Fuzzy rough set over multi-universes and its application in decision making 1

Abstract

This paper presents a general framework for the study of rough approximation of a fuzzy (crisp) concept in multi-universes environment, which is an extension of the original Pawlak rough set theory. The Pawlak rough set is mainly concerned with the approximation of objects confined on the same universe by an arbitrary binary relation. However, the objects (concept) may be related with three or more different universes in reality of the decision-making. This paper presents the rough set model over multi-universes, where the concept approximations are defined by using a multiple relation on the multi-universes. We firstly present the rough approximation of a fuzzy concept over multi-universes by using the multiple fuzzy relation defined on the multi-universes, i.e., the fuzzy rough set over multi-universes. Then, a number of important properties of fuzzy rough set over multi-universes are obtained. It is shown that some of the properties of the fuzzy rough set on the same universe are special instances of those of fuzzy rough set over multi-universes. Furthermore, several special rough set models over multi-universes are derived from the fuzzy rough set over multi-universes. Subsequently, we give a new approach to multiple attribute decision making (MADM) with the characteristic of uncertainty, incomplete and inaccurate available information based on fuzzy rough set over multi-universes by combing the idea of uncertainty risk decision making. The decision steps and the algorithm of the decision method are also given. The proposed approach can obtain an objectively decision result with the data information owned by the decision problem only. Finally, the validity of the decision methods is tested by a numerical example with the background of clinical medical diagnosis decision making. The main contribution of this paper is twofold. One is to present a new perspective of the universe-oriented extension for classical rough set and then establish a new generalized rough set model under the framework of multiple different universes, i.e., fuzzy rough set model over multi-universes. Another is to present an approach to multiple attribute decision making problems based on fuzzy rough set over multi-universes.

Keywords

Rough set theory multiple relation uncertainty decision making multiple attribute decision making (MADM)fuzzy rough set over multi-universes

1 Introduction

Many our traditional tools for formal modeling, reasoning and computing are crisp, deterministic and precise in character. However, most of practical problems within fields as diverse as economics, engineering, environment, social science, medical science involve data that contain uncertainties. We cannot successfully use traditional mathematical tools because of various types of uncertainties existing in these problems. There have been a great amount of researches and applications in the literature concerning some special tools such as probability theory [57], fuzzy set theory [65], vague set theory [13], grey set theory [11], intuitionistic fuzzy set theory [1] and interval mathematics [14]. Rough set theory [29], as a new mathematical theory for handling with the uncertainty and inaccuracy data and information, originated by Pawlak in 1982. It has become a well-established mechanism for uncertainty management in a wide variety of applications related to artificial intelligence [17], pattern recognition and fault diagnostics [35], decision-making under uncertainty [38], and etc. One of the main advantages of rough set theory is that it does not need any preliminary or additional data, such as probability distribution in statistics, basic probability assignment in the Dempster-Shafer theory, grade of membership or the value of possibility in fuzzy set theory. This can be seen in the following paragraph from [30]: “The numerical value of imprecision is not pre-assumed, as it is in probability theory or fuzzy sets-but is calculated on the basis of approximations which are the fundamental concepts used to express imprecision of knowledge.” Originally, the basic concept in Pawlak rough set theory is indiscernibility relation (i.e., indiscernibility relation between objects in information systems induced by different values of attributes characterizing these objects). And the key issues is rough approximation of a given object (concept) based on the indiscernibility relation of the universe of discourse. The equivalence classes induced by indiscernibility relation are the building blocks for the construction of the lower and upper approximations.

As a non-numeric method to represent and manage the uncertainty in various information systems, Pawlak rough set has invoked many attentions of the researchers and practitioners both in the theory and application aspects in the past few years. Meanwhile, there have published a large number of valuable and high quality papers. As is well known, the extensions of the Pawlak rough set model are one of importation and flourish study directions with the different background in reality of management sciences. Generally speaking, classical Pawlak rough set model can most commonly be generalized into four forms according to the requirements of the theories and applications. One is to relax the restriction of the equivalence relation of the universe of discourse. The equivalence relation is usually generalized to binary relation [49], similarity relation [21], tolerance relation [45] and dominance relation [15, 16] over the universe of discourse. At the same time, a large number of original and innovative generalized models of the classical Pawlak rough set were derived from this idea. Secondly, a lack of consideration for the degree of overlap between an equivalence class and the set to be approximated unnecessarily limits the applications of rough set and has motivated another generalization of Pawlak rough set: variable precision rough set [63], which is to introduce a precision parameter to the lower and upper approximations. Furthermore, considering the similar limitation as like as the Ziarko’s variable precision rough set with respect to the classical Pawlak rough set model, Wong and Ziarko [54] introduce the probabilistic approximation space to the studies of rough set and then presented the concept of probabilistic rough set in 1987. Subsequently, Yao et al. [62] proposed a more general probabilistic rough set called decision-theoretic rough set. Then another perspective to deal with the degree of overlap of an equivalence class with the set to be approximated was given, and an approach to select the needed parameters in lower and upper approximations was presented systematically. And then a good deal of research to investigate probabilistic generalization [59] are established of the classical Pawlak rough set. Thirdly, the generalization of the Pawlak rough set theory by combing the other mathematical theories such as fuzzy set theory [10 , 51], Dempster-Shafer theory of evidence [5, 52], intuitionistic fuzzy set theory [64], and etc. Recently, there also have been many rough fuzzy set models and fuzzy rough set models proposed by combining fuzzy set and rough set. This is also the most flourishing researched field of rough set theory in past years and many generalized models and interesting application results were given [4 , 58]. Finally, the generalization of the Pawlak rough set is focus on the extension of the discussed universe of discourse, which the discussed objects are extended from the single universe to two different but related closely universes [53, 60]. Subsequently, there are many studies with respect to the theory and application of rough set over two universes [23–26 , 61] under the framework of the interval structure established by Wong et al. [53, 60]. Meanwhile, several important conclusions have been proposed in this field. All of the aforementioned generalizations of the classical Pawlak rough set are expanded the existing studies both in the theories and applications. Moreover, many complicatedly problems in reality have solved by using the classical Pawlak rough set theory and its extensions.

Though the theory of rough set over two universes, as one of important generalizations of the classical Pawlak rough set theory, can deal with many complicatedly decision-making problems with the uncertainty and inaccuracy of the available information in the management science. Some complexity uncertainty and risk decision-making problems existed in reality still may invalidate the current rough set theory over two universes, then making the research on the rough set approaches over multi-universes a necessity. In order to elaborate the motivation and the application background of the rough set model over multi-universes in the decision-making problems of reality, we present an example of the disease diagnosis decision-making problem in clinic. As is well known, the uncertainty and high risk are the outstandingly characters of medical diagnosis in clinics [12, 44]. The symptoms of a patient are the first uncertainty index of whether a disease may occur or not. The results of clinical examination of a patient are another important index of whether a disease may occur or not. It is easy to know that the basic symptoms and the results of clinical examination are two main facts for a concretely disease which a patient may be owned. That is, the doctor could judge what kind of the disease may be occur for the considered patient according to the symptoms and the results of clinical examination. So, in a doctor’s specific group of the patients, each patient may show many symptoms and also have different results of clinical examination, just as each disease could have many basic symptoms and several concretely results of clinical examination in clinic. Then the decision-making for a doctor is to decide which disease the patient has according to the symptoms and the results of clinical examination in clinic. It can be easily seen that both the symptoms and the results of clinical examination are related to the disease for a concretely patient but them belongs to three different universes of discourse, respectively, i.e., the universe of symptom, the universe of the results of clinical examination and the universe of disease. That is, the characters of the above decision-making problem of disease diagnosis are described by three different universes of discourse. Thus, the multiple different possible universes and their interrelations may invalidate current rough set theory when practical decision making is concerned. So, it is a interesting and valuable topic of research on the rough set theory and methodology over multi-universes.

