Pythagorean uncertain linguistic partitioned Bonferroni mean operators and their application in multi-attribute decision making

Abstract

In this paper, a new extension of linguistic term sets called Pythagorean uncertain linguistic sets (PULSs) is defined, which is based on uncertain linguistic term sets and Pythagorean fuzzy sets (PFSs) originally introduced by Yager [41]. This new fuzzy linguistic set has a greater membership representation space and a more powerful expression ability than the intuitionistic uncertain linguistic sets (IULSs) proposed by Liu [18]. Based on the defined operations of PULSs, the Pythagorean uncertain linguistic Bonferroni mean (PULBM) operator and its weighted form are developed to capture the interrelationship among attributes. Taking into account that the classical BM operator is based on the assumption that all attributes are interrelated, and such assumption does not always exist in most of the practical decision-making situations, we further present a Pythagorean uncertain linguistic Partitioned Bonferroni mean (PULPBM) operator and its weighted form (WPULPBM) to solve such situations where all attributes may be divided into several different categories based on specific correlation characteristics and members in the same category are interrelated while no correlation exists among different categories. Then, based on the WPULPBM operator, an approach for MADM problems with Pythagorean uncertain linguistic information is proposed. Finally, a practical example is given to illustrate the application of the proposed approach and comparison analysis is investigated with other existing methods to show the effectiveness of the proposed approach.

Keywords

Multi-attribute decision making pythagorean uncertain linguistic set bonferroni mean partitioned Bonferroni mean

1 Introduction

Multi-attribute decision making (MADM) problems widely exist in human social life. In the process of decision-making, decision makers are usually invited to provide their assessment information against a set of alternatives over multiple attributes and then the best one(s) or ranking order result will be obtained by means of some MADM methods. Thus, how to represent the assessment information is the basis of any decision-making methods. Since the classical fuzzy set was proposed by Zadeh [44] to express the uncertain and imprecise decision information, a lot of fruitful research achievements have been achieved. Meanwhile, many different forms of extensions of fuzzy sets have been put forward to depict decision-making objects, such as intuitionistic fuzzy sets(IFSs) [1, 5], interval-valued intuitionistic fuzzy sets [2, 6], type-2 fuzzy sets [7 , 11], hesitant fuzzy set [29], etc. However, in many practical decision-making problems, it may be suitable for decision makers to evaluate objects by means of linguistic terms rather than exact quantitative forms. Thus, in 1975, Zadeh [45] first present a new fuzzy linguistic approach to deal with MADM problems, in which linguistic terms were used to express the assessment values. Later, the study on linguistic MADM has received a lot of attention and made a large number of research results in different fields.

(1) Linguistic information representation models.To more accurately and objectively describe the linguistic assessment information,several different and feasible linguistic representation models have been proposed in recent years,including 2-tuple linguistic model [15], intuitionistic linguistic set (ILS) [31], type-2 fuzzy linguistic set [30], hesitant fuzzy linguistic term set (HFLTS) [28], etc. Among these representation models, the concept of intuitionistic linguistic set was originally defined by Wang and Li [31], which is constructed by a linguistic variable and an intuitionistic fuzzy number [40]. It can simultaneously express the membership degree, non-membership degree and hesitation degree for a specific linguistic variable. Later, Liu and Jin [18] further extended the ILS and presented the concept of intuitionistic uncertain linguistic set.

(2) Linguistic information aggregation operators. The aggregation operators play a crucial role in the decision-making process. Many scholars pay close attention to this field over the last decades and many achievements have been obtained to fuse the linguistic information [4 , 48]. For example, 2-tuple linguistic weighted averaging operator (2TWLA), 2-tuple linguistic OWA operator (2TLOWA) [15], intuitionistic uncertain linguistic weighted geometric average operator (IULWGA), intuitionistic uncertain linguistic ordered weighted geometric operator (IULOWG) [18], hesitant fuzzy linguistic weighted averaging (HFLWA) operator,hesitant fuzzy linguistic weighted geometric (HFLWG) operator and generalized hesitant fuzzy linguistic weighted averaging (GHFLWA) operator [48] have been proposed to aggregate different linguistic assessment information. Noted that these linguistic aggregation operators assume that the input arguments are mutually independent. However, in many practical decision situations, interrelationships often exist among attributes. Therefore, many scholars investigated and extended the classical Bonferroni mean (BM) to aggregate different forms of decision information. The most important characteristic of the BM is that it can capture the interrelationship between the input arguments. For instance, Yager [43] first investigated the BM and introduced some generalizations of the BM. Wei etc [33] developed two uncertain linguistic Bonferroni mean operators. Liu and Jin [22] proposed some trapezoid fuzzy linguistic Bonferroni mean operators.

Recently, Yager [41, 42] introduced a new concept of Pythagorean fuzzy set (PFS), which is an effective extension of the traditional IFSs. Similar to the IFS, the PFS is also characterized by the membership degree u and the non-membership degree v satisfying the requirement that u² + v² ≤ 1. Obviously, the fundamental difference between PFS and IFS is their different membership degree constraint conditions. In other words, compared with IFS, PFS reduces the constraint conditions of membership degree such that it has stronger modeling capability to characterize the imprecise and uncertain decision information. Since the concept of PFS is proposed, many scholars focused on MADM methods with Pythagorean fuzzy information and achieved some fruitful research results [14 , 47]. Motivated by the PFSs and uncertain linguistic variables,a new extension of linguistic term sets named Pythagorean uncertain linguistic sets (PULSs) is introduced in this paper, which integrate the advantages of PFSs and linguistic variables. The desirable advantage of PULSs is that they can describe two aspects of an object: a uncertain linguistic variable and a Pythagorean fuzzy number (PFN). The former describes an uncertain linguistic assessment value. The latter depicts the hesitancy of the given uncertain linguistic variable and expresses the membership degree u_p and nonmembership degree v_p associated with the given specific uncertain linguistic variable satisfying the condition $u_{p}^{2} + v_{p}^{2} \leq 1$ . Obviously, PULSs have more powerful representation ability than IULSs to character the fuzziness and uncertainty in some practical situations.

Furthermore, although the classical BM has been successfully extended and applied to a variety of decision-making environments to model the interrelationship among input arguments, it is based on a fundamental theoretical hypothesis that all attributes are interrelated to each other. However, interrelationships do not always exist between all of the attributes. In fact, all attributes may be grouped into several different categories and the attributes of the same category are related. Nevertheless, there is no relationship between any two attributes belonging to different categories. In this context, the classical BM does not consider the partition between attributes and represent the specific partition interrelationship between attributes. Recently, Dutta and Guha [12] originally investigated a new extension of the ordinary BM, named as Partitioned Bonferroni mean (PBM). The most attractive advantage of PBM is that it can exactly model the partition situations mentioned above and remove the effect of the correlation between unrelated attributes on the aggregation results. However, Dutta and Guha only discussed the situations where the argument values take the form of 2-tuples.

