A new method for prioritized multi-criteria group decision making with triangular intuitionistic fuzzy numbers

Abstract

The triangular intuitionistic fuzzy number (TIFN) is a special intuitionistic fuzzy set defined on the real number set. The multi-criteria group decision making (MCGDM) with the prioritization among the criteria is called the prioritized MCGDM. The aim of this paper is to develop some prioritized geometric operators for TIFNs and apply to the prioritized MCGDM problems with TIFNs. Firstly, the weighted possibility means of TIFNs are defined and hereby a new lexicographic approach is given to rank the TIFNs sufficiently considering the risk preference of decision maker. Then, three kinds of prioritized geometric operators for TIFNs are developed including the prioritized weighted geometric (TIFPWG) operator, prioritized ordered weighted geometric (TIFPOWG) operator and prioritized hybrid weighted geometric (TIFPHWG) operator of TIFNs. Utilized the TIFPWG operator, the criteria values of alternatives are aggregated into the individual overall values of alternatives, which are further integrated into the collective ones by the TIFPHWG operator. The ranking order of alternatives is generated according to the collective overall values of alternatives. Finally, the system analyst selection example is analyzed to demonstrate the applicability and validity of the proposed method.

Keywords

Multi-criteria group decision making triangular intuitionistic fuzzy number intuitionistic fuzzy set prioritized geometric operators weighted possibility mean

1 Introduction

Compared with the fuzzy set [35], the intuitionistic fuzzy set (IFS) [1] and interval-valued intuitionistic fuzzy set (IVIFS) [2] through adding an additional non-membership degree may express more abundant and flexible information. Many researchers have paid great attention to applications of IFS and IVIFS to the field of decision analysis ([14–16 , 38]. Since the real-life decision problems often involve many incomplete information and relate to many complex factors, there are lots of fuzzy, uncertain and ill-known quantities inherent in decision problems. The intuitionistic fuzzy numbers (IFNs) seem to suitably describe these ill-known quantities.

As a generalization of fuzzy numbers, IFN is defined with the universe as real line based on the concept of IFS. Shu et al. [10] introduced the concept of triangular IFN (TIFN), defined the operations of TIFNs and employed to intuitionistic fuzzy fault tree analysis. Li [6] corrected some errors in the four arithmetic operations over the TIFNs in [10]. Li [7] further presented the ranking method based on the ratio of the value index to the ambiguity index and employed to multi-attribute decision making (MADM) with TIFNs. Li et al. [8] defined the values and ambiguities of the membership and non-membership degrees for TIFNs as well as the value-index and ambiguity-index. Hereby a value and ambiguity based method is developed to rank TIFNs. Nan et al. [9] defined the ranking order relations of TIFNs and employed to matrix games with payoffs of TIFNs. Wan et al. [17] extended the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method for multi-attribute group decision making (MAGDM) with TIFNs. Wan and Li [13] proposed possibility mean and variance based method for MADM with TIFNs. Wang et al. [20] proposed new arithmetic operations and logic operators for TIFNs and applied them to fault analysis of a printed circuit board assembly system. Wan and Dong [18] developed the possibility method for MAGDM with TIFNs.

As the extensions of the TIFNs, Wang [19] defined trapezoidal IFN (TrIFN) and interval-valued trapezoidal IFN (IVTrIFN). Wang and Zhang [21] investigated the weighted arithmetic and weighted geometric averaging operators on TrIFNs and their applications to multicriteria decision making (MCDM) problems. Wu and Cao [23] developed some families of geometric aggregation operators with TrIFNs and applied to MAGDM problems. Zhang et al. [37] proposed a grey relational projection method for MAGDM based on TrIFNs. Wan [11] developed power average operators of TrIFNs and application to MAGDM. Wu and Liu [24] investigated the attitudinal score and accuracy expected functions for IVTrIFNs, defined some geometric operators for IVTrIFNs, and then proposed a method for MAGDM with IVTrIFNs.

The above research about IFNs mainly focuses on the operation laws, aggregation operators, ranking methods, decision making methods. Compared with the IFSs, TIFNs are defined by using triangular fuzzy numbers expressing their membership and non-membership functions. Hence, TIFNs may better reflect the uncertain and ill-known quantities of decision problems than IFSs. The existing methods about IFNs assume the attributes (or criteria) and the DMs are at the same priority level respectively. They are characterized by the ability to trade off between attributes. Using the above decision making methods, a decrease in satisfaction in one criterion can be compensated by an increase of another criterion.

However, in some applications, we may not allow this kind of compensation between criteria [26–30 , 33]. A typical example is in the case of buying a car based upon the criteria of safety and cost. In this case, usually we may not allow compensation between cost and safety. The prioritization relationship between the criteria is that safety has a higher priority than cost, which indicates that we are not willing to tradeoff satisfaction of criterion cost until perhaps we attain some level of satisfaction of safety [26 , 29]. Another example would be that the general president usually has a higher priority than vice president and the department manager in the decision making of a company. Consequently, it is necessary for us to pay attention to the multi-criteria group decision making (MCGDM) problems considering the prioritization among the criteria. Such kind of MCGDM is called prioritized MCGDM. Yager [26] first investigated prioritized MCDM problems using both the Bellman-Zadeh paradigm for MCDM and the ordered weighted averaging (OWA) operator method. Yager [27] introduced a prioritized scoring operator, a closely related prioritized average operator, the prioritized “anding” and “oring” operators for real numbers. Yager [28] proposed the prioritized OWA operator. Yan et al. [29] proposed the prioritized weighted aggregation operator based on OWA operator and triangular norms. These researches about prioritized aggregation operators [4 , 26–29] are only appropriate for the case in which the attribute values and weights are real numbers. To extend the prioritized aggregation operators to suit intuitionistic fuzzy situations, Yu and Xu [33] developed a prioritized intuitionistic fuzzy aggregation operator. Yu et al. [30] proposed the interval-valued intuitionistic fuzzy prioritized weighted average operator and prioritized weighted geometric operator. Yu [32] defined the triangular intuitionistic fuzzy prioritized weighted average and weighted geometric operators.

The motivations of this paper are summarized as follows: 1) The triangular intuitionistic fuzzy set studied in Yu [32] should be called the fuzzy number intuitionistic fuzzy set defined by Wang [22], whereas TIFNs studied in this paper come from Shu [10]. The fuzzy number intuitionistic fuzzy set and TIFN have significant distinction in essence. They are two different concepts; 2) Yu [32] only studied the prioritized arithmetic operators for fuzzy number intuitionistic fuzzy set. There is no investigation for the prioritized geometric aggregation operators of TIFNs; 3) The aforementioned methods can not be directly applied to prioritized MCGDM with TIFNs. The information integrating of TIFNs is very critical for solving prioritized MCGDM with TIFNs. Thus, we develop three kinds of prioritized geometric operators of TIFNs to integrate triangular intuitionistic fuzzy information which can sufficiently consider the prioritization among the criteria.

The most differences between Wan and Li [13] and this paper lie in two aspects: 1) The former did not consider the prioritization relationship between the criteria and studied the MADM, while the latter considers the prioritization relationship between the criteria and studied the prioritized MCGDM; 2) The ranking method of TIFNs proposed by the former is based on the ratios of the possibility mean to the possibility standard deviation. It ignored the risk preference of DM. In fact, introducing risk preference of DM to rank TIFNs is very necessary and reasonable. Thus, this paper incorporates risk preference of DM to define weighted possibility means and present a novel ranking method of TIFNs.

The main achievements of this work are outlined as follows: 1) Develop a new lexicographic approach to ranking the TIFNs sufficiently considering the risk preference of DM; 2) Investigate three kinds of prioritized geometric operators of TIFNs; 3) Propose a new method for prioritized MCGDM with TIFNs.

The rest of this paper unfolds as follows. In Section 2, the weighted possibility means of TIFNs are defined and thereby a new lexicographic approach is developed to rank the TIFNs. In Section 3, three kinds of prioritized geometric operators of TIFNs are investigated. A new method for the prioritized MCGDM problems with TIFNs is developed in Section 4. An example is analyzed in section 5. Short conclusions are made in Section 6.

2 The weighted possibility means and new ranking approach of TIFNs

This section is devoted to defining the weighted possibility means and ranking approach of TIFNs. The definition and the operation laws for positive TIFNs are referred to Li [7] in detail. The TIFNs discussed in this paper are all positive TIFNs. Denote the set of all positive TIFNs by Ω.

2.1 The weighted possibility mean of TIFNs

Definition 1. Let $\tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ be a TIFN and k ≥ 0. Then the power operation for TIFN is defined as ${\tilde{a}}^{k} = ((a_{-}^{k}, a^{k}, {\bar{a}}^{k}); ω_{\tilde{a}}, u_{\tilde{a}})$ .

The following properties hold:

${\tilde{a}}_{1} {\tilde{a}}_{2} = {\tilde{a}}_{2} {\tilde{a}}_{1}$ , $λ ({\tilde{a}}_{1} + {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} + λ {\tilde{a}}_{2}$ ;

$({\tilde{a}}_{1} {\tilde{a}}_{2})^{k} = ({\tilde{a}}_{1})^{k} ({\tilde{a}}_{2})^{k}$ , ${\tilde{a}}^{k_{1} + k_{2}} = {\tilde{a}}^{k_{1}} \cdot {\tilde{a}}^{k_{2}}$ , where k₁, k₂ ≥ 0

Definition 2. [7] Let $0 \leq α \leq ω_{\tilde{a}}$ , $u_{\tilde{a}} \leq β \leq 1$ and 0 ≤ α + β ≤ 1. A α- cut set and β- cut set of a TIFN $\tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ are respectively defined as ${\tilde{a}}_{α} = {x | μ_{\tilde{a}} (x) \geq α}$ and ${\tilde{a}}_{β} = {x | υ_{\tilde{a}} (x) \leq β}$ , which can be calculated as follows: $\begin{matrix} {\tilde{a}}_{α} & = & [a_{α}^{l}, a_{α}^{u}] \\ = & [\underline{a} + \frac{(a - \underline{a}) α}{ω_{\tilde{a}}}, \bar{a} - \frac{(\bar{a} - a) α}{ω_{\tilde{a}}}], \end{matrix}$ (1) $\begin{matrix} {\tilde{a}}_{β} & = & [a_{β}^{l}, a_{β}^{u}] \\ = & [\frac{(1 - β) a + (β - u_{\tilde{a}}) \underline{a}}{1 - u_{\tilde{a}}}, \frac{(1 - β) a + (β - u_{\tilde{a}}) \bar{a}}{1 - u_{\tilde{a}}}] . \end{matrix}$ (2)

Definition 3. [17] The f weighted lower and upper possibility means of membership function for a TIFN $\tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ are respectively defined as: $\begin{matrix} {\underline{m}}_{μ} (\tilde{a}) & = & \int_{0}^{ω_{\tilde{a}}} f (Pos [\tilde{a} \leq a_{α}^{l}]) a_{α}^{l} d α \\ = & \int_{0}^{ω_{\tilde{a}}} f (α) a_{α}^{l} d α, \end{matrix}$ (3) $\begin{matrix} {\bar{m}}_{μ} (\tilde{a}) & = & \int_{0}^{ω_{\tilde{a}}} f (Pos [\tilde{a} \geq a_{α}^{u}]) a_{α}^{u} d α \\ = & \int_{0}^{ω_{\tilde{a}}} f (α) a_{α}^{u} d α, \end{matrix}$ (4) where Pos means possibility [3, 5] and $f : [0, ω_{\tilde{a}}] \to R$ is a non-negative, monotone increasing weighting function satisfying the conditions: $\int_{0}^{ω_{\tilde{a}}} f (α) d α = ω_{\tilde{a}}$ and f (0) =0.

Definition 4. [17] The g weighted lower and upper possibility means of non-membership function for a TIFN $\tilde{a} = ((\underline{a}, a, \bar{a}); w_{\tilde{a}}, u_{\tilde{a}})$ respectively are defined as: $\begin{matrix} {\underline{m}}_{ν} (\tilde{a}) & = & \int_{u_{\tilde{a}}}^{1} g (Pos [\tilde{a} \leq a_{β}^{l}]) {\tilde{a}}_{β}^{l} d β \\ = & \int_{u_{\tilde{a}}}^{1} g (β) a_{β}^{l} d β, \end{matrix}$ (5) $\begin{matrix} {\bar{m}}_{ν} (\tilde{a}) & = & \int_{u_{\tilde{a}}}^{1} g (Pos [\tilde{a} \geq a_{β}^{u}]) a_{β}^{u} d β \\ = & \int_{u_{\tilde{a}}}^{1} g (β) a_{β}^{u} d β, \end{matrix}$ (6) where $g : [u_{\tilde{a}}, 1] \to R$ is a non-negative, monotone decreasing weighting function satisfying the conditions: $\int_{u_{\tilde{a}}}^{1} g (β) d β = 1 - u_{\tilde{a}}$ and g (1) =1.

Definition 5. For a TIFN $\tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ , the f weighted possibility means of membership function and g weighted possibility mean of non-membership function are, respectively, defined as:

$m_{μ} (\tilde{a}, θ) = (1 - θ) {\underline{m}}_{μ} (\tilde{a}) + θ {\bar{m}}_{μ} (\tilde{a}),$ (7) $m_{ν} (\tilde{a}, θ) = (1 - θ) {\underline{m}}_{ν} (\tilde{a}) + θ {\bar{m}}_{ν} (\tilde{a}),$ (8) where θ ∈ [0, 1] is the risk preference parameter of DM and can reflect different importance to the lower and upper possibility means. Different DMs have different preferences for the lower and upper possibility means. θ ∈ (0.5, 1] implies DM prefers the upper possibility mean, namely DM is pessimistic; θ ∈ [0, 0.5) shows DM prefers the lower possibility mean, namely DM is optimistic; θ = 0.5 indicates DM is indifferent between the lower and upper possibility means, namely DM is preference neutral.

If θ = 0, then $m_{μ} (\tilde{a}, 0) = {\underline{m}}_{μ} (\tilde{a})$ , $m_{ν} (\tilde{a}, 0) = {\underline{m}}_{ν} (\tilde{a})$ ; If θ = 1, then $m_{μ} (\tilde{a}, 1) = {\bar{m}}_{μ} (\tilde{a})$ and $m_{ν} (\tilde{a}, 1) = {\bar{m}}_{ν} (\tilde{a})$ ; If θ = 0.5, then $m_{μ} (\tilde{a}, 0.5) = \frac{1}{2} [{\underline{m}}_{μ} (\tilde{a}) + {\bar{m}}_{μ} (\tilde{a})]$ and $m_{ν} (\tilde{a}, 0.5) = \frac{1}{2} [{\underline{m}}_{ν} (\tilde{a}) + {\bar{m}}_{ν} (\tilde{a})]$ . If TIFN $\tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ degenerates to the triangular fuzzy number $(\underline{a}, a, \bar{a})$ , i.e., $ω_{\tilde{a}} = 1$ and $u_{\tilde{a}} = 0$ , then, $m_{μ} (\tilde{a}, 0.5)$ is just the f weighted possibility mean of fuzzy number defined in Definition 2 of [5]. Therefore, the g weighted possibility mean of fuzzy number defined in [5] is just a special case of that defined in this paper.

Obviously, $m_{μ} (\tilde{a}, θ)$ ( $m_{ν} (\tilde{a}, θ)$ ) synthetically characterizes the information on every membership (non-membership) degree, and $m_{μ} (\tilde{a}, 0.5)$ ( $m_{ν} (\tilde{a}, 0.5)$ ) may be viewed as a central value that represents from the angle of membership (non-membership) function.

Example 1. If f and g are respectively selected as:

$f (α) = 2 α / ω_{\tilde{a}} (α \in [0, ω_{\tilde{a}}]),$ (9) $g (β) = 2 (1 - β) / (1 - u_{\tilde{a}}) (β \in [u_{\tilde{a}}, 1]),$ (10) then, according to Equations (3–8), we have $m_{μ} (\tilde{a}, θ) = [(1 - θ) (\underline{a} + 2 a) + θ (2 a + \bar{a})] ω_{\tilde{a}} / 3,$ (11) $\begin{matrix} m_{ν} (\tilde{a}, θ) & = & [(1 - θ) (\underline{a} + 2 a) + θ (2 a + \bar{a})] (1 - u_{\tilde{a}}) / 3 . \end{matrix}$ (12)

Theorem 1. Assume that $\tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ and $\tilde{b} = ((\underline{b}, b, \bar{b}); ω_{\tilde{b}}, u_{\tilde{b}})$ are two TIFNs with $ω_{\tilde{a}} = ω_{\tilde{b}}$ and $u_{\tilde{a}} = u_{\tilde{b}}$ . Then for any λ, γ ∈ R, we have: $\begin{matrix} m_{μ} (λ \tilde{a} + γ \tilde{b}, θ) & = & λ m_{μ} (\tilde{a}, θ) + γ m_{μ} (\tilde{b}, θ), \\ m_{ν} (λ \tilde{a} + γ \tilde{b}, θ) & = & λ m_{ν} (\tilde{a}, θ) + γ m_{ν} (\tilde{b}, θ) . \end{matrix}$

Proof. It can be easily proven by Definition 3 and 7.

2.2 New lexicographic ranking method of TIFNs

A new lexicographic ranking method between two TIFNs ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ can be defined as follows:

If $m_{μ} ({\tilde{a}}_{1}, θ) < m_{μ} ({\tilde{a}}_{2}, θ)$ , then ${\tilde{a}}_{1}$ is smaller than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ ;

If $m_{μ} ({\tilde{a}}_{1}, θ) > m_{μ} ({\tilde{a}}_{2}, θ)$ , then ${\tilde{a}}_{1}$ is bigger than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

If $m_{μ} ({\tilde{a}}_{1}, θ) = m_{μ} ({\tilde{a}}_{2}, θ)$ , then

If $m_{v} ({\tilde{a}}_{1}, θ) < m_{v} ({\tilde{a}}_{2}, θ)$ , then ${\tilde{a}}_{1}$ is smaller than ${\tilde{a}}_{2}$ , denoted by ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ ;

If $m_{v} ({\tilde{a}}_{1}, θ) > m_{v} ({\tilde{a}}_{2}, θ)$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

If $m_{v} ({\tilde{a}}_{1}, θ) = m_{v} ({\tilde{a}}_{2}, θ)$ , then ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ represent the same information, denoted by ${\tilde{a}}_{1} = {\tilde{a}}_{2}$ .

Theorem 2. Let ${\tilde{a}}_{i} = (({\underline{a}}_{i}, a_{i}, {\bar{a}}_{i}); ω_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}})$ (i = 1, 2) be two TIFNs. If ${\underline{a}}_{1} \leq {\underline{a}}_{2}$ , a₁ ≤ a₂, ${\bar{a}}_{1} \leq {\bar{a}}_{2},$ $ω_{{\tilde{a}}_{1}} \leq ω_{{\tilde{a}}_{2}}$ , $u_{{\tilde{a}}_{1}} \geq u_{{\tilde{a}}_{2}}$ , then ${\tilde{a}}_{1} \leq {\tilde{a}}_{2}$ .

Proof. According to Equations (11) and (12), we have $\begin{matrix} [(1 - θ) ({\underline{a}}_{1} + 2 a_{1}) + θ (2 a_{1} + {\bar{a}}_{1})] ω_{{\tilde{a}}_{1}} / 3 \\ \leq [(1 - θ) ({\underline{a}}_{2} + 2 a_{2}) + θ (2 a_{2} + {\bar{a}}_{2})] ω_{{\tilde{a}}_{2}} / 3, \\ [(1 - θ) ({\underline{a}}_{1} + 2 a_{1}) + θ (2 a_{1} + {\bar{a}}_{1})] (1 - u_{{\tilde{a}}_{1}}) / 3 \\ \leq [(1 - θ) ({\underline{a}}_{2} + 2 a_{2}) + θ (2 a_{2} + {\bar{a}}_{2})] (1 - u_{{\tilde{a}}_{2}}) / 3 . \end{matrix}$

Namely, $m_{μ} ({\tilde{a}}_{1}, θ) \leq m_{μ} ({\tilde{a}}_{2}, θ)$ , $m_{v} ({\tilde{a}}_{1}, θ) \leq m_{v} ({\tilde{a}}_{2}, θ)$ . Thus, ${\tilde{a}}_{1} \leq {\tilde{a}}_{2}$ .

Remark 1. Nan et al. [9] utilized the membership function average index $S_{μ} (\tilde{a}) = ω_{\tilde{a}} (\underline{a} + 2 a + \bar{a}) / 4$ and the non-membership function average index $S_{v} (\tilde{a}) = (1 - u_{\tilde{a}}) (\underline{a} + 2 a + \bar{a}) / 4$ of TIFN $\tilde{a}$ to rank TIFNs. It is easily seen from Equations (11) and (12) that $S_{μ} (\tilde{a})$ and $S_{v} (\tilde{a})$ are respectively equal to $m_{μ} (\tilde{a}, θ)$ and $m_{v} (\tilde{a}, θ)$ if $θ = (3 \bar{a} - 2 a - \underline{a}) / [4 (\bar{a} - \underline{a})]$ . Therefore, the ranking approach [9] may be regarded as a special case of the proposed approach in this paper.

Example 2. Consider two TIFNs ${\tilde{a}}_{1} = ((3, 6, 8); 0.62, 0.22)$ and ${\tilde{a}}_{2} = ((5, 7, 9); 0.52, 0.26)$ . By Equations (11) and (12), we obtain ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ if θ > 0.5686; ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ if θ < 0.5686. However, using the ranking approach of [9], we can calculate $S_{μ} ({\tilde{a}}_{1}) = 3.6$ , $S_{v} ({\tilde{a}}_{1}) = 4.6$ , $S_{μ} ({\tilde{a}}_{2}) = 3.6$ , $S_{v} ({\tilde{a}}_{2}) = 5.2$ . Because $S_{μ} ({\tilde{a}}_{1}) < S_{μ} ({\tilde{a}}_{2})$ , the ranking order obtained by [9] is ${\tilde{a}}_{1} < {\tilde{a}}_{2}$ , which is the same as the ranking order of the case θ < 0.5686. Namely, the ranking result obtained by [9] is just a special case of that obtained by this paper.

Example 2 implies that when choosing different risk preference parameter θ we can get different ranking orders. The proposed new ranking approach sufficiently takes into the risk preference of DM account, which can make the ranking result more reasonable. In contrast, the ranking approaches [7 –9] failed to consider the risk preference of DM.

3 Some prioritized geometric operators of TIFNs

In this section, suppose that all TIFNs ${\tilde{a}}_{i} = (({\underline{a}}_{i}, a_{i}, {\bar{a}}_{i}); ω_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}})$ are normalized, i.e., $0 \leq {\underline{a}}_{i} \leq a_{i} \leq {\bar{a}}_{i} \leq 1$ (i = 1, 2, …, n), then the maximum TIFN is ${\tilde{a}}^{max} = ((1, 1, 1); 1, 0)$ and the minimum TIFN is ${\tilde{a}}^{min} = ((0, 0, 0); 0, 1)$ . Some prioritized geometric operators of TIFNs will be developed and their desirable properties are also investigated in detail.

3.1 Prioritized weighted geometric operator of TIFNs

Yager [27] introduced a class of aggregation operators called prioritized average (PA) operators for real numbers as follows:

Definition 6. [27] Let C = {c₁, c₂, …, c_n} be a collection of criteria with the linear ordering c₁ ≻ c₂ ≻ … ≻ c_n (i.e., c_j has a higher priority than c_k if j < k). The value c_j (x) is the satisfaction of any alternative x under c_j and satisfies c_j (x) ∈ [0, 1]. A PA operator is a function: $PA (c_{1} (x), c_{2} (x), \dots, c_{n} (x)) = \sum_{j = 1}^{n} w_{j} c_{j} (x),$ (13) where $w_{j} = T_{j} / \sum_{i = 1}^{n} T_{i}$ , $T_{j} = \prod_{k = 1}^{j} c_{k - 1} (x)$ , T₁ = c₀ (x) =1.

Motivated by the PA operator and geometric aggregation operator, we extend the PA operator to define the prioritized weighted geometric (TIFPWG) operator of TIFNs. Firstly, suppose that the TIFN ${\tilde{a}}_{i}$ is considered as the satisfaction of alternative x under criterion c_i. There is a linear priority order c₁ ≻ c₂ ≻ … ≻ c_n.

Definition 7. A prioritized weighted geometric operator of TIFNs is mapping TIFPWG: Ωⁿ → Ω and ${TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{j = 1}^{n} ({\tilde{a}}_{j})^{w_{j}},$ where w = (w₁, w₂, …, w_n) ^T is the priority based weight vector and can be determined as follows:

Step 1: Calculate the weighted possibility mean $m_{μ} ({\tilde{a}}_{k}, θ)$ of membership function of ${\tilde{a}}_{k}$ . Due to the fact that all TIFNs ${\tilde{a}}_{k}$ are already normalized, it yields by Equation (11) that $m_{μ} ({\tilde{a}}_{k}, θ) \in [0, 1]$ (k = 1, 2, …, n).

Step 2: Compute the un-normalized priority based weights: T₁ = 1, $T_{j} = \prod_{k = 1}^{j} m_{μ} ({\tilde{a}}_{k - 1}, θ)$ (j = 2, 3, …, n).

Step 3: Normalize the T_j (j = 1, 2, …, n) to obtain the normalized priority based weights: $w_{j} = T_{j} / \sum_{i = 1}^{n} T_{i} (j = 1, 2, \dots, n) .$ (14)

Theorem 3. The aggregated value for TIFNs ${\tilde{a}}_{i}$ (i = 1, 2, …, n) by using TIFPWG operator is also a TIFN, and

$\begin{matrix} {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = ((\prod_{j = 1}^{n} ({\underline{a}}_{j})^{w_{j}}, \prod_{j = 1}^{n} (a_{j})^{w_{j}}, \prod_{j = 1}^{n} ({\bar{a}}_{j})^{w_{j}}); \\ \underset{j}{\land} {ω_{{\tilde{a}}_{j}}}, \underset{j}{\lor} {u_{{\tilde{a}}_{j}}}) . \end{matrix}$ (15)

The TIFPWG operator has some desirable properties as follows.

Theorem 4. (Idempotency). If all TIFNs ${\tilde{a}}_{i}$ (i = 1, 2, …, n) are equal, i.e., ${\tilde{a}}_{1} = {\tilde{a}}_{2} = \dots = {\tilde{a}}_{n} = \tilde{a}$ , then ${TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ .

Proof. Since $w_{j} = T_{j} / \sum_{i = 1}^{n} T_{i}$ , we have $\sum_{j = 1}^{n} w_{j} = 1$ . Thus, ${TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{j = 1}^{n} {\tilde{a}}_{j}^{w_{j}} = \prod_{j = 1}^{n} {\tilde{a}}^{w_{j}} = \tilde{a} .$

It directly follows from Theorem 4 that the following Corollary 1 holds.

Corollary 1. If all TIFNs ${\tilde{a}}_{i} = {\tilde{a}}^{max}$ (i = 1, 2, …, n), then ${TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\tilde{a}}^{max}$ .

Corollary 2. (Non-compensatory). If TIFN ${\tilde{a}}_{1} = {\tilde{a}}^{min}$ , then ${TIFPWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\tilde{a}}^{min}$ .

Proof. Using Equation (11), we get $\begin{matrix} m_{μ} ({\tilde{a}}_{1}, θ) = m_{μ} ({\tilde{a}}^{min}, θ) \\ = \frac{1}{3} [(1 - θ) (\underline{a} + 2 a) + θ (2 a + \bar{a})] ω_{{\tilde{a}}^{min}} = 0 . \end{matrix}$

Since T₁ = 1, $T_{j} = \prod_{k = 1}^{j} m_{μ} ({\tilde{a}}_{k - 1}, θ) = 1 \times 0 \times_{μ} ({\tilde{a}}_{2}, θ) \times \dots \times m_{μ} ({\tilde{a}}_{j - 1}, θ) = 0$ for j = 2, 3, …, n, we have w₁ = 1, $w_{j} = T_{j} / \sum_{i = 1}^{n} T_{i} = 0$ for j = 2, 3, …, n. Thus, ${TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{j = 1}^{n} ({\tilde{a}}_{j})^{w_{j}} = {\tilde{a}}_{1} = {\tilde{a}}^{min} .$

Corollary 2 indicates that when the satisfaction ${\tilde{a}}_{1}$ to the highest order criterion c₁ which owns the highest priority is the smallest TIFN, we cannot get any compensation from other criteria even if they are satisfied. Therefore, poor satisfaction to any higher criterion reduces the ability for compensation by lower priority criteria. This is of course the fundamental feature of the prioritization relationship [27].

Theorem 5. (Boundedness). Let $\begin{matrix} {\tilde{a}}^{+} & = & ((max {{\underline{a}}_{i}}, max {a_{i}}, max {{\bar{a}}_{i}}); \\ max {ω_{{\tilde{a}}_{i}}}, min {u_{{\tilde{a}}_{i}}}), \\ {\tilde{a}}^{-} & = & ((min {{\underline{a}}_{i}}, min {a_{i}}, min {{\bar{a}}_{i}}); \\ min {ω_{{\tilde{a}}_{i}}}, max {u_{{\tilde{a}}_{i}}}) . \end{matrix}$

Then, ${\tilde{a}}^{-} \leq {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ .

Proof. Since $w_{j} = T_{j} / \sum_{i = 1}^{n} T_{i}$ , we have $\sum_{j = 1}^{n} w_{j} = 1$ and $\begin{matrix} min {{\underline{a}}_{j}} & = & \prod_{j = 1}^{n} (min {{\underline{a}}_{j}})^{w_{j}} \leq \prod_{j = 1}^{n} \underset{-_{j}^{w_{j}}}{a} \\ \leq & \prod_{j = 1}^{n} (max {{\underline{a}}_{j}})^{w_{j}} = max {\underset{-_{j}}{a}}, \\ min {a_{j}} & = & \prod_{j = 1}^{n} (min {a_{j}})^{w_{j}} \leq \prod_{j = 1}^{n} a_{j}^{w_{j}} \\ \leq & \prod_{j = 1}^{n} (max {a_{j}})^{w_{j}} = max {a_{j}}, \\ min {{\bar{a}}_{j}} & = & \prod_{j = 1}^{n} (min {{\bar{a}}_{j}})^{w_{j}} \leq \prod_{j = 1}^{n} {\bar{a}}_{j}^{w_{j}} \\ \leq & \prod_{j = 1}^{n} (max {{\bar{a}}_{j}})^{w_{j}} = max {{\bar{a}}_{j}} . \end{matrix}$

Let ${TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a} = ((\underline{a}, a, \bar{a}); ω_{\tilde{a}}, u_{\tilde{a}})$ , then by Theorem 3, it yields that $\underline{a} = \prod_{j = 1}^{n} ({\underline{a}}_{j})^{w_{j}}$ , $a = \prod_{j = 1}^{n} (a_{j})^{w_{j}}$ , $\bar{a} = \prod_{j = 1}^{n} ({\bar{a}}_{j})^{w_{j}}$ , $ω_{\tilde{a}} = \underset{j}{\land} {ω_{{\tilde{a}}_{j}}}$ , $u_{\tilde{a}} = \underset{j}{\lor} {u_{{\tilde{a}}_{j}}}$ .

Since $\begin{matrix} m_{μ} (\tilde{a}, θ) & = & \frac{1}{3} [(1 - θ) (\underline{a} + 2 a) + θ (2 a + \bar{a})] ω_{\tilde{a}} \\ \leq & \frac{1}{3} [(1 - θ) (max {{\underline{a}}_{j}} + 2 max {a_{j}}) \\ + θ (2 max {a_{j}} + max {{\bar{a}}_{j}})] max {ω_{{\tilde{a}}_{i}}} \\ = & m_{μ} ({\tilde{a}}^{+}, θ), \\ m_{μ} (\tilde{a}, θ) & = & \frac{1}{3} [(1 - θ) (\underline{a} + 2 a) + θ (2 a + \bar{a})] ω_{\tilde{a}} \\ \geq & \frac{1}{3} [(1 - θ) (min {{\underline{a}}_{j}} + 2 min {a_{j}}) \\ + θ (2 min {a_{j}} + min {{\bar{a}}_{j}})] min {ω_{{\tilde{a}}_{i}}} \\ = & m_{μ} ({\tilde{a}}^{-}, θ), \end{matrix}$ thus ${\tilde{a}}^{-} \leq \tilde{a} \leq {\tilde{a}}^{+}$ , which completes the proof.

Theorem 6. (Monotonicity) For two sequence s of ${\tilde{a}}_{i} = (({\underline{a}}_{i}, a_{i}, {\bar{a}}_{i}); ω_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}})$ and ${\tilde{b}}_{i} = (({\underline{b}}_{i}, b_{i}, {\bar{b}}_{i}); ω_{{\tilde{b}}_{i}}, u_{{\tilde{b}}_{i}})$ (i = 1, 2, …, n), if ${\underline{a}}_{j} \leq {\underline{b}}_{j}$ , a_j ≤ b_j, ${\bar{a}}_{j} \leq {\bar{b}}_{j}$ , $ω_{{\tilde{a}}_{j}} \leq ω_{{\tilde{b}}_{j}}$ , $u_{{\tilde{a}}_{j}} \geq u_{{\tilde{b}}_{j}}$ (j = 1, 2, …, n), then, $\begin{matrix} {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {TIFPWG}_{w} ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}), \end{matrix}$ where the associated weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (14).

Proof. Since ${\underline{a}}_{j} \leq {\underline{b}}_{j}$ , a_j ≤ b_j, ${\bar{a}}_{j} \leq {\bar{b}}_{j}$ , $ω_{{\tilde{a}}_{j}} \leq ω_{{\tilde{b}}_{j}}$ , $u_{{\tilde{a}}_{j}} \geq u_{{\tilde{b}}_{j}}$ and w_j ≥ 0, we have $\begin{matrix} \frac{1}{3} [(1 - θ) (\prod_{j = 1}^{n} {\underline{a}}_{j}^{w_{j}} + 2 \prod_{j = 1}^{n} a_{j}^{w_{j}}) \\ + θ (2 \prod_{j = 1}^{n} a_{j}^{w_{j}} + \prod_{j = 1}^{n} {\bar{a}}_{j}^{w_{j}}] ω_{{\tilde{a}}_{j}} \\ \leq \frac{1}{3} [(1 - θ) (\prod_{j = 1}^{n} {\underline{b}}_{j}^{w_{j}} + 2 \prod_{j = 1}^{n} b_{j}^{w_{j}}) \\ + θ (2 \prod_{j = 1}^{n} b_{j}^{w_{j}} + \prod_{j = 1}^{n} {\bar{b}}_{j}^{w_{j}})] ω_{{\tilde{b}}_{j}} . \end{matrix}$

Thus, it yields from Theorem 3 that $\prod_{j = 1}^{n} ({\tilde{a}}_{σ (j)})^{w_{j}} \leq \prod_{j = 1}^{n} ({\tilde{b}}_{σ (j)})^{w_{j}}$ , which has proven Theorem 6.

Theorem 7. For two sequence s of TIFNs ${\tilde{a}}_{i}$ and ${\tilde{b}}_{i}$ (i = 1, 2, …, n), we have $\begin{matrix} {TIFPWG}_{w} ({\tilde{a}}_{1} {\tilde{b}}_{1}, {\tilde{a}}_{2} {\tilde{b}}_{2}, \dots, {\tilde{a}}_{n} {\tilde{b}}_{n}) \\ = {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \cdot {TIFPWG}_{w} ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}), \end{matrix}$ where the priority based weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (14).

Proof. By Theorem 3 and Definitions 1 and 7, $\begin{matrix} {TIFPWG}_{w} ({\tilde{a}}_{1} {\tilde{b}}_{1}, {\tilde{a}}_{2} {\tilde{b}}_{2}, \dots, {\tilde{a}}_{n} {\tilde{b}}_{n}) = \prod_{j = 1}^{n} ({\tilde{a}}_{j} {\tilde{b}}_{j})^{w_{j}} \\ = ((\prod_{j = 1}^{n} ({\underline{a}}_{j} {\underline{b}}_{j})^{w_{j}}, \prod_{j = 1}^{n} (a_{j} b_{j})^{w_{j}}, \prod_{j = 1}^{n} ({\bar{a}}_{j} {\bar{b}}_{j})^{w_{j}}); \\ min {\underset{j}{\land} {ω_{{\tilde{a}}_{j}}}, \underset{j}{\land} {ω_{{\tilde{b}}_{j}}}}, max {\underset{j}{\lor} {u_{{\tilde{a}}_{j}}}, \underset{j}{\lor} {u_{{\tilde{b}}_{j}}}}) \\ = ((\prod_{j = 1}^{n} ({\underline{a}}_{j})^{w_{j}}, \prod_{j = 1}^{n} (a_{j})^{w_{j}}, \prod_{j = 1}^{n} ({\bar{a}}_{j})^{w_{j}}); \underset{j}{\land} {ω_{{\tilde{a}}_{j}}}, \underset{j}{\lor} {u_{{\tilde{a}}_{j}}}) \\ \cdot ((\prod_{j = 1}^{n} ({\underline{b}}_{j})^{w_{j}}, \prod_{j = 1}^{n} (b_{j})^{w_{j}}, \prod_{j = 1}^{n} ({\bar{b}}_{j})^{w_{j}}); \underset{j}{\land} {ω_{{\tilde{b}}_{j}}}, \underset{j}{\lor} {u_{{\tilde{b}}_{j}}}) \\ = {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \cdot {TIFPWG}_{w} ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}) . \end{matrix}$

Hence, the proof of Theorem 7 is completed.

Theorem 8. Let $\tilde{b} = ((\underline{b}, b, \bar{b}); ω_{\tilde{b}}, u_{\tilde{b}})$ be another TIFN. Then, $\begin{matrix} {TIFPWG}_{w} ({\tilde{a}}_{1} \tilde{b}, {\tilde{a}}_{2} \tilde{b}, \dots, {\tilde{a}}_{n} \tilde{b}) \\ = {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \cdot \tilde{b}, \end{matrix}$ where the priority based weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (14).

Proof. By Theorem 3 and Definitions 1 and 7, we have $\begin{matrix} {TIFPWG}_{w} ({\tilde{a}}_{1} \tilde{b}, {\tilde{a}}_{2} \tilde{b}, \dots, {\tilde{a}}_{n} \tilde{b}) = \prod_{j = 1}^{n} ({\tilde{a}}_{j} \tilde{b})^{w_{j}} \\ = ((\prod_{j = 1}^{n} ({\underline{a}}_{j} \underline{b})^{w_{j}}, \prod_{j = 1}^{n} (a_{j} b)^{w_{j}}, \prod_{j = 1}^{n} ({\bar{a}}_{j} \bar{b})^{w_{j}}); \\ min {\underset{j}{\land} {ω_{{\tilde{a}}_{j}}}, ω_{\tilde{b}}}, max {\underset{j}{\lor} {u_{{\tilde{a}}_{j}}}, u_{\tilde{b}}}) \\ = ((\prod_{j = 1}^{n} (a_{j})^{w_{j}}, \prod_{j = 1}^{n} (a_{j})^{w_{j}}, \prod_{j = 1}^{n} ({\bar{a}}_{j})^{w_{j}}); \\ \underset{j}{\land} {ω_{{\tilde{a}}_{j}}}, \underset{j}{\lor} {u_{{\tilde{a}}_{j}}}) \cdot ((\underline{b}, b, \bar{b}); ω_{\tilde{b}}, u_{\tilde{b}}) \\ = {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \cdot \tilde{b}, \end{matrix}$ which ends the proof of Theorem 8.

Theorem 9. For λ ≥ 0, we have $\begin{matrix} {TIFPWG}_{w} (λ {\tilde{a}}_{1}, λ {\tilde{a}}_{2}, \dots, λ {\tilde{a}}_{n}) \\ = λ \cdot {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}), \\ {TIFPWG}_{w} ({\tilde{a}}_{1}^{λ}, {\tilde{a}}_{2}^{λ}, \dots, {\tilde{a}}_{n}^{λ}) \\ = ({TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}))^{λ}, \end{matrix}$ where the priority based weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (14).

Proof. By Theorem 3 and Definition 1, we have $\begin{matrix} {TIFPWG}_{w} (λ {\tilde{a}}_{1}, λ {\tilde{a}}_{2}, \dots, λ {\tilde{a}}_{n}) = \prod_{j = 1}^{n} (λ {\tilde{a}}_{j})^{w_{j}} \\ = λ \prod_{j = 1}^{n} ({\tilde{a}}_{j})^{w_{j}} = λ \cdot {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}), \\ {TIFPWG}_{w} ({\tilde{a}}_{1}^{λ}, {\tilde{a}}_{2}^{λ}, \dots, {\tilde{a}}_{n}^{λ}) = \prod_{j = 1}^{n} ({\tilde{a}}_{j}^{λ})^{w_{j}} \\ = (\prod_{j = 1}^{n} ({\tilde{a}}_{j})^{w_{j}})^{λ} = ({TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}))^{λ} . \end{matrix}$

Hence, the proof of Theorem 9 is completed.

Combined with Theorems 6–9, the following Theorem 10 can be easily proven.

Theorem 10. Let ${\tilde{a}}_{i}$ and ${\tilde{b}}_{i}$ (i = 1, 2, …, n) be two sequence s of TIFNs and $\tilde{b} = ((\underline{b}, b, \bar{b}); ω_{\tilde{b}}, u_{\tilde{b}})$ be another TIFN. If λ ≥ 0, then, $\begin{matrix} 1) & {TIFPWG}_{w} (λ {\tilde{a}}_{1} {\tilde{b}}_{1}, {\tilde{a}}_{2} {\tilde{b}}_{2}, \dots, {\tilde{a}}_{n} {\tilde{b}}_{n}) \\ = λ {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \cdot {TIFPWG}_{w} ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}), \end{matrix}$ $\begin{matrix} 2) & {TIFPWG}_{w} (λ {\tilde{a}}_{1} \tilde{b}, λ {\tilde{a}}_{2} \tilde{b}, \dots, λ {\tilde{a}}_{n} \tilde{b}) \\ = λ \cdot {TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \cdot \tilde{b}, \end{matrix}$ $\begin{matrix} 3) & {TIFPWG}_{w} (({\tilde{a}}_{1} {\tilde{b}}_{1})^{λ}, ({\tilde{a}}_{2} {\tilde{b}}_{2})^{λ}, \dots, ({\tilde{a}}_{n} {\tilde{b}}_{n})^{λ}) \\ = ({TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}))^{λ} \\ ({TIFPWG}_{w} ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}))^{λ}, \end{matrix}$ $\begin{matrix} 4) & {TIFPWG}_{w} (({\tilde{a}}_{1} \tilde{b})^{λ}, ({\tilde{a}}_{2} \tilde{b})^{λ}, \dots, ({\tilde{a}}_{n} \tilde{b})^{λ}) \\ = ({TIFPWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}))^{λ} \cdot {\tilde{b}}^{λ} . \end{matrix}$

3.2 Prioritized ordered weighted geometric operator of TIFNs

Based on the PA operator, Yager [28] further introduced the prioritized OWA (POWA) operator.

Definition 8. [28] Let σ (j) be the index of the jth largest of the c_j (x) ∈ [0, 1] (j = 1, 2, …, n). A prioritized ordered weighted average operator is a function:

$POWA (c_{1} (x), c_{2} (x), \dots, c_{n} (x)) = \sum_{j = 1}^{n} w_{j} c_{σ (j)} (x),$ where w = (w₁, w₂, …, w_n) ^T is the associated OWA weight vector.

Motivated by the POWA operator, the prioritized ordered weighted geometric (TIFPOWG) operator of TIFNs is developed below.

Definition 9. A prioritized ordered weighted geometric operator of TIFNs is mapping TIFPOWG: Ωⁿ → Ω and

${TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{j = 1}^{n} ({\tilde{a}}_{σ (j)})^{w_{j}},$ where w = (w₁, w₂, …, w_n) ^T is the associated weight vector with TIFPOWG and can be determined involving the following steps:

Step 1: Let $S_{i} = m_{μ} ({\tilde{a}}_{i}, θ)$ for each criterion c_i.

Step 2: The un-normalized priority based importance weights are obtained as T₁ = 1, $T_{j} = \prod_{k = 1}^{j} S_{k - 1}$ for j = 2, 3, …, n, the normalized priority based importance weights are obtained as $r_{j} = T_{j} / \sum_{i = 1}^{n} T_{i}$ (j = 1, 2, …, n).

Step 3: Let R₀ = 0, $R_{k} = \sum_{j = 1}^{k} r_{σ (j)}$ (k = 1, 2, …, n), where σ (j) is the index of the jth largest of the TIFNs ${\tilde{a}}_{i}$ (i = 1, 2, …, n) and f is a basic unit-interval monotonic (BUM) function, Thus, the associated weight vector w can be calculated, where $w_{k} = f (R_{k}) - f (R_{k - 1}) (k = 1, 2, \dots, n) .$ (16)

Similar to Theorem 3, we have the following result.

Theorem 11. Let ${\tilde{a}}_{σ (j)} = (({\underline{a}}_{σ (j)}, a_{σ (j)}, {\bar{a}}_{σ (j)}); ω_{{\tilde{a}}_{σ (j)}}, u_{{\tilde{a}}_{σ (j)}})$ be the jth largest of the TIFNs ${\tilde{a}}_{i}$ (i = 1, 2, …, n), then the aggregated value by using TIFPOWG operator is also a TIFN as follows:

$\begin{matrix} {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = ((\prod_{j = 1}^{n} ({\underline{a}}_{σ (j)})^{w_{j}}, \prod_{j = 1}^{n} (a_{σ (j)})^{w_{j}}, \\ \prod_{j = 1}^{n} ({\bar{a}}_{σ (j)})^{w_{j}}); \underset{j}{\land} {ω_{{\tilde{a}}_{σ (j)}}}, \underset{j}{\lor} {u_{{\tilde{a}}_{σ (j)}}}) . \end{matrix}$ (17)

The TIFPOWG operator has some desirable properties given below.

Theorem 12. (Idempotency). If all TIFNs ${\tilde{a}}_{i}$ (i = 1, 2, …, n) are equal, i.e., ${\tilde{a}}_{1} = {\tilde{a}}_{2} = \dots = {\tilde{a}}_{n} = \tilde{a}$ , then ${TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ .

Proof. Since $r_{j} = T_{j} / \sum_{i = 1}^{n} T_{i}$ and $R_{k} = \sum_{j = 1}^{k} r_{σ (j)}$ , we have $R_{n} = \sum_{j = 1}^{n} r_{σ (j)} = 1$ . Since w_k = f (R_k) - f (R_k-1) , we get $\sum_{j = 1}^{n} w_{j} = 1 .$ Thus, if ${\tilde{a}}_{1} = {\tilde{a}}_{2} = \dots = {\tilde{a}}_{n} = \tilde{a}$ , then $\begin{matrix} {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \prod_{j = 1}^{n} ({\tilde{a}}_{σ (j)})^{w_{j}} = \prod_{j = 1}^{n} {\tilde{a}}^{w_{j}} = \tilde{a}, \end{matrix}$ which completes the proof of Theorem 12.

Theorem 13. (Boundedness) TIFNs ${\tilde{a}}^{-}$ and ${\tilde{a}}^{+}$ are defined as in Theorem 5 , t hen, ${\tilde{a}}^{-} \leq {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+} .$

Proof. The proof is similar to Theorem 5.

Theorem 14. (Commutativity) If is any permutation of $({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n})$ , then $\begin{array}{l} {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2},, {\tilde{a}}_{1 n}) \\ = {TIFPOWG}_{w} ({\overset{\dot{~}}{a}}_{1}, {\overset{\dot{~}}{a}}_{2},, {\overset{\dot{~}}{a}}_{n}) . \end{array}$

Theorem 15. (Monotonicity) Denote ${\tilde{a}}_{σ (j)} = (({\underline{a}}_{σ (j)}, a_{σ (j)}, {\bar{a}}_{σ (j)}); ω_{{\tilde{a}}_{σ (j)}}, u_{{\tilde{a}}_{σ (j)}})$ and ${\tilde{b}}_{σ (j)} = (({\underline{b}}_{σ (j)}, b_{σ (j)}, {\bar{b}}_{σ (j)}); ω_{{\tilde{a}}_{σ (j)}}, u_{{\tilde{b}}_{σ (j)}})$ be the jth largest of the TIFNs ${\tilde{a}}_{i}$ and ${\tilde{b}}_{i}$ (i = 1, 2, …, n). If ${\underline{a}}_{σ (j)} \leq {\underline{b}}_{σ (j)}$ , a_σ(j) ≤ b_σ(j), ${\bar{a}}_{σ (j)} \leq {\bar{b}}_{σ (j)}$ , $ω_{{\tilde{a}}_{σ (j)}} \leq ω_{{\tilde{b}}_{σ (j)}}$ , $u_{{\tilde{a}}_{σ (j)}} \geq u_{{\tilde{b}}_{σ (j)}}$ (j = 1, 2, …, n), then, $\begin{matrix} {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq {TIFPOWG}_{w} ({\tilde{b}}_{1}, {\tilde{b}}_{2}, \dots, {\tilde{b}}_{n}), \end{matrix}$ where the associated weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (16).

Proof. Since ${\underline{a}}_{σ (j)} \leq {\underline{b}}_{σ (j)}$ , a_σ(j) ≤ b_σ(j), ${\bar{a}}_{σ (j)} \leq {\bar{b}}_{σ (j)}$ , $ω_{{\tilde{a}}_{σ (j)}} \leq ω_{{\tilde{b}}_{σ (j)}}$ , $u_{{\tilde{a}}_{σ (j)}} \geq u_{{\tilde{b}}_{σ (j)}}$ and w_j ≥ 0, we have $\begin{matrix} \frac{1}{3} [(1 - θ) (\prod_{j = 1}^{n} ({\underline{a}}_{σ (j)})^{w_{j}} + 2 \prod_{j = 1}^{n} (a_{σ (j)})^{w_{j}}) \\ + θ (2 \prod_{j = 1}^{n} (a_{σ (j)})^{w_{j}} + \prod_{j = 1}^{n} ({\bar{a}}_{σ (j)})^{w_{j}})] ω_{{\tilde{a}}_{σ (j)}} \\ \leq \frac{1}{3} [(1 - θ) (\prod_{j = 1}^{n} ({\underline{b}}_{σ (j)})^{w_{j}} + 2 \prod_{j = 1}^{n} (b_{σ (j)})^{w_{j}}) \\ + θ (2 \prod_{j = 1}^{n} (b_{σ (j)})^{w_{j}} + \prod_{j = 1}^{n} ({\bar{b}}_{σ (j)})^{w_{j}})] ω_{{\tilde{b}}_{σ (j)}} . \end{matrix}$

Thus, $\prod_{j = 1}^{n} ({\tilde{a}}_{σ (j)})^{w_{j}} \leq \prod_{j = 1}^{n} ({\tilde{b}}_{σ (j)})^{w_{j}}$ , which has proven Theorem 15.

Theorem 16. Let $\tilde{b} = ((\underline{b}, b, \bar{b}); ω_{\tilde{b}}, u_{\tilde{b}})$ be another TIFN. If $ω_{{\tilde{a}}_{i}} = ω_{\tilde{b}}$ , $u_{{\tilde{a}}_{i}} = u_{b}$ . (i = 1, 2, …, n), then, $\begin{matrix} {TIFPOWG}_{w} ({\tilde{a}}_{1} \tilde{b}, {\tilde{a}}_{2} \tilde{b}, \dots, {\tilde{a}}_{n} \tilde{b}) \\ = {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \cdot \tilde{b} . \end{matrix}$

Proof. First, we prove that if ${\tilde{a}}_{i} \leq {\tilde{a}}_{j}$ , then ${\tilde{a}}_{i} \tilde{b} \leq {\tilde{a}}_{j} \tilde{b}$ .

Since ${\tilde{a}}_{i} \leq {\tilde{a}}_{j}$ , we have $\begin{matrix} m_{μ} ({\tilde{a}}_{i}, θ) = \frac{1}{3} [(1 - θ) ({\underline{a}}_{i} + 2 a_{i}) + θ (2 a_{i} + {\bar{a}}_{i})] ω_{{\tilde{a}}_{i}} \\ \leq m_{μ} ({\tilde{a}}_{j}, θ) = \frac{1}{3} [(1 - θ) ({\underline{a}}_{j} + 2 a_{j}) \\ + θ (2 a_{j} + {\bar{a}}_{j})] ω_{{\tilde{a}}_{j}}, \\ m_{μ} ({\tilde{a}}_{i} \tilde{b}, θ) = \frac{1}{3} {(1 - θ) [({\underline{a}}_{i} \underline{b}) + 2 (a_{i} b)] \\ + θ [2 (a_{i} b) + ({\bar{a}}_{i} \bar{b})]} \cdot {ω_{{\tilde{a}}_{i}} \land ω_{\tilde{b}}} \\ \leq \frac{1}{3} {(1 - θ) [({\underline{a}}_{j} \underline{b}) + 2 (a_{j} b)] + θ [2 (a_{j} b) + ({\bar{a}}_{j} \bar{b})]} \\ \cdot {ω_{{\tilde{a}}_{i}} \land ω_{\tilde{b}}} = m_{μ} ({\tilde{a}}_{j} \tilde{b}, θ) . \end{matrix}$

Then, $m_{μ} ({\tilde{a}}_{i} \tilde{b}, θ) \leq m_{μ} ({\tilde{a}}_{j} \tilde{b}, θ)$ and thus ${\tilde{a}}_{i} \tilde{b} \leq {\tilde{a}}_{j} \tilde{b}$ , which implies that if ${\tilde{a}}_{σ (j)}$ is the jth largest of the TIFNs ${\tilde{a}}_{i}$ (i = 1, 2, …, n), then ${\tilde{a}}_{σ (j)} \tilde{b}$ is the jth largest of the TIFNs ${\tilde{a}}_{i} \tilde{b}$ (i = 1, 2, …, n).

Therefore, by Definitions 1 and 9, we have $\begin{matrix} {TIFPOWG}_{w} ({\tilde{a}}_{1} \tilde{b}, {\tilde{a}}_{2} \tilde{b}, \dots, {\tilde{a}}_{n} \tilde{b}) = \prod_{j = 1}^{n} ({\tilde{a}}_{σ (j)} \tilde{b})^{w_{j}} \\ = \prod_{j = 1}^{n} {\tilde{a}}_{σ (j)}^{w_{j}} \cdot \prod_{j = 1}^{n} {\tilde{b}}^{w_{j}} = \prod_{j = 1}^{n} {\tilde{a}}_{σ (j)}^{w_{j}} \cdot \tilde{b} = \prod_{j = 1}^{n} {\tilde{a}}_{σ (j)}^{w_{j}} \tilde{b} . \end{matrix}$

Consequently, $\begin{matrix} {TIFPOWG}_{w} ({\tilde{a}}_{1} \tilde{b}, {\tilde{a}}_{2} \tilde{b}, \dots, {\tilde{a}}_{n} \tilde{b}) \\ = {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \cdot \tilde{b} . \end{matrix}$

The proof of Theorem 16 is completed.

Theorem 17. For λ ≥ 0, we have $\begin{matrix} {TIFPOWG}_{w} (λ {\tilde{a}}_{1}, λ {\tilde{a}}_{2}, \dots, λ {\tilde{a}}_{n}) \\ = λ \cdot {TIFPOWG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}), \\ {TIFPOWG}_{w} ({\tilde{a}}_{1}^{λ}, {\tilde{a}}_{2}^{λ}, \dots, {\tilde{a}}_{n}^{λ}) \\ = ({TIFPWOG}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}))^{λ}, \end{matrix}$ where the associated weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (16).

Proof. The proof is similar to Theorem 9.

3.3 Prioritized hybrid weighted geometric operator of TIFNs

It is worth noticing that the TIFPWG operator gives the importance of each TIFN argument, while the TIFPOWG operator gives the importance of ordered position of each TIFN argument. To avoid this limitation, we define a new prioritized hybrid weighted geometric operator of TIFNs which emphasizes importance of both the given arguments and ordered positions of arguments.

Definition 10. A prioritized hybrid weighted geometric operator of TIFNs is a mapping TIFPHWG: Ωⁿ → Ω and ${TIFPHWG}_{w, η} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \prod_{j = 1}^{n} ({\tilde{a}}_{σ (j)}^{'})^{w_{j}},$ where the associated weight vector w = (w₁, w₂, …, w_n) ^T is determined by Equation (16), ${\tilde{a}}_{σ (k)}^{'}$ is the k-th largest of the weighted TIFNs ${\tilde{a}}_{i}^{'}$ (i = 1, 2, …, n), ${\tilde{a}}_{i}^{'} = ({\tilde{a}}_{i})^{n η_{i}}$ , η = (η₁, η₂, …, η_n) ^T is the weight vector of ${\tilde{a}}_{i}$ , satisfying that 0 ≤ η_i ≤ 1 and $\sum_{i = 1}^{n} η_{i} = 1$ , n is the balancing coefficient.

Especially, if η = (1/n, 1/n, …, 1/n) ^T, then the TIFPHWG operator reduces to the TIFPOWG operator. In this case, the TIFPHWG operator has the same properties as the TIFPOWG operator.

Theorem 18. If ${\tilde{a}}_{σ (k)}^{'} = (({\underline{a^{'}}}_{σ (k)}, a_{σ (k)}^{'}, {\bar{a}}_{σ (k)}^{'}); ω_{{\tilde{a}}_{σ (k)}^{'}}, u_{{\tilde{a}}_{σ (k)}^{'}})$ is the k-th largest of the weighted TIFNs ${\tilde{a}}_{i}^{'} = ({\tilde{a}}_{i})^{n η_{i}}$ (i = 1, 2, …, n), then the aggregated value by using TIFPHWG operator is also a TIFN as follows:

$\begin{matrix} {TIFPHWG}_{w, η} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = ((\prod_{j = 1}^{n} a_{σ (j)}^{'}^{w_{j}}, \prod_{j = 1}^{n} a_{σ (j)}^{'}^{w_{j}}, \prod_{j = 1}^{n} {\bar{a}}_{σ (j)}^{'}^{w_{j}}); \\ \underset{j}{\land} {ω_{{\tilde{a}}_{σ (j)}^{'}}}, \underset{j}{\lor} {u_{{\tilde{a}}_{σ (j)}^{'}}}) . \end{matrix}$ (18)

The above three kinds of prioritized average operators can take the prioritization among criteria into consideration. The TIFPWG operator emphasizes the importance of each argument, the TIFPOWG operator emphasizes the importance of ordered position of each argument, while the TIFPHWG operator reflects the important degrees of both the given arguments and ordered positions of arguments.

The existing weighted arithmetic and geometric aggregation techniques, such as the TrIFNs weighted arithmetic and geometric average operators [21] and the TrIFNs weighted geometric operator and hybrid geometric operator [23], can not reflect the prioritization relationship among criteria, i.e., these aggregation techniques did not take into account the prioritization of arguments to be aggregated. However, these three kinds of prioritized geometric operators of TIFNs have non-linear properties since the weight of argument ${\tilde{a}}_{k}$ , $w_{k} = T_{k} / \sum_{i = 1}^{n} T_{i}$ in Equation (14) or w_k = f (R_k) - f (R_k-1) in Equation (16), depends on the input arguments ${\tilde{a}}_{i}$ (i = 1, 2, …, k) and allows for the inclusion of priority between arguments in the aggregation process. There often exists the prioritization among criteria in some practical MCGDM problems. The fundamental characteristic of these operators is that the associated weights are determined by sufficiently considering the priority over arguments. Hence, applying these operators to prioritized MCGDM field can make the decision results more reasonable and practical.

4 Application to prioritized MCGDM problems with TIFNs information

In this section, a new method for the prioritized MCGDM problems with TIFNs is presented based on the TIFPWG and TIFHWG operators.

4.1 Description of prioritized MCGDM problems

A MCGDM problem is to find the best compromise solution from all feasible alternatives assessed on multiple criteria. Denote an alternative set by {A₁, A₂, …, A_m} and a criterion set by C = {c₁, c₂, …, c_n}. There exists a linear ordering c₁ ≻ c₂ ≻ … ≻ c_n among the criteria. Assume that there are p DMs participating in decision making, denote the set of DMs by E = {e₁, e₂, …, e_p}. The weight vector of DMs is V = (v₁, v₂, …, v_p) ^T, satisfying that $\sum_{k = 1}^{p} v_{k} = 1$ and 0 ≤ v_k ≤ 1 (k = 1, 2 . … , p). Suppose that there is also a linear ordering e₁ ≻ e₂ ≻ … ≻ e_p between the DMs (For example, the president has the highest priority in the process of decision process for personnel selection). The rating of an alternative A_i on a criterion c_j given by the DM e_k is a TIFN ${\tilde{a}}_{ij}^{k} = (({\underline{a}}_{ij}^{k}, a_{ij}^{k}, {\bar{a}}_{ij}^{k}); ω_{{\tilde{a}}_{ij}^{k}}, u_{{\tilde{a}}_{ij}^{k}})$ , where $ω_{{\tilde{a}}_{ij}^{k}}$ and $u_{{\tilde{a}}_{ij}^{k}}$ denote respectively the maximum membership degree and the minimum non-membership degree of alternative A_i on criterion c_j given by the DM e_k, satisfying $0 \leq ω_{{\tilde{a}}_{ij}^{k}} \leq 1$ , $0 \leq u_{{\tilde{a}}_{ij}^{k}} \leq 1$ and $0 \leq ω_{{\tilde{a}}_{ij}^{k}} + u_{{\tilde{a}}_{ij}^{k}} \leq 1$ . Hence, we can elicit decision matrices ${\tilde{A}}^{k} = ({\tilde{a}}_{ij}^{k})_{m \times n}$ (k = 1, 2, …, p) to represent the MCGDM problem.

To eliminate the impact of different dimensions on the decision results, the matrix ${\tilde{A}}^{k} = ({\tilde{a}}_{ij}^{k})_{m \times n}$ needs to be normalized into ${\tilde{R}}^{k} = ({\tilde{r}}_{ij}^{k})_{m \times n}$ , where ${\tilde{r}}_{ij}^{k} = (({\underline{r}}_{ij}^{k}, r_{ij}^{k}, {\bar{r}}_{ij}^{k}); ω_{{\tilde{r}}_{ij}^{k}}, u_{{\tilde{r}}_{ij}^{k}})$ , $ω_{{\tilde{r}}_{ij}^{k}} = ω_{{\tilde{a}}_{ij}^{k}}$ and $u_{{\tilde{r}}_{ij}^{k}} = u_{{\tilde{a}}_{ij}^{k}}$ .

For benefit criteria, ${\tilde{r}}_{ij}^{k} = (({\underline{a}}_{ij}^{k} / {\bar{a}}_{j}^{+}, a_{ij}^{k} / {\bar{a}}_{j}^{+}, {\bar{a}}_{ij}^{k} / {\bar{a}}_{j}^{+}); ω_{{\tilde{r}}_{ij}^{k}}, u_{{\tilde{r}}_{ij}^{k}});$ (19)

For cost criteria, ${\tilde{r}}_{ij}^{k} = (({\underline{a}}_{j}^{-} / {\bar{a}}_{ij}^{k}, {\underline{a}}_{j}^{-} / a_{ij}^{k}, {\underline{a}}_{j}^{-} / {\underline{a}}_{ij}^{k}); ω_{{\tilde{r}}_{ij}^{k}}, u_{{\tilde{r}}_{ij}^{k}}),$ (20) where ${\bar{a}}_{j}^{+} = max {{\bar{a}}_{ij}^{k} | i = 1, 2, \dots . m; k = 1, 2, \dots, p}$ and ${\underline{a}}_{j}^{-} = min {{\underline{a}}_{ij}^{k} | i = 1, 2, \dots . m; k = 1, 2, \dots, p}$ (j = 1, 2, …, n).

4.2 Proposed decision method

In sum, an algorithm and process for solving the prioritized MCGDM problems with TIFNs are described step by step as follows.

Step 1: Normalize the decision matrix ${\tilde{A}}^{k}$ into ${\tilde{R}}^{k}$ (k = 1, 2, …, p) according to Equations (19) and (20);

Step 2: By the TIFPWG operator, the individual overall value of alternative A_i given by DM e_k can be obtained as ${\tilde{r}}_{i}^{k} = {TIFPWG}_{W_{i}^{k}} ({\tilde{r}}_{i 1}^{k}, {\tilde{r}}_{i 2}^{k}, \dots, {\tilde{r}}_{in}^{k}) = \prod_{j = 1}^{n} ({\tilde{r}}_{ij}^{k})^{w_{ij}^{k}},$ (21) where the normalized priority based importance weight vector $W_{i}^{k} = (w_{i 1}^{k}, w_{i 2}^{k}, \dots, w_{in}^{k})^{T}$ can beobtained by Equation (14) as $T_{i, 1}^{k} = 1$ , $T_{ij}^{k} = \prod_{t = 1}^{j} m_{μ} ({\tilde{r}}_{i, t - 1}^{k}, θ_{k})$ (j = 2, 3, …, n), $w_{ij}^{k} = T_{ij}^{k} / \sum_{j = 1}^{n} T_{ij}^{k}$ (j = 1, 2, …, n), where θ_k is the preference parameter of DM e_k.

Step 3: Combined the DMs’ weight vector V = (v₁, v₂, …, v_p) ^T with the TIFPHWG operator, the collective overall value of alternative A_i is calculated as:

$\begin{matrix} {\tilde{r}}_{i} & = & ((\underset{-_{i}}{r}, r_{i}, {\bar{r}}_{i}); ω_{{\tilde{r}}_{i}}, u_{{\tilde{r}}_{i}}) \\ = & {TIFPHWG}_{w_{i}, V} ({\tilde{r}}_{i}^{1}, {\tilde{r}}_{i}^{2}, \dots, {\tilde{r}}_{i}^{p}) \\ = & \prod_{k = 1}^{p} ({\tilde{r}}_{i}^{'}^{σ (k)})^{w_{i}^{k}} (i = 1, 2, \dots, m) \end{matrix}$ (22) where ${\tilde{r}}_{i}^{' k} = ({\tilde{r}}_{i}^{k})^{{pv}_{i}}$ , ${\tilde{r}}_{i}^{'}^{σ (k)}$ is the kth largest of the ${\tilde{r}}_{i}^{'}^{t}$ (t = 1, 2, …, p), the associated weight vector $w_{i} = (w_{i}^{1}, w_{i}^{2}, \dots, w_{i}^{p})^{T}$ can be determined by Equation (16).

Step 4: By using Equations (11) and (12), the weighted possibility means of membership and non-membership functions for TIFN ${\tilde{r}}_{i}$ are computed as: $\begin{matrix} m_{μ} ({\tilde{r}}_{i}, θ) \\ = \frac{1}{3} [(1 - θ) ({\underline{r}}_{i} + 2 r_{i}) + θ (2 r_{i} + {\bar{r}}_{i})] ω_{{\tilde{r}}_{i}}, \end{matrix}$ (23) $\begin{matrix} m_{ν} ({\tilde{r}}_{i}, θ) \\ = \frac{1}{3} [(1 - θ) ({\underline{r}}_{i} + 2 r_{i}) + θ (2 r_{i} + {\bar{r}}_{i})] (1 - u_{{\tilde{r}}_{i}}), \end{matrix}$ (24) where $θ = \sum_{k = 1}^{p} v_{k} θ_{k}$ is the preference parameter of the decision group.

Step 5: The ranking orders of alternatives are generated by $m_{μ} ({\tilde{r}}_{i}, θ)$ and $m_{ν} ({\tilde{r}}_{i}, θ)$ (i = 1, 2, …, m) by the ranking approach developed in Subsection 2.2.

5 A system analyst selection example analysis

In this section, a system analyst selection example is analyzed to demonstrate the applicability and implementation process of the proposed method. The comparison analysis is also conducted to show the superiority of the proposed method.

5.1 A system analyst selection example

The proposed method is illustrated with a personnel selection problem adapted from Li et al. [8]. Suppose that a software company desires to hire a system analyst from three candidates A₁, A₂ and A₃. The decision making committee consists of three DMs: president e₁, vice-president e₂ and department manager e₃. They assesses the three candidates on the basis of five criteria: emotional steadiness c₁, oral communication skills c₂, personality c₃, past experience c₄ and self-confidence c₅. After the negotiation and investigation, the decision making committee agrees that there is a prioritization between the criteria expressed by the linear ordering c₁ ≻ c₂ ≻ c₃ ≻ c₄ ≻ c₅. Obviously, the president e₁ has the highest priority in the process of decision process and thus there is also a linear order e₁ ≻ e₂ ≻ e₃ among the DMs. The weight vector of DMs is V = (0 . 40, 0 . 35, 0 . 25) ^T. After the data acquisition and statistical treatment, the ratings of the candidates with respect to criteria can be represented by TIFNs as in Tables 1–3. For example, ((5.7,7.7,9.3); 0.7,0.2) in Table 1 is an TIFN which indicates that the mark of the candidate A₁ on criterion c₁ is about 7.7 with the maximum satisfaction degree is 0.7, while the minimum dissatisfaction degree is 0.2. In other words, the hesitation degree is 0.1. Other TIFNs in Tables 1–3 are explained similarly [8].

Step 1: The five criteria are all benefit criteria. According to Equations (19) and (20), the normalized TIFN decision matrixes are obtained (omitted).

Step 2: Assume the preference parameter of DM e₁ is θ₁ = 0.5. Utilized the TIFPWG operator, the individual overall value of alternative A₁ given by DM e₁ can be computed by Equation (21) as ${\tilde{r}}_{1}^{1} = ((0 . 55, 0 . 75, 0 . 91); 0 . 6, 0 . 4)$ . Assume that the preference parameters of DMs e₂ and e₃ are θ₂ = 0.5 and θ₃ = 0.5, respectively. In a similar way, the individual overall values of the other alternatives are obtained.

Step 3: Suppose that the BUM function is f (x) = x². Combined the DMs’ weight vector V = (0 . 40, 0 . 35, 0 . 25) ^T with the TIFPHWG operator, the collective value of alternative A₁ is calculated by Equation (22) as ${\tilde{r}}_{1} = ((0 . 40, 0 . 66, 0 . 90); 0 . 2, 0 . 7)$ . Analogously, the collective overall values of alternatives A₂ and A₃ are respectively obtainedas ${\tilde{r}}_{2} = ((0 . 65, 0 . 85, 0 . 97); 0 . 3, 0 . 6)$ and ${\tilde{r}}_{3} = ((0.57, 0.81, 0.96); 0.3, 0.6)$ .

Step 4: By using Equations (23) and (24), the weighted possibility means for the TIFNs ${\tilde{r}}_{i}$ (i = 1, 2, 3) are computed as: $\begin{matrix} m_{μ} ({\tilde{r}}_{1}, θ) & = & 0.13, m_{ν} ({\tilde{r}}_{1}, θ) \\ = & 0.20, m_{μ} ({\tilde{r}}_{2}, θ) = 0.25, \\ m_{ν} ({\tilde{r}}_{2}, θ) & = & 0.33, m_{μ} ({\tilde{r}}_{3}, θ) \\ = & 0.24, m_{ν} ({\tilde{r}}_{3}, θ) = 0.32, \end{matrix}$ where θ = v₁θ₁ + v₂θ₂ + v₃θ₃ = 0.5.

Step 5: Since $m_{μ} ({\tilde{r}}_{2}, θ) > m_{μ} ({\tilde{r}}_{3}, θ) > m_{μ} ({\tilde{r}}_{1}, θ)$ , the ranking order of alternatives is generated as A₂ ≻ A₃ ≻ A₁ and the best candidate is A₂.

For any other preference parameter values, in the same way, we can obtain the collective overall values and ranking order of candidates listed in Table 4. It can be seen from Table 4 that, for different preference parameter values, the ranking orders of candidates are also not completely the same. For instance, if θ₁ = θ₂ = θ₃ = 0, then the ranking order is A₁ ≻ A₂ ≻ A₃, the best is A₁; if θ₁ = θ₂ = θ₃ = 0.8, then the ranking order is A₃ ≻ A₂ ≻ A₁, the best is A₃.

The above analysis suggests that the DMs’ preferences indeed play an important role in the decision making. Since TIFN is a special kind of IFS, involving DM’s preference to rank the TIFNs is very reasonable and necessary. When the preference parameter values are different, the corresponding decision results may be different.

5.2 Comparison analysis

Since TIFN $\tilde{a} = ((a, b, c); ω_{\tilde{a}}, u_{\tilde{a}})$ can be written as TrIFN $\tilde{a} = ([a, b, b, c]; ω_{\tilde{a}}, u_{\tilde{a}})$ , if we rewrite all the TIFNs in Tables 1–3 as TrIFNs, then the above personnel selection problem is changed as a MAGDM problem with TrIFNs. Wu and Cao [23] proposed an approach to solve the TrIFN MAGDM problem without considering the prioritization between the criteria and DMs. Suppose that the weight vector of criteria is W = (0.20, 0.18, 0.22, 0.40) ^T, we use the method [23] to solve the above personnel selection problem with TrIFNs. The distances between the collective overall preference values and the positive ideal solution are calculated as follows: $d ({\tilde{r}}_{1}, {\tilde{r}}^{+}) = 0.22, d ({\tilde{r}}_{2}, {\tilde{r}}^{+}) = 0.44, d ({\tilde{r}}_{3}, {\tilde{r}}^{+}) = 0.51 .$

Therefore, the ranking order of alternatives obtained by [23] is A₁ ≻ A₂ ≻ A₃ and the best is candidate A₁, which is remarkably different from that obtained by this paper. Compared with method [23], the proposed method has some advantages as follows:

The method [23] did not consider the prioritization between the criteria and DMs while this paper sufficiently considers the linear order between the criteria and DMs. The geometric aggregation operators developed in [23] are assumed that the criteria are independent. These operators did not consider the relationship among the criteria. Consequently, the method [23] is not applicable to prioritized MADM and MAGDM problems.

The ranking method of TrIFNs adapted in [23] simply calculated the distances between the TrIFNs and positive ideal solution. Such a ranking method is a single-index approach, which is not always feasible and effective. This paper, however, introduces the concepts of weighted lower and upper possibility means for TrIFN as well as weighted possibility mean. Since it takes into consideration the weighted possibility means of membership and non-membership functions of TrIFN, the ranking method of TIFNs proposed in this paper is a two-index approach and more reasonable than that of [23].

This paper sufficiently considers the different preference between the weighted lower and upper possibility means for DMs, which makes the decision results more consistent with actual situation, while the method [23] did not consider the DM’s preference (namely it assumes all DMs are risk neutral).

The method [23] researched the TrIFN MAGDM problems without considering the prioritization relationship among the criteria. The proposed method in this paper researched the prioritized TIFN MCGDM problems without the need of criteria weights. At the same time, it can be applicable to the MCDM and MCGDM problems in which the weights are completely unknown. Therefore, it is of universality and flexibility.

6 Conclusions

The MCGDM problems often require the inclusion of information about importance associated with the different criteria. Importance information plays a fundamental role in the comparison between alternatives by overseeing tradeoffs between the respective satisfactions to different criteria. In this work, we consider the situation in which the information regarding the importance of the individual criterion is captured by a prioritization of the criteria.

The main contributions of this paper are threefold:

The weighted possibility means of TIFN are defined and a new lexicographic approach is proposed to rank the TIFNs sufficiently considering the risk preference of DM.

Three kinds of triangular intuitionistic fuzzy prioritized geometric operators are developed including the TIFPWG, TIFPOWG and TIFPHWG operators. The significant feature of the proposed operators is that they take the prioritization among the criteria into consideration. Some of their desirable properties are also investigated in detail.

By utilizing the TIFPWG operator, the criterion values of alternatives are aggregated into the individual overall values of alternatives, which are further integrated into the collective ones by the TIFPHWG operator. The ranking order of alternatives is generated according to the collective overall values of alternatives. Thereby, a new method is proposed for solving the prioritized MAGDM problems with TIFNs. It sufficiently considers the different risk preferences of different DMs and the priority relationship among the criteria and DMs, which can make the decision results more reasonable and consistent with the reality.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos. 71061006, 61263018 and 11461030), the Natural Science Foundation of Jiangxi Province of China (Nos. 20114BAB201012 and 20142BAB201011), the Science and Technology Project of Jiangxi province educational department of China (Nos. GJJ15265 and GJJ15267), Young scientists Training object of Jiangxi province (No. 20151442040081) and the Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.

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