Inspired by the existing studies of the classical Pawlak rough set theory and its extensions, this paper considers the rough approximation of a concept (or object with the characterize of fuzzy or crisp) with respect to multiple different universes of discourse. Similar to the classical Pawlak rough set theory, we first present the original definition of the multiple relation over multi-universes. We then give the lower and upper approximations of a fuzzy concept of the universe of discourse based on the multiple relation over three universes according to the constructive approach, i.e., the fuzzy rough set over three universes. Several interesting properties of the fuzzy rough approximation operators over three different universes are investigated in detail. Also, the interrelationship of the fuzzy rough set over three universes with the existing rough set models are investigated. At the same time, some other rough set models over three universes are established based on the fuzzy rough set over three universes. Without lack of generality, we then present the rough approximation of a fuzzy concept over n different universes based on the concept of multiple relation. That is, the fuzzy rough set over multi-universes.

As is well known, the increasing complexity of the socio-economic environment makes it less and less possible for a single attribute (or criteria) to consider all relevant aspects of a decision making problem. So, a large number of decision making problems in the practice of management science, operational research, and industrial engineering are usually conducted by multiple attribute decision making (MADM) [20]. So far there are many approaches to solve the multiple attribute decision making problem such as analytic hierarchy process (AHP) [35, 36], the monte carlo method (or random simulation) [27, 46], the data envelopment analysis (DEA) [2], and etc. Recently, several decision making models with new perspective were proposed by Cabrerizo et al. [6, 7] and P $\overset{´}{e}$ rez et al. [32, 33] as well as there are other scholars discussions for the related directions [8 , 55]. Cabrerizo et al. [6] explore the consensus or group decision making problems by using the fuzzy set theory systematically. Based on the consensus degrees and similarity measures, P $\overset{´}{e}$ rez et al. present a new consensus model for heterogeneous group decision making problems. The proposed model also present a new feedback mechanism that adjusts the amount of advice required by each expert depending on his/her own relevance or importance level [33]. Meanwhile, Cabrerizo et al. [7] discuss the group decision making problems with linguistic information defined in heterogeneous contexts by using the method of granular computing. Considering two important characteristics in the decision making problems of reality: 1) mobile technologies are applied in the decision process and 2) the set of solution alternatives could change throughout the decision-making process, P $\overset{´}{e}$ rez et al. [33] establish a dynamic decision making model. All the existing methods have made a reasonable and effectively solution for multiple attribute decision making problems from one of aspects of the considered problems. This paper tries to make a new perspective to explore and discuss the multiple attribute decision making problems with the characteristic of uncertainty, incomplete and inaccurate available information based on fuzzy rough set model over multi-universes. The basic idea of the proposed method is transform the multiple attribute decision making problem into a uncertainty decision making problem over multiple different universes. We then present the approach to multiple attribute decision making (MADM) with the characteristic of uncertainty, incomplete and inaccurate available information by combing the new fuzzy rough set model given in this paper and the principle of the risk decision-making of operational research [16]. Thus, the contribution of this study is twofold. One is to present a new way to solve the multiple attribute decision making problems with the characteristic of uncertainty, incomplete and inaccurate available information. Another is to provide a reasonable interpretation of the motivation for discussing fuzzy rough set theory over multi-universes.

The paper is organized as follows. Section 2 briefly introduces fuzzy set theory, Pawlak rough set theory and the rough set model over two universes. Section 3 establishes fuzzy rough set theory over multi-universes and also discusses some properties for this model in detail. In Section 4, we present a new method to multiple attribute decision making (MADM) by using the fuzzy rough set theory over multi-universes. We present the decision making principle and process using a medical diagnosis decision making problem of management science. At last we conclude our research and set further research directions in Section 5.

2 Preliminaries

In this section, we outline some basic concepts in fuzzy set [65] and some rough set models, such as Pawlak’s rough set model [29], rough set model on two universes [53, 60] and the fuzzy rough set over two universes [29, 38].

2.1 Fuzzy set theory

Let U be a non-empty finite universe, for any A ⊆ U, we introduce a characteristic function A (x) =1_A (x) =1 if x ∈ A and A (x) =1_A (x) =0 if x ∉ A .

The characteristic function A (x) is a mapping from U to {0, 1} . Then any characteristic function on universe U determines a classical subset of U . That is, A = {x ∈ U|A (x) =1} .

In following, we present the concept of fuzzy set by using the definition of classical crisp set.

Definition 2.1. [56] Let U be a non-empty finite universe. A fuzzy set of universe U is defined by the mapping A (•) : U ⟶ [0, 1] , where the A (x) denotes the membership of element x (x ∈ U) with fuzzy set A .

It can be easily seen that the classical crisp set is a special case of fuzzy set.

For any A ∈ F (U) (where F (U) denotes all fuzzy subsets of universe U), the α-level and the strong α-level of A will be denoted by A_α and A_α+, respectively. That is, A_α = {x ∈ U|A (x) ≥ α} and A_α+ = {x ∈ U|A (x) > α} , where α ∈ I = [0, 1] the unit interval, A₀ = U and A₁₊ = ∅ .

Definition 2.2. [38] Let U be a non-empty finite universe and R ∈ F (U × U) , then

R is reflexive, if R (u, u) =1, for any u∈ U ;

R is symmetric, if R (u₁, u₂) = R (u₁, u₂) , for any u₁, u₂ ∈ U,

R is transitive, if R ∘ R ≤ R .

For binary relation R ∈ F (U × U) , if R is reflexive, symmetric, then R is called a fuzzy similarity relation on U, if R is reflexive, symmetric, and transitive, then R is called a fuzzy equivalence relation on U .

2.2 Pawlak’s rough set model

In this section, we briefly review the concept of rough set theory [29, 30].

Let U be a universe of discourse, for binary relation R on U, we call R an equivalence relation onU, if.

R is reflexive if for all x ∈ U, xRx ;

R is symmetric if for all x, y ∈ U, xRy implies yRx ;

R is transitive if for all x, y, z ∈ U, xRy and yRz implies xRz .

An equivalence relation is a reflexive, symmetric and transitive relation. The equivalence relation R partitions U into disjoint subsets (or equivalence classes). Let U/R denote the family of all equivalence classes of R . For every object x ∈ U, let [x] _R denote the equivalence class of relation R that contains element x, called the equivalence class of x under relation R . Based on the definition of equivalence relation R over the universe U . We present the Pawlak’s rough set model as follows.

Definition 2.3. [29, 30] Let U be non-empty finite universe and R an equivalence relation on U . We call (U, R) the Pawlak approximation space. For any X ⊆ U, we can describe X by a pair of lower and upper approximations defined as follows:

$\begin{matrix} \underline{R} (X) = {x \in U | [x]_{R} \subseteq X}, \\ \bar{R} (X) = {x \in U | [x]_{R} \cap X \neq \emptyset} . \end{matrix}$

$\underline{R} (X)$ is called the lower approximation of X, which is the union of all the equivalence classes which contain in X, and $\bar{R} (X)$ is called the upper approximation of X, which is the union of all equivalence classes which have non-empty intersection with X . Then $(\underline{R} (X), \bar{R} (X))$ is called the rough sets of X about the Pawlak approximation space (U, R) .

2.3 Rough set model over two universes

In this section, we shall review some basic concepts and properties of the rough set model on two universes. Detailed description of the model can be found in [38 , 59].

