Medicines selection via fuzzy upward β -covering rough sets

Abstract

The notions of fuzzy upward β-covering, the fuzzy upward β-neighborhood, upward β-neighborhood and fuzzy complement β-neighborhood are introduced and several related properties are studied. Furthermore, multigranulation optimistic/pessimistic fuzzy rough sets based on fuzzy upward β-covering are initiated and their fundamental properties are investigated. We also find the upward β-neighborhood in the fuzzy upward covering approximation space and present the optimistic/pessimistic multigranulation rough sets to further enrich the presented notions. The medicine selection via fuzzy upward β-covering rough sets in medical diagnosis is another main contribution of the present work. It is also explored that which medicine can be prescribed for which particular symtom(s) and which disease.

1 Introduction

Most of the theoretical work on decision making under uncertainty takes a certain type of the individual behavior as a primitive and the determines the preference functional that represents the behavior. In general the individual behavior is their private and professional life. The critical problem of decision making under uncertainty is how to deal with the individual behavior and the reach to final optimal objective. However, because of the complexity, inaccuracy and unstructured of the decision making problems under uncertainty, and the limitations of knowledge and cognitive for individual decision maker, it is difficult to acquire a reasonable and scientific decision making with only single decision maker under uncertainty in reality. In order to a reasonable and reliable optimal result, several experts come from different fields with different specialities are invited to constitute a group and handle the decision making problems together. So far the idea and principle of group decision making is used in many decision making problems emerged in management sciences, engineering management, and the social sciences. In the past few years, a large number of real world case studies and several new approaches and decision theories of multiple attribute group decision making problems in different domains are reported such as energy [28], logistics [21], safety management [18], facility location [1], business process management [17], supplier selection [19], etc. Another theory and methodology named as granular computing is introduced into multiple attribute group decision making problems and presents several interesting and valuable models and methods. Granular computing, established by Zadeh [48], as a new perspective and way to handle of the uncertainty. Granular computing is referred an umbrella term to cover several theories, methodology, techniques, and make use of information granules in complex problem solving [46 , 54–56]. Since the inception of rough set [30], number of generalizations have been proposed in terms of different demands. For example, rough sets based on set valued mapping [10], generalized rough sets based on relations [57], decision theoretic rough sets [9], fuzzy rough set/rough fuzzy sets [10, 43], decision theoretic rough fuzzy sets [37], variable precision rough sets [63], probabilistic rough sets [44], dominance based rough set approach [12 –14], multigranulation rough sets [32, 33], multigranulation decision theoretic rough sets [34], dominance based multigranulation rough sets [2], soft dominance based multigranulation rough sets and their application in conflict [35]. Covering based rough set theory was introduced by Zakowski et al. [49] as an extension of classical rough set theory. As the importance of covering based rough set theory grew, an increasing number of scholars emphasized many of its features. The lower and upper approximations of an arbitrary set are constructed in [3 , 59]. Pomykala [31] proposed two different types of covering based rough sets. Some researchers studied the covering based rough sets and the general covering based rough sets in [40, 62]. They also put forward topological approaches. To interpret the various aspects of covering based rough sets, several works were proposed by [24 , 62]. In additional investigations such as [25 , 59], other types of covering based rough sets are proposed and their relationships are discussed. D’eer et al. [7] combined Pawlak’s rough sets and covering based rough sets and proposed a semantically appealing approach to them. In recent years, initial efforts have been done to extend covering based rough set models to the fuzzy setting [22]. Some researchers [8 , 27] investigated fuzzy covering rough sets. De Cock et al. [5] gave the definition of fuzzy rough sets based on the R-foresets of all objects in a universe of discourse with respect to a fuzzy binary relation. When R is a fuzzy serial relation, the family of all R-foresets forms a fuzzy covering of the universe of discourse. A method using a novel fuzzy rough set based information entropy was put forward in [50]. Although fuzzy coverings were used by Li and Ma in [23], they only employed two special logical operators. It is therefore necessary to construct more general fuzzy rough sets based on fuzzy coverings. Following this method, D’eer et al. [6] have done remarkable efforts to generalize fuzzy rough sets based on fuzzy relations by using the concept of a fuzzy covering. The fuzzy covering based fuzzy rough sets and coverings based fuzzy rough sets were investigated by many researchers, such as [39 , 47]. In 2016, Ma [27] defined two types of fuzzy covering rough set models which appear to draw a bridge between covering rough set theory and fuzzy rough set theory. This work generalized the models and their matrix representations to L-fuzzy covering rough sets too. Yang and Hu [41, 42] and D’eer et al. [6] further studied fuzzy covering based on rough sets, and they proposed three types of fuzzy covering-based rough set models based on Ma’s models [27]. Rough set theory has diverse applications in medical sciences. Several authors applied this theory to solve problems in medical sciences (for example [4, 36]). The original definition of a fuzzy covering is defined in [22].

1.1 Motivation and significance

Let $U$ be the universe of discourse and $F (U)$ denotes the collection of all fuzzy subsets of $U$ . Let ${A_{1}, A_{2}, . . ., A_{n}}$ with $A_{i} \in F (U)$ (i = 1, 2, . . . , n), is a fuzzy covering of $U$ . Then $(⋃_{i = 1}^{n} A_{i}) (x) = 1$ for each $x \in U$ . Let $U = {x_{1}, x_{2}, x_{3}}$ be the set of medicines and ${K_{1}, K_{2}, K_{3}}$ be the set of criteria which represent first test, second test and third test. For these criteria ${C (K_{1}), C (K_{2}), C (K_{3})}$ , where $C (K_{1}) = \frac{0.5}{x_{1}} + \frac{0.5}{x_{2}} + \frac{0.3}{x_{3}},$ $C (K_{2}) = \frac{0.6}{x_{1}} + \frac{0.4}{x_{2}} + \frac{0.4}{x_{3}},$ $C (K_{3}) = \frac{0.8}{x_{1}} + \frac{0.7}{x_{2}} + \frac{0.4}{x_{3}},$ where $C (K_{i}) (x_{j})$ denotes the efficiency of the medicine x_j for the test $K_{i}$ . If the doctor wants to choose only one medicine of $U$ , then a natural trouble arises due to the fact that the value given by the doctor is critical. This problem arises in a standard evaluation context. It is not difficult to realize that the doctor would not be able to select the right medicine for disease by the recourse to fuzzy covering evaluation procedures. To overcome these limits, Ma [27]. generalized the fuzzy covering to fuzzy β-covering by replacing 1 with a parameter β (0 < β ≤ 1). Subsequently, Ma defined two new types of fuzzy covering based rough set models by introducing the new concept of fuzzy β-neighborhood. More work on this topic can be seen in [20 , 51–53]. But there are several shortcomings, for example, the above example isβ-covering for (0 < β ≤ 0.4). If the required critical value β = 0.5, then how is it possible to makeβ-covering for (0 < β ≤ 0.5) ? In this paper we present a more generalized form of fuzzy β-covering, namely the fuzzy upward β-covering. Based on this idea, we introduce new type of fuzzy upward covering based rough set model by introducing the concept of fuzzy upward β-neighborhood and their applications in medicine selection for a disease. Furthermore, we propose multigranulation optimistic/pessimistic fuzzy rough sets based on fuzzy upward β-covering and investigate some of their properties. Finally, we find upward β-neighborhood in the fuzzy upward covering approximation space and present optimistic/pessimistic multigranulation rough sets and their basic properties are discussed. The medicine selection via fuzzy upward β-covering rough sets in medical diagnosis is another main contribution of the present work. It is also seen that which medicine can be prescribed for which particular symptom(s) and which disease.

This paper consists of six sections. After the introductory section, the Section 2 consists of preliminary results that are needed for the rest of sections. Section 3 highlights the fuzzy upward β-covering fuzzy rough sets. Section 4 studies multigranulation fuzzy rough sets based on fuzzy upward β-covering and their fundamental proporties. Section 5 provides comparative analysis among the proposed techniques and other existing methods. Section 6 concludes this paper, where manly the outcomes of the proposed techniques are given.

2 Preliminaries

Definition 1. [15] A fuzzy preference relation ( $FPR$ ) $R$ is a fuzzy set on $U \times U$ , which is a membership function $μ_{R} : U \times U \to [0, 1]$ . For $U = {x_{1}, x_{2}, . . ., x_{n}}$ , the $FPR$ can also be represented by an n × n matrix (r_ij) _n×n, $R = {(r_{i j})}_{n \times n} = \begin{array}{l} \begin{array}{l} x_{1} \\ x_{2} \\ \cdot \\ \cdot \\ \cdot \\ x_{n} \end{array} \begin{array}{l} \begin{array}{l} x_{1} & x_{2} & \dots & x_{n} \end{array} \\ (\begin{array}{l} r_{11} & r_{12} & \dots & r_{n} \\ r_{21} & r_{22} & \dots & r_{2 n} \\ \cdot & \cdot & \dots & \cdot \\ \cdot & \cdot & \dots & \cdot \\ \cdot & \cdot & \dots & \cdot \\ r_{n 1} & r_{n 2} & \dots & r_{n n} \end{array}) \end{array} \end{array}$ where r_ij interpreted as the preference degree of feasible alternative x_i over feasible alternative x_j, r_ij ∈ [0, 1], r_ij + r_ji = 1, for all i, j∈ { 1, 2, . . . , n }. Especially, r_ij = 0.5 indicates that there is no difference between feasible alternative x_i and feasible alternative x_j; r_ij > 0.5 shows feasible alternative x_i is preferred to feasible alternative x_j; r_ij = 1 means feasible alternative x_i is absolutely preferred to feasible alternative x_j; the r_ij < 0.5 shows feasible alternative x_j is preferred to feasible alternative x_i; r_ij = 0 means feasible alternative x_j is absolutely preferred to feasible alternative x_i.

In the above definition, the $FPR$ is considered, r_ij merely presents the degree of preference of feasible alternative x_i is prior to the feasible alternative x_j. However, in practical applications, we need to show the degree of feasible alternative x_i is poor than the feasible alternative x_j. In order to satisfy all the two cases, we call the $FPR$ as upward fuzzy preference relation $(UFPR)$ and the other downward fuzzy preference relation $(DFPR)$ . The $UFPR$ denote as $R^{↑} = {(r_{ij}^{↑})}_{n \times n}$ and $DFPR$ as $R^{↓} = {(r_{ij}^{↓})}_{n \times n}$ . In general, $r_{ij}^{↑} + r_{ij}^{↓} = 1$ . For $DFPR,$ $r_{ij}^{↓} = 0.5$ indicates that there is no difference between the feasible alternative x_i and feasible alternative x_j; $r_{ij}^{↓} > 0.5$ shows feasible alternative x_i is poor than feasible alternative x_j; $r_{ij}^{↓} = 1$ means feasible alternative x_i is absolutely poor than feasible alternative x_j; $r_{ij}^{↓} < 0.5$ shows feasible alternative x_j is poor than feasible alternative x_i; $r_{ij}^{↓} = 0$ means feasible alternative x_j is absolutely poor than feasible alternative x_i.

