Soft ordered based multi-granulation rough sets and incomplete information system

Abstract

This study proposes the multi attribute group decision making in the presence of incomplete multi attribute and incomplete multi decision while making a decision with preferences in an incomplete information system. We then consider resolving the problem in an incomplete information system by using two different approximation strategies, that is seeking the common reserving difference and common rejecting difference, four kinds of soft dominance based multi-granulation rough sets namely soft dominance based optimestic multi-granulation rough sets and soft dominance based pessimistic multi-granulation rough sets are presented. Another worth mentioning contibution of this paper is to disclose the ideas of two kinds of approximate precision, rough degree, approximate quality, maximal and minimal rough member ships and their mutual relationships. Finally the validity of these concepts are proved by constructing two algorithms and applying them in solving incomplete multi-agent conflict analysis problem.

Keywords

Rough set soft set preference relation multi-granulation rough sets

1 Introduction

The rough set theory, introduced by Pawlak [46] has been conceived as a useful tool to conceptualize and analyze various types of data. It has diverse applications in various fields such as artificial intelligence, cognitive sciences, medical, engineering, management sciences and economics etc. In rough set theory, it is assumed that a collection of objects represented by values of many attributes/parameters is given. In this theory, all attributes are implicitly assumed to be nominal. However, in the real world applications, one may encounter cases that some attributes are ordinal. For example, weights, evaluations of quality, test scores, etc. can be considered as ordinal attributes, that is, for those attributes, we may have inequality or preference relations on their attribute values. However, in many real world data sets, it may happen that some attribute values of an object are missing. For example, some measurement data of a patient in the clinical data set are missing, since the expense or difficulty of obtaining certain results. An information system with missing values is called an incomplete information system [30, 32]. A simple way of handling such systems is to transform the incomplete information system into a complete one, of which missing values can be deleted or filled with some values.

1.1 Importance of the study

Decision making is one of the key component to achieve objectives in many areas, particularly in a field which obligates analyzing the conflict. Conflict analysis is one of the fields whose importance is increasing nowadays as huge social networks based on computers, cell phones, tablets and other gadgets systems of computers are starting to play a significant role in the societies [43]. Conflict analysis plays a paramount role in business, governmental, politics, legal disputes, labor-management negotiation, military operations economic and games. In short, such analysis is always needed whenever people have difference of opinions. In a conflict situation, there is always an uncertainty about agreement, neutrality and disagreement among agents. In such situations, the main problem is that how to find a way to model uncertainty.

1.1.1 Rough set theory and conflict analysis

Applications of rough set theory in case of conflict analysis were introduced by Pawlak in [47] where he discussed a mathematical formulation of conflict situations based on three binary relations, that is, agreement, disagreement and neutrality, and given the axioms for agreement and disagreement relations. He also introduced a conflict graph model by representing the conflict situation with discernibility. Regarding conflict problem using rough sets, the model introduced by Deja [11] is an extension of the model proposed by Pawlak by adding to the model some local aspects of conflicts. Subsequently Deja put forward three basic questions which are related to conflict analysis model: “What are the intrinsic reasons for the conflict?”, “How can a feasible consensus strategy be found?” and “Is it possible to satisfy all the agents?”

The concept of dominance based rough sets initiated by Greco et al. [18] and their applications in multi criteria decision analysis [19], methodology for sorting problem of multi attributes and criteria [20], robust ordinal regression [22]. Many extended models of the dominance based rough set approach have been proposed by various authors including monotonic variable consistency rough set [7], graded dominance interval-based fuzzy objective information systems [23], dominance based rough set model in intuitionistic fuzzy information system [24], multi criteria decision making method based on dominance relation and variable precision rough set [25], variable precision dominance based rough set approach and attribute reduction [26], stochastic dominance-based rough set model for ordinal classification [29] and α-Dominance relation and rough sets in interval-valued information systems [60]. The hybridization of dominance based rough set approach and other mathematical tools have been created and applied to multi criteria decision analysis focussing a new fuzzy multicriteria decision making approach: an application for European quality award assessment [5], incremental update of approximations in dominance-based rough sets approach under the variation of attribute values [33], qualitative and quantitative combinations of crisp and rough clustering schemes using dominance relations [37].

