Multi-attribute group decision-making method based on time-series q-rung orthopair fuzzy sets

Abstract

The q-rung orthopair fuzzy set is a powerful and useful tool to deal with uncertainty, but in actual decision-making process, decision-makers are usually required to analyze the actual problem dynamically. Therefore in this paper, we consider the time-series q-rung orthopair fuzzy decision making. First, we introduce the new cosine similarity measure of q-ROFS which combines the cosine similarity measure and the Euclidean distance measure. Then, we combine the advantages of projection method and grey correlation degree, establishing the nonlinear programming model to calculate the weights of attributes. Furthermore, we use the exponential decay model to get the weights formulas of q-ROFS at different times. Then we replace the distance function with grey relational projection and extend TOPSIS method. Based on these, we propose a new MAGDM approach to deal with time-series q-rung orthopair fuzzy problem not only from the point of view of geometry but also from the point of view of algebra. Finally, we give a practical example to illustrate effectiveness and feasibility of the new method.

Keywords

q-rung orthopair fuzzy set time-series grey correlation degree cosine distance measure

1 Introduction

Decision-making is the process of selecting an optimal decision from a series of alternatives that are evaluated based on multiple attributes. Multiple attribute decision making (MADM) is a tool to help decision makers in the ranking of listed alternatives. MADM theory has been widely studied as an important branch of management science and engineering disciplines, and has been applied to supplier selection, investment plan selection, medical diagnosis and other aspects. However, due to the uncertainty and complexity of the decision-making environment, it is difficult to express actual information with crisp numbers. To overcome this problem, Zadeh [1] proposed the fuzzy set with the help of membership function.

Subsequently, Atanassov [2] proposed the use of intuitionistic fuzzy sets (IFSs) characterized by a membership function, a non-membership function, and a hesitancy function for evaluating an alternative; these functions can express the decision-makers’ satisfaction degree (SD), dissatisfaction degree (DSD), and hesitancy degree (HD), respectively. IFSs are flexible and practical for dealing with ambiguity and uncertainty and widely used in decision-making [3 –7]. For an IFS, the membership degree and nonmembership degree satisfy. However, if a decision-maker provides the orthopair information < 0.8,0.6>, then the sum of the SD and DSD exceeds one. Apparently, the use of IFSs cannot solve this problem. Therefore, Yager expanded the concept of IFS and proposed the concept of Pythagorean fuzzy set (PFS), given by P = < x_i, (μ_P (x_i) , v_P (x_i)) |x_i ∈ X > with $0 ⩽ μ_{P}^{2} (x_{i}) + v_{P}^{2} (x_{i}) ⩽ 1$ . The orthopair < 0.8,0.6 > is a Pythagorean membership grade, not an IFS (since 0 . 8² + 0 .6² < 1). However, if the DSD is 0.7, this information cannot be described as an IFS or a PFS because 0 . 8² + 0 .7² > 1. To solve this problem, Yager [8] and coworkers proposed a new concept of fuzzy set on the basis of the IFS and PFS, namely, the q-rung orthopair fuzzy set (q-ROFS), which can be expressed as Q = < x_i, (μ_Q (x_i), v_Q (x_i)) _q|x_i∈ X >, with $0 ⩽ μ_{Q}^{q} (x_{i}) + v_{Q}^{q} (x_{i}) ⩽ 1$ . When q = 1 and q = 2, the q-ROFS can be Represented as IFS and PFS.

In the above example,<0.8,0.7 > is a q-rung orthopair membership grade. As the degree of membership and non-membership increase, the space of uncertain information becomes larger. In other words, the q-ROFS is more flexible and suitable than the PFS and or IFS for uncertain information. The difference between them is shown in Fig. 1. Accordingly, the decision-maker can freely choose the parameter q to determine the information expression range [9].

Fig. 1

Comparison of the space range between IFNs, PFNs, and q-ROFNs.

Studies on the q-ROFS have mainly focused on the basic theory, extensions of this concept, and MADM methods [10 –17]. Zeng S [18] investigated aggregation operators such as the order weighted average operator and Choquet integral aggregation operator. On the basis of these studies, Wang and Yang [18] developed power Maclaurin symmetric mean operators and Archimedean Bonferroni operators for the q-rung orthopair fuzzy set number (q-ROFN) to solve MADM problems [19, 20]. An example of a study on the basic theory of q-ROFS is that of Jie Gao [21], in which the continuities, derivatives, and differentials of q-ROFN functions were discussed. On the basis of cosine similarity measures, Donghai Liu [22] proposed distance measures between q-ROFSs.

The above studies basically investigated q-ROFSs from a static perspective. In an actual decision-making process, decision-makers are usually required to analyze the actual problem dynamically. Therefore, it is imperative to introduce a time parameter in the q-ROFN and extend it to the time-series q-ROFS [23 –25]. Accordingly, the objective of this study was to develop a method for dynamic MAGDM when information on the q-ROFS is available.

The rest of this paper is ordered as follows: Section 2 briefly reviews basic definitions of q-ROFN, operation laws, and distance measures. Sections 3 and 4 discuss the calculation of attribute weights through the construction of a nonlinear programming model on the basis of the grey correlation degree and a technique for order preference by similarity to ideal solution (TOPSIS) method. Furthermore, these sections describe an investigation of weight formulas of the q-ROFN at different times performed using the exponential decay model. Sections 5 and 6 propose a new MAGDM approach based on time-series q-ROFN and present an analysis of a practical example, performed to verify the reasonability and feasibility of the approach. Finally, Section 7 summarizes the conclusions.

2 Preliminaries

2.1 q-Rung orthopair fuzzy set

Some basic definitions pertaining to the q-ROFS are reviewed in this section. Throughout this paper, X = {x₁, x₂, ⋯ , x_n} is assumed to be a finite, discrete, and nonempty discourse set.