Therefore,the purpose of this paper is to introduce a new class of linguistic fuzzy set called Pythagorean uncertain linguistic sets (PULSs) and some novel BM operators will also be developed to deal with Pythagorean uncertain linguistic variables.These new operators can accurately depict the interrelationship between arguments, where input arguments are partitioned into several different categories according to their specific correlation characteristic. Then, these proposed operators are used to present a new MADM method. To do this, the rest of this paper is organized as follows. In Section 2, some basic concepts related to this paper are briefly reviewed, the concept of PULSs and their operation rules are introduced and the comparison method of any two PULVs is also defined. Sections 3 and 4 introduce two kinds of Pythagorean uncertain linguistic Bonferroni mean operators. Some desirable properties and special cases of these proposed operators are analyzed and studied. Based on the proposed operators, Section 5 proposes a new approach to MADM with Pythagorean uncertain linguistic information. In Section 6, a practical example is provided to illustrate the proposed MADM method and a comparison analysis is conducted and discussed with other Pythagorean uncertain linguistic MADM methods. Finally, conclusions are presented in Section 7.

2 Preliminaries

Suppose that S = {s_i|i = 1, 2, …, m} is a discrete linguistic term set with odd cardinality. In general, m is set to 5, 7 or 9. For instance, if m = 7, then S could be represented as follows: S = {s₁ = very poor, s₂ = poor, s₃ = slightly poor, s₄ = fair, s₅ = slightly good, s₆ = good, s₇ = verygood}. Usually, S should satisfy the following conditions:

The set is ordered: s_i > s_j, if i > j;

There is the negation operator: Neg (s_i) = s_j, such that j = t - i;

Max operator: max {s_i, s_j} = s_i, if i ≥ j; Min operator: min {s_i, s_j} = s_j, if i ≥ j.

To preserve the evaluation information in the calculation process, Xu [37] extended the discrete linguistic term set S to a continuous linguistic term set $\tilde{S} = {{\tilde{s}}_{i} | i \in [0, r]}$ which also satisfies the above characteristics and r (r > m) is a sufficiently large positive integer. Moreover, Xu [37] introduced the concept of uncertain linguistic sets to describe the situations where the linguistic evaluation value may be situated between two of the predefined linguistic terms.

Definition 1. [37] Let $\tilde{s} = [s_{α}, s_{β}], s_{α}, s_{β} \in \bar{S}$ and 0 ≤ α ≤ β, s_α is the low limit of $\tilde{s}$ and s_β is the upper limit of $\tilde{s}$ , then $\tilde{s}$ is called an uncertain linguistic variable.

Let ${\tilde{s}}_{1} = [s_{α_{1}}, s_{β_{1}}]$ and ${\tilde{s}}_{2} = [s_{α_{2}}, s_{β_{2}}]$ be any two uncertain linguistic variables, then the operational rules are defined as follows:

${\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [s_{α_{1}}, s_{β_{1}}] \oplus [s_{α_{2}}, s_{β_{2}}] = [s_{α_{1} + α_{2}}, s_{β_{1} + β_{2}}]$ ;

${\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = [s_{α_{1}}, s_{β_{1}}] \otimes [s_{α_{2}}, s_{β_{2}}] = [s_{α_{1} \times α_{2}}, s_{β_{1} \times β_{2}}]$ ;

$λ {\tilde{s}}_{1} = λ [s_{α_{1}}, s_{β_{1}}] = [s_{λ α_{1}}, s_{λ β_{1}}]$ ;

$({\tilde{s}}_{1})^{λ} = ([s_{α_{1}}, s_{β_{1}}])^{λ} = [(s_{α_{1}})^{λ}, (s_{β_{1}})^{λ}] = [s_{α_{1}^{λ}}, s_{β_{1}^{λ}}]$ .

Definition 2. Let X be a no empty finite set. A Pythagorean fuzzy set (PFS) proposed by Zhang and Xu [46] is expressed by the form: P = {< x, P (μ_p (x) , ν_p (x)) > |x ∈ X}, where μ_p, ν_p : X → [0, 1] denote the degree of membership and nonmembership of the element x ∈ X to the set P, respectively, satisfying the condition that 0 ≤ (μ_p (x)) ² + (ν_p (x)) ² ≤ 1. The hesitant degree of x ∈ X is expressed as: $π_{p} (x) = \sqrt{1 - μ_{p}^{2} (x) - ν_{p}^{2} (x)}$ .

Definition 3. Let X = {x₁, x₂, …, x_n} be a universal set. A Pythagorean uncertain linguistic set (PULS) A in X is expressed by $A = {〈 x_{i} | ([s_{θ (x_{i})}, s_{τ (x_{i})}], (μ_{p} (x_{i}), ν_{p} (x_{i}))) 〉 | x_{i} \in X},$ where $s_{θ (x_{i})}, s_{τ (x_{i})} \in \tilde{S}, μ_{p} : X \to [0, 1]$ and ν_p : X → [0, 1], respectively represent the degrees of membership and nonmembership of the element x_i ∈ X to the uncertain linguistic variable [s_{θ(x_i)}, s_{τ(x_i)}], satisfying the condition $0 \leq μ_{p}^{2} (x_{i}) + ν_{p}^{2} (x_{i}) \leq 1$ for any x_i ∈ X. For each PULS in X, if $π_{p} (x_{i}) = \sqrt{1 - μ_{p}^{2} (x_{i}) - ν_{p}^{2} (x_{i})}$ , ∀x_i ∈ X, then π_p (x) is called the indeterminacy degree of x to the uncertain variable [s_θ(x), s_τ(x)]. For simplicity, α =< [s_{θ(x_i)}, s_{τ(x_i)}] , (μ_p (x_i) , ν_p (x_i)) > is called a Pythagorean uncertain linguistic variable (PULV).

When 0 ≤ μ_p (x_i) + ν (x_i) ≤1 for each x_i ∈ X, then the PULS reduces to the intuitionistic uncertain linguistic set (IULS). Obviously,the main difference between PULS and IULS is their different constraint conditions and the PULS is more general and powerful than the IULS. Further, if s_{θ(x_i)} = s_{τ(x_i)}, then the PULS reduces to the Pythagorean linguistic set (PLS). If s_{θ(x_i)} = s_{τ(x_i)} and 0 ≤ μ_p (x_i) + ν (x_i) ≤1, then the PULS reduces to the intuitionistic linguistic set (ILS).

