Pythagorean fuzzy uncertain linguistic TODIM method and their application to multiple criteria group decision making

Abstract

In this paper, with respect to multiple criteria group decision making (MCGDM) problems in which criteria values are expressed by Pythagorean fuzzy uncertain linguistic variables (PFULVs), we propose an extended TODIM method. Firstly, we define the Pythagorean fuzzy uncertain linguistic set, and propose the operational laws, Hamming distance, score function and accuracy function of PFULVs. Then an extended TODIM method is presented to solve the MCGDM problems under the Pythagorean fuzzy uncertain linguistic environment, and a numerical example with the Pythagorean fuzzy uncertain linguistic information is given to show the effectiveness of the proposed method. Further, we analyze the influence of the different parameter on the MCGDM, and test the practicality of the proposed method.

Keywords

Multiple criteria group decision making Pythagorean fuzzy uncertain linguistic set TODIM

1 Introduction

Multiple criteria decision making (MCDM) or MCGDM means that the optimal alternative is selected from the finite alternatives set according to the multiple criteria, which can be regard as cognitive processing. MCDM (or MCGDM) is an important branch of the decision making theory which has been widely applied in human activities [9 , 17–19]. Because the real decision making problems were frequently produced from the complicated environment, the evaluation information is usually fuzzy. In general, the fuzzy information takes two forms: one quantitative and one qualitative. The quantitative fuzzy information can be expressed by fuzzy set (FS) [36], intuitionistic fuzzy set (IFS) [1], Pythagorean fuzzy set (PFS) [34] and so on. FS theory proposed by Zadeh [36] has been used to describe fuzzy quantitative information which contains only a membership degree. Based on this, Atanassov [1, 2] presented IFS which consists of a membership degree and a non-membership degree and satisfies the constraint condition that the sum of two degrees is equal to or less than 1. However, sometimes the two degrees don’t meet the restriction, but the square sum of the two degrees is equal to or less than 1. Yager [34, 35] introduced the PFS in which the square sum of membership degree and non-membership degree is equal to or less than 1. In some situation, the PFS has the greater ability to express the fuzzy information than the IFS. For example, if an expert gives the membership about the support to a matter is 0.8 and the non-membership for against is 0.6. Obviously, IFS is invalid to describe this decision information, but it can be effectively described by PFS. In practical decision making, many problems are so complex and indistinct that the quantitative fuzzy information could do no more. In order to overcome these issues, it is best to choose qualitative fuzzy information which can be in the form of linguistic variables or uncertain linguistic variables. Wang and Li [33] combined the IFS with linguistic set and proposed intuitionistic linguistic set (ILS). Based on this, Liu and Jin [13] further defined the intuitionistic uncertain linguistic set (IULS) because the uncertain linguistic set describes fuzzy information better than linguistic set. Similar to the extension of IUFS, we combine the PFS with uncertain linguistic set and propose the Pythagorean fuzzy uncertain linguistic set (PFULS). PFULS can conquer the shortcomings for the PFS which only can describe the criteria’s membership and non-membership to a particular concept and for uncertain linguistic variables in which membership degree is 1, and the non-membership degree or hesitation degree is neglected. Moreover, because the scope of PFS is superior to IFS, the application range of PFULS is wider than IULS. So, the PFULS describes fuzzy evaluation information more appropriately than PFS and IULS. For example, when an evaluator assesses the car performance, he/she thinks that it may be lower than ‘very good’ (s₆) but higher than ‘fair’ (s₃). The evaluation information comes from the evaluator’s subjective feeling, at the same time, he can respectively give that the membership degree to [s₃, s₆] is 0.8 and the non-membership degree to [s₃, s₆] is 0.6. The evaluation result can be regard as < [s₃, s₆] , (0.8, 0.6)>. Certainly, we can easily find that the sum of the two degrees is 0.8 + 0.6 = 1.4 > 1 and their square sum is 0 . 8² + 0 .6² = 1. Due to the square sum of the membership degree and the on-membership degree is equal to or less than 1 in PFULS, the PFULS exhibits wider applied ranges than IULS in dealing with the real MCDM or MCGDM problems. In exceptional cases, if the upper and the lower of the uncertain linguistic part in PFULS are equal, the PFULS reduces to the Pythagorean fuzzy linguistic set (PFLS) presented by Peng and Yang [24]. In other word, the PFULS is the extension of uncertain linguistic variables, the PFS and the PFLS, which can more easily and precisely describe the uncertain information.

In previous research, many decision making methods under uncertain environment were proposed to deal with subjective evaluation information in real decision making problems, such as TOPSIS [38, 39], VIKOR [20, 31], MOORA [27, 37], ELECTRE [8, 32], PROMETHEE [3, 10], COPRAS [15, 30], OCRA [25], ARAS [11, 22], SWARA [21, 28], EDAS [23, 29] and so on. Devi [4] provided the extended VIKOR method under the intuitionistic fuzzy environment where the weighted vector was expressed by triangular intuitionistic fuzzy set (TrIFS). Peng and Yang [24] used the TOPSIS method to solve the MCGDM problem with the PFS information. Based on prospect theory, Li and Zhao [16] introduced a grey-VIKOR method to deal with MCDM problem where the evaluation information was grey numbers. Stanujkic et al. [27] proposed a new MOORA method to handle lots of decision making problems in real world. Many decision making tools and techniques don’t consider the DMs’ risk attitude. TODIM method based on prospect theory takes the DMs’ psychological behavior into account so that DMs can make a more rational choice under risk. Since the IULS can appropriately describe the uncertain information, Liu and Teng [14] presented an extended TODIM method under the IULS environment. Ren, Xu and Gou [26] further proposed a Pythagorean fuzzy TODIM method which was applied to solve the MCDM problems with PFS information. Due to the shortcoming of the PFS which only can express the criteria’s membership and non-membership to a particular concept, we extend the TODIM method to Pythagorean fuzzy uncertain linguistic environment. In this paper, we introduce an extended TODIM method based on PFULS to solve MCGDM problems in which the criteria values are in the form of Pythagorean fuzzy uncertain linguistic numbers (PFULNs) and the weights of experts and criteria are crisp numbers.

In order to realize the above purpose, the remainder of this paper is constructed as follows: Section 2 briefly introduces the concepts of the PFS, PFLS and TODIM method. Then, we give the related presentations about the PFULS. Section 3 proposes the Pythagorean fuzzy uncertain linguistic TODIM method and describes the procedure of the proposed method in detail. Section 4 gives an illustrative example to demonstrate the effectiveness of the new method and the influence of the attenuation factor of the losses θ. Section 5 gives the concluding remarks.

2 Preliminaries

2.1 Uncertain linguistic set

Let S = {s_i|i = 0, 1, …, l - 1} be a linguistic term set with odd cardinality in which s_i represents the possible value for the ith linguistic variable in S. In general, l = 3, 5, 7, 9, etc. For example, if l = 7, then S = {verypoor, poor, slightlypoor, fair, slightlygood, good, verygood}. The linguistic variables s_i and s_j should meet the following condition [6, 7]:

Ordered linguistic set: if i > j, then s_i > s_j;

Max operator: if s_i ≥ s_j, then max(s_i, s_j) = s_i;

Min operator: if s_i ≤ s_j, then min(s_i, s_j) = s_i;

Negation operator: neg (s_i) = s_j, where j = l - 1 - i.

In order to minimize the loss of linguistic information, the discrete linguistic term set S = {s_i|i = 0, 1, …, l - 1} is extended into the continuous linguistic set $\bar{S} = {s_{α} | α \in R^{+}}$ . About the operational laws of linguistic variables, please refer to references [6, 7].

Suppose $\tilde{s} = [s_{a}, s_{b}]$ , $s_{a}, s_{b} \in \bar{S}$ , a ≤ b, s_a, s_b are the lower limit and upper limit respectively, then $\bar{s}$ is called an uncertain linguistic variable. $\tilde{S}$ is the set of all uncertain linguistic variables.

