A fuzzy measure theoretical approach for multi criteria decision making problems containing sub-criteria

Abstract

The main goal of the present paper is to study the general structure and theoretical properties of a particular type of a fuzzy measure that can be used to model multi criteria decision making problems in which there exist some sub criteria. After constructing the general form of the non-additive set function, we deal with the interaction coefficient, Möbius representation and dual measure related to proposed measure. Finally, we are concerned with the usage of this type of fuzzy measures in multi criteria decision making problems in which at least one of the criteria contains some sub-criteria.

Keywords

Fuzzy measure nonadditive measure sub-criteria multicriteria decision making Möbius representation

1. Introduction

Multi Criteria Decision Making (MCDM) is a process that ranks the alternatives or selects the best alternative under the conflicting criteria. Many researchers [9 , 36] focused on the fuzzy studies related to the varied MCDM methods in order to obtain more realistic results since the concepts of decision making under fuzzy environments were proposed by Bellman and Zadeh [1]. Today, we meet numerous fuzzy-based MCDM studies [6 –8]. A part of these studies deals with the fuzzy measure that is used as a tool to identify the weights of the criteria in many MCDM methods.

1.1. Related works

It is known that a fuzzy measure produces more factual results than additive ones. Thus, identification of a particular fuzzy measure is an important issue. In the literature, there are remarkable studies which consider fuzzy measure theory in MCDM problems [10 , 41]. Although a fuzzy measure is such a useful tool for modelling the interaction of criteria in MCDM environment, the process of determining the measures of exponentially growing number of subsets of a finite set is complicated. Many authors studied fuzzy measure identification to obtain particular fuzzy measures [12 , 39]. Besides, some authors used evolutionary or genetic algorithm to relieve this complexity [2 , 38]. There are some general studies as well. For instance, Grabisch [13] introduced the concept of k-order additive fuzzy measures to control the calculation process of the measures of exponentially growing number of subsets.

Many popular MCDM problems such as supplier selection process, personnel selection process, website evaluation process, robot selection process etc. can include sub-criteria. To make clear the concept of sub-criterion, let us consider three main criteria: cost, acceleration, and quality as ranking the alternatives for a car selection problem. Should the cost criterion is divided into two criteria such as the cost of procurement and cost of maintenance, these two criteria are called the sub-criteria of the cost criterion and these structures are constructed to do more sensitive analysis in MCDM processes. Such problems can be evaluated with Analytic Hierarchy Process (AHP) [32] or Analytic Network Process (ANP) [33]. Besides, there are some works that study fuzzy measure theory in MCDM problems with sub-criteria. However, in these studies the authors have constructed known fuzzy measures to solve the problem. For instance, Yang et al. [42] considered λ-fuzzy measures in vendor selection problem that has some sub-criteria by obtaining the weights via AHP. Similarly, λ-fuzzy measures were considered in a supplier evaluation problem with sub-criteria by obtaining the weights via Fuzzy Analytic Hierarchy Process (FAHP) in [15]. Moreover, Takahagi [36] presented fuzzy measure identification methods by using the Ordered Weighted Average (OWA) operator and the λ-fuzzy measure for a hierarchy diagram. Wu and Beliakov [40] proposed the nonadditivity index to quantify the degree and kind of nonadditivity of a capacity, discussed some properties of this index, and presented some tools to help decision makers to determine the nonadditivity index of given subset. Krishnan et. al. [17] proposed an alternate version of λ⁰-measure identification method that synchronously compensates the shortcomings associated with each existing method. A supplier selection problem was used to demonstrate the feasibility of the method. Gong and Lei [14] integrated moving average with non-additive measures with σ - λ rules under fuzzy environment. There is not any direct study evaluating MCDM problems that contains sub-criteria in the realm of fuzzy measure theory so far. The main goal of this study is to construct the general form of a fuzzy measure that models MCDM problems allowing sub-criteria. The structure of the proposed set function is directly obtained from sub-criteria. In this context, this study fills a theoretical gap. Furthermore, many MCDM problems have multiple decision makers. Besides, many of these problems have large scale of decision makers. Moreover, in the large-scale group decision making, decision makers usually express their preferences using heterogeneous preference representation structures [43]. If we face such a problem the proposed measure still can be applicable. Further information about group decision making can be found in [23–25 , 43].

