Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators

doi:10.1016/j.asoc.2013.12.010

Applied Soft Computing

Volume 17, April 2014, Pages 90-104

https://doi.org/10.1016/j.asoc.2013.12.010 Get rights and content

Highlights

•
We proposed ILPGWA operator and ILPGOWA operator.
•
We investigated some desirable properties of the ILPGWA and ILPGOWA operators.
•
We studied some special cases of the generalized parameters in these operators.
•
We proposed two multiple attribute group decision making approaches based these operators.

Abstract

With respect to multiple attribute group decision making (MADM) problems in which attribute values take the form of intuitionistic linguistic numbers, some new group decision making methods are developed. Firstly, some operational laws, expected value, score function and accuracy function of intuitionistic linguistic numbers are introduced. Then, an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and an intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator are developed. Furthermore, some desirable properties of the ILPGWA and ILPGOWA operators, such as commutativity, idempotency and monotonicity, etc. are studied. At the same time, some special cases of the generalized parameters in these operators are analyzed. Based on the ILPGWA and ILPGOWA operators, two approaches to multiple attribute group decision making with intuitionistic linguistic information are proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.

Graphical abstract

Introduction

Multiple attribute decision making (MADM) problems are the important research parts of decision theory. Since the object things are fuzzy and uncertain, the attributes involved in the decision problems are not always expressed as crisp numbers, and some of them are more suitable to be denoted by fuzzy numbers, such as interval number, linguistic variable, intuitionistic fuzzy number etc. Since the fuzzy set theory, which was proposed by Zadeh [1], has been a rapid development and a wide range of applications [2], [3], [4]. However, the fuzzy set is used to character the fuzziness just by membership degree. Atanassov [5], [6] proposed the intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set. Obviously, the intuitionistic fuzzy set (IFS) can describe and character the fuzzy essence of the objective world more exquisitely [5], and it has received more and more attention since its appearance. Later, Atanassov and Gargov [7], Atanassov [8] further introduced the interval-valued intuitionistic fuzzy set (IVIFS), and Xu [9], Wang [10] proposed the decision-making methods based on IVIFS. Shu, Cheng and Chang [11] gave the definition of the intuitionistic triangular fuzzy number, and Zhang and Liu [12] defined the triangular intuitionistic fuzzy number, and they proposed the relevant decision making methods separately. Wang [13] gave the definition of intuitionistic trapezoidal fuzzy number and interval intuitionistic trapezoidal fuzzy number, then some decision making methods based on the intuitionistic triangular fuzzy number had been proposed [14], [15], [16], [17], [18]. Furthermore, Wang and Li [19] proposed intuitionistic linguistic sets, intuitionistic linguistic numbers, intuitionistic two-semantics and the Hamming distance between two intuitionistic two-semantics, and rank the alternatives by calculating the comprehensive membership degree to the ideal solution for each alternative.

