluation values for the conditional probability and loss functions of the three-way decisions model. The aggregating results based on cloud model are shown as $\begin{matrix} Y (\hat{Pr (C | [x_{i}])}) & = & w_{1} Y_{i 1}^{1} + w_{2} Y_{i 1}^{2} \\ + \dots + w_{k} Y_{i 1}^{k} \end{matrix}$ (22) $\begin{matrix} Y (\hat{λ_{• •}}) & = & w_{1} Y_{i 2}^{1} + w_{2} Y_{i 2}^{2} + \dots \\ + w_{k} Y_{i 2}^{k} (• = P, B, N) \end{matrix}$ (23)

Utilizing the aggregating Equations (22, 23), we can obtain the final evaluation results of the conditional probability and loss functions in the DTRSs. The thresholds values, α₃, β₃, and γ₃ of every alternative are calculated based on Equations (16–18). Comparing the thresholds values, we judge the decision results belonging to the decision rules (P3^′-N3^′) or ( P3 ^″ -N3 ^″). According to the comparison results of the conditional probability and the thresholds, α₃, β₃, and γ₃, we determine which region the alternative belongs to.

5 Medical care product investment based on three-way group decisions with linguistic DTRSs

5.1 Background

In this section, we use three-way group decisions with linguistic DTRSs based on cloud to support the investment decisions of new medical care products and illustrate the decision process of our proposed model.

In investment decisions for new medical care products, the two states Ω ={ C, ¬ C } indicate whether a new product possesses a good market prospect or not. The set of actions for new product x is given by A ={ a_P, a_B, a_N }, where a_P, a_B, and a_N represent invest, further investigate, and reject investment, respectively. In Model 3, seven attributes are presented, the conditional probability $\hat{Pr (C | [x_{i}])}$ denotes the probability of product x_i possesses a good market prospect, and x_i is described by its equivalence class x_i (i = 1, 2, 3, 4, 5, 6, 7). $\hat{λ_{PP}}$ , $\hat{λ_{BP}}$ , and $\hat{λ_{NP}}$ are the losses incurred for taking actions of investment, further investigation, and investment rejection, respectively, when the new product possesses a good market prospect. $\hat{λ_{PN}}$ , $\hat{λ_{BN}}$ and $\hat{λ_{NN}}$ are the losses incurred for investing, further investigating, and rejecting investment, respectively, when the new product possesses a bad market prospect. To comprehensively evaluate the new products, at the initial evaluation and selection, all decision makers are asked to use linguistic term set H = {h_-4 = Absolutelylow, h_-3 = Very low, h_-2 = Low, h_-1 = Fairlylow, h₀ = Medium, h₁ = Fairlyhigh, h₂ = High, h₃ = Veryhigh, h₄ = Abso - lutelyhigh} to evaluate the two parameters in DTRSs for each new product. Then, three experts e_k (k = 1, 2, 3) are invited to give their evaluations based on related information about the new products. Next, the experts and managers met to make a final decision according to the collective results. Consequently, after a heated discussion, they came to a consensus on the final evaluation information showed in Tables 3–5.

Table 3
Evaluation information from expert e₁

X $\hat{Pr (C | [x_{i}])}$ $\hat{λ_{PP}}$ $\hat{λ_{BP}}$ $\hat{λ_{NP}}$ $\hat{λ_{PN}}$ $\hat{λ_{BN}}$ $\hat{λ_{NN}}$

x ₁ Fairly high Fairly low Medium High High Fairly high Low

x ₂ High Low Medium Fairly high High Medium Fairly low

x ₃ Very high Very low Medium Fairly high High Fairly high Fairly low

x ₄ Very High Low Fairly high High Very high High Medium

x ₅ Fairly low Very low Medium High Absolutely high Very high Fairly high

x ₆ Fairly high Absolutely low Fairly low Fairly high Very high Medium Fairly low