1.2. Theoretical background

In this section we recall some definitions that are used in the present paper. Throughout the paper we use the notation AB instead of A ∪ B for any two sets A and B.

Definition 1. Let X be a finite set. A set function μ : 2^X → [0, 1] is said to be a fuzzy measure over X if

i) μ (∅) =0 and μ (X) =1

ii) μ (A) ≤ μ (B) whenever A ⊂ B ⊂ X (monotonicity)

where 2^X is the family of all subsets of X [11].

Note that the concept of fuzzy measure is a generalization of the concept of additive measures.

The following definition recalls the concept of Möbius representation [4] which is not only a useful tool to investigate the properties of the fuzzy measures but also characterizes them.

Definition 2. Let X be a finite set and let f be a set function over X. The Möbius representation of f is a set function $m : 2^{X} \to ℝ$ which is defined by $m (G) : = \sum_{K \subset G} (- 1)^{| G ∖ K |} f (K) .$

Grabisch [13] relaxed the concept of additivity by introducing the concept of k-order additivity. The motivation behind this new concept of additivity is to calculate the measures of at most k elements subsets to define a fuzzy measure. Now let us recall the definition of this concept:

Definition 3. Let X be a finite set, let μ be a fuzzy measure over X and let 2 ≤ k ≤ |X|. Then μ is said to be k-order additive if m (A) =0 for all A ⊂ X such that |A| > k and there exists at least one subset A of k elements such that m (A) ¬ =0 where m is the Möbius representation of μ [13].

Note here that k-additive fuzzy measures aresomehow between additive measures and general fuzzy measures, i.e., any fuzzy measure is k-order additive for some 1 ≤ k ≤ |X| and the worst scenario is the |X|-additivity of a fuzzy measure.

Another important concept to represent a fuzzy measure is interaction index [13]. Let X be a finite set and let μ be a fuzzy measure over X. The interaction index I (T) of a subset T of X is defined by $I (T) : = \sum_{k = 0}^{n - | T |} ϑ_{k}^{| T |} \underset{| K | = k}{\sum_{K \subset X ∖ T}} \sum_{L \subset T} (- 1)^{| T ∖ L |} μ (LK)$ where $ϑ_{k}^{p} : = \frac{(n - k - p)! k!}{(n - p + 1)!} .$

The remain of the paper is organized as follows: In Section 2, after introducing the fuzzy measure we investigate some properties of this measure. We calculate interaction coefficients, Möbius representation and dual measure related to the proposed measure. Section 3 is devoted for a hypothetical application. Finally, we conclude the paper in Section 4.

2. Main results

In this section we give the general structure and properties of a fuzzy measure that can be used in MCDM problems which contains sub-criteria. First of all, we introduce the theoretical part and then we give the connection of the theory with MCDM problems that allows sub-criteria.

Theorem 1. Consider $X_{j} = {x_{1}^{(j)}, x_{2}^{(j)}, . . ., x_{τ (j)}^{(j)}}$ for j = 1, 2, …, n. Let ${α_{j}}_{j = 1}^{n}$ be a finite sequence of positive real numbers and let μ_j be a fuzzy measure over X_j for each j. Then the set function μ : 2^S → [0, 1] that is defined by

μ (A) = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j} (X_{j} \cap A)

(1)

is a fuzzy measure over $S : = ⋃_{j = 1}^{n} X_{j}$ where $Γ : = \sum_{j = 1}^{n} α_{j}$ .