The information aggregation operators are an interesting research topic, which is receiving increasing concern. Yager [20] proposed the generalized ordered weighted averaging (GOWA) operator which is an extension of the OWA operator. Li [21], Zhao et al. [22] further proposed the generalized aggregation operators for intuitionistic fuzzy sets. Merigó and Casanovas [23] presented the generalized hybrid averaging (GHA) operator. It is able to generalize a wide range of mean operators such as the HA, the hybrid geometric averaging (HGA), the hybrid quadratic averaging (HQA), the generalized ordered weighted averaging (GOWA) operator and the weighted generalized mean (WGM). A key feature in GHA operator is that it is able to deal with the weighted average and the ordered weighted averaging (OWA) operator in the same formulation. Merigó and Casanovas [24] introduced the fuzzy generalized hybrid averaging (FGHA) operator for the multi-attribute decision-making problems in which the attribute values take the form of the fuzzy number; this expanded the application scope of GHA operator. However, most of the existing aggregation operators do not consider the relationship between the values being aggregated. Yager [25] proposed a power-average (PA) operator and a power OWA (POWA) operator, which weighting vectors depend on the input data and allow values being fused to support and reinforce each other. Xu and Yager [26] developed a power-geometric (PG) operator, a power-ordered-geometric (POG) operator and a power-ordered-weighted-geometric (POWG) operator, and studied some properties of these operators. Then, an uncertain PG (UPG) operator and its weighted form, and an uncertain power-ordered-weighted-geometric (UPOWG) operator are proposed to aggregate the input arguments taking the form of interval values, and the approaches to group decision making are developed based on these operators. Xu [27] developed a series of operators for aggregating the intuitionistic fuzzy numbers, and studied various properties of these power aggregation operators, and then some approaches to multiple attribute group decision making with intuitionistic fuzzy information and interval-valued intuitionistic fuzzy information were proposed. Xu and Wang [28] developed some linguistic aggregation operators, included 2-tuple linguistic power average (2TLPA) operator, 2-tuple linguistic weighted power average (2TLWPA) operator, and 2-tuple linguistic power-ordered-weighted average (2TLPOWA) operator, then studied some properties of these operators, such as idempotency, boundary, etc. Moreover, two approaches to deal with group decision making problems under linguistic environment were developed. Zhou et al. [29] proposed a generalized power ordered weighted average (GPOWA) operator, an uncertain generalized power average (UGPA) operator, an uncertain generalized power ordered weighted average (UGPOWA) operator, an generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy power ordered weighted averaging (GIFPOWA) operator. Moreover, some properties of these operators are studied and the new approaches on the basis of the proposed operators were developed for the MADM problems. Xu et al. [30] proposed the linguistic power average (LPA) operator, the linguistic weighted PA operator, the LPOWA operator, the uncertain linguistic weighted PA operator, and the ULPOWA operator. Then some approaches, which are based on ULWPA and ULPOWA operators, were developed to solve group decision-making problems under an uncertain linguistic environment.

The intuitionistic linguistic variables are very suitable to be used for depicting uncertain or fuzzy information, and the existing methods based on intuitionistic linguistic variables do not consider the relationships of the aggregated arguments. Motivated by the idea of power aggregation operator proposed by Yager [25] and the generalized aggregation operators proposed by Yager [20] and Zhao et al. [22], this paper is to propose some operators, such as an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and an intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator. They can not only provide the generalized aggregation functions, but also provide the power aggregation functions which consider the relationships of the aggregated arguments. Furthermore, some desirable properties of the ILPGWA and ILPGOWA operators, such as commutativity, idempotency and monotonicity, are studied, and some special cases of these operators are analyzed. To do this, the structure of this paper is shown as follows. In Section 2, we briefly review some basic concepts of the intuitionistic fuzzy set, linguistic variables, intuitionistic linguistic variables, GOWA operator, and power operators. Section 3 proposes an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and an intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator, some desirable properties and special cases of these operators are discussed. In Section 4, we develop two methods for multi-criteria group decision making based on the proposed operators. Section 5 gives an example to illustrate the decision steps and discusses the influence of different parameters on the decision-making results, and compare with existing method. In Section 6, we give the conclusions and future research directions.

Section snippets

The intuitionistic fuzzy set

Definition 1

[5], [6]

An IFS A in X is given by $A = {< x, u_{A} (x), v_{A} (x) > x \in X}$ where u_A : X → [0, 1] and $v_{A} : X \to [0,1]$ , meet the condition $0 \leq u_{A} (x) + v_{A} (x) \leq 1$ , ∀x ∈ X. The numbers u_A(x) and v_A(x) represent the membership degree and non-membership degree of the element x to the set A, respectively. Furthermore, $π (x) = 1 - u_{A} (x) - v_{A} (x)$ is called a hesitancy degree of x to the set A. It is obvious that 0 ≤ π(x) ≤ 1, ∀x ∈ X.