Definition 1. [8] A q-ROFS in a finite universe of discourse X is given by $A = {< x, μ_{A} (x), v_{A} (x) > | x \in X},$ where μ_A (x): X → [0, 1] and v_A (x) : X → [0, 1] represent the membership and nonmembership degrees of x ∈ X, respectively. For each x ∈ X, the condition $0 ⩽ μ_{A}^{q} (x_{i}) + v_{A}^{q} (x_{i}) ⩽ 1$ where q < 1 holds. The hesitancy degree is given by $π_{A} (x) = (μ_{A}^{q} (x_{i}) + v_{A}^{q} (x_{i}) - μ_{A}^{q} (x_{i}) v_{A}^{q} (x_{i}))^{1 / q} .$

Apparently, 0 ⩽ π_A (x) ⩽1 for x ∈ X. If the set X has only one element, then the q-ROFS is reduced to A = μ_A (x) , v_A (x) _q. For convenience, we call A = μ_A (x) , v_A (x) _q a q-ROFN.

Definition 2. [22] Let a₁ = (μ₁, v₁) _q and a₂ = (μ₂, v₂) _q be two q-ROFNs. Then, the following operation laws hold.

a₁ ∨ a₂ = (max {μ₁, μ₂} , min {v₁, v₂})

a₁ ∧ a₂ = (min {μ₁, μ₂} , max {v₁, v₂})

$a_{1} \oplus a_{2} = ((μ_{1}^{q} + μ_{2}^{q} - μ_{1}^{q} μ_{2}^{q})^{1 / q}, v_{1} v_{2})$

$a_{1} \otimes a_{2} = (u_{1} u_{2} (v_{1}^{q} + v_{2}^{q} - v_{1}^{q} v_{2}^{q})^{1 / q})$

$λ a_{1} = (1 - (1 - μ_{1}^{q})^{λ}, v_{1}^{λ}), λ > 0$

$a_{1}^{λ} = (u_{1}^{λ}, (1 - (1 - v_{1}^{q})^{λ}), λ > 0$

Definition 3. [23] Let a = (μ_a, v_a) _q be a q-ROFN. Then, a score function S of a can be defined as $S (a) = μ_{a}^{q} - v_{a}^{q},$ (1) where S (a) ∈ [-1, 1]. The larger the score S (a), the greater the value of a.

Definition 4. [23] Let a = (μ_a, v_a) _q be a q-ROFN. Then, an accuracy function H of a can be defined as $H (a) = μ_{a}^{q} + v_{a}^{q},$ (2) where H (a) ∈ [0, 1]. The larger the score H (a), the greater the value of a.

Lemma 1. [23] Let a₁ = (μ₁, v₁) _q and a₂ = (μ₂, v₂) _q be any two q-ROFNs and let S (a₁) and S (a₂) be their score functions. Then, the following relations hold.

If S (a₁) > S (a₂), then a₁ > a₂.

If S (a₁) > S (a₂), then there are two cases as follows.

If H (a₁) > H (a₂), then a₁ > a₂.

If H (a₁) = H (a₂), then a₁ = a₂.

2.2 Cosine distance measure between q-ROFSs

For accuracy purposes, we used the cosine similarity measures and distance measures between q-ROFNs proposed by Donghai Liu [10], for representing the distance and similarity between variables.

Definition 5. [21] Let X = {x₁, x₂, ⋯ , x_n} be a fixed set and let a₁ = (μ₁ (x_i) , v₁ (x_i)) _q and a₂ = (μ₂ (x_i) , v₂ (x_i)) _q be any two q-ROFNs in X. Then, the cosine similarity measure is given by $\begin{matrix} C_{qROFN} (a_{1}, a_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} \frac{μ_{1}^{q} (x_{i}) μ_{2}^{q} (x_{i}) + v_{1}^{q} (x_{i}) v_{2}^{q} (x_{i})}{\sqrt{μ_{1}^{2 q} (x_{i}) + v_{1}^{2 q} (x_{i})} \sqrt{μ_{2}^{2 q} (x_{i}) + v_{2}^{2 q} (x_{i})}} . \end{matrix}$ (3)

The Euclidean distance measure d_qROFN (a₁, a₂) between any two q-ROFNs can be defined as Definition 6.

Definition 6. [19] Let X = {x₁, x₂, ⋯ , x_n} be a fixed set and let a₁ = (μ₁ (x_i) , v₁ (x_i)) _q and a₂ = (μ₂ (x_i) , v₂ (x_i)) _q be any two q-ROFNs in X. Then, the Euclidean distance measure d_qROFN (a₁, a₂) is defined as

$d_{q R O F N} (a_{1}, a_{2}) = {(\frac{1}{2} \sum_{x_{i} \in X} (| μ_{1}^{q} (x_{i}) - μ_{2}^{q} (x_{i}) |^{2} + | v_{1}^{q} (x_{i}) - v_{2}^{q} (x_{i}) |^{2}))}^{1 / 2} .$ (4)

By combining the aforementioned cosine similarity and Euclidean distance measures, Donghai Liu proposed a new similarity measure (Definition 7) that satisfies the axiom of similarity measure.

Definition 7. [21] Let X = {x₁, x₂, ⋯ , x_n} be a fixed set and let a₁ = (μ₁ (x_i) , v₁ (x_i)) _q and a₂ = (μ₂ (x_i) , v₂ (x_i)) _q be any two q-ROFNs in X, then a new similarity measure M_qROFN (a₁, a₂) can be defined as $\begin{matrix} M_{qROFN} (a_{1}, a_{2}) = \\ \frac{C_{qROFN} (a_{1}, a_{2}) + 1 - d_{qROFN} (a_{1}, a_{2})}{2} . \end{matrix}$ (5)

Definition 8. [21] Let X = {x₁, x₂, ⋯ , x_n} be a fixed set and let a₁ = (μ₁ (x_i) , v₁ (x_i)) _q and a₂ = (μ₂ (x_i) , v₂ (x_i)) _q be any two q-ROFNs in X, then the distance measure D_qROFN (a₁, a₂) is defined as $\begin{matrix} D_{qROFN} (a_{1}, a_{2}) = \\ 1 - M_{qROFN} (a_{1}, a_{2}) \\ = \frac{1 - C_{qROFN} (a_{1}, a_{2}) + d_{qROFN} (a_{1}, a_{2})}{2}, \end{matrix}$ (6)

