Fuzzy Linguistic Induced Ordered Weighted Averaging Operator and Its Application

Abstract

With respect to multiple attribute group decision making (MAGDM) problems, in which the attribute weights take the form of real numbers, and the attribute values take the form of fuzzy linguistic scale variables, a decision analysis approach is proposed. In this paper, we develop a new fuzzy linguistic induce OWA (FLIOWA) operator and analyze the properties of it by utilizing some operational laws of fuzzy linguistic scale variables. A method based on the FLIOWA operators for multiple attribute group decision making is presented. Finally, a numerical example is used to illustrate the applicability and effectiveness of the proposed method.

1. Introduction

In multiple attribute group decision making (MAGDM) analysis with linguistic information, the aggregation operators are required to deal with the aggregation of linguistic information and rankings of alternatives. Since Yager and Filev [1] developed the induced ordered weighted averaging (IOWA) operator, which takes as its argument pairs, called OWA pairs, in which one component is used to induce an ordering over the second components which are exact numerical values and then aggregated, many operators have been developed to aggregate linguistic information in group decision making such as the induced ordered weighted geometric (IOWG) operators that is based on the IOWA operator and the geometric mean [2, 3], the Quasi-IOWA operator based on the quasi-arithmetic means [4], and the induced ordered weighted harmonic averaging (IOWHA) operator based the IOWA operator and the harmonic mean [5], the induced trapezoidal fuzzy ordered weighted harmonic averaging (ITFOWHA) operator that is based on the IOWHA operator with the trapezoidal fuzzy information [6] using the idea of [7], we propose a fuzzy linguistic induce order weight averaging (FLIOWA) operator based on the extended ordered weighted averaging (EWOA) operator [8–10], which aims to resolve the input arguments taking the form of fuzzy linguistic scale information for the EOWA operator.

The rest of this paper is organized as follows. In Section 2, we briefly describe some basic concepts about the fuzzy number. In Section 3 we present the FLIOWA operator and its properties. A method based on the FLIOWA operators for multiple attribute group decision making (MAGDM) is presented in Section 4. In Section 5, we develop a numerical example of the new approach. Finally, in Section 6 we summarize the main conclusions of the paper.

2. Preliminaries

For the convenience of analysis, we give several definitions in this section. These definitions are used throughout the paper.

Definition 2.1. A triangular fuzzy number 𝑎 can be defined by a triplet 𝑎=(𝑎₁,𝑎₂,𝑎₃), 𝑎₁,𝑎₂,𝑎₃∈ℜ. The membership function [11, 12] 𝜇_𝑎(𝑥) is defined as 𝜇_𝑎(𝑥)=0,𝑥<𝑎₁,𝑥−𝑎₁𝑎₂−𝑎₁,𝑎₁≤𝑥≤𝑎₂,𝑎₃−𝑥𝑎₃−𝑎₂,𝑎₂≤𝑥≤𝑎₃,0,𝑥>𝑎₃, (2.1) where 𝑎₁≤𝑎₂≤𝑎₃, if 𝑎₂−𝑎₁=𝑎₃−𝑎₂, 𝑎 is a symmetrical triangular fuzzy number. If 𝑎₁=𝑎₂=𝑎₃, then 𝑎 is a real number, that is, 𝑎=𝑎₁=𝑎₂=𝑎₃. If 𝑎₁>0, we call 𝑎 as a positive triangular fuzzy number.

Definition 2.2 (see [11]). Let 𝑎=(𝑎₁,𝑎₂,𝑎₃) and 𝑏̃=(𝑏₁,𝑏₂,𝑏₃) be any two positive triangular fuzzy numbers, 𝜆∈ℜ, the basic operational laws related to triangular fuzzy numbers are as follows: (1)𝑎⊕𝑏̃=𝑎₁+𝑏₁,𝑎₂+𝑏₂,𝑎₃+𝑏₃,(2)𝑎⊗𝑏̃=𝑎₁×𝑏₁,𝑎₂×𝑏₂,𝑎₃×𝑏₃,(3)𝜆⊙𝑎=𝜆𝑎₁,𝜆𝑎₂,𝜆𝑎₃,(4)1𝑎=1𝑎₃,1𝑎₂,1𝑎₁. (2.2)

Definition 2.3. Let 𝑎=(𝑎₁,𝑎₂,𝑎₃) be the positive triangular fuzzy number, the graded mean value representation of a triangular fuzzy number 𝑎 is defined as [13] 𝑃(𝑎)=𝑎₁+4𝑎₂+𝑎₃6. (2.3)

Definition 2.4. Let 𝑎=(𝑎₁,𝑎₂,𝑎₃) and 𝑏̃=(𝑏₁,𝑏₂,𝑏₃) be any two positive triangular fuzzy numbers, if 𝑃(𝑎)≤𝑃(𝑏̃), then 𝑎≤𝑏̃.

Suppose that 𝑆={𝑠₁,𝑠₂,…,𝑠_𝑇} is the preestablished finite and totally ordered discrete linguistic term set with odd cards, where 𝑠_𝑖 denotes the 𝑖th linguistic term of 𝑆. It can be seen that 𝑇 is the cardinality of 𝑆. An alternative possibility for reducing the complexity of defining a grammar consists of directly supplying the term set by considering all terms as primary ones and distributed on a scale on which a total order is defined [14–16]. For example, a set of nine terms 𝑆 could be given as follows: 𝑆=𝑠₁=none(N),𝑠₂=extremelylow(EL),𝑠₃(),𝑠=verylowVL₄=low(L),𝑠₅=medium(M),𝑠₆=High(H),𝑠₇(),𝑠=veryhighVH₈=extremelyhigh(EH),𝑠₉=perfect(P), (2.4) in which 𝑠_𝑖<𝑠_𝑗 if and only if 𝑖<𝑗. Usually, in these cases, it is often required that the linguistic term set satisfies the following additional characteristics.(1)There is a negation operator, for example, Neg(𝑠_𝑖)=𝑠_𝑗, 𝑗=𝑇−1(𝑇+1 is the cardinality).(2)Maximization and minimization operator: Max(𝑠_𝑖,𝑠_𝑗)=𝑠_𝑖 if 𝑠_𝑖≥𝑠_𝑗, Min (𝑠_𝑖,𝑠_𝑗)=𝑠_𝑖 if 𝑠_𝑖≤𝑠_𝑗.

The fuzzy linguistic scale 𝑆 is used by the decision making to evaluate the performance of decisions, and these and fuzzy linguistic terms are denoted by triangular fuzzy numbers as follows: 𝑆=𝑠₉=𝑠_(0.8,0.9,1),𝑠₈=𝑠_{(0.7,0.8,0.9)},𝑠₇=𝑠_{(0.6,0.7,0.8)},𝑠₆=𝑠_{(0.5,0.6,0.7)},𝑠₅=𝑠_{(0.4,0.5,0.6)},𝑠₄=𝑠_{(0.3,0.4,0.5)},𝑠₃=𝑠_{(0.2,0.3,0.4)},𝑠₂=𝑠_{(0.1,0.2,0.3)},𝑠₁=𝑠_(0,0.1,0.2). (2.5)

Definition 2.5. Let 𝑎=(𝑎₁,𝑎₂,𝑎₃) and 𝑏̃=(𝑏₁,𝑏₂,𝑏₃) be any two positive triangular fuzzy numbers, 𝑠_𝑎,𝑠_̃𝑏∈𝑆, if 𝑎≤𝑏̃, then 𝑠_𝑎≤𝑠_̃𝑏.

According to Definition 2.5, there is obviously 𝑠₉>𝑠₈>𝑠₇>𝑠₆>𝑠₅>𝑠₄>𝑠₃>𝑠₂>𝑠₁. (2.6)

Definition 2.6 (see [9]). Let 𝑠_𝛼,𝑠_𝛽∈𝑆, 𝑦,𝑧∈[0,1], then: (1)𝑠_𝛼⊕𝑠_𝛽=𝑠_𝛼+𝛽=𝑠_𝛽⊕𝑠_𝛼,(2)𝑦⊙𝑠_𝛼=𝑠_𝑦𝛼,(3)𝑦⊙𝑠_𝛼⊕𝑠_𝛽=𝑦⊙𝑠_𝛼⊗𝑦⊙𝑠_𝛽=𝑠_𝑦𝛼_+𝑦𝛽,(4)𝑦+𝑧⊙𝑠_𝛼=𝑦⊙𝑠_𝛼⊕𝑧⊙𝑠_𝛼=𝑠_(𝑦+𝑧)𝛼. (2.7)

3. Fuzzy Linguistic Induced Ordered Weighted Averaging (FLIOWA) Operator

Inspired by Mitchell and Estrakh's work [17], Yager and Filev introduced in [18] a more general type of OWA operator, which they named the induced ordered weighted averaging (IOWA) operator.

Definition 3.1. An IOWA operator of dimension 𝑛 is a function Φ_𝑊∶(𝑅×𝑅)^𝑛→𝑅, to which a weighting vector is associated, 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛), such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, and it is defined to aggregate the set of second arguments of a list of 𝑛 pairs {(𝑢₁,𝑝₁),(𝑢₂,𝑝₂),…,(𝑢_𝑛,𝑝_𝑛)} according to the following expression: Φ_𝑊𝑢₁,𝑝₁,𝑢₂,𝑝₂,…,𝑢_𝑛,𝑝_𝑛=∑^𝑛_𝑖=1𝑤_𝑖𝑝_𝜎(𝑖), (3.1) where 𝜎∶(1,2,…,𝑛)→(1,2,…,𝑛) is a permutation such that 𝑢_𝜎(𝑖)≥𝑢_𝜎(𝑖+1),forall𝑖=1,2,…,𝑛−1,: that is, (𝑢_𝜎(𝑖),𝑝_𝜎(𝑖)) is the pair with 𝑢_𝜎(𝑖) the 𝑖th highest value in the set {𝑢₁,𝑢₂,…,𝑢_𝑛}. 𝑝₁,𝑝₂,…,𝑝_𝑛 is induced by the ordering of the values 𝑢₁,𝑢₂,…,𝑢_𝑛 associated with them. Because of this use of the set of values 𝑢₁,𝑢₂,…,𝑢_𝑛, Yager and Filev have called them the values of an order inducing variable and 𝑝₁,𝑝₂,…,𝑝_𝑛 the values of the argument variable [19].

Xu et al. [8–10] developed the extended ordered weighted averaging (EOWA) operator.

Definition 3.2. An extended ordered weighted averaging (EOWA) operator of dimension 𝑛 is a mapping, Φ∶𝑆^𝑛→𝑆, that has an associated weighting vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 with the properties 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, such that: Φ_𝑤𝑠_𝛼1,𝑠_𝛼2,…,𝑠_𝛼𝑛=𝑤₁⊙𝑠_𝛽1⊕⋯⊕𝑤_𝑛⊙𝑠_𝛽𝑛=𝑠_𝛽, (3.2) where 𝛽=∑^𝑛_𝑖=1𝑤_𝑖𝛽_𝑖, (𝑠_𝛽1,𝑠_𝛽2,…,𝑠_𝛽𝑛)→(𝑠_𝛼1,𝑠_𝛼2,…,𝑠_𝛼𝑛) being a permutation such that 𝑠_𝛽𝑖≥𝑠_𝛽𝑖+1,forall𝑖=1,2,…,𝑛−1, that is, 𝑠_𝛽𝑖 is the 𝑖th highest value in the set (𝑠_𝛼1,𝑠_𝛼2,…,𝑠_𝛼𝑛).

Definition 3.3. Let 𝑎_𝑖=(𝑎_𝑖1,𝑎_𝑖2,𝑎_𝑖3),(𝑖=1,2,…,𝑛)  be a collection of triangular fuzzy numbers. A fuzzy linguistic induced ordered weighted averaging (FLIOWA) operator of dimension 𝑛 is a function, FLIOWA∶𝑆^𝑛→𝑆, that has associated a set of weights or weighting vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 to it, so that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1 and is defined to aggregate a list of values (𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛) according to the following expression: FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑤₁𝛽⊙𝑠₁⊕⋯⊕𝑤_𝑛𝛽⊙𝑠_𝑛=𝑠_𝛽, (3.3) where 𝛽=∑^𝑛_𝑖=1𝑤_𝑖𝛽_𝑖, 𝛽(𝑠₁𝛽,𝑠₂𝛽,…,𝑠_𝑛)→(𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛) being a permutation such that 𝑠𝛽_𝑖𝛽≥𝑠_𝑖+1,forall𝑖=1,2,…,𝑛−1, that is, 𝑠𝛽_𝑖 is the 𝑖th highest value in the set (𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛).
The FLIOWA operator has the following properties.