2.2 The Pythagorean fuzzy set (PFS)

Definition 1. [34, 35] Let X be an universe of discourse. A Pythagorean fuzzy set (PFS) P in X is defined as $P = {< x, (μ_{P} (x), ν_{P} (x)) > | x \in X}$ (1) where μ_P : X → [0, 1] denotes the membership degree, ν_P : X → [0, 1] denotes the non-membership degree and 0 ≤ μ_P (x) ² + ν_P (x) ² ≤ 1. The indeterminacy degree is $π_{P} (x) = \sqrt{1 - μ_{P} (x)^{2} - ν_{P} (x)^{2}}$ . For convenience, a Pythagorean fuzzy number (PFN) can be expressed as p =< (μ_p, ν_p) >.

Based on the Definition 1, Zhang and Xu [38] further proposed the distance between p₁ =< (μ_{p
₁}, ν_{p
₁}) > and p₂ =< (μ_{p
₂}, ν_{p
₂}) >, which was given as follows:

$\begin{matrix} d (p_{1}, p_{2}) \\ = \frac{1}{2} (| (μ_{p_{1}})^{2} - (μ_{p_{2}})^{2} | + | (ν_{p_{1}})^{2} - (ν_{p_{2}})^{2} | \\ + | (π_{p_{1}})^{2} - (π_{p_{2}})^{2} |) \end{matrix}$ (2)

The operational rules of the PFNs can refer to references [34, 35].

2.3 The Pythagorean fuzzy linguistic Set (PFLS)

Definition 2. [24] Let $s_{θ_{\tilde{A}} (x)} \in \bar{S}$ , X be an universe of discourse, A Pythagorean fuzzy linguistic set (PFLS) $\tilde{A}$ in X is defined as $\tilde{A} = {< x, (s_{θ_{\tilde{A}} (x)}, μ_{\tilde{A}} (x), ν_{\tilde{A}} (x)) > | x \in X}$ (3) where $μ_{\tilde{A}} : X \to [0, 1]$ denotes the membership degree, $ν_{\tilde{A}} : X \to [0, 1]$ denotes the non-membership degree satisfying $0 \leq μ_{\tilde{A}} (x)^{2} + ν_{\tilde{A}} (x)^{2} \leq 1$ . The indeterminacy degree is $π_{\tilde{A}} (x) = \sqrt{1 - μ_{\tilde{A}} (x)^{2} - ν_{\tilde{A}} (x)^{2}}$ . For convenience, a Pythagorean fuzzy linguistic number (PFLN) can be expressed as $\tilde{a} = < s_{θ (\tilde{a})}, (μ (\tilde{a}), ν (\tilde{a})) >$ .

The operational rules of the PFLNs can refer to the reference [24].

3 The Pythagorean fuzzy uncertain linguistic set (PFULS)

Based on the uncertain linguistic set and Pythagorean fuzzy set, in this section, we propose the Pythagorean fuzzy uncertain linguistic set (PFULS) and define the operational laws, distance, and comparison method of the PFULS.

Definition 3. Let $[s_{θ_{\tilde{P}} (x)}, s_{τ_{\tilde{P}} (x)}] \in \bar{S}$ , X be an universe of discourse, A Pythagorean fuzzy uncertain linguistic set (PFULS) $\tilde{A}$ in X is defined as

$\begin{matrix} \tilde{P} & = & {< x, ([s_{θ_{\tilde{P}} (x)}, s_{τ_{\tilde{P}} (x)}], \\ (μ_{\tilde{P}} (x), ν_{\tilde{P}} (x))) > | x \in X} \end{matrix}$ (4) where $μ_{\tilde{P}} : X \to [0, 1]$ denotes the membership degree, $ν_{\tilde{P}} : X \to [0, 1]$ denotes the non-membership degree, satisfying the condition that: $0 \leq μ_{\tilde{P}} (x)^{2} + ν_{\tilde{P}} (x)^{2} \leq 1$ . The indeterminacy degree is $π_{\tilde{P}} (x) = \sqrt{1 - μ_{\tilde{P}} (x)^{2} - ν_{\tilde{P}} (x)^{2}}$ . For convenience, a Pythagorean fuzzy uncertain linguistic number (PFLN) can be expressed as $\tilde{p} = < [s_{θ (\tilde{p})}, s_{τ (\tilde{p})}], (μ (\tilde{p}), ν (\tilde{p})) >$ .

Definition 4. Let ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}],$ $(μ ({\tilde{p}}_{1}),$ $ν ({\tilde{p}}_{1})) >$ and ${\tilde{p}}_{2} = < [s_{θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{2})}], (μ ({\tilde{p}}_{2}), ν ({\tilde{p}}_{2})) >$ be any two PFULNs and λ ≥ 0, the operational rules based on t-norm (T) and s-norm (S) are defined as follows: $\begin{matrix} {\tilde{p}}_{1} \tilde{\oplus} {\tilde{p}}_{2} \\ = 〈 [s_{θ ({\tilde{p}}_{1}) + θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{2})}], \\ (S (μ ({\tilde{p}}_{1}), μ ({\tilde{p}}_{2})), T (ν ({\tilde{p}}_{1}), ν ({\tilde{p}}_{2}))) 〉 \\ = 〈 [s_{θ ({\tilde{p}}_{1}) + θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{2})}], \\ (h^{- 1} (h (μ ({\tilde{p}}_{1})) + h (μ ({\tilde{p}}_{2}))), \\ (g^{- 1} (g (ν ({\tilde{p}}_{1})) + g (ν ({\tilde{p}}_{2})))) 〉 \end{matrix}$ (5) $\begin{matrix} {\tilde{p}}_{1} \tilde{\otimes} {\tilde{p}}_{2} \\ = 〈 [s_{θ ({\tilde{p}}_{1}) \times θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) \times τ ({\tilde{p}}_{2})}], \\ (T (μ ({\tilde{p}}_{1}), μ ({\tilde{p}}_{2})), S (ν ({\tilde{p}}_{1}), ν ({\tilde{p}}_{2}))) 〉 \\ = 〈 [s_{θ ({\tilde{p}}_{1}) \times θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) \times τ ({\tilde{p}}_{2})}], \\ (g^{- 1} (g (μ ({\tilde{p}}_{1})) + g (μ ({\tilde{p}}_{2}))), \\ h^{- 1} (h (ν ({\tilde{p}}_{1})) + h (ν ({\tilde{p}}_{2})))) 〉 \end{matrix}$ (6) $\begin{matrix} λ \tilde{⊙} {\tilde{p}}_{1} \\ = 〈 [s_{λ θ ({\tilde{p}}_{1})}, s_{λ τ ({\tilde{p}}_{1})}], \\ (h^{- 1} (λ h (μ ({\tilde{p}}_{1}))), g^{- 1} (λ g (ν ({\tilde{p}}_{1})))) 〉 \end{matrix}$ (7) $\begin{matrix} {\tilde{p}}_{1}^{λ} = 〈 [s_{θ ({\tilde{p}}_{1})^{λ}}, s_{τ ({\tilde{p}}_{1})^{λ}}], \\ (g^{- 1} (λ g (μ ({\tilde{p}}_{1}))), h^{- 1} (λ h (v ({\tilde{p}}_{1})))) 〉 \end{matrix}$ (8) where g (x) is a monotonically decreasing function, h (x) is monotonically increasing function, and h^-1 (x) = g^-1 (1 - x).

Based on Definition 4, when we set some specific values of function g (x), then different operational rules can be obtained. For simplicity we set g (x) = - log x² so that we can get $g^{- 1} (x) = \sqrt{e^{- x}}$ , h (x) = - log(1 - x²), $h^{- 1} (x) = \sqrt{1 - e^{- x}}$ . Then the operational rules based on Algebra t-norm and s-norm can be defined as follows: $\begin{matrix} {\tilde{p}}_{1} \oplus {\tilde{p}}_{2} & = & 〈 [s_{θ ({\tilde{p}}_{1}) + θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{2})}], \\ (\sqrt{μ ({\tilde{p}}_{1})^{2} + μ ({\tilde{p}}_{2})^{2} - μ ({\tilde{p}}_{1})^{2} μ ({\tilde{p}}_{2})^{2}}, ν ({\tilde{p}}_{1}) ν ({\tilde{p}}_{2})) 〉 \end{matrix}$ (9) $\begin{matrix} {\tilde{p}}_{1} \otimes {\tilde{p}}_{2} & = & 〈 [s_{θ ({\tilde{p}}_{1}) \times θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) \times τ ({\tilde{p}}_{2})}], \\ (μ ({\tilde{p}}_{1}) μ ({\tilde{p}}_{2}), \sqrt{ν ({\tilde{p}}_{1})^{2} + ν ({\tilde{p}}_{2})^{2} - ν ({\tilde{p}}_{1})^{2} ν ({\tilde{p}}_{2})^{2}}) 〉 \end{matrix}$ (10) $λ {\tilde{p}}_{1} = 〈 [s_{λ θ ({\tilde{p}}_{1})}, s_{λ τ ({\tilde{p}}_{1})}], (\sqrt{1 - {(1 - μ ({\tilde{p}}_{1})^{2})}^{λ}}, ν ({\tilde{p}}_{1})^{λ}) 〉$ (11) ${\tilde{p}}_{1}^{λ} = 〈 [s_{θ ({\tilde{p}}_{1})^{λ}}, s_{τ ({\tilde{p}}_{1})^{λ}}], (μ ({\tilde{p}}_{1})^{λ}, \sqrt{1 - {(1 - ν ({\tilde{p}}_{1})^{2})}^{λ}}) 〉$ (12)