Where $\begin{matrix} C_{qROFN} (a_{1}, a_{2}) = \\ \frac{1}{n} \sum_{i = 1}^{n} \frac{μ_{1}^{q} (x_{i}) μ_{2}^{q} (x_{i}) + v_{1}^{q} (x_{i}) v_{2}^{q} (x_{i})}{\sqrt{μ_{1}^{2 q} (x_{i}) + v_{1}^{2 q} (x_{i})} \sqrt{μ_{2}^{2 q} (x_{i}) + v_{2}^{2 q} (x_{i})}} \end{matrix}$ and

$d_{q R O F N} (a_{1}, a_{2}) = {(\frac{1}{2} \sum_{x_{i} \in X} (| μ_{1}^{q} (x_{i}) - μ_{2}^{q} (x_{i}) |^{2} + | v_{1}^{q} (x_{i}) - v_{2}^{q} (x_{i}) |^{2}))}^{1 / 2} .$

3 Dynamic MAGDM method based on TOPSIS method and grey correlation degree

In this section, we propose a new method for dynamic MAGDM with the evaluation value are q-ROFNs and the weights of attributes and the weights of the time series are unknown.

Let S (X, C, T) be a dynamic q-ROFN MAGDM system, X = {x₁, x₂, ⋯ , x_m} be a finite set of alternatives, C = {c₁, c₂, ⋯ , c_n} be a set of attributes, and T = {t₁, t₂, ⋯ , t_p} be a set of time series. Let W = {w₁, w₂, ⋯ , w_n} be the weight vector of the attributes, with $\sum_{j = 1}^{n} w_{j} = 1$ , and let V = {v₁, v₂, ⋯ , v_p} be the vector of the time series. The evaluation information A = (a_ij (t_k)) _m×n = ((μ_ij (t_k) , v_ij (t_k))) _m×n over the alternative X_i with respect to the attribute C_j for the period t_k can be expressed as a q-ROFN (μ_ij (t_k) , v_ij (t_k)), where i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, t = 1, 2, ⋯ , p. The parameter a_ij (t_k) denotes the degree of agreement between X_i and C_j, with a_ij (t_k) = (μ_ij (t_k) , v_ij (t_k)) during the period t_k.

Initially, we normalize the evaluation information since the decision matrix might contain benefit and cost attribute. The values of the two attribute have opposite indications: a larger value of the benefit attribute (cost criterion) indicates better (worse) performance. Therefore, to ensure that all attributes are compatible, the following formula was used for converting cost standards into benefit attributes: ${\bar{a}}_{ij} (t_{k}) = {\begin{matrix} \begin{matrix} a_{ij} (t_{k}), & C_{j} \begin{matrix} \begin{matrix} is & a \end{matrix} & benefit & attribute, \end{matrix} \end{matrix} \\ \begin{matrix} 1 - a_{ij} (t_{k}), & C_{j} \begin{matrix} \begin{matrix} is & a \end{matrix} & a cos t & attribute . \end{matrix} \end{matrix} \end{matrix}$ (7)

This equation can be used to obtain the normalized q-ROFN matrix $\bar{A} (t_{k}) = {({\bar{a}}_{ij} (t_{k}))}_{m \times n}$ . For convenience, we continue to represent the matrix as A (t_k) = (a_ij (t_k)) _m×n.

3.1 Model of dynamic MAGDM

Under q-ROFN information, some researchers have improved the TOPSIS method and proposed a new projection algorithm [25 –29]. It is well known that a projection algorithm can determine the proximity of data sequences at the macro level [30 –35]. However, it has defects in reflecting the shape similarity between the data curves. Therefore, to overcome this deficiency, we introduce the grey relation degree, which can precisely reflect the data curve geometry similarity. Based on these, we integrate TOPSIS and grey relation degree to develop a new type of relative closeness that can be used as a standard for evaluating a scheme.

3.1.1 Decision-making model

We first eliminate the impact of different dimensions on the outcome of the decision and then obtain the normalized q-ROFN matrix A (t_k) = (a_ij (t_k)) _m×n.

Next, at time t_k, for all alternatives with identical attributes, we can obtain the positive ideal solution $A^{+} (t_{k}) = {A_{1}^{+} (t_{k}), A_{2}^{+} (t_{k}), \dots, A_{n}^{+} (t_{k})}$ and the negative ideal solution $A^{-} (t_{k}) = {A_{1}^{-} (t_{k}), A_{2}^{-} (t_{k}), \dots, A_{n}^{-} (t_{k})}$ , which are determined by the score and accuracy functions defined by Definitions 3 and 4: $A_{j}^{+} (t_{k}) = (r_{1 j}^{+} (t_{k}), r_{2 j}^{+} (t_{k}), \dots, r_{nj}^{+} (t_{k})),$ (8) $A_{j}^{-} (t_{k}) = (r_{1 j}^{-} (t_{k}), r_{2 j}^{-} (t_{k}), \dots, r_{nj}^{-} (t_{k})),$ (9) where $r_{ij}^{+} (t_{k}) = max_{j} {S (r_{ij} (t_{k}))}$ and $r_{ij}^{-} (t_{k}) = min_{j} {S (r_{ij} (t_{k}))} .$

If $S (r_{1 j} (t_{k})) = S (r_{2 j} (t_{k})) = \dots = S (r_{nj} (t_{k}))$ , then we can compare the values of the accuracy scores $H (r_{ij} (t_{k}))$ .

We calculate the similarity measure between the alternative set and the ideal solutions in different time periods t_k using Definition 7 then obtain the similarity measure matrices $M^{+} (t_{k}) = M (m_{ij}^{+} (t_{k}))$ , $M^{-} (t_{k}) = M (m_{ij}^{-} (t_{k}))$ . Further, using Definition 8, we acquire the distance measure matrices $D^{+} (t_{k}) = D (d_{ij}^{+} (t_{k}))$ , $D^{-} (t_{k}) = D (d_{ij}^{-} (t_{k}))$ .