Theorem 3.4 (commutativity). Let (𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛) be any permutation of the fuzzy linguistic scale data (𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛), then FLIOWA_𝑤𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛=FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛. (3.4)

Proof. Let FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑤₁𝛽⊙𝑠₁⊕𝑤₂𝛽⊙𝑠₂⊕⋯⊕𝑤_𝑛𝛽⊙𝑠_𝑛,FLIOWA_𝑤𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛=𝑤₁⊙𝑠^∗𝛽₁⊕𝑤₂⊙𝑠^∗𝛽₂⊕⋯⊕𝑤_𝑛⊙𝑠^∗𝛽_𝑛. (3.5)
Since (𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛) is a permutation of (𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛), using the Definitions 2.3–2.6, we have 𝑠𝛽_𝑖=𝑠^∗𝛽_𝑖,  𝑖∈𝑁, then FLIOWA_𝑤𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛=FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛. (3.6)

Theorem 3.5 (idempotency). If 𝛼_𝑖=𝛼 for all 𝑖, and for all𝑠𝛼_𝑖,𝑠_𝛼∈𝑆, then FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑠_𝛼. (3.7)

Proof. Since 𝛼_𝑖=𝛼   for all 𝑖, according to the Definition 2.5, 𝑠𝛼_𝑖=𝑠_𝛼 for all 𝑖, then FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑤₁𝛽⊙𝑠₁⊕𝑤₂𝛽⊙𝑠₂⊕⋯⊕𝑤_𝑛𝛽⊙𝑠_𝑛=𝑤₁⊙𝑠_𝛼⊕𝑤₂⊙𝑠_𝛼⊕⋯⊕𝑤_𝑛⊙𝑠_𝛼=𝑤₁+𝑤₂+⋯+𝑤_𝑛⊙𝑠_𝛼=𝑠_𝛼. (3.8)

Theorem 3.6 (monotonicity). If 𝛼_𝑖≤𝛼_𝑖^∗,𝑠𝛼_𝑖,𝑠𝛼_𝑖^∗∈𝑆, for all 𝑖, then FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛≤FLIOWA_𝑤𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛. (3.9)

Proof. Since 𝛼_𝑖≤𝛼_𝑖^∗ for all 𝑖, according to the Definition 2.5, 𝑠𝛼_𝑖≤𝑠𝛼_𝑖^∗ for all 𝑖. Let FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑤₁𝛽⊙𝑠₁⊕⋯⊕𝑤_𝑛𝛽⊙𝑠_𝑛=𝑠_𝛽,FLIOWA_𝑤𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛=𝑤₁𝛽⊙𝑠^∗₁⊕⋯⊕𝑤_𝑛𝛽⊙𝑠^∗_𝑛𝛽=𝑠^∗, (3.10) where 𝛽=∑^𝑛_𝑖=1𝑤_𝑖𝛽_𝑖,𝛽^∗=∑^𝑛_𝑖=1𝑤_𝑖𝛽^∗_𝑖, since 𝛽(𝑠₁𝛽,𝑠₂𝛽,…,𝑠_𝑛) is a permutation of (𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛), and 𝛽(𝑠^∗₁𝛽,𝑠^∗₂𝛽,…,𝑠^∗_𝑛) is a permutation of (𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛), respectively, so, we have 𝑠𝛽_𝑖𝛽≤𝑠^∗_𝑖 for all 𝑖, then 𝑠_𝛽𝛽≤𝑠^∗, that is, FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛≤FLIOWA_𝑤𝑠𝛼^∗₁,𝑠𝛼^∗₂,…,𝑠𝛼^∗_𝑛. (3.11)