Molodtsov’s soft set theory [41] was proposed as a general mathematical tool to deal with uncertainty. The rational behind soft sets is founded on the idea of parameterization, which suggests that complicated objects should be perceived from various points of view. Without the curb caused by inadequacy of parametrization tools, this theory comes with an ability to represent and manipulate data in a convenient and meaningful way [3]. Maji et al. [39] introduced several algebraic operations in soft set theory and examined their basic properties. Ali et al. [4] proposed some new operations in soft set theory to further consolidate the algebraic basis of soft sets.

1.2 Motivation and Novelty

In recent years, some hybrid uncertain models occur such as fuzzy rough sets and rough fuzzy sets [12], rough soft sets, soft rough sets [16], soft rough fuzzy sets and soft fuzzy rough sets [40] to handle the uncertainty. Karavidić and Projović, applied rough set theory to the security forces operations models [28]. Pamuĉar et al. introduced group multi-criteria decision making based on interval rough numbers [44]. Pamuĉar et al. discussed the best-worst and MABAC methods based on interval valued fuzzy rough numbers [45]. It is noted that all these hybrid systems have their own benefits and drawbacks. Qian et al. [51], extended Pawlak’s single granulation rough set model to a multi-granulation rough set model, where the approximations are defined by using multi equivalence relations on the single universe. Kumar and Inbarani applied multi-granulation rough set approach in medical diagnoses [31]. Lin et al. put forward the idea of neighborhood based multi-granulation rough sets and its applications [34]. Subsequently, Lin et al. in [35] combined multi-granulation rough sets with evidence theory while in [36] Lin et al. discussed fuzzy multigranulation decision theoretic approach for multi source fuzzy information system. Yang et al. in [61] introduced the concept of multi-granulation rough set in incomplete information system. Ali et al. [1] defined a new hybrid structure called soft preference relation and their application in conflict analysis problems.

1.2.1 Research questions, targets and our contributions

The foremost research questions are: whether the idea of soft preference relation can be applied to the incomplete information system? Whether Deja’s questions be answered using soft preference relation in the case of incomplete information system? The main target of this paper is to put forward the novel idea of multigranulation rough sets based on hybrid structure (soft preference relation) and their applications in incomplete information system. Another target of the present paper is to answer the Deja’s questions using the proposed model. Further contributions of the current paper are: (i) to present the idea of soft preference relation and soft dominance relation in an incomplete information system to solve a multi-agent conflict analysis problem and try to answer Deja’s first question in a best manner which is related to conflict analysis; (ii) another worth mentioning contribution of the present paper is to define two types of optimistic approximations with the help of soft dominance (soft dominating/soft dominated) classes and apply these to discuss various properties of the approximations in an incomplete information system. The results on labor management negotiation problem show that the proposed algorithm based on optimistic approximations is much effective and efficient for feasible consensus strategy; (iii) one of the main contributions of this paper is to put forward the idea of two new kinds of pessimistic approximations with the help of soft dominance (soft dominating/soft dominated) classes and applied these to discuss related properties of the approximations in an incomplete information system; (iv) another worth mentioning contribution of this paper is to disclose the ideas of two kinds of approximate precision, rough degree, approximate quality, maximal and minimal rough member ships and their mutual relationship.

1.3 The organization of this paper

The arrangement of this article is as follows: Section 2 focuses mainly on the problem statement. Section 3 highlights the literature review needed for the subsequent article. In Section 4, we present the idea of soft preference and soft dominance relations in an incomplete information system to solve a multi agent conflict analysis labor-management negotiations problem. In Section 5, we focus our attention on the development of feasible consensus strategy for labor-management negotiation conflict analysis problem based on optimistic approximations in an incomplete information system. Further in this section, we present another idea for the development of feasible consensus strategy in labor-management negotiations conflict analysis problem based on soft dominance multi-granulation rough sets called pessimistic rough sets. In addition, several uncertainty measures, such as approximate precision, rough degree, approximate quality, maximal and minimal rough member ships and their mutual relationships are discussed in an incomplete information system.

2 Problem statement

The incomplete multiple decision problems with preference relations have been studied in this paper. In general these may be the incomplete multi criteria group decision problems or incomplete multi criteria and incomplete multi decision with preference choice problem. The later is comprises of two parts, the first is the incomplete multi criteria with predefined evaluations for every action and the other is the incomplete multi decision with a predefined preference for every action. An incomplete decision problem may be considered as an S^∗ : = (A, C, D, E), where A is a finite set of actions a_i, i : =1, 2, 3, ... , |A|, C is a finite set of conditional attributes C_j, j : =1, 2, ... , |C|, D is a finite set of decision attributes D_k, k : =1, 2, ... , |D| and E is a finite set of the domain for the information functions f (a_i, C_j) and g (a_i, D_k). It may happen that some attribute values of an action are missing. To indicate such a situation, a distinguished value is often assigned. We denote special symbol “*” to indicate the missing attribute value. For an information system, if there exist a_i ∈ A, C_j ∈ C and D_k ∈ D such that f (a_i, C_j) =∗ or g (a_i, D_k) =∗.