Definition 4. Let $\bar{α} = < [s_{θ (\bar{α})}, s_{τ (\bar{α})}], (μ (\bar{α}), ν (\bar{α})) >$ and $\bar{β} = < [s_{θ (\bar{β})}, s_{τ (\bar{β})}], (μ (\bar{β}), ν (\bar{β})) >$ be any two PULVs and λ > 0, then some operational laws of $\bar{α}$ and $\bar{β}$ are defined as follows: $\begin{matrix} (1) \bar{α} \oplus \bar{β} = 〈 [s_{θ (\bar{α}) + θ (\bar{β})}, s_{τ (\bar{α}) + τ (\bar{β})}, \\ (\sqrt{μ^{2} (\bar{α}) + μ^{2} (\bar{β}) - μ^{2} (\bar{α}) μ^{2} (\bar{β})}, ν (\bar{α}) ν (\bar{β})) 〉; \end{matrix}$ (1) $\begin{matrix} (2) \bar{α} \otimes \bar{β} = 〈 [s_{θ (\bar{α}) \times τ (\bar{β})}], \\ (μ (\bar{α}) μ (\bar{β}), \sqrt{ν^{2} (\bar{α}) + ν^{2} (\bar{β}) - ν^{2} (\bar{α}) ν^{2} (\bar{β})}) 〉; \end{matrix}$ (2) $\begin{matrix} (3) λ \bar{α} = 〈 [s_{λ \times θ (\bar{α})}, s_{λ \times τ (\bar{β})}], \\ (\sqrt{1 - (1 - μ^{2} (\bar{α}))^{λ}}, (ν (\bar{α}))^{λ}) 〉; \end{matrix}$ (3) $\begin{matrix} (4) {\bar{α}}^{λ} = 〈 [s_{(θ (\bar{α}))^{λ}}, s_{(τ (\bar{β}))^{λ}}], ((μ (\bar{α}))^{λ}, \\ \sqrt{1 - (1 - ν^{2} (\bar{α}))^{λ}}) 〉 . \end{matrix}$ (4)

Definition 5. Let $\bar{α} = < [s_{θ (\bar{α})}, s_{τ (\bar{α})}], (μ (\bar{α}), ν (\bar{α})) >$ be a PULV. An expected value $E (\bar{α})$ and an accuracy function of $\bar{α}$ can be defined as follows, respectively: $\begin{matrix} E (\bar{α}) & = & \frac{μ^{2} (\bar{α}) + 1 - ν^{2} (\bar{α})}{2} \times s_{(θ (\bar{α}) + τ (\bar{α})) / 2} \\ = & s_{((θ (\bar{α}) + τ (\bar{α})) \times (μ^{2} (\bar{α}) + 1 - ν^{2} (\bar{α}))) / 4}, \\ H (\bar{α}) & = & (μ^{2} (\bar{α}) + ν^{2} (\bar{α})) \times s_{(θ (\bar{α}) + θ (\bar{β})) / 2} \\ = & s_{((μ^{2} (\bar{α}) + ν^{2} (\bar{α})) \times (θ (\bar{α}) + τ (\bar{α}))) / 2} . \end{matrix}$

Motivated by Ref [38], we present a comparison method for any two PULVs.

Definition 6. Let $\bar{α} = < [s_{θ (\bar{α})}, s_{τ (\bar{α})}], (μ (\bar{α}), ν (\bar{α})) >$ and $\bar{β} = < [s_{θ (\bar{β})}, s_{τ (\bar{β})}], (μ (\bar{β}), ν (\bar{β})) >$ be any two PULVs, then $\begin{matrix} (1) & If E (\bar{α}) > E (\bar{β}), then \bar{α} ≻ \bar{β} . \\ (2) & If E (\bar{α}) = E (\bar{β}), then : \\ If H (\bar{α}) > H (\bar{β}), then \bar{α} ≻ \bar{β} . \\ If H (\bar{α}) = H (\bar{β}), then \bar{α} = \bar{β} . \end{matrix}$

3 Pythagorean uncertain linguistic Bonferroni means

Definition 7. Let α_i = 〈 [s_{θ(α_i)}, s_{τ(α_i)}] , (u_{α
_i}, v_{α
_i}) 〉 (i = 1, 2, …, n) be a collection of PULVs. For any p, q ≥ 0 with p + q > 0, if ${PULBM}^{(p, q)} = (\frac{1}{n (n - 1)} (\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} α_{i}^{p} \otimes α_{j}^{q}))^{\frac{1}{p + q}}$ (5) then PULBM^(p,q) is called the Pythagorean uncertain linguistic Bonferroni mean operator.

Theorem 1. Let α_i = 〈 [s_{θ(α_i)}, s_{τ(α_i)}] , (u_{α
_i}, v_{α
_i}) 〉 (i = 1, 2, …, n) be a collection of PULVs. For p ≥ 0, q ≥ 0 with p + q > 0 . Then, the aggregated value by using the PULBM is still a PULV, and

$\begin{matrix} {PULBM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = 〈 [s_{(\frac{\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} θ^{p} (α_{i}) θ^{q} (α_{j})}{n (n - 1)})^{\frac{1}{p + q}}}, s_{(\frac{\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (τ (α_{i}))^{p} (τ (α_{j}))^{q}}{n (n - 1)})^{\frac{1}{p + q}}}], \\ (1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - u_{α_{i}}^{2 p} u_{α_{j}}^{2 q})^{\frac{1}{n (n - 1)}})^{\frac{1}{2 (p + q)}}, ((1 - (1 - \\ \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - (1 - v_{α_{i}}^{2})^{p} (1 - v_{α_{j}}^{2})^{q})^{\frac{1}{n (n - 1)}})^{\frac{1}{p + q}})^{\frac{1}{2}}) 〉 \end{matrix}$ (6)

Theorem 2. Let α_i = 〈 [s_{θ(α_i)}, s_{τ(α_i)}] , (u_{α
_i}, v_{α
_i}) 〉 (i = 1, 2, …, n) be a collection of PULVs. For p ≥ 0, q ≥ 0 with p + q > 0. Then, the following properties hold:

(1) Idempotency: If α_i = α for all i, then

PULBM^(p,q) (α₁, α₂, …, α_n) = α

(2) Commutativity: If ${\tilde{α}}_{i}$ is any permutation of α_i, then $\begin{matrix} {PULBM}^{(p, q)} (α_{1}, α_{2}, \dots, α_{n}) \\ = {PULBM}^{(p, q)} ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \end{matrix}$

(3) Boundedness: Let $α^{-} = 〈 [min_{i} s_{θ (α_{i})}, min_{i} s_{τ (α_{i})}], (min_{i} u_{α_{i}}, max_{i} v_{α_{i}}) 〉,$ $α^{+} = 〈 [max_{i} s_{θ (α_{i})}, max_{i} s_{τ (α_{i})}], (max_{i} u_{α_{i}}, min_{i} v_{α_{i}}) 〉$ then $α^{-} \leq {PULBM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \leq α^{+}$

It should be noted that the PULBM operator is not monotonic.

Now, based on different values of the parameters p and q, we can obtain some special cases of the PULBM:

Case 1. If q → 0, then by Equation (6), we have: $\begin{matrix} lim_{q \to 0} {PULBM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = 〈 [s_{(\frac{1}{n} \sum_{i = 1}^{n} (θ (α_{i}))^{p})^{\frac{1}{p}}}, s_{(\frac{1}{n} \sum_{i = 1}^{n} (τ (α_{i}))^{p})^{\frac{1}{p}}}], \\ ((1 - \prod_{i = 1}^{n} (1 - u_{α_{i}}^{2 p})^{\frac{1}{n}})^{\frac{1}{2 p}}, ((1 - (1 - (\prod_{i = 1}^{n} \\ (1 - (1 - v_{α_{i}}^{2})^{p}))^{\frac{1}{n}})^{\frac{1}{p}})^{\frac{1}{2}}) 〉 = (\frac{1}{n} (\sum_{i = 1}^{n} α_{i}^{p}))^{\frac{1}{p}}, \end{matrix}$ which we call the generalized Pythagorean uncertain linguistic mean.

Case 2. If p = 1 and q → 0, then Equation (6) reduces to th Pythagorean uncertain linguistic average: $\begin{matrix} lim_{q \to 0} {PULBM}^{1, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = 〈 [s_{\frac{\sum_{i = 1}^{n} θ (α_{i})}{n}}, s_{\frac{\sum_{i = 1}^{n} τ (α_{i})}{n}}], ((1 - \prod_{i = 1}^{n} (1 - u_{α_{i}}^{2})^{\frac{1}{n}})^{\frac{1}{2}}, \\ \prod_{i = 1}^{n} v_{α_{i}}^{\frac{1}{n}}) 〉 = \frac{1}{n} \sum_{i = 1}^{n} α_{i}, \end{matrix}$

Case 3. If p = 2 and q → 0, then Equation (6) is transformed as

$\begin{matrix} lim_{q \to 0} {PULBM}^{2, 0} (α_{1}, α_{2}, \dots, α_{n}) = (\frac{1}{n} (\sum_{i = 1}^{n} α_{i}^{2}))^{\frac{1}{2}} \\ = 〈 [s_{(\frac{1}{n} \sum_{i = 1}^{n} (θ (α_{i}))^{p})^{\frac{1}{p}}}, s_{(\frac{1}{n} \sum_{i = 1}^{n} (τ (α_{i}))^{p})^{\frac{1}{p}}}], \\ ((1 - \prod_{i = 1}^{n} (1 - u_{α_{i}}^{2 p})^{\frac{1}{n}})^{\frac{1}{2 p}}, ((1 - (1 - (\prod_{i = 1}^{n} \\ (1 - (1 - v_{α_{i}}^{2})^{p}))^{\frac{1}{n}})^{\frac{1}{p}})^{\frac{1}{2}}) 〉 = (\frac{1}{n} (\sum_{i = 1}^{n} α_{i}^{p}))^{\frac{1}{p}}, \end{matrix}$

which we call the Pythagorean uncertain linguistic square mean.

Case 4. If p = q = 1, the Equation (6) reduces to the following:

$\begin{matrix} {PULBM}^{1, 1} (α_{1}, α_{2}, \dots, α_{n}) \\ = (\frac{1}{n (n - 1)} (\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} α_{i} \otimes α_{j}))^{\frac{1}{2}} \\ = 〈 [s_{(\frac{\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} θ (α_{i}) θ (α_{j})}{n (n - 1)})^{\frac{1}{2}}}, s_{(\frac{\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} τ (α_{i}) τ (α_{j})}{n (n - 1)})^{\frac{1}{2}}}], \\ ((1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - u_{α_{i}}^{2} u_{α_{j}}^{2})^{\frac{1}{n (n - 1)}})^{\frac{1}{4}}, (1 - (1 - \\ \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - (1 - v_{α_{i}}^{2}) (1 - v_{α_{j}}^{2}))^{\frac{1}{n (n - 1)}})^{\frac{1}{2}})^{\frac{1}{2}}) 〉, \end{matrix}$ (7)

which we call the Pythagorean uncertain linguistic interrelated square mean.

In the following,we present the weighted Pythagorean uncertain linguistic Bonferroni mean (WPULBM) to consider the weight vector of input arguments.

Definition 8. Let α_i = 〈 [s_{θ(α_i)}, s_{τ(α_i)}] , (u_{α
_i}, v_{α
_i}) 〉 (i = 1, 2, …, n) be a collection of PULVs. w_i is the weight of α_i, satisfying the conditions: w_i > 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . For p ≥ 0, q ≥ 0 with p + q > 0 . if

$\begin{matrix} {WPULBM}^{p, q} \\ = (\frac{1}{n (n - 1)} (\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (w_{i} α_{i})^{p} \otimes (w_{j} α_{j})^{q}))^{\frac{1}{p + q}}, \end{matrix}$ (8) then WPULBM^(p,q) is called the weighted Pythagorean uncertain linguistic Bonferroni mean operator.

4 Partitioned Pythagorean uncertain linguistic Bonferroni means

It is well known that the BM operator is a useful tool to capture the interrelationship between aggregated arguments, which has been successfully applied to solve decision making problems. However, the BM is based on the assumption that each attribute has relationship with the rest of aggregated arguments. Obviously, this situation may not often appear in practical decision making problems. In fact, in many real problems, all of the attributes may be divided into several different categories. Attributes of each categories may be interrelated to other attributes of the same categories, but there are no interrelationship between the attributes in different categories. To deal with such situations, Dutta and Guha [12] developed a new extended Bonferroni mean to fuse the input arguments with such relationship among input arguments, which is referred as partitioned Bonferroni mean (PBM).

Assume C = {c₁, c₂, …, c_n} is a collection of attributes, A = (a₁, a₂, …, a_n) is the collection of attribute values with respect to C and a_i (i = 1, 2, …, n) are nonnegative real numbers. Suppose the attribute set C can be partitioned into t different categories, i.e., {P₁, P₂, …, P_t}, and satisfies the conditions: P_i∩ P_j = ∅ and $\cup_{k = 1}^{t} P_{k} = C$ . In addition, the attributes of each partition P_k (k = 1, 2, …, t) are related to other attributes of the same category and there is no relationship among the attributes of any two different partitions P_i, P_j (i, j ∈ {1, 2, …, t} , i ≠ j). With these conditions, Dutta and Guha [12] developed the partitioned Bonferroni mean (PBM) to fuse these input arguments defined as follows:

Definition 9. For any p, q ≥ 0 with p + q > 0, if

$\begin{matrix} {PBM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{t} (\sum_{k = 1}^{t} (\frac{1}{| P_{k} |} \sum_{i \in P_{k}} a_{i}^{p} (\frac{1}{| P_{k} | - 1} \sum_{\begin{matrix} j \in P_{k} \\ j \neq i \end{matrix}} a_{j}^{q}))^{\frac{1}{p + q}}), \end{matrix}$ (9) then PBM^p,q is called the partitioned Bonferroni mean, where t represents the numbers of partitions, |P_k| denotes the cardinality of the partitioned attribute set P_k (k = 1, 2, …, t) and |P_k|≥2.