Theorem 3.7 (boundedness). Let 𝑠_𝛼=min_𝑖(𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛),𝑠_𝛽=min_𝑖(𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛), then 𝑠_𝛼≤FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛≤𝑠_𝛽. (3.12)

Proof. Since 𝑠_𝛼≤𝑠𝛼_𝑖≤𝑠_𝛽 for all 𝑖, using the Theorems 3.5 and 3.6, then FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑤₁𝛽⊙𝑠₁⊕𝑤₂𝛽⊙𝑠₂⊕⋯⊕𝑤_𝑛𝛽⊙𝑠_𝑛≥𝑤₁⊙𝑠_𝛼⊕𝑤₂⊙𝑠_𝛼⊕⋯⊕𝑤_𝑛⊙𝑠_𝛼=𝑤₁+𝑤₂+⋯+𝑤_𝑛⊙𝑠_𝛼=𝑠_𝛼,FLIOWA_𝑤𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛=𝑤₁𝛽⊙𝑠₁⊕𝑤₂𝛽⊙𝑠₂⊕⋯⊕𝑤_𝑛𝛽⊙𝑠_𝑛≤𝑤₁⊙𝑠_𝛽⊕𝑤₂⊙𝑠_𝛽⊕⋯⊕𝑤_𝑛⊙𝑠_𝛽=𝑤₁+𝑤₂+⋯+𝑤_𝑛⊙𝑠_𝛽=𝑠_𝛽. (3.13)
So 𝑠_𝛼≤FLIOWA_𝑤(𝑠𝛼₁,𝑠𝛼₂,…,𝑠𝛼_𝑛)≤𝑠_𝛽.

Especially, if 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then FLIOWA is reduced to the fuzzy linguistic extended arithmetical averaging (FEAA) operator; if 𝛼_𝑖1=𝛼_𝑖2=𝛼_𝑖3 for all 𝑖, then 𝛼_𝑖=𝛼_𝑖, the FLIOWA is reduced to the extended order weighted averaging (EOWA) operator.

4. A New Approach to Decision Making with Fuzzy Linguistic Scale Information

In the following, we are going to develop a new approach in a decision making problem about selection of strategies with fuzzy linguistic scale information. Let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑘} be a discrete set of alternatives, we suppose that the decision makers compare these six companies with respect to a single criterion by using the fuzzy linguistic scale 𝑆=𝑠₉=𝑃,𝑠₈=𝐸𝐻,𝑠₇=𝑉𝐻,𝑠₆=𝐻,𝑠₅=𝑀,𝑠₄=𝐿,𝑠₃=𝑉𝐿,𝑠₂=𝐸𝐿,𝑠₁=𝑁. (4.1)

In the following, we shall apply the FLIOWA operator to decision making with fuzzy linguistic scale information.

Step 1. Construct a fuzzy linguistic scale decision making matrix 𝑅=(𝑟̃_𝑖𝑗)_𝑘×𝑘 according to the fuzzy linguistic scale evaluation for the alternative 𝑥_𝑖∈𝑋, (𝑖=1,2,…,𝑘) under the attribute 𝑢_𝑖∈𝑈, (𝑖=1,2,…,𝑘).

Step 2. Calculate the overall preference triangular fuzzy values of the alternative by utilizing the FLIOWA operator 𝑠𝛼_𝑖=𝑠_{(𝛼𝑖1,𝛼𝑖2,𝛼𝑖3)}=FLIOWA_𝑤𝑟̃_𝑖1,𝑟̃_𝑖2,…,𝑟̃_𝑖𝑘,(𝑖=1,2,…,𝑘). (4.2)

Step 3. Calculate the graded mean values of triangular fuzzy number 𝛼_𝑖, (𝑖=1,2,…,𝑘)𝑃𝑎_𝑖=𝑎_𝑖1+4𝑎_𝑖2+𝑎_𝑖36. (4.3)

Step 4. Rank all the alternatives 𝑥_𝑖(𝑖=1,2,…,𝑘) and select the best one(s) in accordance with by using Definition 2.5.

Step 5. End.