In order to show the incomplete decision problem clearly, an example of a conflict situation for labor-managment negotiations is presented in Table 1. There are five issues (conditionalattributes) and four agents (decisionattributes) with twelve feasible actions A : ={ a_i : i : =1, 2, ... , 12 }. The issues may be C₁ : = employees incomes, C₂ : = working conditions, C₃ : = factory profits, C₄ : = housing facility and C₅ : = children education, while D₁, D₂, D₃ and D₄ are decision makers to handle the conflict situation for labor-managment negotiations. The association of the integers are defined as follows: 0-small (orbad) , 1-medium (oraverage), 2-high (orgood), 3-highest (orexcellent).

Table 1
Incomplete multi criteria and multi decision information table

A C ₁ C ₂ C ₃ C ₄ C ₅ D ₁ D ₂ D ₃ D ₄

a ₁ 1 0 1 ∗ 2 2 ∗ 1 ∗

a ₂ 1 ∗ 1 1 2 1 3 0 1

a ₃ 1 0 2 0 2 3 1 0 3

a ₄ 1 1 0 0 2 1 1 ∗ 2

a ₅ ∗ 1 1 0 2 0 2 0 1

a ₆ 1 1 2 1 2 1 0 2 0

a ₇ 1 1 2 2 2 3 2 3 2

a ₈ 1 2 ∗ 0 2 0 ∗ 0 2

a ₉ 1 2 1 1 2 2 2 1 ∗

a ₁₀ 1 2 0 0 ∗ ∗ 3 0 3

a ₁₁ 1 2 1 0 1 0 1 2 0

a ₁₂ 1 2 1 1 1 2 1 1 1

A	C ₁	C ₂	C ₃	C ₄	C ₅	D ₁	D ₂	D ₃	D ₄
a ₁	1	0	1	∗	2	2	∗	1	∗
a ₂	1	∗	1	1	2	1	3	0	1
a ₃	1	0	2	0	2	3	1	0	3
a ₄	1	1	0	0	2	1	1	∗	2
a ₅	∗	1	1	0	2	0	2	0	1
a ₆	1	1	2	1	2	1	0	2	0
a ₇	1	1	2	2	2	3	2	3	2
a ₈	1	2	∗	0	2	0	∗	0	2
a ₉	1	2	1	1	2	2	2	1	∗
a ₁₀	1	2	0	0	∗	∗	3	0	3
a ₁₁	1	2	1	0	1	0	1	2	0
a ₁₂	1	2	1	1	1	2	1	1	1

Table 1 describes the incomplete multi attribute and incomplete multi decision makers with preference for making a decision in the case of labor-management negotiations conflict problem.

3 Proposed conflict analysis model

A simple conflict occurs when two persons have different points of view about a thing or event. Analysis of conflict described in [47] is restricted to outermost conclusions, such as finding the most conflicting attributes or the coalitions of agents if more than two take part in the conflict. Because in the Pawlak model (see [47]) the reason of the conflict cannot be determined, there is no way to specify the situation to avoid the conflict. Moreover, we cannot be sure that the issues the agents vote represent the issues each agent takes care of. Though the Pawlak conflict analysis model has proven to be an effective method in practice, yet Deja in [10], put forward three basic (below given) questions which are not answered by the Pawlak’s conflict analysis model:

(i) What are the intrinsic reasons for the conflict?

(ii) How can a feasible consensus strategy be found?

(iii) Is it possible to satisfy all the agents?

Let $P_{C_{j}}$ be an outranking relation on a universe A with reference to criteria C_j ∈ C such that a_iC_ja_j which means “a_i is at least as good as a_j with respect to criteria C_j.” Suppose that $P_{C_{j}}$ is a complete preorder. We employed dominance relation for the study of the problem as follows: denote a_i ⪰ a_j by f (a_i, C_j) ≥ f (a_j, C_j) according to increasing preference, where C_j ∈ C and a_i, a_j ∈ A. For any subset of the conditional attributes $C \subseteq C$ , $a_{i} ⪰_{C} a_{j}$ means that a_i ⪰ _{C
_j}a_j for all $C_{j} \in C$ ; that is, a_i dominates a_j with respect to all attributes in $C$ . The intersection of complete preorders is a partial preorder and $P_{C} = ⋂_{C_{j} \in C} P_{C_{j}},$ the dominance relation $P_{C}$ is a partial preorder.