Using the distance measure matrix $D^{+} (t_{k}) = D (d_{ij}^{+} (t_{k}))$ , we calculate the grey correlation coefficient of each period $r_{ij}^{+} (t_{k})$ , that is, $r_{ij}^{+} (t_{k}) = \frac{d_{min}^{+} (t_{k}) + ζ d_{max}^{+} (t_{k})}{d_{ij}^{+} (t_{k}) + ζ d_{max}^{+} (t_{k})}, ζ \in [0, 1],$ (10) where ζ is the distinguishing coefficient (generally considered to be 0.5), $d_{min}^{+} (t_{k})$ is the minimum value of $d_{ij}^{+} (t_{k})$ , and $d_{max}^{+} (t_{k})$ is the maximum value of $d_{ij}^{+} (t_{k})$ . Using the grey correlation coefficients, we can obtain the grey correlation matrices $R^{+} (t_{k}) = (r_{ij}^{+} (t_{k}))$ and $R^{-} (t_{k}) = (r_{ij}^{-} (t_{k}))$ . R⁺ (t_k) and R^- (t_k) are then weighted to obtain the weighted grey correlation matrices ${\bar{R}}^{+} (t_{k}) = (ω_{j} r_{ij}^{+} (t_{k}))$ and ${\bar{R}}^{-} (t_{k}) = (ω_{j} r_{ij}^{-} (t_{k}))$ . If each alternative is regarded as a row vector, the cosine of the angle between an alternative and the positive ideal solution is $cos θ^{+} = \frac{\sum_{j = 1}^{m} ω_{j} r_{ij}^{+} (t_{k}) ω_{j}}{\sqrt{\sum_{j = 1}^{n} ω_{j}^{2}} \sqrt{\sum_{j = 1}^{n} {[ω_{j} r_{ij}^{+} (t_{k})]}^{2}}},$ (11) with the norm of alternative A_i being $| A_{i} | = \sqrt{\sum_{j = 1}^{n} {[ω_{j} r_{ij}^{+} (t_{k})]}^{2}}$ .

We can now calculate the projection value ${Pr}_{i}^{+} (t_{i})$ of A_i on $A_{i}^{+}$ : ${Pr}_{i}^{+} (t_{k}) = | A_{i} | cos θ = \frac{\sum_{j = 1}^{n} ω_{j}^{2} r_{ij}^{+} (t_{k})}{\sqrt{\sum_{j = 1}^{n} ω_{j}^{2}}} = \sum_{j = 1}^{n} {\bar{ω}}_{j} r_{ij}^{+} (t_{k}),$ (12) where $\bar{ω} = ω_{j}^{2} / \sqrt{\sum_{j = 1}^{n} ω_{j}^{2}}$ . Analogously, the project value ${Pr}_{i}^{-} (t_{i})$ of A_i on $A_{i}^{-}$ is obtained as ${Pr}_{i}^{-} (t_{k}) = | A_{i} | cos θ = \frac{\sum_{j = 1}^{n} ω_{j}^{2} r_{ij}^{-} (t_{k})}{\sqrt{\sum_{j = 1}^{n} ω_{j}^{2}}} = \sum_{j = 1}^{n} {\bar{ω}}_{j} r_{ij}^{-} (t_{k}) .$ (13)

With these values, we can calculate the closeness index ρ_i (t_k) of alternatives A_i: $ρ_{i} (t_{k}) = \frac{{Pr}_{i}^{+} (t_{k})}{{Pr}_{i}^{+} (t_{k}) + {Pr}_{i}^{-} (t_{k})} .$

Evidently, at time t_k, the smaller the value of the closeness coefficient ρ_i (t_k), the better the alternative A_i.

3.1.2 Calculation of weights of attributes

The weights of the attributes are calculated by constructing a nonlinear programming model for the smallest total deviation of grey correlation degree. At time t_k, the deviation of the grey correlation between the alternative A_i and positive ideal solution $A_{i}^{+}$ is $(1 - r_{ij}^{+} (t_{k}))$ . To eliminate the effects of positive and negative signs, we calculate the sum of squares of deviations, namely, $\sum_{k = 1}^{p} \sum_{j = 1}^{m} (1 - r_{ij}^{+} (t_{k}) ω_{j})^{2},$ (14) where ω_j denotes the weights of attributes and $ω_{j} ⩾ 0, \sum_{j = 1}^{m} ω_{j} = 1$ .

Here is no preference relationship between the alternatives. Therefore, the sum of the weighted biases of the grey correlations for all the schemes is $C = \sum_{i = 1}^{n} \sum_{k = 1}^{p} \sum_{j = 1}^{m} (1 - r_{ij}^{+} (t_{k}) ω_{j})^{2} .$ (15)

The following model can then be established: $\begin{matrix} min C = \sum_{i = 1}^{n} \sum_{k = 1}^{p} \sum_{j = 1}^{m} (1 - r_{ij}^{+} (t_{k}) ω_{j})^{2} \\ s . t {\begin{matrix} \sum_{j = 1}^{m} ω_{j} = 1 \\ 0 ⩽ ω_{j} ⩽ 1 \end{matrix} . \end{matrix}$ (16)

We also construct the Lagrange function

$L (ω_{j}, λ) = \sum_{i = 1}^{n} \sum_{k = 1}^{p} \sum_{j = 1}^{m} (1 - r_{ij}^{+} (t_{k}) ω_{j})^{2} + 2 λ (\sum_{j = 1}^{m} ω_{j} - 1) .$ (17)

For the conditions under which this function attains the extreme value, we calculate the partial derivative of the function:

${\begin{matrix} \frac{\partial L}{\partial ω_{j}} = 2 \sum_{i = 1}^{n} \sum_{k = 1}^{p} [(1 - r_{ij}^{+} (t_{k}))^{2} ω_{j} + (1 - r_{ij}^{+} (t_{k}))] + 2 λ = 0 \\ \frac{\partial L}{\partial λ} = \sum_{j = 1}^{m} ω_{j} - 1 = 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} .$ (18)

We obtain the solution

${\begin{matrix} λ = - {[\sum_{j = 1}^{m} {(\sum_{i = 1}^{n} \sum_{k = 1}^{p} (1 - r_{ij}^{+} (t_{k}))^{2})}^{- 1}]}^{- 1} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \\ ω_{j} = {[\sum_{j = 1}^{m} {(\sum_{i = 1}^{n} \sum_{k = 1}^{p} (1 - r_{ij}^{+} (t_{k}))^{2})}^{- 1}]}^{- 1} \times {(\sum_{i = 1}^{n} \sum_{k = 1}^{p} (1 - r_{ij}^{+} (t_{k}))^{2})}^{- 1} . \end{matrix}$ (19)

Subsequently, the weighted projections are obtained.

3.1.3 Calculation of weights of time series

Because the decision-making problem is dynamic, we should consider the weights of time series. Sometimes, the decision-makers will obtain an increasing amount of information over time, and the impact of this acquisition on the final decision-making result will be significant. Many studies have shown that the temporal change in the weights of a time series can be described by an exponential decay model. Therefore, we use an exponential decay model to determine the weights of the time series.

Let T = {t₁, t₂, ⋯ , t_p} be a set of time series and v_k be the weight of time t_k. Then, we can write $\begin{matrix} v_{k} {= C}_{0} e^{λ (t_{k} - t_{p})}, \\ k = 1, 2, \dots, p, \end{matrix}$ (20) where C₀ is a constant and λ (0 < λ < 1) is the attenuation coefficient.

The weights should satisfy $\sum_{k = 1}^{p} v_{k} = \sum_{k = 1}^{p} C_{0} e^{λ (t_{k} - t_{p})} = 1 .$

From this expression, we have $C_{0} = \frac{1}{\sum_{k = 1}^{p} e^{λ (t_{k} - t_{p})}},$ (21) and hence $\begin{matrix} v_{k} = \frac{e^{λ (t_{k} - t_{p})}}{\sum_{k = 1}^{p} e^{λ (t_{k} - t_{p})}}, \\ k = 1, 2, \dots, p . \end{matrix}$ (22)

In a real dynamic MADM problem, λ reflects the cumulative rate of change in the information held by the decision-maker with time, and therefore the decision-maker can determine the specific value according to the actual situation.

Finally, the grey correlation projection values of each alternative in each period are weighted and integrated to obtain the comprehensive grey correlation projection values z_i: $z_{i} = \sum_{t = 1}^{p} ρ_{i} (t_{k}) \cdot v_{k} .$

The larger the value of z_i, the better the alternative A_i.

3.2 Steps in dynamic MAGDM

Dynamic MAGDM comprises the following steps(see Fig. 2):

Fig. 2

The steps of in Dynamic MAGDM.

Step 1. Construct the q-ROFN decision matrix A = (a_ij (t_k)) _m×n = ((μ_ij (t_k) , v_ij (t_k))) _m×n for a different time and normalize it. If the attribute is of benefit type, no action is required. If it is of cost type, then the cost type should be converted to benefit type by following the function (7) We then obtain λ (t_k) = (r_ij (t_k)) _m×n.

Step 2. Obtain the positive ideal solution $A^{+} (t_{k}) = {A_{1}^{+} (t_{k}), A_{2}^{+} (t_{k}), \dots, A_{n}^{+} (t_{k})}$ and the negative ideal solution $A^{-} (t_{k}) = {A_{1}^{-} (t_{k}), A_{2}^{-} (t_{k}), \dots, A_{n}^{-} (t_{k})}$ for all alternatives under the same attributes in period t_k.

Step 3. Calculate the similarity measure between the alternative set and the ideal solutions in different time periods t_k and obtain the similarity measure matrices $M^{+} (t_{k}) = M (m_{ij}^{+} (t_{k}))$ and $M^{-} (t_{k}) = M (m_{ij}^{-} (t_{k})) .$

Next, acquire the distance measure matrices $D^{+} (t_{k}) = D (d_{ij}^{+} (t_{k}))$ and $D^{-} (t_{k}) = D (d_{ij}^{-} (t_{k}))$ .

Step4. According to the grey correlation coefficient of each period, obtain the grey correlation matrices R⁺ (t_k), R^- (t_k) and the weighted grey correlation matrices ${\bar{R}}^{+} (t_{k}) = (ω_{j} r_{ij}^{+} (t_{k}))$ and ${\bar{R}}^{-} (t_{k}) = (ω_{j} r_{ij}^{-} (t_{k}))$ .

Step5. Calculate the weight vector W = (ω₁, ω₂, ⋯ , ω_m) by using the Equation (19), and then calculate the project values ${Pr}_{i}^{+} (t_{i})$ , ${Pr}_{i}^{-} (t_{i})$ and the closeness index ρ_i (t_k).

Step6. Calculate the weight vector of the time, V = {v₁, , ⋯ , v_p}.

Step7. Rank the alternatives by the weighted values of the closeness coefficient, $z_{i} = \sum_{t = 1}^{p} ρ_{i} (t_{k}) \cdot v_{k}$ .

4 Application example

To show the application of the proposed method to MADM, we consider a practical decision-making example—university ranking to simulate. Currently, universities pay considerable attention to their ranking because it significantly affects government investment, social donations, recruitment, etc. In China, there are many systems to rank the universities. Here we give four more influential and authoritative evaluation systems A₁, A₂, A₃, and A₄. To evaluate the stability of these four evaluation systems, we obtained experts’ opinions. Eventually we chose four attributes, namely, stability of the indicator system (C1), stability of weight setting (C2), correlation of historical ranking data (C3), and consistency of data trends (C4). On the basis of the data of the four evaluation systems for 2017, 2018, and 2019, the experts provided an evaluation matrix of an alternative A_i with respect to attributes C_i, where the value of the q-ROFN, the weight of each attribute, and the timing weight were unknown.

Step 1. Normalize the q-ROFN decision-making matrices. Because the attributes are of benefit type, transform them into normalized q-ROFN decision-making matrices;

Step 2. Obtain the positive ideal solution $A_{i}^{+} = (A_{i 1}^{+}, A_{i 2}^{+}, A_{i 3}^{+}, A_{i 4}^{+})$ , where i = 1, 2, 3, and the negative ideal solution $A_{i}^{-} = (A_{i 1}^{-}, A_{i 2}^{-}, A_{i 3}^{-}, A_{i 4}^{-})$ , where i = 1, 2, 3: $A_{1}^{+} = ((0.7, 0.1), (0.6, 0.3), (0.6, 0.2), (0.4, 0.3)),$ $A_{2}^{+} = ((0.7, 0.1), (0.6, 0.3), (0.6, 0.2), (0.4, 0.3)),$ $A_{3}^{+} = ((0.8, 0.1), (0.7, 0.2), (0.5, 0.2), (0.6, 0.1)),$ $A_{1}^{-} = ((0.4, 0.3), (0.5, 0.4), (0.4, 0.5), (0.3, 0.4)),$ $A_{2}^{-} = ((0.4, 0.3), (0.4, 0.2), (0.4, 0.4), (0.3, 0.3)),$ $A_{3}^{-} = ((0.5, 0.2), (0.5, 0.3), (0.3, 0.2), (0.5, 0.4)) .$

Steps 3–5. Calculate the cosine similarity measure and Euclidean distance measure and then obtain the similarity measures $M^{+} (t_{k}) = M (m_{ij}^{+} (t_{k}))$ and $M^{-} (t_{k}) = M (m_{ij}^{-} (t_{k}))$ . Next, acquire the distance measure matrices $D^{+} (t_{k}) = D (d_{ij}^{+} (t_{k}))$ and $D^{-} (t_{k}) = D (d_{ij}^{-} (t_{k}))$ . On the basis of the grey correlation coefficient of each period, obtain the grey correlation matrices R⁺ (t_k), R^- (t_k) and the weighted grey correlation matrices ${\bar{R}}^{+} (t_{k}) = (ω_{j} r_{ij}^{+} (t_{k}))$ and ${\bar{R}}^{-} (t_{k}) = (ω_{j} r_{ij}^{-} (t_{k}))$ . Use R⁺ (t_k) and obtain the weight vector of the attributes, which is obtained as ω = (0.210, 0.222, 0.316, 0.252).

Step 6. Calculate the weight of the time, which is obtained as V = (0.186, 0.308, 0.506).

To consider the influence of parameter q on the decision ranking result, we calculate the results for different q-values; the results are presented in Table 4 and Fig. 3.

Table 1
q-ROFN decision-making matrix for 2017

C ₁ C ₂ C ₃ C ₄

A ₁ (0.5,0.2)_q (0.4,0.2)_q (0.5,0.4)_q (0.3,0.3)_q

A ₂ (0.7,0.1)_q (0.6,0.3)_q (0.4,0.3)_q (0.4,0.4)_q

A ₃ (0.7,0.3)_q (0.5,0.4)_q (0.4,0.5)_q (0.3,0.4)_q

A ₄ (0.4,0.2)_q (0.5,0.3)_q (0.6,0.2)_q (0.4,0.3)_q

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.5,0.2)_q	(0.4,0.2)_q	(0.5,0.4)_q	(0.3,0.3)_q
A ₂	(0.7,0.1)_q	(0.6,0.3)_q	(0.4,0.3)_q	(0.4,0.4)_q
A ₃	(0.7,0.3)_q	(0.5,0.4)_q	(0.4,0.5)_q	(0.3,0.4)_q
A ₄	(0.4,0.2)_q	(0.5,0.3)_q	(0.6,0.2)_q	(0.4,0.3)_q

Table 2

q-ROFN decision-making matrix for 2018

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.6,0.2)_q	(0.4,0.2)_q	(0.5,0.4)_q	(0.4,0.2)_q
A ₂	(0.7,0.1)_q	(0.6,0.1)_q	(0.5,0.2)_q	(0.4,0.3)_q
A ₃	(0.6,0.2)_q	(0.5,0.2)_q	(0.4,0.4)_q	(0.3,0.4)_q
A ₄	(0.4,0.3)_q	(0.5,0.3)_q	(0.6,0.2)_q	(0.4,0.2)_q

Table 3

q-ROFN decision-making matrix for 2019

	C ₁	C ₂	C ₃	C ₄
A ₁	(0.7,0.2)_q	(0.5,0.3)_q	(0.5,0.3)_q	(0.3,0.2)_q
A ₂	(0.6,0.2)_q	(0.7,0.2)_q	(0.5,0.2)_q	(0.5,0.4)_q
A ₃	(0.8,0.1)_q	(0.6,0.2)_q	(0.6,0.2)_q	(0.3,0.5)_q
A ₄	(0.5,0.2)_q	(0.7,0.1)_q	(0.5,0.3)_q	(0.5,0.2)_q

Table 4

Decision results based on closeness coefficient Z for different q-values

	Z ₁	Z ₂	Z ₃	Z ₄	Ranking
q=1	0.505	0.493	0.496	0.517	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=2	0.533	0.489	0.486	0.558	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=3	0.569	0.498	0.474	0.601	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=4	0.603	0.511	0.463	0.638	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=5	0.631	0.527	0.456	0.667	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=6	0.653	0.540	0.450	0.687	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=7	0.669	0.552	0.445	0.702	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=8	0.680	0.561	0.441	0.713	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=9	0.688	0.569	0.439	0.721	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=10	0.694	0.575	0.437	0.726	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=20	0.708	0.591	0.432	0.738	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=30	0.710	0.592	0.431	0.739	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=40	0.710	0.593	0.431	0.739	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=50	0.710	0.593	0.431	0.739	A₂ ≻ A₃ ≻ A₁ ≻ A₄
q=100	0.710	0.593	0.431	0.739	A₂ ≻ A₃ ≻ A₁ ≻ A₄

Fig. 3

Curves of the attributes for different q-values.

When q = 1, we calculate the projective values ${Pr}_{i}^{+} (t_{i})$ , ${Pr}_{i}^{-} (t_{i})$ and then obtain the closeness index ρ_i (t_k). The alternatives are ranked according to the weighted closeness index values, namely, z₁ = 0.505, z₂ = 0.493, z₃ = 0.496, and z₄ = 0.517. The ranking result is A₂ ≻ A₃ ≻ A₁ ≻ A₄. Thus, the most stable university evaluation system is A₂. To verify the feasibility and effectiveness of the proposed method, we simulated the example for q = 1 to q = 100. The results obtained are presented in Table 4.

Clearly, the order of the results is identical for different q-values. Furthermore, the curve of each alternative becomes smooth after q = 10. It is known that when the q-ROFN is an intuitionistic fuzzy number, for q = 2, the q-ROFN is a Pythagorean fuzzy number. The space of acceptable orthopairs becomes larger as the rung q increases.

Since the q-ROFN has a wider range than the IFS and PFS, it can provide decision-makers more freedom to express their belief about the membership degree.

Our method extends the existing method by not only considering new cosine similarity and distance measures but also using projection and the grey correlation degree. In other words, our method combines the relevance and location of the attributes from both algebraic and geometric viewpoints.

5 Conclusion and future work

In real group decision making environment and in process of dynamic decision-making, the value of attributes is expressed as q-ROFNs. To solve the dynamic group decision-making based on q-ROFNs, there are many DM methods discussed [36 –43]. This paper mainly pertains to time-series q-rung orthopair fuzzy decision-making. We first introduce a new cosine similarity measure for the q-ROFN; the measure was devised using a cosine similarity measure and a Euclidean distance measure. We then present a nonlinear programming model based on a TOPSIS method and the grey correlation degree to calculate the weights of attributes. Furthermore, on the basis of an exponential-decay-model-based investigation of the weight formulas of the q-ROFN for different times, we propose a new MAGDM approach based on the extended TOPSIS method and grey relational projection. Finally, a numerical example is presented to show the feasibility and effectiveness of the method.

This study considered group MADM, extended an q-ROFN method based on time-series q-rung orthopair fuzzy sets, investigated the application of the traditional MADM method, and realized the virtuous circle of theory guiding practice and practice testing theory. In the future research work, we extend the beneficial model for q-ROFN MADM problems which decision attributes are correlated. We use the fuzzy measure to develop some new aggregation operators and propose some new decision making technique for dynamic decision making.

Footnotes

Acknowledgments

This work was supported by the Doctoral Foundation of China (2017M622169) and the Young Scholars “Future Plan” of Shandong University (2017WLJH17).

References

Zadeh

L.A.

, Fuzzy sets, Information Control 8 (1965), 338–353.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

Liu

, Chen

and Peng

, Interval-valued intuitionistic fuzzy ordered weighted cosine similarity measure and its application in investment decision-making, Complexity 1 (2017), 1–11.

and Yager

R.R.

, Some geometric aggregation operators based on intuitionistic fuzzy sets, Int J Gen Syst 35 (2006), 417–433.

Zhu

, Topological approaches to covering rough sets, Information Sciences 177 (2007), 1499–1508.

, Intuitionistic fuzzy aggregation operators, EEE Transactions on Fuzzy Systems 15 (2007), 1179–1187.

Yager

R.R.

, Pythagorean Membership Grades in Multicriteria Decision Making, EEE Transactions on Fuzzy Systems 22 (2014), 958–965.

Yager

R.R.

, Pythagorean membership grades in multicriteria decision making, IEEE Trans Fuzzy Syst 22 (2014), 958–965.

Yager

R.R.

, Generalized orthopair fuzzy sets, IEEE Trans Fuzzy Syst 25 (2017), 1222–1230.

10.

Ali

M.I.

, Another view on q-rung orthopair fuzzy sets, Int J Intell Syst 33 (2018), 2139–2153.

11.

Garg

, Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process, Int J Intell Syst 32 (2017), 597–630.

12.

Wei

G.W.

and Lu

, Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making, Int J Intell Syst 33 (2018), 1043–1070.

13.

Zeng

S.Z.

, Chen

J.P.

and LI

X.S.

, A hybrid method for Pythagorean fuzzy multiple-criteria decision making, Int J Inf Tech nol Decis 14 (2015), 1–20.

14.

Wei

G.W.

, Gao

and Wei

, Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making, Int J Intell Syst 33 (2018), 1426–1458.

15.

Z.M.

and Xu

Z.S.

, Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems, Int J Intell Syst 31 (2016), 1198–1219.

16.

Peng

X.D.

and Yang

, Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators, Int J Intell Syst 31 (2016), 444–487.

17.

Yager

R.R.

, Pythagorean membership grades in multicriteria decision making, IEEE Trans Fuzzy Syst 22 (2014), 958–965.

18.

Zeng

, Pythagorean Fuzzy Multiattribute Group Decision Making with Probabilistic Information and OWA Approach, International Journal of Intelligent Systems 32 (2017), 1136–1150.

19.

Wang

, Wei

and Wei

, Dual Hesitant q-Rung Orthopair Fuzzy Muirhead Mean Operators in Multiple Attribute Decision Making, IEEE Access 7 (2019), 67139–67166.

20.

Yang

and Pang

, New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making, Int J Intell Syst 34 (2019), 439–476.

21.

Gao

, Liang

, Shang

and Xu

, Continuities, Derivatives, and Differentials of q-Rung Orthopair Fuzzy Functions, IEEE Transactions on Fuzzy Systems 27 (2019), 1687–1620.

22.