Proof 1. As μ_j is a fuzzy measure for each j we have $μ (S) = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j} (X_{j}) = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} = 1$ and $μ (\emptyset) = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j} (\emptyset) = 0 .$ On the other hand the monotonicity of μ_j and positivity of α_j for each j imply that $α_{j} μ_{j} (X_{j} \cap A) \leq α_{j} μ_{j} (X_{j} \cap B)$ whenever A ⊂ B ⊂ S. Hence we get μ (A) ≤ μ (B) which proves the monotonicity.

The following example shows that μ given in Theorem is not additive in general.

Example 1.Let X₁ ={ x₁, x₂ } and let X₂ = { x₃, x₄ }. Assume that μ₁ and μ₂ are fuzzy measures over X₁ and X₂, respectively. Consider the set function μ over S = X₁X₂ defined with (1). Now for A = X₁ we have $μ (A) = \frac{1}{Γ} α_{1} μ_{1} (X_{1}) = \frac{α_{1}}{Γ} .$ On the other hand $μ (x_{1}) = \frac{1}{Γ} α_{1} μ_{1} (x_{1})$ and $μ (x_{2}) = \frac{1}{Γ} α_{1} μ_{1} (x_{2}) .$ As μ₁ is a fuzzy measure over X₁ it does not have to be additive. Therefore, μ₁ (X₁) does not have to equal to μ₁ (x₁) + μ₁ (x₂) which implies that μ does not have to be additive.

Remark 1. Note that a fuzzy measure μ defined with (1) is additive with respect to the family ${X_{j}}_{j = 1}^{n}$ , i.e., for any A ⊂ S we can write $μ (A) = \sum_{A \cap X_{j} \neg = \emptyset} μ (A \cap X_{j}) .$ This fact makes μ a k-order additive set function. We explain this property μ after we calculate the Möbius representation.

Remark 2. If μ_j is additive for each j then μ is additive. Indeed, for any disjoint A, B ⊂ S we have $\begin{matrix} μ (A \cup B) & = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j} (X_{j} \cap (AB)) \\ = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j} ((X_{j} \cap A) (X_{j} \cap B)) \\ = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} [μ_{j} (X_{j} \cap A) + μ_{j} (X_{j} \cap B)] \\ = μ (A) + μ (B) . \end{matrix}$

Remark 3. AHP is an additive example of a fuzzy measure defined in (1). This is clear from Remark. Note that this factor is the motivation of this study.

The following theorem gives the Möbius representation of a fuzzy measure μ given in Theorem with (1).

Theorem 2. The set function m over 2^S which is defined by

m (T) = {\begin{matrix} [c] cc \frac{1}{Γ} α_{j} m_{j} (T) & , if T \subset X_{j} \\ 0 & , otherwise \end{matrix}

is the Möbius representation of the fuzzy measure μ given in Theorem where m_j is the Möbius representation of μ_j for each j.