Definition 2. A $FPR$ $R = {(r_{ij})}_{n \times n}$ is called an additive consistent $FPR$ , if r_ij = r_ik - r_kj + 0.5, for all i, j, k∈ { 1, 2, . . . , n }.

Hu et al. [16] adopted the well-known logis transfer function $\frac{1}{1 + e^{k (g (x_{i}, a) - g (x_{j}, a))}}$ to compute the fuzzy preference degree of the feasible alternative x_i to the feasible alternative x_j $r_{ij}^{↑} = \frac{1}{1 + e^{- k (g (x_{i}, a) + g (x_{j}, a))}}$ $r_{ij}^{↓} = \frac{1}{1 + e^{k (g (x_{i}, a) - g (x_{j}, a))}}$ where k is a positive constant. Pan et al. [29] point out that this transfer fuzzy preference degree is not additive consistent and they suggest another transfer function. The fuzzy preference degree of the feasible alternative x_i to the feasible alternative x_j $r_{ij}^{↑} = 0.5 \times (\frac{g (x_{i}, a) - \land_{i = 1}^{n} g (x_{i}, a)}{\land_{i = 1}^{n} g (x_{i}, a) - \lor_{i = 1}^{n} g (x_{i}, a)} - \frac{g (x_{j}, a) - \land_{i = 1}^{n} g (x_{i}, a)}{\land_{i = 1}^{n} g (x_{i}, a) - \lor_{i = 1}^{n} g (x_{i}, a)} + 1)$ (1) $r_{ij}^{↓} = 0.5 \times (\frac{g (x_{j}, a) - \land_{i = 1}^{n} g (x_{i}, a)}{\land_{i = 1}^{n} g (x_{i}, a) - \lor_{i = 1}^{n} g (x_{i}, a)} - \frac{g (x_{i}, a) - \land_{i = 1}^{n} g (x_{i}, a)}{\land_{i = 1}^{n} g (x_{i}, a) - \lor_{i = 1}^{n} g (x_{i}, a)} + 1)$ (2) when $\land_{i = 1}^{n} g (x_{i}, a) \neq \lor_{i = 1}^{n} g (x_{i}, a)$ , otherwise $r_{i j}^{↑} = r_{i j}^{↓} = 0.5$ ; where ∧ and ∨ are the minimum and maximum value of g (x_i, a), respectively. The upward and downward fuzzy preference classes $A_{i}^{R^{↑}}$ and $A_{i}^{R^{↓}}$ of x_i induced by the upward and downward additive fuzzy preference relation $(UAFPR)$ $R^{↑}$ and $(DAFPR)$ $R^{↓}$ are defined by $A_{i}^{R^{↑}} = \frac{r_{i 1}^{↑}}{x_{1}} + \frac{r_{i 2}^{↑}}{x_{2}} + . . . + \frac{r_{in}^{↑}}{x_{n}}$ $A_{i}^{R^{↓}} = \frac{r_{i 1}^{↓}}{x_{1}} + \frac{r_{i 2}^{↓}}{x_{2}} + . . . + \frac{r_{in}^{↓}}{x_{n}}$ where ‘+’ means the union operation. The $UAFPR$ and $DAFPR$ generate a family of fuzzy information granules from the universe, which composes the upward additive fuzzy preference granular structure $(UAFPGS)$ and downward additive fuzzy preference granular structure $(DAFPGS)$ , written by $P (R^{↑}) = {A_{1}^{R^{↑}}, A_{2}^{R^{↑}}, . . ., A_{n}^{R^{↑}}},$ and $P (R^{↓}) = {A_{1}^{R^{↓}}, A_{2}^{R^{↓}}, . . ., A_{n}^{R^{↓}}} .$

Definition 3. $U$ be an arbitrary universal set and $P (R^{↑})$ be an $UAFPGS$ . Then for each β ∈ (0, 1], $P (R^{↑})$ a fuzzy upward β-covering of $U$ , if $(⋃_{i = 1}^{n} A_{i}^{R ↑}) (x) \geq β$ for each $x \in U$ . The pair $(U, P (R^{↑}))$ is called fuzzy upward covering approximation space $(FUCAS)$ .

Definition 4. Let $(U, P (R^{↑}))$ be an $FUCAS$ , where $P (R^{↑})$ is a fuzzy upward β-covering of $U$ . For each $x \in U$ , we define the fuzzy β-upward neighborhood $N_{x}^{↑ β}$ of x as $N_{x}^{↑ β} = ⋂ {A_{i}^{R^{↑}} \in P (R^{↑}) : A_{i}^{R^{↑}} (x) \geq β} .$

Definition 5. $(U, P (R^{↑}))$ be an $FUCAS$ , where $P (R^{↑})$ is a fuzzy upward β-covering of $U$ . For each $x \in U$ , we define the fuzzy complementary β-upward neighborhood $M_{x}^{↑ β}$ of x as: $M_{x}^{↑ β} (y) = N_{y}^{↑ β} (x) for all y \in U .$

Proposition 1. For each $x \in U$ , $N_{x}^{↑ β} (x) \geq β$ .

Proof Let $x \in U$ . Then $N_{x}^{↑ β} (x) = (⋂_{A_{i}^{R^{↑}} (x) \geq β} A_{i}^{R^{↑}}) (x) = ⋀_{A_{i}^{R^{↑}} (x) \geq β} A_{i}^{R^{↑}} (x) \geq β .$ □

Proposition 2. For all x, y, $z \in U$ , if $N_{x}^{↑ β} (y) \geq β$ and $N_{y}^{↑ β} (z) \geq β$ , then $N_{x}^{↑ β} (z) \geq β$ .

Proof For $N_{x}^{↑ β} (y) \geq β$ and for each i∈ I = { 1, 2, . . . , n }, if $A_{i}^{R^{↑}} (x) \geq β$ , then $A_{i}^{R^{↑}} (y) \geq β$ . Similarly if $N_{y}^{↑ β} (z) \geq β$ which implies $A_{i}^{R^{↑}} (y) \geq β$ , and thus $A_{i}^{R^{↑}} (z) \geq β$ . Hence, for each i ∈ I, $A_{i}^{R^{↑}} (x) \geq β$ implies $A_{i}^{R^{↑}} (z) \geq β$ . Therefore $N_{x}^{↑ β} (z) \geq β$ . □

Proposition 3. If β₁ ≤ β₂, then $N_{x}^{↑ β_{1}} \subseteq N_{x}^{↑ β_{2}}$ for all $x \in U$ .

Proof. Let β₁ ≤ β₂ for each $x \in U$ . ${A_{i}^{R^{↑}} :$ $A_{i}^{R^{↑}} (x) \geq β_{1}}$

$\supseteq {A_{i}^{R^{↑}} : A_{i}^{R^{↑}} (x) \geq β_{2}}$ . Hence $N_{x}^{↑ β_{1}} = ⋂$ ${A_{i}^{R^{↑}} : A_{i}^{R^{↑}} (x) \geq β_{1}}$

$\subseteq ⋂ {A_{i}^{R^{↑}} : A_{i}^{R^{↑}} (x) \geq β_{2}} = N_{x}^{↑ β_{2}}$ . □

3 Fuzzy upward β-covering fuzzy rough set

In this section, we discuss the rough approximation of a fuzzy concept with respect to fuzzy upward β-covering environment. In the following we propose fuzzy rough set based on fuzzy upward β-covering and investigate their fundamental properties.

Definition 6. $(U, P (R^{↑}))$ be a $FUCAS$ , where $P (R^{↑})$ is a fuzzy upward β-covering of $U$ . For each fuzzy subset μ of $U$ , the lower approximation ${(LA μ)}^{↑}$ and the upper approximation ${(UA μ)}^{↑}$ of μ are given respectively by: ${(LA μ)}^{↑} (x) = ⋀_{y \in U} {(1 - N_{x}^{↑ β} (y)) \lor μ (y)}, x \in U,$ ${(UA μ)}^{↑} (x) = ⋁_{y \in U} {N_{x}^{↑ β} (y) \land μ (y)}, x \in U .$

Proposition 4. Let $(U, P (R^{↑}))$ be an $FUCAS$ where $P (R^{↑})$ is a fuzzy upward β-covering of $U$ for some β ∈ (0, 1]. Then for any α, $λ \in F (U)$ , we have

${(LA α^{c})}^{↑} = {({(UA α)}^{↑})}^{c}$ , ${(UA α^{c})}^{↑} = {({(LA α)}^{↑})}^{c}$ , where α^c (x) = 1 - α (x),

${(LAU)}^{↑} = U$ , ${(UA \emptyset)}^{↑} = \emptyset$ ,

${(LA (α \cap λ))}^{↑} = {(LA α)}^{↑} \cap {(LA λ)}^{↑}$ ,

${(UA (α \cup λ))}^{↑} = {(UA α)}^{↑} \cup {(UA λ)}^{↑}$ ,

If α ⊆ λ, then ${(LA α)}^{↑} \subseteq {(LA λ)}^{↑}$ and ${(UA α)}^{↑} \subseteq {(UA λ)}^{↑}$ ,

${(LA α)}^{↑} \cup {(LA λ)}^{↑} \subseteq {(LA (α \cup λ))}^{↑}$ ,

${(UA (α \cap λ))}^{↑} \subseteq {(UA α)}^{↑} \cap {(UA λ)}^{↑}$ ,

If $1 - N_{x}^{↑ β} (x) \leq α (x) \leq N_{x}^{↑ β} (x)$ for all $x \in U$ , then ${(LA α)}^{↑} \subseteq α \subseteq {(UA α)}^{↑}$ .

Proof. The proof is straightforward. □

Proposition 5. Let $(U, P (R^{↑}))$ be an $FUCAS$ where $P (R^{↑})$ is a fuzzy upward β-covering of $U$ for some β ∈ (0, 1]. Then for any $α \in F (U)$ and δ ∈ [0, 1], we have

${(LA (α \cup δ_{U}))}^{↑} = {(LA α)}^{↑} \cup δ_{U}$ ,

${(UA (α \cap δ_{U}))}^{↑} = {(UA α)}^{↑} \cap δ_{U}$ , where $δ_{U}$ is the constant fuzzy set i.e. $δ_{U} (x) = δ$ for each $x \in U$ .

Proof. The proof is straightforward. □

We now give a practical example of the fuzzy upward covering based rough set model.

Example 1. While treatment a disease, we usually combine some kinds of medicines to treat the disease.

Let $U = {x_{j} : j = 1, 2, . . ., n}$ be the universe of n kinds of medicines, $V = {y_{i} : i = 1, 2, . . ., m}$ be m main symptoms (for example dizzy giddy, cough, fever, etc.) of a disease A, and E be a finite set of the domain for the information function g (x_i, a). In this study, the value of information function g (x_i, a) is belonging to [0, 1], which shows the degree of recommendation of medicine x_i by the doctor a. $A_{i}^{R_{a} ↑} (x_{j})$ denote the efficacy value of the medicine x_j for the symptom y_i (i = 1, 2, . . . , m, j = 1, 2, . . . , n). For a critical value β suppose that for each medicine $x_{j} \in U$ , there is at least one symptoms $y_{i} \in V$ such that the efficacy value of the medicine x_j for the symptom y_i is not less than β, and $P (R_{a}^{↑})$ is a fuzzy upward β-covering of $U$ . Then the fuzzy upward β-neighborhood $a N_{x_{j}}^{↑ β}$ of x_j with respect to a is a fuzzy set given by $a N_{x_{j}}^{↑ β} (x_{k}) = [⋂ {A_{i}^{R_{a} ↑} : A_{i}^{R_{a} ↑} (x_{j}) \geq β}] (x_{k}), k = 1, 2, . . ., n,$ which denotes the minimum value among all the efficacy values of each medicine x_k for treating the symptoms. If a fuzzy set μ denotes the ability of all medicines in $U$ to cure the disease A, since the inaccuracy of μ, then we can take it approximate evaluation according to the lower and upper approximation of μ.

Let $U = {x_{j} : j = 1, 2, . . ., 9}$ be the set of medicines and a be the criterion. The evaluation of $U$ by the a is given in Table 1.

Table 1
Information Table

x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈ x ₉

a ₁ 0.8 0.3 0.2 0.6 0.4 0.2 0.3 0.3 0.3

	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈	x ₉
a ₁	0.8	0.3	0.2	0.6	0.4	0.2	0.3	0.3	0.3

Using the relation (1) to compute the fuzzy preference degree of the alternative x_i (i = 1, 2, . . . , 9) to the x_j (j = 1, 2, . . . , 9), we have $\begin{matrix} R_{a_{1}}^{↑} (x_{i}, x_{j}) = (\begin{matrix} 0.5000 & 0.9167 & 1 & 0.6667 & 0.8333 & 1 & 0.9167 & 0.9167 & 0.9167 \\ 0.0833 & 0.5000 & 0.5833 & 0.2500 & 0.4167 & 0.5833 & 0.5000 & 0.5000 & 0.5000 \\ 0 & 0.4167 & 0.5000 & 0.1667 & 0.3333 & 0.5000 & 0.4167 & 0.4167 & 0.4167 \\ 0.3333 & 0.7500 & 0.8333 & 0.5000 & 0.6667 & 0.8333 & 0.7500 & 0.7500 & 0.7500 \\ 0.1667 & 0.5833 & 0.6667 & 0.3333 & 0.5000 & 0.6667 & 0.5833 & 0.5833 & 0.5833 \\ 0 & 0.4167 & 0.5000 & 0.1667 & 0.3333 & 0.5000 & 0.4167 & 0.4167 & 0.4167 \\ 0.0833 & 0.5000 & 0.5888 & 0.2500 & 0.4167 & 0.5833 & 0.5000 & 0.5000 & 0.5000 \\ 0.0833 & 0.5000 & 0.5888 & 0.2500 & 0.4167 & 0.5833 & 0.5000 & 0.5000 & 0.5000 \\ 0.0833 & 0.5000 & 0.5888 & 0.2500 & 0.4167 & 0.5833 & 0.5000 & 0.5000 & 0.5000 \end{matrix}) . \end{matrix}$ The upward fuzzy preference classes $A_{i}^{R_{a_{1}} ↑}$ are defined by: $\begin{matrix} A_{1}^{R_{a_{1}} ↑} & = & \frac{0.5000}{x_{1}} + \frac{0.9167}{x_{2}} + \frac{1.0000}{x_{3}} + \frac{0.6667}{x_{4}} + \frac{0.8333}{x_{5}} \\ + \frac{1.0000}{x_{6}} + \frac{0.9167}{x_{7}} + \frac{0.9167}{x_{8}} + \frac{0.9167}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{2}^{R_{a_{1}} ↑} & = & \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5833}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} \\ + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{3}^{R_{a_{1}} ↑} & = & \frac{0.0000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.3333}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{4}^{R_{a_{1}} ↑} & = & \frac{0.3333}{x_{1}} + \frac{0.7500}{x_{2}} + \frac{0.8333}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.6667}{x_{5}} \\ + \frac{0.8333}{x_{6}} + \frac{0.7500}{x_{7}} + \frac{0.7500}{x_{8}} + \frac{0.7500}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{5}^{R_{a_{1}} ↑} & = & \frac{0.1667}{x_{1}} + \frac{0.5833}{x_{2}} + \frac{0.6667}{x_{3}} + \frac{0.3333}{x_{4}} + \frac{0.5000}{x_{5}} \\ + \frac{0.6667}{x_{6}} + \frac{0.5833}{x_{7}} + \frac{0.5833}{x_{8}} + \frac{0.5833}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{6}^{R_{a_{1}} ↑} & = & \frac{0.0000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.3333}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{7}^{R_{a_{1}} ↑} & = & \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5888}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} \\ + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{8}^{R_{a_{1}} ↑} & = & \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5888}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} \\ + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}}; \end{matrix}$ $\begin{matrix} A_{9}^{R_{a_{1}} ↑} & = & \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5888}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} \\ + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}} . \end{matrix}$ We see that $P (R_{a_{1}}^{↑})$ $= {\begin{matrix} A_{1}^{R_{a_{1}} ↑}, A_{2}^{R_{a_{1}} ↑}, A_{3}^{R_{a_{1}} ↑}, A_{4}^{R_{a_{1}} ↑}, A_{5}^{R_{a_{1}} ↑}, A_{6}^{R_{a_{1}} ↑}, A_{7}^{R_{a_{1}} ↑}, \\ A_{8}^{R_{a_{1}} ↑}, A_{9}^{R_{a_{1}} ↑} \end{matrix}}$ is a fuzzy upward β-covering of $U$ (0 < β ≤ 0.5). Thus $a_{1} N_{x_{j}}^{↑ 0.5}$ of x_j (j = 1, 2, . . . , 9) are: $a_{1} N_{x_{1}}^{↑ 0.5} = \frac{0.5000}{x_{1}} + \frac{0.9167}{x_{2}} + \frac{1.0000}{x_{3}} + \frac{0.6667}{x_{4}} + \frac{0.8333}{x_{5}} + \frac{1.0000}{x_{6}} + \frac{0.9167}{x_{7}} + \frac{0.9167}{x_{8}} + \frac{0.9167}{x_{9}};$ $a_{1} N_{x_{2}}^{↑ 0.5} = \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5833}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}};$ $a_{1} N_{x_{3}}^{↑ 0.5} = \frac{0.0000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.3333}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}};$ $a_{1} N_{x_{4}}^{↑ 0.5} = \frac{0.3333}{x_{1}} + \frac{0.7500}{x_{2}} + \frac{0.8333}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.6667}{x_{5}} + \frac{0.8333}{x_{6}} + \frac{0.7500}{x_{7}} + \frac{0.7500}{x_{8}} + \frac{0.7500}{x_{9}};$ $a_{1} N_{x_{5}}^{↑ 0.5} = \frac{0.1667}{x_{1}} + \frac{0.5833}{x_{2}} + \frac{0.6667}{x_{3}} + \frac{0.3333}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.6667}{x_{6}} + \frac{0.5833}{x_{7}} + \frac{0.5833}{x_{8}} + \frac{0.5833}{x_{9}};$ $a_{1} N_{x_{6}}^{↑ 0.5} = \frac{0.0000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.3333}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}};$ $a_{1} N_{x_{7}}^{↑ 0.5} = \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5833}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}};$ $a_{1} N_{x_{8}}^{↑ 0.5} = \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5833}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}};$ $a_{1} N_{x_{9}}^{↑ 0.5} = \frac{0.0833}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5833}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.4167}{x_{5}} + \frac{0.5833}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}};$ Let $\begin{matrix} μ & = & \frac{0.3}{x_{1}} + \frac{0.4}{x_{2}} + \frac{0.7}{x_{3}} + \frac{0.5}{x_{4}} + \frac{0.6}{x_{5}} + \frac{0.1}{x_{6}} + \frac{0.8}{x_{7}} + \frac{0.1}{x_{8}} \\ + \frac{0.4}{x_{9}} . \end{matrix}$ Then $\begin{matrix} {(LA μ)}^{↑} & = & \frac{0.1000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.3333}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}}, \end{matrix}$

$\begin{matrix} {(UA μ)}^{↑} & = & \frac{0.8000}{x_{1}} + \frac{0.5833}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.7500}{x_{4}} + \frac{0.6667}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.5833}{x_{7}} + \frac{0.5833}{x_{8}} + \frac{0.5833}{x_{9}} . \end{matrix}$ The comparison of the above approximations From Figure 1, we can obtain the following results by analyzing ${(LA μ)}^{↑}$ , ${(UA μ)}^{↑}$ and μ under critical value β = 0.5.

(i). Since μ (x_j) ≥ 0.5, ${(LA μ)}^{↑} (x_{j}) \geq 0.5$ and ${(UA μ)}^{↑} (x_{j}) \geq 0.5$ (j = 3), it is concluded that the medicine x₃ is the most important for treatment of the disease A.

Fig.1

Graphical representation.

(ii). Since μ (x_j) ≥ 0.5 and ${(UA μ)}^{↑} (x_{j}) \geq 0.5$ (j = 4, 5, 7), it is concluded that the medicines x₄, x₅ and x₇ are also important for treatment of the disease A.

(iii). Since μ (x_j) < 0.5 (j = 2, 3, 6, 8, 9), ${(LA μ)}^{↑} (x_{j}) \geq 0.5$ (j = 6) ${(UA μ)}^{↑} (x_{j}) \geq 0.5$ (j = 2, 6, 8, 9), hence the medicine x₂, x₆, x₈ and x₉ are less important than x₃, x₄, x₅ and x₇ for treatment of the disease A.

Remark 1. Let $(U, P (R^{↑}))$ be an $FUCAS$ where $P (R^{↑})$ is a fuzzy upward β-covering of $U$ for some β ∈ (0, 1]. Then for any $α \in F (U)$ , utilizing Definitions 2 and 3, the following can be deduced:

${(UA (U))}^{↑} = U$ ,

${(LA \emptyset)}^{↑} = \emptyset$ ,

${(LA {(LA μ)}^{↑})}^{↑} = {(LA μ)}^{↑}$ ,

${(UA {(UA μ)}^{↑})}^{↑} = {(UA μ)}^{↑}$ ,

${(LA {({(LA μ)}^{↑})}^{c})}^{↑} = {({(LA μ)}^{↑})}^{c}$ ,

${(UA {({(UA μ)}^{↑})}^{c})}^{↑} = {({(UA μ)}^{↑})}^{c}$ ,

${(LA μ)}^{↑} \subseteq {(UA μ)}^{↑}$ .

To illustrate this remark, let us see an example.

Example 2. (Continued from Example 3) Let $\begin{matrix} μ & = & \frac{0.3}{x_{1}} + \frac{0.4}{x_{2}} + \frac{0.7}{x_{3}} + \frac{0.5}{x_{4}} + \frac{0.6}{x_{5}} + \frac{0.1}{x_{6}} + \frac{0.8}{x_{7}} + \frac{0.1}{x_{8}} \\ + \frac{0.4}{x_{9}} . \end{matrix}$ Then $\begin{matrix} {(LA μ)}^{↑} & = & \frac{0.1000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.3333}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}}, \end{matrix}$ $\begin{matrix} {(UA μ)}^{↑} & = & \frac{0.8000}{x_{1}} + \frac{0.5833}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.7500}{x_{4}} + \frac{0.6667}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.5833}{x_{7}} + \frac{0.5833}{x_{8}} + \frac{0.5833}{x_{9}}, \end{matrix}$ $\begin{matrix} {(LA {(LA μ)}^{↑})}^{↑} & = & \frac{0.3333}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.3333}{x_{4}} \\ + \frac{0.4167}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} \\ + \frac{0.5000}{x_{9}}, \end{matrix}$

$\begin{matrix} {(UA {(UA μ)}^{↑})}^{↑} & = & \frac{0.6667}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.6667}{x_{4}} \\ + \frac{0.5833}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} \\ + \frac{0.5000}{x_{9}}, \end{matrix}$ $\begin{matrix} {({(LA μ)}^{↑})}^{c} & = & \frac{0.9000}{x_{1}} + \frac{0.5833}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.8333}{x_{4}} + \frac{0.6667}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.5833}{x_{7}} + \frac{0.5833}{x_{8}} + \frac{0.5833}{x_{9}}, \end{matrix}$ $\begin{matrix} {({(UA μ)}^{↑})}^{c} & = & \frac{0.2000}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.3333}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}}, \end{matrix}$ $\begin{matrix} {(LA {({(LA μ)}^{↑})}^{c})}^{↑} & = & \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.5000}{x_{4}} \\ + \frac{0.5000}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} \\ + \frac{0.5000}{x_{9}}, \end{matrix}$ $\begin{matrix} {(UA {({(UA μ)}^{↑})}^{c})}^{↑} & = & \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.5000}{x_{4}} \\ + \frac{0.5000}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} \\ + \frac{0.5000}{x_{9}}, \end{matrix}$ $U = \frac{1}{x_{1}} + \frac{1}{x_{2}} + \frac{1}{x_{3}} + \frac{1}{x_{4}} + \frac{1}{x_{5}} + \frac{1}{x_{6}} + \frac{1}{x_{7}} + \frac{1}{x_{8}} + \frac{1}{x_{9}},$ $\begin{matrix} {(UA (U))}^{↑} & = & \frac{1}{x_{1}} + \frac{0.5833}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.8333}{x_{4}} + \frac{0.6667}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.5833}{x_{7}} + \frac{0.5833}{x_{8}} + \frac{0.5833}{x_{9}}, \end{matrix}$ $\emptyset = \frac{0}{x_{1}} + \frac{0}{x_{2}} + \frac{0}{x_{3}} + \frac{0}{x_{4}} + \frac{0}{x_{5}} + \frac{0}{x_{6}} + \frac{0}{x_{7}} + \frac{0}{x_{8}} + \frac{0}{x_{9}},$ $\begin{matrix} {(LA \emptyset)}^{↑} & = & \frac{0}{x_{1}} + \frac{0.4167}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.1667}{x_{4}} + \frac{0.4167}{x_{5}} \\ + \frac{0.5000}{x_{6}} + \frac{0.4167}{x_{7}} + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}} . \end{matrix}$ It is easy to find that $(UA (U))^{↑} \neq U$ , $(LA \emptyset)^{↑}$ ≠∅, $(LA (LA μ)^{↑})^{↑} \neq (LA μ)^{↑}$ , $(UA (UA μ)^{↑})^{↑} \neq$ $(UA μ)^{↑}$ , $(LA ((LA μ)^{↑})^{c})^{↑} \neq ((LA μ)^{↑})^{c}$ , $(UA ((UA$ $μ)^{↑})^{c})^{↑} \neq ((UA μ)^{↑})^{c}$ , $(LA μ)^{↑} ⊊ (UA μ)^{↑}$ .

4 Multigranulation fuzzy rough sets basedon fuzzy upward β-covering

The idea of optimistic multigranulation rough set reflect the decision making of risk preferring decision maker in practice of medical sciences. Generally speaking, in the practice of decision making of medical sciences, there are many non determined decision making problems due to the difficult structure of the decision making problem itself, the complexity of the decision making environment and the inaccuracy and incompleteness available information. Also, different patterns of decision making occur because of the different risk preferences of decision makers. In this section, we propose multigranulation optimistic fuzzy rough set based on fuzzy upward β-covering and investigate some of their properties.

Definition 7. $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for each fuzzy subset μ of $U$ , we define the optimistic lower approximation ${(\sum_{k = 1}^{m} LA μ)}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{m} UA μ)}^{o ↑}$ of μ as: $\begin{matrix} {(\sum_{k = 1}^{m} LA μ)}^{o ↑} (x) \\ = ⋁_{k = 1}^{m} {⋀_{y \in U} {(1 -_{a_{k}} N_{x}^{↑ β} (y)) \lor μ (y)}}, x \in U, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} UA μ)}^{o ↑} (x) \\ = ⋀_{k = 1}^{m} {⋁_{y \in U} {a_{k} N_{x}^{↑ β} (y) \land μ (y)}}, x \in U . \end{matrix}$ It is easy to see that ${(\sum_{k = 1}^{m} LA μ)}^{o ↑}$ and ${(\sum_{k = 1}^{m} UA μ)}^{o ↑}$ are two fuzzy sets in the universe $U$ . Further, μ is called a definable fuzzy set on multigranulation fuzzy upward approximation space if ${(\sum_{k = 1}^{m} LA μ)}^{o ↑} = {(\sum_{k = 1}^{m} UA μ)}^{o ↑}$ . Otherwise, $({(\sum_{k = 1}^{m} LA μ)}^{o ↑}, {(\sum_{k = 1}^{m} UA μ)}^{o ↑})$ is called optimistic multigranulation fuzzy upward rough set.

Theorem 1. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for any μ, $λ \in F (U)$ , such that μ ⊆ λ the following hold:

${(\sum_{k = 1}^{m} LA μ)}^{o ↑} \subseteq U$ ;

${(\sum_{k = 1}^{m} UA μ)}^{o ↑} \subseteq U$ ;

${(\sum_{k = 1}^{m} LA μ^{c})}^{o ↑} = {({(\sum_{k = 1}^{m} UA μ)}^{o ↑})}^{c}$ ;

${(\sum_{k = 1}^{m} UA μ^{c})}^{o ↑} = {({(\sum_{k = 1}^{m} LA μ)}^{o ↑})}^{c}$ ;

${(\sum_{k = 1}^{m} LA μ)}^{o ↑} \subseteq {(\sum_{k = 1}^{m} LA λ)}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA μ)}^{o ↑} \subseteq {(\sum_{k = 1}^{m} UA λ)}^{o ↑}$ ;

Theorem 2. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , where $P (R_{a_{m}}^{↑})$ are m fuzzy upward β-covering of $U$ . For any $μ_{i} \in F (U)$ , where (i = 1, 2, . . . , n), the following hold:

${(\sum_{k = 1}^{m} LA (⋂_{i = 1}^{n} μ_{i}))}^{o ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (μ_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA (⋂_{i = 1}^{n} μ_{i}))}^{o ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (μ_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} LA (⋃_{i = 1}^{n} μ_{i}))}^{o ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (μ_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA (⋃_{i = 1}^{n} μ_{i}))}^{o ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (μ_{i}))}^{o ↑}$ .

Pessimistic multigranulation fuzzy rough set model describes the decision making process of conservative type decision makers or risk-averse decision makers. We propose multigranulation pessimistic fuzzy rough set based on fuzzy upward β-covering and investigate some of their properties.

Definition 8. $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , where $P (R_{a_{m}}^{↑})$ are m fuzzy upward β-covering of $U$ . Then for each fuzzy subset μ of $U$ , we define the pessimistic lower approximation ${(\sum_{k = 1}^{m} LA μ)}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{m} UA μ)}^{p ↑}$ of μ as: $\begin{matrix} {(\sum_{k = 1}^{m} LA μ)}^{p ↑} (x) = \\ ⋀_{k = 1}^{m} {⋀_{y \in U} {(1 -_{a_{k}} N_{x}^{↑ β} (y)) \lor μ (y)}}, x \in U, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} UA μ)}^{p ↑} (x) = \\ ⋁_{k = 1}^{m} {⋁_{y \in U} {a_{k} N_{x}^{↑ β} (y) \land μ (y)}}, x \in U . \end{matrix}$ It is easy to see that ${(\sum_{k = 1}^{m} LA μ)}^{p ↑}$ and ${(\sum_{k = 1}^{m} UA μ)}^{p ↑}$ are two fuzzy sets in the universe $U$ . Further, μ is called a definable fuzzy set on multigranulation fuzzy upward approximation space if ${(\sum_{k = 1}^{m} LA μ)}^{p ↑} = {(\sum_{k = 1}^{m} UA μ)}^{p ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} LA μ)}^{p ↑}, {(\sum_{k = 1}^{m} UA μ)}^{p ↑})$ is called pessimistic multigranulation fuzzy upward rough set.

Theorem 3. $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for any μ, $λ \in F (U)$ , such that μ ⊆ λ the following hold:

${(\sum_{k = 1}^{m} LA μ)}^{p ↑} \subseteq U$ ;

${(\sum_{k = 1}^{m} UA μ)}^{p ↑} \subseteq U$ ;

${(\sum_{k = 1}^{m} LA μ^{c})}^{p ↑} = {({(\sum_{k = 1}^{m} UA μ)}^{p ↑})}^{c}$ ;

${(\sum_{k = 1}^{m} UA μ^{c})}^{p ↑} = {({(\sum_{k = 1}^{m} LA μ)}^{p ↑})}^{c}$ ;

${(\sum_{k = 1}^{m} LA μ)}^{p ↑} \subseteq {(\sum_{k = 1}^{m} LA λ)}^{p ↑}$ ;

${(\sum_{k = 1}^{m} UA μ)}^{p ↑} \subseteq {(\sum_{k = 1}^{m} UA λ)}^{p ↑}$ .

Theorem 4. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for any $μ_{i} \in F (U)$ , where (i = 1, 2, . . . , n), we have

${(\sum_{k = 1}^{m} LA (⋂_{i = 1}^{n} μ_{i}))}^{p ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (μ_{i}))}^{p ↑}$ ,

${(\sum_{k = 1}^{m} UA (⋂_{i = 1}^{n} μ_{i}))}^{p ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (μ_{i}))}^{p ↑}$ ,

${(\sum_{k = 1}^{m} LA (⋃_{i = 1}^{n} μ_{i}))}^{p ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (μ_{i}))}^{p ↑}$ ,

${(\sum_{k = 1}^{m} UA (⋃_{i = 1}^{n} μ_{i}))}^{p ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (μ_{i}))}^{p ↑}$ .

The concept of level set of a fuzzy set provides an effective method to transform a fuzzy set into a crisp set. In the following, we find upward β-neighborhood in the fuzzy upward covering approximation space and then present optimistic/pessimistic multigranulation rough sets and discussed their fundamental properties.

Definition 9. Let $(U, P (R^{↑}))$ be an $FUCAS$ and $P (R^{↑})$ be a fuzzy upward β-covering of $U$ for some β ∈ (0, 1]. Then for each $x \in U$ , we define the β-neighborhood $N_{x}^{↑ β}$ of x as: $N_{x}^{↑ β} = {y \in U : N_{x}^{↑ β} (y) \geq β} .$

Example 3. (Continued from Example 3) Using Definition 4, we have $N_{x_{1}}^{↑ β} = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}},$ $N_{x_{2}}^{↑ β} = {x_{2}, x_{3}, x_{6}, x_{7}, x_{8}, x_{9}}, N_{x_{3}}^{↑ β} = {x_{3}, x_{6}},$ $\begin{matrix} N_{x_{4}}^{↑ β} = {x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}, \\ N_{x_{9}}^{↑ β} = {x_{2}, x_{3}, x_{6}, x_{7}, x_{8}, x_{9}}, \end{matrix}$ $\begin{matrix} N_{x_{5}}^{↑ β} = {x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}}, \\ N_{x_{6}}^{↑ β} = {x_{3}, x_{6}}, \end{matrix}$ $\begin{matrix} N_{x_{7}}^{↑ β} = {x_{2}, x_{3}, x_{6}, x_{7}, x_{8}, x_{9}}, \\ N_{x_{8}}^{↑ β} = {x_{2}, x_{3}, x_{6}, x_{7}, x_{8}, x_{9}} . \end{matrix}$

Definition 10. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . For each crisp subset $X$ of $U$ , we define the optimistic lower approximation ${(\sum_{k = 1}^{m} LA (X))}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{m} UA (X))}^{o ↑}$ of $X$ as: $\begin{matrix} {(\sum_{k = 1}^{m} LA (X))}^{o ↑} = {x \in U :_{a_{1}} N_{x}^{↑ β} \subseteq X \lor_{a_{2}} \\ N_{x}^{↑ β} \subseteq X \lor . . . \lor_{a_{m}} N_{x}^{↑ β} \subseteq X}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} UA (X))}^{o ↑} = {x \in U :_{a_{1}} N_{x}^{↑ β} \cap X \neq \emptyset \land_{a_{2}} \\ N_{x}^{↑ β} \cap X \neq \emptyset \land . . . \land_{a_{m}} N_{x}^{↑ β} \cap X \neq \emptyset} . \end{matrix}$ It is easy to see that ${(\sum_{k = 1}^{m} LA (X))}^{o ↑}$ and ${(\sum_{k = 1}^{m} UA (X))}^{o ↑}$ are two crisp sets of universe $U$ . Further, $X$ is called a definable set on multigranulation upward approximation space if ${(\sum_{k = 1}^{m} LA (X))}^{o ↑} = {(\sum_{k = 1}^{m} UA (X))}^{o ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} LA (X))}^{o ↑}, {(\sum_{k = 1}^{m} UA (X))}^{o ↑})$ is called optimistic multigranulation upward rough set.

Theorem 5. Let $(U, P (R_{a_{m}}^{↑}))$ be a $FUCAS$ , where $P (R_{a_{m}}^{↑})$ are m fuzzy upward β-covering of $U$ . Then for all $X_{1}, X_{2}, . . ., X_{n} \subseteq U$ , the optimistic multigranulation upward rough set satisfy the following properties:

${(\sum_{k = 1}^{m} LA (X_{i}))}^{o ↑} \subseteq X_{i} \subseteq {(\sum_{k = 1}^{m} UA (X_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA (U))}^{o ↑} = U = {(\sum_{k = 1}^{m} LA (U))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA (\emptyset))}^{o ↑} = \emptyset = {(\sum_{k = 1}^{m} LA (\emptyset))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} LA {(X)}^{c})}^{o ↑} = {({(\sum_{k = 1}^{m} UA (X))}^{o ↑})}^{c}$ ;

${(\sum_{k = 1}^{m} UA {(X)}^{c})}^{o ↑} = {({(\sum_{k = 1}^{m} LA (X))}^{o ↑})}^{c}$ ;

$X_{1} \subseteq X_{2}$ implies ${(\sum_{k = 1}^{m} LA (X_{1}))}^{o ↑} \subseteq {(\sum_{k = 1}^{m} LA (X_{2}))}^{o ↑}$ ;

$X_{1} \subseteq X_{2}$ implies ${(\sum_{k = 1}^{m} UA (X_{1}))}^{o ↑} \subseteq {(\sum_{k = 1}^{m} UA (X_{2}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} LA (⋂_{i = 1}^{n} (X_{i})))}^{o ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (X_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA (⋂_{i = 1}^{n} (X_{i})))}^{o ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (X_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} LA (⋃_{i = 1}^{n} (X_{i})))}^{o ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (X_{i}))}^{o ↑}$ ;

${(\sum_{k = 1}^{m} UA (⋃_{i = 1}^{n} (X_{i})))}^{o ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (X_{i}))}^{o ↑}$ .

Definition 11. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for each crisp subset $X$ of $U$ , we define the pessimistic lower approximation ${(\sum_{k = 1}^{m} LA (X))}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{m} UA (X))}^{p ↑}$ of $X$ as: $\begin{matrix} {(\sum_{k = 1}^{m} LA (X))}^{p ↑} = {x \in U :_{a_{1}} N_{x}^{↑ β} \subseteq X \land_{a_{2}} \\ N_{x}^{↑ β} \subseteq X \land . . . \land_{a_{m}} N_{x}^{↑ β} \subseteq X}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} UA (X))}^{p ↑} = {x \in U :_{a_{1}} N_{x}^{↑ β} \cap X \neq \emptyset \lor_{a_{2}} \\ N_{x}^{↑ β} \cap X \neq \emptyset \lor . . . \lor_{a_{m}} N_{x}^{↑ β} \cap X \neq \emptyset} . \end{matrix}$ It is easy to see that ${(\sum_{k = 1}^{m} LA (X))}^{p ↑}$ and ${(\sum_{k = 1}^{m} UA (X))}^{p ↑}$ are two crisp subsets of universe $U$ . Further, $X$ is called a definable set on multigranulation upward approximation space if ${(\sum_{k = 1}^{m} LA (X))}^{p ↑} = {(\sum_{k = 1}^{m} UA (X))}^{p ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} LA (X))}^{p ↑}, {(\sum_{k = 1}^{m} UA (X))}^{p ↑})$ is called pessimistic multigranulation upward rough set.

Theorem 6. Let $(U, P (R_{a_{m}}^{↑}))$ be a $FUCAS$ , where $P (R_{a_{m}}^{↑})$ are m fuzzy upward β-covering of $U$ . Then for all $X_{1}, X_{2}, . . ., X_{n} \subseteq U$ , the pessimistic multigranulation upward rough set has the following properties:

${(\sum_{k = 1}^{m} LA (X_{i}))}^{p ↑} \subseteq X_{i} \subseteq {(\sum_{k = 1}^{m} UA (X_{i}))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} UA (U))}^{p ↑} = U = {(\sum_{k = 1}^{m} LA (U))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} UA (\emptyset))}^{p ↑} = \emptyset = {(\sum_{k = 1}^{m} LA (\emptyset))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} LA {(X)}^{c})}^{p ↑} = {({(\sum_{k = 1}^{m} UA (X))}^{p ↑})}^{c}$ ;

${(\sum_{k = 1}^{m} UA {(X)}^{c})}^{p ↑} = {({(\sum_{k = 1}^{m} LA (X))}^{p ↑})}^{c}$ ;

$X_{1} \subseteq X_{2}$ implies ${(\sum_{k = 1}^{m} LA (X_{1}))}^{p ↑} \subseteq {(\sum_{k = 1}^{m} LA (X_{2}))}^{p ↑}$ ;

$X_{1} \subseteq X_{2}$ implies ${(\sum_{k = 1}^{m} UA (X_{1}))}^{p ↑} \subseteq {(\sum_{k = 1}^{m} UA (X_{2}))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} LA (⋂_{i = 1}^{n} (X_{i})))}^{p ↑} = ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (X_{i}))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} UA (⋂_{i = 1}^{n} (X_{i})))}^{p ↑} \subseteq ⋂_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (X_{i}))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} LA (⋃_{i = 1}^{n} (X_{i})))}^{p ↑} \supseteq ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} LA (X_{i}))}^{p ↑}$ ;

${(\sum_{k = 1}^{m} UA (⋃_{i = 1}^{n} (X_{i})))}^{p ↑} = ⋃_{i = 1}^{n} {(\sum_{k = 1}^{m} UA (X_{i}))}^{p ↑}$ .

Definition 12. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ , and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for any $X \subseteq U$ , The upward approximate precision $ρ^{p ↑} (X)$ of $X$ is defined by: $ρ^{p ↑} (X) = \frac{| {(\sum_{k = 1}^{m} LA (X))}^{p ↑} |}{| {(\sum_{k = 1}^{m} UA (X))}^{p ↑} |}$ where $X \neq \emptyset$ and | . | denotes the cardinality of a set. $σ^{p ↑} (X) = 1 - ρ^{p ↑} (X)$ , is called the rough degree of $X$ .

The following theorem describe the relationship of the precision $ρ^{p ↑} (X)$ and also the rough degree $σ^{p ↑} (X)$ for the intersection and union of any subsets $X$ and $Y$ of the universe $U$ .

Theorem 7. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ and $X$ , $Y \subseteq U$ . Then the rough degree and precision of the subsets $X$ , $Y$ $X \cup Y$ and $X \cap Y$ satisfy the following relations:

$σ^{p ↑} (X \cup Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑}$ $\cup {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | \leq σ^{p ↑} (X) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} | + σ^{p ↑} (Y) | {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | - σ^{p ↑} (X \cap Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} \cap {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} |$ .

$ρ^{p ↑} (X \cup Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑}$ $\cup {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | \geq ρ^{p ↑} (X) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} | + ρ^{p ↑} (Y) | {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | - ρ^{p ↑} (X \cap Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} \cap {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} |$ .

Definition 13. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ and $P (R_{a_{m}}^{↑})$ be m fuzzy upward β-covering of $U$ . Then for any $X \subseteq U$ , the upward approximate quality $γ^{p ↑} (X)$ of $X$ is defined by: $γ^{p ↑} (X) = \frac{| {(\sum_{k = 1}^{m} LA (X))}^{p ↑} |}{| U |} .$ Clearly $0 \leq γ^{p ↑} (X) \leq 1$ .

The following theorem describe the relationship of the approximate quality $γ^{p ↑} (X)$ and also the rough degree $σ^{p ↑} (X)$ for the intersection and union of any subsets $X$ and $Y$ of the universe $U$

Theorem 8. Let $(U, P (R_{a_{m}}^{↑}))$ be an $FUCAS$ and $P (R_{a_{m}}^{↑})$ are m fuzzy upward β-covering of $U$ and $X$ , $Y \subseteq U$ . Then the rough degree and approximate quality, also precision and approximate quality, of the subsets $X$ , $Y$ $X \cup Y$ and $X \cap Y$ satisfy the following relations.

$σ^{p ↑} (X \cup Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} \cup {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | \leq | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} | + | {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | - | U | {γ^{p ↑} (X) + γ^{p ↑} (Y)} - σ^{p ↑} (X \cap Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} \cap {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} |$ .

$ρ^{p ↑} (X \cup Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} \cup {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} | \geq | U | {γ^{p ↑} (X) + γ^{p ↑} (Y)} - ρ^{p ↑} (X \cap Y) | {(\sum_{k = 1}^{m} UA (X))}^{p ↑} \cap {(\sum_{k = 1}^{m} UA (Y))}^{p ↑} |$ .

Proof. It is analogous to Theorem 4. □

Example 4. (Continued from Example 3)Let $U = {x_{j} : j = 1, 2, . . ., 9}$ be the set of medicines and a_i be the criteria. The evaluation of $U$ by the a_i are given in Table 2.

Table 2
Information Table

x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈ x ₉

a ₁ 0.8 0.3 0.2 0.6 0.4 0.2 0.3 0.3 0.3

a ₂ 0.1 0.5 0.1 0.3 0.4 0.3 0.3 0.4 0.2

a ₃ 0.2 0.2 0.6 0.5 0.3 0.5 0.6 0.3 0.4

	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈	x ₉
a ₁	0.8	0.3	0.2	0.6	0.4	0.2	0.3	0.3	0.3
a ₂	0.1	0.5	0.1	0.3	0.4	0.3	0.3	0.4	0.2
a ₃	0.2	0.2	0.6	0.5	0.3	0.5	0.6	0.3	0.4

Utilizing equation (1) to compute the fuzzy preference degree of the alternative x_i (i = 1, 2, . . . , 9) to the x_j (j = 1, 2, . . . , 9), we have $R_{a_{2}}^{↑} (x_{i}, x_{j}) = (\begin{matrix} 0.5000 & 0.0000 & 0.5000 & 0.2500 & 0.1250 & 0.2500 & 0.2500 & 0.1250 & 0.3750 \\ 1.0000 & 0.5000 & 1.0000 & 0.7500 & 0.6250 & 0.7500 & 0.7500 & 0.6250 & 0.8750 \\ 0.5000 & 0.0000 & 0.5000 & 0.2500 & 0.1250 & 0.2500 & 0.2500 & 0.1250 & 0.3750 \\ 0.7500 & 0.2500 & 0.7500 & 0.5000 & 0.3750 & 0.5000 & 0.5000 & 0.3750 & 0.6250 \\ 0.8750 & 0.3750 & 0.8750 & 0.6250 & 0.5000 & 0.6250 & 0.6250 & 0.5000 & 0.7500 \\ 0.7500 & 0.2500 & 0.7500 & 0.5000 & 0.3750 & 0.5000 & 0.5000 & 0.3750 & 0.6250 \\ 0.7500 & 0.2500 & 0.7500 & 0.5000 & 0.3750 & 0.5000 & 0.5000 & 0.3750 & 0.6250 \\ 0.8750 & 0.3750 & 0.8750 & 0.6250 & 0.5000 & 0.6250 & 0.6250 & 0.5000 & 0.7500 \\ 0.6250 & 0.1250 & 0.6250 & 0.3750 & 0.2500 & 0.3750 & 0.3750 & 0.2500 & 0.5000 \end{matrix}) .$ The $a_{2} N_{x_{j}}^{↑ 0.5}$ of x_j (j = 1, 2, . . . , 9) are as follows:

$a_{2} N_{x_{1}}^{↑ 0.5} = \frac{0.5000}{x_{1}} + \frac{0.0000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.1250}{x_{5}} + \frac{0.2500}{x_{6}} + \frac{0.2500}{x_{7}} + \frac{0.1250}{x_{8}} + \frac{0.3750}{x_{9}};$ $a_{2} N_{x_{2}}^{↑ 0.5} = \frac{1.0000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{1.0000}{x_{3}} + \frac{0.7500}{x_{4}} + \frac{0.6250}{x_{5}} + \frac{0.7500}{x_{6}} + \frac{0.7500}{x_{7}} + \frac{0.6250}{x_{8}} + \frac{0.8750}{x_{9}};$ $a_{2} N_{x_{3}}^{↑ 0.5} = \frac{0.5000}{x_{1}} + \frac{0.0000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.1250}{x_{5}} + \frac{0.2500}{x_{6}} + \frac{0.2500}{x_{7}} + \frac{0.1250}{x_{8}} + \frac{0.3750}{x_{9}};$ $a_{2} N_{x_{4}}^{↑ 0.5} = \frac{0.7500}{x_{1}} + \frac{0.2500}{x_{2}} + \frac{0.7500}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.3750}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.3750}{x_{8}} + \frac{0.6250}{x_{9}};$ $a_{2} N_{x_{5}}^{↑ 0.5} = \frac{0.8750}{x_{1}} + \frac{0.3750}{x_{2}} + \frac{0.8750}{x_{3}} + \frac{0.6250}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.6250}{x_{6}} + \frac{0.6250}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.7500}{x_{9}};$ $a_{2} N_{x_{6}}^{↑ 0.5} = \frac{0.7500}{x_{1}} + \frac{0.2500}{x_{2}} + \frac{0.7500}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.3750}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.3750}{x_{8}} + \frac{0.6250}{x_{9}};$ $a_{2} N_{x_{7}}^{↑ 0.5} = \frac{0.7500}{x_{1}} + \frac{0.2500}{x_{2}} + \frac{0.7500}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.3750}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.3750}{x_{8}} + \frac{0.6250}{x_{9}};$ $a_{2} N_{x_{8}}^{↑ 0.5} = \frac{0.8750}{x_{1}} + \frac{0.3750}{x_{2}} + \frac{0.8750}{x_{3}} + \frac{0.6250}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.6250}{x_{6}} + \frac{0.6250}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.7500}{x_{9}};$ $a_{2} N_{x_{9}}^{↑ 0.5} = \frac{0.6250}{x_{1}} + \frac{0.1250}{x_{2}} + \frac{0.6250}{x_{3}} + \frac{0.3750}{x_{4}} + \frac{0.2500}{x_{5}} + \frac{0.3750}{x_{6}} + \frac{0.3750}{x_{7}} + \frac{0.2500}{x_{8}} + \frac{0.5000}{x_{9}} .$

Similarly we use the relation (1) to compute the fuzzy preference degree of the alternative x_i (i = 1, 2, . . . , 9) to the x_j (j = 1, 2, . . . , 9), we have

$R_{a_{3}}^{↑} (x_{i}, x_{j}) = (\begin{matrix} 0.5000 & 0.5000 & 0.0000 & 0.1250 & 0.3750 & 0.1250 & 0.0000 & 0.3750 & 0.2500 \\ 0.5000 & 0.5000 & 0.0000 & 0.1250 & 0.3750 & 0.1250 & 0.0000 & 0.3750 & 0.2500 \\ 1.0000 & 1.0000 & 0.5000 & 0.6250 & 0.8750 & 0.6250 & 0.5000 & 0.8750 & 0.7500 \\ 0.8750 & 0.8750 & 0.3750 & 0.5000 & 0.7500 & 0.5000 & 0.3750 & 0.7500 & 0.6250 \\ 0.6250 & 0.6250 & 0.1250 & 0.2500 & 0.5000 & 0.2500 & 0.1250 & 0.5000 & 0.3750 \\ 0.8750 & 0.8750 & 0.3750 & 0.5000 & 0.7500 & 0.5000 & 0.3750 & 0.7500 & 0.6250 \\ 1.0000 & 1.0000 & 0.5000 & 0.6250 & 0.8750 & 0.6250 & 0.5000 & 0.8750 & 0.7500 \\ 0.6250 & 0.6250 & 0.1250 & 0.2500 & 0.5000 & 0.2500 & 0.1250 & 0.5000 & 0.3750 \\ 0.7500 & 0.7500 & 0.2500 & 0.3750 & 0.6250 & 0.3750 & 0.2500 & 0.6250 & 0.5000 \end{matrix}) .$

The $a_{3} N_{x_{j}}^{↑ 0.5}$ of x_j (j = 1, 2, . . . , 9) are as follows: $a_{3} N_{x_{1}}^{↑ 0.5} = \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.0000}{x_{3}} + \frac{0.1250}{x_{4}} + \frac{0.3750}{x_{5}} + \frac{0.1250}{x_{6}} + \frac{0.0000}{x_{7}} + \frac{0.3750}{x_{8}} + \frac{0.2500}{x_{9}};$ $a_{3} N_{x_{2}}^{↑ 0.5} = \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.0000}{x_{3}} + \frac{0.1250}{x_{4}} + \frac{0.3750}{x_{5}} + \frac{0.1250}{x_{6}} + \frac{0.0000}{x_{7}} + \frac{0.3750}{x_{8}} + \frac{0.2500}{x_{9}};$ $a_{3} N_{x_{3}}^{↑ 0.5} = \frac{1.0000}{x_{1}} + \frac{1.0000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.6250}{x_{4}} + \frac{0.8750}{x_{5}} + \frac{0.6250}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.8750}{x_{8}} + \frac{0.7500}{x_{9}};$ $a_{3} N_{x_{4}}^{↑ 0.5} = \frac{0.8750}{x_{1}} + \frac{0.8750}{x_{2}} + \frac{0.3750}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.7500}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.3750}{x_{7}} + \frac{0.7500}{x_{8}} + \frac{0.6250}{x_{9}};$ $a_{3} N_{x_{5}}^{↑ 0.5} = \frac{0.6250}{x_{1}} + \frac{0.6250}{x_{2}} + \frac{0.1250}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.2500}{x_{6}} + \frac{0.1250}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.3750}{x_{9}};$ $a_{3} N_{x_{6}}^{↑ 0.5} = \frac{0.8750}{x_{1}} + \frac{0.8750}{x_{2}} + \frac{0.3750}{x_{3}} + \frac{0.5000}{x_{4}} + \frac{0.7500}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.3750}{x_{7}} + \frac{0.7500}{x_{8}} + \frac{0.6250}{x_{9}};$ $a_{3} N_{x_{7}}^{↑ 0.5} = \frac{1.0000}{x_{1}} + \frac{1.0000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.6250}{x_{4}} + \frac{0.8750}{x_{5}} + \frac{0.6250}{x_{6}} + \frac{0.5000}{x_{7}} + \frac{0.8750}{x_{8}} + \frac{0.7500}{x_{9}};$ $a_{3} N_{x_{8}}^{↑ 0.5} = \frac{0.6250}{x_{1}} + \frac{0.6250}{x_{2}} + \frac{0.1250}{x_{3}} + \frac{0.2500}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.2500}{x_{6}} + \frac{0.1250}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.3750}{x_{9}};$ $a_{3} N_{x_{9}}^{↑ 0.5} = \frac{0.7500}{x_{1}} + \frac{0.7500}{x_{2}} + \frac{0.2500}{x_{3}} + \frac{0.3750}{x_{4}} + \frac{0.6250}{x_{5}} + \frac{0.3750}{x_{6}} + \frac{0.2500}{x_{7}} + \frac{0.6250}{x_{8}} + \frac{0.5000}{x_{9}} .$ Let $\begin{matrix} μ & = & \frac{0.3}{x_{1}} + \frac{0.4}{x_{2}} + \frac{0.7}{x_{3}} + \frac{0.5}{x_{4}} + \frac{0.6}{x_{5}} + \frac{0.1}{x_{6}} + \frac{0.8}{x_{7}} \\ + \frac{0.1}{x_{8}} + \frac{0.4}{x_{9}} . \end{matrix}$ Using the optimistic lower approximation ${(\sum_{k = 1}^{3} LA μ)}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{3} UA μ)}^{o ↑}$ of μ are $\begin{matrix} {(\sum_{k = 1}^{3} LA μ)}^{o ↑} & = & \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.3000}{x_{4}} \\ + \frac{0.3750}{x_{5}} + \frac{0.6000}{x_{6}} + \frac{0.4167}{x_{7}} \\ + \frac{0.4167}{x_{8}} + \frac{0.4167}{x_{9}}, \end{matrix}$

$\begin{matrix} {(\sum_{k = 1}^{3} UA μ)}^{o ↑} & = & \frac{0.4000}{x_{1}} + \frac{0.4000}{x_{2}} + \frac{0.5000}{x_{3}} + \frac{0.6000}{x_{4}} \\ + \frac{0.5000}{x_{5}} + \frac{0.5000}{x_{6}} + \frac{0.5833}{x_{7}} \\ + \frac{0.5000}{x_{8}} + \frac{0.5833}{x_{9}} . \end{matrix}$

The comparison of the above approximations.

From Figure 2, we obtain the following results by analyzing ${(\sum_{k = 1}^{3} LA μ)}^{o ↑}$ , ${(\sum_{k = 1}^{3} UA μ)}^{o ↑}$ and μ under critical value β = 0.5.

Since μ (x_j) ≥ 0.5, ${(\sum_{k = 1}^{3} LA μ)}^{o ↑} (x_{j}) \geq 0.5$ and ${(\sum_{k = 1}^{3} UA μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 3), we conclude that the medicine x₃ is the most important for the treatment of disease A.

Since μ (x_j) ≥0.5 and ${(\sum_{k = 1}^{3} UA μ)}^{o ↑}$ (x_j) ≥ 0.5 (j = 4, 5, 7), we conclude that the medicine x₄, x₅ and x₇ are also important for treatment of the disease A.

Since μ (x_j) < 0.5 (j = 1, 2, 6, 8, 9), ${(\sum_{k = 1}^{3} LA μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 1, 2, 6) ${(\sum_{k = 1}^{3} UA μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 6, 8, 9), thus we conclude that the medicine x₁, x₂, x₆, x₈ and x₉ are less important than x₃, x₄, x₅ and x₇ for treatment of the disease A.

Fig.2

Graphical representation.

Now the pessimistic lower approximation ${(\sum_{k = 1}^{3} LA μ)}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{3} UA μ)}^{p ↑}$ of μ are $\begin{matrix} {(\sum_{k = 1}^{3} LA μ)}^{p ↑} & = & \frac{0.1000}{x_{1}} + \frac{0.2500}{x_{2}} + \frac{0.1250}{x_{3}} + \frac{0.1667}{x_{4}} \\ + \frac{0.3000}{x_{5}} + \frac{0.2500}{x_{6}} + \frac{0.1250}{x_{7}} + \frac{0.3000}{x_{8}} \\ + \frac{0.3000}{x_{9}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{3} UA μ)}^{p ↑} & = & \frac{0.8000}{x_{1}} + \frac{0.7500}{x_{2}} + \frac{0.6000}{x_{3}} + \frac{0.7500}{x_{4}} \\ + \frac{0.7000}{x_{5}} + \frac{0.7000}{x_{6}} + \frac{0.7000}{x_{7}} + \frac{0.7000}{x_{8}} \\ + \frac{0.6250}{x_{9}} . \end{matrix}$ The comparison of the above approximations.

From Figure 3, the following results can be obtained by analyzing ${(\sum_{k = 1}^{3} LA μ)}^{p ↑}$ , ${(\sum_{k = 1}^{3} UA μ)}^{p ↑}$ and μ under critical value β = 0.5.

Since μ (x_j) ≥0.5 and ${(\sum_{k = 1}^{3} UA μ)}^{p ↑}$ (x_j) ≥ 0.5 (j = 3, 4, 5, 7), we conclude that the medicine x₃, x₄, x₅ and x₇ are important for treatment of the disease A.

Since μ (x_j) < 0.5 (j = 1, 2, 6, 8, 9), ${(\sum_{k = 1}^{3} UA μ)}^{p ↑} (x_{j}) \geq 0.5$ (j = 1, 2, 6, 8, 9) we conclude that the medicine x₂, x₆, x₈ and x₉ are less important than x₃, x₄, x₅ and x₇ for treatment of the disease A.

Fig.3

Graphical representation.

One naturally asks the following interestingquestion.

Question: Which particular medicine be prescribed for special symptom(s)?

The answer to the above question is given in the following.

Using Definition 2, it can see the fuzzy complementary β-upward neighborhood $M_{x_{j}}^{↑ β}$ of x_j is a fuzzy set $M_{x_{j}}^{↑ β} (x_{k}) = N_{x_{k}}^{↑ β} (x_{j}), k = 1, 2, . . . n,$ which denotes the minimum value among all the efficacy values of each symptom $y \in {y_{i} : A_{i}^{R_{a} ↑} (x_{k}) \geq β}$ treated by the medicines x_k. The multigranulation rough set based on fuzzy complementary β-upward neighborhood $M_{x}^{↑ β}$ of x, for each fuzzy subset μ of $U$ , we define the optimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{C} μ)}^{o ↑}$ of μ as: $\begin{matrix} {(\sum_{k = 1}^{m} {LA}_{C} μ)}^{o ↑} (x) = \\ ⋁_{k = 1}^{m} {⋀_{y \in U} {(1 -_{a_{k}} M_{x}^{↑ β} (y)) \lor μ (y)}}, x \in U, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} {UA}_{C} μ)}^{o ↑} (x) = \\ ⋀_{k = 1}^{m} {⋁_{y \in U} {a_{k} M_{x}^{↑ β} (y) \land μ (y)}}, x \in U . \end{matrix}$

It is easy to see that ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{o ↑}$ and ${(\sum_{k = 1}^{m} {UA}_{C} μ)}^{o ↑}$ are two fuzzy sets in universe $U$ . Further, μ is called a definable fuzzy set on multigranulation fuzzy upward approximation space if ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{o ↑} = {(\sum_{k = 1}^{m} {UA}_{C} μ)}^{o ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} {LA}_{C} μ)}^{o ↑}, {(\sum_{k = 1}^{m} {UA}_{C} μ)}^{o ↑})$ is called optimistic multigranulation fuzzy upward rough set. Similarly the pessimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{C} μ)}^{p ↑}$ of μ as: $\begin{matrix} {(\sum_{k = 1}^{m} {LA}_{C} μ)}^{p ↑} (x) = \\ ⋀_{k = 1}^{m} {⋀_{y \in U} {(1 -_{a_{k}} M_{x}^{↑ β} (y)) \lor μ (y)}}, x \in U, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} {UA}_{C} μ)}^{p ↑} (x) = \\ ⋁_{k = 1}^{m} {⋁_{y \in U} {a_{k} M_{x}^{↑ β} (y) \land μ (y)}}, x \in U . \end{matrix}$

It is easy to see that ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{p ↑}$ and ${(\sum_{k = 1}^{m} {UA}_{C} μ)}^{p ↑}$ are two fuzzy sets in universe $U$ . Further, μ is called a definable fuzzy set on multigranulation fuzzy upward approximation space if ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{p ↑} = {(\sum_{k = 1}^{m} {UA}_{C} μ)}^{p ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} {LA}_{C} μ)}^{p ↑}, {(\sum_{k = 1}^{m} {UA}_{C} μ)}^{p ↑})$ is called pessimistic multigranulation fuzzy upward rough set. Using the optimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{C} μ)}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{C} μ)}^{o ↑}$ of μ are $\begin{matrix} {(\sum_{k = 1}^{3} {LA}_{C} μ)}^{o ↑} & = & \frac{0.1667}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.6250}{x_{3}} \\ + \frac{0.5000}{x_{4}} + \frac{0.4000}{x_{5}} + \frac{0.5000}{x_{6}} \\ + \frac{0.6250}{x_{7}} + \frac{0.4000}{x_{8}} + \frac{0.5000}{x_{9}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{3} {UA}_{C} μ)}^{o ↑} & = & \frac{0.4000}{x_{1}} + \frac{0.4000}{x_{2}} + \frac{0.5000}{x_{3}} \\ + \frac{0.5000}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.5833}{x_{6}} \\ + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5833}{x_{9}} . \end{matrix}$

The comparison of the above approximations.

From Figure 4, the following results can be obtained by analyzing ${(\sum_{k = 1}^{3} {LA}_{C} μ)}^{o ↑}$ , ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{o ↑}$ and μ under critical value β = 0.5.

Since μ (x_j) ≥ 0.5, ${(\sum_{k = 1}^{3} {LA}_{C} μ)}^{o ↑} (x_{j}) \geq 0.5$ and ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 3, 4, 7), we concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be effectively treated by the medicine x₃, the symptoms y_i (i = 1, 2, 3, 4, 5, 6, 7, 8) can be effectively treated by the medicine x₄ and x₇.

Since μ (x_j) ≥ 0.5 and ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{o ↑}$ (x_j) ≥ 0.5 (j = 5), that the symptom y₅ can be effectively treated by the medicine x₅.

Since μ (x_j) < 0.5 (j = 2, 6, 8, 9), ${(\sum_{k = 1}^{3} {LA}_{C} μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 2, 6, 9) ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 6, 8, 9), it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be effectively treated by the medicines x₂, x₆ and x₉. The symptoms y₅ and y₈ can be effectively treated by the medicines x₈.

Since μ (x₁) < 0.5, ${(\sum_{k = 1}^{3} {LA}_{C} μ)}^{o ↑} (x_{1}) \leq 0.5$ , ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{o ↑} (x_{j}) \leq 0.5$ , it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can not be treated by the medicine x₁.

Fig.4

Graphical representation.

Now the pessimistic lower approximation ${(\sum_{k = 1}^{3} {LA}_{C} μ)}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{p ↑}$ of μ are $\begin{matrix} {(\sum_{k = 1}^{3} {LA}_{C} μ)}^{p ↑} & = & \frac{0.1250}{x_{1}} + \frac{0.1250}{x_{2}} + \frac{0.1250}{x_{3}} \\ + \frac{0.3333}{x_{4}} + \frac{0.2500}{x_{5}} + \frac{0.3000}{x_{6}} \\ + \frac{0.3000}{x_{7}} + \frac{0.2500}{x_{8}} + \frac{0.2500}{x_{9}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{3} {UA}_{C} μ)}^{p ↑} & = & \frac{0.8000}{x_{1}} + \frac{0.8000}{x_{2}} + \frac{0.7500}{x_{3}} \\ + \frac{0.6250}{x_{4}} + \frac{0.8000}{x_{5}} + \frac{0.6250}{x_{6}} \\ + \frac{0.6000}{x_{7}} + \frac{0.8000}{x_{8}} + \frac{0.7500}{x_{9}} . \end{matrix}$ The comparison of the above approximations.

From Figure 5, the following results can be obtained by analyzing ${(\sum_{k = 1}^{3} {LA}_{C} μ)}^{p ↑}$ , ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{p ↑}$ and μ under critical value β = 0.5.

Since μ (x_j) ≥ 0.5 and ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{p ↑}$ (x_j) ≥ 0.5 (j = 3, 4, 5, 7), it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be effectively treated by the medicines x₃, x₅ and x₇. The symptoms y_i (i = 1, 2, . . . , 8) can be effectively treated by the medicine x₄.

Since μ (x_j) < 0.5 (j = 1, 2, 6, 8, 9), ${(\sum_{k = 1}^{3} {UA}_{C} μ)}^{p ↑} (x_{j}) \geq 0.5$ (j = 1, 2, 6, 8, 9)it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be less effectively treated by the medicines x₁, x₂, x₆, x₈ and x₉.

Fig.5

Graphical representation.

We define the optimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{o ↑}$ of μ as: $\begin{matrix} {(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{o ↑} (x) = \end{matrix}$ $\begin{matrix} ⋁_{k = 1}^{m} {⋀_{y \in U} {(1 -_{a_{k}} M_{x}^{↑ β} (y)) \\ \lor (1 -_{a_{k}} N_{x}^{↑ β} (y)) \lor μ (y)}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{o ↑} (x) = ⋀_{k = 1}^{m} \\ {⋁_{y \in U} {a_{k} M_{x}^{↑ β} (y) \land_{a_{k}} N_{x}^{↑ β} (y) \land μ (y)}}, x \in U . \end{matrix}$

It is easy to see that ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{o ↑}$ and ${(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{o ↑}$ are two fuzzy sets in universe $U$ . Further, μ is called a definable fuzzy set on multigranulation fuzzy upward approximation space if ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{o ↑} = {(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{o ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{o ↑}, {(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{o ↑})$ is called optimistic multigranulation fuzzy upward rough set. Similarly the pessimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{p ↑}$ of μ as: $\begin{matrix} {(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{p ↑} (x) = \\ ⋀_{k = 1}^{m} {⋀_{y \in U} {(1 -_{a_{k}} M_{x}^{↑ β} (y)) \\ \lor (1 -_{a_{k}} N_{x}^{↑ β} (y)) \lor μ (y)}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{p ↑} (x) = ⋁_{k = 1}^{m} \\ {⋁_{y \in U} {a_{k} M_{x}^{↑ β} (y) \land_{a_{k}} N_{x}^{↑ β} (y) \land μ (y)}}, x \in U . \end{matrix}$ It is easy to see that ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{p ↑}$ and ${(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{p ↑}$ are two fuzzy sets in universe $U$ . Further, μ is called a definable fuzzy set on multigranulation fuzzy upward approximation space if ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{p ↑} = {(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{p ↑}$ . Otherwise, the pair $({(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{p ↑}, {(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{p ↑})$ is called pessimistic multigranulation fuzzy upward rough set.

Using the optimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{o ↑}$ and the optimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{o ↑}$ of μ are $\begin{matrix} {(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{o ↑} \end{matrix}$ $\begin{matrix} = \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.6250}{x_{3}} \\ + \frac{0.5000}{x_{4}} + \frac{0.5833}{x_{5}} + \frac{0.5000}{x_{6}} \\ + \frac{0.6250}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{o ↑} & = \frac{0.3000}{x_{1}} + \frac{0.4000}{x_{2}} + \frac{0.5000}{x_{3}} \\ + \frac{0.5000}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.5000}{x_{6}} \\ + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.3750}{x_{9}} . \end{matrix}$

The comparison of the above approximations.

From Figure 6, the following results can be obtained by analyzing ${(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{o ↑}$ , ${(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{o ↑}$ and μ under critical value β = 0.5.

Since μ (x_j) ≥0.5, ${(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{o ↑} (x_{j}) \geq$ 0.5 and ${(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 3,4, 5, 7), we concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be effectively treated by the medicine x₃ and x₅, the symptoms y_i (i = 1, 2, 3, 4, 5, 6, 7, 8) can be effectively treated by the medicine x₄ and x₇ and most important for the treatment of disease A.

Since μ (x_j) < 0.5 (j = 1, 2, 6, 8, 9), ${(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 1, 2, 6, 8,9), ${(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{o ↑} (x_{j}) \geq 0.5$ (j = 6, 8), it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be effectively treated by the medicines x₁, x₂, x₆ and x₉. The symptoms y₅ and y₈ can be effectively treated by the medicines x₈ and important for the treatment of disease A.

Fig.6

Graphical representation.

Using the pessimistic lower approximation ${(\sum_{k = 1}^{m} {LA}_{CN} μ)}^{p ↑}$ and the pessimistic upper approximation ${(\sum_{k = 1}^{m} {UA}_{CN} μ)}^{p ↑}$ of μ are $\begin{matrix} {(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{p ↑} & = & \frac{0.4000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} \\ + \frac{0.5000}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.5000}{x_{6}} \\ + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}}, \end{matrix}$ $\begin{matrix} {(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{p ↑} & = & \frac{0.5000}{x_{1}} + \frac{0.5000}{x_{2}} + \frac{0.5000}{x_{3}} \\ + \frac{0.5000}{x_{4}} + \frac{0.5000}{x_{5}} + \frac{0.5000}{x_{6}} \\ + \frac{0.5000}{x_{7}} + \frac{0.5000}{x_{8}} + \frac{0.5000}{x_{9}} . \end{matrix}$

The comparison of the above approximations.

From Figure 7, the following results can be obtained by analyzing ${(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{p ↑}$ , ${(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{p ↑}$ and μ under critical value β = 0.5.

Since μ (x_j) ≥0.5, ${(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{p ↑} (x_{j}) \geq$ 0.5 and ${(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{p ↑} (x_{j}) \geq 0.5$ (j = 3, 4, 5, 7), it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be effectively treated by the medicines x₃, x₅ and x₇. The symptoms y_i (i = 1, 2, . . . , 8) can be effectively treated by the medicine x₄ and most important for the treatment of disease A.

Since μ (x_j) <0.5 (j = 1, 2, 6, 8, 9), ${(\sum_{k = 1}^{3} {LA}_{CN} μ)}^{p ↑} (x_{j}) \geq 0.5$ , (j = 2, 6, 8, 9)and ${(\sum_{k = 1}^{3} {UA}_{CN} μ)}^{p ↑} (x_{j}) \geq 0.5$ (j = 1, 2, 6,8, 9) it is concluded that the symptoms y_i (i = 1, 2, . . . , 9) can be less effectively treated by the medicines x₁, x₂, x₆, x₈ and x₉ and important for the treatment of disease A.

Fig.7

Graphical representation.

5 Comparison and discussion

A comparative analysis among the methods of Ma [27] and Yang & Hu [42] with our proposed method provides the merits of the proposed techniques in this section. On one hand, in light of the numerical example of the previous section, we compare the methods of Ma,Yang & Hu, with our proposed method. On the other hand, for the drawbacks of the above mentioned methods that can not make a decision in some situations for example when β = 0.5, we find that proposed method can make up for this defect. In the study of multiple attributes decision making (MADM) problems with fuzzy information, there are many decision making methods based on a fuzzy binary relation. However, not all MADM problems can be characterized by a fuzzy binary relation. For this reason, we set methods to solve MADM problems with fuzzy information based on the optimistic/pessimistic multigranulation fuzzy upward rough set based on fuzzy upward β-covering. Furthermore, by comparative analysis, we find that our proposed method is more widely used than the above mentioned methods based on a fuzzy binary relation.

Table 3
Comparison of different methods when β is 0.5

Methods Applicable/Work on Example 4

1 Ma [27] No

2 Yang &Hu [42] No

3 Proposed Yes

	Methods	Applicable/Work on Example 4
1	Ma [27]	No
2	Yang &Hu [42]	No
3	Proposed	Yes

6 Conclusion

In this paper we have developed a more generalized form of fuzzy β-covering, namely the fuzzy upward β-covering. Based on this idea, we introduce new type of fuzzy upward covering based fuzzy rough set model by introducing the concept of fuzzy upward β-neighborhood and their applications in medicine selection for a disease. Furthermore, we propose multigranulation optimistic/pessimistic fuzzy rough sets based on fuzzy upward β-covering and investigate some of their properties. Finally, we find upward β-neighborhood in the fuzzy upward covering approximation space and present optimistic/pessimistic multigranulation rough sets and their basic properties are discussed. In addition, this paper aims to present several uncertainty measures, such as approximate precision, rough degree approximate quality and their mutual relationships are discussed.

In our future study of fuzzy upward β-covering rough sets, may be the following topics to be considered:

Fuzzy upward additive consistency and its application in conflict analysis.

Generalized multigranulation fuzzy rough sets based on upward additive consistency.

Competing interests

The authors declare that they have no competing interests.

Footnotes

Acknowledgments

S. Y. Jang has been supported by the Research Fund of University of Ulsan, 2018.

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Medicines selection via fuzzy upward β -covering rough sets

Abstract

1 Introduction

1.1 Motivation and significance

2 Preliminaries

Table 1 Information Table x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 a 1 0.8 0.3 0.2 0.6 0.4 0.2 0.3 0.3 0.3

Table 2 Information Table x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 a 1 0.8 0.3 0.2 0.6 0.4 0.2 0.3 0.3 0.3 a 2 0.1 0.5 0.1 0.3 0.4 0.3 0.3 0.4 0.2 a 3 0.2 0.2 0.6 0.5 0.3 0.5 0.6 0.3 0.4

Table 3 Comparison of different methods when β is 0.5 Methods Applicable/Work on Example 4 1 Ma [27] No 2 Yang &Hu [42] No 3 Proposed Yes

Competing interests

Footnotes

Acknowledgments

References

Table 1
Information Table

x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈ x ₉

a ₁ 0.8 0.3 0.2 0.6 0.4 0.2 0.3 0.3 0.3

Table 2
Information Table

x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈ x ₉

a ₁ 0.8 0.3 0.2 0.6 0.4 0.2 0.3 0.3 0.3

a ₂ 0.1 0.5 0.1 0.3 0.4 0.3 0.3 0.4 0.2

a ₃ 0.2 0.2 0.6 0.5 0.3 0.5 0.6 0.3 0.4

Table 3
Comparison of different methods when β is 0.5

Methods Applicable/Work on Example 4

1 Ma [27] No

2 Yang &Hu [42] No

3 Proposed Yes