x ₇ Very low Very low Fairly low Medium High Medium Low

X	$\hat{Pr (C \| [x_{i}])}$	$\hat{λ_{PP}}$	$\hat{λ_{BP}}$	$\hat{λ_{NP}}$	$\hat{λ_{PN}}$	$\hat{λ_{BN}}$	$\hat{λ_{NN}}$
x ₁	Fairly high	Fairly low	Medium	High	High	Fairly high	Low
x ₂	High	Low	Medium	Fairly high	High	Medium	Fairly low
x ₃	Very high	Very low	Medium	Fairly high	High	Fairly high	Fairly low
x ₄	Very High	Low	Fairly high	High	Very high	High	Medium
x ₅	Fairly low	Very low	Medium	High	Absolutely high	Very high	Fairly high
x ₆	Fairly high	Absolutely low	Fairly low	Fairly high	Very high	Medium	Fairly low
x ₇	Very low	Very low	Fairly low	Medium	High	Medium	Low

Table 4

Evaluation information from expert e₂

X	$\hat{Pr (C \| [x_{i}])}$	$\hat{λ_{PP}}$	$\hat{λ_{BP}}$	$\hat{λ_{NP}}$	$\hat{λ_{PN}}$	$\hat{λ_{BN}}$	$\hat{λ_{NN}}$
x ₁	Medium	Very low	Fairly low	Medium	Absolutely high	High	Medium
x ₂	Fairly high	Low	Medium	High	Very high	Fairly high	Fairly low
x ₃	High	Low	Fairly high	Very high	Fairly high	Medium	Low
x ₄	Fairly low	Fairly low	Fairly high	Medium	High	Fairly low	Low
x ₅	Very high	Fairly low	Medium	High	Very high	Fairly high	Medium
x ₆	Very low	Absolutely low	Medium	High	Very high	Medium	Fairly low
x ₇	Low	Fairly low	Medium	Very high	Absolutely high	High	Very low

Table 5

Evaluation information from expert e₃

X	$\hat{Pr (C \| [x_{i}])}$	$\hat{λ_{PP}}$	$\hat{λ_{BP}}$	$\hat{λ_{NP}}$	$\hat{λ_{PN}}$	$\hat{λ_{BN}}$	$\hat{λ_{NN}}$
x ₁	Fairly high	Low	Fairly low	Fairly high	Very high	Fairly high	Fairly low
x ₂	High	Very low	Fairly low	Fairly high	High	Medium	Fairly low
x ₃	Low	Absolutely low	Very low	Medium	Very high	Fairly high	Medium
x ₄	Absolutely low	Fairly low	High	Absolutely high	Absolutely high	Very high	Fairly high
x ₅	Low	Medium	Fairly high	High	Fairly high	Medium	Low
x ₆	Very low	Low	Medium	Fairly high	Medium	Fairly low	Very low
x ₇	High	Low	Medium	Very high	High	Fairly high	Fairly low

5.2 Illustration of the proposed model

After receiving the assessment information for these new medical care products, the three-way group decision making procedure based on cloud regarding the aforementioned problem is shown asfollows:

Step 1. Convert the three-way decision-making information into clouds.

Given universe X = [X_min, X_max] = [0, 10], the linguistic term set is transformed to 10-point evaluation profile. According to the conversion methods in Section 2.3, the assessment information represented by clouds is showed in Tables 6–8.

Table 6
Evaluation information represented by clouds from expert e₁

X $Y (\hat{Pr (C | [x_{i}])})$ $Y (\hat{λ_{PP}})$ $Y (\hat{λ_{BP}})$ $Y (\hat{λ_{NP}})$ $Y (\hat{λ_{PN}})$ $Y (\hat{λ_{BN}})$ $Y (\hat{λ_{NN}})$

x ₁ (5.70,1.932,0.466) (4.30,1.932,0.466) (5,1.667,0.554) (6.69,2.27,0.353) (6.69,2.27,0.353) (5.70,1.932,0.466) (3.31,2.27,0.353)

x ₂ (6.69,2.27,0.353) (3.31,2.27,0.353) (5,1.667,0.554) (5.70,1.932,0.466) (6.69,2.27,0.353) (5,1.667,0.554) (4.30,1.932,0.466)

x ₃ (8.07,2.75,0.193) (1.93,2.75,0.193) (5,1.667,0.554) (5.70,1.932,0.466) (6.69,2.27,0.353) (5.70,1.932,0.466) (4.30,1.932,0.466)