Theorem 1.Suppose ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ , ${\tilde{p}}_{2} = < [s_{θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{2})}], (μ ({\tilde{p}}_{2}), ν ({\tilde{p}}_{2})) >$ and ${\tilde{p}}_{3} = < [s_{θ ({\tilde{p}}_{3})}, s_{τ ({\tilde{p}}_{3})}], (μ ({\tilde{p}}_{3}), ν ({\tilde{p}}_{3})) >$ are any three PFULNs, λ, λ₁, λ₂ ≥ 0, the operational laws have the following properties: $(1) {\tilde{p}}_{1} \oplus {\tilde{p}}_{2} = {\tilde{p}}_{2} \oplus {\tilde{p}}_{1}$ (13) $(2) {\tilde{p}}_{1} \otimes {\tilde{p}}_{2} = {\tilde{p}}_{2} \otimes {\tilde{p}}_{1}$ (14) $(3) λ ({\tilde{p}}_{1} \oplus {\tilde{p}}_{2}) = λ {\tilde{p}}_{1} \oplus λ {\tilde{p}}_{2}$ (15) $(4) {\tilde{p}}_{1}^{λ} \otimes {\tilde{p}}_{2}^{λ} = ({\tilde{p}}_{1} \otimes {\tilde{p}}_{2})^{λ}$ (16) $(5) λ_{1} {\tilde{p}}_{1} \oplus λ_{2} {\tilde{p}}_{1} = (λ_{1} + λ_{2}) {\tilde{p}}_{1}$ (17) $(6) {\tilde{p}}_{1}^{λ_{1}} \otimes {\tilde{p}}_{1}^{λ_{2}} = {\tilde{p}}_{1}^{λ_{1} + λ_{2}}$ (18) $(7) {\tilde{p}}_{1} \oplus {\tilde{p}}_{2} \oplus {\tilde{p}}_{3} = {\tilde{p}}_{1} \oplus ({\tilde{p}}_{2} \oplus {\tilde{p}}_{3})$ (19) $(8) {\tilde{p}}_{1} \otimes {\tilde{p}}_{2} \otimes {\tilde{p}}_{3} = {\tilde{p}}_{1} \otimes ({\tilde{p}}_{2} \otimes {\tilde{p}}_{3})$ (20)

Proof.

According to the operational rule (1) expressed by the formula (9), it is obvious that the formula (13) is right.

According to the operational rule (2) expressed by the formula (10), it is obvious that the formula (14) is right.

For the formula (15), we can get

\begin{matrix} λ {\tilde{p}}_{1} \\ = 〈 [s_{λ θ ({\tilde{p}}_{1})}, s_{λ τ ({\tilde{p}}_{1})}], (\sqrt{1 - {(1 - μ ({\tilde{p}}_{1})^{2})}^{λ}}, ν ({\tilde{p}}_{1})^{λ}) 〉 \\ λ ({\tilde{p}}_{1} \oplus {\tilde{p}}_{2}) = λ 〈 [s_{θ ({\tilde{p}}_{1}) + θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{2})}], \\ (\sqrt{μ ({\tilde{p}}_{1})^{2} + μ ({\tilde{p}}_{2})^{2} - μ ({\tilde{p}}_{1})^{2} μ ({\tilde{p}}_{2})^{2}}, ν ({\tilde{p}}_{1}) ν ({\tilde{p}}_{2})) 〉 \\ = 〈 [s_{λ (θ ({\tilde{p}}_{1}) + θ ({\tilde{p}}_{2}))}, s_{λ (τ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{2}))}], \\ (\sqrt{1 - {(1 - μ ({\tilde{p}}_{1})^{2})}^{λ} {(1 - μ ({\tilde{p}}_{2})^{2})}^{λ}}, ν ({\tilde{p}}_{1})^{λ} ν ({\tilde{p}}_{2})^{λ}) 〉 . \end{matrix}

(21)

and $\begin{matrix} λ {\tilde{p}}_{1} \oplus λ {\tilde{p}}_{2} = 〈 [s_{λ θ ({\tilde{p}}_{1})}, s_{λ τ ({\tilde{p}}_{1})}], \\ (\sqrt{1 - {(1 - μ ({\tilde{p}}_{1})^{2})}^{λ}}, ν ({\tilde{p}}_{1})^{λ}) 〉 \\ \oplus 〈 [s_{λ θ ({\tilde{p}}_{2})}, s_{λ τ ({\tilde{p}}_{2})}], (\sqrt{1 - {(1 - μ ({\tilde{p}}_{2})^{2})}^{λ}}, ν ({\tilde{p}}_{2})^{λ}) 〉 \\ = 〈 [s_{λ θ ({\tilde{p}}_{1}) + λ θ ({\tilde{p}}_{2})}, s_{λ τ ({\tilde{p}}_{1}) + λ τ ({\tilde{p}}_{2})}], \\ (\sqrt{1 - {(1 - μ ({\tilde{p}}_{1})^{2})}^{λ} {(1 - μ ({\tilde{p}}_{2})^{2})}^{λ}}, ν ({\tilde{p}}_{1})^{λ} ν ({\tilde{p}}_{2})^{λ}) 〉 \end{matrix}$ (22)

So, we can get $λ ({\tilde{p}}_{1} \oplus {\tilde{p}}_{2}) = λ {\tilde{p}}_{1} \oplus λ {\tilde{p}}_{2}$ , which finishes the proof of formula (15).

Similar to the proof of formula (15), it is easy to prove that formulas (15–18) hold, which are omitted here.

According to the operational rule (1) expressed by the formula (9), it is obvious that the formula (19) is right.

According to the operational rule (2) expressed by the formula (10), it is obvious that the formula (20) is right.

Definition 5. Let ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ be one PFLN, and then the expectation function $E ({\tilde{p}}_{1})$ of a PFULN can be expressed asfollows: $E ({\tilde{p}}_{1}) = s_{\frac{(μ^{2} ({\tilde{p}}_{1}) + 1 - ν^{2} ({\tilde{p}}_{1})) \times (θ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{1}))}{4}}$ (23)

Definition 6. Let ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ be one PFLN, and then the score function $S ({\tilde{p}}_{1})$ of a PFULN can be expressed asfollows: $S ({\tilde{p}}_{1}) = s_{\frac{(μ^{2} ({\tilde{p}}_{1}) - ν^{2} ({\tilde{p}}_{1})) \times (θ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{1}))}{4}}$ (24)

Definition 7. Let ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ be any one PFLN, and then the accuracy function $A ({\tilde{p}}_{1})$ of a PFULN can be expressed as follows: $A ({\tilde{p}}_{1}) = s_{\frac{(μ^{2} ({\tilde{a}}_{i}) + ν^{2} ({\tilde{a}}_{i})) \times (θ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{1}))}{2}}$ (25)

Definition 8. Let ${\tilde{p}}_{1}$ and ${\tilde{p}}_{2}$ be any two PFULNs, then

If $E ({\tilde{p}}_{1}) > E ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ ;

If $E ({\tilde{p}}_{1}) = E ({\tilde{p}}_{2})$ , then

If $S ({\tilde{p}}_{1}) > S ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ ;

If $E ({\tilde{p}}_{1}) = E ({\tilde{p}}_{2})$ and $S ({\tilde{p}}_{1}) = S ({\tilde{p}}_{2})$ , then

If $A ({\tilde{p}}_{1}) > A ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ ;

If $A ({\tilde{p}}_{1}) = A ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} = {\tilde{p}}_{2}$ .

Suppose ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ , ${\tilde{p}}_{2} = < [s_{θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{2})}], (μ ({\tilde{p}}_{2}), ν ({\tilde{p}}_{2})) >$ and ${\tilde{p}}_{3} = < [s_{θ ({\tilde{p}}_{3})}, s_{τ ({\tilde{p}}_{3})}], (μ ({\tilde{p}}_{3}), ν ({\tilde{p}}_{3})) >$ are any three PFULNs. It’s noted that when ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ and ${\tilde{p}}_{2} > {\tilde{p}}_{3}$ then ${\tilde{p}}_{1} > {\tilde{p}}_{3}$ , which mean that PFULNs have transitivity.

Definition 9. Let ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ and ${\tilde{p}}_{2} = < [s_{θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{2})}], (μ ({\tilde{p}}_{2}), ν ({\tilde{p}}_{2})) >$ be any two PFULNs, then the Hamming distance between ${\tilde{p}}_{1}$ and ${\tilde{p}}_{2}$ is: $d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) = \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \end{matrix})$ (26)

Theorem 2.Let ${\tilde{p}}_{1} = < [s_{θ ({\tilde{p}}_{1})}, s_{τ ({\tilde{p}}_{1})}], (μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1})) >$ , ${\tilde{p}}_{2} = < [s_{θ ({\tilde{p}}_{2})}, s_{τ ({\tilde{p}}_{2})}], (μ ({\tilde{p}}_{2}), ν ({\tilde{p}}_{2})) >$ and ${\tilde{p}}_{3} = < [s_{θ ({\tilde{p}}_{3})}, s_{τ ({\tilde{p}}_{3})}], (μ ({\tilde{p}}_{3}), ν ({\tilde{p}}_{3})) >$ be any three PFULNs, the Hamming distance $d ({\tilde{p}}_{1}, {\tilde{p}}_{2})$ satisfies the following condition:

$0 \leq d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) \leq 1$ ;

If $d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) = 0$ , then ${\tilde{p}}_{1} = {\tilde{p}}_{2}$ ;

$d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) = d ({\tilde{p}}_{2}, {\tilde{p}}_{1})$ ;

$d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) + d ({\tilde{p}}_{2}, {\tilde{p}}_{3}) \geq d ({\tilde{p}}_{1}, {\tilde{p}}_{3})$ .

Proof. According to the Definition 3, we can know $θ ({\tilde{p}}_{1}), θ ({\tilde{p}}_{2}) \in [0, l - 1]$ , $μ ({\tilde{p}}_{1}), ν ({\tilde{p}}_{1}) \in [0, 1]$ $μ ({\tilde{p}}_{2}), ν ({\tilde{p}}_{2}) \in [0, 1]$ , $0 \leq μ ({\tilde{p}}_{1})^{2} + ν ({\tilde{p}}_{1})^{2} \leq 1$ , and $0 \leq μ ({\tilde{p}}_{2})^{2} + ν ({\tilde{p}}_{2})^{2} \leq 1$ , then $\begin{matrix} (1) d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) & = & \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \end{matrix}) \end{matrix}$ (27)

$\begin{matrix} \leq & \frac{1}{4 (l - 1)} (\begin{matrix} μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) + μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) + ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) + ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) \\ + π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) + π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) + μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) + μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) \\ + ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) + ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) + π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) + π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) \end{matrix}) \\ = & \frac{1}{4 (l - 1)} (\begin{matrix} μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) + μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) + ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) + ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) \\ + (1 - μ ({\tilde{p}}_{1})^{2} - ν ({\tilde{p}}_{1})^{2}) θ ({\tilde{p}}_{1}) + (1 - μ ({\tilde{p}}_{2})^{2} - ν ({\tilde{p}}_{2})^{2}) θ ({\tilde{p}}_{2}) \\ + μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) + μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) + ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) + ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) \\ + (1 - μ ({\tilde{p}}_{1})^{2} - ν ({\tilde{p}}_{1})^{2}) τ ({\tilde{p}}_{1}) + (1 - μ ({\tilde{p}}_{2})^{2} - ν ({\tilde{p}}_{2})^{2}) τ ({\tilde{p}}_{2}) \end{matrix}) \\ = & \frac{1}{4 (l - 1)} (θ ({\tilde{p}}_{1}) + θ ({\tilde{p}}_{2}) + τ ({\tilde{p}}_{1}) + τ ({\tilde{p}}_{2})) \leq \frac{1}{4 (l - 1)} ((l - 1) + (l - 1) + (l - 1) + (l - 1)) = 1 \end{matrix}$ (28)

According to the formula (17), it is easy to find that the Hamming distance $d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) \geq 0$ . So we can get $0 \leq d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) \leq 1$ .

$(2) If d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) = \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \end{matrix}) = 0 .$ (29)

We can get the following equations:

$\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | = 0, \\ | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | = 0, \\ | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | = 0, \\ | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | = 0, \\ | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | = 0, \\ | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | = 0 . \end{matrix}$ (30)

In order to satisfy the above equations, the solution is the following condition:

$\begin{matrix} θ ({\tilde{p}}_{1}) & = & θ ({\tilde{p}}_{2}), τ ({\tilde{p}}_{1}) = τ ({\tilde{p}}_{2}), μ ({\tilde{p}}_{1}) = μ ({\tilde{p}}_{2}), \\ ν ({\tilde{p}}_{1}) & = & ν ({\tilde{p}}_{2}), π ({\tilde{p}}_{1}) = π ({\tilde{p}}_{2}) \end{matrix}$ (31)

So, we can get ${\tilde{p}}_{1} = {\tilde{p}}_{2}$ .

$\begin{matrix} (3) d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) & = & \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | \end{matrix}) \\ = & \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) | + | ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) | \\ + | π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) | + | μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) | \\ + | ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) | + | π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) | \end{matrix}) \\ = & d ({\tilde{p}}_{2}, {\tilde{p}}_{1}) \end{matrix}$ (32)

$\begin{matrix} (4) d ({\tilde{p}}_{1}, {\tilde{p}}_{3}) & = & \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \end{matrix}) \\ = & \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) + μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - μ ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) + ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - ν ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) + π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - π ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) + μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - μ ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) + ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - ν ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \\ + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) + π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - π ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \end{matrix}) \\ \leq & \frac{1}{4 (l - 1)} (\begin{matrix} | μ ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | μ ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - μ ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | ν ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | ν ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - ν ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | π ({\tilde{p}}_{1})^{2} θ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) | + | π ({\tilde{p}}_{2})^{2} θ ({\tilde{p}}_{2}) - π ({\tilde{p}}_{3})^{2} θ ({\tilde{p}}_{3}) | \\ + | μ ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | μ ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - μ ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \\ + | ν ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | ν ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - ν ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \\ + | π ({\tilde{p}}_{1})^{2} τ ({\tilde{p}}_{1}) - π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) | + | π ({\tilde{p}}_{2})^{2} τ ({\tilde{p}}_{2}) - π ({\tilde{p}}_{3})^{2} τ ({\tilde{p}}_{3}) | \end{matrix}) \\ = & d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) + d ({\tilde{p}}_{2}, {\tilde{p}}_{3}) . \end{matrix}$ (33)

So, $d ({\tilde{p}}_{1}, {\tilde{p}}_{2}) + d ({\tilde{p}}_{2}, {\tilde{p}}_{3}) \geq d ({\tilde{p}}_{1}, {\tilde{p}}_{3})$ is hold.

Q.E.D.

4 The Pythagorean fuzzy uncertain linguistic TODIM method

The TODIM method [5] is based on the prospect theory (PT), which can consider the risk attitude of DMs. However, so far, the existing TODIM methods cannot solve decision making problems under the Pythagorean fuzzy uncertain linguistic environment. In this part, we will extend the TODIM method to solve the MCGDM problems in which the attributes take the form of the Pythagorean fuzzy uncertain linguistic information.

4.1 The origin TODIM method

The TODIM method is inspired by the prospect theory (PT), which could consider the bounded rationality of the decision makers (DMs). In other word, TODIM method not only takes the risk averting attitude of the DMs in the face of gain into account but

also pays attention to the risk seeking attitude of the DMs in the face of loss. The method can give the corresponding dominance between each alternative and others with respect to a particular criterion.

Suppose {A₁, A₂, …, A_m} is the set of alternatives, {C₁, C₂, …, C_n} is the set of criteria, and (w₁, w₂, …, w_n) ^T is the weighted vector of the criteria, which satisfies the condition that w_j ∈ [0, 1] (j = 1, 2, …, n) and $\sum_{j = 1}^{n} w_{j} = 1$ . So the decision matrix A = [a_ij] _m×n can be constructed by m alternatives and n criteria, where a_ij is the evaluation value of the ith alternative under the jth criterion. The steps of the TODIM method can be expressed as follows:

Step 1. Normalize the decision matrix A = [a_ij] _m×n into the matrix $\tilde{A} = [{\tilde{a}}_{ij}]_{m \times n}$ .

Step 2. Calculate the relative weight w_ir of the criterion C_j to C_r. $w_{jr} = \frac{w_{j}}{w_{r}} where w_{r} = max {w_{j} | j = 1, 2, \dots, n}$ (34)

Step 3. Calculate the dominance degree of the alternative A_i over each alternative A_t with respect to the criterion C_j.

$\begin{matrix} φ_{j} (A_{i}, A_{t}) \\ = {\begin{matrix} \sqrt{\frac{w_{jr} ({\tilde{a}}_{ij} - {\tilde{a}}_{tj})}{\sum_{j = 1}^{n} w_{jr}}} & if {\tilde{a}}_{ij} - {\tilde{a}}_{tj} > 0 \\ 0 & if {\tilde{a}}_{ij} - {\tilde{a}}_{oj} = 0 \\ - \frac{1}{θ} \sqrt{\frac{(\sum_{j = 1}^{n} w_{jr}) ({\tilde{a}}_{ij} - {\tilde{a}}_{tj})}{w_{jr}}} & if {\tilde{a}}_{ij} - {\tilde{a}}_{tj} < 0 \end{matrix} \end{matrix}$ (35) where the parameter θ is the attenuation factor of the losses.

Step 4. Calculate the overall dominance degree of the alternative A_i over each alternative A_t. $δ (A_{i}, A_{t}) = \sum_{j = 1}^{n} φ_{j} (A_{i}, A_{t}) (i, t = 1, 2, \dots, m)$ (36)

Step 5. Calculate the overall value of the alternative A_i. $δ (A_{i}) = \frac{\sum_{t = 1}^{m} δ (A_{i}, A_{t}) - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}{max_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})} - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}$ (37)

Step 6. Rank all the alternatives by the overall values δ (A_i) (i = 1, 2, …, m). The bigger δ (A_i) is, the better the alternative A_i is.

4.2 Description of the MCGDM problems

For a MCGDM problem with the Pythagorean fuzzy uncertain linguistic information, let A = {A₁, A₂, …, A_m} be the set of alternatives, C = {C₁, C₂, …, C_n} be the set of criteria. w = (w₁, w₂, …, w_n) ^T is the weighted vector of the criteria, satisfying w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . E = {E₁, E₂, …, E_q} is the set of DMs, λ = (λ₁, λ₂, …, λ_q) ^T is the weighted vector of DMs, where λ_k ∈ [0, 1] and $\sum_{k = 1}^{q} λ_{k} = 1$ . Suppose that the decision making matrix is $R^{k} = [r_{ij}^{k}]_{m \times n}$ , where $r_{ij}^{k} = 〈 [s_{θ (r_{ij}^{k})}, s_{τ (r_{ij}^{k})}], (μ (r_{ij}^{k}), ν (r_{ij}^{k})) 〉$ is the evaluation information of alternative A_i under the criteria C_j given by the DM E_k, which takes the form of the PFULN, satisfying that $0 \leq μ (r_{ij}^{k})^{2} + ν (r_{ij}^{k})^{2} \leq 1$ , $[s_{θ (r_{ij}^{k})}, s_{τ (r_{ij}^{k})}] \in \bar{S}$ . $\bar{S} = {s_{0}, s_{1}, \dots, s_{l - 1}}$ is the linguistic term set.

4.3 Procedure of the Pythagorean Fuzzy Uncertain Linguistic TODIM method

The extended TODIM method has been effectively applied to lots of MCGDM problems, which can conduct the sensitivity analysis through the different parameter θ. Based on PT, the Pythagorean Fuzzy Uncertain Linguistic TODIM method can make effective choices in risked environment according to the evaluation information expressed by PFULNs. In this sub-section, the steps of the proposed TODIM are described in detail.

Step 1. Normalize the decision matrix $R = [r_{ij}^{k}]_{m \times n}$ into the matrix $\tilde{R} = [{\tilde{r}}_{ij}^{k}]_{m \times n}$ .

For benefit criteria: ${\tilde{r}}_{ij}^{k} = 〈 [s_{θ (r_{ij}^{k})}, s_{τ (r_{ij}^{k})}], (μ (r_{ij}^{k}), ν (r_{ij}^{k})) 〉$ (38)

For cost criteria: ${\tilde{r}}_{ij}^{k} = 〈 [neg (s_{τ (r_{ij}^{k})}), neg (s_{θ (r_{ij}^{k})})], (ν (r_{ij}^{k}), μ (r_{ij}^{k})) 〉 .$ (39)

Step 2. Calculate the relative weight w_ir of the criteria C_j to C_r. $w_{jr} = \frac{w_{j}}{w_{r}} where w_{r} = max {w_{j} | j = 1, 2, \dots, n}$ (40)

Step 3. Calculate the dominance degree of the alternative A_i over each alternative A_t with respect to the DM E_k under the criteria C_j.

In this step, it is essential to measure the difference between alternative A_i and alternative A_t according to the criterion C_j. Normally, there are two methods for selection. In one way, the PFULNs can be translated into specific crisp numbers by the score function. But this method generally results in the loss of evaluation information. Another way is to measure the distance between different alternatives, and it doesn’t damage the integrity of information. As a result, we select the second way to calculate the difference between different alternatives. $φ_{j}^{k} (A_{i}, A_{t}) = {\begin{matrix} \sqrt{\frac{w_{jr} d ({\tilde{r}}_{ij}^{k} - {\tilde{r}}_{tj}^{k})}{\sum_{j = 1}^{n} w_{jr}}} & if {\tilde{r}}_{ij}^{k} > {\tilde{r}}_{tj}^{k} \\ 0 & if {\tilde{r}}_{ij}^{k} = {\tilde{r}}_{tj}^{k} \\ - \frac{1}{θ} \sqrt{\frac{(\sum_{j = 1}^{n} w_{jr}) d ({\tilde{r}}_{ij}^{k} - {\tilde{r}}_{tj}^{k})}{w_{jr}}} & if {\tilde{r}}_{ij}^{k} < {\tilde{r}}_{tj}^{k} \end{matrix}$ (41) where the parameter θ is the attenuation factor of the losses, and $d ({\tilde{r}}_{ij}^{k} - {\tilde{r}}_{tj}^{k})$ is used to measure the distance between ${\tilde{r}}_{ij}^{k}$ and ${\tilde{r}}_{tj}^{k}$ by the formula (24). When ${\tilde{r}}_{ij}^{k} > {\tilde{r}}_{tj}^{k}$ , $φ_{j}^{k} (A_{i}, A_{j})$ means a gain; when ${\tilde{r}}_{ij}^{k} < {\tilde{r}}_{tj}^{k}$ , $φ_{j}^{k} (A_{i}, A_{j})$ means a loss; when ${\tilde{r}}_{ij}^{k} = {\tilde{r}}_{tj}^{k}$ , $φ_{j}^{k} (A_{i}, A_{j})$ is neither a gain or a loss.

Step 4. Calculate the overall dominance degree of the alternative A_i over each alternative A_t with respect to the DM E_k.

$\begin{matrix} δ^{k} (A_{i}, A_{t}) = \sum_{j = 1}^{n} φ_{j}^{k} (A_{i}, A_{t}) \\ (i, t = 1, 2, \dots, m; k = 1, 2, \dots, q) \end{matrix}$ (42)

Step 5. Calculate the collective overall dominance degree of the alternative A_i over each alternative A_t. $δ (A_{i}, A_{t}) = \sum_{k = 1}^{q} λ_{k} δ^{k} (A_{i}, A_{t}) (i, t = 1, 2, \dots, m)$ (43)

Step 6. Calculate the overall value of the alternative A_i. $δ (A_{i}) = \frac{\sum_{t = 1}^{m} δ (A_{i}, A_{t}) - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}{max_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})} - min_{i} {\sum_{t = 1}^{m} δ (A_{i}, A_{t})}}$ (44)

Step 7. Rank all the alternatives by the overall values δ (A_i)(i = 1, 2, …, m). The bigger δ (A_i) is, the better the alternative A_i is.

5 An illustrative example

In this part, we use a numerical example to demonstrate the effectiveness of the proposed method, which is an adaptation of numerical example in [24]. An investment company wants to invest a software project, the potential four alternatives are {A₁, A₂, …, A_m}. In the process of selecting ideal alternative, there are three criteria: C₁ the technical feasibility; C₂ the economic feasibility; C₃ the operational feasibility. The weighted vector of these criteria is w = (0.39, 0.26, 0.35) ^T. The evaluation information of alternatives A_i (i = 1, 2, 3, 4) given by three DMs E_k (k = 1, 2, 3) with respect to the criteria C_j (j = 1, 2, 3) are used to constructed the decision making matrices $R^{k} = [r_{ij}^{k}]_{m \times n}$ which are listed in Tables 1–3. Suppose the weighted vector of DMs is λ = (0.2, 0.5, 0.3) ^T, and the linguistic term set is $\bar{S} = {s_{0}, s_{1}, \dots, s_{6}}$ .

Table 1
The evaluation values from expert E₁

C ₁ C ₂ C ₃

A ₁ < [s₁, s₃] , (0.8, 0.6)> < [s₂, s₄] , (0.7, 0.3)> < [s₃, s₅] , (0.6, 0.2)>

A ₂ < [s₂, s₄] , (0.7, 0.6)> < [s₃, s₅] , (0.8, 0.3)> < [s₁, s₃] , (0.4, 0.2)>

A ₃ < [s₃, s₅] , (0.6, 0.2)> < [s₁, s₃] , (0.6, 0.4)> < [s₂, s₄] , (0.8, 0.2)>

A ₄ < [s₁, s₃] , (0.7, 0.4)> < [s₀, s₂] , (0.7, 0.5)> < [s₃, s₅] , (0.9, 0.2)>

	C ₁	C ₂	C ₃
A ₁	< [s₁, s₃] , (0.8, 0.6)>	< [s₂, s₄] , (0.7, 0.3)>	< [s₃, s₅] , (0.6, 0.2)>
A ₂	< [s₂, s₄] , (0.7, 0.6)>	< [s₃, s₅] , (0.8, 0.3)>	< [s₁, s₃] , (0.4, 0.2)>
A ₃	< [s₃, s₅] , (0.6, 0.2)>	< [s₁, s₃] , (0.6, 0.4)>	< [s₂, s₄] , (0.8, 0.2)>
A ₄	< [s₁, s₃] , (0.7, 0.4)>	< [s₀, s₂] , (0.7, 0.5)>	< [s₃, s₅] , (0.9, 0.2)>

Table 2

The evaluation values from expert E₂

	C ₁	C ₂	C ₃
A ₁	< [s₃, s₅] , (0.7, 0.6)>	< [s₁, s₃] , (0.7, 0.6)>	< [s₂, s₄] , (0.6, 0.4)>
A ₂	< [s₀, s₂] , (0.8, 0.3)>	< [s₂, s₄] , (0.7, 0.3)>	< [s₃, s₅] , (0.6, 0.3)>
A ₃	< [s₁, s₃] , (0.6, 0.2)>	< [s₃, s₅] , (0.6, 0.4)>	< [s₂, s₄] , (0.7, 0.3)>
A ₄	< [s₂, s₄] , (0.6, 0.4)>	< [s₁, s₃] , (0.8, 0.5)>	< [s₄, s₆] , (0.8, 0.2)>

Table 3

The evaluation values from expert E₃

	C ₁	C ₂	C ₃
A ₁	< [s₂, s₄] , (0.5, 0.6)>	< [s₁, s₃] , (0.6, 0.3)>	< [s₆, s₆] , (0.6, 0.4)>
A ₂	< [s₃, s₅] , (0.5, 0.6)>	< [s₂, s₄] , (0.8, 0.3)>	< [s₀, s₂] , (0.8, 0.2)>
A ₃	< [s₁, s₃] , (0.5, 0.2)>	< [s₂, s₄] , (0.8, 0.4)>	< [s₃, s₅] , (0.5, 0.2)>
A ₄	< [s₃, s₅] , (0.7, 0.5)>	< [s₁, s₃] , (0.7, 0.6)>	< [s₂, s₄] , (0.7, 0.2)>

5.1 The evaluation steps

Step 1. Because all criteria are benefit types and the evaluation information is expressed by the PFULNs, the normalization of the three criteria can be neglected in this example. So, $\tilde{R} = [{\tilde{r}}_{ij}^{k}]_{m \times n} = R = [r_{ij}^{k}]_{m \times n} .$ (45)

Step 2. Calculate the relative weight w_ir of the criteria C_j to C_r. $\begin{matrix} w_{r} & = & max {0.39, 026, 0.35} = 0.39 \\ w_{jr} & = & (1.000, 0.6667, 0.8974)^{T} \end{matrix}$ (46)

Step 3. Calculate the dominance degree of the alternative A_i over each alternative A_t with respect to the DM E_k under the criteria C_j (suppose θ = 2.25), which are shown as follows: $\begin{matrix} φ_{1}^{1} & = & [\begin{matrix} 0 & - 0.2045 & - 0 . 3698 & - 0 . 2431 \\ 0 . 1803 & 0 & 0 . 3122 & 0.2208 \\ 0.3245 & 0.3122 & 0 & 0.2746 \\ 0.2133 & - 0.2516 & - 0.3129 & 0 \end{matrix}], \\ φ_{2}^{1} & = & [\begin{matrix} 0 & - 0.2934 & 0 . 1558 & 0 . 2153 \\ 0 . 1717 & 0 & 0 . 2143 & 0.2571 \\ - 0.2663 & - 0.3664 & 0 & 0.1487 \\ - 0.3681 & - 0.4394 & - 0.2541 & 0 \end{matrix}], \\ φ_{3}^{1} & = & [\begin{matrix} 0 & 0.2415 & 0 . 2391 & - 0 . 4115 \\ - 0 . 3067 & 0 & - 0 . 3275 & - 0.4337 \\ - 0.3036 & 0.2579 & 0 & - 0.2844 \\ 0.3240 & 0.3416 & 0.2240 & 0 \end{matrix}], \\ φ_{1}^{2} & = & [\begin{matrix} 0 & 0.3122 & 0 . 3225 & 0 . 2951 \\ - 0 . 3558 & 0 & - 0 . 2306 & - 0.2905 \\ - 0.3675 & 0.2024 & 0 & - 0.2054 \\ - 0.3363 & 0.2550 & 0.1803 & 0 \end{matrix}], \\ φ_{2}^{2} & = & [\begin{matrix} 0 & - 0.3468 & - 0 . 3799 & - 0 . 1949 \\ 0 . 2029 & 0 & - 0 . 2710 & 0.1779 \\ 0.2223 & 0.1585 & 0 & 0.2413 \\ 0.1140 & - 0.3040 & - 0.3664 & 0 \end{matrix}], \\ φ_{3}^{2} & = & [\begin{matrix} 0 & - 0.2415 & - 0 . 1915 & - 0 . 3470 \\ 0 . 1902 & 0 & 0 . 1839 & - 0.3443 \\ 0.1508 & - 0.2336 & 0 & - 0.3172 \\ 0.2733 & 0.2711 & 0.2498 & 0 \end{matrix}], \\ φ_{1}^{3} & = & [\begin{matrix} 0 & - 0.2054 & 0 . 2259 & - 0 . 2474 \\ 0 . 1803 & 0 & 0 . 2662 & - 0.2847 \\ - 0.2574 & - 0.3033 & 0 & - 0.3456 \\ 0.2171 & 0.2498 & 0.3033 & 0 \end{matrix}], \\ φ_{2}^{3} & = & [\begin{matrix} 0 & - 0 . 3163 & - 0 . 3558 & 0 . 1862 \\ 0 . 1850 & 0 & 0 . 0954 & 0.2029 \\ 0.2082 & - 0.1631 & 0 & 0.1791 \\ - 0.3183 & - 0.3468 & - 0.3061 & 0 \end{matrix}], \\ φ_{3}^{3} & = & [\begin{matrix} 0 & 0 . 3819 & 0 . 2791 & 0 . 2958 \\ - 0 . 4849 & 0 & - 0 . 3775 & - 0.3067 \\ - 0.3544 & 0.2973 & 0 & 0.2379 \\ - 0.3756 & 0.2415 & - 0.3021 & 0 \end{matrix}] . \end{matrix}$ (47)

