2-Tuple linguistic harmonic operators and their applications in group decision making

doi:10.1016/j.knosys.2013.01.006

Knowledge-Based Systems

Volume 44, May 2013, Pages 10-19

https://doi.org/10.1016/j.knosys.2013.01.006 Get rights and content

Abstract

Harmonic mean is reciprocal of arithmetic mean of reciprocal, which is a conservative average to be used to provide for aggregation lying between max and min operators. In this paper, we develop some new linguistic aggregation operators such as 2-tuple linguistic harmonic (2TLH) operator, 2-tuple linguistic weighted harmonic (2TLWH) operator, 2-tuple linguistic ordered weighted harmonic (2TLOWH) operator, and 2-tuple linguistic hybrid harmonic (2TLHH) operator, which can be utilized to aggregate preference information taking the form of linguistic variables, and then study some desirable properties of the operators. Based on the 2TLWH and 2TLHH operators, we present an approach to multiple attribute decision making with 2-tuple linguistic information. Finally, illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness by comparing with the existing approaches.

Introduction

Group decision making (i.e., multi-expert) is a typical decision making activity where utilizing several experts alleviate some of the decision making difficulties due to the problem’s complexity and uncertainty. In the real world, the uncertainty, constraints, and even unclear knowledge of the experts imply that decision makers cannot provide exact numbers to express their opinions. Linguistic variables are a very useful tool to express a decision maker’s preference information over objects in process of decision making under uncertain or vague environments [1], [2]. In order to get a decision result, an important step is the aggregation of linguistic variables. Over the last decades, various linguistic aggregation operators have been proposed. We classify these operators into the following categories: (1) the linguistic aggregation operators are based on the semantic model, such as the linguistic approximation operator [3], linguistic OWA operator [4], [5], [6], [7], [8], [9], linguistic weighted OWA operator [10] and inverse-LOWA operator [11], these operators use linguistic terms as labels for fuzzy numbers while the computations over them are done directly over those fuzzy numbers; (2) the linguistic aggregation operators based on the symbolic model [5], [12], [13], which make computations on the indexes of the linguistic labels; (3) the linguistic aggregation operators, which compute with words directly, such as the linguistic weighted averaging (LWA) operator [14], extended ordered weighted averaging (EOWA) operator [15], extended ordered weighted geometric (EOWG) operator [15], linguistic weighted arithmetic averaging (LWAA) operator [16], [17], linguistic weighted geometric averaging (LWGA) operator [18], linguistic ordered weighted geometric averaging (LOWGA) operator [18], linguistic hybrid geometric averaging (LHGA) operator [18], uncertain LWA (ULWA) operator [19], [20], [21], uncertain linguistic hybrid aggregation (ULHA) operator [19], induced uncertain LOWA (IULOWA) operator [22], uncertain linguistic ordered weighted geometric (ULOWG) operator [23], induced uncertain linguistic ordered weighted geometric (IULOWG) operator [23], induced linguistic generalized ordered weighted averaging (ILGOWA) operator [24], induced linguistic generalized hybrid averaging (ILGHA) operator [24], induced linguistic quasi-arithmetic OWA (Quasi-ILOWA) operator [24] and linguistic power aggregation operator [25]; (4) the linguistic aggregation operators based on the 2-tuple linguistic representation model [26], [27], which represent the linguistic information with a pair of values called 2-tuple, composed by a linguistic term and number, including 2-tuple weighted averaging operator [26], 2-tuple OWA operator [26], 2-tuple weighted geometric averaging (TWGA) operator [28], 2-tuple ordered weighted geometric averaging (TOWGA) operator [28] and 2-tuple hybrid geometric averaging (THGA) operator [28]. The operators in (1), (2) develop some approximation processes to express the results in initial expression domain, which produce the consequent loss of information and hence bring about the lack of precision, while the operators in (3), (4) allow a continuous representation of the linguistic information on its domain, and thus they can represent any counting of information obtained in an aggregation process without any loss of information.