x ₄ (8.07,2.75,0.193) (3.31,2.27,0.353) (5.70,1.932,0.466) (6.69,2.27,0.353) (8.07,2.75,0.193) (6.69,2.27,0.353) (5,1.667,0.554)

x ₅ (4.30,1.932,0.466) (1.93,2.75,0.193) (5,1.667,0.554) (6.69,2.27,0.353) (10,3.01,0.107) (8.07,2.75,0.193) (5.70,1.932,0.466)

x ₆ (5.70,1.932,0.466) (0,3.01,0.107) (4.30,1.932,0.466) (5.70,1.932,0.466) (8.07,2.75,0.193) (5,1.667,0.554) (4.30,1.932,0.466)

x ₇ (1.93,2.75,0.193) (1.93,2.75,0.193) (4.30,1.932,0.466) (5,1.667,0.554) (6.69,2.27,0.353) (5,1.667,0.554) (3.31,2.27,0.353)

X	$Y (\hat{Pr (C \| [x_{i}])})$	$Y (\hat{λ_{PP}})$	$Y (\hat{λ_{BP}})$	$Y (\hat{λ_{NP}})$	$Y (\hat{λ_{PN}})$	$Y (\hat{λ_{BN}})$	$Y (\hat{λ_{NN}})$
x ₁	(5.70,1.932,0.466)	(4.30,1.932,0.466)	(5,1.667,0.554)	(6.69,2.27,0.353)	(6.69,2.27,0.353)	(5.70,1.932,0.466)	(3.31,2.27,0.353)
x ₂	(6.69,2.27,0.353)	(3.31,2.27,0.353)	(5,1.667,0.554)	(5.70,1.932,0.466)	(6.69,2.27,0.353)	(5,1.667,0.554)	(4.30,1.932,0.466)
x ₃	(8.07,2.75,0.193)	(1.93,2.75,0.193)	(5,1.667,0.554)	(5.70,1.932,0.466)	(6.69,2.27,0.353)	(5.70,1.932,0.466)	(4.30,1.932,0.466)
x ₄	(8.07,2.75,0.193)	(3.31,2.27,0.353)	(5.70,1.932,0.466)	(6.69,2.27,0.353)	(8.07,2.75,0.193)	(6.69,2.27,0.353)	(5,1.667,0.554)
x ₅	(4.30,1.932,0.466)	(1.93,2.75,0.193)	(5,1.667,0.554)	(6.69,2.27,0.353)	(10,3.01,0.107)	(8.07,2.75,0.193)	(5.70,1.932,0.466)
x ₆	(5.70,1.932,0.466)	(0,3.01,0.107)	(4.30,1.932,0.466)	(5.70,1.932,0.466)	(8.07,2.75,0.193)	(5,1.667,0.554)	(4.30,1.932,0.466)
x ₇	(1.93,2.75,0.193)	(1.93,2.75,0.193)	(4.30,1.932,0.466)	(5,1.667,0.554)	(6.69,2.27,0.353)	(5,1.667,0.554)	(3.31,2.27,0.353)