Definition 2.4. [43] Let U and V be two universes, and R be a binary relation from U to V, i.e. a subset of U × V . R is said to be compatibility, or a compatibility relation, if for any u∈ U ; v∈ V ; there exist t∈ V ; s ∈ U such that (u, t) , (s, v) ∈ R .

Definition 2.5. [53, 60] Let U and V be two universes, and R be a compatibility relation from U to V . The mapping F : U → 2^V, u ↦ {v ∈ V| (u, v) ∈ R} is called the mapping induced by R .

Obviously, the above-defined binary relation R can uniquely determine the mapping F, and vice versa. Then the rough set over two universes is defined as follows:

Let U and V be two universes, and R be a compatibility relation from U to V . The ordered triple (U, V, R) is called a (two-universe) approximation space. The lower and upper approximations of Y ⊆ V are, respectively, defined as follows [31 , 60]: $\begin{matrix} \underline{apr} (Y) & = & {x \in U | F (x) \subseteq Y}; \\ \bar{apr} (Y) & = & {x \in U | F (x) \cap Y \neq \emptyset} . \end{matrix}$

The ordered set-pair $(\underline{apr} (Y), \bar{apr} (Y))$ is called a generalized rough set, and the ordered operator-pair $(\underline{apr}, \bar{apr})$ is an interval structure. Particularly, Y is called definable with (U, V, R) if $\underline{apr} (Y) = \bar{apr} (Y) .$ Otherwise, Y is indefinable set. Meanwhile, the model present by Definition 2.4 is called rough set over two universes.

2.4 Fuzzy rough set over two universes

As mentioned earlier, due to the need of many management decisions in real-world problems, similar to the research on the single universe, fuzzy rough set over two universe has been proposed by many scholars in past years. In this section, we only present the definition of fuzzy rough set over two universes and also give a fraction of conclusions of the model used in the following sections.

Firstly, we give the definition of the fuzzy binary relation over two universes [38 , 51].

Definition 2.6. [38 , 51] We call fuzzy subset R ∈ F (U × V) the binary fuzzy relation from U to V . R (x, y) is the related degree between elements x and y, where (x, y) ∈ U × V . Particularly, for any x ∈ U, R is called serial fuzzy binary relation from U to V if there exists element y ∈ V and satisfies R (x, y) =1 .

By the fuzzy binary relation, we give the definition of fuzzy rough set over two universes, it is consistent with the definition in reference [22, 24].

Let U, V be two non-empty finite universes. R be binary fuzzy relation from U to V . The 3-triple set (U, V, R) is called fuzzy approximation space over two universes. For any fuzzy set A ∈ F (V) , the lower approximation $\underline{R} (A)$ and upper approximation $\bar{R} (A)$ of A with (U, V, R) , respectively, are two fuzzy subsets of universe U . For any x ∈ U, the membership function can be calculated as follows: $\begin{matrix} \underline{R} (A) (x) & = & ⋀_{y \in V} [(1 - R (x, y)) \lor A (y)] x \in U, \\ \bar{R} (A) (x) & = & ⋁_{y \in V} [R (x, y) \land A (y)], x \in U . \end{matrix}$

The ordered set-pair $(\underline{R} (A), \bar{R} (A))$ is called fuzzy rough set over two universes.

The lower and upper approximations will be changed into the following forms when U and V are infinite set. $\begin{matrix} \underline{R} (A) (x) & = & inf_{y \in V} [(1 - R (x, y)) \lor A (y)] x \in U, \\ \bar{R} (A) (x) & = & sup_{y \in V} [R (x, y) \land A (y)], x \in U . \end{matrix}$

Like on the single universe, the lower and upper approximation operators pair $\underline{R} (A)$ and $\bar{R} (A)$ also have some similar properties. However, the binary fuzzy relation R defined on two universes U and V has not the same properties on the single universe. So, the relationship $\underline{R} (A) \subseteq A \subseteq \bar{R} (A)$ holds for the fuzzy rough set on the single universe but it could not satisfies on two universes.

3 Fuzzy rough set model over multi-universes and its properties

This section will discuss the theory of fuzzy rough set model over multi-universes. We firstly present the rough approximation of a fuzzy concept over three universes, i.e., fuzzy rough set model over three universes. Several special cases of the established model are obtained. Meanwhile, some interesting properties for this model will be investigated in detail. We then present a general forms for the rough approximation of a fuzzy concept over the multiple universes by extending the proposed fuzzy rough set over three universes. That is, the definition of fuzzy rough set over multi-universes.

3.1 Fuzzy rough set over three universes and its properties

First of all, we present some concepts which will be used in the following sections.

Definition 3.1. Let U, V and W be three non-empty finite universes. We call a fuzzy subset $R \in F (U \times V \times W)$ the multiple fuzzy relation over universe U, V and W, i.e., for any u ∈ U, v ∈ V, w ∈ W, there is R (u, v, w) ∈ [0, 1] .

Based on the above definition of multiple fuzzy relation over three different universes, the following properties are clear.

Property 3.1. Let U, V and W be three non-empty finite universes, and $R \in F (U \times V \times W)$ be a multiple fuzzy relation. Then

$R$ is pseudo reflexive, if $R (u, v, w) = 1 \Leftrightarrow R (v, w, u) = 1, R (w, u, v) = 1,$ for any u ∈ U, v ∈ V, w∈ W ;

$R$ is pseudo symmetric, if $R (u, v, w) = R (v, w, u) = R (w, u, v)$ for any u ∈ U, v ∈ V, w∈ W ;

$R$ is multiplicative, if $R (u, v, w) = R_{1} (u, v) \circ R_{2} (v, w) (where \circ denotes a binary operator),$ for any u ∈ U, v ∈ V, w ∈ W, and R₁ ∈ F (U × V) , R₂ ∈ F (V × W) .

Particularly, if U = V, then $R$ will be the multiple fuzzy relation over U and W . Furthermore, if U = V = W, then $R$ will be the multiple fuzzy relation over U . So, the multiple fuzzy relation $R$ over three different universes U, V and W is a directly generalization of the binary fuzzy relation over the single universe of discourse.

Example 3.1. Here, we employ an example to illustrate the concept of multiple fuzzy relation over three different universes. We consider a medical diagnosis decision-making problem. Let U = {u₁, u₂, ⋯ , u_m} be the universe of symptom, V = {v₁, v₂, ⋯ , v_n} be the universe of the results of clinical examination, and W = {w₁, w₂, ⋯ , w_l} be the universe of several different diseases. Then, for any u_i ∈ U, v_j ∈ V and w_k ∈ W, i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n ; k = 1, 2, ⋯ , l ; the multiple fuzzy relation $R (u_{i}, v_{j}, w_{k})$ means that the related degree of the disease w_k with the symptom u_i and the clinical examination result v_j for a given patient. Generally speaking, the more the value of $R (u_{i}, v_{j}, w_{k}),$ the possible of the disease w_k for a given patient. So, this makes it very easy for a doctor to decide which disease the patient has.

By the concept of multiple fuzzy relation over multiple different universes of discourse, we present the concept of approximation space over threeuniverses.

Definition 3.2. Let U, V and W be three non-empty finite universes, and $R \in F (U \times V \times W)$ be a multiple fuzzy relation. We call the 4-triple $(U, V, W, R)$ the fuzzy approximation space over three universes.

In the following, we present the rough approximation of a fuzzy concept in approximation space over three universes.