In addition, we can call each two-tuple $(u_{A} (x), v_{A} (x))$ an intuitionistic fuzzy number (IFN), and for convenience, denoted an IFN by $a$

The intuitionistic linguistic power generalized aggregation operators

Definition 10

Let ${\tilde{a}}_{j} = 〈 s_{θ ({\tilde{a}}_{j})}, (u ({\tilde{a}}_{j}), v ({\tilde{a}}_{j})) 〉 (j = 1,2, \dots, n)$ be a collection of the ILNs, and ILPGA : Ωⁿ → Ω, if $ILPGA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {(\frac{\sum_{j = 1}^{n} (1 + T ({\tilde{a}}_{j})) {\tilde{a}}_{j}^{λ}}{\sum_{j = 1}^{n} (1 + T ({\tilde{a}}_{j}))})}^{1 / λ}$ where Ω is the set of all intuitionistic linguistic numbers, and $T ({\tilde{a}}_{j}) = \sum_{\begin{array}{l} i = 1 \\ i \neq j \end{array}}^{n} S u p ({\tilde{a}}_{j}, {\tilde{a}}_{i})$ , λ is a parameter such that λ ∈ (0, + ∞), then IPLGA is called the intuitionistic linguistic power generalized aggregation operator.

The ILPGA operator satisfies the following properties:

Theorem 1 (1, Commutativity).

Let $({\tilde{a^{'}}}_{1}, {\tilde{a^{'}}}_{2}, \dots, {\tilde{a^{'}}}_{n})$ be any permutation of $({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n})$ , then $ILPGA (a$

An approach to group decision making based on the intuitionistic linguistic numbers

Consider a multiple attribute group decision making with intuitionistic linguistic information: let A = {A₁, A₂, …, A_m} be a discrete set of alternatives, and C = {C₁, C₂, …, C_n} be the set of attributes, ω = (ω₁, ω₂, …, ω_n)^T is the weighting vector of the attribute C_j(j = 1, 2, …, n), where $ω_{j} \geq 0, j = 1,2, \dots, n, \sum_{j = 1}^{n} ω_{j} = 1,$ . Let D = {D₁, D₂, …, D_p} be the set of decision makers, and γ = (γ₁, γ₂, …, γ_p) is the expert weight, with $γ_{k} \geq 0 (k = 1,2, \dots, p), \sum_{k = 1}^{p} γ_{k} = 1$ . Suppose that ${\tilde{R}}^{k} = {[{\tilde{r}}_{i j}^{k}]}_{m \times n}$ is the decision matrix, where ${\tilde{r}}_{i}$

Example

Suppose that an investment company wants to invest a sum of money to one of four possible alternatives which are shown as follows.

(1)
A₁ is a car company;
(2)
A₂ is a computer company;
(3)
A₃ is a TV company;
(4)
A₄ is a food company.

There are four attributes (suppose that their weight vector is ω = (0.32, 0.26, 0.18, 0.24)^T) which are shown as follows.

(1)
C₁ is the risk analysis;
(2)
C₂ is the growth analysis;
(3)
C₃ is the social–political impact analysis;
(4)
C₄ is the environmental impact analysis.

The four possible alternatives {A

Conclusion

The traditional power average operators and the generalized aggregation operators are generally suitable for aggregating the information taking the form of real values, and yet they will fail in dealing with intuitionistic linguistic variables. In this paper, with respect to multiple attribute group decision making (MAGDM) problems in which both the attribute weights and the expert weights take the form of real numbers and attribute values take the form of intuitionistic linguistic numbers,

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (No. 71271124), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), the Natural Science Foundation of Shandong Province (No. ZR2011FM036), Shandong Provincial Social Science Planning Project (No. 13BGLJ10), and graduate education innovation projects in Shandong Province (SDYY12065). The author also would like to express appreciation to the anonymous reviewers and

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Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

The intuitionistic fuzzy set

[5], [6]

The intuitionistic linguistic power generalized aggregation operators

An approach to group decision making based on the intuitionistic linguistic numbers

Example

Conclusion

Acknowledgments

Applied Soft Computing

Applied Soft Computing

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Microelectronics Reliability

Expert Systems with Applications

Knowledge-Based Systems

Applied Soft Computing

Computers & Industrial Engineering

Applied Mathematical Modelling

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy sets

Information and Control

Applications of linear differential inclusion-based criterion to a nonlinear chaotic system: a critical review

Journal of Vibration and Control

More on intuitionistic fuzzy sets

Fuzzy Sets and Systems

Models for multiple attribute decision-making with intuitionistic fuzzy information

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems