Uncertainty measure of Z -soft covering rough models based on a knowledge granulation

Abstract

Z-soft rough covering models introduced by zhan et al are important generalizations of classical rough set theory to deal with more complex problems of real world. So far, the existing studies mainly focus on constructing various forms of approximation operators and their related properties by means of neighborhoods. In this paper, we introduce different kinds of uncertainty measures related to Z-soft rough covering sets and discuss their limitations. An axiomatic definition of knowledge granulation for soft covering approximations space is introduced. Some main theoretical results are obtained and investigated with the help of examples. Finally, a fully developed example describing the application of the proposed theory in multicriteria decision making is constructed.

Keywords

Z-soft rough covering sets,uncertainty,knowledge granulation,axiomation,decision making

1 Introduction

Rough set theory introduced by Z. Pawlak [22] is an excellent mathematical tool to handle with the given information and it is an access to tackle the problems of equivocalness and ambiguity. The main significance of the rough set theory is that it does not involve any additional information about the data like in fuzzy sets we need membership function for values of data. The classical rough set theory is based on the equivalence relations or partitions. But in many situations the equivalence relations or partitions is very restricted to use. Therefore many extensions andgeneralizations of rough set theory have been proposed such as similarity relation based rough sets, arbitrary binary relations based rough sets, tolerance relation based rough sets, covering based sough sets, soft covering based rough sets and so on [5 , 40].

Molodtsov [20] introduced the concept of soft sets to deal with the problems involving uncertainty and attributes. This concept not only changed the role of above theories as the sole representative of multi attributes but also used in many disciplines to tackle problems of uncertainty and vagueness [2 , 28]. By connecting the soft sets to covering based rough sets, a useful drastic theory of soft covering based rough sets has been proposed by [30 , 36] by connecting the covering soft sets to rough sets. A novel granulation structure, soft coveringapproximation space and soft covering rough approximation space are defined and some of their basic properties has been investigated. A number of applications given by many researchers, in multi-attributes decision making problems, attributes reduction problems, data labeling problems, data mining problems and knowledge based systems can be seen in [1 , 38]. Zhan et al [34] introduced the notion of Z-soft rough fuzzy sets and gave a corresponding decision making.

Uncertainty including vagueness, fuzziness, randomness, incompleteness, inconsistency and imprecision exists in almost every spheres of life. Measuring the uncertain knowledge has become a challenging issue in almost every field of life. A good measure of uncertainty can improve the efficiency and accuracy of the information for a decision support system. To evaluate the uncertainty measurement in any information system, Shannon [25] introduced the notion of entropy which is an important, effective and suitable tool for characterizing information content in different models. The extension and generalization of entropy and its modifications were endorsed for rough sets in [4 , 28].

Z-soft rough covering models are important generalizations of classical rough set theory to deal with more complex problems of real world. So far, the existing study fucus on constructing various forms of approximation operators and their related properties. In this paper, we have introduced different kinds of uncertainty measures related to Z-soft rough covering sets and have discussed their limitations. An axiomatic definition of knowledge granulation for soft covering approximations space is given under which the improved versions of the above said measures are introduced. Finally, a fully developed example describing the application of the proposed theory in multicriteria decision making is constructed.

The present article is completed in the following sections. In section 2, some basic concepts are revised. Section 3 is about basic definitions and characterization of roughness measure and accuracy measure for Z-soft rough covering sets where basic theory is discussed with examples. Section 4 is devoted to knowledge granulation soft covering approximation spaces. Some important properties of these uncertainty measures are investigated and the relationship between such measures are established. These properties will help to understand the essence of uncertainty measurement and in measuring the quality of a decision rule. Uncertainty measures of Z-soft rough covering sets based on knowledge granulation are discussed in section 5. In section 6, applications of Z-soft rough covering sets is given and an example is constructed.

2 Preliminaries

In this section, the basic ideas behind the soft sets, soft rough sets are introduced which will help us in the rest of sections.

Definition 1. [18] Let E be the set of parameters and K ⊆ E. A pair (δ, K) is called a soft set over the set U of universe, where δ : K → P (U) is a set valued mapping and P (U) is the power set of U.

Definition 2. [12] Let S = (δ, K) be a soft set over U . Then, the pair $P = (U, S)$ is called soft approximation space. Based on the soft approximation space $P$ , we define

${\underline{appr}}_{P} (X) = {u \in U : \exists a \in K, [u \in δ (a) \subseteq X]},$ ${\bar{appr}}_{P} (X) = {u \in U : \exists a \in K, [u \in δ (a), δ (a) \cap X \neq \emptyset]}$

assigning to any set X ⊆ U, the sets ${\underline{appr}}_{P} (X)$ and ${\bar{appr}}_{P} (X),$ are called soft $P$ - lower approximation and soft $P$ - upper approximation of X, respectively.

The sets $Pos (X) = {\underline{appr}}_{P} (X),$ $Neg (X) = - {\underline{appr}}_{P} (X),$ $and Bnd (X) = {\bar{appr}}_{P} (X) - {\underline{appr}}_{P} (X)$ are called the soft $P$ -positive region, the soft $P$ -negative region, and the soft $P$ -boundary region of X, respectively. If ${\bar{appr}}_{P} (X) = {\underline{appr}}_{P} (X),$ X is said to be soft $P$ -definable; otherwise X is called a soft $P$ -rough set.

3 Roughness measure and accuracy measure for Z-Soft rough covering sets

The concept of soft neighborhood of an element and Z-soft rough covering sets is introduced by Zhan et al [36]. In this section, we introduce the concept of uncertainty measurements in Z-soft rough covering sets. We introduce the notion of roughness measure and accuracy measure for Z-soft rough covering sets. Some basic properties are investigated and related examples are constructed.

Definition 3. Let (β, M) be a soft set, where M is the set of parameters, β : M → P (U) is a set valued mapping and P (U) is the power set of U . If ⋃_θ∈Mβ (θ) = U, then (β, M) is called full soft set.

Definition 4. Let (β, M) be a full soft set. Then (β, M) is called covering soft set if β (θ)≠ ∅ for all θ ∈ M . In this case (β, M) is called covering soft set over U, denoted by $C_{U} .$ Denote by $W = (U, C_{U})$ and call it soft covering approximation space.

Definition 5. Let $W = (U, C_{U})$ be a soft covering approximation space and x ∈ U, then the set

$C_{W} (x) = {β (θ) \in ℭ_{U} : x \in β (θ)}$ is called the soft association of x .

$N_{W} (x) = \cap {β (θ) \in ℭ_{U} : x \in β (θ)} = \cap C_{W} (x)$ is called soft neighborhood of x.

Definition 6. Let $W = (U, C_{U})$ be a soft covering approximation space. For any X ⊆ U, the soft covering lower and soft covering upper approximation operators are respectively, defined as: $\underline{Z_{W}} (X) = {x \in U : N_{W} (x) \subseteq X} = \underline{Z_{W}} (X)$ $\bar{Z_{W}} (X) = {x \in U : N_{W} (x) \cap X \neq \emptyset} = \bar{Z_{W}} (X)$

If $\underline{Z_{W}} (X) = \bar{Z_{W}} (X),$ then X is called Z-soft covering definable set. In opposite case if $\underline{Z_{W}} (X) = \bar{Z_{W}} (X),$ then X is called Z-soft rough covering set.

Definition 7. Let $W = (U, C_{U})$ be a soft covering approximation space. Then the accuracy measure of X ⊆ U is given by $α_{W} (X) = | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} |$

Definition 8. Let $W = (U, C_{U})$ be a soft covering approximation space. Then the roughness measure of X ⊆ U is given by $ρ_{W} (X) = 1 - | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} | = 1 - α_{W} (X)$ Thus if $\bar{Z_{W}} (X) = \underline{Z_{W}} (X)$ then $ρ_{W} (X) = 0$ then X is Z-soft covering definable. Clearly,

Example 1. Let U ={ x₁, x₂, x₃, x₄, x₅, x₆, x₇ } be a finite universe and let M ={ θ₁, θ₂, θ₃, σ₄ } be the set of parameters such that (β, M) is covering soft set over U, see Table 1 and β (θ₁) = { x₁, x₄, x₇ } , β (θ₂) = { x₂, x₃, x₅, x₆ } , β (θ₃) = { x₃, x₄, x₆, x₇ } , β (θ₄) = { x₂, x₃, x₅ } .

Table 1
Tabular representation of soft set

x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇

θ ₁ 1 0 0 1 0 0 1

θ ₂ 0 1 1 0 1 1 0

θ ₃ 0 0 1 1 0 1 1

θ ₄ 0 1 1 0 1 0 0

	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇
θ ₁	1	0	0	1	0	0	1
θ ₂	0	1	1	0	1	1	0
θ ₃	0	0	1	1	0	1	1
θ ₄	0	1	1	0	1	0	0

Let $W = (U, C_{U})$ be a soft covering approximation space and x ∈ U . Then the soft neighborhood of x is $N_{W} (x) = \cap {δ (σ) \in C_{V} : x \in δ (σ)}$ , shows $N_{W} (x_{1}) = {x_{1}, x_{4}, x_{7}}, N_{W} (x_{2}) = {x_{2}, x_{3}, x_{5}} = N_{W} (x_{5}),$

$N_{W} (x_{3}) = {x_{3}}, N_{W} (x_{4}) = {x_{4}, x_{7}} = N_{W} (x_{7}), N_{W} (x_{6}) = {x_{3}, x_{6}} .$ Let X = { x₂, x₄, x₇ } ⊆ U, then $\underline{Z_{W}} (X) = {x \in U : N_{W} (x) \subseteq X} = {x_{4}, x_{7}}$ $\bar{Z_{W}} (X) = {x \in U : N_{W} (x) \cap X \neq 0} .$ Also, the measure of roughness of X ⊆ V is given by $ρ_{W} (X) = 1 - | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} | = 0.667.$

Definition 9. Let $W = (U, C_{U})$ and $W^{*} = (U, C_{U}^{*})$ be two soft covering approximation spaces with

For all $β (θ) \in C_{W} (x),$ there exists $β^{*} (θ) \in C_{W^{*}} (x)$ such that β (θ) ⊆ β^∗ (θ) and

For all $β^{*} (θ) \in C_{W^{*}} (x),$ there exists $β (θ) \in C_{W} (x)$ such that β (θ) ⊆ β^∗ (θ) . Then we say $C_{U}$ is finer than $C_{U}^{*}$ and denote it by $C_{U} ⪯ C_{U}^{*}$ .

In the following proposition, we show that $(C_{U}, ⪯)$ is a partial ordered.

Proposition 1. Let U be a finite universe and let $W = (U, C_{U})$ be a soft covering approximation space. Then $(C_{U}, ⪯)$ is a partial ordered.

Proof. (i) Since β (θ) ⊆ β (θ) for all $β (θ) \in C_{W} (x)$ and |β (θ) | = |β (θ) |, therefore $C_{U} ⪯ C_{U} .$

(ii) Suppose $C_{U} ⪯ C_{U}^{*}$ and $C_{U}^{*} ⪯ C_{U},$ where $W^{*} = (U, C_{U}^{*})$ is a soft covering approximation space. Then by definition 10, we can obtain that;

$C_{U} ⪯ C_{U}^{*}$ if and only if β (θ) ⊆ β^∗ (θ) for all θ ∈ M, where $β (θ) \in C_{W} (x)$ and $β^{*} (θ) \in C_{W^{*}} (x) .$

$C_{U}^{*} ⪯ C_{U}$ if and only if β^∗ (θ) ⊆ β (θ) for all θ ∈ M, where $β^{*} (θ) \in C_{W^{*}} (x)$ and $β (θ) \in C_{W} (x) .$

Therefore, β (θ) ⊆ β^∗ (θ) ⊆ β (θ) . That is, β (θ) = β^∗ (θ) . Hence $C_{U} = C_{U}^{*} .$

(iii) Suppose $C_{U} ⪯ C_{U}^{*}$ and $C_{U}^{*} ⪯ C_{U}^{* *},$ where $W^{* *} = (U, C_{U}^{* *})$ is a soft covering approximation space. Then by definition 10, we can obtainthat;

$C_{U} ⪯ C_{U}^{*}$ if and only if β (θ) ⊆ β^∗ (θ) for all θ ∈ M, where $β (θ) \in C_{W} (x)$ and $β^{*} (θ) \in C_{W *} (x) .$

$C_{U}^{*} ⪯ C_{U}^{* *}$ if and only if β^∗ (θ) ⊆ β^∗∗ (θ) for all θ ∈ M, where $β^{*} (θ) \in C_{W^{*}} (x)$ and $β^{* *} (θ) \in C_{W^{* *}} (x) .$

Thus we have, (a) For all $β (θ) \in C_{W} (x),$ there exists $β^{* *} (θ) \in C_{W^{* *}} (x)$ such that β (θ) ⊆ β^∗∗ (θ) and

(b) For all $β^{* *} (θ) \in C_{W^{* *}} (x),$ there exists $β (θ) \in C_{W} (x)$ such that β (θ) ⊆ β^∗∗ (θ) . Then we say $C_{U}$ is finer than $C_{U}^{* *} .$ That is, $C_{U} ⪯ C_{U}^{* *}$ . Therefore, $(C_{U}, ⪯)$ is a partial ordered.

Proposition 2. Let U be a finite universe and let $W = (U, C_{U})$ and $W^{*} = (U, C_{U}^{*})$ be two soft vertex covering approximation spaces such that $C_{U} ⪯ C_{U}^{*}$ . Then the following always hold:

$N_{W} (x) \subseteq N_{W^{*}} (x)$

$\underline{Z_{W^{*}}} (X) \subseteq$ $\underline{Z_{W}} (X)$

$\bar{Z_{W}} (X) \subseteq$ $\bar{Z_{W^{*}}} (X)$

α_W (X) ≥ α_W^* (X)

ρ_W (X) ≤ ρ_W^* (X)

Proof. (i) Suppose $z \in N_{W} (x) = \cap {β (θ) \in C_{U} : x \in β (θ)} = \cap C_{R} (x)$ . Then z ∈ β (θ) for any $β (θ) \in C_{W} (x) .$ By Definition 10, for any $β^{*} (θ) \in C_{W^{*}} (x),$ there exists $β (θ) \in C_{W} (x)$ such that z ∈ β (θ) ⊆ β^∗ (θ) . Hence z ∈ β^∗ (θ) , for any $β^{*} (θ) \in C_{R^{*}} (x) .$ Thus $z \in \cap C_{W^{*}} (x) = N_{W^{*}} (x) .$ Consequently, $N_{W} (x) \subseteq N_{W^{*}} (x) .$

(ii) Let $x \in \underline{Z_{W^{*}}} (X) = {x \in U : N_{W^{*}} (x) \subseteq X} .$ Then $N_{W^{*}} (x) \subseteq X .$ Since $C_{U} ⪯ C_{U}^{*}$ , so by part (i) $N_{W} (x) \subseteq N_{W^{*}} (x) \subseteq X$ $\Rightarrow N_{W} (x) \subseteq X .$ $\Rightarrow x \in \underline{Z_{W}} (X)$ showing that $\underline{Z_{W^{*}}} (X) \subseteq$ $\underline{Z_{W}} (X) .$

(iii) $x \in \bar{Z_{W}} (X)$ then there exists a z ∈ U such that $x \in N_{W} (z)$ and $N_{W} (z) \cap X \neq \emptyset .$ Since $C_{U} ⪯ C_{U}^{*}$ , so by part (i) , $N_{W} (z) \subseteq N_{W^{*}} (z) .$ Thus $z \in N_{W^{*}} (z)$ and $N_{W^{*}} (z) \cap X \neq \emptyset .$ Which shows x∈ $\bar{Z_{W^{*}}} (X) .$ Thus $\bar{Z_{W}} (X) \subseteq$ $\bar{Z_{W^{*}}} (X) .$

(iv) From part (ii) and (iii) , $α_{W} (X) = | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} | \geq | \frac{\underline{Z_{W *}} (X)}{\bar{Z_{W *}} (X)} | = α_{W *} (X)$ Consequently

(v) $ρ_{W} (X) = 1 - α_{W} (X) \leq 1 - α_{W *} (X) = ρ_{W *} (X) .$ So, $ρ_{W} (X) \leq ρ_{W *} (X) .$ □

In the following example we show that for two soft covering approximation spaces which are not exactly same but the same accuracy measure and same roughness measure is obtained.

Example 2. Let U ={ x₁, x₂, x₃, x₄, x₅, x₆ } be a finite universe and let M ={ θ₁, θ₂, θ₃, θ₄, θ₅ } be the set of parameters such that (β, M) and (β^∗, M) are two covering soft sets over U, see Tables 2and 3.

Table 2

Tabular representation of soft set (β, M)

(β, M)	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
θ ₁	1	0	1	0	0	0
θ ₂	0	1	0	0	0	1
θ ₃	0	0	1	0	0	0
θ ₄	0	0	0	1	1	0
θ ₅	1	0	0	0	1	1

Table 3

Tabular representation of soft set (β^∗, M)

(β^∗, M)	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
θ ₁	1	0	1	0	0	0
θ ₂	0	1	1	0	0	1
θ ₃	0	0	1	0	0	0
θ ₄	0	0	0	1	1	0
θ ₅	1	1	0	0	1	1

Let $W = (U, C_{U})$ and $W^{*} = (U, C_{U}^{*})$ be two soft covering approximation spaces then we can see $C_{U} ⪯ C_{U}^{*} .$ Here $N_{W} (x_{1}) = {x_{1}}, N_{W} (x_{2}) = {x_{2}, x_{6}}, N_{W} (x_{3}) = {x_{3}}, N_{W} (x_{4}) = {x_{4}, x_{5}}, N_{W} (x_{5}) = {x_{5}}, N_{W} (x_{6}) = {x_{6}} .$ Also $N_{W *} (x_{1}) = {x_{1}}, N_{W *} (x_{2}) = {x_{2}, x_{6}}, N_{W *} (x_{3}) = {x_{3}}, N_{W *} (x_{4}) = {x_{4}, x_{5}}, N_{W *} (x_{5}) = {x_{5}}, N_{W *} (x_{6}) = {x_{2}, x_{6}}$

Let X = { x₂, x₄, x₆ } , then $\underline{Z_{W}} (X) = \underline{Z_{W *}} (X) = {x_{2}, x_{6}}$ $\bar{Z_{W}} (X) = \bar{Z_{W *}} (X) = {x_{2}, x_{4,} x_{6}} = \bar{Z_{W *}} (X)$ Since the upper and lower approximations are not equal so X is Z-soft rough covering set in both approximation spaces $W = (U, C_{U})$ and $W^{*} =$ $(U, C_{U}^{*}) .$

Then the accuracy measure of X ⊆ U is given by $α_{W} (X) = | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} | = \frac{2}{3} = | \frac{\underline{Z_{W *}} (X)}{\bar{Z_{W *}} (X)} | = α_{W *} (X) .$ And the roughness measure of X ⊆ U is given by $\begin{array}{l} ρ_{W} (X) & = & 1 - | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} | = \frac{1}{3} \\ = & 1 - | \frac{\underline{Z_{W *}} (X)}{\bar{Z_{W *}} (X)} | = ρ_{W *} (X) . \end{array}$

Remark 1. We have seen that for two soft covering approximation spaces $W = (U, C_{U})$ and $W^{*} = (U, C_{U}^{*})$ inclusion relation exists. But the same accuracy measure and same roughness measure is obtained for Z-soft rough covering set X . Therefore, a more accurate and effective measure for Z-soft rough covering sets is needed.

4 Knowledge granulation of soft covering approximation spaces

In section, we introduce an axiomatic definition of knowledge granulation. It is proved that the above granulation measure is a special form of the axiomatic definition. Denote the set of all soft neighborhood of x by $Z S_{C_{U}}$ and call is Z-soft neighborhood of x .

Definition 10. Let U be a finite universe and let $W = (U, C_{U})$ be a soft covering approximation space. Let $G_{W} : P (C_{U}) \to ℝ$ be a mapping from the power set of $C_{U}$ to the set $ℝ$ of real numbers. We say G_W is a knowledge granulation in $W = (U, C_{U})$ if G_W satisfies the following conditions:

(i) $C \subseteq$ $C_{U}$ (Non - negativity)

(ii) $G_{W} (C_{1}) = G_{W} (C_{2})$ for any $C_{1}, C_{2} \subseteq C_{U}$ if there exists a bijective mapping $δ : Z S_{C_{1}} \to Z S_{C_{2}}$ such that $| N_{W_{C_{1}}} (x_{j}) | = | δ (N_{W_{C_{1}}} (x_{j})) |$ for all j ∈ {1, 2, ..., |U|} where $Z S_{C_{1}} = {N_{W_{C_{1}}} (x_{1}), N_{W_{C_{1}}} (x_{2}), \dots, N_{W_{C_{1}}} (x_{| U |})}$ and $Z S_{C_{2}} = {N_{C_{2}} (x_{1}), N_{C_{2}} (x_{2}), \dots, N_{C_{2}} (x_{| U |})};$ and

(iii) $G_{W} (C_{1}) \leq G_{W} (C_{2})$ for any $C_{1}, C_{2} \subseteq C_{U}$ with $C_{1} ⪯ C_{2}$ (Monotonicity) .

Definition 11. In [15], a different kind of knowledge granulation is introduced, which is as follows: $G_{W} K (C_{U}) = \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W} (x_{j}) |$ Clearly, when $W = (U, C_{U})$ is a soft covering approximation space then $\frac{1}{{| U |}^{2}} \leq G_{W} K (C) \leq 1, where C \subseteq C_{U} .$

Proposition 3. The knowledge granulation in definition 12 is a knowledge granulation under definition 11.

Proof. We have to show G_W meets all the conditions of definition 11.

(i) Clearly, G_W K is non-negative.

(ii) Let $C_{1}, C_{2} \subseteq C_{U}$ and let $δ : Z S_{C_{1}} \to Z S_{C_{2}}$ be a bijective mapping such that $| N_{W_{C_{1}}} (x_{j}) | = | δ (N_{W_{C_{1}}} (x_{j})) |$ for all $j \in {1, 2, \dots, | U |}$ where $Z S_{C_{1}} = {N_{W_{C_{1}}} (x_{1}), N_{W_{C_{1}}} (x_{2}), \dots, N_{W_{C_{1}}} (x_{| U |})}$ and $Z S_{C_{2}} = {N_{C_{2}} (x_{1}), N_{C_{2}} (x_{2}), \dots, N_{C_{2}} (x_{| U |})}$ . Let $δ (N_{W_{C_{1}}} (x_{j})) = N_{W_{C_{2}}}^{*} (x_{j}),$ where $N_{W_{C_{2}}}^{*} (x_{j}) \in {N_{W_{C_{2}}} (x_{1}), N_{W_{C_{2}}} (x_{2}), \dots, N_{W_{C_{2}}} (x_{| U |})} .$ Then, we have $\begin{array}{l} = \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W_{C_{1}}} (x_{j}) | = \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W_{C_{2}}}^{*} (x_{j}) | \\ = \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W_{C_{2}}} (x_{j}) | = G_{W} K (C_{2}) . \end{array}$

(iii) Let $C_{1}, C_{2} \subseteq C_{U}$ such that $C_{1} ⪯ C_{2} .$ Then for all $j \in {1, 2, \dots | U |}, N_{W_{C_{1}}} (x_{j}) \subseteq N_{W_{C_{2}}} (x_{j}),$ which shows that

Therefore $\begin{array}{l} G_{W} K (C_{1}) & = & \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W_{C_{1}}} (x_{j}) | \\ \leq & \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W_{C_{2}}} (x_{j}) | = G_{W} K (C_{2}) . \end{array}$

Example 3. Let U = {x₁, x₂, x₃, x₄, x₅, x₆} be a finite universe and let M ={ θ₁, θ₂, θ₃, θ₄ } be the set of parameters such that $C_{U} =$ (β, M) is a covering soft sets over U, see Table 4

Table 4
Tabular representation of soft set (β, M)

(β, M) x ₁ x ₂ x ₃ x ₄ x ₅ x ₆

θ ₁ 1 1 0 0 1 1

θ ₂ 0 1 1 1 0 1

θ ₃ 1 0 0 1 1 1

θ ₄ 0 0 1 1 0 1

(β, M)	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆
θ ₁	1	1	0	0	1	1
θ ₂	0	1	1	1	0	1
θ ₃	1	0	0	1	1	1
θ ₄	0	0	1	1	0	1

Now $N_{W_{C}} (x_{1}) = {x_{1}, x_{5}, x_{6}}, N_{W_{C}} (x_{2}) = {x_{2}, x_{6}}, N_{W_{C}} (x_{3}) = {x_{3}, x_{4}, x_{6}}, N_{W_{C}} (x_{4}) = {x_{4}, x_{6}}, N_{W_{C}} (x_{5}) = {x_{1}, x_{5}, x_{6}}, N_{W_{C}} (x_{6}) = {x_{6}}$ Let G_W $P (C_{U}) \to ℝ$ then

$\begin{array}{l} G_{W} K (C_{U}) & = & \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W} (x_{j}) | \\ = & \frac{1}{6^{2}} [3 + 2 + 3 + 2 + 3 + 1] = \frac{14}{36} = \frac{7}{18} \end{array}$

5 Uncertainty measure of z-soft rough covering sets based on knowledge granulation

In this section, a new measure of uncertainty based on knowledge granulation for Z-soft rough covering sets is introduced. It is shown that this new and improved measure is more effective for evaluating the roughness and accuracy of a set in soft covering approximation space.

Definition 12. Let $W = (U, C_{U})$ be a soft covering approximation space and let X be a non-empty subset of U. Then the improved roughness of X is defined as: $R o u g h n e s s_{W} (X) = ρ_{W} (X) G_{W} K (X)$ where $ρ_{W} (W) = 1 - | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} |$ . Clearly the improved roughness of X not only involve the roughness of itself but with the uncertainty caused by the soft covering approximation spaces. Furthermore, the degree of uncertainty is more accurately reflected by this definition, for Z-soft rough covering sets, based on knowledge granulation.

Proposition 4. Let U be a finite universe and let $W = (U, C_{U})$ a soft covering approximation space. Then for a non-empty subset a X of U, we have

Proof. Since $R o u g h n e s s_{W} (X) = ρ_{W} (X) G_{W} K (X)$ with $0 \leq ρ_{W} (X) \leq 1$ and $\frac{1}{| U |} \leq G_{W} K (X) \leq 1$ so we have

That is, $0 \leq R o u g h n e s s_{W} (X) \leq 1.$ □

Proposition 5. Let $W = (U, C_{U})$ be a soft covering approximation space and X ⊆U. If (X) = 1, then ρ_W (X) = 1 = G_W K (X)

Proof. Straightforward. □

Proposition 6. Let U be a finite universe and let $W = (U, C_{U})$ be a soft covering approximation space. Then Roughness_W (X) = 0 if and only if X is Z-soft covering definable.

Proof. Suppose Roughness_W (X) = 0. This implies $ρ_{W} (X) G_{W} K (X) = 0$ . Which is possible only when ρ_W (X) = 0 because $\frac{1}{| U |} \leq G_{W} K (X) \leq 1$ . But $ρ_{W} (W) = 1 - | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} |$ . So ρ_W (X) = 0 means $\underline{Z_{W}} (X) = \bar{Z_{W}} (X)$ . That is, Z-soft covering definable. Conversely suppose that Z-soft covering definable. So $\underline{Z_{W}} (X) = \bar{Z_{W}} (X)$ or $| \underline{Z_{W}} (X) | = | \bar{Z_{W}} (X) |$ showing that $1 - | \frac{\underline{Z_{W}} (X)}{\bar{Z_{W}} (X)} | = 0$ or Therefore, Roughness_W (X) = ρ_W (X) G_W K(X) = 0. □

Definition 13. Let $W = (U, C_{U})$ be a soft covering approximation space and let X be a non-empty subset of U. Then the improved accuracy of X is defined as: $A c c u r a c y_{W} (X) = 1 - ρ_{W} (X) G_{W} K (X)$

Proposition 7. Let U be a finite universe and let $W = (U, C_{U})$ and $W^{*} = (U, C_{U}^{*})$ be two soft covering approximation spaces such that $C_{U} ⪯ C_{U}^{*} .$ Then

(i) $R o u g h n e s s_{W} (X) \leq R o u g h n e s s_{W *} (X)$

(ii) $A c c u r a c y_{W} (X) \geq A c c u r a c y_{W *} (X) .$

Proof. Straightforward.□

Example 4. Let U = {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈} be a finite universe and let M ={ θ₁, θ₂, θ₃, θ₄ } be the set of parameters. Let $W = (U, C_{U}),$ $W^{*} =$ $(U, C_{U}^{*})$ and $W^{* *} = (U, C_{U}^{* *})$ be three covering soft approximation spaces such that the covering soft sets $C_{U} = (β_{1}, M),$ $C_{U}^{*} = (β_{2}, M)$ and $C_{U}^{* *} = (β_{3}, M)$ over U are shown in Tables 5, 6 and 7 respectively.

Table 5
Tabular representation of soft set (β₁, M)

(β₁, M) x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈

θ ₁ 1 0 1 1 1 0 0 1

θ ₂ 0 1 0 1 0 1 1 1

θ ₃ 1 1 1 0 1 0 0 1

θ ₄ 0 0 0 1 1 0 1 0

θ ₅ 0 0 1 1 0 1 1 0

(β₁, M)	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
θ ₁	1	0	1	1	1	0	0	1
θ ₂	0	1	0	1	0	1	1	1
θ ₃	1	1	1	0	1	0	0	1
θ ₄	0	0	0	1	1	0	1	0
θ ₅	0	0	1	1	0	1	1	0

Table 6

Tabular representation of soft set (β₂, M)

(β₂, M)	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
θ ₁	1	0	1	1	1	0	0	0
θ ₂	0	1	0	0	0	1	1	1
θ ₃	0	0	1	0	1	0	0	1
θ ₄	0	0	0	1	0	0	1	0
θ ₅	0	0	1	1	0	1	0	0

Table 7

Tabular representation of soft set (β₃, M)

(β₃, M)	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈
θ ₁	1	0	0	1	1	0	0	0
θ ₂	0	1	0	0	0	1	1	0
θ ₃	0	0	1	0	0	0	0	1
θ ₄	0	0	0	1	0	0	0	0
θ ₅	0	0	0	1	0	1	0	0

$\begin{array}{l} N_{W} (x_{1}) & = & {x_{1}, x_{3}, x_{5}, x_{8}}, N_{W} (x_{2}) = {x_{2}, x_{8}}, N_{W} (x_{3}) = {x_{3}}, N_{W} (x_{4}) = {x_{4}}, \\ N_{W} (x_{5}) & = & {x_{5}}, N_{W} (x_{6}) = {x_{6}, x_{7}}, N_{W} (x_{7}) = {x_{4}, x_{7}}, N_{W} (x_{8}) = {x_{8}}, \\ N_{W *} (x_{1}) & = & {x_{1}, x_{3}, x_{4}, x_{5}}, N_{W *} (x_{2}) = {x_{2}, x_{6}, x_{7}, x_{8}}, N_{W *} (x_{3}) = {x_{3}}, N_{W *} (x_{4}) = {x_{4}} \\ N_{W *} (x_{5}) & = & {x_{3}, x_{5}}, N_{W *} (x_{6}) = {x_{6}}, N_{W *} (x_{7}) = {x_{7}}, N_{W *} (x_{2}) = {x_{8}} \\ N_{W * *} (x_{1}) & = & {x_{1}, x_{4}, x_{5}}, N_{W * *} (x_{2}) = {x_{2}, x_{6}, x_{7}}, N_{W * *} (x_{3}) = {x_{3}, x_{8}}, N_{W * *} (x_{4}) = {x_{4}}, \\ N_{W * *} (x_{5}) & = & {x_{1}, x_{4}, x_{5}}, N_{W * *} (x_{6}) = {x_{6}}, N_{W * *} (x_{7}) = {x_{2}, x_{6}, x_{7}}, N_{W * *} (x_{1}) = {x_{3}, x_{8}} \end{array}$ Let X = {x₁, x₂, x₃, x₈} ⊆ U . Then $\begin{array}{l} \underline{Z_{W}} (X) & = & {x \in U : N_{W} (x) \subseteq X} = {x_{3}, x_{4}} \\ = & \underline{Z_{W *}} (X) = \underline{Z_{W * *}} (X) \\ \bar{Z_{W}} (X) & = & {x \in U : N_{W} (x) \cap X \neq 0} \\ = & {x_{1}, x_{2}, x_{3}, x_{4}} = \bar{Z_{W *}} (X) = \bar{Z_{W * *}} (X) \end{array}$ Also, $ρ_{W} (X) = ρ_{W *} (X) = ρ_{W * *} (X) = 1 - \frac{3}{4} = \frac{1}{4}$ Now

$\begin{array}{l} G_{W} K (X) & = & \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W} (x_{j}) | \\ = & \frac{1}{8^{2}} [3 + 3 + 1 + 3 + 1 + 3 + 3 + 1] = 0.281 \\ G_{W} K (X) & = & \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W *} (x_{j}) | \\ = & \frac{1}{8^{2}} [1 + 2 + 1 + 3 + 1 + 3 + 3 + 1] = 0.234 \\ G_{W} K (X) & = & \frac{1}{{| U |}^{2}} \sum_{j = 1}^{| U |} | N_{W * *} (x_{j}) | \\ = & \frac{1}{8^{2}} [3 + 3 + 1 + 3 + 2 + 1 + 1 + 1] = 0.234 \end{array}$

Thus Roughness_W* (X) = Roughness_W** (X) < Roughness_W (X).

6 Applications of Z-soft rough covering sets

One of the important applications of rough sets is decision making and after combining with soft sets, it has promoted to multicriteria group decision making. Many applications of multicriteria group decision making are available in literature which can be seen in [13 , 38]. In this section, the applications of Z-soft rough covering sets is given. We put an algorithm based on Z-soft rough covering sets.

A TEST EXAMPLE Suppose ten applicants x₁, x₂, . . . , x₁₀ appears in an interview conducted by a company, for the job of stock market analyst. Suppose there are five characteristics θ₁, θ₂, . . . , θ₅ of applicants, to be considered by company, where θ₁ represents "educational background", θ₂ represents "experience", θ₃ represents "command on different languages", θ₄ represents "strain capability", θ₅ represents "tolerance". Let U ={ x₁, x₂, . . . , x₁₀ } be the set of applicants(universe) and T ={ θ₁, θ₂, . . . , θ₅ } be the set of characteristics(parameters). Let (β, T) and (γ, T) be two soft sets on U and U × U respectively, defined by; $β (θ_{i}) = {x_{j} \in U : x_{j} may have the characteristics θ_{i}}$ for i = 1, 2, . . .,5, j = 1, 2, 3, . . . , 10, and $\begin{matrix} γ (θ_{i}) = & {(x_{k}, x_{j}) \in U \times U : x_{k} is better than x_{j} \\ with respect to characteristics θ_{i}}, \end{matrix}$ For convenience, we will represent x_kx_j in place of (x_k, x_j) . Let $\begin{matrix} β (θ_{1}) = & {x_{2}, x_{3}, x_{5}, x_{6}, x_{7}, x_{9}, x_{10}}, \\ β (θ_{2}) = & {x_{1}, x_{3}, x_{4}, x_{5}, x_{8}, x_{10}}, \\ β (θ_{3}) = & {x_{3}, x_{7}, x_{8}, x_{10}}, β (θ_{4}) = {x_{1}, x_{2}, x_{5}, x_{9}}, \\ β (θ_{5}) = & {x_{2}, x_{5}, x_{6}, x_{8}, x_{10}} and \\ γ (θ_{1}) = & {x_{2} x_{3}, x_{2} x_{6}, x_{2} x_{9}, x_{10} x_{2}, x_{5} x_{9}, x_{7} x_{9}, \\ x_{7} x_{10}, x_{5} x_{4}, x_{6} x_{7}}, \\ γ (θ_{2}) = & {x_{3} x_{1}, x_{1} x_{4}, x_{4} x_{5}, x_{8} x_{9}, x_{1} x_{5}, x_{8} x_{5}, \\ x_{3} x_{5}, x_{9} x_{3}}, \\ γ (θ_{3}) = & {x_{3} x_{10}, x_{3} x_{7}, x_{3} x_{8}, x_{10} x_{8}}, \\ γ (θ_{4}) = & {x_{1} x_{9}, x_{1} x_{5}, x_{5} x_{2}, x_{5} x_{9}, x_{1} x_{2}, x_{9} x_{2}}, \\ γ (θ_{5}) = & {x_{5} x_{6}, x_{5} x_{8}, x_{2} x_{6}, x_{2} x_{10}, x_{10} x_{8}} \end{matrix}$

$\begin{matrix} So N_{W} (x_{1}) & = {x_{1}, x_{5}}, N_{W} (x_{2}) = {x_{2}, x_{5}}, \\ N_{W} (x_{3}) & = {x_{3}, x_{5}, x_{10}}, \\ N_{W} (x_{4}) & = {x_{1}, x_{3}, x_{4}, x_{5}, x_{8}, x_{10}} \\ N_{W} (x_{5}) & = {x_{5}}, N_{W} (x_{6}) = {x_{2}, x_{5}, x_{6}, x_{10}}, \\ N_{W} (x_{7}) & = {x_{3}, x_{7}, x_{10}}, N_{W} (x_{8}) = {x_{8}, x_{10}}, \\ N_{W} (x_{9}) & = {x_{2}, x_{5}, x_{9}}, N_{W} (x_{10}) = {x_{10}} \end{matrix}$

Here in each directed graph G_i, as shown in Figure 1, the direction from vertex x_k to x_j means x_k is better than x_j with respect to characteristics θ_i. Suppose $W = (U, C_{U})$ be a soft covering approximation space. For basic evaluation, Suppose a panel of three experts $D = {E_{1}, E_{2}, E_{3}}$ who gave their opinion about the selection of applicants. Let $(π, D)$ be the evaluation soft set for three experts set over U, given by

Fig.1
$\begin{matrix} π (E_{t}) = & {x_{j} \in U : the expert specialist E_{t} suggested \\ x_{j} possessing some attribute characteristics θ_{i}} \end{matrix}$

Let $\begin{matrix} π (E_{1}) & = & {x_{1}, x_{4}, x_{5}, x_{8}, x_{10}}, \\ π (E_{2}) & = & {x_{2}, x_{5}, x_{6}, x_{8}, x_{10}} and \\ π (E_{3}) & = & {x_{3}, x_{7}, x_{8}, x_{9}, x_{10}} . \end{matrix}$

Now,

$\begin{matrix} \underline{Z_{W}} (E_{1}) & = {x \in U : N_{W} (x) \subseteq E_{1}} = {x_{1}, x_{5}, x_{8}, x_{10}} \\ \underline{Z_{W}} (E_{2}) & = {x \in U : N_{W} (x) \subseteq E_{2}} = {x_{2}, x_{5}, x_{6},, x_{8}, x_{10}} \\ \underline{Z_{W}} (E_{3}) & = {x \in U : N_{W} (x) \subseteq E_{3}} = {x_{7}, x_{8}, x_{10}} . \end{matrix}$

Also, $\begin{matrix} \bar{Z_{W}} (E_{1}) & = {x \in U : N_{W} (x) \cap E_{1} \neq \emptyset} \\ = U = \bar{Z_{W}} (E_{2}) \\ \bar{Z_{W}} (E_{3}) & = {x_{3}, x_{4}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}} \end{matrix}$ Suppose $Ω_{\underline{π (D)}} (x_{j})$ and $Ω_{\bar{π (D)}} (x_{j})$ be two fuzzy sets for optimality measure and possibly measure of optimality, respectively on U, for each object x_j such that $\begin{matrix} Ω_{\underline{π (D)}} (x_{j}) & = \frac{1}{3} \underset{t = 1}{\sum^{3}} χ \underline{π (E_{t})} (x_{j}) and \\ Ω_{\bar{π (D)}} (x_{j}) & = \frac{1}{3} \underset{t = 1}{\sum^{3}} χ \bar{π (E_{t})} (x_{j}) . \end{matrix}$ Where $χ_{\underline{π (E_{t})}}$ and $χ_{\bar{π (E_{t})}}$ are a kind of indicator functions, defined by $χ_{\underline{π (E_{t})}} (x_{j}) = {\begin{array}{l} 1 if x_{j} \in \underline{Z_{W}} (E_{t}) \\ 0, otherwise \end{array}$ and $χ_{\bar{π (E_{t})}} (x_{j}) = {\begin{array}{l} 1 if x_{j} \in \bar{Z_{W}} (E_{t}) \\ 0, otherwise \end{array} .$ Clearly $Ω_{\underline{π (D)}} (x_{j})$ and $Ω_{\bar{π (D)}} (x_{j})$ represents the optimality and possible optimality of each object x_j, according to each expert. Now consider the betterment of applicant x_p with vertex x_q and vice versa. The marginal weight function φ for each x_p can be computed by; $φ (x_{j}) = \frac{1}{| U |} [φ_{r} (x_{j}) + φ_{c} (x_{j})]$ for j = 1, 2, 3, . . . , 10, where $φ_{r} (x_{p}) = \underset{p = 1}{\sum^{10}} χ_{E} (x_{p} x_{q}),$ is the measures the betterment of x_pthan x_q, and $φ_{c} (x_{p}) = \underset{q = 1}{\sum^{10}} χ_{E} (x_{q} x_{p}),$ is the measures the betterment of x_q than x_p . Where χ_E is an indicator function on U × U, defined by $χ_{E} (x_{p} x_{q}) = {\begin{matrix} 1 if x_{p} x_{q} \in U \times U \\ 0, otherwise \end{matrix} .$ The values of the marginal weight function φ can seen in Table 8

Table 8
Table for values of the marginal weight function φ

U x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈ x ₉ x ₁₀ Total

x ₁ 0 1 0 1 1 0 0 0 1 0 4/10

x ₂ 0 0 0 0 0 1 0 0 1 1 3/10

x ₃ 1 1 0 0 1 0 1 1 0 0 5/10

x ₄ 0 0 0 0 1 0 0 0 0 0 1/10

x ₅ 0 1 0 1 0 1 0 1 1 0 5/10

x ₆ 0 0 0 0 0 0 1 0 0 0 1/10

x ₇ 0 0 0 1 0 0 0 0 0 1 2/10

x ₈ 0 0 0 0 1 0 0 0 1 0 2/10

x ₉ 0 1 1 0 0 0 0 0 0 0 2/10

x ₁₀ 0 1 0 0 0 0 0 1 0 0 2/10

Total 1/10 5/10 1/10 3/10 4/10 2/10 2/10 3/10 4/10 2/10

Finally an evaluation function ψ is defined on U by $ψ (x_{p}) = \frac{1}{2} [Ω_{\underline{π (D)}} (x_{p}) + Ω_{\bar{π (D)}} (x_{p})] ϕ (x_{p})$ Here $Ω_{\underline{π (D)}} (x_{j}) = {\begin{matrix} \frac{1}{3} for j = 1, 2, 6, 7 \\ 1 for j = 8, 10 \\ \frac{2}{3} for j = 5 \\ 0, for other values of j \end{matrix}$

and $Ω_{\bar{π (D)}} (x_{j}) = {\begin{matrix} \frac{2}{3} for j = 1, 2, 5 \\ 1, for other values of \end{matrix}$ The marginal weight function φ for each x_j can be computed by; $φ (x_{j}) = \frac{1}{| U |} [φ_{r} (x_{j}) + φ_{c} (x_{j})]$

That is

$\begin{matrix} φ (x_{1}) = \frac{5}{10}, φ (x_{2}) = \frac{8}{10}, φ (x_{3}) = \frac{6}{10}, \\ φ (x_{4}) = \frac{4}{10}, φ (x_{5}) = \frac{9}{10}, φ (x_{6}) = \frac{3}{10}, \\ φ (x_{7}) = \frac{4}{10}, φ (x_{8}) = \frac{5}{10}, φ (x_{9}) = \frac{6}{10}, φ (x_{10}) = \frac{4}{10} \end{matrix}$

Finally an evaluation function ψ is defined on U by $ψ (x_{j}) = \frac{1}{2} [Ω_{\underline{π (D)}} (x_{j}) + Ω_{\bar{π (D)}} (x_{j})] φ (x_{j})$ That is

$\begin{matrix} ψ (x_{1}) = \frac{1}{4}, ψ (x_{2}) = \frac{2}{5}, ψ (x_{3}) = \frac{3}{10}, \\ ψ (x_{4}) = \frac{1}{5}, ψ (x_{5}) = \frac{3}{5}, ψ (x_{6}) = \frac{2}{5}, \\ ψ (x_{7}) = \frac{4}{15}, ψ (x_{8}) = \frac{1}{2}, ψ (x_{9}) = \frac{3}{10}, ψ (x_{10}) = \frac{2}{5} \end{matrix}$

Hence from the above calculations, the best optimal applicant is x₅ . The pseudocode of the above algorithm is presented below;

Pseudo code

(i) Consider a soft covering approximation space $W = (U, C_{U})$ and an evaluation soft set $(π, D)$ .

(ii) Find lower and upper Z-soft covering approximations of each π (E_t) .

(iii) Compute the fuzzy functions and $Ω_{\underline{π (D)}} (x_{p})$ and $Ω_{\bar{π (D)}} (x_{p})$ given by $\begin{matrix} Ω_{\underline{π (D)}} (x_{p}) & = \frac{1}{m} \underset{t = 1}{\sum^{m}} χ \underline{π (E_{t})} (x_{p}) and \\ Ω_{\bar{π (D)}} (x_{p}) & = \frac{1}{m} \underset{t = 1}{\sum^{m}} χ \bar{π (E_{t})} (x_{p}) . \end{matrix}$ (iv) Calculate the weights φ (x_p) for each x_p, given by $φ (x_{p}) = \frac{1}{| U |} [φ_{r} (x_{j}) + φ_{c} (x_{j})]$ (v) Finally calculate the evaluation function ψ (x_p) given by $ψ (x_{p}) = \frac{1}{2} [Ω_{\underline{π (D)}} (x_{p}) + Ω_{\bar{π (D)}} (x_{p})] φ (x_{p})$ The person x_k is most optimal if $ψ (x_{k}) = max_{p} {ψ (x_{p})} .$
7 Conclusion

U	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇	x ₈	x ₉	x ₁₀	Total
x ₁	0	1	0	1	1	0	0	0	1	0	4/10
x ₂	0	0	0	0	0	1	0	0	1	1	3/10
x ₃	1	1	0	0	1	0	1	1	0	0	5/10
x ₄	0	0	0	0	1	0	0	0	0	0	1/10
x ₅	0	1	0	1	0	1	0	1	1	0	5/10
x ₆	0	0	0	0	0	0	1	0	0	0	1/10
x ₇	0	0	0	1	0	0	0	0	0	1	2/10
x ₈	0	0	0	0	1	0	0	0	1	0	2/10
x ₉	0	1	1	0	0	0	0	0	0	0	2/10
x ₁₀	0	1	0	0	0	0	0	1	0	0	2/10
Total	1/10	5/10	1/10	3/10	4/10	2/10	2/10	3/10	4/10	2/10

In our present paper, a new kind of partial order is proposed for Z-soft rough covering sets. Some uncertain measures like roughness and accuracy measures for Z-soft rough covering sets are proposed and their basic characteristics are explored. We have studied the uncertainty measurement issues in Z-soft rough covering sets based on knowledge granulation. An application is developed for the applicability of the proposed theory. We hope our results will prove a strong base for understanding the essence of uncertainty measurement and in measuring the quality of a decision rule.

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Uncertainty measure of Z -soft covering rough models based on a knowledge granulation

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Roughness measure and accuracy measure for Z-Soft rough covering sets

Table 1 Tabular representation of soft set x 1 x 2 x 3 x 4 x 5 x 6 x 7 θ 1 1 0 0 1 0 0 1 θ 2 0 1 1 0 1 1 0 θ 3 0 0 1 1 0 1 1 θ 4 0 1 1 0 1 0 0

Table 4 Tabular representation of soft set (β, M) (β, M) x 1 x 2 x 3 x 4 x 5 x 6 θ 1 1 1 0 0 1 1 θ 2 0 1 1 1 0 1 θ 3 1 0 0 1 1 1 θ 4 0 0 1 1 0 1

Table 5 Tabular representation of soft set (β1, M) (β1, M) x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 θ 1 1 0 1 1 1 0 0 1 θ 2 0 1 0 1 0 1 1 1 θ 3 1 1 1 0 1 0 0 1 θ 4 0 0 0 1 1 0 1 0 θ 5 0 0 1 1 0 1 1 0

References

Table 1
Tabular representation of soft set

x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇

θ ₁ 1 0 0 1 0 0 1

θ ₂ 0 1 1 0 1 1 0

θ ₃ 0 0 1 1 0 1 1

θ ₄ 0 1 1 0 1 0 0

Table 4
Tabular representation of soft set (β, M)

(β, M) x ₁ x ₂ x ₃ x ₄ x ₅ x ₆

θ ₁ 1 1 0 0 1 1

θ ₂ 0 1 1 1 0 1

θ ₃ 1 0 0 1 1 1

θ ₄ 0 0 1 1 0 1

Table 5
Tabular representation of soft set (β₁, M)

(β₁, M) x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇ x ₈

θ ₁ 1 0 1 1 1 0 0 1

θ ₂ 0 1 0 1 0 1 1 1

θ ₃ 1 1 1 0 1 0 0 1

θ ₄ 0 0 0 1 1 0 1 0

θ ₅ 0 0 1 1 0 1 1 0