Table 7

Evaluation information represented by clouds from expert e₂

X	$Y (\hat{Pr (C \| [x_{i}])})$	$Y (\hat{λ_{PP}})$	$Y (\hat{λ_{BP}})$	$Y (\hat{λ_{NP}})$	$Y (\hat{λ_{PN}})$	$Y (\hat{λ_{BN}})$	$Y (\hat{λ_{NN}})$
x ₁	(5,1.667,0.554)	(1.93,2.75,0.193)	(4.30,1.932,0.466)	(5,1.667,0.554)	(10,3.01,0.107)	(6.69,2.27,0.353)	(5,1.667,0.554)
x ₂	(6.69,2.27,0.353)	(3.31,2.27,0.353)	(5,1.667,0.554)	(6.69,2.27,0.353)	(8.07,2.75,0.193)	(5.70,1.932,0.466)	(4.30,1.932,0.466)
x ₃	(5.70,1.932,0.466)	(3.31,2.27,0.353)	(5.70,1.932,0.466)	(8.07,2.75,0.193)	(5.70,1.932,0.466)	(5,1.667,0.554)	(3.31,2.27,0.353)
x ₄	(4.30,1.932,0.466)	(4.30,1.932,0.466)	(5.70,1.932,0.466)	(5,1.667,0.554)	(6.69,2.27,0.353)	(4.30,1.932,0.466)	(3.31,2.27,0.353)
x ₅	(8.07,2.75,0.193)	(4.30,1.932,0.466)	(5,1.667,0.554)	(6.69,2.27,0.353)	(8.07,2.75,0.193)	(5,1.667,0.554)	(4.30,1.932,0.466)
x ₆	(1.93,2.75,0.193)	(0,3.01,0.107)	(5,1.667,0.554)	(5.70,1.932,0.466)	(8.07,2.75,0.193)	(5,1.667,0.554)	(4.30,1.932,0.466)
x ₇	(3.31,2.27,0.353)	(4.30,1.932,0.466)	(5,1.667,0.554)	(8.07,2.75,0.193)	(10,3.01,0.107)	(8.07,2.75,0.193)	(1.93,2.75,0.193)

Table 8

Evaluation information represented by clouds from expert e₃

X	$Y (\hat{Pr (C \| [x_{i}])})$	$Y (\hat{λ_{PP}})$	$Y (\hat{λ_{BP}})$	$Y (\hat{λ_{NP}})$	$Y (\hat{λ_{PN}})$	$Y (\hat{λ_{BN}})$	$Y (\hat{λ_{NN}})$
x ₁	(5.70,1.932,0.466)	(3.31,2.27,0.353)	(4.30,1.932,0.466)	(5.70,1.932,0.466)	(8.07,2.75,0.193)	(5.70,1.932,0.466)	(4.30,1.932,0.466)
x ₂	(6.69,2.27,0.353)	(1.93,2.75,0.193)	(4.30,1.932,0.466)	(5.70,1.932,0.466)	(6.69,2.27,0.353)	(5,1.667,0.554)	(4.30,1.932,0.466)
x ₃	(3.31,2.27,0.353)	(0,3.01,0.107)	(1.93,2.75,0.193)	(5,1.667,0.554)	(8.07,2.75,0.193)	(5.70,1.932,0.466)	(5,1.667,0.554)
x ₄	(0,3.01,0.107)	(4.30,1.932,0.466)	(6.69,2.27,0.353)	(10,3.01,0.107)	(10,3.01,0.107)	(8.07,2.75,0.193)	(5.70,1.932,0.466)
x ₅	(3.31,2.27,0.353)	(5,1.667,0.554)	(5.70,1.932,0.466)	(6.69,2.27,0.353)	(5.70,1.932,0.466)	(5,1.667,0.554)	(3.31,2.27,0.353)
x ₆	(8.07,2.75,0.193)	(3.31,2.27,0.353)	(5,1.667,0.554)	(5.70,1.932,0.466)	(5,1.667,0.554)	(4.30,1.932,0.466)	(1.93,2.75,0.193)
x ₇	(6.69,2.27,0.353)	(3.31,2.27,0.353)	(5,1.667,0.554)	(8.07,2.75,0.193)	(6.69,2.27,0.353)	(5.70,1.932,0.466)	(4.30,1.932,0.466)

Step 2. Apply the forward generator to generate 2000 cloud drops $(x_{α}^{d}, y_{α}^{d}) (d = 1, 2, \dots, 2000)$ for each cloud Y_α.

Step 3. Utilize Equation (16) to calculate the comprehensive distance between each cloud Y_α (α = -4, …, 4) based on the 2000 cloud drops generated by the forward generator.

Step 4. Calculate the weight vector of each expert e^k (k = 1, 2, 3). First, according to Equation (20) in Section 4, the derivation degrees of the three experts’ evaluations can be obtained: g₁ = 182.228, g₂ = 200.844, g₃ = 198.790. Then, based on Equation (21), the weight vector of each expert e^k can be calculated: w₁ = 0.354, w₂ = 0.321, w₃ = 0.325.