Definition 3.3. Let $(U, V, W, R)$ be fuzzy approximation space over three universes, R₁ ∈ F (U × W) and R₂ ∈ F (V × W) be two binary fuzzy relations over universes U, V and W . For any fuzzy set A ∈ F (W) and ∀w ∈ W, the lower and upper approximations of A with respect to multiple fuzzy relation $R$ are as follows, respectively. $\begin{matrix} \bar{R} (A) (w) & = & [⋁_{u \in U} R_{1} (u, w) \land A (w)] \\ \land [⋁_{v \in V} R_{2} (v, w) \land A (w)]; \\ \underline{R} (A) (w) & = & [⋀_{u \in U} ((1 - R_{1} (u, w)) \lor A (w)] \land \\ [⋀_{v \in V} ((1 - R_{2} (v, w)) \lor A (w)] . \end{matrix}$

We call $\underline{R} (A)$ the multiple fuzzy lower approximation and $\bar{R} (A)$ the multiple fuzzy upper approximation of fuzzy set A ∈ F (W) with respect to fuzzy approximation space over three universes, respectively. Similar to the existing results [3 , 53], we call the interval $(\underline{R} (A), \bar{R} (A))$ the fuzzy rough set over three universes when $\underline{R} (A) \neq \bar{R} (A));$ Otherwise, we call the fuzzy set A ∈ F (W) the definable concept with respect to fuzzy approximation space over three universes.

Especially, the lower and upper approximations given in Definition 3.3 will be changed into the following forms when U, V and W are infinite set. For any w ∈ W, there are $\begin{matrix} \bar{R} (A) (w) & = & [sup_{u \in U} R_{1} (u, w) \land A (w)] \land \\ [sup_{v \in V} R_{2} (v, w) \land A (w)]; \\ \underline{R} (A) (w) & = & [inf_{u \in U} ((1 - R_{1} (u, w)) \lor A (w)] \land \\ [inf_{v \in V} ((1 - R_{2} (v, w)) \lor A (w)] . \end{matrix}$

Remark 3.1. From the definition of the multiple fuzzy lower approximation $\underline{R} (A)$ and multiple fuzzy upper approximation $\bar{R} (A),$ we can easily present a directly explanation by using the background of the management decision-making problem given in Example 3.1: For any patient A of a doctor, it is easy to know that the patient A is a fuzzy set with respect to the universe W which includes all possible disease w ∈ W (this means that there exists various possibility of patent A may be has every diseases of universe W, i.e., A ∈ F (W)); the multiple fuzzy rough approximation $\underline{R} (A)$ is the conservatively (or pessimistic) judgement of the doctor according to the characteristic of symptoms R₁ (u, w) and the results of clinical examination R₂ (v, w) of the patient A about the concretely disease w (w∈ W) ; the multiple fuzzy rough approximation $\bar{R} (A)$ is the riskily (or optimistic) judgement of the doctor according to the characteristic of symptoms R₁ (u, w) and the results of clinical examination R₂ (v, w) of the patient A about the concretely disease w (w ∈ W) .

Remark 3.2. As is well known, the existing fuzzy rough sets over two universes define a pair of fuzzy approximation operators between two different universes of discourse (i.e., $\underline{R}, \bar{R} : F (U) \to F (V) (or F (V) \to F (U))$ ) [22 , 56]. However, the fuzzy rough set model over three universes given in Definition 3.3 defines a pair of fuzzy approximation operators on the same universe which included in the three universes of discourse. For this characteristic of the fuzzy approximation operators over three universes (i.e., $\underline{R}, \bar{R} : F (W) \to F (W)$ ), we can present a kind of interpretation and then show the rationality and usefulness for this definition by using the background of the management decision-making problem given in Example 3.1: For a given patient A of a doctor, A ∈ F (W) means that there exists various possibility of patent A may be has every diseases of universe W, this is the original judgement (or cognitive) of the doctor according to the available limited information provided by the patient. Subsequently, the doctor will update the original judgement (or cognitive) after making several clinical examinations (i.e., R₂ ∈ F (V × W)) and the further clinical symptoms analysis (i.e., R₁ ∈ F (U × W)) for the patient A . Then, the lower fuzzy approximation $\underline{R} (A) \in F (W)$ and upper fuzzy approximation $\bar{R} (A) \in F (W)$ are the update judgement (or cognitive) of the doctor. So, the definition of the fuzzy rough approximation operators over three universes also describe the cognitive process of humans for the uncertainty decision-making problems.

In the following, we present an numerical example to illustrate the multiple fuzzy lower approximation $\underline{R}$ and multiple fuzzy upper approximation $\bar{R}$ in fuzzy approximation space over three universes.

Example 3.2. (Continued From Example 3.1) Let us consider a concretely numerical for the background of the management decision-making problem given in Example 3.1. Tables 1 and 2 present the values of binary relation R₁ ∈ F (U × W) and R₂ ∈ F (V × W) .

Let the fuzzy set A over the universe W as follows: $A = \frac{0.2}{w_{1}} + \frac{0.8}{w_{2}} + \frac{0.6}{w_{3}} + \frac{0.3}{w_{4}} + \frac{0.6}{w_{5}} + \frac{0.1}{w_{6}} \in F (W)$

Then, we can obtain the multiple fuzzy rough lower approximation $\underline{R} (A)$ and multiple fuzzy rough upper approximation $\bar{R} (A)$ are as follows, respectively.

$\bar{R} (A) (w_{1}) = [(0.3 \land 0.2) \lor (0.1 \land 0.2) \lor (0.4 \land 0.2) \lor (0.4 \land 0.2) \lor (0.1 \land 0.2) \lor (0.1 \land 0.2) \lor (0.5 \land 0.2)] \land [(0.4 \land 0.2) \lor (0.7 \land 0.2) \lor (0.3 \land 0.2) \lor (0.1 \land 0.2) \lor (0.1 \land 0.2)] = 0.2;$

$\underline{R} (A) (w_{1}) = [((1 - 0.3) \lor 0.2) \land ((1 - 0.1) \lor 0.2) \land ((1 - 0.4) \lor 0.2) \land ((1 - 0.4) \lor 0.2) \land ((1 - 0.1) \lor 0.2) \land ((1 - 0.1) \lor 0.2) \land ((1 - 0.5) \lor 0.2)] \land [((1 - 0.4) \lor 0.2) \land ((1 - 0.7) \lor 0.2) \land ((1 - 0.3) \lor 0.2) \land ((1 - 0.1) \lor 0.2) \land ((1 - 0.1) \lor 0.2)] = 0.3 .$

The others membership degree of w₂, w₃, ⋯ , w₆ with respect to the multiple fuzzy rough lower and upper approximations can be obtained as similar as the way of w₁ . So, we have $\begin{matrix} \bar{R} (A) & = & \frac{0.2}{w_{1}} + \frac{0.5}{w_{2}} + \frac{0.6}{w_{3}} + \frac{0.3}{w_{4}} \\ + \frac{0.4}{w_{5}} + \frac{0.1}{w_{6}} \in F (W); \\ \underline{R} (A) & = & \frac{0.3}{w_{1}} + \frac{0.8}{w_{2}} + \frac{0.6}{w_{3}} + \frac{0.5}{w_{4}} \\ + \frac{0.6}{w_{5}} + \frac{0.4}{w_{6}} \in F (W) . \end{matrix}$

By this numerical example, it is easy to see that the relationship $\underline{R} (A) \subseteq A \subseteq \bar{R} (A)$ not hold. Because the binary fuzzy relation R₁ and R₂ over universe U and W, V and W are not symmetric and transitive.

