Multifactorial decision making based on type-2 fuzzy sets and factor space approach

Abstract

In this paper, a new method for the representation of concept in factor space by using type-2 fuzzy sets, and the composition method of states of factors is developed. A general model of multifactorial decision making based on type-2 fuzzy sets is formulated, and the comprehensive evaluation problem based on type-2 fuzzy sets is analyzed. Finally, two applied instances are given to demonstrate the effectiveness of the proposed methods.

Keywords

Factor space type-2 fuzzy sets multifactorial decision making comprehensive evaluation

1 Introduction

The original concept of “factor space” was proposed by Wang Pei-Zhuang in 1981, which was used to explain the source of randomness and the essence of probability laws [1]. And in [2], an axiomatic definition of factor space was given. The introducing of factor space aims at deepening and broadening the depth and breadth of fuzzy set theory in concept describing, judging, reasoning and decision making and so on, which provides a mathematical framework to solve the structure and selection problems of domain and variable [1]. Variants of factor space have found many applications in areas such as artificial intelligence [3] and fuzzy information processing [1, 4] etc.

From factor space point of view, multiple objective decision making (MOD) problems can be classified into multiple criteria decision making (MCD) and multiple attribute decision making (MAD), both of which are referred to as multifactorial decision making problems. A general model of multifactorial decision making was presented by Li et al. in [4], which is widely used in practical problems. Also with the purpose of providing a synthetic evaluation of an object relative to an objective in a fuzzy decision environment with many factors, the theoretical foundation of CE based on type-1 fuzzy sets was presented [4].

However, the decision making results of many phenomena in daily life are often not absolutely positive or negative, the relationship between the variables and the object is always difficult to be predicted by precise mathematics, so it is more reasonable to use fuzzy judgments in decision than crisp comparisons. With in-depth and expansion of researches on the fuzzy comprehensive evaluation (FCE) theory, the FCE method has rapidly penetrated into and widely applied to many fields of engineering, economy and finance, transportation, social, meteorology, etc. [5, 6]. Numerous practical problems have been successfully solved by using various FCE methods. Among others, Liu et al. [6] presented a FCE method to assess motion performance of autonomous underwater vehicles. Li et al. [7] integrated weighted average and FCE method to measure the performance evaluation of a company’s achievement in health, safety and environment management. Jamshidi et al. [8] proposed a new comprehensive risk-based prioritization framework for selecting the best maintenance strategy of medical devices. Wei et al. [9] imported trustworthy degree to FCE and proposed an automatic hotel service quality assessment method using online comments. Chen et al. [10] presented a novel framework for teaching performance evaluation based on the combination of fuzzy Analytic Hierarchy Process (AHP) and FCE method. It is noted that in all these contents the fuzzy comprehensive evaluation index system is based on type-1 fuzzy sets.

Type-2 fuzzy sets, which were initially proposed by Zadeh [12], can be regarded as an extension of the concept of type-1 fuzzy sets. The main difference between the two kinds of fuzzy sets lies in that the membership functions of type-1 fuzzy sets are crisp sets whereas the membership functions of type-2 fuzzy sets are fuzzy sets [13]. Membership functions of type-1 fuzzy sets are two-dimensional, whereas membership functions of type-2 fuzzy sets are three-dimensional. It is the new third-dimension of type-2 fuzzy sets that provides additional degrees of freedom that make it possible to directly model uncertainties [6]. Therefore, type-2 fuzzy sets are more suitable to present uncertainties than type-1 fuzzy sets, and the study of type-2 fuzzy sets achieves more and more attentions [14 –20]. Among others, Zhou et al. [14] proposed rule-reduction methods for constructing parsimonious type-2 fuzzy systems, where four novel indexes were used to rank the relative contribution of type-2 fuzzy rules. And followed in [15] they further presented a new type of type-2 ordered weighted averaging (OWA) operator to aggregate the linguistic opinions or preferences in decision making modeled by type-2 fuzzy sets. Sun et al. [16] proposed a new approach by formulating route evaluation based on type-2 fuzzy sets, where type-2 fuzzy sets are introduced to represent linguistic values, manage linguistic uncertainty effectively and make the evaluation process realistic and reliable. Torshizi et al. [17] developed a new method of type-2 fuzzy c-means, where the type-2 fuzzy cluster validity index is used for finding the optimal number of clusters. Ganjefar et al. [18] presented a new method of controller design based on a type-2 fuzzy wavelet neural network structure, where the system is constructed on the basis of type-2 membership functions to handle uncertainties associated with information and data in the knowledge base. Rena et al. [19] presented a nouvelle cutting tool wear assessment in high precision turning process based on using type-2 fuzzy uncertainty estimation in acoustic emission. Kilic et al. [20] given a decision making methodology based on type-2 fuzzy sets for investment project evaluation, where the type-2 fuzzy AHP and the type-2 fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) methods were used in the proposed model. Due to the intensive computation of general type-2 fuzzy sets, interval type-2 fuzzy sets have been successfully applied in MOD process, and a series of decision making methods based on interval type-2 fuzzy sets are established [21 –25].

To the best of our knowledge, the decision making methods based on type-2 fuzzy sets in the literature mostly aimed at a particular problem or under specific conditions, and little work has been done on general model of multifactorial decision making based on type-2 fuzzy sets. It is also worth noting that the research of CE based on type-2 fuzzy sets has not been seen until recently. Therefore, this work aims at presenting a general model of multifactorial decision making and providing theoretical analysis with applications of CE in the framework of type-2 fuzzy set.

This paper is organized as follows. In Section 2 the axiomatic definitions of factor spaces and fuzzy sets are reviewed. In Section 3 a new representation of concept description based on type-2 fuzzy sets is presented and the composition method of states of factors is given. In Section 4 a general model of multifactorial decision making based on type-2 fuzzy sets is formulated and the theoretical analysis of comprehensive evaluation based on type-2 fuzzy sets is given, and two examples are proposed to illustrate the application of the proposed method. At last, the conclusions are presented in Section 5.

2 Preliminaries

This section reviews some fundamental notions of factor space theory provided by Li et al. [1] and fuzzy sets defined by Zadeh [11, 12].

2.1 Axiomatic definition of factor spaces

Definition 1. For a given left-matched pair (U, V], let U be universe and V be a set of factors, and let F = {f ∣ f : U → X (f)} be a family of mappings on U, and F ⊂ V. The family {X (f)} _(f∈F) is called a factor space on U if it satisfies the following axioms:

F = F (∨ , ∧ , c, 1, 0) is a complete Boolean algebra;

X (0) = {θ};

If ∀T ⊂ F, factors T independent, then $⋁_{f \in T} f = \prod_{f \in T} f$ , where $\prod_{f \in T} f$ can beviewed as the direct product of mappings (That is to say, a factor f can be regarded as a mapping.).

We call F the set of factors, f ∈ F a factor, X (f) the state space of f, 1 the complete factor, and X (1) the complete space, respectively.

From the mathematics point of view, factors can be considered as mappings from the universe U to their state spaces X (f).

Definition 2. Let C = {α, β, γ, …} be a group of concepts under the common universe U, and V be a family of factors such that U and V form a left-matched pair (U, V]. Also, let F be a set of factors of V such that F is sufficient on universe U, i.e., $(\forall u_{1}, u_{2} \in U) (\exists f \in F) (f (u_{1}) \neq f (u_{2})) .$

Then, the triple (U, C, F] or (U, C, {X (f)} _(f∈F)] is called a description frame of C.

Note that, for a given description frame (U, C, F], the complete factor 1 must be an injection [1]. For any u₁, u₂ ∈ U, it follows from the definition of “sufficiency” in the description frame that there exists a factor f ∈ F such that f (u₁) ≠ f (u₂).

2.2 Fuzzy sets

Definition 3. A type-1 fuzzy set A on universe U is a fuzzy set whose membership grades are crisp numbers, i.e., μ_A (u)∈ [0,1], and is expressed as follows: $A = {(u, μ_{A} (u)) | μ_{A} (u) \in [0, 1], \forall u \in U} .$

All type-1 fuzzy sets on universe U be written as $F (U)$ .

Definition 4. A type-2 fuzzy set $\tilde{A}$ on universe U is a fuzzy set whose membership grades are themselves fuzzy, i.e., $μ_{\tilde{A}} (u)$ is a type-1 fuzzy set on [0,1] for all u, i.e., $\tilde{A} = {(u, μ_{\tilde{A}} (u)) | μ_{\tilde{A}} (u) \in F ([0, 1]), \forall u \in U} .$ where, $F ([0, 1])$ is the set of fuzzy sets on [0,1]. All type-2 fuzzy sets on universe U be written as $\tilde{F} (U)$ .

It can be seen that the membership degree of an element u with respect to a type-1 fuzzy set A on common universe U is a crisp number μ_A (u) in [0,1], whereas the membership degree of an element u with respect to a type-2 fuzzy set $\tilde{A}$ on common universe U is a type-1 fuzzy set $μ_{\tilde{A}} (u)$ on [0,1], namely $μ_{\tilde{A}} (u) \in F ([0, 1])$ . Obviously, type-2 fuzzy set is an extension of the type-1 fuzzy set, and type-2 fuzzy set is degraded into type-1 fuzzy set if the fuzzy membership degree becomes clear membership degree.

3 Representation of concept based on type-2 fuzzy sets and composition of states of factor

3.1 The representation of concept based on type-2 fuzzy sets

Given a description frame (U, C, F], for any α ∈ C represented by type-2 fuzzy set, the extension of the concept α is a type-2 fuzzy set $\tilde{A} \in \tilde{F} (U)$ on U with membership function expressed by: $\begin{matrix} μ_{\tilde{A}} : & U \to F ([0, 1]) \\ u \mapsto μ_{\tilde{A}} (u) \end{matrix}$ where, $μ_{\tilde{A}} (u)$ is called the membership degree of u with respect to α or $\tilde{A}$ . Then, for the description frame (U, C, F], X (f) is the representation universe of C with respect to f, X (1) is called the complete representation universe. For any factor f ∈ F, according to Zadeh’s extension principle, f can be extended as follows: $\begin{matrix} f : & F (U) \to F (X (f)) \\ \tilde{A} \mapsto f (\tilde{A}) \\ μ_{\tilde{A}} : & X (f) \to F ([0, 1]) \\ x \mapsto μ_{\tilde{A}} (x) ≜ ⋁_{f (u) = x} μ_{\tilde{A}} (u) \end{matrix}$ where $f (\tilde{A})$ is called the representation extension of concept α based on type-2 fuzzy sets in the representation universe X (f).

This means that a concept or its extension on U can be transformed to the representation universe X (f) and represented by its states of factor f on the representation universe X (f). As indicated by Li et al. in [4], a concept can be decomposed and modeled by {X (f)} _(f∈F) and the “decomposed” concept can be always represented.

3.2 The composition of states of factor

In a given description frame (U, C, {X (f)} _(f∈F)], for any f ∈ F, if the factor f is quite complex, then the state space X (f) of f will be difficult to be determine. When a complex factor is decomposed into a group of simple and independent factors, then the state space of the complex factor can be determined by such state spaces of simple factors. Let {f_t} _(t∈T) ⊂ F be a family of mutually independent factors and define f = ∨ _t∈Tf_t. Then the composition of states of factors used in [4] is adapted, i.e. $\begin{matrix} X (f) & = \prod_{t \in T} X (f_{t}) \\ = {ϖ : T \to ⋃_{t \in T} X (f_{t}) ∣ (\forall t \in T) \\ (ϖ (t) \in X (f_{t}))} . \end{matrix}$

In practice, the index set T is most likely a finite set, and we assume that T = {1, 2, …, m}. Then $f = ⋁_{j = 1}^{m} f_{j}$ , and $\begin{matrix} X (f) \\ = \prod_{j = 1}^{m} X (f_{j}) \\ = {(x_{1}, x_{2}, \dots, x_{m}) ∣ x_{j} \in X (f_{j}), 1 \leq j \leq m} . \end{matrix}$

Thus, for any u ∈ U, if f_j (u) (j = 1, 2, …, m) is known, one have $f (u) = (f_{1} (u), f_{2} (u), \dots, f_{m} (u)) .$ (1)

In order to transform Equation (1) into an easier operational form without a loss of its informational content, the “dimensionality reduction” method is developed, i.e. a low-dimensional state space, say X′ (f), is used to approximate the high-dimensional state space $X (f) = \prod_{j = 1}^{m} X (f_{j})$ , as illustrated in Fig. 1.

The effect of the mapping M_m is to synthesize an m-dimensional vector into a one-dimensional scalar, functionally represented by $M_{m} (f (u)) = M_{m} (f_{1} (u), f_{2} (u), \dots, f_{m} (u)) .$ where, M_m is been called multifactorial function.

When a type-2 fuzzy set is used to represent a concept α in description frame (U, C, F], for every u ∈ U, the membership degree of u with respect to $\tilde{A}$ is a type-1 fuzzy set on [0,1]. Accordingly, for representation universe X (f), the representation extension membership degree of every factor state is also a type-1 fuzzy set on [0,1]. In order to synthesize representation extension membership, the multifactorial functions are defined as follows:

$\begin{matrix} (1) M_{m} : [F ([0, 1])]^{m} \to F ([0, 1]) \\ (A_{1}, \dots, A_{m}) \mapsto M_{m} (A_{1}, \dots, A_{m}) ≜ ⋂_{j = 1}^{m} A_{j} . \\ (2) M_{m} : [F ([0, 1])]^{m} \to F ([0, 1]) \\ (A_{1}, \dots, A_{m}) \mapsto M_{m} (A_{1}, \dots, A_{m}) ≜ ⋃_{j = 1}^{m} A_{j} . \end{matrix}$

For w_j ∈ [0, 1] with $\sum_{j = 1}^{m} w_{j} = 1$ , we define

$\begin{matrix} (3) M_{m} : [F ([0, 1])]^{m} \to F ([0, 1]) \\ (A_{1}, \dots, A_{m}) \mapsto M_{m} (A_{1}, \dots, A_{m}) ≜ \sum_{j = 1}^{m} (w_{j} A_{j}), \\ μ_{\sum_{j = 1}^{m} (w_{j} A_{j})} (u) = \sum_{j = 1}^{m} (w_{j} μ_{A_{j}} (u)), u \in [0, 1] . \\ (4) M_{m} : [F ([0, 1])]^{m} \to F ([0, 1]) \\ (A_{1}, \dots, A_{m}) \mapsto M_{m} (A_{1}, \dots, A_{m}) ≜ ⋃_{j = 1}^{m} (w_{j} A_{j}), \\ μ_{⋃_{j = 1}^{m} (w_{j} A_{j})} (u) = ⋁_{j = 1}^{m} (w_{j} μ_{A_{j}} (u)), u \in [0, 1] . \\ (5) M_{m} : [F ([0, 1])]^{m} \to F ([0, 1]) \\ (A_{1}, \dots, A_{m}) \mapsto M_{m} (A_{1}, \dots, A_{m}) ≜ ⋃_{j = 1}^{m} (w_{j} \land A_{j}), \\ μ_{⋃_{j = 1}^{m} (w_{j} \land A_{j})} (u) = ⋁_{j = 1}^{m} (w_{j} \land μ_{A_{j}} (u)), u \in [0, 1] . \end{matrix}$

Note. For a complex problem, especially which contains a concept of uncertainty, the more accurate description of the problem and suitable selected model, the better in solving the problem. In multifactorial decision making problems, the concept of the objectives or standards usually contain uncertainties. When type-1 fuzzy sets are used to represent the concept of uncertainty, the decision results are crisp numbers. While in case of type-2 fuzzy sets, the decision results will be represented by type-1 fuzzy sets. In comparison, the decision results represented by type-1 fuzzy sets are more reasonable than that by crisp numbers.

4 Multifactorial decision making problems based on type-2 fuzzy sets

4.1 A general model of multifactorial decision making based on type-2 fuzzy sets

Let U be a set of decision alternatives and f₁, f₂, …, f_m be mutually independent basic factors of U with X (f_j) (j = 1, 2, …, m) denoting the state space. Let’s assume the family of atomic factors π ≜ {f₁, f₂, …, f_m}, and define $F ≜ P (π)$ , ∨ =∪, ∧ =∩, θ = φ, 1 = π, and - =\. Then (F, ∨ , ∧ , c, 1, 0) is a complete Boolean algebra. For any factor f ∈ F, there must exist a family of factors {f_jk} _(1≤k≤r) ⊂ π, where 1 ≤ r ≤ m and $f = ⋁_{k = 1}^{r} f_{jk}$ . Let $X (f) = \prod_{j = 1}^{r} X (f_{jk})$ . Then {X (f)} _(f∈F) forms a factor space. Each strategy u ∈ U, u can be completely determined by the state 1 (u) = (f₁ (u) , f₂ (u) , …, f_m (u)), where 1 (u) is called a decision variable. The state space (complete), say $X (1) = \prod_{j = 1}^{m} X (f_{j})$ is called the space of decision variables. Every factor f ∈ F is called an objective, and 1 is called the complete objective. In general, suppose we have p complete objective statements or sentences: e₁, e₂, …, e_p, where e_k (k = 1, 2, …, p) is the k - th objective expressed in words or sentences. The set E = {e₁, e₂, …, e_p} is called the set of complete objective statements.

Corresponding to each objective f_j, there is an objective function which is based on type-2 fuzzy set, i.e., $φ_{j}^{(k)} : X (f_{j}) \to F ([0, 1]), (1 \leq j \leq m; 1 \leq k \leq p) .$

Thus, the complete objective function is obtained: $\begin{matrix} φ^{(k)} : & X (1) = \prod_{j = 1}^{m} X (f_{j}) \to [F ([0, 1])]^{m} \\ X = (x_{1}, x_{2}, \dots, x_{m}) \mapsto φ^{(k)} (X) \\ = (φ_{1}^{(k)} (x_{1}), φ_{2}^{(k)} (x_{2}), \dots, φ_{m}^{(k)} (x_{m})) . \end{matrix}$

In this way, a decision problem is transformed into an optimization problem on (φ^(k) ∘ 1) (U). The most direct way is to project φ^(k) (1 (u)) on [0,1] performed by the multifactorial function, as shown in Fig. 2. By means of 1, φ^(k), M_m, a decision function $J_{m}^{(k)} (k = 1, 2, \dots, p)$ can be determined as follows: $J_{m}^{(k)} ≜ M_{m} \circ φ^{(k)} \circ 1 : U \to F ([0, 1]) .$

For every decision alternative u ∈ U and every decision function $J_{m}^{(k)}$ , the functional value $J_{m}^{(k)} (u) (k = 1, 2, \dots, p)$ can be defined. Since E is finite, the fuzzy set J^(u) can be expressed in vector form: $\begin{matrix} J^{(u)} & = (J^{(u)} (e_{1}), J^{(u)} (e_{2}), \dots, J^{(u)} (e_{p})) \\ = (J_{m}^{(1)} (u), J_{m}^{(2)} (u), \dots, J_{m}^{(p)} (u)) . \end{matrix}$

According to the principle of the maximum membership, if there exists an index k ∈ {1, 2, …, p} such that $J^{(u)} (e_{k}) = \max {J^{(u)} (e_{1}), J^{(u)} (e_{2}), \dots, J^{(u)} (e_{p})},$ then u can be regarded to satisfy the complete objective statement e_k.

The detail steps of multifactorial decision making based on type-2 fuzzy sets can be described as follows:

Define a strategies set U = {u₁, u₂, …, u_n};

Determine the complete objective statements set E = {e₁, e₂, …, e_p}, i.e., a set of evaluation phases;

Determine the set of atomic factors π = {f₁, f₂, …, f_m} with respect to U and its factor space {X (f_j)} _1≤j≤m;

Construct the object function $φ_{j}^{(k)} (1 \leq j \leq p; 1 \leq k \leq m)$ , which is based on type-2 fuzzy set;

Select multifactorial function M_m and attain the decision function $J_{m}^{(k)} (k = 1, 2, \dots, p)$ ;

Calculate the fuzzy set J^(u) of u with respect to the decision function $J_{m}^{(k)} (k = 1, 2, \dots, p)$ ;

Defuzzy the fuzzy set J^(u) according to the principle of the maximum membership, and define u corresponding the objective statements e_k.

4.2 The theoretical analysis of comprehensive evaluation based on type-2 fuzzy sets

Comprehensive evaluation based on type-2 fuzzy sets is a special case of multifactorial decision making. In the literature, the objective function of comprehensive evaluation is represented by the membership function of type-1 fuzzy sets, and the function values are real numbers. However, the evaluation process contains a large number of subjective factors, it will be more reasonable to use fuzzy sets to represent the function values.

Let U be a set of objects for evaluation, π = {f₁, f₂, …, f_m} be the set of basic factors in the evaluation system (or process), and let E = {e₁, e₂, …, e_p} be a set of letter grades or qualitative classes used in the evaluation. According to Section 4.1, for every object u ∈ U, there are m × p values of objective functions: $φ_{j}^{(k)} (f_{j} (u)), j = 1, 2, \dots, m; k = 1, 2, \dots, p .$ where $φ_{j}^{(k)} (f_{j} (u))$ is represented by a type-1 fuzzy set. The m × p fuzzy sets can be expressed in the matrix form: ${\tilde{R}}^{(u)} = [\begin{matrix} φ_{1}^{(1)} (f_{1} (u)) & φ_{1}^{(2)} (f_{1} (u)) & \dots & φ_{1}^{(p)} (f_{1} (u)) \\ φ_{2}^{(1)} (f_{2} (u)) & φ_{2}^{(2)} (f_{2} (u)) & \dots & φ_{2}^{(p)} (f_{2} (u)) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ φ_{m}^{(1)} (f_{m} (u)) & φ_{m}^{(2)} (f_{m} (u)) & \dots & φ_{m}^{(p)} (f_{m} (u)) \end{matrix}] .$

Let ${\tilde{r}}_{jk} (u) ≜ φ_{j}^{(k)} (f_{j} (u))$ , then the matrix ${\tilde{R}}^{(u)}$ can be written as ${\tilde{R}}^{(u)} = [\begin{matrix} {\tilde{r}}_{11} (u) & {\tilde{r}}_{12} (u) & \dots & {\tilde{r}}_{1 p} (u) \\ {\tilde{r}}_{21} (u) & {\tilde{r}}_{22} (u) & \dots & {\tilde{r}}_{2 p} (u) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{r}}_{m 1} (u) & {\tilde{r}}_{m 2} (u) & \dots & {\tilde{r}}_{mp} (u) \end{matrix}],$ where, every element in the matrix ${\tilde{R}}^{(u)}$ is a type-1 fuzzy set.

Actually the matrix ${\tilde{R}}^{(u)}$ may also be viewed as a fuzzy relation between π and E, i.e., ${\tilde{R}}^{(u)} \in \tilde{F} (π \times E)$ . Define ${\tilde{R}}_{j}^{(u)} ≜ {{\tilde{r}}_{j 1} (u), {\tilde{r}}_{j 2} (u), \dots, {\tilde{r}}_{jp} (u)} .$

Then this is the membership degree of the object u with respect to a factor f_j on the set of letter grades e₁, e₂, …, e_p. The matrix ${\tilde{R}}_{j}^{(u)}$ is called the single-factor evaluation vector. Since the matrix ${\tilde{R}}^{(u)}$ consists of the row vectors ${\tilde{R}}_{1}^{(u)}, {\tilde{R}}_{2}^{(u)}, \dots, {\tilde{R}}_{m}^{(u)}$ , then the matrix ${\tilde{R}}^{(u)}$ can be written as ${\tilde{R}}^{(u)} = [\begin{matrix} {\tilde{R}}_{1}^{(u)} \\ {\tilde{R}}_{2}^{(u)} \\ ⋮ \\ {\tilde{R}}_{m}^{(u)} \end{matrix}],$ which is called the single-factor evaluation matrix based on type-2 fuzzy sets of object u.

For every object u ∈ U and each object function φ^(k) (k = 1, 2, …, p), the decision function $J_{m}^{(k)} (k = 1, 2, \dots, p)$ can be obtained as follows: $\begin{matrix} J^{(u)} (e_{k}) \\ = J_{m}^{(k)} (u) \\ = (M_{m} \circ φ^{(k)} \circ 1) (u) \\ = (M_{m} \circ φ^{(k)}) (1 (u)) \\ = (M_{m} \circ φ^{(k)}) (f_{1} (u), f_{2} (u), \dots, f_{m} (u)) \\ = M_{m} (φ^{(k)} (f_{1} (u), f_{2} (u), \dots, f_{m} (u))) \\ = M_{m} (φ_{1}^{(k)} (f_{1} (u)), φ_{2}^{(k)} (f_{2} (u)), \dots, φ_{m}^{(k)} (f_{m} (u))) \\ = M_{m} ({\tilde{r}}_{1 k} (u), {\tilde{r}}_{2 k} (u),, {\tilde{r}}_{mk} (u)) . \end{matrix}$

And in general, the multifactorial function M_m can be chosen as $\begin{matrix} M_{m} (A_{1}, A_{2}, \dots, A_{m}) ≜ ⋃_{j = 1}^{m} (w_{j} \land A_{j}), \\ μ_{⋃_{j = 1}^{m} (w_{j} \land A_{j})} (u) = ⋁_{j = 1}^{m} (w_{j} \land μ_{A_{j}} (u)), u \in [0, 1] . \end{matrix}$

Hence J^(u) (e_k) can be expressed as $J^{(u)} (e_{k}) = ⋃_{j = 1}^{m} (w_{j} \land {\tilde{r}}_{jk} (u)),$ (2) where w_j ∈ [0, 1], and $\sum_{j = 1}^{m} w_{j} = 1$ .

Let $\begin{matrix} J^{(u)} & = (J^{(u)} (e_{1}), J^{(u)} (e_{2}), \dots, J^{(u)} (e_{p})) \\ = (J_{m}^{(1)} (u), J_{m}^{(2)} (u), \dots, J_{m}^{(p)} (u)) . \end{matrix}$

Then a general representation can be expressed as: $J^{(u)} = w \circ {\tilde{R}}^{(u)} .$ (3) where, the product “∘” is defined in Equation (2), and w = (w₁, w₂, …, w_m) is called weight vector.

The general model of comprehensive evaluation based on type-2 fuzzy sets is denoted by $\tilde{M} (T^{*}, T)$ , where the most common model of comprehensive evaluation based on type-2 fuzzy sets is $\tilde{M} (\cup, \land)$ defined by Equation (2).

In conclusion, a comprehensive evaluation model based on type-2 fuzzy sets requires three basic elements: (1)

a family of basic factors, π = {f₁, f₂, …, f_m};

a set of evaluation phases, E = {e₁, e₂, …, e_p};

for every object u ∈ U, there is a single-factor evaluation matrix ${\tilde{R}}^{(u)} = {({\tilde{r}}_{jk} (u))}_{m \times p}$ .

With the preceding three elements, for a given u ∈ U, its evaluation result $J^{(u)} \in \tilde{F} (E)$ can be derived as: $J^{(u)} = w \circ {\tilde{R}}^{(u)} .$

Hence, the comprehensive evaluation based on type-2 fuzzy sets can be viewed as a set-valued mapping $ξ : U \to \tilde{F} (E), u \mapsto ξ (u) ≜ J^{(u)} = w \circ {\tilde{R}}^{(u)} .$

4.3 Applied instances

Example 1. A SCHEME SELECTION PROBLEM. Let U = {u₁, u₂, u₃} be the set of candidates, E = {e} be the set of evaluation statement, i.e., “the optimal”. Let π = {f₁, f₂, f₃, f₄} be the set of basic factors, the factor space X (f_j) = [0, 100] , (j = 1, 2, 3, 4). The states of u_i (i = 1, 2, 3) on f_j (j = 1, 2, 3, 4) are shown in Table 1.

Taking the commonly used Gaussian type-2 fuzzy sets as an example to illustrate the proposed model. It should be pointed out that the primary membership of a Gaussian type-2 fuzzy set can be followed any shape, but the secondary membership must be Gaussian-type membership function. In order to compare with the type-1 fuzzy sets based results in [4], the value of states f_j (u_i) are kept the same, and the basic objective function φ_j (x) is taken as the primary membership function, $φ_{j} (x) = {\begin{matrix} 1, & 90 < x \leq 100 \\ \frac{x - 80}{10}, & 80 \leq x \leq 90 \\ 0, & 0 \leq x < 80 \end{matrix}$

Construct the objective function based on type-2 fuzzy sets as follows: $φ (u) = e^{- \frac{{(u - φ_{j} (x))}^{2}}{2 σ^{2}}}, (j = 1, 2, 3, 4) .$ where σ is a parameter that needs to be determined. And the objective function values φ_j (f_j (u_i)) represented by type-1 fuzzy sets are shown in Table 2.

Take the multifactorial function as $M_{m} (A_{1}, A_{2}, \dots, A_{m}) ≜ ⋃_{j = 1}^{m} A_{j}, j = 1, 2, 3, 4 .$ then the decision function J₄ can be described as follows: $J_{4} (u_{i}) = ⋃_{j = 1}^{4} φ_{j} (f_{j} (u_{i})) .$

The operation rules about type-1 fuzzy sets are presented in [4], and the decision results of J₄ (u₁) , J₄ (u₂) , J₄ (u₃) are shown in Fig. 3.

By means of the centroid defuzzification method, the defuzzification values of ${\hat{J}}_{4} (u_{1}), {\hat{J}}_{4} (u_{2}), {\hat{J}}_{4} (u_{3})$ are given as follows: ${\hat{J}}_{4} (u_{1}) = 0.7078, {\hat{J}}_{4} (u_{2}) = 0.8792, {\hat{J}}_{4} (u_{3}) = 0.6439$

According to the principle of the maximum membership, u₂ is recognized as the best.

When type-1 fuzzy set is used to describe the concept of scheme optimization problem, the evaluation result is a crisp number. By means of the principle of maximum membership, we conclude that the optimal solution is u₂, but it fails to provide an explanation on the optimality of u₂. However, when type-2 fuzzy set is used to describe the same problem, the result of the comprehensive evaluation is a type-1 fuzzy set. From Fig. 3 it can be easily seen that u₂ is optimal. The difference between the single factor evaluation values illustrated in scheme u₁ and scheme u₃ is large, which results in the irregularity of the indicators. But the difference between the single factor evaluation values in scheme u₂ is small, the single factor evaluation indicator is good. Therefore, the CE result of u₂ is optimal.

In addition, the basic objective functions φ_j (x) in [4] are based on type-1 fuzzy sets. It is easy to find that when state value belongs to [90, 100] then the objective function value is 1, the difference between state values f₁ (u₂) =98 and f₁ (u₃) =90 can’t be distinguished by the basic objective function. Actually, the phrase “state value belong to [90, 100]” contain uncertainty, only with the value of 1 can’t reflect the uncertainty of the fuzzy concept of “state value belong to [90, 100]”. However, when type-2 fuzzy sets are used to represent the concept, the uncertainty contained in the concept can be easily expressed.

When state value belong to [90,100], in order to more accurate represent the concept, the parameters can be flexibly adjust to choose a more suitable objective function, for example, takeing the objective function as $Φ (u) = e^{- \frac{{(u - φ_{j} (x))}^{2}}{2 σ^{2}}}, (j = 1, 2, 3, 4) .$ where $σ = {\begin{matrix} 0.1, & 0 < φ_{j} (x) < 1, \\ 0.2, & φ_{j} (x) = 1 . \end{matrix}$

And the new calculation values illustrated by fuzzy sets $J_{4}^{'} (u_{1})$ , $J_{4}^{'} (u_{2})$ , $J_{4}^{'} (u_{3})$ are shown in Fig. 4.

By means of the centroid defuzzification method, the defuzzification values of ${\hat{J^{'}}}_{4} (u_{1}), {\hat{J^{'}}}_{4} (u_{2}), {\hat{J^{'}}}_{4} (u_{3})$ are given as follows: ${\hat{J^{'}}}_{4} (u_{1}) = 0.7286, {\hat{J^{'}}}_{4} (u_{2}) = 0.8448, {\hat{J^{'}}}_{4} (u_{3}) = 0.6772$

According to the principle of the maximum membership, u₂ is recognized as the best, and the reason for this is more intuitive.

When type-2 fuzzy sets are used to represent the concept, not only the model itself but also the third dimension of type-2 fuzzy sets can be used to reflect the uncertainties in the concept. However, this can’t be achieved by applying type-1 fuzzy sets, which is also the main reason that the interval type-2 fuzzy sets are extensively used in MCD and MAD. In fact, the CE result represented by type-1 fuzzy set should be more reasonable than that expressed by a single number since whatever standards are used to evaluate the process, there exists many subjective factors contained in the evaluation results, which causes a relative final conclusion. However, when type-1 fuzzy set is used to replace a crisp number, the evaluation result will be more in accordance with people’s psychology, and the judgement could explain the evaluation result easily. This is the reason that the concept in the CE represented by type-2 fuzzy sets is more reasonable than that expressed by type-1 fuzzy sets.

When type-2 fuzzy sets are applied in CE, and the basic objective function of CE is represented by means of the membership function of type-2 fuzzy sets, it not only gets the same result by type-1 fuzzy sets but also receives some extra information. Next, an example in [4] is used to illustrate the proposed method.

Example 2. A CLOTH SELECTION PROBLEM. Assume the basic factors of interest in the cloth selection consist of f₁ = style, f₂ = quality, and f₃ = price, i.e., π = {f₁, f₂, f₃}. Define a set of evaluation phases used for the selection as E = {e₁, e₂, e₃, e₄}, where e₁ = best, e₂ = good, e₃ = fair, e₄ = poor. For a particular piece of cloth, u, the single-factor evaluation may be carried out by professionals or customers. For example, if the survey results on the “style” factor are 60% for the best, 30% for the good, 10% for the fair, and none for the poor.

As in the previous example, the single-factor evaluation vector of u with respect to f₁ is presented as ${\tilde{R}}_{1}^{(u)} = [\begin{matrix} e^{- \frac{(u - 0.6)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.3)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.1)^{2}}{2 σ^{2}}}, e^{- \frac{u^{2}}{2 σ^{2}}} \end{matrix}] .$

Similarly, the following single-factor evaluation vectors for f₂, and f₃ can be obtained as follows: ${\tilde{R}}_{2}^{(u)} = [\begin{matrix} e^{- \frac{(u - 0.2)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.4)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.3)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.1)^{2}}{2 σ^{2}}} \end{matrix}],$ ${\tilde{R}}_{3}^{(u)} = [\begin{matrix} e^{- \frac{(u - 0.1)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.3)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.4)^{2}}{2 σ^{2}}}, e^{- \frac{(u - 0.2)^{2}}{2 σ^{2}}} \end{matrix}] .$

Based on the single-factor evaluation vectors of ${\tilde{R}}_{1}^{(u)}$ , ${\tilde{R}}_{2}^{(u)}$ , ${\tilde{R}}_{3}^{(u)}$ , we can acquire the single-factor evaluation matrix ${\tilde{R}}^{(u)}$ : ${\tilde{R}}^{(u)} = [\begin{matrix} e^{- \frac{(u - 0.6)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.3)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.1)^{2}}{2 σ^{2}}} & e^{- \frac{u^{2}}{2 σ^{2}}} \\ e^{- \frac{(u - 0.2)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.4)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.3)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.1)^{2}}{2 σ^{2}}} \\ e^{- \frac{(u - 0.1)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.3)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.4)^{2}}{2 σ^{2}}} & e^{- \frac{(u - 0.2)^{2}}{2 σ^{2}}} \end{matrix}] .$

If a customer’s weight vector with respect to the three factors is w = (0.5, 0.3, 0.2), and we choose the comprehensive evaluation model as ${\tilde{M}}_{m} (\cup, \land)$ , where $\begin{matrix} M_{m} (A_{1}, A_{2}, \dots, A_{m}) ≜ ⋃_{j = 1}^{m} (w_{j} \land A_{j}), \\ μ_{⋃_{j = 1}^{m} (w_{j} \land A_{j})} (u) = ⋁_{j = 1}^{m} (w_{j} \land μ_{A_{j}} (u)), u \in [0, 1] . \end{matrix}$

And the comprehensive computed evaluation results based on type-1 fuzzy sets as shown in Fig. 5.

When the centroid defuzzification method is applied, the defuzzification values of the CE results can be obtained: $\begin{matrix} J^{(u)} (e_{1}) = 0.4591, J^{(u)} (e_{2}) = 0.2857, \\ J^{(u)} (e_{3}) = 0.2574, J^{(u)} (e_{4}) = 0.1733 . \end{matrix}$

According to the principle of the maximum membership, u is recognized as the evaluation grade of e₁.

From Fig. 5, it can be observed that the results based on type-2 fuzzy sets not only contains the same results based on type-1 fuzzy sets, but also include a lot of additional information. For example, the specific cloth’s evaluation grade is regard as the best. The main factor influences the clothing popularity is the style, and next is the quality. While the influence of price for some groups can be neglected, the additional information can’t be obtained in the previous method. Obviously, the additional information can provide a beneficial reference for cloth designing and pricing.

5 Conclusions

In this paper, a novel approach by using type-2 fuzzy sets to represent the concept in factor space has been developed. Five kinds of membership degree synthesis methods are given by defining the multifactorial function. By applying factor space theory, a general model of CE based on type-2 fuzzy sets is formulated, and analyzed, which provides a theoretical foundation for its application. From the results, one can conclude that type-2 fuzzy sets is efficient and reasonable in representing the concept.

The study of CE based on type-2 fuzzy sets will promote its application in the practical problems. With the improvement of science and technology, more and more information can be collected, and the CE status are also more complex than before. When type-2 fuzzy set is used in CE to represent the concept, the evaluation result can be presented by a type-1 fuzzy set. Besides the conclusion obtained by using type-1 fuzzy sets, it can also achieve some other valuable information, which is helpful to make an appropriate decision. In actual situation, the final result of CE mostly depends on the judger’s subjective attitude. While it is more reasonable to obtain an evaluation result expressed by type-1 fuzzy set, which is also more in accordance with people’s psychology, and conducive to the judger to make an explanation on the obtained results.

Footnotes

Acknowledgments

This work was supported in part by the NationalNatural Science Foundation of China (Nos.: 61473327, 61374118).

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