Step 5. Utilize WA operator to aggregate the evaluation values of all experts. The aggregation results in Table 9 can be obtained according to Equations (22, 23).

Table 9

Group decision matrix

X	$Y (\hat{Pr (C \| [x_{i}])})$	$Y (\hat{λ_{PP}})$	$Y (\hat{λ_{BP}})$	$Y (\hat{λ_{NP}})$	$Y (\hat{λ_{PN}})$	$Y (\hat{λ_{BN}})$	$Y (\hat{λ_{NN}})$
x ₁	(5.475,1.851,0.496)	(3.217,2.329,0.360)	(4.548,1.843,0.499)	(5.826,1.982,0.462)	(8.201,2.681,0.245)	(6.018,2.047,0.433)	(4.174,1.982,0.462)
x ₂	(6.690,2.270,0.353)	(2.862,2436,0.310)	(4.773,1.758,0.527)	(6.018,2.047,0.433)	(7.753,2.531,0.297)	(5.225,1.756,0.527)	(4.174,1.982,0.462)
x ₃	(5.762,2.356,0.351)	(1.746,2.697,0.239)	(4.227,2.154,0.436)	(6.232,2.157,0.460)	(6.821,2.341,0.359)	(5.475,1.851,0.496)	(4.210,1.969,0.465)
x ₄	(4.237,2.611,0.294)	(3.950,2.058,0.429)	(6.022,2.048,0.433)	(7.223,2.379,0.383)	(8.254,2.697,0.239)	(6.371,2.341,0.355)	(4.685,1.962,0.468)
x ₅	(5.188,2.329,0.360)	(3.689,2.186,0.427)	(5.228,1.758,0.527)	(6.690,2.270,0.353)	(7.983,2.617,0.294)	(6.087,2.115,0.460)	(4.474,2.048,0.433)
x ₆	(5.260,2.491,0.359)	(1.076,2.791,0.220)	(4.752,1.765,0.525)	(5.700,1.932,0.466)	(7.072,2.451,0.353)	(4.773,1.758,0.527)	(3.530,2.231,0.398)
x ₇	(3.920,2.451,0.306)	(3.139,2.356,0.351)	(5.752,1.765,0.525)	(6.983,2.423,0.364)	(7.203,2.374,0.384)	(6.213,2.150,0.437)	(3.189,2.338,0.356)

Step 6. Calculate the conditional probability and the thresholds α₃, β₃, and γ₃ based on Equations (16–18). Compare the values of conditional probability and three thresholds α₃, β₃, and γ₃, deciding which type of three-way decision rules each product belongs to. Table 10 provides the final decision results based on three-way group decisions.

Table 10

Three-way decision results of each new medical care product

X	$\hat{s} (Y (\hat{Pr (C \| [x_{i}])}))$	α₃	β₃	γ ₃	Results
x ₁	3.894	6.186	5.890	6.027	NEG(C)
x ₂	4.708	5.685	4.569	5.360	BND(C)
x ₃	4.097	3.419	3.863	3.726	POS(C)
x ₄	3.018	4.731	5.761	5.214	NEG(C)
x ₅	3.640	5.383	5.166	5.367	NEG(C)
x ₆	3.701	3.850	5.566	4.312	NEG(C)
x ₇	2.777	2.746	7.015	5.071	NEG(C)

According to the final results, investors should invest on new product x₃ for immediate development, further investigate additional market and product information of x₂, and reject investing on the five other new medical care products.

5.3 Comparison analysis

To validate the feasibility of the proposed three-way group-decisions model with linguistic DTRSs based on cloud, a comparison analysis is conducted with other linguistic methods, in which the linguistic information uses the subscript values of the linguistic evaluation information [18]. The analysis is based on the same illustrative example.

We utilize the weight vector of experts w = (0.354, 0.321, 0.325) ^T calculated through our method in subsection 4.1. The calculation results of the conditional probability and the thresholds α₃, β₃, and γ₃ are obtained in Table 11.