Proof 2. Assume that T ⊂ X_{j
₀}. Then from the definition of Möbius representation we have $\begin{matrix} m (T) & = \sum_{K \subset T} (- 1)^{| T ∖ K |} μ (K) \\ = \frac{1}{Γ} \sum_{K \subset T} (- 1)^{| T ∖ K |} \sum_{j = 1}^{n} α_{j} μ_{j} (X_{j} \cap K) \\ = \frac{α_{j_{0}}}{Γ} \sum_{K \subset T} (- 1)^{| T ∖ K |} μ_{j_{0}} (K) \\ = \frac{α_{j_{0}}}{Γ} m_{j_{0}} (T) . \end{matrix}$ Now if T ⊄ X_j for all j then we have $\begin{matrix} m (T) & = \frac{1}{Γ} \sum_{K \subset T} (- 1)^{| T ∖ K |} μ (K) \\ = \frac{1}{Γ} \underset{T \cap X_{j} \neg = \emptyset}{T ⊄ X_{j} \sum} α_{j} (\sum_{K \subset T \cap X_{j}} (- 1)^{| T ∖ K |} μ_{j} (K) \\ + \underset{K ⊄ X_{j}}{K \cap X_{j} \neg = \emptyset \sum_{K \subset T}} (- 1)^{| T ∖ K |} μ_{j} (X_{j} \cap K)) . \end{matrix}$ . Thus one can write $\begin{matrix} m (T) & = \frac{1}{Γ} \underset{T \cap X_{j} \neg = \emptyset}{T ⊄ X_{j} \sum} α_{j} ((- 1)^{| T ∖ X_{j} |} m_{j} (T \cap X_{j}) \\ \begin{matrix} [c] c + \sum_{L \subset X_{j} \cap T} \sum_{i = 1}^{| T ∖ X_{j} |} {(- 1)^{| (T \cap X_{j}) ∖ L |} \\ \times (- 1)^{| T ∖ X_{j} | - i} | T ∖ X_{j} | i μ_{j} (L)} \end{matrix}) \\ = \frac{1}{Γ} \underset{T \cap X_{j} \neg = \emptyset}{T ⊄ X_{j} \sum} α_{j} m_{j} (T \cap X_{j}) [(- 1)^{| T ∖ X_{j} |} \\ + \sum_{i = 1}^{| T ∖ X_{j} |} (- 1)^{| T ∖ X_{j} | - i} | T ∖ X_{j} | i] . \end{matrix}$ (2) On the other hand for any j we have $\begin{matrix} \sum_{i = 1}^{| T ∖ X_{j} |} (- 1)^{| T ∖ X_{j} | - i} | T ∖ X_{j} | i \\ = (- 1)^{| T ∖ X_{j} |} \sum_{i = 1}^{| T ∖ X_{j} |} | T ∖ X_{j} | i (- 1)^{i} \\ = - (- 1)^{| T ∖ X_{j} |} m_{j} (T \cap X_{j}) . \end{matrix}$ which imply with from (2) that m (T) =0.

Note that the Möbius representation of a set which is not a subset of one of the X_j’s vanishes. This is not an unexpected result due to the Remark 1. Now we can say that a fuzzy measure defined with (1) is a k-order additive fuzzy measure for $k = max_{j} τ_{j}$ .

It is possible to express the Shapley value of an element of S and interaction index of a subset of S with respect to μ in the terms of the Shapley values of an element of S and interaction indices of a subset of S, respectively, with respect to μ_j for some j. For this purpose we recall the following result of Grabisch [13]:

Theorem 3. Let X be finite set, let m be the Möbius representation and let I be the interaction representation of a fuzzy measure over X then we have for any T ⊂ X that

I (T) = \sum_{k = 0}^{| X ∖ T |} \frac{1}{k + 1} \underset{| K | = k}{\sum_{K \subset X ∖ T}} m (TK)

and

m (T) = \sum_{k = 0}^{| X ∖ T |} ρ_{k} \underset{| K | = k}{\sum_{K \subset X ∖ T}} I (TK)

where (ρ_k) is iterated sequence defined with

ρ_{k} = - \sum_{l = 0}^{k - 1} \frac{α_{l}}{k - l + 1} k l

and α₀ = 1.

Theorem 4. Let S, μ_j, α_j be as in Theorem 1. Then we have

\begin{matrix} I (T) & = \frac{1}{Γ} \sum_{k = 0}^{| S ∖ T |} \frac{1}{k + 1} \underset{| K | = k}{\sum_{K \subset X_{j_{0}} ∖ T}} α_{j} \\ \sum_{r = 0}^{| X_{j_{0}} ∖ (KT) |} ρ_{r} \underset{L = r}{\sum_{L \subset X_{j_{0}} ∖ (KT)}} I_{j_{0}} (KTL) \end{matrix}

if T ⊂ X_{j
₀} and zero otherwise where I is the interaction representation of μ and I_{j
₀} is the interaction representation of μ_{j
₀}.

