Dynamic intuitionistic fuzzy multiattribute decision making based on evidential reasoning and MDIFWG operator

Abstract

The present work is focused on dynamic intuitionistic fuzzy multi-attribute decision making (DIF-MADM) problem, while "dynamic" means the decision-related information may be collected at different periods, a situation commonly happened in many of real world MADM problems. After the review and analysis of some drawbacks on the existing DIF-MADM methods, on the one hand, we propose a new DIF-MADM method based on the evidential reasoning (ER) algorithm in order to address some of those limits; on the other hand, and a new dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator is introduced, named modified dynamic intuitionistic fuzzy weighted geometric (MDIFWG) operator, then a MDIFWG-based DIF-MADM method is also proposed to address some other limits of the existing methods. Some numerical examples are provided to illustrate the practicality and feasibility of the proposed two methods through, the comparative analysis with the existing DIF-MADM methods, along with some sensitivity analyses also carried out to analyse the distinct features of the proposed methods.

Keywords

DIF-MADM MDIFWG operator Evidential reasoning algorithm

1. Introduction

The aim of decision making is to achieve some goal(s) through a series of decisions. In many situations, the alternatives among which one must choose or make decision need to be evaluated in light of multiple criteria. But due to the various constraints in day-today life, decision makers may give their judgements under the uncertain and imprecise in nature. Thus, there always exists a degree of hesitancy between the preferences of the decision making and hence, the analysis conducted under such circumstances is not ideal and hence does not tell the exact information to the system analyst. As an important extension of fuzzy set, Intuitionistic Fuzzy Set (IFS) [1] is characterized by three parameters at the same time, namely, a membership degree (MD), a nonmembership degree (NMD) and an indeterminacy degree (ID) are adopted at the same time. Therefore, IFS is considered to be more appropriate to represent and deal with imprecise, uncertain and vague information in some decision making problems (DMPs). In last few years, some fuzzy multi-attribute decision making (MADM) methods based on IFS have been proposed, e.g., [3 , 43–45], among others. All these researches are concentrated up on the DMPs in which all the decision-related information are provided at the same period, however, those information are usually collected at different periods in many decision problems. To handle this type of situation, Xu and Yager [40] investigated dynamic intuitionistic fuzzy multi-attribute decision making (DIF-MADM) problems where all the attribute values are expressed as intuitionistic fuzzy numbers (IFNs) collected at different periods. Regardless of the MADM problem based on IFS or DIF-MADM problem, aggregation of information is always one of key research issues. Accordingly, many aggregation operators have been introduced under IF environment and applied to different MADM problems, e.g., as far as IFS is concerned, intuitionistic fuzzy weighted aggregation (IFWA) operator [36], intuitionistic fuzzy ordered weighted aggregation (IFOWA) operator [36], intuitionistic fuzzy hybrid aggregation (IFHA) operator [41], intuitionistic fuzzy weighted geometric (IFWG) operator [32, 39], intuitionistic fuzzy ordered weighted geometric (IFOWG) operator [39], intuitionistic fuzzy hybrid geometric (IFHG) operator [39] and other induced aggregation operators [23 , 37]. In addition, different aggregation operators have been also introduced and applied into different intuitionistic fuzzy multiatttribute decision making methods [2 , 48], e.g., DIFWA operator [40], uncertain dynamic intuitionistic fuzzy weighted (UDIFWA) operator [40], DIFWG operator [30 , 42], uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operators [30 , 42], dynamic intuitionistic fuzzy aggregation Operators (DIFWA^ϵ and DIFWG^ϵ [22]) based on Einstein Operations.

Although different aggregation operators have been introduced, they still cannot help to surmount the drawback of some existing DIF-MADM methods which result in unreasonable preference orders of alternatives in some decision situations [22 , 40]. Motivated by this limitation in some existing DIF-MADM methods, this paper aims at proposing new DIF-MADM strategy and new aggregation operators and evaluates their feasibility and performance compared with the existing work. In order to improve the DIF-MADM method, we proposed to use new strategy based on evidential reasoning (ER) methodology. Based on D-S Theory [10, 11], Yang and Xu [46] proposed an ER algorithm for MADM under uncertainty. Since then, ER methodology/algorithms have been successfully used in different DM problems [6 , 49]. Specially, Yang et al. [47] presented an ER approach for MADA. Chen et al. [47] took the advantage of the ER methodology and the representation capability of IFSs to propose a new fuzzy MADM method based on the ER methodology. Chen et al. [8] also proposed a new method for fuzzy MADM based on the transformation techniques between intuitionistic fuzzy numbers (IFNs) and rightangled triangular fuzzy numbers along with a new IFGWA operators of IFNs. The ER methodology has shown its potential capability in MADM and the likability to be incorporated with the DIF-MADM method, this is one of main focus of the present work.

Now that aggregation operators plays the key role in DIF-MADM method, in order to surmount the drawbacks of some existing DIF-MADM methods, the second focus of the present work is on introducing and evaluating the new aggregation operators. Accordingly, a new dynamic intuitionistic fuzzy weighted geometric aggregation operator (MDIFWG) is proposed along with the corresponding DIF-MADM method. The main contributions of this work are focused on the following aspects: (1) some drawbacks on the existing DIF-MADM methods are reviewed and analysed; (2) we propose a new DIF-MADM method based on the evidential reasoning (ER) algorithm in order to address some of those drawbacks, whilst, a new dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator is also introduced to address some other limits of the existing methods; (3) Some numerical examples are provided to illustrate the practicality, feasibility and robustness of the proposed two methods through, the comparative analysis with the existing DIF-MADM methods, along with some sensitivity analyses also carried out to analyse the distinct features of the proposed methods.

The remaining of the paper is organized as follows: Section 2 includes preliminary concepts and definitions relevant, such as IFS and intuitionistic fuzzy variable, score function, and evidential reasoning algorithm. In Section 3, we provide the formal description of DIF-MADM problems and review and analyse some drawbacks of existing DIF-MADM methods. In Section 4, a new DIF-MADM methods based on the ER algorithm is proposed first (denoted as Method I) and then a new DIFWG operator named MDIFWG operator introduced along with the MDIFWG-based DIF-MADM method (denoted as Method II). In Section 5 focuses on the evaluation of the feasibility and validity of the proposed DIF-MADM methods through some numerical examples and comparative analysis with some existing DIF-MADM method, along with some sensitivity analysis. This paper is concluded in Section 6.

2. Preliminaries

In this section, firstly some basic concepts related to IFS and DIFS are reviewed, along with an overview of the evidential reasoning algorithm [46], which are the basis of the present work.

2.1. IFSs and IFV

Definition 2.1. [1] Let X = {x₁, x₂, ⋯ , x_n} be a finite universe of discourse, an intuitionistic fuzzy set (IFS) A in X characterized by a membership function μ_A : X → [0, 1] and a non-membership function ν_A : X → [0, 1], which satisfies the condition 0 ≤ μ_A (x) + ν_A (x) ≤1. An IFS A can be expressed as $A = {〈 x, (μ_{A} (x), ν_{A} (x)) 〉 | x \in X} .$ π_A (x) =1 - μ_A (x) - ν_A (x) is called the degree of indeterminacy, π_A (x) represents the degree of hesitance of x to A and is also called intuitionistic index. For convenience, called (μ_A (x) , ν_A (x)) is an intuitionistic fuzzy number (IFN) and denoted by (μ_A, ν_A).

For an IFS A on the universe X, A will be reduced to a fuzzy set under the condition that intuitionistic index π_A (x) =0 for any x ∈ X.

Refer to [42], the intuitionistic fuzzy number (ν_A, μ_A) is the complement of a IFN (μ_A, ν_A), denoted as (μ_A, ν_A) ^C = (ν_A, μ_A).

Definition 2.2. [40] Let t be a time variable, then α (t) = (μ_α(t), ν_α(t)) is called an intuitionistic fuzzy variable (IFV), where μ_α(t) ∈ [0, 1] , ν_α(t) ∈ [0, 1] and μ_α(t) + ν_α(t) ∈ [0, 1].

For an IFV α (t), if t = t₁, t₂, ⋯ , t_k, then α_{t
₁}, ⋯ , α_{t
_k} indicate k IFNs collected at p different periods.

2.2. Score function of decision-making problem

Score function, an important tool to evaluate IFNs in order to obtain the best alternative in decision making problem, is needed to convert IFNs into real numbers in order to become easier to compare with each other.

In the intuitionistic fuzzy MADM problem, as far as the score function is concerned, an effective score function has the following properties [33]: (1) the MD, NMD and ID (hesitation) of IFS should be considered; (2) it should have higher precision; and (3) it should also have stronger selection ability.

Wang [33] proposed a new score functions based on the cross entropy of MD from the NMD to address these limitations. The cross-entropy [33] based on IFS is defined as follows.