Table 11
Values of the conditional probability and the three thresholds for each new product

X $I (\hat{Pr (C | [x_{i}])})$ α₃ β₃ γ ₃ Results

x ₁ 4.207 5.423 5.485 5.444 NEG(C)

x ₂ 4.514 2.772 3.324 2.933 POS(C)

x ₃ 4.248 1.705 3.457 2.344 POS(C)

x ₄ 3.074 4.934 3.320 4.173 NEG(C)

x ₅ 3.983 5.767 4.757 5.367 NEG(C)

x ₆ 4.107 1.630 4.052 2.058 POS(C)

x ₇ 3.455 5.130 4.713 4.940 NEG(C)

X	$I (\hat{Pr (C \| [x_{i}])})$	α₃	β₃	γ ₃	Results
x ₁	4.207	5.423	5.485	5.444	NEG(C)
x ₂	4.514	2.772	3.324	2.933	POS(C)
x ₃	4.248	1.705	3.457	2.344	POS(C)
x ₄	3.074	4.934	3.320	4.173	NEG(C)
x ₅	3.983	5.767	4.757	5.367	NEG(C)
x ₆	4.107	1.630	4.052	2.058	POS(C)
x ₇	3.455	5.130	4.713	4.940	NEG(C)

The final decision results are the same except for new products x₂ and x₆, indicating that our proposed method is feasible. The three-way decision results for x₂ and x₆ are investment using the method proposed in [18]. In our proposed method, the decision selection for x₂ is further investigation and for x₆ is rejection investment. The different results is probably because we utilize the cloud model to translate the linguistic evaluation information to quantitative information, and the model considers the randomness and fuzziness of linguistic information given by decision makers. On the contrary, utilizing the subscript values of linguistic variables can neither describe fuzziness nor randomness of linguistic information to handle three-way decision problems, possibly causing loss of linguistic information. This loss ignores the uncertainty of linguistic information, making decision results radical. Therefore, the proposed model based on cloud can not only overcome these deficiencies but also ensure the practicality and effectiveness of the decision results.

6 Conclusions

We considered the advantage of cloud model in converting qualitative concepts to quantitative ones and the DTRS model in quantifying risk loss in investment decision. In this study, a new three-way group decisions model with cloud-based linguistic DTRSs was proposed to handle problem in medical care product investment decision. We discussed the linguistic DTRS model based on conversion methods between clouds and linguistic variables and proposed corresponding three-way decision rules in Section 3. In addition, a new distance between different clouds based on score function was defined to determine the weights of experts in our proposed model. Finally, a medical care product investment problem was presented to illustrate the validity and feasibility of the model. A comparison analysis was also conducted to validate the effectiveness of our proposed model. This study provides a method for determining thresholds α, β, and γ in linguistic DTRSs from the viewpoint of cloud model, which extends the application range of cloud model. Future research may explore three-way group decision models with uncertain linguistic DTRSs and corresponding three-way decision rules. Methods for weight determination problems can also be discussed in three-way group decisions.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 71371196 and 71210003). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper.

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Three-way linguistic group decisions model based on cloud for medical care product investment

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 DTRSs

Table 1 Loss functions of actions in different states C (P) ¬C (N) a P λ PP λ PN a B λ BP λ BN a N λ NP λ NN

2.3 Conversion between linguistic variables and cloud

3.1 Three-way decisions with the cloud interpretation of the conditional probability

Table 2 Cost loss functions of actions with cloud variables C (P) ¬C (N) a P Y ( λ PP ˆ ) Y ( λ PN ˆ ) a B Y ( λ BP ˆ ) Y ( λ BN ˆ ) a N Y ( λ NP ˆ ) Y ( λ NN ˆ )

4.1 Determination of experts’ weights

5.1 Background

Footnotes

Acknowledgments

References

Table 1
Loss functions of actions in different states

C (P) ¬C (N)

a _P λ _PP λ _PN

a _B λ _BP λ _BN

a _N λ _NP λ _NN

Table 2
Cost loss functions of actions with cloud variables

C (P) ¬C (N)

a _P $Y (\hat{λ_{PP}})$ $Y (\hat{λ_{PN}})$

a _B $Y (\hat{λ_{BP}})$ $Y (\hat{λ_{BN}})$

a _N $Y (\hat{λ_{NP}})$ $Y (\hat{λ_{NN}})$