In the following, we shall investigate the PBM under Pythagorean uncertain linguistic environments and propose the Pythagorean uncertain linguistic partitioned Bonferroni mean operator (PULPBM) to aggregate Pythagorean uncertain linguistic information.

Definition 10. Let ${\bar{α}}_{i} = 〈 [s_{θ ({\bar{α}}_{i})}, s_{τ ({\bar{α}}_{i})}], (u_{{\bar{α}}_{i}}, v_{{\bar{α}}_{i}}) 〉 (i = 1, 2, \dots, n)$ be a collection of PULVs partitioned into t distinct categories P₁, P₂, …, P_t. For any p, q ≥ 0 with p + q > 0, If

$\begin{matrix} {PULPBM}^{p, q} ({\bar{α}}_{1}, {\bar{α}}_{2}, \dots, {\bar{α}}_{n}) \\ = \frac{1}{t} (\sum_{k = 1}^{t} (\frac{1}{| P_{k} |} \sum_{i \in P_{k}} {\bar{α}}_{i}^{p} (\frac{1}{| P_{k} | - 1} \sum_{\begin{matrix} j \in P_{k} \\ j \neq i \end{matrix}} {\bar{α}}_{j}^{q}))^{\frac{1}{p + q}}) \\ = \frac{1}{t} (\sum_{k = 1}^{t} (\frac{\sum_{\begin{matrix} i, j \in P_{k} \\ i \neq j \end{matrix}} {\bar{a}}_{i}^{p} {\bar{a}}_{j}^{q}}{| P_{k} | (| P_{k} | - 1)})^{\frac{1}{p + q}}), \end{matrix}$ (10) then PULPBM^p,q is called the Pythagorean uncertain linguistic partitioned Bonferroni mean, where t represents the numbers of partitions, |P_k| denotes the cardinality of the partitioned attribute set P_k (k = 1, 2, …, t) and |P_k|≥2.

Theorem 3. Let ${\bar{α}}_{i} = 〈 [s_{θ ({\bar{α}}_{i})}, s_{τ ({\bar{α}}_{i})}], (u_{{\bar{α}}_{i}}, v_{{\bar{α}}_{i}}) 〉 (i = 1, 2, \dots, n)$ be a collection of PULVs partitioned into t distinct categories P₁, P₂, …, P_t. For any p, q ≥ 0 with p + q > 0, the aggregated value by using the PULPBM is also a PULV, and

$\begin{matrix} {PULPBM}^{p, q} ({\bar{α}}_{1}, {\bar{α}}_{2}, \dots, {\bar{α}}_{n}) \\ = 〈 [s_{θ (\bar{a})}, s_{τ (\bar{a})}], (u (\bar{a}), v (\bar{a})) 〉, \end{matrix}$ (11) where $\begin{matrix} θ (\bar{α}) & = & \frac{1}{t} (\sum_{k = 1}^{t} (\frac{\sum_{\begin{matrix} i, j \in P_{k} \\ i \neq j \end{matrix}} θ ({\bar{α}}_{i})^{p} θ ({\bar{α}}_{j})^{q}}{| P_{k} | (| P_{k} | - 1)})^{\frac{1}{p + q}}), \\ τ (\bar{α}) & = & \frac{1}{t} (\sum_{k = 1}^{t} (\frac{\sum_{\begin{matrix} i, j \in P_{k} \\ i \neq j \end{matrix}} τ ({\bar{α}}_{i})^{p} τ ({\bar{α}}_{j})^{q}}{| P_{k} | (| P_{k} | - 1)})^{\frac{1}{p + q}}), \\ u (\bar{α}) & = & (1 - (\prod_{k = 1}^{t} (1 - (1 - \prod_{\begin{matrix} i, j \in P_{k} \\ j \neq i \end{matrix}} (1 - u ({\bar{α}}_{i})^{2 p} \\ u ({\bar{α}}_{j})^{2 q})^{\frac{1}{| P_{k} | (| P_{k} | - 1)}})^{\frac{1}{p + q}}))^{\frac{1}{t}})^{\frac{1}{2}}, \\ v (\bar{α}) & = & (\prod_{k = 1}^{t} (1 - (1 - \prod_{\begin{matrix} i, j \in P_{k} \\ j \neq i \end{matrix}} (1 - (1 - v ({\bar{α}}_{i})^{2})^{p} \\ (1 - v ({\bar{α}}_{j})^{2})^{q})^{\frac{1}{| P_{k} | (| P_{k} | - 1)}})^{\frac{1}{p + q}}))^{\frac{1}{2 t}} . \end{matrix}$

Now, based on the number of partitioned categories and different values of the parameters p and q, we can obtain some special cases of PULPBM:

(1) When all attributes belong to the same category and interrelationship exists among all attributes,then the above PULPBM operator with the parameter t = 1 and |P₁| = n reduces to the PULBM.

$\begin{matrix} {PULPBM}^{p, q} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) \\ = (\frac{1}{| P_{1} |} \sum_{i \in P_{1}} {\bar{a}}_{i}^{p} (\sum_{\begin{matrix} j \in P_{1} \\ j \neq i \end{matrix}} \frac{1}{| P_{1} | - 1} {\bar{a}}_{j}^{q}))^{\frac{1}{p + q}} \\ = (\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ j \neq i \end{matrix}}^{n} {\bar{a}}_{i}^{p} {\bar{a}}_{j}^{q})^{\frac{1}{p + q}} \\ = PULBM ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) . \end{matrix}$ (12)

(2) When some attributes are independent and do not belong to any of the partitioned categories. In this case, we divide all of the attributes into two different type sets C₁ and C₂, satisfying the condition: C₁∩ C₂ = ∅. C₁ includes the attributes which have relationship with other attributes and those independent attributes are summarized into the set C₂. In addition, assume that the attributes of C₁ are divided into several distinct categories according to interrelationship pattern(the number of partitions is t). Then, Equation (10) should be modified as follows:

$\begin{matrix} {PULPBM}^{p, q} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) \\ = \frac{n - | C_{2} |}{n} (\frac{1}{t} (\sum_{k = 1}^{t} (\frac{\sum_{\begin{matrix} i, j \in P_{k} \\ j \neq i \end{matrix}} {\bar{a}}_{i}^{p} {\bar{a}}_{j}^{q}}{| P_{k} | (| P_{k} | - 1)})^{\frac{1}{p + q}})) \\ \oplus \frac{| C_{2} |}{n} (\frac{1}{| C_{2} |} \sum_{i \in C_{2}} {\bar{a}}_{i}) . \end{matrix}$ (13)

It is easy to see that Equation (13) reduces to the PULPBM defined in Equation (10) when C₂ is an empty set.