Definition 2.3. [33] Let α = (μ, ν) be an IFN of an IFS, the cross-entropy of the MD μ from the NMD ν is called cross-entropy based on IFS, which measures the divergence between μ and ν: $H (μ, ν) = {\begin{matrix} \log_{2} \frac{2}{2 - ν}, & μ = 0 \\ \log_{2} \frac{2}{1 + ν}, & μ = 1 \\ μ \log_{2} \frac{2 μ}{(μ + ν)} \\ + (1 - μ) \log_{2} \frac{2 (1 - μ)}{2 - (μ + ν)}, & 0 < μ < 1 \end{matrix}$ (1)

From Definition 2.3, it is obvious that H (μ, ν) ≠ H (ν, μ), that is, H (μ, ν) is not symmetric. Therefore, Definition 2.3 should be modified as: $H_{M} (α) = \frac{H (α) + H (α^{C})}{2} .$ (2)

Theorem 2.1. [33] Let α = (μ, ν) be an IFN, then H_M (α) satisfies the following properties:

(1) H_M (α) ∈ [0, 1];

(2) H_M (α) = H_M (α^C);

(3) If α = (1, 0) or α = (0, 1), then H_M (α) =1;

(4) If α = α^C, then H_M (α) =0.

To determine the best alternative in DMPs, an effective score function is defined as follows:

Definition 2.4. Let α = (μ, ν) be an IFN. The new score function of α is defined as $S (α) = {\begin{matrix} μ - ν + H_{M} (α) π & μ > ν \\ μ - ν - H_{M} (α) π & μ < ν \\ 0^{*} & μ = ν \end{matrix}$ (3) where π = 1 - μ - ν and 0^* means that S is close to 0.

For an IFN α = (μ, ν), the value of unknown degree π = 1 - μ - ν is moderate under the condition μ = ν. As π denotes degree of indeterminacy, hence the degree of accuracy of IFN α will change with π change and indeterminacy of π almost have little influence on score value of α, so the value is close to 0 rather than equal to 0. Only if π = 0, i.e. μ = ν = 0.5, the value of score equal to 0, that is, the degree of indeterminacy is the smallest and the value of accuracy is the largest.

Theorem 2.2. Let α = (μ, ν) be an IFN. Then S (α) satisfies the following properties:

(1) S (α) ∈ [-1, 1];

(2) S (α) =1 if and only if α = (1, 0);

(3) S (α) = -1 if and only if α = (0, 1);

(4) If S (α) =0 if and only if α = (0.5, 0.5).

For any two IFNs α₁, α₂,

(1) if S (α₁) < S (α₂), then α₁ ≺ α₂;

(2) if S (α₁) > S (α₂), then α₁ ≻ α₂;

(3) if S (α₁) > S (α₂), then α₁ ∼ α₂.

2.3. Evidential reasoning algorithm for MADM

In this subsection, we review the ER algorithm for MADM under uncertain environment [47]. Let X = {x₁, x₂, ⋯ , x_m} be a set of alternatives and A = {a₁, a₂, ⋯ , a_p} be a set of attributes. Assume that there are N evaluation grades θ₁, θ₂, ⋯ , θ_N for assessing the attributes of alternatives and denoted by Θ = {θ₁, θ₂, ⋯ , θ_N}, w_i refer to the weight of attribute a_i (i = 1, ⋯ , p), respectively, with w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Let S (a_i (x_j)) denote the evaluation value of attribute a_i of alternative x_j and be defined as follows:

$S (a_{i} (x_{j})) = {(θ_{n}, β_{n, i}) (x_{j}), n = 1, \dots, N},$ (4) where i = 1, ⋯ , p and j = 1, ⋯ , m.

The assessments of the attributes of the alternatives are represented by a decision matrix D = (S (a_i (x_j))) _p×m. Now we aggregate the assessment values of attributes for all alternatives. According to Equation (4), the belief of degree β_{θ_n,i} (x_j) regarding to the ith attribute a_i of alternative x_j can be transformed into basic probability mass (BPM) m_{θ_n,i} (x_j) as follows: $m_{n, i} (x_{j}) = w_{i} β_{n, i} (x_{j});$ (5) $\begin{matrix} m_{Θ, i} (x_{j}) & = 1 - \sum_{n = 1}^{N} m_{θ_{n}, i} (x_{j}) \\ = 1 - w_{i} \sum_{n = 1}^{N} β_{θ_{n}, i} (x_{j}), \end{matrix}$ (6)

where n = 1, ⋯ , N, i = 1, ⋯ , p and j = 1, ⋯ , m.

Below is the results aggregating the criteria (or attribute) by combining the BPMs generated above, where m_n,I(1) (x_j) = m_n,1 (x_j), m_Θ,I(1) (x_j) = m_Θ,1 (x_j),

$\begin{matrix} {θ_{n}} : m_{n, I (i)} (x_{j}) & = K [m_{n, I (i - 1)} (x_{j}) m_{n, i} (x_{j}) \\ + m_{n, I (i - 1)} (x_{j}) m_{Θ, i} (x_{j}) \\ + m_{Θ, I (i - 1)} (x_{j}) m_{n, i} (x_{j})] \end{matrix}$ (7) ${Θ} : m_{Θ, I (i)} (x_{j}) = K [m_{Θ, I (i - 1)} (x_{j}) m_{Θ, i} (x_{j}),$ (8) $\begin{matrix} K = 1 - \sum_{r = 1}^{N} \sum_{t = 1, t \neq r}^{N} m_{m_{r}, I (i - 1)} (x_{j}) m_{t, i} (x_{j}) \\ {θ_{n}} : & β_{n} (x_{j}) = \frac{m_{n, I (p)} (x_{j})}{1 - m_{Θ, I (p)} (x_{j})} . \end{matrix}$ (9)

From Eq. (6), we can obtain another equivalent form:

$β_{n} (x_{j}) = \frac{(1 - β_{Θ} (x_{j})) m_{n, I (p)} (x_{j})}{1 - m_{Θ, I (p)} (x_{j})},$ (10) where $β_{Θ} (x_{j}) = \sum_{i = 1}^{p} w_{i} (1 - \sum_{n = 1}^{N} β_{n, i} (x_{j}))$ .

3. Analysis of the existing DIF-MADM methods

In this section, we will review the formal representation of the typical DIF-MADM problem, and analyse their drawbacks, then in Section 4, we will introduce methods in order to overcome those drawbacks.

3.1. Formal representation of DIF-MADM

DIF-MADM methods aim at handling the MADM problems under dynamic intuitionistic fuzzy environment. A DIF-MADM problem can be formally described as follows:

(1) X = {x₁, x₂, ⋯ , x_m} a set of m alternatives;

(2) A = {a₁, a₂, ⋯ , a_n} the set of n attributes whose weight vector (WV) is w = (w₁, ⋯ , w_n) with w_i > 0 and $\sum_{i = 1}^{n} w_{i} = 1$ ;

(3) There are p periods P = {t₁, t₂, ⋯ , t_p}, whose WV is ω (t) = (ω (t₁) , ⋯ , ω (t_p)) with ω (t_k) >0 (k = 1, 2, ⋯ , p) and $\sum_{k = 1}^{p} ω (t_{k}) = 1$ .

(4) The DMs provide the attribute values of alternative x_i ∈ X (i = 1, 2, ⋯ , m) w.r.t. attribute a_j (j = 1, 2, ⋯ , n) at period t_k (k = 1, 2, ⋯ , p) and construct the intuitionistic fuzzy decision matric (IFDM) $\begin{matrix} D_{t_{k}} = (\begin{matrix} (μ_{11, t_{k}}, ν_{11, t_{k}}) & \dots & (μ_{1 n, t_{k}}, ν_{1 n, t_{k}}) \\ (μ_{21, t_{k}}, ν_{21, t_{k}}) & \dots & (μ_{2 n, t_{k}}, ν_{2 n, t_{k}}) \\ ⋮ & ⋮ & ⋮ \\ (μ_{m 1, t_{k}}, ν_{m 1, t_{k}}) & \dots & (μ_{mn, t_{k}}, ν_{mn, t_{k}}) \end{matrix}) \end{matrix}$ where (μ_{ij,t_k}, ν_{ij,t_k}) is an IFN, μ_{ij,t_k} is the degree that alternative x_i should satisfy the attribute a_j at period t_k, ν_{ij,t_k} is the degree that alternative x_i should not satisfy the attribute a_j at period t_k, and 0 ≤ μ_{ij,t_k}, ν_{ij,t_k} ≤ 1, 0 ≤ μ_{ij,t_k} + ν_{ij,t_k} ≤ 1.

3.2. Analysis of the existing DIF-MADM methods

Although with some interesting and solid results, there are still some drawbacks found in the existing DIF-MADM methods presented in Gumus [22], Xu [40], Wei [35] and Park [30]. In these DIF-MADM methods, different aggregation operators were introduced.

Let α (t₁) , α (t₂) , ⋯ , α (t_p) be a collection of IFNs collected at p different periods t_k (k = 1, 2, ⋯ , p), and λ (t) = (λ (t₁) , λ (t₂) , ⋯ , λ (t_p)) be the WV of the periods t_k(k = 1, 2, ⋯ , p) with λ (t_i) ≥0 and $\sum_{i = 1}^{p} λ (t_{i}) = 1$ . Then a DIFWA operator [40] is defined as follows: $\begin{matrix} DIFWA (α (t_{1}), α (t_{2}), \dots, α (t_{p})) \\ = (1 - \prod_{i = 1}^{p} (1 - μ_{α (t_{i})})^{λ (t_{i})}, \prod_{i = 1}^{p} ν_{α (t_{i})})^{λ (t_{i})}); \end{matrix}$ (11) A DIFWG operator [22, 32] is defined as follows:

$\begin{matrix} DIFWG (α (t_{1}), α (t_{2}), \dots, α (t_{p})) \\ = (\prod_{i = 1}^{p} μ_{α (t_{i})}^{λ (t_{i})}, 1 - \prod_{i = 1}^{p} (1 - ν_{α (t_{i})}))^{λ (t_{i})}) . \end{matrix}$ (12) A DIFWG^ϵ operator [22, 32] is defined as follows:

$\begin{matrix} {DIFWG}^{ϵ} (α (t_{1}), α (t_{2}), \dots, α (t_{p})) \\ = (\frac{2 \prod_{i = 1}^{p} μ_{α (t_{i})}^{λ (t_{i})}}{\prod_{i = 1}^{p} (2 - μ_{α (t_{i})})^{λ (t_{i})} + \prod_{i = 1}^{p} μ_{α (t_{i})}^{λ (t_{i})}}, \\ \frac{\prod_{i = 1}^{p} (1 + ν_{α (t_{i})}))^{λ (t_{i})} - \prod_{i = 1}^{p} (1 - ν_{α (t_{i})}))^{λ (t_{i})}}{\prod_{i = 1}^{p} (1 + ν_{α (t_{i})}))^{λ (t_{i})} + \prod_{i = 1}^{p} (1 - ν_{α (t_{i})}))^{λ (t_{i})}}) . \end{matrix}$ (13)

In the following, we analyse and illustrate some drawbacks about those aggregation operators:

(Drawback A.) For Eq. (11), if there exist μ_α(t₁) = 1, and μ_α(t₂) = ⋯ = μ_{α(t_p)} = 0, then $1 - \prod_{i = 1}^{p} (1 - μ_{α (t_{i})})^{λ (t_{i})} = 1$ , which is incorrect because of the impact of the values μ_α(t₂) = ⋯ = μ_{α(t_p)} = 0 on the aggregation result is not considered. For example, in a DIF-MADM problem, as far as alternative x₁ is concerned, if μ_α(t₁) = 1 for the first attribute at the period t₁, although the MDs and NMD of the first attribute regarding to the alternative x₁ at others periods changed, aggregation result of the first attribute regarding to x₁ will be constant. Therefore, DIFWA operator Eq. (11) is not well defined. Accordingly the DIF-MADM method presented in [40] uses this ill-defined DIFWA operator will get an unreasonable preference order of the alternatives in some situations.

(Drawback B.) Considering Eq. (12) and Eq. (13), if there is only one MD of IFNs is equal to 0, the aggregation MD of IFNs is 0 even if the MDs of n - 1 IFNs are not 0, this seems a rather extreme property. Therefore, the DIF-MADM methods in [22, 35] will again lead to inappropriate preference order of alternatives in some situations.

(Drawback C.) Besides Drawback B about, there is another drawback in the existing DIF-MADM methods presented in [22 , 40]. The example below gives a better illustration.

Example 3.1. This example about invest company which is from [35]), the detailed information refer to [35]). (1) x₁, x₂, x₃ are three alternative companies, (2) a₁, a₂, a₃ are three attributes. The alternatives x_i (i = 1, 2, 3) are to be evaluated using the IFI under the above three attributes at the periods t_k (k = 1, 2, 3), as listed in the following matrix, shown as Tabled 1 and 2. Let $λ (t) = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ be the WV of the periods t_k (k = 1, 2, 3), and $w = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ be the WV of the attributes a_j (j = 1, 2, 3, 4).

Table 1

Individual IF decision matrix D_{t
_k} (k = 1, 2, 3)

years		a ₁	a ₂	a ₃
t ₁	x ₁	(0.25,0.7)	(0.5, 0.2)	(0.7, 0.2)
	x ₂	(0.5, 0.2)	(0.4, 0.5)	(0.4, 0.1)
	x ₃	(0.4, 0.3)	(0.5, 0.3)	(0.6, 0.3)
t ₂	x ₁	(0.6, 0.3)	(0.4, 0.1)	(0.6, 0.1)
	x ₂	(0.7, 0.1)	(0.25, 0.7)	(0.5, 0.3)
	x ₃	(0.5, 0.2)	(0.7, 0.2)	(0.4, 0.5)
t ₃	x ₁	(0.4, 0.5)	(0.7, 0.3)	(0.5, 0.3)
	x ₂	(0.6, 0.3)	(0.6, 0.3)	(0.7, 0.2)
	x ₃	(0.7, 0.1)	(0.6, 0.1)	(0.25, 0.7)

Table 2

Complex IFDM by DIFWA operators

	a ₁	a ₂	a ₃
x ₁	(0.4351, 0.4721)	(0.5515, 0.1820)	(0.6081, 0.1820)
x ₂	(0.6081, 0.1820)	(0.4351, 0.4721)	(0.5515, 0.1820)
x ₃	(0.5515, 0.1820)	(0.6081, 0.1820)	(0.4351, 0.4721)

Calculate the distance between the alternative x_i and IFPIS α⁺ = (1, 0) and the distance between the alternative x_i and IFNIS α^- = (0, 1) by the equations in [40], respectively, we have

$\begin{matrix} d (x_{1}, α^{+}) = d (x_{2}, α^{+}) = d (x_{3}, α^{+}) = 0.4684, \\ d (x_{1}, α^{-}) = d (x_{2}, α^{-}) = d (x_{3}, α^{-}) = 0.7213 . \end{matrix}$

According to [40], the closeness coefficient of each alternative is given by

$c (x_{i}) = \frac{d (x_{i}, α^{-})}{d (x_{i}, α^{-}) + d (x_{i}, α^{+})} .$ It follows that c (x₁) = c (x₂) = c (x₃) =0.6063. Therefore x₁ = x₂ = x₃, which is obviously an incorrect preference orders of alternatives. The same results also can be obtained by using Gumus’s [22] and Wei’s [35] DIF-MADM method based on the DIFWG operators defined in Eqs. (12) and (13).

In Section 4 bellow, two new methods are proposed to surmount the above mentioned drawbacks of the existing DIF-MADM methods.

4. New methods for DIF-MADM problems

In this section, we propose two kinds of DIF-MADM methods to overcome the drawbacks presented in Section 3. It shows that Method I can overcome the drawbacks A, B and C. And Method II can overcome the drawbacks B and C.

4.1. Method I: New DIF-MADM method based on the ER methodology

Suppose that the alternatives are assessed on each attribute using the following two assessment grades: H₁ and H₂, where H₁ stands for satisfying the fuzzy concept "excellence", H₂ stands for not satisfying the fuzzy concept "excellence", and H = {H₁, H₂} stands for the assessment grade indeterminacy. The proposed method for intuitionistic fuzzy MADM based on IFSs and the ER algorithm is now presented as follows:

Step 1. Determine the belief matrix of decision maker w.r.t. attribute a_j of alternative x_i regarding the evaluation grade H₁, H₂ as follows: $\begin{matrix} D_{t_{k}} = (μ_{ij, t_{k}}, ν_{ij, t_{k}})_{m \times n} \end{matrix}$

where (μ_{ij,t_k}, ν_{ij,t_k}) = (β_{1j,t_k} (x_i) , β_{2j,t_k} (x_i)), β_{1j,t_k} (x_i) denotes the degree of belief of decision maker d_l w.r.t. attribute a_j of alternative x_i at period t_k regarding evaluation grade H₁ and β_{2j,t_k} (x_i) represents the degree of belief w. r. t. attribute a_j of alternative x_i at period t_k regarding evaluation grade H₂, 0 ≤ β_{1j,t_k} (x_i) , β_{2j,t_k} (x_i) ≤1 and 0 ≤ β_{1j,t_k} (x_i) + β_{2j,t_k} (x_i) ≤1 (j = 1, 2, ⋯ , n ; i = 1, 2, ⋯ , m ; k = 1, 2, ⋯ , p).