Harmonic mean is a conservative average, which is widely used to aggregate central tendency data. In the existing literature, the harmonic mean is generally considered as a fusion technique of numerical data information. However, in many situations, the input arguments take the form of 2-tuple linguistic variables because of time pressure, lack of knowledge and people’s limited expertise related with problem domain. Therefore, “how to aggregate 2-tuple linguistic variables by using the harmonic mean?” is an interesting research topic and is worth paying attention to. In this paper, we focus our attention on developing some 2-tuple linguistic harmonic (2TLH) operators. To do so, the rest of this paper is organized as follows. The basic concept related to 2-tuple linguistic representation is presented in Section 2. Section 3 develops some 2TLH operators, such as 2-tuple linguistic weighted harmonic (2TLWH) operator, 2-tuple linguistic ordered weighted harmonic (2TLOWH) operator and 2-tuple linguistic hybrid harmonic (2TLHH) operator, and investigate some of their properties. Section 4 presents an approach to multiple attribute group decision making (MAGDM) based on the developed operators. Section 5 illustrates the presented approach with a practical example, and verify and show the advantages of the presented approach and makes a comparative study to the existing approach. Section 6 ends the paper with some concluding remarks.

Section snippets

Preliminaries

The linguistic variables are used in processes of computing with words that imply their fusion, aggregation and comparison, etc. The most often used models dealing with linguistic information are: (1) the semantic model that uses the linguistic terms just as labels for fuzzy numbers, while the computations over them are done directly over those fuzzy numbers, (2) the second one is the symbolic model that uses the order index of the linguistic terms to make direct computation on labels, and (3)

2-Tuple linguistic harmonic operators

Definition 4

[31]

Let $WAA : R^{n} \to R$ , if $WAA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} w_{j} a_{j},$ where $a_{j} (j = 1, 2, \dots, n)$ is a collection of positive real numbers, $w = (w_{1}, w_{2}, \dots, w_{n})^{T}$ is the weight vector of $a_{j} (j = 1, 2, \dots, n)$ , with $w_{j} ⩾ 0$ and $\sum_{j = 1}^{n} w_{j} = 1, R$ is the set of real numbers, then WAA is called the weighted arithmetic averaging (WAA) operator. Especially, if $w_{i} = 1, w_{j} = 0, j \neq i$ , then $WAA (a_{1}, a_{2}, \dots, a_{n}) = a_{i}$ ; if $w = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , then the WAA operator is reduced to the arithmetic averaging (AA) operator, i.e., $AA (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{n} \sum_{j = 1}^{n} a_{j} .$

Definition 5

[32]

Let $WHM : (R^{+})^{n} \to R^{+}$ , if $WHM (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{\sum_{j}}$

An approach to multiple attribute group decision making

Now we consider a MAGDM problem, let $X = {x_{1}, x_{2}, \dots, x_{n}}$ be a discrete set of n feasible alternatives and $G = {G_{1}, G_{2}, \dots, G_{m}}$ be a set of m attributes, whose weight vector is $w = (w_{1}, w_{2}, \dots, w_{m})^{T}$ , where $w_{i} ⩾ 0, i = 1, 2, \dots, m, \sum_{i = 1}^{m} w_{i} = 1$ . Let $D = {d_{1}, d_{2}, \dots, d_{l}}$ be the set of l decision makers, and $v = (v_{1}, v_{2}, \dots, v_{l})^{T}$ be the weight vector of decision makers, where $v_{k} ⩾ 0, k = 1, 2, \dots, l, \sum_{k = 1}^{l} v_{k} = 1$ . Suppose that $A^{(k)} = (a_{ij}^{(k)})_{m \times n}$ is the linguistic decision matrix, where $a_{ij}^{(k)} \in S$ is preference value, which takes the form of linguistic

Illustrative examples

Example 3

Let us suppose an investment company, which wants to invest a sum of money in the best option (adapted from [5]). There is a panel with five possible alternatives in which to invest the money: (1) $x_{1}$ is a car industry; (2) $x_{2}$ is a food company; (3) $x_{3}$ is a computer company; (4) $x_{4}$ is an arms company; (5) $x_{5}$ is a TV company.

The investment company must take a decision according to the following four attributes (suppose that the weighting vector of four attributes is $w = (0.35, 0.15, 0.20, 0.30)^{T}$ ): (1)

Conclusions

In this paper, based on the harmonic mean operator, we have developed several new harmonic aggregation operators including the 2-tuple linguistic weighted harmonic (2TLWH), 2-tuple linguistic ordered weighted harmonic (2TLOWH) and 2-tuple linguistic hybrid harmonic (2TLHH) operators, which can be utilized to aggregate preference information taking the form of linguistic variables. We have studied some desired properties of the developed operators, such as commutativity, idempotency and