(3) When all of the attributes belong to the same categories and they are related to other attributes, i.e., t = 1 and P₁ = n, by taking different values of the parameters p and q, the developed PULPBM operator is transformed into four special cases:

Case 1. If q → 0, then, Equation (10) is transformed as follows: $\begin{matrix} lim_{q \to 0} {PULPBM}^{p, q} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) \\ = lim_{q \to 0} (\frac{1}{| P_{1} |} \sum_{i \in P_{1}} {\bar{a}}_{i}^{p} (\frac{1}{| P_{1} | - 1} \sum_{\begin{matrix} j \in P_{1} \\ j \neq i \end{matrix}} {\bar{a}}_{j}^{q}))^{\frac{1}{p + q}} \\ = (\frac{1}{n} \sum_{i = 1}^{n} {\bar{a}}_{i}^{p})^{\frac{1}{p}} \\ = {PULPBM}^{p, 0} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) . \end{matrix}$ (14) which we call the generalized Pythagorean uncertain linguistic mean (GPULM) operator.

Case 2. If p = 1 and q → 0, then Equation (10) reduces to the Pythagorean uncertain linguistic mean operator.

$\begin{matrix} lim_{q \to 0} {PULPBM}^{p, q} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) \\ = lim_{q \to 0} (\frac{1}{| P_{1} |} \sum_{i \in P_{1}} {\bar{a}}_{i}^{1} (\frac{1}{| P_{1} | - 1} \sum_{\begin{matrix} j \in P_{1} \\ j \neq i \end{matrix}} {\bar{a}}_{j}^{0}))^{\frac{1}{p + q}} \\ = (\frac{1}{n} \sum_{i = 1}^{n} {\bar{a}}_{i}) . \end{matrix}$ (15)

Case 3. If p = 2 and q → 0, then Equation (10) reduces to the following:

$\begin{matrix} lim_{\begin{matrix} p = 2 \\ q \to 0 \end{matrix}} {PULPBM}^{p, q} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) \\ = lim_{\begin{matrix} p = 2 \\ q \to 0 \end{matrix}} (\frac{1}{| P_{1} |} \sum_{i \in P_{1}} {\bar{a}}_{i}^{p} (\frac{1}{| P_{1} | - 1} \sum_{\begin{matrix} j \in P_{1} \\ j \neq i \end{matrix}} {\bar{a}}_{j}^{q}))^{\frac{1}{p + q}} \\ = (\frac{1}{n} \sum_{i = 1}^{n} {\bar{a}}_{i}^{2})^{\frac{1}{2}}, \end{matrix}$ (16) which we call the Pythagorean uncertain linguistic geometric mean (PULGM).

Case 4. If p = q = 1, then Equation (10) reduces to the following:

$\begin{matrix} {PULPBM}^{p, q} ({\bar{a}}_{1}, {\bar{a}}_{2}, \dots, {\bar{a}}_{n}) \\ = (\frac{\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}} {\bar{a}}_{i} {\bar{a}}_{j}}{n (n - 1)})^{\frac{1}{2}}, \end{matrix}$ (17) which we call the Pythagorean uncertain linguistic interrelated square mean.

In the following,we investigate some desirable properties of the developed PULPBM operator.

Theorem 4. Let ${\bar{α}}_{i} = 〈 [s_{θ ({\bar{α}}_{i})}, s_{τ ({\bar{α}}_{i})}], (u_{{\bar{α}}_{i}}, v_{{\bar{α}}_{i}}) 〉 (i = 1, 2, \dots, n)$ be a collection of PULVs, and they are partitioned into t distinct categories P₁, P₂, …, P_t based on the interrelationship pattern. For p ≥ 0, q ≥ 0 with p + q > 0. Then, the following properties hold:

(1) Idempotency. If all ${\bar{α}}_{i}$ are equal and ${\bar{α}}_{i} = \bar{α}$ for all i = 1, 2, …, n. Then, ${PULPBM}^{p, q} ({\bar{α}}_{1}, {\bar{α}}_{2}, \dots, {\bar{α}}_{n}) = \bar{α} .$ (2) Commutativity. Let ${\bar{β}}_{i} = 〈 [s_{θ ({\bar{β}}_{i})}, s_{τ ({\bar{β}}_{i})}], (u ({\bar{β}}_{i}), v ({\bar{β}}_{i})) 〉 (i = 1, 2, \dots, n)$ is any permutation of ${\bar{α}}_{i}$ . If the partitioned structure of ${\bar{β}}_{i}$ and ${\bar{α}}_{i}$ are the same, then

$\begin{matrix} {PULPBM}^{p, q} ({\bar{α}}_{1}, {\bar{α}}_{2}, \dots, {\bar{α}}_{n}) \\ = {PULPBM}^{p, q} ({\bar{β}}_{1}, {\bar{β}}_{2}, \dots, {\bar{β}}_{n}) . \end{matrix}$ (18)(3) Boundedness: Let $\begin{matrix} {\bar{α}}^{-} & = & 〈 [min_{i} s_{θ ({\bar{α}}_{i})}, min_{i} s_{τ ({\bar{α}}_{i})}], \\ (min_{i} u ({\bar{α}}_{i}), max_{i} v ({\bar{α}}_{i})) 〉, \\ {\bar{α}}^{+} & = & 〈 [max_{i} s_{θ ({\bar{α}}_{i})}, max_{i} s_{τ ({\bar{α}}_{i})}], \\ (max_{i} u ({\bar{α}}_{i}), min_{i} v ({\bar{α}}_{i})) 〉, \end{matrix}$ then ${\tilde{a}}^{-} \leq {PULPBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+} .$ (19)

Similar to the PULBM, the PULPBM operator does not have the monotonicity.