Step 1.1. Based on the above step, the intuitionistic fuzzy assessment (μ_{ij,t_k}, ν_{ij,t_k}) can be transformed into the ER belief distribution assessment profiled as

${(H_{1}, β_{1 j, t_{k}} (x_{i})), (H_{2}, β_{2 j, t_{k}} (x_{i})), (H, β_{Hj, t_{k}} (x_{i}))}$ (14)

Transform the degree of belief β_{qj,t_k} (x_i) into BPM ${\tilde{m}}_{qj, t_{k}} (x_{i})$ and ${\tilde{m}}_{H_{j}} (x_{i})$ by the following formulae [47]: ${\tilde{m}}_{qj, t_{k}} (x_{i}) = w_{j} (t_{k}) β_{qj, t_{k}} (x_{i});$ (15) ${\tilde{m}}_{Hj, t_{k}} (x_{i}) = 1 - \sum_{j = 1}^{n} {\tilde{m}}_{qj, t_{k}} (x_{i}) .$ (16) We can then obtain the BPM matrix:

$P_{t_{k}} = ({\tilde{m}}_{1 j, t_{k}} (x_{i}), {\tilde{m}}_{2 j, t_{k}} (x_{i})),$ where $0 \leq {\tilde{m}}_{1 j, t_{k}} (x_{i}), {\tilde{m}}_{2 j, t_{k}} (x_{i}) \leq 1$ and $0 \leq {\tilde{m}}_{1 j, t_{k}} (x_{i}) + {\tilde{m}}_{2 j, t_{k}} (x_{i}) \leq 1 (j = 1, 2, \dots, n; i = 1, 2, \dots, m; k = 1, 2, \dots, p)$ .

Step 1.2. Let the combined probability mass ${\tilde{n}}_{q 1, t_{k}} (x_{i})$ of the decision maker d_l w.r.t. attribute a_j of alternative x_i at period t_k be equal to ${\tilde{m}}_{q 1, t_{k}} (x_{i})$ , that is, ${\tilde{n}}_{q 1, t_{k}} (x_{i}) = {\tilde{m}}_{q 1, t_{k}} (x_{i}) (q = 1, 2)$ . Similarly, ${\tilde{n}}_{H 1, t_{k}} (x_{i}) = {\tilde{m}}_{H 1, t_{k}} (x_{i}) (q = 1, 2)$ . Now, calculate the combined probability mass ${\tilde{n}}_{qj, t_{k}} (x_{i})$ and ${\tilde{n}}_{Hj, t_{k}} (x_{i})$ w. r. t. the attribute a_j of alternative x_i at period t_k by the following equations:

$\begin{matrix} {\tilde{n}}_{qj, t_{k}} (x_{i}) = \frac{a}{b} \\ {\tilde{n}}_{Hj, t_{k}}^{g} (x_{i}) = \\ \frac{{\tilde{n}}_{Hj - 1, t_{k}} (x_{i}) {\tilde{m}}_{Hj, t_{k}} (x_{i})}{1 - \sum_{r = 1}^{2} \sum_{h = 1, h \neq r}^{2} {\tilde{n}}_{rj - 1, t_{k}} (x_{i}) {\tilde{m}}_{hj, t_{k}} (x_{i})} \end{matrix}$ (17) where $\begin{matrix} a = {\tilde{n}}_{qj - 1, t_{k}} (x_{i}) {\tilde{m}}_{qj, t_{k}} (x_{i}) + {\tilde{n}}_{qj - 1, t_{k}} (x_{i}) {\tilde{m}}_{Hj, t_{k}} (x_{i}) \\ + {\tilde{n}}_{Hj - 1, t_{k}} (x_{i}) {\tilde{m}}_{qj, t_{k}} (x_{i}), \\ b = 1 - \sum_{r = 1}^{2} \sum_{h = 1, h \neq r}^{2} {\tilde{n}}_{rj - 1, t_{k}} (x_{i}) {\tilde{m}}_{hj, t_{k}} (x_{i}), \end{matrix}$ j = 1, 2, ⋯ , n ; i = 1, 2, ⋯ , m ; k = 1, 2, ⋯ , p.

Step 1.3. Aggregate the evaluating values of DMs w. r. t. attribute a_j of alternative x_i at period t_k to obtain the belief:

$\begin{matrix} β_{q, t_{k}} (x_{i}) = \frac{(1 - β_{H, t_{k}} (x_{i})) {\tilde{n}}_{qn, t_{k}} (x_{j})}{1 - {\tilde{n}}_{Hn, t_{k}} (x_{i})} \end{matrix}$ (18) with $β_{Hn, t_{k}} (x_{i}) = \sum_{j = 1}^{n} w_{j} (t_{k}) (1 - \sum_{q = 1}^{2} {\tilde{n}}_{qj, t_{k}} (x_{i}))$ . where $\sum_{q = 1}^{2} β_{q, t_{k}} (x_{i}) + β_{Hn, t_{k}} (x_{i}) = 1$ . Let the aggregated value obtained by above equations β_{1,t_k} (x_i), β_{2,t_k} (x_i) form an IFN (β_{1,t_k} (x_i) , β_{2,t_k} (x_i)), where β_{1,t_k} (x_i) and β_{2,t_k} (x_i) are the degree of DM d_l w.r.t. alternative x_i at period t_k regarding evaluation grades H₁ and H₂.

Step 2. Based on Step 1, construct the aggregated decision making matrix Q as follows $\begin{matrix} Q = (β_{1, t_{k}} (x_{i}), β_{2, t_{k}} (x_{i}))_{m \times p} \end{matrix}$ where (β_{1,t_k} (x_i) , β_{2,t_k} (x_i)) is an IFN, β_{1,t_k} (x_i) is the degree of belief with respect to alternative x_i at period t_k regarding evaluation grade H₁ and β_{2,t_k} (x_i) represents the degree of belief w. r. t. alternative x_i at period t_k regarding evaluation grade H₂, 0 ≤ β_{1,t_k} (x_i) , β_{2,t_k} (x_i) ≤1 and 0 ≤ β_{1,t_k} (x_i) + β_{2,t_k} (x_i) ≤1 (i = 1, 2, ⋯ , m ; k = 1, 2, ⋯ , p).

Step 2.1. Based on the above step, the intuitionistic fuzzy assessment (β_{1,t_k} (x_i) , β_{2,t_k} (x_i)) can be transformed into the ER belief distribution assessment profiled by

${(H_{1}, β_{1} (x_{i})), (H_{2}, β_{2} (x_{i})), (H, β_{H} (x_{i}))} .$ (19)

Transform the degree of belief β_{q,t_k} (x_i) into BPM ${\tilde{m}}_{q, t_{k}} (x_{i})$ and ${\tilde{m}}_{H, t_{k}} (x_{i})$ by the following formulae: ${\tilde{m}}_{q, t_{k}} (x_{i}) = w_{k} β_{q, t_{k}} (x_{i});$ (20) ${\tilde{m}}_{H, t_{k}} (x_{i}) = 1 - \sum_{k = 1}^{p} {\tilde{m}}_{q, t_{k}} (x_{i}) .$ (21) We can obtain the BPM matrix $\begin{matrix} Q = ({\tilde{m}}_{1, t_{k}} (x_{i}), {\tilde{m}}_{2, t_{k}} (x_{i})) \end{matrix}$ where $0 \leq {\tilde{m}}_{1, t_{k}} (x_{i}), {\tilde{m}}_{2, t_{k}} (x_{i}) \leq 1$ and $0 \leq {\tilde{m}}_{1, t_{k}}$ $(x_{i}) + {\tilde{m}}_{2, t_{k}} (x_{i}) \leq 1 (i = 1, 2, \dots, m; k = 1, 2, \dots,$ p).

Step 2.2. Let the combined probability mass ${\tilde{n}}_{q, t_{1}} (x_{i})$ w.r.t. alternative x_i at period t₁ be equal to ${\tilde{m}}_{q, t_{1}} (x_{i})$ , that is, $n_{q, t_{1}} (x_{i}) = {\tilde{m}}_{q, t_{1}} (x_{i}) (q = 1, 2)$ . Similarly, $n_{C, t_{1}} (x_{i}) = {\tilde{m}}_{C, t_{1}} (x_{i}) (q = 1, 2)$ . Now, calculate the combined probability mass ${\tilde{n}}_{q, t_{k}} (x_{i})$ and ${\tilde{n}}_{C, t_{k}} (x_{i})$ w. r. t. alternative x_i at period t_k by the following equations: ${\tilde{n}}_{q, t_{k}} (x_{i}) = \frac{\tilde{a}}{\tilde{b}}$ (22) $\begin{matrix} {\tilde{n}}_{H, t_{k}} (x_{i}) = \\ \frac{{\tilde{n}}_{H, t_{k - 1}} (x_{i}) {\tilde{m}}_{H, t_{k}} (x_{i})}{1 - \sum_{r = 1}^{2} \sum_{h = 1, h \neq r}^{2} {\tilde{n}}_{r, t_{k - 1}} (x_{i}) {\tilde{m}}_{h, t_{k}} (x_{i})}, \end{matrix}$

where $\begin{matrix} \tilde{a} = {\tilde{n}}_{q, t_{k - 1}} (x_{i}) {\tilde{m}}_{q, t_{k}} (x_{i}) + {\tilde{n}}_{q, t_{k - 1}} (x_{i}) {\tilde{m}}_{H, t_{k}} (x_{i}) \\ + {\tilde{n}}_{H, t_{k - 1}} (x_{i}) {\tilde{m}}_{q, t_{k}} (x_{i}) \\ \tilde{b} = 1 - \sum_{r = 1}^{2} \sum_{h = 1, h \neq r}^{2} {\tilde{n}}_{r, t_{k - 1}} (x_{i}) {\tilde{m}}_{h, t_{k}} (x_{i}), \end{matrix}$

k = 2, ⋯ , p ; i = 1, 2, ⋯ , m.

Step 2.3. Aggregate the evaluating values of DMs w. r. t. alternative x_i to obtain the belief:

$β_{q} (x_{i}) = \frac{(1 - β_{H} (x_{i})) {\tilde{n}}_{q} (x_{j})}{1 - {\tilde{n}}_{H} (x_{i})}, q = 1, 2 .$ (23) with $β_{H} (x_{i}) = \sum_{k = 1}^{p} w_{k} (1 - \sum_{q = 1}^{2} {\tilde{n}}_{q, t_{k}} (x_{i}))$ . where $\sum_{q = 1}^{2} β_{q} (x_{i}) + β_{H} (x_{i}) = 1$ . Let the aggregated value be obtained by above equations β₁ (x_i) , β₂ (x_i) form an IFN (β₁ (x_i) , β₂ (x_i)), where β₁ (x_i) and β₂ (x_i) are the degree decision makers of alternative x_j regarding evaluation grades H₁ and H₂, respectively.

Step 3. Calculate the scores of IFN (β₁ (x_i) , β₂ (x_i)) obtained by the aggregation result of Step 3. Let α_i = (β₁ (x_i) , β₂ (x_i)) (i = 1, 2, ⋯ , m).

Step 3.1. According to Eq. (1), calculate H (α_i) and $H (α_{i}^{C})$ , where $α_{i}^{C} = (β_{2} (x_{i}), β_{1} (x_{i})), (i = 1, 2, \dots, m)$ ;

Step 3.2. Calculate the score of α_i and denote as S (α_i);

Step 4. Determine the ranking of alternatives according to Step 3.2. The larger the value S (α_i), the better the order of alternative x_i (i = 1, 2, ⋯ , m).

4.2. Method II: New DIF-MADM method based on MDIFWG operators

In this section, we propose new operators in IFNs and further propose the new DIFWG operator, and then introduce a new DIF-MADM method which can overcome some drawbacks analysed in Section 3.

4.2.1 New aggregation operator for DIF-MADM problems

In this section, a new aggregation operator named MDIFWG operator will be introduced.

Definition 4.1. Let α (t₁) , α (t₂) , ⋯ , α (t_p) be a collection of IFVs collected at p different periods t_k(k = 1, 2, ⋯ , p), and λ (t) = (λ (t₁) , λ (t₂) , ⋯ , λ (t_p)) be the weight vector of the periods t_k(k = 1, 2, ⋯ , p) with λ (t_i) ≥0 and $\sum_{i = 1}^{p} λ (t_{i}) = 1$ . Then a MDIFWG operator is defined as follows:

$\begin{matrix} MDIFWG (α (t_{1}), \dots, α (t_{p})) = \\ (1 - \prod_{i = 1}^{p} (1 - μ_{α (t_{i})}))^{λ (t_{i})}, \\ \prod_{i = 1}^{p} (1 - μ_{α (t_{i})})^{λ (t_{i})} - \prod_{i = 1}^{p} (1 - μ_{α (t_{i})} - ν_{α (t_{i})})^{λ (t_{i})}) . \end{matrix}$ (24)

4.2.2 New DIF-MADM method based on MDIFWG operators

In this section, we design a new method for DIF-MADM based on the proposed MDIFWG operator presented in Section 4.1.1. The details of this method are described as follows:

Step 1. Aggregate all the intuitionistic fuzzy decision matrices (IFDMs) D_{t
_k} = (α_{ij,t_k}) _m×n (k = 1, 2, ⋯ , p) into a complex IFDM D = (α_ij) _m×n based on the MDIFWG operator: $α_{ij} = MDIFWG ((μ_{ij, t_{1}}, ν_{ij, t_{1}}), \dots, (μ_{ij, t_{p}}, ν_{ij, t_{p}})),$ (25)

where α_ij = (μ_ij, v_ij) is an IFN obtained by Eq. (24) or Eq. (25).

Step 2- Step 6 are the same as Xu’s method [40].

Step 2. Define the IFIS $α^{+} = (α_{1}^{+}, \dots, α_{m}^{+})$ and the IFNIS $α^{-} = (α_{1}^{-}, \dots, α_{m}^{-})$ , respectively, where $α_{i}^{+} = (1, 0) (i = 1, 2, \dots, n)$ are the n largest IFNs and $α_{i}^{+} = (0, 1) (i = 1, 2, \dots, n)$ are the n smallest IFNs.

Step 3. Calculate the distance between x_i and the IFIS α⁺, IFNIS α^-, respectively:

$d (x_{i}, α^{+}) = \sum_{j = 1}^{n} w_{j} (1 - μ_{ij})$ ,

$d (x_{i}, α^{-}) = \sum_{j = 1}^{n} w_{j} (1 - ν_{ij})$ .

Step 4. Calculate the closeness coefficient of each alternative $\begin{matrix} c (x_{i}) = \frac{d (x_{i}, α^{-})}{d (x_{i}, α^{+}) + d (x_{i}, α^{-})} . \end{matrix}$

Step 5. Determine the preference orders of all the alternatives x_i according to c (x_i) (i = 1, 2, ⋯ , n).

5. Case study

In this section, we use some examples to illustrate and compare the proposed methods with some existing DIF-MADM methods.

5.1. Examples and comparison analysis

Example 5.1. This example is from [30] to evaluate university faculty for tenure and promotion. There are three attributes (A₁: teaching, A₂: research, and A₃: service) and five faculty alternatives x₁, x₂, ⋯ , x₅ in the three years t₁, t₂, t₃. According to above information, construct the IFDMs D_{t
_k} (k = 1, 2, 3). Let λ (t) = (0.2, 0.3, 0.5) be the WV of the years t_k and w = (0.3, 0.4, 0.3) be the WV of the attributes A_j (j = 1, 2, 3) .

Table 3
Individual IF decision matrix D_{t
_k} (k = 1, 2, 3)

years a ₁ a ₂ a ₃

t ₁ x ₁ (0.8, 0.1) (0.9, 0.1) (0.7, 0.2)

x ₂ (0.7, 0.3) (0.6, 0.2) (0.6, 0.3)

x ₃ (0.5, 0.4) (0.7, 0.3) (0.6, 0.3)

x ₄ (0.9, 0.1) (0.7, 0.2) (0.8, 0.2)

x ₅ (0.6, 0.1) (0.8, 0.2) (0.5 0.1)

t ₂ x ₁ (0.9, 0.1) (0.8, 0.2) (0.8, 0.1)

x ₂ (0.8, 0.2) (0.5, 0.1) (0.7, 0.2)

x ₃ (0.5, 0.5) (0.7, 0.2) (0.8, 0.2)

x ₄ (0.9, 0.1) (0.9, 0.1) (0.7, 0.3)

x ₅ (0.5, 0.2) (0.6, 0.3) (0.6, 0.2)

t ₃ x ₁ (0.7, 0.1) (0.9, 0.1) (0.9, 0.1)

x ₂ (0.9, 0.1) (0.6, 0.2) (0.5, 0.2)

x ₃ (0.4, 0.5) (0.8, 0.1) (0.7, 0.1)

x ₄ (0.8, 0.1) (0.7, 0.2) (0.9, 0.1)

x ₅ (0.6, 0.3) (0.8, 0.2) (0.7, 0.2)

years		a ₁	a ₂	a ₃
t ₁	x ₁	(0.8, 0.1)	(0.9, 0.1)	(0.7, 0.2)
	x ₂	(0.7, 0.3)	(0.6, 0.2)	(0.6, 0.3)
	x ₃	(0.5, 0.4)	(0.7, 0.3)	(0.6, 0.3)
	x ₄	(0.9, 0.1)	(0.7, 0.2)	(0.8, 0.2)
	x ₅	(0.6, 0.1)	(0.8, 0.2)	(0.5 0.1)
t ₂	x ₁	(0.9, 0.1)	(0.8, 0.2)	(0.8, 0.1)
	x ₂	(0.8, 0.2)	(0.5, 0.1)	(0.7, 0.2)
	x ₃	(0.5, 0.5)	(0.7, 0.2)	(0.8, 0.2)
	x ₄	(0.9, 0.1)	(0.9, 0.1)	(0.7, 0.3)
	x ₅	(0.5, 0.2)	(0.6, 0.3)	(0.6, 0.2)
t ₃	x ₁	(0.7, 0.1)	(0.9, 0.1)	(0.9, 0.1)
	x ₂	(0.9, 0.1)	(0.6, 0.2)	(0.5, 0.2)
	x ₃	(0.4, 0.5)	(0.8, 0.1)	(0.7, 0.1)
	x ₄	(0.8, 0.1)	(0.7, 0.2)	(0.9, 0.1)
	x ₅	(0.6, 0.3)	(0.8, 0.2)	(0.7, 0.2)

(1) Method I:

Step 1.1. Based on the DMs D_{t
₁}, D_{t
₂} and D_{t
₃}, the WV of attributes and Eq. (15), Eq. (16), we can obtain the BPM P_{t
₁}, P_{t
₂} and P_{t
₃}: $\begin{matrix} P_{t_{1}} = (\begin{matrix} (0.24, 0.03) & (0.36, 0.04) & (0.21, 0.06) \\ (0.21, 0.09) & (0.24, 0.08) & (0.18, 0.09) \\ (0.15, 0.12) & (0.28, 0.12) & (0.18, 0.09) \\ (0.27, 0.03) & (0.28, 0.08) & (0.24, 0.06) \\ (0.18, 0.03) & (0.32, 0.08) & (0.15, 0.03) \end{matrix}) \\ P_{t_{2}} = (\begin{matrix} (0.27, 0.03) & (0.32, 0.08) & (0.24, 0.03) \\ (0.24, 0.06) & (0.2, 0.04) & (0.21, 0.06) \\ (0.15, 0.15) & (0.28, 0.08) & (0.24, 0.06) \\ (0.27, 0.03) & (0.36, 0.04) & (0.21, 0.09) \\ (0.15, 0.06) & (0.24, 0.12) & (0.18, 0.06) \end{matrix}) \\ P_{t_{3}} = (\begin{matrix} (0.21, 0.03) & (0.36, 0.04) & (0.27, 0.03) \\ (0.27, 0.03) & (0.24, 0.08) & (0.15, 0.06) \\ (0.12, 0.15) & (0.32, 0.04) & (0.21, 0.03) \\ (0.24, 0.03) & (0.28, 0.08) & (0.27, 0.03) \\ (0.18, 0.09) & (0.32, 0.08) & (0.21, 0.06) \end{matrix}) \end{matrix}$ Step 1.2. We can obtain the combined probability based on Eq. (17) $\begin{matrix} {\tilde{n}}_{13, t_{1}} (x_{1}) = 0.5911, {\tilde{n}}_{23, t_{1}} (x_{1}) = 0.0686, {\tilde{n}}_{13, t_{1}} (x_{2}) = 0.457, \\ {\tilde{n}}_{23, t_{1}} (x_{2}) = 0.1598, {\tilde{n}}_{13, t_{1}} (x_{3}) = 0.4341, {\tilde{n}}_{23, t_{1}} (x_{3}) = 0.2053, \\ {\tilde{n}}_{13, t_{1}} (x_{4}) = 0.568, {\tilde{n}}_{23, t_{1}} (x_{4}) = 0.0928, {\tilde{n}}_{13, t_{1}} (x_{5}) = 0.5016, \\ {\tilde{n}}_{23, t_{1}} (x_{5}) = 0.0896, {\tilde{n}}_{13, t_{2}} (x_{1}) = 0.5971, {\tilde{n}}_{23, t_{2}} (x_{1}) = 0.0755, \\ {\tilde{n}}_{13, t_{2}} (x_{2}) = 0.4891, {\tilde{n}}_{23, t_{2}} (x_{2}) = 0.0978, {\tilde{n}}_{13, t_{2}} (x_{3}) = 0.4754, \\ {\tilde{n}}_{23, t_{2}} (x_{3}) = 0.1709, {\tilde{n}}_{13, t_{2}} (x_{4}) = 0.5998, {\tilde{n}}_{23, t_{4}} (x_{4}) = 0.0814, \\ {\tilde{n}}_{13, t_{2}} (x_{5}) = 0.4261, {\tilde{n}}_{23, t_{2}} (x_{5}) = 0.1576, {\tilde{n}}_{13, t_{3}} (x_{1}) = 0.6128, \\ {\tilde{n}}_{23, t_{3}} (x_{1}) = 0.0523, {\tilde{n}}_{13, t_{3}} (x_{2}) = 0.4953, {\tilde{n}}_{23, t_{3}} (x_{2}) = 0.1022, \\ {\tilde{n}}_{13, t_{3}} (x_{3}) = 0.48, {\tilde{n}}_{23, t_{3}} (x_{3}) = 0.1294, {\tilde{n}}_{13, t_{3}} (x_{4}) = 0.5742, \\ {\tilde{n}}_{23, t_{4}} (x_{2}) = 0.0772, {\tilde{n}}_{13, t_{5}} (x_{3}) = 0.5147, {\tilde{n}}_{23, t_{3}} (x_{5}) = 0.133 \end{matrix}$

We can obtain the remaining combined probability based on Eq. (18) $\begin{matrix} {\tilde{n}}_{H 3, t_{1}} (x_{1}) = 0.34, {\tilde{n}}_{H 3, t_{1}} (x_{2}) = 0.3832, {\tilde{n}}_{H 3, t_{1}} (x_{3}) = 0.3606, \\ {\tilde{n}}_{H 3, t_{1}} (x_{4}) = 0.3391, {\tilde{n}}_{H 3, t_{1}} (x_{5}) = 0.4087; \\ {\tilde{n}}_{H 3, t_{2}} (x_{1}) = 0.3275, {\tilde{n}}_{H 3, t_{2}} (x_{2}) = 0.4131, {\tilde{n}}_{H 3, t_{2}} (x_{3}) = 0.3537, \\ {\tilde{n}}_{H 3, t_{2}} (x_{4}) = 0.3187, {\tilde{n}}_{H 3, t_{2}} (x_{5}) = 0.4163; \\ {\tilde{n}}_{H 3, t_{3}} (x_{1}) = 0.3349, {\tilde{n}}_{H 3, t_{3}} (x_{2}) = 0.4024, {\tilde{n}}_{H 3, t_{3}} (x_{3}) = 0.3905, \\ {\tilde{n}}_{H 3, t_{3}} (x_{4}) = 0.3486, {\tilde{n}}_{H 3, t_{3}} (x_{5}) = 0.3522 . \end{matrix}$

Step 1.3. Aggregate the evaluating values w. r. t. attribute a₁, a₂, a₃ of alternative x₁, x₂, x₃ at the periods t₁, t₂, t₃ to obtain the belief distributions based on Eq. (19). Let the aggregated value obtained by the belief distributions β_{1,t_k} (x_i) , β_{2,t_k} (x_i) form the intuitionistic fuzzy values (β_{1,t_k} (x_i) , β_{2,t_k} (x_i)) as follows: $\begin{matrix} (β_{1, t_{1}} (x_{1}), β_{2, t_{1}} (x_{1})) = (0.8422, 0.0978), \\ (β_{1, t_{1}} (x_{2}), β_{2, t_{1}} (x_{2})) = (0.6595, 0.2305), \\ (β_{1, t_{1}} (x_{3}), β_{2, t_{1}} (x_{3})) = (0.6381, 0.3019), \\ (β_{1, t_{1}} (x_{4}), β_{2, t_{1}} (x_{4})) = (0.8251, 0.1349), \\ (β_{1, t_{1}} (x_{5}), β_{2, t_{1}} (x_{5})) = (0.6702, 0.1198); \\ (β_{1, t_{2}} (x_{1}), β_{1, t_{2}} (x_{1})) = (0.8611, 0.1089), \\ (β_{1, t_{2}} (x_{2}), β_{1, t_{2}} (x_{2})) = (0.675, 0.135), \\ (β_{1, t_{2}} (x_{3}), β_{1, t_{2}} (x_{3})) = (0.7061, 0.2539), \\ (β_{1, t_{2}} (x_{4}), β_{1, t_{2}} (x_{4})) = (0.8805, 0.1195), \\ (β_{1, t_{2}} (x_{5}), β_{1, t_{2}} (x_{5})) = (= 0.5913, 0.2187); \\ (β_{1, t_{3}} (x_{1}), β_{1, t_{3}} (x_{1})) = (0.866, 0.074), \end{matrix}$ $\begin{matrix} (β_{1, t_{3}} (x_{2}), β_{1, t_{3}} (x_{2})) = (0.688, 0.142), \\ (β_{1, t_{3}} (x_{3}), β_{1, t_{3}} (x_{3})) = (0.6853, 0.1847), \\ (β_{1, t_{3}} (x_{4}), β_{1, t_{3}} (x_{4})) = (0.82, 0.1102) \\ (β_{1, t_{3}} (x_{5}), β_{1, t_{3}} (x_{5})) = (0.7469, 0.1931) \end{matrix}$

Step 2. Based on Step 1, construct the aggregation DM Q as follows

$\begin{matrix} Q = \\ (\begin{matrix} (0.8422, 0.0978) & (0.8611, 0.1089) & (0.866, 0.074) \\ (0.6595, 0.2305) & (0.675, 0.135) & (0.688, 0.142) \\ (0.6381, 0.3019) & (0.7061, 0.2536) & (0.6853, 0.1847) \\ (0.8251, 0.1349) & (0.8805, 0.1195) & (0.8198, 0.1102) \\ (0.6702, 0.1198) & (0.5913, 0.2187) & (0.7469, 0.1931) \end{matrix}) \end{matrix}$

Step 2.1. Based on Step 2 and Eq. (21), we can obtain the BPM matrix $\begin{matrix} Q = (\begin{matrix} (0.1684, 0.0196) & (0.2583, 0.0327) & (0.433, 0.037) \\ (0.132, 0.0461) & (0.2025, 0.0405) & (0.344, 0.071) \\ (0.1276, 0.0604) & (0.2118, 0.0762) & (0.3426, 0.0924) \\ (0.165, 0.027) & (0.26416, 0.0358) & (0.41, 0.0551) \\ (0.134, 0.024) & (0.1774, 0.0656) & (0.3735, 0.0965) \end{matrix}) \end{matrix}$

Step 2.2. Based on Eq. (22), we can calculate the combined probability mass ${\tilde{n}}_{q, t_{k}} (x_{i})$ and ${\tilde{n}}_{H, t_{k}} (x_{i})$ w.r.t. alternative x_i at the period t_k as follows: $\begin{matrix} n_{1, 3} (x_{1}) = 0.635, n_{2, 3} (x_{1}) = 0.0465, n_{1, 3} (x_{2}) = 0.5171, \\ n_{2, 3} (x_{2}) = 0.0959, n_{1, 3} (x_{3}) = 0.5051, n_{2, 3} (x_{3}) = 0.1372, \\ n_{1, 3} (x_{4}) = 0.6169, n_{2, 3} (x_{4}) = 0.0634, n_{1, 3} (x_{5}) = 0.5214, \\ n_{2, 3} (x_{5}) = 0.1163, n_{H, 3} (x_{1}) = 0.3185, n_{H, 3} (x_{2}) = 0.3871, \\ n_{H, 3} (x_{3}) = 0.3576, n_{H, 3} (x_{4}) = 0.3197, n_{H, 3} (x_{5}) = 0.3622 . \end{matrix}$

Step 2.3. Aggregate the evaluating values of DMs w.r.t alternative x_i to obtain the belief distributions based on Eq. (23). Let the aggregated value obtained by the belief distributions β₁ (x_i) , β₂ (x_i) form the intuitionistic fuzzy values (β₁ (x_i) , β₂ (x_i)) asfollows: $\begin{matrix} (β_{1} (x_{1}), β_{2} (x_{1})) = (0.8842, 0.0648), \\ (β_{1} (x_{2}), β_{2} (x_{2})) = (0.7053, 0.1307), \\ (β_{1} (x_{3}), β_{2} (x_{3})) = (0.7163, 0.1947), \\ (β_{1} (x_{4}), β_{2} (x_{4})) = (0.8679, 0.0891), \\ (β_{1} (x_{5}), β_{2} (x_{5})) = (0.7121, 0.1589) . \end{matrix}$

Step 3. Calculate the scores of IFN (β₁ (x_i) , β₂ (x_i)) obtained by the aggregation result of Step 3. Let α_i = (β₁ (x_i) , β₂ (x_i)) (i = 1, 2, 3, 4, 5)

Step 3.1. According to Eq. (2), calculate H_M (α_i) (i = 1, 2, 3, 4, 5) as follows:

H_M (α₁) =0.5665, H_M (α₂) =0.2634, H_M (α₃)= 0.2085, H_M (α₄) =0.5, H_M (α₅) =0.2391.

Step 3.2. Calculate the score of α_i and denote as S (α_i) (i = 1, 2, 3) according to Eq. (3): S (α₁) =0.8484, S (α₂) =0.6177, S (α₃) =0.5402, S (α₄) =0.8, S (α₅) =0.584.

Step 4. Determine the ranking of alternatives according to Step 3. The preference order of alternatives is obtained as follows: x₁ > x₄ > x₂ > x₅ > x₃, that is, x₁ is the desirable one.

(2) Method II: We utilize the MDIFWG which is presented in Section 4.2.2.

Step 1. Aggregate all the IFDMs D_{t
_k} = (α_{ij,t_k}) _5×3 (k = 1, 2, 3) into a complex IFDM

D = (α_ij) _5×3 by using MDIFWG operator, where α_ij = (μ_ij, v_ij) is an IFN obtained by Eq. (25). $\begin{matrix} D = \\ (\begin{matrix} (0.801, 0.199) & (0.8769, 0.1231) & (0.8466, 0.1534) \\ (0.8466, 0.1534) & (0.5723, 0.1815) & (0.5898, 0.237) \\ (0.4523, 0.5477) & (0.7551, 0.2449) & (0.7186, 0.2814) \\ (0.8586, 0.1414) & (0.7842, 0.2158) & (0.8403, 0.1597) \\ (0.5723, 0.2545) & (0.7538, 0.2462), & (0.6378, 0.1998) \end{matrix}) \end{matrix}$

Step 2-3. Calculate the distance between the alternative x_i and α⁺, α^-, respectively, and list as follows: $\begin{matrix} d (x_{1}, α^{+}) = 0.1549, d (x_{2}, α^{+}) = 0.3402, \\ d (x_{3}, α^{+}) = 0.3467, \\ d (x_{4}, α^{+}) = 0.1766, d (x_{5}, α^{+}) = 0.3355; \\ d (x_{1}, α^{-}) = 0.8451, \\ d (x_{2}, α^{-}) = 0.8103, d (x_{3}, α^{-}) = 0.6533, \\ d (x_{4}, α^{-}) = 0.8234, \\ d (x_{5}, α^{-}) = 0.7652 . \end{matrix}$ Step 4. Calculate the closeness coefficient of each alternative $\begin{matrix} c (x_{1}) = 0.8451, c (x_{2}) = 0.7043, c (x_{3}) = 0.6533, \\ c (x_{4}) = 0.8234, c (x_{5}) = 0.6952 . \end{matrix}$ Step 5. According to Step 4, we can obtain the preference order as x₁ > x₄ > x₂ > x₅ > x₃, which coincides with the order obtained by using Method I.

Table 4 shows a comparison of the preference order of the alternatives for different methods for Example 5.1.

Table 4

A comparison of preference order for different methods

Methods	preference order
DIFWA [40]	x₁ > x₄ > x₂ > x₅ > x₃
DIFWG [35]	x₁ > x₄ > x₂ > x₅ > x₃
DIFWG^ϵ [22]	x₁ > x₄ > x₂ > x₅ > x₃
Extended VIKOR based on DIFWG [30]	x₁ > x₄ > x₅ > x₂ > x₃
The proposed method based MDIFWG	x₁ > x₄ > x₂ > x₅ > x₃
The proposed method based ER algorithm	x₁ > x₄ > x₂ > x₅ > x₃

It follows from the Table 4 that the preference order of alternatives obtained by our proposed method are the same with the preference order obtained by Xu’s [40], Gumus’s [22] and Wei’s [35] methods. It is also shown that our proposed methods based on ER algorithm and MDIFWG operators are valid.

Now, the following two examples will be used to show the our proposed methods can overcome effectively the Drawbacks A, B and C listed in Section 3.

Example 5.2. Considering Example 3.1, we illustrate how this example can be solved by using the proposed two methods in Section 4.

(1) Method I. Utilizing the ER algorithm. Computation process is the same with Example 5.1, so we omit it. According to Eq. (2), calculate H_M (α_i) (i = 1, 2, 3) and shown as follows:

H_M (α₁) =0.06, H_M (α₁) =0.06266, H_M (α₁) =0.06367.

Furthermore, calculate the score of α_i and denote S (α_i) (i = 1, 2, 3) according to Eq. (3): S (α₁) =0.2923, S (α₂) =0.2988, S (α₃) =0.3013.

Therefore, determine the ranking of alternatives according to Step 3. The preference order of alternatives is x₃ > x₂ > x₁, that is, x₃ is the desirable one.

We can see from Exa. 5.2 that the DIF-MADM methods proposed by Xu [40], Gumus [22] and Wei [35] can not distinguish the preference order of alternatives x₁, x₂, x₃. However, we can see from above Method I that our DIF-MADM method based on ER algorithm can distinguish the preference order of alternatives x₁, x₂, x₃. It is also shown that our method based on ER algorithm can overcome the Drawback C. That is, Drawback C is not the drawback anymore in this new method based on ER algorithm.

(2) Method II. We utilized the MDIFWG which is presented in Section 4.2.2. for Example 5.2. Computation process is the same with Example 5.1, so we omit it. Calculate the closeness coefficient of each alternative $\begin{matrix} c (x_{1}) = 0.3646, c (x_{2}) = 0.3893, c (x_{3}) = 0.396 . \end{matrix}$

According to Step 4, we can obtain the preference order is x₃ > x₂ > x₁, which coincided with the order obtained by using Method I as detailed above.

The different preference order of the alternatives for different methods for Example 5.2 is list in Table 5.