Liu

, Chen

and Peng

, Some cosine similarity measures and distance measures between q-Rung orthopair fuzzy sets, Int J Intell Syst 34 (2019), 1572–1587.

23.

Liu

P.D.

and Liu

, Some q-Rung orthopair fuzzy Bonferroni mean operators and their application to multi-attribute group decision making, Int J Intell Syst 33 (2018), 315–347.

24.

Shih

H.-S.

, Shyur

H.-J.

and Lee

E.S.

, An extension of TOPSIS for group decision making, Math Comput Model 45 (2007), 801–813.

25.

Hussaina

, Irfan Alib

and Mahmood

, Covering based q-rung orthopair fuzzy rough set model hybrid with TOPSIS for multi-attribute decision making, Journal of Intelligent & Fuzzy Systems 37 (2019), 981–993.

26.

Zhang

, Dong

and Herrera-Viedma

, Group Decision Making with Heterogeneous Preference Structures: An Automatic Mechanism to Support Consensus Reaching, Group Decision and Negotiation 28 (2019), 585–617.

27.

Zhang

and Xu

, Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets, International Journal of Intelligent Systems 29 (2014), 1061–1078.

28.

Lourenzutti

and Krohling

R.A.

, A generalized TOPSIS method for group decision making with heterogeneous information in a dynamic environment, Information Sciences 330 (2016), 1–18.

29.

Zhang

and Xu

, Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets, International Journal of Intelligent Systems 29 (2014), 1061–1078.

30.

Zaoli

, Xin

and Zehong

, Q-rung orthopair normal fuzzy aggregation operators and their application in multi-attribute decision-making, Mathematics 7 (2019), 1142–1168.

31.

Zhang

, A Novel Approach Based on Similarity Measure for Pythagorean Fuzzy Multiple Criteria Group Decision Making, International Journal of Intelligent Systems 31 (2016), 593–611.

32.

Goodarzian

and Hosseini-Nasab

, Applying a fuzzy multi-objective model for a production-distribution network design problem by using a novel self-adoptive evolutionary algorithm, International Journal of Systems Science: Operations & Logistics 8 (2019), 1–22.

33.

Peng

H.-G.

and Wang

J.-Q.

, A Multicriteria Group Decision-Making Method Based on the Normal Cloud Model With Zadeh’s Z-Numbers, EEE Transactions on Fuzzy Systems 26 (2018), 3246–3260.

34.

Zhou

, Su

, Balezentis

and Streimikiene

, Multiple criteria group decision-making considering symmetry with regards to the positive and negative ideal solutions via the Pythagorean normal cloud model for application to economic decisions, Symmetry 10 (2018), 140–155.

35.

Peng

, Neutrosophic Reducible Weighted Maclaurin Symmetric Mean for Undergraduate Teaching Audit and Evaluation, IEEE Access 7 (2019), 18634–18648.

36.

Peng

, Algorithm for Pythagorean Fuzzy Multi-criteria Decision Making Based on WDBA with New Score Function, Fundamenta Informaticae 165 (2019), 99–137.

37.

Nie

R.-X.

, Tian

Z.-P.

, Wang

J.-Q.

and Hu

J.-H.

, Pythagorean fuzzy multiple criteria decision analysis based on Shapley fuzzy measures and partitioned normalized weighted Bonferroni mean operator, International Journal of Intelligent Systems 34 (2019), 297–324.

38.

, Zhou

, Chen

and Tao

, Additive Consistency of Hesitant Fuzzy Linguistic Preference Relation With a New Expansion Principle for Hesitant Fuzzy Linguistic Term Sets, EEE Transactions on Fuzzy Systems 27 (2019), 716–730.

39.

Liu

, Jin

and Zhu

, A Multi-Criteria Group Decision Making Model for Green Supplier Selection under the Ordered Weighted Hesitant Fuzzy Environment, Symmetry-Basel 11 (2019), 17–33.

40.

Yager

R.R.

, Alajlan

and Bazi

, Aspects of generalized orthopair fuzzy sets, International Journal of Intelligent Systems 33 (2018), 2154–2174.

41.

Liu

and Liu

, Some q-Rung Orthopai Fuzzy Bonferroni Mean Operators and Their Application to Multi-Attribute Group Decision Making, International Journal of Intelligent Systems 33 (2018), 315–347.

42.

Peng

, Neutrosophic Reducible Weighted Maclaurin Symmetric Mean for Undergraduate Teaching Audit and Evaluation, Access 7 (2019), 18634–18648.

43.

Muhammad Jabir

, Poom

and Meshal

, Knowledge measure for the q-rung orthopair fuzzy sets, International Journal of Intelligent Systems 36 (2021), 628–655.

Multi-attribute group decision-making method based on time-series q-rung orthopair fuzzy sets

Abstract

Keywords

1 Introduction

2.1 q-Rung orthopair fuzzy set

3.1.1 Decision-making model

Table 1 q-ROFN decision-making matrix for 2017 C 1 C 2 C 3 C 4 A 1 (0.5,0.2)q (0.4,0.2)q (0.5,0.4)q (0.3,0.3)q A 2 (0.7,0.1)q (0.6,0.3)q (0.4,0.3)q (0.4,0.4)q A 3 (0.7,0.3)q (0.5,0.4)q (0.4,0.5)q (0.3,0.4)q A 4 (0.4,0.2)q (0.5,0.3)q (0.6,0.2)q (0.4,0.3)q

Footnotes

Acknowledgments

References

Table 1
q-ROFN decision-making matrix for 2017

C ₁ C ₂ C ₃ C ₄

A ₁ (0.5,0.2)_q (0.4,0.2)_q (0.5,0.4)_q (0.3,0.3)_q

A ₂ (0.7,0.1)_q (0.6,0.3)_q (0.4,0.3)_q (0.4,0.4)_q

A ₃ (0.7,0.3)_q (0.5,0.4)_q (0.4,0.5)_q (0.3,0.4)_q

A ₄ (0.4,0.2)_q (0.5,0.3)_q (0.6,0.2)_q (0.4,0.3)_q