Definition 11. Let ${\bar{α}}_{i} = 〈 [s_{θ ({\bar{α}}_{i})}, s_{τ ({\bar{α}}_{i})}], (u_{{\bar{α}}_{i}}, v_{{\bar{α}}_{i}}) 〉 (i = 1, 2, \dots, n)$ be a collection of PULVs partitioned into t different categories P₁, P₂, …, P_t according to correlation structure. W = (w₁, w₂, …, w_n) ^T is the weight vector of ${\bar{α}}_{i} (i = 1, 2, \dots, n)$ , where w_i indicates the importance degree of ${\bar{α}}_{i}$ and satisfies the conditions: w_i ∈ [0, 1] (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ . For p ≥ 0, q ≥ 0 with p + q > 0 . if

$\begin{matrix} {WPULPBM}^{p, q} ({\bar{α}}_{1}, {\bar{α}}_{2}, \dots, {\bar{α}}_{n}) \\ = \frac{1}{t} (\sum_{k = 1}^{t} (\frac{1}{| P_{k} | (| P_{k} | - 1)} \sum_{\begin{matrix} i, j \in P_{k} \\ i \neq j \end{matrix}} (w_{i} {\bar{α}}_{i})^{p} \otimes (w_{j} {\bar{α}}_{j})^{q})^{\frac{1}{p + q}}), \end{matrix}$ (20)

then WPULPBM^p,q is called the weighted Pythagorean uncertain linguistic partitioned Bonferroni mean.

5 An approach to multiple attribute group decision making with Pythagorean uncertain linguistic information

For a MADM problem with Pythagorean uncertain linguistic information,let X = {x₁, x₂, …, x_m} be a set of alternatives, and Y = {y₁, y₂, …, y_n} be a set of attributes whose weight vector is W = (w₁, w₂, …, w_n) ^T, satisfying w_i ≥ 0 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ . Suppose ${\tilde{R}}_{ij} = [{\tilde{r}}_{ij}]_{m \times n}$ is the decision matrix given by the decision maker, where the performance values ${\tilde{r}}_{ij} = 〈 [s_{θ ({\tilde{α}}_{ij})}, s_{τ ({\tilde{α}}_{ij})}], (μ_{{\tilde{α}}_{ij}}, ν_{{\tilde{α}}_{ij}}) 〉$ take the form of the PULVs given by the decision maker for an alternative x_i (i = 1, 2, …, m) under the attribute y_j (j = 1, 2, …, n), such that $s_{θ ({\tilde{α}}_{ij})}, s_{τ ({\tilde{α}}_{ij})}$ belong to the predefined linguistic term set, $μ_{{\tilde{α}}_{ij}} \in [0, 1], ν_{{\tilde{α}}_{ij}} \in [0, 1]$ and $μ_{{\tilde{α}}_{ij}}^{2} + ν_{{\tilde{α}}_{ij}}^{2} \leq 1$ .

Furthermore, suppose that according to the interrelationship pattern among the attributes, the attribute set Y is divided into t (t ≥ 1) distinct categories P₁, P₂, …, P_t such that P_i∩ P_j = ∅ and $\cup_{k = 1}^{t} P_{k} = Y$ . The attributes of each category P_i are interrelated to other attributes in the same category whereas there are no interrelationship among the attributes belong to any two different categories. Now our aim is to choose the optimal alternative(s) among m alternatives depending on the decision matrix ${\tilde{R}}_{ij}$ . The detailed steps of the proposed approach are presented as follows:

Step 1. Transform the given Pythagorean uncertain linguistic decision matrix ${\tilde{R}}_{ij} = [{\tilde{r}}_{ij}]_{m \times n}$ into the normalized Pythagorean uncertain linguistic decision matrix R_ij = [r_ij] _m×n. The transformation method is presented as follows [39]: $r_{ij} = {\begin{matrix} {\tilde{r}}_{ij} & for benefit attribute of Y_{j} \\ ({\tilde{r}}_{ij})^{c} & for cost attribute Y_{j} \end{matrix}$ (21) where $({\tilde{r}}_{ij})^{c} (i = 1, 2, \dots, m; j = 1, 2, \dots, n .)$ is the complement of ${\tilde{r}}_{ij}$ and $\begin{matrix} ({\tilde{r}}_{ij})^{c} & = & 〈 [Neg (s_{τ ({\tilde{α}}_{ij})}), Neg (s_{θ ({\tilde{α}}_{ij})})], \\ (ν ({\tilde{α}}_{ij}), μ ({\tilde{α}}_{ij})) 〉 \end{matrix}$

Step 2. Calculate the overall aggregated value r_i (i = 1, 2, …, m) corresponding to the alternative x_i. Utilize the WPULPBM (t ≥ 2) or WPULBM (t = 1, i.e., no partition exists among the attributes) operator to aggregate all the performance values r_ij (i = 1, 2, …, m ; j = 1, 2, …, n) to each ith line and get the overall performance value r_i of the alternative x_i.

If the partition number t ≥ 2, the WPULPUB operator (Equation (10)) can be utilized to aggregate the performance values as follows:

$\begin{matrix} r_{i} & = & {WPULPBM}^{p, q} (r_{i 1}, r_{i 2}, \dots, r_{in}) \\ = & \frac{1}{t} (\sum_{k = 1}^{t} (\frac{\sum_{\begin{matrix} d, g \in P_{k} \\ d \neq g \end{matrix}} (w_{d} r_{id})^{p} (w_{g} r_{ig})^{q}}{| P_{k} | (| P_{k} | - 1)})^{\frac{1}{p + q}}), \end{matrix}$ (22) where p, q ≥ 0 and p + q > 0

If there is no partition among the attribute set Y, i.e., t = 1 and all attributes belong to the same partition, then the WPULBM operator (Equation (8)) can be utilized to aggregate the performance values as follows:

$\begin{matrix} r_{i} = {WPULBM}^{p, q} (r_{i 1}, r_{i 2}, \dots, r_{in}) \\ = (\frac{1}{n (n - 1)} (\sum_{\begin{matrix} d, g = 1 \\ d \neq g \end{matrix}}^{n} (w_{d} r_{id})^{p} \otimes (w_{g} r_{ig})^{q}))^{\frac{1}{p + q}}, \end{matrix}$ (23) where p, q ≥ 0 and p + q > 0.

Step 3. Calculate the expected values and the accuracy degrees E (r_i) and H (r_i) of the overall PULVs r_i (i = 1, 2, 3, 4) according to Definition 5, respectively.

Step 4. Use E (r_i) and H (r_i) to rank the overall performance values r_i (i = 1, 2, …, m) in descending order according to the ranking method in Definition 6 and select the best alternative(s) with the largest overall performance value.