Similar to the existing fuzzy rough set theory [36 , 51], we also can present the multiple rough approximation for the α-level set of a fuzzy set with respect to the approximation space over three universes. Let $(U, V, W, R)$ be a fuzzy approximation space over three universes, R₁ ∈ F (U × W) and R₂ ∈ F (V × W) be two binary fuzzy relations over universes U, V and W . For any fuzzy set A ∈ F (W) and ∀α ∈ [0, 1] , w ∈ W the lower and upper approximations of A_α with respect to multiple fuzzy relation $R$ are as follows, respectively. $\begin{matrix} \bar{R} (A_{α}) (w) & = & max_{u \in U} R_{1} (u, w) \land max_{v \in V} R_{2} (v, w); \\ \underline{R} (A_{α}) (w) & = & min_{u \in U} ((1 - R_{1} (u, w)) \\ \land min_{v \in V} ((1 - R_{2} (v, w)) . \end{matrix}$

The $\bar{R} (A_{α})$ and $\underline{R} (A_{α})$ present a way to rough approximation of a crisp concept on approximation space over three universes. Here, the parameter α ∈ [0, 1] can be regarded as the threshold value of the risk preference of a decision-maker for the uncertainty decision-making problems.

Based on the Definition 3.3, the following properties are clear.

Remark 3.3. Let $(U, V, W, R)$ be an approximation space over three universes. If U = V, for any w ∈ W we have $\begin{matrix} \bar{R} (A) (w) & = & ⋁_{u \in U} R (u, w) \land A (w); \\ \underline{R} (A) (w) & = & ⋀_{u \in U} ((1 - R (u, w)) \lor A (w) . \end{matrix}$ where R ∈ F (U × V) is a binary fuzzy relation between universe U and W .

Though the fuzzy rough approximations $\underline{R} (A)$ and $\bar{R} (A)$ are defined on fuzzy approximation space over two universes $(U, W, R),$ there presents a pair of fuzzy approximation operators $\underline{R}$ and $\bar{R}$ on the universe W . So, we establish another way to construct the fuzzy rough set model over two different universes of discourse based on the interpretation given in Remark 3.2.

Remark 3.4. In general, if A is a crisp set (or concept) of the universe W, then the lower and upper approximations of A and ∀w ∈ W with respect to multiple fuzzy relation $R$ are as follows, respectively. $\begin{matrix} \bar{R} (A) (w) & = & [⋁_{u \in U} R_{1} (u, w)] \land [⋁_{v \in V} R_{2} (v, w)]; \\ \underline{R} (A) (w) & = & [⋀_{u \in U} (1 - R_{1} (u, w))] \\ \land [⋀_{v \in V} (1 - R_{2} (v, w))] . \end{matrix}$

By the above definition of the fuzzy rough set on the approximation space over three universes, we can easily obtained the properties for the multiple fuzzy lower approximation operator $\underline{R}$ and multiple upper approximation operator $\bar{R}$ are as following, respectively.

Theorem 3.1. Let $(U, V, W, R)$ be a fuzzy approximation space over three universes. The lower and upper fuzzy rough approximation operators, $\underline{R}$ and $\bar{R},$ satisfy the properties: for any fuzzy subsets A, B ∈ F (W) ,

$A \subseteq B \Rightarrow \underline{R} (A) \subseteq \underline{R} (B),$ $\bar{R} (A) \subseteq \bar{R} (B);$

$\underline{R} (A \cap B) \subseteq \underline{R} (A) \cap \underline{R} (B),$ $\bar{R} (A \cup B) \supseteq \bar{R} (A) \cup \bar{R} (B);$

$\underline{R} (A \cup B) \supseteq \underline{R} (A) \cup \underline{R} (B),$ $\bar{R} (A \cap B) \subseteq \bar{R} (A) \cap \bar{R} (B) .$

Proof. It can be easily verified by the Definition 3.3.

So far we have established the fuzzy rough set model over three universes. Actually, more than three universes of the discussed objects may be included for a given decision-making problem in reality. Then we present the general form of the fuzzy rough set model over multiple different universes in the following section.

3.2 Fuzzy rough set over multi-universes

Similar to the discussion in Section 3.1, we first present the concept of the multiple fuzzy relation over multiple different universes.

Definition 3.4. Let U₁, U₂, ⋯ , U_N be N different non-empty finite universes. We call a fuzzy subset $R \in F (U_{1} \times U_{2} \times \dots \times U_{N})$ the multiple fuzzy relation over universe U₁, U₂, ⋯ , U_N i.e., for any $u_{j}^{1} \in U_{1}, u_{k}^{2} \in U_{2}, \dots, u_{n}^{N} \in U_{N},$ there is $R (u_{j}^{1}, u_{k}^{2}, \dots, u_{n}^{N}) \in [0, 1] .$

Based on the definition of multiple fuzzy relation over three different universes, the following properties are clear.

Property 3.2. Let U₁, U₂, ⋯ , U_N be N different non-empty finite universes, and $R \in F (U_{1} \times U_{2} \times \dots \times U_{N})$ be a multiple fuzzy relation. Then

$R$ is pseudo reflexive, if $R (u_{j}^{1}, u_{k}^{2}, \dots, u_{n}^{N}) = 1 \Leftrightarrow R (u_{k}^{2}, u_{h}^{3}, \dots, u_{n}^{N}, u_{j}^{1}) = 1, \dots, R (u_{n}^{N}, u_{j}^{1}, \dots, u_{m}^{N - 1}) = 1,$ $\forall u_{j}^{1} \in U_{1},$ $u_{k}^{2} \in U_{2}, \dots, u_{n}^{N} \in U_{N};$

$R$ is pseudo symmetric, if $R (u_{j}^{1}, u_{k}^{2}, \dots, u_{n}^{N}) = R (u_{k}^{2}, u_{h}^{3}, \dots, u_{n}^{N}, u_{j}^{1}) = \dots = R (u_{n}^{N}, u_{j}^{1}, \dots, u_{m}^{N - 1})$ for any $u_{j}^{1} \in U_{1}, u_{k}^{2} \in U_{2}, \dots, u_{n}^{N} \in U_{N};$

$R$ is multiplicative, if $R (u_{j}^{1}, u_{k}^{2}, \dots, u_{n}^{N}) = R_{1} (u_{j}^{1}, u_{k}^{2}) \circ R_{2} (u_{k}^{2}, u_{h}^{3}) \circ \dots \circ R_{N} (u_{n}^{N}, u_{j}^{1}),$ for any $u_{j}^{1} \in U_{1},$ $u_{k}^{2} \in U_{2}, \dots, u_{n}^{N} \in U_{N};$ and R₁ ∈ F (U₁ × U₂) , R₂ ∈ F (U₂ × U₃) , ⋯, R_N ∈ F (U_N × U₁) .

Where j, k, h, m, n ∈ Γ are the indices set. Particularly, if U₁ = U₂ = ⋯ = U_N, then $R$ will be the multiple fuzzy relation over U .

We then present the concept of multi-universes approximation space.

Definition 3.5. Let U₁, U₂, ⋯ , U_N be N different non-empty finite universes, and $R \in F (U_{1} \times U_{2} \times \dots \times U_{N})$ be a multiple fuzzy relation. We call the N + 1-triple $(U_{1}, U_{2}, \dots, U_{N}, R R)$ fuzzy approximation space over multi-universes.

In following, we present the rough approximation of a fuzzy concept over multi-universes approximation space.