5. Numerical Example

In this section, we are going to give a brief illustrative example of the new approach in a decision making problem. Let us suppose an investment company that it is operating in Europe and North America is analyzing its general policy for the next year, which wants to invest strategies in the best option [17]. In order to evaluate these strategies, the group of experts of the company considers that the key factor for the next year is the economic situation. Then, depending on the situation, the expected benefits for the company will be different. The experts have considered 6 possible situations for the next year. There is a panel with six possible alternatives to invest the strategies One main criterion used is the growth analysis. In this example, we assume that the experts use the following weighting vector for the FLIOWA operators: 𝑊=(0.2,0.2,0.2,0.2,0.1,0.1)^𝑇. Then, we utilize the approach developed to get the most desirable alternative(s).

Firstly, construct a fuzzy linguistic scale decision making matrix 𝑅=(𝑟̃_𝑖𝑗)_𝑛×𝑛. We suppose that the decision makers compare these six alternatives with respect to the criterion growth analysis by using the fuzzy linguistic scale and the fuzy linguistic decision matrix 𝑅=(𝑟̃_𝑖𝑗)_6×6 is in Table 1 [20].

Table 1 

Fuzzy linguistic scale decision matrix.

Secondly, calculate the overall preference triangular fuzzy values of the alternative by utilizing the FLIOWA operator 𝑠𝛼₁=FLIOWA_𝑤𝑠₆,𝑠₆,𝑠₈,𝑠₆,𝑠₈,𝑠₅=𝑠_{(0.57,0.67,0.77)},𝑠𝛼₂=FLIOWA_𝑤𝑠₅,𝑠₇,𝑠₆,𝑠₇,𝑠₆,𝑠₇=𝑠_{(0.55,0.65,0.75)},𝑠𝛼₃=FLIOWA_𝑤𝑠₄,𝑠₄,𝑠₉,𝑠₅,𝑠₇,𝑠₈=𝑠_{(0.56,0.66,0.76)},𝑠𝛼₄=FLIOWA_𝑤𝑠₇,𝑠₆,𝑠₇,𝑠₇,𝑠₄,𝑠₆=𝑠_{(0.54,0.64,0.74)},𝑠𝛼₅=FLIOWA_𝑤𝑠₅,𝑠₆,𝑠₇,𝑠₆,𝑠₅,𝑠₇=𝑠_{(0.52,0.62,0.72)},𝑠𝛼₆=FLIOWA_𝑤𝑠₄,𝑠₆,𝑠₈,𝑠₃,𝑠₆,𝑠₆=𝑠_{(0.49,0.59,0.69)}. (5.1)

Thirdly, calculate the grade mean value of triangular fuzzy number 𝛼_𝑖 as follows: 𝑃𝑎₁=0.67,𝑃𝑎₂=0.65,𝑃𝑎₃𝑃𝑎=0.66,₄=0.64,𝑃𝑎₅=0.62,𝑃𝑎₆=0.58333, (5.2) then we have 𝑎₁>𝑎₃>𝑎₂>𝑎₄>𝑎₅>𝑎₆.

Finally, rank all the alternatives 𝑥_𝑖(𝑖=1,2,3,4,5,6). According to the grade mean value of the linguistic scale and the Definition 2.5, we have 𝑠𝑎₁>𝑠𝑎₃>𝑠𝑎₂>𝑠𝑎₄>𝑠𝑎₅>𝑠𝑎₆, and thus the most desirable alternative is 𝑥₁.

6. Conclusion

The traditional induced aggregation operators are generally suitable for aggregating the information taking the form of numerical values, and yet they will fail in dealing with fuzzy linguistic scale information. In this paper, we develop a new FLIOWA operator and study some desirable properties of the operator, such as commutativity, idempotency, and monotonicity and applied the FIOWA operators to decision making with fuzzy linguistic scale information. The proposed operator can be used to aggregate the fuzzy linguistic scale information in MAGDM problems. It also enriches the existing approaches to aggregating fuzzy linguistic information. Finally an illustrative example has been given to show the developed method.

Acknowledgments

The author express his gratitude to the editor and the anonymous reviewers for their valuable and constructive comments. This work was supported by The Science and Technology Research Program of Chongqing Municipal Educational Committee (Grant no. KJ120515) and National Science Foundation of Chongqing University of Posts and Telecommunications (Grant no. A2010-18).