Step 4. Calculate the overall dominance degree of the alternative A_i over each alternative A_t with respect to the DM E_k, which are shown as follows. $\begin{matrix} δ^{1} (A_{i}, A_{t}) \\ = [\begin{matrix} 0 & - 0.2574 & 0.0251 & - 0.4392 \\ 0.0452 & 0 & 0.1991 & 0.0441 \\ - 0.2454 & 0.2038 & 0 & 0.1388 \\ 0.1693 & - 0.3495 & - 0.3431 & 0 \end{matrix}] \\ δ^{2} (A_{i}, A_{t}) \\ = [\begin{matrix} 0 & - 0.2761 & - 0.2490 & - 0.2468 \\ 0.0372 & 0 & - 0.3177 & - 0.4569 \\ 0.0056 & 0.1273 & 0 & - 0.3084 \\ 0.0509 & 0.2220 & 0.0638 & 0 \end{matrix}] \\ δ^{3} (A_{i}, A_{t}) \\ = [\begin{matrix} 0 & - 0.1398 & 0.1491 & 0.2346 \\ - 0.1196 & 0 & - 0.0159 & - 0.3885 \\ - 0.4036 & - 0.1691 & 0 & 0.0713 \\ - 0.4768 & 0.1445 & - 0.3049 & 0 \end{matrix}] . \end{matrix}$ (48)

Step 5. Calculate the collective overall dominance degree of the alternative A_i over each alternative A_t, which is shown as follows. $\begin{matrix} δ (A_{i}, A_{t}) \\ = [\begin{matrix} 0 & - 0.2315 & - 0.0747 & - 0.1408 \\ - 0.0082 & 0 & - 0.1238 & - 0.3362 \\ - 0.1674 & 0.0537 & 0 & - 0.1050 \\ - 0.0837 & 0.0845 & - 0.1282 & 0 \end{matrix}] . \end{matrix}$ (49)

Step 6. Calculate the overall value of the alternative A_i. $\begin{matrix} δ (A_{1}) & = & 0.0621, δ (A_{2}) = 0, \\ δ (A_{3}) & = & 0.7322, δ (A_{4}) = 1 . \end{matrix}$ (50)

Step 7. Rank all the alternatives by the overall values δ (A_i), and select the best alternative.

Based on the Step 6, we can know that δ (A₄) > δ (A₃) > δ (A₁) > δ (A₂) so that the ranking of the alternatives is A₄ ≻ A₃ ≻ A₁ ≻ A₂. Certainly, the alternative A₄ is the best.

5.2 Analyzing the effect of the parameter θ

Based on the above introduction of the proposed TODIM method, we can know that the parameter θ is crucial to the ranking of alternatives in the process of the decision making. θ is the attenuation factor of the losses, which represents the characteristic of being steeper for losses than for gains. So, in this part, we give the further analysis about the influence on the ranking results of different parameter value θ, which are shown in Figs. 1–3.

Fig.1

The Overall Values With θ Varying From 1 to 2.

Fig.2

The Overall Values With θ Varying From 2 to 3.

Fig.3

The Overall Values With θ Varying From 3 to 4.

Based on the δ above Figs. 1–3, we can get an ordering rule which is listed in Table 4. According to the Table 4, we can find that the ranking results have changed with different parameter value θ. When the parameter 1 ≤ θ ≤ 1.1, the ranking result is A₃ ≻ A₄ ≻ A₂ ≻ A₁; when the parameter 1.2 ≤ θ ≤ 1.8, the ranking result is A₄ ≻ A₃ ≻ A₂ ≻ A₁; when the parameter 1.9 ≤ θ ≤ 4, the ranking result is A₄ ≻ A₃ ≻ A₁ ≻ A₂. With the change of the θ parameter, the best alternative also changes.

Table 4

Ranking Results of the Alternatives By the Different Parameter value θ

θ	Overall values δ (A_i)	order
θ = 1	δ (A₁) =0, δ (A₂) =0.1812, δ (A₃) =1, δ (A₄) =0.9614	A₄ ≻ A₃ ≻ A₁ ≻ A₂
θ = 1.1	δ (A₁) =0, δ (A₂) =0.1608, δ (A₃) =1, δ (A₄) =0.9941	A₄ ≻ A₃ ≻ A₁ ≻ A₂
θ = 1.2	δ (A₁) =0, δ (A₂) =0.1363, δ (A₃) =0.9735, δ (A₄) =1	A₄ ≻ A₃ ≻ A₂ ≻ A₁
θ = 1.8	δ (A₁) =0, δ (A₂) =0.0086, δ (A₃) =0.8101, δ (A₄) =1	A₄ ≻ A₃ ≻ A₂ ≻ A₁
θ = 1.9	δ (A₁) =0.0092, δ (A₂) =0, δ (A₃) =0.7892, δ (A₄) =1	A₃ ≻ A₄ ≻ A₂ ≻ A₁
θ = 2	δ (A₁) =0.0258, δ (A₂) =0, δ (A₃) =0.7713, δ (A₄) =1	A₃ ≻ A₄ ≻ A₂ ≻ A₁
θ = 4	δ (A₁) =0.2060, δ (A₂) =0, δ (A₃) =0.5771, δ (A₄) =1	A₃ ≻ A₄ ≻ A₂ ≻ A₁

5.3 Comparison analysis and discussions

In order to demonstrate the effectiveness of the proposed method in this part, we compare the ranking results in this paper with the results of two existing methods in reference [24]. We should first transform the PFUNs to Pythagorean fuzzy linguistic numbers (PFLNs) by means of substituting each uncertain linguistic part with the mean value of its upper and lower limits, which are precisely the evaluation information of numerical example in [24]. Then two existing methods in [24] are used to deal with evaluation information take the form of PFLNs. The first ranking result computed by PFLWA operator is A₄ ≻ A₃ ≻ A₂ ≻ A₁, which is the same with the result used by the proposed method in 1.2 ≤ θ ≤ 1.8. The second order got by the TOPSIS method is A₄ ≻ A₁ ≻ A₃ ≻ A₂. According to contrast between these results, we find that the best alternative is consistent. Therefore, it can demonstrate the efficiency of the new proposed method. Here, the advantages of the new method are listed as follows:

The PFULS can more precisely express the uncertainty in the MCGDM than the PFLS. In other words, PFULS is the extension of the PFLS. If the upper and the lower of the uncertain linguistic part in PFILS are equal, the PFULS can degenerate into PFLS. So, we can know that the PFULS has more extensive application prospect than the PFLS.

The proposed TODIM method considers the DMs’ psychological behavior, which can help DMs to effectively make decision in the face of risk by changing the attenuation factor of the losses. Neither the PFLWA operator nor TOPISIS method in reference [24] takes the subjective judgment in the face of the gain/loss into account. So, the proposed TODIM method with PFULS information is more practical in handling the MCGDM problems.

The method proposed in this paper is the optimization of the existing method in reference [14]. The new TODIM method under the PFULS environment can deal with more MCGDM problems than the TODIM method with IULS information. Because the square sum of membership degree and non-membership degree is less than or equal 1 in PFULS, but the sum of membership degree and non-membership degree is less than or equal 1 in IULS, the proposed TODIM method is better than the existing method in reference [14].

6 Conclusion

In this paper, we propose an extended TODIM method under the Pythagorean fuzzy uncertain linguistic environment. The PFULS is the combination of PFS with the uncertain linguistic set, which can express the subjective evaluation information more precisely. At the same time, DMs can give the membership degree and the non-membership degree belonging to the subjective evaluation information according to their knowledge accumulation. So, the PFULS can more suitably portray the uncertainty in the real MCGDM. The classical TODIM method is based on prospect theory, which takes the DMs’ psychological behavior into account by using the different parameter θ. Further, we proposed an extended TODIM method to cope with the MCGDM problem under the PFULS environment. In order to demonstrate the effectiveness of the proposed method, we gave a numerical example expressed by PFULS. In order to consider the influence of the attenuation factor of the losses θ, we used different parameters θ to analyze and compare the different ranking results with the two existing methods in reference [24]. As a result, we can prove that the proposed TODIM method is useful and precisely in the process of handling the MCGDM problems. The proposed method not only can portray the uncertainty in real decision making, but also can present the DMs’ psychological behavior in the face of risk. The future studies should focus on widening the scope of the proposed method in the real world. In addition, we should make further effort to improve and perfect the proposed TODIM method so that it can handle uncertain information presented by different forms.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71771140, 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project (Nos. 15BGLJ06, 16CGLJ31 and 16CKJJ27) the Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (2015Z057).

References

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.

Atanassov

K.T.

, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems33 (1989), 37–46.

Chen

Y.H.

, Wang

T.C.

and Wu

C.Y.

, Strategic decisions using the fuzzy PROMETHEE for IS outsourcing, Expert Systems with Applications38(10) (2011), 13216–13222.

Devi

, Extension of VIKOR method in intuitionistic fuzzy environment for robot selection, Expert Systems with Applications38(11) (2011), 14163–14168.

Gomes

and Lima

, TODIM: Basics and application to multicriteria ranking of projects with environmental impacts, Foundations of Computing and Decision Sciences16(4) (1992), 113–127.

Herrera

and Herrera Viedma

, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems115 (2000), 67–82.

Herrera

, Herrera Viedma

and Verdegay

J.L.

, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems79 (1996), 73–87.

Hashemi

S.S.

, Hajiagha

S.H.R.

, Zavadskas

E.K.

and Mahdiraji

H.A.

, Multi-criteria group decision making with ELECTRE III method based on interval-valued intuitionistic fuzzy information, Applied Mathematical Modelling40(2) (2016), 1554–1564.

Kou

, Ergu

, Lin

and Chen

, Pairwise comparison matrix in multiple criteria decision making, Technological and Economic Development of Economy22(5) (2016), 738–765.

10.

Kuang

, Kilgour

D.M.

and Hipel

K.W.

, Grey-based PROMETHEE II with application to evaluation of source water protection strategies, Information Sciences294 (2015), 376–389.

11.

Karabasevic

, Zavadskas

E.K.

, Turskis

and Dragisa

, The framework for the selection of personnel basedon the SWARA and ARAS methods under uncertainties, Informatica27(1) (2016), 49–65.

12.

Liu

P.D.

, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to Group Decision Making, IEEE Transactions on Fuzzy Systems22(1) (2014), 83–97.

13.

Liu

P.D.

and Jin

, Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making, Information Sciences205 (2012), 58–71.

14.

Liu

P.D.

and Teng

, An extended TODIM method for multiple attribute group decision making based on intuitionistic uncertain linguistic variables, Journal of Intelligent & Fuzzy Systems29(2) (2015), 701–711.

15.

Liou

J.J.H.

, Tamošaitienė

, Zavadskas

E.K.

and Tzeng

G.H.

, New hybrid COPRAS-G MADM model for improving and selecting suppliers in green supply chain management, International Journal of Production Research54(1) (2016), 114–134.

16.

and Zhao

, Stochastic interval-grey number VIKOR method based on prospect theory, Grey Systems: Theory and Application5(1) (2015), 105–116.

17.

Merigó

J.M.

, Fuzzy decision making using immediate probabilities, Computers & Industrial Engineering58(4) (2010), 651–657.

18.

Merigó

J.M.

, Gil-Lafuente

A.M.

and Yager

R.R.

, An overview of fuzzy research with bibliometric indicators, Applied Soft Computing27 (2015), 420–433.

19.

Mardani

, Jusoh

, Nor

K.M.D.

, Khalifah

, Zakwan

and Valipouret

, Multiple criteria decision-making techniques and their applications–a review of the literature from 2000 to 2014, Economic Research-Ekonomska Istraživanja28(1) (2015), 516–571.

20.

Mardani

, Zavadskas

E.K.

, Govindan

, Senin

A.A.

and Jusoh

, VIKOR technique: A systematic review of the state of the art literature on methodologies and applications, Sustainability8(1) (2016), 38.

21.

Nakhaei

, Bitarafan

, Lale Arefi

and Kapliński

, Model for rapid assessment of vulnerability of office buildings to blast using SWARA and SMART methods (a case study of swiss re tower), Journal of Civil Engineering and Management22(6) (2016), 831–843.

22.

Nguyen

H.T.

, Dawal

S.Z.M.

, Nukman

, Rifai

P.A.

and Aoyama

, An integrated MCDM model for conveyor equipment evaluation and selection in an FMC based on a fuzzy AHP and fuzzy ARAS in the presence of vagueness, PLoS One11(4) (2016), 26.

23.

Peng

, Dai

and Yuan

, Interval-valued fuzzy soft decision making methods based on MABAC, Similarity measure and EDAS, Fundamenta Informaticae152(4) (2017), 373–396.

24.

Peng

X.D.

and Yang

, Multiple attribute group decision making methods based on Pythagorean fuzzy linguistic set, Computer Engineering and Applications52(23) (2016), 50–54.

25.

Ruddy

, Eduardo

and Edoardo

, Application of the OCRA Method in the sugar cane harvest and its repercussion on the workers’ health, Work41 (2012), 3981–3983.

26.

Ren

, Xu

and Gou

, Pythagorean fuzzy TODIM approach to multi-criteria decision making, Applied Soft Computing42 (2016), 246–259.

27.

Stanujkic

, Kazimieras

, Brauers

W.K.M.

and Karabasevic

, An extension of the MULTIMOORA method for solving complex decision-making problems based on the use of interval-valued triangular fuzzy numbers, Transformations in Business & Economics14(2) (2015), 355–375.

28.

Stanujkic

, Karabasevic

and Zavadskas

E.K.

, A framework for the selection of a packaging design based on the SWARA method, Engineering Economics26(2) (2015), 181–187.

29.

Stanujkic

, Zavadska

E.K.

, Ghorabaee

M.K.

and Turskis

, An extension of the EDAS method based on the use of interval grey numbers, Studies in Informatics and Control26(1) (2016), 5–12.

30.

Turanoglu Bekar

, Cakmakci

and Kahraman

, Fuzzy COPRAS method for performance measurement in total productive maintenance: A comparative analysis, Journal of Business Economics and Management17(5) (2016), 663–684.

31.

Tavana

, Mavi

R.K.

, Santos-Arteaga

F.J.

and Doust

E.R.

, An extended VIKOR method using stochastic data and subjective judgments, Computers & Industrial Engineering97 (2016), 240–247.

32.

Veeramachaneni

and Kandikonda

, An ELECTRE approach for multi-criteria interval-valued intuitionistic trapezoidal fuzzy group decision making problems, Advances in Fuzzy Systems2016 (2016), 17.

33.

Wang

J.Q.

and Li

J.J.

, The multi-criteria group decision making method based on multi-granularity intuitionistic two semantics, Science & Technology Information33(1) (2009), 8–9.

34.

Yager

R.R.

, Pythagorean fuzzy subsets, In Proc Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, 2013, pp. 57–61.

35.

Yager

R.R.

, Pythagorean membership grades in multi-criteria decision making, IEEE Transactions on Fuzzy Systems22(4) (2014), 958–965.

36.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 338–356.

37.

Zavadskas

E.K.

, Antucheviciene

, Razavi Hajiagha

S.H.

and Hashemi

S.S.

, The interval-valued intuitionistic fuzzy MULTIMOORA method for group decision making in engineering, Mathematical Problems in Engineering2015 (2015), 13.

38.

Zhang

X.L.

and Xu

Z.S.

, Extension of TOPSIS to multiple criteria decision making with pythagorean fuzzy sets, International Journal of Intelligent Systems29(12) (2014), 1061–1078.

39.

Zavadskas

E.K.

, Mardani

, Turskis

, Jusoh

and Nor

K.M.D.

, Development of TOPSIS method to solve complicated decision-making problems— An overview on developments from 2000 to 2015, International Journal of Information Technology & Decision Making15(3) (2016), 645–682.