6 Illustrative example

Let us consider a practical MADM problem(revised from [12]) to demonstrate the application of the developed approach. An investment company intends to select and invest the most desirable investment area from four possible alternatives {x₁, x₂, x₃, x₄} by using the idle capital. After preliminary investigation and research, five factors are selected as evaluation attributes, including c1: the risk of loss of investment capital; c2: the possibility of capital investment gains; c3: the vulnerability of capital sum to modification by inflation; c₄ : prospect of the market development; c₅ : industry growth potential, in which c1 and c₃ are the cost-type attribute and others are the benefit-type attribute. The five attributes are assigned to the corresponding weight w = {0.3, 0.10, 0.25, 0.15, 0.2} ^T. Suppose the decision maker provides the assessments by using PULVs ${\tilde{r}}_{ij} = 〈 [s_{θ ({\tilde{α}}_{ij})}, s_{τ ({\tilde{α}}_{ij})}], (μ_{{\tilde{α}}_{ij}}, ν_{{\tilde{α}}_{ij}}) 〉$ . The linguistic term set S is the same as defined in Section 2. The given linguistic assessments of these four alternatives are presented in Table 1.

Assume that the abovementioned five attributes are divided into two distinct categories: D1 = {c₁, c₂, c₃} and D2 = {c₄, c₅} according to the correlation among attributes. Now, we need to select the best investment area(s) based on the above decision conditions.

6.1 Decision making steps

Step 1. Using Equation (21) to transform the given Pythagorean uncertain linguistic ${\tilde{R}}_{ij}$ into the normalized Pythagorean uncertain linguistic R_ij presented in Table 2.

Step 2. Since these five attributes are divided into two categories {D1, D2} and their corresponding weighting vector are previously known, we choose the WPULPBM (Equation 22) to aggregate all the preference PULVs r_ij (j = 1, 2, …, 5) into the collective PULVs r_i (i = 1, 2, 3, 4) shown as follows (without loss of generality, suppose p = q = 1): $\begin{matrix} r_{1} = < [s_{0.7088}, s_{0.9509}], (0.2669, 0.8609) >, \\ r_{2} = < [s_{0.6536}, s_{0.9473}], (0.222, 0.8258) >, \\ r_{3} = < [s_{0.6477}, s_{0.9308}], (0.2915, 0.8643) >, \\ r_{4} = < [s_{0.6914}, s_{0.9308}], (0.3552, 0.8251) > . \end{matrix}$

Step 3. Calculate the expected values E (r_i) of PULVs r_i (i = 1, 2, 3, 4), which are listed as follows: $\begin{matrix} E (r_{1}) = 0.1370, E (r_{2}) = 0.1495, \\ E (r_{3}) = 0.1324, E (r_{4}) = 0.1523 . \end{matrix}$

Step 4. Since E (r₄) > E (r₂) > E (r₁) > E (r₃), according to Definition 6, we can obtain the ranking of x_i (i = 1, 2, 3, 4): x₄ ≻ x₂ ≻ x₁ ≻ x₃. Thus, the best alternative is x₄.

In the above computation process, the parameters p and q are taken as one and the proposed approach produces the ranking of all the alternatives as x₄ ≻ x₂ ≻ x₁ ≻ x₃. If we take different values of p and q, the final ranking result of these alternatives may be slightly different as shown in Table 3. For example, if we take p = 5 and q = 1, then the expected values of these alternatives are obtained: E (r_i) =0.1604, E (r₂) =0.1681, E (r₃) =0.1665, E (r₄) =0.1894. By Definition 6, we get the ranking of these alternatives as x₄ ≻ x₂ ≻ x₃ ≻ x₁. Thus, this result is slightly different from that obtained by taking p = q = 1. That is, the ranking of x₁ and x₃ is reversed.

Based on the above analysis, we can easily see that the values of p and q play a crucial role to the ranking result. Different ranking results may be produced when different values are assigned to p and q. In general, the larger the values of p and q, the more computational effort are needed to aggregate the evaluation values and more interactions among attributes are emphasized. In some special situations,if p or q takes the value of 0, then the WPULPBM cannot reflect the interrelationship among attributes. Noted that if p = 0 or q = 0 and no partition pattern exists among the attribute set, the WPULPBM reduces to the generalized Pythagorean uncertain linguistic mean (GPULM) operator. Hence, we generally recommend decision makers take the values of p = q = 1, which can not only accurately depict the correlation between attributes, but also appropriately simplify the calculation during the aggregation process.

6.2 Comparison analyses and discussions

In the following, we compare these two types of aggregation operators, i.e., the WPULBM operator defined in Definition 8 and the WPULPBM operator defined in Definition 11 and analyze the effects of partition structure on the ranking results. In the above approach, if we utilize the WPULBM operator to aggregate the evaluation information, then the ranking results of alternatives are presented in Table 4.

From Tables 3 and 4, we can easily find that these two abovementioned aggregation operators may generate different ranking results. For example, when p = 2 and q = 1, the ranking result obtained by the WPULPBM operator is X₄ ≻ X₂ ≻ X₁ ≻ X₃. However, the WPULBM operator generates the result as X₄ ≻ X₂ ≻ X₃ ≻ X₁. Although the best alternative is still X₄ and X₂ is the second best alternative, the ranking positions of X₃ and X₁ are reverse. The main reason causing this ranking result difference is that the WPULPBM operator can more accurately model the relationships between attributes by introducing the partition structure and it can avoid the effects of calculated value among unrelated aggregated arguments. However, the WPULBM operator does not consider such actual situations that some of arguments may not be related to all other input arguments.

7 Conclusions

In this paper, to express the qualitative assessment values, based on the uncertain linguistic term sets and PFSs, we have presented a new type of linguistic term sets named Pythagorean uncertain linguistic sets (PULSs). The operational laws and a comparison method of any two PULVs are defined. Then, two kinds of Bonferroni mean (BM) operators for PULVs have been proposed to capture the interrelationship among the input arguments, i.e., the PULBM and the WPULBM, the PULPBM and the WPULPBM. Furthermore, based on the WPULPBM operator, a new approach is developed to solve MADM problems under Pythagorean uncertain linguistic environment. Finally, we have utilized the proposed approach to solve a practical MADM problem and a detailed comparison analyses are also made with other existing methods.

The main contribuitions of this paper are listed as follows: (1) We have presented a new type of linguistic term set,named Pythagorean uncertain linguistic set (PULS), which has much more stronger expression ability than the intuitionistic uncertain linguistic set (IULS) to represent qualitative information. (2) We have defined some operational laws, the expected values of PLVs and proposed a comparison method for any two PULVs. (3) We have developed two kinds of Bonferroni mean operators. The proposed PULPBM and WPULPBM operators can model the exact interrelationship among attributes and remove the effect of correlation among unrelated attributes on the final decision making results by introducing partitioned structure pattern. (4) We have developed a approach to solve the MADM problems with PULVs by applying the proposed operator.

Footnotes

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (Nos. 71271124, 71471172), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), the Shandong Provincial Natural Science Foundation (No. ZR2013GQ011), the Shandong Province Higher Educational Science and Technology Program (No. J16LN25).

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