Proof 3. If T ⊄ X_j for all j then from Theorem one can have I (T) =0. If there exists j₀ such that T ⊂ X_{j
₀} then we can write from Theorem 3 that $\begin{matrix} I (T) & = \sum_{k = 0}^{| S ∖ T |} \frac{1}{k + 1} \underset{| K | = k}{\sum_{K \subset S ∖ T}} m (TK) \\ = \frac{1}{Γ} \sum_{k = 0}^{| S ∖ T |} \frac{1}{k + 1} \underset{| K | = k}{\sum_{K \subset X_{j_{0}} ∖ T}} α_{j_{0}} m_{j_{0}} (TK) . \end{matrix}$ (3) On the other hand, again from Theorem 3 we have for any K ⊂ X_{j
₀} ∖ T that $m_{j_{0}} (TK) = \sum_{r = 0}^{| X_{j_{0}} ∖ (KT) |} ρ_{r} \underset{L = r}{\sum_{L \subset X_{j_{0}} ∖ (KT)}} I_{j_{0}} (KTL) .$ (4) Thus from (3) and (4) proof is concluded.

Let X be a finite set and let μ be a fuzzy measure on X. Then the set function μ^∗ defined with $μ^{*} (A) = 1 - μ (A^{c})$ is said to be the dual of μ. It is easy to see that μ^∗ is again a fuzzy measure on X. The concept of dual fuzzy measure has an important role in the classification of fuzzy measures (see e.g., [19]). Now we are concerned with the representation of dual of fuzzy measure μ given in Theorem 1.

Proposition 1. Let S, μ_j, α_j and μ be as in Theorem 1. Then we have for any A ⊂ S that

μ^{*} (A) = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j}^{*} (X_{j} \cap A) .

Proof 4. From the definition of dual fuzzy measure we get for every A ⊂ S that $\begin{matrix} μ^{*} (A) & = 1 - μ (A^{c}) \\ = \frac{1}{Γ} (\sum_{j = 1}^{n} α_{j} - \sum_{j = 1}^{n} α_{j} μ_{j} (X_{j} \cap A)) \\ = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} (1 - μ_{j} (X_{j} \cap A)) \\ = \frac{1}{Γ} \sum_{j = 1}^{n} α_{j} μ_{j}^{*} (X_{j} \cap A) . \end{matrix}$ Hence the proof is completed.

3. An application

In this section, we give a hypothetical application in order to show the applicability of the proposed fuzzy measure on MCDM problems. Let us consider a personnel selection problem consisting of three main evaluation criteria and some sub-criteria: communication skills (x₁) with sub-criteria (x₁₁, x₁₂); professional experience (x₂); educational background (x₃) with sub-criteria (x₃₁, x₃₂) (see Fig. 1). In the problem, the objective is to determine the best appropriate candidate based on these benefit criteria. For this purpose we deal with establishing a fuzzy measure on the set of criteria taking into account the hierarchy given in Fig. 1.

Fig. 1

Hierarchy of considered criteria for a personnel selection problem.

For this MCDM problem, we can use a fuzzy measure that is given as (1). First of all we need a sequence ${(α_{j})}_{j = 1}^{3}$ and a fuzzy measure over the set of sub-criteria of each main criterion. Let α_j be the weight of criteria x_j for each j = 1, 2, 3. These weights can be evaluated with some previous methods such as AHP, ANP, FAHP, any known fuzzy measure identification method etc. For example we can use a known fuzzy measure identification over {x₁, x₂, x₃} to get the weights by considering the measures of singletons as weights. Note that the fuzzy measure is identified just for singletons. Assume that α₁ = 0.3, α₂ = 0.5 and α₃ = 0.4 are obtained. We can construct fuzzy measures over sets of sub-criteria. Consider the set functions μ_j : 2^{X
_j} → [0, 1] given in Table 1 where X₁ = { x₁₁, x₁₂ }, X₂ ={ x₂ } and X₃ ={ x₃₁, x₃₂ }. It is obvious that the set functions μ₁, μ₂ and μ₃ are fuzzy measures over X₁, X₂ and X₃, respectively. These fuzzy measures can be identified by a fuzzy measure identification method that is already existing in the literature. Now if use to construct a fuzzy measure over $S : = ⋃_{j = 1}^{3} X_{j}$ we obtain the set function μ given in Table 1.

Table 1

Fuzzy measures μ₁, μ₂, μ₃ and μ t1

μ₁ (∅) =0	μ₁ (X₁) =1	μ₁ ({ x₁₁ }) =0.6
μ₁ ({ x₁₂ }) =0.3	μ₂ (∅) =0	μ₂ (X₂) =1
μ₃ (∅) =0	μ₃ (X₃) =1	μ₃ ({ x₃₁ }) =0.5
μ₃ ({ x₃₂ }) =0.7	μ (∅) =0	μ (S) =1
μ ({ x₁₁ }) =0.15	μ ({ x₁₂ }) =0.075	μ ({ x₂ }) =0.416
μ ({ x₃₁ }) =0.166	μ ({ x₃₂ }) =0.233	μ ({ x₁₁, x₁₂ }) =0.25
μ ({ x₁₁, x₂ }) =0.566	μ ({ x₁₁, x₃₁ }) =0.316	μ ({ x₁₁, x₃₂ }) =0.383
μ ({ x₁₂, x₂ }) =0.491	μ ({ x₁₂, x₃₁ }) =0.241	μ ({ x₁₂, x₃₂ }) =0.308
μ ({ x₂, x₃₁ }) =0.582	μ ({ x₂, x₃₂ }) =0.649	μ ({ x₃₁, x₃₂ }) =0.333
μ ({ x₁₁, x₁₂, x₂ }) =0.666	μ ({ x₁₁, x₁₂, x₃₁ }) =0.416	μ ({ x₁₁, x₁₂, x₃₂ }) =0.483
μ ({ x₁₂, x₂, x₃₁ }) =0.657	μ ({ x₁₂, x₂, x₃₂ }) =0.724	μ ({ x₂, x₃₁, x₃₂ }) =0.746
μ ({ x₁₁, x₂, x₃₁ }) =0.732	μ ({ x₁₁, x₂, x₃₂ }) =0.799	μ ({ x₁₁, x₃₁, x₃₂ }) =0.483
μ ({ x₁₂, x₃₁, x₃₂ }) =0.408	μ ({ x₁₁, x₁₂, x₂, x₃₁ }) =0.832	μ ({ x₁₁, x₁₂, x₂, x₃₂ }) =0.899
μ ({ x₁₁, x₂, x₃₁, x₃₂ }) =0.899	μ ({ x₁₁, x₁₂, x₃₁, x₃₂ }) =0.583	μ ({ x₁₂, x₂, x₃₁, x₃₂ }) =0.824

Clearly, the function μ is a fuzzy measure over S and it is not additive. However, it is 2-order additive, i.e., we can obtain the measure of a set that has at least three elements by adding up the measures of some smaller sets. Hence, it is enough to calculate the measures of pairs to determine the fuzzy measure. For example we see that μ ({ x₁₁, x₁₂, x₂, x₃₂ }) = μ ({ x₁₁, x₁₂ }) + μ ({ x₂ }) + μ ({ x₃₂ }). Note that defining the fuzzy measure μ we only calculate the measures of dbinom50 + dbinom51 + dbinom52 - 2 number of subsets.

4. Conclusion

This study introduces a new approach to use fuzzy measure theory in MCDM problems considering sub-criteria. Despite the previous works deal with fuzzy measure theory in MCDM problems containing sub-criteria, we give the general structure of a fuzzy measure which is not introduced before. First of all, we are concerned with a particular type of fuzzy measure that can be directly used in MCDM problems that contains sub-criteria. Then we calculate the interaction index, Möbius representation and dual fuzzy measure. We support our results with a numerical example.

Footnotes

Acknowledgement

We are thankful to the referees for their careful reading of the paper and several valuable suggestions which improved the presentation of the paper.

References

R.E.

Bellman and

L.A.

Zadeh , Decision-making in a fuzzy environment, Management Science 17(4) (1970), 141–164.

T.Y.

Chen ,

J.C.

Wang and

G.H.

Tzeng , Identification of general fuzzy measures by genetic algorithms based on partial information, IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics) 30 (2000), 517–528.

T.Y.

Chen and

J.C.

Wang , Identification of k-fuzzy measures using sampling design and genetic algorithms, Fuzzy Sets and Systems 123(3) (2001), 321–341.

Chateauneuf and

J.Y.

Jaffray , Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion, Mathematical Social Sciences 17 (1989), 263–283.

E.F.

Combarro and

Miranda , Identification of fuzzy measures from sample data with genetic algorithms, Computers and Operations Research 33 (2006), 3046–3066.

Garg and

Kumar , Distance measures for connection number sets based on set pair analysis and its applications to decision-making process, Applied Intelligence (2018), 1–14.

Garg , Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision-making process, International Journal of Intelligent Systems 33(6) (2018), 1234–1263.

Garg and

Arora , Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of the Operational Research Society (2018), 1–14.

Garg , Generalized intuitionistic fuzzy entropy-based approach for solving multi-attribute decision-making problems with unknown attribute weights, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2017, pp. 1–11.

10.

Grabisch , Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69(3) (1995), 279–298.

11.

Grabisch , The application of fuzzy integrals in multi criteria decision making, European Journal of Operational Research 89(3) (1996), 445–456.

12.

Grabisch , A new algorithm for identifying fuzzy measures and its application to pattern recognation, Int Joint Conf Of the 4th IEEE International Conference Fuzzy Systems and the 2nd International Fuzzy Engineering Symposium, Yokohama, Japan, 1995, pp. 145–150.

13.

Grabisch , k-Order additive discrete fuzzy measures and their representation, Fuzzy Sets and Systems 92(2) (1997), 167–189.

14.

Z.T.

Gong and

W.J.

Lei , The weighted moving averages for a series of fuzzy numbers based on non-additive measures with a – k rules, Journal of Computational Analysis Applications 27 (1) (2019), 882–891.

15.

Hamidi and

Samouei , A non-additive fuzzy hybrid model for supplier evaluation and prioritization: A case study of automotive brake system manufacturer, Journal of Industrial Strategic Management 1 (1) (2016), 1–18.

16.

Ishii and

Sugeno , A model of human evaluation process using fuzzy measure, International Journal of Man-Machine Studies 22 (1) (1985), 19–38.

17.

Krishnan ,

Wahab ,

Kasim and

Bakar , An alternate method to determine k0-measure values prior to applying Choquet integral in a multi-attribute decision making environment, Decision Science Letters 8 (2) (2019), 193–210.

18.

Y.C.

Kuo ,

S.T.

Lu ,

G.H.

Tzeng ,

Y.C.

Lin and

Y.S.

Huang , Using fuzzy integral approach to enhance site selection assessment-A Case Study of the Optoelectronics Industry, First International Conference on Information Technology and Quantitative Management, Procedia Computer Science, 17, 2013, pp. 306–313.

19.

M.T.

Lamata and

Moral , Classification of fuzzy measures, Fuzzy Sets and Systems 33 (2) (1989), 243–253.

20.

Larbani ,

Huang and

Tzeng , A novel method for fuzzy measure identification, International Journal of Fuzzy Systems 13(1) (2011), 24–34.

21.

K.M.L.

Lee and

Leekwang , Identification of k-fuzzy measure by genetic algorithms, Fuzzy Sets and Systems 75 (1995), 301–309.

22.

Liginlal and

Ow , On policy capturing with fuzzy measures, European Journal of Operational Research 2005 (167), 461–474.

23.

Liu , Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators, Computers Industrial Engineering 108 (2017), 199–212.

24.

Liu and

S.M.

Chen , Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers, IEEE Transactions on Cybernetics 47 (9) (2017), 2514–2530.

25.

Liu and

S.M.

Chen , Multiple attribute group decision making based on intuitionistic fuzzy interaction partitioned Bonferroni mean operators, Information Sciences 411 (2017), 98–121.

26.

Mikenina and

H.J.

Zimmermann , Improved feature selection and classification by the 2-additive fuzzy measure, Fuzzy Sets and Systems 107 (1999), 195–218.

27.

Miranda and

Grabisch , Optimization issues for fuzzy measures, International Journal of Uncertainty, Fuzziness, and Knowledge Based Systems 7 (6) (1999), 545–560.

28.

Miranda ,

E.F.

Combarro and

Gil , Extreme points of some families of non-additive measures, European Journal of Operational Research 174 (2006), 1865–1884.

29.

Murillo ,

Guillaume ,

Tapia and

Bulacio , Revised HLMS: A useful algorithm for fuzzy measure identification, Information Fusion 14 (2013), 532–540.

30.

Ozcelik ,

Unver and

Temel Gencer , Evaluation of the global warming impacts using a hybrid method based on fuzzy techniques: A case study in Turkey, Gazi Journal of Science 29 (4) (2016), 883–894.

31.

I.J.

Perez ,

F.J.

Cabrerizo ,

Alonso and

Herrera-Viedma , A new consensus model for group decision making problems with non homogeneous experts, IEEE Transactions on Systems, Man, and Cybernetics: Systems 44 (4) (2014), 494–498.

32.

T.L.

Saaty , The analytic hierarchy process, McGraw-Hill, New York, 1980.

33.

T.L.

Saaty , Decision making with dependence and feedback: The analytic network process. Pittsburgh, RWS Publications (revised), 2001.

34.

Sekita , A consideration of identifying fuzzy measures, Osaka University Economy 25 (4) (1976), 133–138.

35.

D.Y.

Shee ,

G.H.

Tzeng and

T.I.

Tang , AHP, Fuzzy measure and fuzzy integral approaches for the appraisal of information service providers in Taiwan, Journal of Global Information Technology Management 6(1) (2003), 8–30.

36.

Takahagi , Fuzzy measure identification methods for a multilevel hierarchy diagram by assignment functions, IEEE In 2017 Joint 17th World Congress of International Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (IFSA-SCIS), 2017, pp. 1–6.

37.

Torra and

Narukawa , Fuzzy measures and integrals in evaluation of strategies, Information Sciences 177 (21) (2007), 4686–4695.

38.

J.C.

Wang and

T.Y.

Chen , Experimental analysis of k-fuzzy measure identification by evolutionary algorithms, International Journal of Fuzzy Systems 7(1) (2005), 1–10.

39.

S.T.

Wierzchon , An algorithm for identification of fuzzy measure, Fuzzy Sets and Systems 9 (1983), 69–78.

40.

J.Z.

Wu and

Beliakov , Nonadditivity index and capacity identification method in the context of multicriteria decision making, Information Sciences 467 (2018), 398–406.

41.

R.R.

Yager , Modeling multi-criteria objective functions using fuzzy measures, Information Fusion 2016 29, 105–111.

42.

J.L.

Yang ,

H.N.

Chiu ,

G.H.

Tzeng and

R.H.

Yeh , Vendor selection by integrated fuzzy MCDM techniques with independent and interdependent relationships, Information Sciences 178 (21) (2008), 4166–4183.

43.

Zhang ,

Dong and

Herrera-Viedma , Consensus building for the heterogeneous large-scale GDM with the individual concerns and satisfactions, IEEE Trans On Fuzzy Systems 26(2) (2018), 884–898.