Table 5

A comparison of preference order for different methods for Example 5.2

Methods	Preference order
DIFWA [40]	x₁ = x₂ = x₃
DIFWG [35]	x₁ = x₂ = x₃
DIFWG^ϵ [22]	x₁ = x₂ = x₃
Extended VIKOR based on DIFWG [30]	x₁ = x₂ = x₃
The proposed method based on MDIFWG	x₁ < x₂ < x₃
The proposed method based on the ER algorithm	x₁ < x₂ < x₃

We can see from Table 5 that the DIF-MADM methods proposed by Xu [40], Gumus [22] and Wei [35] can not distinguish the preference order of alternatives. The same problem is also obtained by extend VIKOR method based on DIFWG, the root of this problem is the related aggregation operators (or the definition of operation of dynamic intuitionistic fuzzy numbers). However, we can see from above Table 5 that our DIF-MADM method based on ER algorithm and MDIFWG operator can distinguish the preference order of alternatives x₁, x₂, x₃. It is also shown that our methods based on ER algorithm and MDIFWG operator can overcome effectively the Drawback C.

The following example can show the proposed methods can surmount the drawbacks A and B of existing methods analysed in Section 3.

Example 5.3. In this example, we consider the following decision problem. There are three alternatives x_i (i = 1, 2, 3) and three attributes (1) A₁, A₂, A₃. The three possible alternatives x_i (i = 1, 2, 3) are to be evaluated using the IFI by the DM at the periods t_k (k = 1, 2, 3), as listed in the following matrix, shown as Table 6. Let $λ (t) = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ be WV of the periods t_k (k = 1, 2, 3), and $w = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ be WV of the attributes A_j (j = 1, 2, 3).

Table 6

Individual IF decision matrix D_{t
_k} (k = 1, 2, 3)

years		A ₁	A ₂	A ₃
t ₁	x ₁	(0.6, 0.1)	(0.5, 0.2)	(0.7, 0.2)
	x ₂	(0, 0.1)	(0.4, 0.5)	(0.4, 0.1)
	x ₃	(0.4, 0.3)	(0.5, 0.1)	(0.6, 0.3)
t ₂	x ₁	(0.6, 0.3)	(0.7, 0.1)	(0.6, 0.1)
	x ₂	(0.7, 0.2)	(0.1, 0.3)	(0.5, 0.3)
	x ₃	(0.5, 0.2)	(0.7, 0.2)	(0, 0.1)
t ₃	x ₁	(0.4, 0.5)	(0.6, 0.3)	(0.5, 0.4)
	x ₂	(0.5, 0.4)	(0.6, 0.1)	(0.7, 0.2)
	x ₃	(0.7, 0.1)	(0.5, 0.4)	(0.1, 0.5)

The results obtained by the proposed methods based on Method I and Method II are listed in Table 7. The details of process are the same as the ones in Example 5.1 and Example 5.2, so skipped. Whilst, Table 7 also shows the comparisons with some existing methods.

Table 7

A comparison of preference order for different methods for Example 5.3

Methods	Preference order
DIFWA [40]	x₁ > x₃ > x₂
DIFWG [35]	x₂ = x₃ > x₁
DIFWG^ϵ [22]	x₂ = x₃ > x₁
The proposed method based MDIFWG	x₁ > x₃ > x₂
The proposed method based ER algorithm	x₁ > x₃ > x₂

We can see from Table 7 that the DIF-MADM methods Gumus [22] and Wei [35] can not distinguish the preference order of alternatives x₂, x₃. The reason is that there is only one MD of IFNs is equal to 0, the aggregation MD of IFNs is 0 even if the MDs of n - 1 IFNs are not 0, which leads to inappropriate preference order of alternatives in this situation. However, we can see from above Table 7 that our DIF-MADM methods based on ER algorithm and MDIFWG operator can distinguish the preference order of alternatives x₁, x₂, x₃. It is also shown that our methods based on ER algorithm and MDIFWG operator can overcome effectively theDrawback B.

In Exa 5.3., if μ_α(t₁) (x₂) =1 at the period t₁, μ_α(t₂) (x₂) = μ_α(t₃) (x₂) =0, the modified decision matrix is shown as follows (Table 8):

Table 8

Modified individual IF decision matrix D_{t
_k} (k = 1, 2, 3)

years		A ₁	A ₂	A ₃
t ₁	x ₁	(0.2, 0.6)	(0.5, 0.2)	(0.7, 0.2)
	x ₂	(1, 0)	(0.4, 0.5)	(0.4, 0.1)
	x ₃	(0.4, 0.3)	(0.5, 0.1)	(0.6, 0.3)
t ₂	x ₁	(0.4, 0.3)	(0.7, 0.1)	(0.6, 0.1)
	x ₂	(0, 0.2)	(0.1, 0.3)	(0.5, 0.3)
	x ₃	(0.5, 0.2)	(0.7, 0.2)	(0, 0.1)
t ₃	x ₁	(0.2, 0.5)	(0.6, 0.3)	(0.5, 0.4)
	x ₂	(0, 0.4)	(0.6, 0.1)	(0.7, 0.2)
	x ₃	(0.5, 0.4)	(0.5, 0.4)	(0.1, 0.5)

The results obtained by the proposed methods based on Method I is x₂ > x₁ > x₃. The preference order of alternatives is the same with order obtained by Xu’s method based on DIFWA [40]. However, the preference order of alternatives obtained by Xu’s method [40] will be the same no matter how the non-MDs of A₁ regarding on x₂ change at period t₂, t₃, this situation is shown in Fig.1. Obviously, it is unreasonable.

As far as our proposed method I is concerned, the preference order will be changed with the NMD of A₁ regarding on x₂ change. For example, when the non-MDs of A₁ regarding on x₂ change at period t₂, t₃, the preference order of alternatives obtained by our proposed method I is shown in Fig. 2.

We can see from the above analysis, Fig. 1 and Fig. 2 that our proposed method I can overcome the Drawback A.

Fig.1

Ranking order sensitivity to the NMDs of x₂ with respect to the first attribute A₁ by Xu’s Method [40].

Fig.2

Ranking order sensitivity to the NMDs of x₂ with respect to the first attribute A₁ by our Method I.

Fig.3

Ranking order sensitivity to the MD and NMDs w.r.t. the first attribute A₁.

5.2. Sensitivity analysis

Baird pointed out that sensitivity analysis (SA) is the investigation of some potential changes and errors of rating values and their impact on the final ranking order. In this sub-section, we conduct some sensitivity analyses to analyze the impact of changing the membership and NMDs of the rating values on the alternatives ranking order based on Method I (DIF-MADM based on the ER algorithm).

For the original membership and NMDs α_{t
_k} = (μ_{ij,t_k}, ν_{ij,t_k}), because the sum of MD and the NMD of a IFN is not more than 1, so we can assume it is updated as (μ_{ij,t_k} + Δ_{ij,t_k}, ν_{ij,t_k} - Δ_{ij,t_k}), where μ_{ij,t_k} + Δ_{ij,t_k}, ν_{ij,t_k} - Δ_{ij,t_k} ∈ [0, 1]. Therefore, we can determine the step size Δ_{ij,t_k} according to the condition μ_{ij,t_k} + Δ_{ij,t_k}, ν_{ij,t_k} - Δ_{ij,t_k} ∈ [0, 1].

Now, we take Example 5.3 (Section 5.1) as an example, we can obtain the preference order of the alternatives by changing the membership and NMDs of three attributes, the details are shown in Figures 3-5, which also show the desirable alternatives will remain constant when the variation values of the membership and NMDs with respect to the three attributes vary in the range from 0.1 to 1. But regarding the range of MD and NMD, the ranking order of the two alternatives A₂ and A₃ will change with the membership and NMDs. It demonstrates that the alternatives A₂ and A₃ are more sensitive to membership and NMDs than A₁.

Fig.4

Ranking order sensitivity to the MD and NMDs w.r.t. the second attribute A₂.

Fig.5

Ranking order sensitivity to the MD and NMDs w.r.t. the third attribute A₃.

6. Conclusions

In this paper, the main findings of this present review are:

•we some drawbacks reviewed and analysed on the existing DIF-MADM methods;

•we propose a new DIF-MADM method based on the evidential reasoning (ER) algorithm in order to address some of those drawbacks, whilst, a new dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator is also introduced to address some other limits of the existing methods;

•some numerical examples are provided to illustrate the practicality, feasibility and robustness of the proposed two methods through, the comparative analysis with the existing DIF-MADM methods, along with some sensitivity analyses also carried out to analyse the distinct features of the proposed methods.

From the experimental results of several examples shown in Tables 3, 5, 7 and the comparative analysis, we can concluded that the proposed methods can surmount the drawbacks of some existing DIF-MADM methods, so have shown the good potential in handling DIF-MADM problem.

Many dynamic consensus methodological approaches have appeared in recent years, which have been reviewed and analysed in terms of their associated advantages and drawbacks by Dong et al [50, 51]. Whilst, Dong et al [50] have proposed some open problems on dynamic consensus process. So our future research works is to try to drive these open problems in group decision making.

Footnotes

Acknowledgments

This work is supported by National Natural Science Foundation of P.R.China (Grant no. 61673320, 61305074); Sichuan Province Youth Science and Technology Innovation Team (No. 2019JDTD0015); The Application Basic Research Plan Project of Sichuan Province (No.2017JY0199); The Scientific Research Project of Department of Education of Sichuan Province (18ZA0273, 15TD0027); The Scientific Research Project of Neijiang Normal University (18TD08).

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