Definition 3.6. Let $(U_{1}, U_{2}, \dots, U_{N}, R R)$ be a fuzzy approximation space over multi-universes, R₁ ∈ F (U₁ × U_N) , R₂ ∈ F (U₂ × U_N) , ⋯, R_N−1 ∈ F (U_N−1 × U_N) be N − 1 binary fuzzy relations over universes U₁, U₂, ⋯ , U_N . For any fuzzy set $A \in F (U_{N}), u_{n}^{N} \in U_{N}$ the lower and upper approximations of A with respect to multiple fuzzy relation $R$ are as follows, respectively. $\begin{matrix} \bar{R} (A) (u_{n}^{N}) & = & ⋀_{i = 1}^{N - 1} [⋁_{u_{k}^{i} \in U_{i}} R_{i} (u_{k}^{i}, u_{n}^{N}) \land A (u_{n}^{N})]; \\ \underline{R} (A) (u_{n}^{N}) & = & ⋀_{i = 1}^{N - 1} [⋀_{u_{k}^{i} \in U_{i}} (1 - R_{i} (u_{k}^{i}, u_{n}^{N})) \\ \lor A (u_{n}^{N})] . \end{matrix}$

Similarly, We call $\underline{R} (A)$ the multiple fuzzy lower approximation and $\bar{R} (A)$ the multiple fuzzy upper approximation of fuzzy set A ∈ F (W) with respect to fuzzy approximation space over multi-universes, respectively. Furthermore, we call the interval $(\underline{R} (A), \bar{R} (A))$ the fuzzy rough set over multi-universes when $\underline{R} (A) \neq \bar{R} (A));$ Otherwise, we call the fuzzy set A ∈ F (W) the definable concept with respect to the multi-universes approximation space.

Meanwhile, we have the lower and upper approximations of A with respect to multiple fuzzy relation $R$ and $\forall u_{n}^{N} \in U_{N}$ are as follows when U₁, U₂, ⋯ , U_N are infinite set, respectively. $\begin{matrix} \bar{R} (A) (u_{n}^{N}) & = & ⋀_{i = 1}^{N - 1} [sup_{u_{k}^{i} \in U_{i}} R_{i} (u_{k}^{i}, u_{n}^{N}) \land A (u_{n}^{N})], \\ \underline{R} (A) (u_{n}^{N}) & = & ⋀_{i = 1}^{N - 1} [inf_{u_{k}^{i} \in U_{i}} (1 - R_{i} (u_{k}^{i}, u_{n}^{N})) \\ \lor A (u_{n}^{N})] . \end{matrix}$

Furthermore, we also have the following results for the α-level set of A ∈ F (W) with respect to multiple fuzzy relation $R$ and $\forall u_{n}^{N} \in U_{N},$ respectively. $\begin{matrix} \bar{R} (A_{α}) (u_{n}^{N}) & = & ⋀_{i = 1}^{N - 1} (max_{u_{k}^{i} \in U_{k}^{i}} R_{i} (u_{k}^{i}, u_{n}^{N})); \\ \underline{R} (A_{α}) (u_{n}^{N}) & = & ⋀_{i = 1}^{N - 1} (min_{u_{k}^{i} \in U_{i}} (1 - R_{i} (u_{k}^{i}, u_{n}^{N}))) . \end{matrix}$

It is easy to know that the multiple rough lower and upper approximations of a crisp concept with respect to the multi-universes approximation space $(U_{1}, U_{2}, \dots, U_{N}, R R)$ have the same formulation with the α-level set of a fuzzy concept on the universe of discourse by the discussion of Section 3.1.

Remark 3.5. If N = 3, then the multiple fuzzy rough lower approximation $\underline{R} (A)$ and multiple fuzzy rough upper approximation $\bar{R} (A)$ in Definition 3.6 are boil down to the results given in Definition 3.3. Moreover, we also can present s similar explanation for the multiple fuzzy rough lower and upper approximations of A with respect to fuzzy approximation space over multi-universes $(U_{1}, U_{2}, \dots, U_{N}, R R) .$

By the above definition of the fuzzy rough set over multi-universes approximation space, we can easily obtained the properties for the multiple fuzzy lower approximation operator $\underline{R}$ and multiple upper approximation operator $\bar{R}$ are as following, respectively.

Theorem 3.2. Let $(U_{1}, U_{2}, \dots, U_{N}, R R)$ be fuzzy approximation space over multi-universes. The lower and upper fuzzy approximation operators, $\underline{R}$ and $\bar{R},$ satisfy the properties: for any fuzzy subsets A_i ∈ F (U_N) (i = 1, 2, ⋯ , Γ) ,

$\underline{R} (\cap_{i = 1}^{Γ} A_{i}) \subseteq \cap_{i = 1}^{Γ} \underline{R} (A_{i}),$

$\bar{R} (\cup_{i = 1}^{Γ} A_{i}) \supseteq \cup_{i = 1}^{Γ} \bar{R} (A_{i});$

$\underline{R} (\cup_{i = 1}^{Γ} A_{i}) \supseteq \cup_{i = 1}^{Γ} \underline{R} (A_{i}),$

$\bar{R} (\cap_{i = 1}^{Γ} A_{i}) \subseteq \cap_{i = 1}^{Γ} \bar{R} (A_{i}) .$

Proof. It can be easily verified by the Definition 3.6.

Therefore, we established the fuzzy rough set over multi-universes. The proposed lower and upper approximations gives basic framework to investigate the rough approximation of a uncertainty concept in approximation space over multi-universes. As reviewed in former, the idea of rough set models over multi-universes comes from the uncertainty decision making of management science in reality. So, we present a new approach to uncertainty decision making based on fuzzy rough set over multi-universes in next section.

4 Fuzzy rough set over multi-universes based decision making

In this section, we will establish a new approach to multiple attribute decision making (MADM) with the characteristic of uncertainty, incomplete and inaccurate available information based on the fuzzy rough set model over multi-universes given in Section 3 in detail.

Since classical rough set was introduced by Pawlak in 1982, Pawlak rough set and its various extensions have been applied in dealing with decision making problems [30]. Like most of the decision making problems by using the existing theories and methods such as probability theory, fuzzy set theory, Dempster-Shafer theory, intuitionistic fuzzy set theory, soft set theory and so on, the existing results of classical rough set and its extensions forms based decision making are focus on the description of the uncertainty and inaccurate available information for a decision objects. Some of there problems are essentially humanistic and thus subjective in nature (e,g. human understanding and vision systems). In general, there actually does not exist a unique or uniform criterion to evaluate the superiority and the limitations of a decision making methods. Therefore, every existing decision approach could inevitably have their limitations and advantages more or less. In fact, all the existing approaches to multiple attribute decision making (MADM) with incomplete and inaccurate available information based on the aforementioned uncertainty mathematic theories and its extension forms have solved kinds of decision problem effectively. Practically, the key issue of the decision process for complexity, uncertainty and inaccurate of available information of a multiple attribute decision making problem in management science is considering all possible affected factors and then obtain a reasonable and optimal decision making alternative. Just as the aforementioned clinic medical diagnosis decision-making problem, an exactly and scientific decision-making of a doctor is based on considering both the symptoms and the results of clinical examination of a given patient.

In this paper, we propose an new approach to multiple attribute decision making (MADM) with the characteristic of uncertainty, incomplete and inaccurate available information based on fuzzy rough set theory over multi-universes. The basic idea of the proposed approach to decision making is considering all attributes (or factors) of the decision problem as much as possible under the framework of multiple different universes (or all related attributes (factors) with the decision making problem are described by several different universes according to their properties and classifications). Then the multiple attribute decision making problem is transformed into a uncertainty decision making problem over multiple different universes. And then the limited decision information provided by the considered objects are represented and updated by using the theory of fuzzy rough set over multi-universes. Therefore, a further and overall recognition of the given decision objects is obtained. Finally, we can present the decision rules and computing methods for the proposed method by combining the risk decision making principle of classical operational research. This approach will using the data information provided by the decision making problem only and does not need any additional available information provided by decision makers or other ways. Also, the subjective judgement and risk preference information of a decision-maker for the decision problems are included in the results. So, the decision results could be more objectively and match the practically scenario as well as possible and also could avoid the paradox results for the same decision problem of the existing methods.

In the following subsections, we will present the decision steps and the algorithm for the new approach, respectively, in detail.

4.1 Steps for fuzzy rough set over multi-universes based decision making

Considering a multiple attribute decision making problem with uncertainty, incomplete and inaccurate available information of management science of reality.

First of all. we present the mainly decision process and the basic idea of the proposed method under the framework of fuzzy rough set theory over multi-universes as follows.

Firstly, we consider all attributes in multiple attribute decision making as the characteristic factor description set of one aspect of the decision making problem according to their properties and classifications. For example, it is easy to know that the aforementioned clinical medical diagnosis decision making problem is a multiple attribute decision making (MADM), then the three decision attributes clinical symptoms, the results of clinical examination and the possible diseases can be represented by three different but closely related universes.

Secondly, we present, respectively, the quantitative numerical descriptions of the elements between different characteristic factor universes for the multiple attribute decision making (MADM) problem. That is, the related degree of the elements between different characteristic factor universes. Broadly speaking, the quantitative numerical between the elements of different universes are given by using the existing results or the historical statistics data.

Meanwhile, we present a fuzzy representation for the decision making object with one of the characteristic factors universe according to the uncertainty, incomplete and inaccurate available information. Just as the clinical medical diagnosis decision making, a given patient is a fuzzy subset over the disease universe according to the original uncertainty information.

Finally, we construct the approximation space over multiple different universes based on the former two processes. Then we will obtain a pair of new fuzzy subsets related to the decision making object by using the theory of fuzzy rough set over multi-universes. Furthermore, we can establish the decision rules based on the two new fuzzy sets and the risk decision making principle of classical operational research.

As is well known, the subjective judgment and the risk preference of the decision-makers in face of uncertainty and inaccurate decision making environment are very important for the decision results. So, the key role of the decision-makers are included by introducing the idea of risk decision making principle for the uncertainty and inaccurate multiple attribute decision making.

Next, according to the mainly decision making process described in the above, we present the detailed decision making steps for multiple attribute decision making (MADM) by using the fuzzy rough set theory over multi-universes given in Section 3. We present the general formulation of the proposed decision method based on fuzzy rough set over multi-universes as follows.

Suppose that U₁, U₂, ⋯ , U_N are N different finite attribute sets of a multiple attribute decision making (MADM). Then we will obtain the quantitative numerical descriptions of the elements among the universe U₁, U₂, ⋯ , U_N according to the existing results or the historical statistics data of the similar decision making problem. That is, the binary fuzzy relation R₁ ∈ F (U₁ × U_N) , R₂ ∈ F (U₂ × U_N) , ⋯, R_N−1 ∈ F (U_N−1 × U_N) , i.e., we obtain the multiple fuzzy relation $R (u_{j}^{1}, u_{k}^{2}, \dots, u_{n}^{N})$ (where j, k, ⋯ , n are indices sets) over the N different finite universes.

Let A be a given objects about the multiple attribute decision making (MADM). As the aforementioned, the decision-maker only can present a fuzzy description for A about the characteristic factor set U_i (i = 1, 2, ⋯ , N) since there is incomplete and inaccurate available information. Then, A is a fuzzy set of universe U_i (i = 1, 2, ⋯ , N) , i.e., A ∈ F (U_i) (i = 1, 2, ⋯ , N) .

So, we transform a multiple attribute decision making (MADM) into a uncertainty decision making process over multiple different universes. Moreover, we have constructed the multi-universes decision information systems $(U_{1}, U_{2}, \dots, U_{N}, R, A) .$

Now we present the multiple fuzzy lower approximation $\underline{R} (A)$ and multiple fuzzy upper approximation $\bar{R} (A)$ of decision object A with respect to multi-universes decision information systems $(U_{1}, U_{2}, \dots, U_{N}, R, A),$ for any $u_{k}^{i} \in U_{i}, R_{l} \in F (U_{l} \times U_{i}), l \neq i,$ there are $\bar{R} (A) (u_{k}^{i}) = ⋀_{l = 1}^{N - 1} [⋁_{u_{k}^{i} \in U_{i}} R_{l} (u_{h}^{l}, u_{k}^{i}) \land A (u_{k}^{i})],$ (1) $\underline{R} (A) (u_{k}^{i}) = ⋀_{l = 1}^{N - 1} [⋀_{u_{k}^{i} \in U_{i}} (1 - R_{l} (u_{h}^{l}, u_{k}^{i})) \lor A (u_{k}^{i})],$ (2)

Actually, from the point of view of risk decision-making principle of classical operational research, we can present the new interpretation for multiple fuzzy lower approximation $\underline{R} (A)$ and multiple fuzzy upper approximation $\bar{R} (A)$ of decision object A as follows:

$\underline{R} (A)$ is the max − min decision criterion (or pessimistic decision rule) in risk decision-making principle of classical operational research [34].

$\bar{R} (A)$ is the max − max decision criterion (or optimistic decision rule) in risk decision-making principle of classical operational research [34].

Meanwhile, from the former discussion, we know that both $\underline{R} (A)$ and $\bar{R} (A)$ are the fuzzy subset of universe U_i . Moreover, the semantic interpretation about a given decision making problem of $\underline{R} (A)$ and $\bar{R} (A)$ are same to the Remark 3.2 completely. At the same time, the relationship $\underline{R} (A) (u_{k}^{i}) \subseteq \bar{R} (A) (u_{k}^{i})$ for any $u_{k}^{i} \in U_{i}$ may be not hold according to the Example 3.2.

Therefore, we define a preference difference function δ_k for any objects $u_{k}^{i} \in U_{i}$ as follows: $δ_{k} = | \bar{R} (A) (u_{k}^{i}) - \underline{R} (A) (u_{k}^{i}) | u_{k}^{i} \in U_{i},$ (3)

Because $\underline{R} (A) (u_{i}) \subseteq \bar{R} (A) (u_{k}^{i})$ for any $u_{k}^{i} \in U_{i},$ then we know that δ_k ≥ 0 hold. The preference difference function $δ_{k} (u_{k}^{i} \in U_{i})$ describes the distance between pessimistic decision criterion and optimistic decision criterion for a decision-maker about the given decision making problem. So, the smaller of preference difference function $δ_{k} (u_{k}^{i}),$ the better of the object $u_{k}^{i}$ for given decision making problem. Thus, we obtain the optimal decision rule for multi-universes decision information systems $(U_{1}, U_{2}, \dots, U_{N}, R, A)$ as follows: $O_{A} = u_{t}^{i}, where t \in Γ = {k | min {δ_{k} | u_{k}^{i} \in U_{i}}},$ (4)

Remark 4.1. In general, if index sets Γ is not single-point set, anyone can be taken, i.e., there are multiple optimal decision making for given decision making problem.

Remark 4.2. From the procedure of decision making given in this section, it is easy to known that the method of decision making is based on the complete information, i.e., the values of the multiple fuzzy relation defined on the multiple attribute sets are available for the considered decision making problems. However, as pointed out by Ure $\tilde{n}$ a et al. [47], many required information for the decision making problems may missing because several alternatives choose from and/or conflicting and dynamic sources of information or the experts may not possess a precise or sufficient level of knowledge of the whole problem to tackle, including the ability to discriminate the degree up to which some options are better than others, i.e., the available information for considered decision making problems are missing data. Then, some approaches to decision making problems with incomplete information are proposed by many scholars. So, the future work will be attempted to handle to the multiple fuzzy relation with missing values for the decision making problems by using the idea and methods of the existing approaches [47].

4.2 Algorithm for fuzzy rough set over multi-universes based decision making

In this subsection, we present the algorithm for the approach to the decision making problem based on fuzzy rough set model over multi-universes.

Let U₁, U₂, ⋯ , U_N be N different finite universe of the discourse, and R₁ ∈ F (U₁ × U_N) , R₂ ∈ F (U₂ × U_N) , ⋯, R_N−1 ∈ F (U_N−1 × U_N) be N − 1 binary fuzzy relations. A is a fuzzy set of universe U_i (i = 1, 2, ⋯ , N) . Then we present the decision algorithm for multi-universes decision information systems $(U_{1}, U_{2}, \dots, U_{N}, R, A)$ as follows:

Input. Multi-universes decision information systems $(U_{1}, U_{2}, \dots, U_{N}, R, A)$

Output. The optimal decision making for A .

Step 1. Computing multiple fuzzy lower approximation $\underline{R} (A)$ and multiple fuzzy upper approximation $\bar{R} (A)$ of fuzzy subset A of universe U_i about fuzzy approximation space over multi-universes $(U_{1}, U_{2}, \dots, U_{N}, R);$

Step 2. Computing preference difference function δ_k ;

Step 3. Computing index set Γ ;

Step 4. Using the decision rule $O_{A} = u_{t}^{i}$ presents the optimal decision making.

4.3 A numerical example

In this subsection, we will show the principal and steps of the approach to decision making proposed in this paper by using the numerical Example 3.2 given in Section 3. The basic characteristic of the medical diagnosis decision making problem is described as follows:

Suppose that U = {fever (u₁) , headache (u₂) , stomachache (u₃) , cough (u₄) , myalgia (u₅) , regurgitation (u₆) , diarrhea (u₇)} is the symptom set, and V = {electrocardiography (v₁) , magnetic resonance imaging (v₂, peritoneoscope (v₃) , percutaneous umbilical blood sampling (v₄) , stomachachescope (v₅)} is the results of examination for the patient A . Let W = {cold (w₁) , pneumonia (w₂) , gastricism (w₃) , dysentery (w₄) , typhoid (w₅) , acute appendicitis (w₆)} is the disease set.

The related degree R₁ ∈ F (U × W) and R₂ ∈ F (V × W) of the elements between the three universes can be obtained by using the existing results or the historical statistics data. The numerical value of R₁ (u, w) and R₂ (v, w) are presented in Tables 1 and 2. The beginning judgement of the patient A by the doctor according to the basic clinical symptoms are described as follows: $A = \frac{0.2}{w_{1}} + \frac{0.8}{w_{2}} + \frac{0.6}{w_{3}} + \frac{0.3}{w_{4}} + \frac{0.6}{w_{5}} + \frac{0.1}{w_{6}} \in F (W) .$

Thus, we construct the medical diagnosis decision information system $(U, V, W, R, A) .$

So, based on the numerical results given in Example 3.2, we have $\begin{matrix} \underline{R} (A) & = & \frac{0.2}{w_{1}} + \frac{0.5}{w_{2}} + \frac{0.6}{w_{3}} + \frac{0.3}{w_{4}} \\ + \frac{0.4}{w_{5}} + \frac{0.1}{w_{6}} \in F (W); \\ \bar{R} (A) & = & \frac{0.3}{w_{1}} + \frac{0.8}{w_{2}} + \frac{0.6}{w_{3}} + \frac{0.5}{w_{4}} \\ + \frac{0.6}{w_{5}} + \frac{0.4}{w_{6}} \in F (W) . \end{matrix}$

Then, using the formula $δ_{k} = | \bar{R} (A) (w_{k}) - \underline{R} (A) (w_{k}) |, w_{k} \in W,$ we can compute the preference difference function δ_k (k = 1, 2, ⋯ , 6) as follows, respectively. $\begin{matrix} δ_{1} & = & 0.3 - 0.2 = 0.1, δ_{2} = 0.8 - 0.5 = 0.3, \\ δ_{3} & = & 0.6 - 0.6 = 0, δ_{4} = 0.5 - 0.3 = 0.2, \\ δ_{5} & = & 0.6 - 0.4 = 0.2, δ_{6} = 0.4 - 0.1 = 0.3 . \end{matrix}$

So, using the optimal decision rule O_A = w_t, t ∈ Γ = {k| min {δ_k|w_k ∈ W}} for the medical diagnosis decision information systems $(U, V, W, R, A),$ we can obtain the optimal decision making as: O_A = w₃ .

Therefore, the doctor gives the decision is that the patient A has the disease w₃ (gastricism) .

The numerical example with the background of medical diagnosis decision making in clinic shows the basic steps of the proposed decision method. It can be easily seen that the risk preference of decision-makers were included successfully for a uncertainty decision making problem of reality. So, the decision results will more reasonable and scientific and more suitable for the scenarios of reality.

Remark 4.3. Comparing the existing approaches to decision making under uncertainty based on vary rough set theories over two universes [24 , 38–40],the multi-universes rough set theory based decision making method provides an effectiveness way to deal with the decision making problems under uncertainty which close to the characteristics and background as possible as the reality in management sciences. In Ref. [24], the author proposes a multiple criteria decision making method by combining the fuzzy rough set over two universes and the principle of maximum membership of Zadeh fuzzy set theory [65]. Actually, the method given in Ref. [24] can be regarded as the Max-Max decision rule (i.e., optimistic decision rule) of our model. It is easy to know that the proposed method in this paper can decrease the influence of the subjective judgment of decision-makers as much as possible by using the preference difference function δ_k, which compromises two extreme cases (i.e., optimistic decision rule and pessimistic decision rule) and then can give a reasonable result. Though the above advantages can be easily verified for the proposed decision making method, it does not deal with the decision making problems with incomplete information [47] and also there may exist an extreme case that all values of the preference difference function are equal. Then we should improve or modify the values of multiple fuzzy relations over multiple universes and recalculated the results.

5 Conclusions and remarks

The generalization of the classical Pawlak rough set model always is an important research topic and several interesting and valuable extended models have established in the past few years. This paper studies the rough set model over multiple different universes. We first define the multiple fuzzy lower and upper approximations of a fuzzy concept with respect to fuzzy approximation space over three universes. We then establish a general formulation for multiple fuzzy lower and upper approximations of a fuzzy concept with respect to fuzzy approximation space over multi-universes, i.e., fuzzy rough set over multi-universes. Meanwhile, several interesting properties have discussed in detailed. Subsequently, we apply the fuzzy rough set method to multiple attribute decision making problems with the characteristic of uncertainty, incomplete and inaccurate available information and then propose a new approach to multiple attribute decision making (MADM) based on the fuzzy rough set model over multi-universes. The detailed decision steps, decision rules and algorithm of the proposed method are given and also the validity of proposed model are illustrated by using a numerical example with the background of medical diagnosis in clinic.

Decision making is a procedure to find the best alternative among a set of feasible alternatives. The solution of decision making problems can be complex as well as simple. The proposed method for multiple attribute decision making (MADM) with the characteristic of uncertainty, incomplete and inaccurate available information based on fuzzy rough set over multi-universes not only overcomes the inconsistencies of the quantitative semantics in the traditional multiple attribute evaluation method, but also fully considers the judgments about risk preference that are made by decision-makers during real-time scenarios with incomplete and inaccurate available information. Therefore, it shows exactly how the actors making real-time or on-line decisions with uncertainty and inaccurate available information make these decisions by adapting to the characteristics faced in reality. Clearly, there is no optimal decision-making method due to various kinds of practical problems under the conditional of uncertainty and an imprecise environment. Thus, a suitable and adaptable approach can be the best decision-making method in practice.

For further study, the primary theory and characterization of fuzzy rough set over multi-universes are needed. Although this paper focuses on the basic theory and decision-making principal steps with the fuzzy rough sets over multi-universes, it is recommended that real-life data be used to test the approach established in this paper.

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