We present the idea of soft preference and soft dominance relation in an incomplete information system to solve a multi-agent conflict analysis problem.

According to Deja [10], the conflict analysis decision task is proposed into three problems (as discussed in Introduction). In the present paper, we initiate the notion of soft preference and soft dominance relation and applied these to solve the problems/questions posed by Deja.

Definition 1. A pair $R = (F, C)$ is called a soft set over a set A, where C is a set of parameters and $F : C \to P (A)$ is a set valued mapping, where $P (A)$ is the set of all subsets of A.

Definition 2. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with multi decisions information system and $F_{k}^{*} : C \to P (A \times A)$ be a set valued mapping from C to $P (A \times A)$ , where C denotes the set of parameters or attributes and $P (A \times A)$ the set of all subsets of A × A. If $F_{k}^{*} (C_{j})$ is a preference relation for all C_j ∈ C then $(F_{k}^{*}, C)$ is called a soft preference relation corresponding to the decision attributes D_k, k = 1, 2, ... , |D|, where $F_{k}^{*} (C_{j}) = {\begin{matrix} a_{m} ⪰ a_{n} ∣ f (a_{m}, C_{j}) \geq f (a_{n}, C_{j}) or f (a_{m}, C_{j}) = * or f (a_{n}, C_{j}) = * \\ and g (a_{m}, D_{k}) \geq g (a_{n}, D_{k}) or g (a_{m}, D_{k}) = * or g (a_{n}, D_{k}) = * \end{matrix}} .$

Denote a_m ⪰ a_n by (a_m, a_n). In an incomplete multi attributes with incomplete multi decisions information system $S^{*} = (A, C, D, E)$ , we answer to the first question of Deja’s which is related to conflict analysis model, that is,“What are the intrinsic reasons for the conflict ?” This means what issues are focussed by every agent or feasible action and have different attitudes in a conflict corresponding to a specific decision maker. For any issue C_j ∈ C, we obtain attitude information for every agent with respect to issue C_j ∈ C by using $F_{k}^{*} (C_{j})$ which is a preference relation corresponding to decision maker D_k, where j = 1, 2, ... , |C| and k = 1, 2, ... , |D|. Thus we have answered the first question of Deja’s which is related to conflict analysis model.

Definition 3. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and $(F_{k}^{*}, C)$ be a soft preference relation corresponding to the decision attributes D_k, k = 1, 2, ... , |D| over A. Then there is dominance relation associated with which can be denoted by $D om (F_{k}^{*}, C)$ and is defined by: $D om (F_{k}^{*}, C) = ⋂_{C_{j} \in C} F_{k}^{*} (C_{j})$ where j = 1, 2, ... , |C| and k = 1, 2, ... , |D|.

It is worth mentioning that in an incomplete information system $S^{*}$ , the $D om (F_{k}^{*}, C)$ depicts that which agent(s) is/are preferred by decision maker (D_k) on all criteria by other agents. But still we do not have sufficient information about the agent(s) on which all decision maker(s) are agree. So by Deja’s second question which is related to conflict analysis, “How can a feasible consensus strategy be found”? which means that to find feasible alternative(s) on which all the optimal decision makers agreed or at least sub optimal decision makers. If all the decision makers do not agree on any feasible alternative(s), then in order to develop feasible consensus strategy some authors remove certain decision maker(s) from the information system (e.g. see [57]).

Definition 4. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and $D om (F_{k}^{*}, C)$ be a soft dominance relation. For a ∈ A, define the soft dominance classes by ${[a]}_{D om (F_{k}^{*}, C)}^{+} = {a_{i} \in A : a_{i} ⪰_{D om (F_{k}^{*}, C)} a}$ and ${[a]}_{D om (F_{k}^{*}, C)}^{-} = {a_{i} \in A : a_{i} ⪯_{D om (F_{k}, C)} a}$ which represent $(F_{k}^{*}, C)$ -dominating set and $(F_{k}^{*}, C)$ -dominated set with respect to a, over the set of conditional attributes C corresponding to the decision attributes D_k, k = 1, 2, ... , |D| of $S^{*} = (A, C, D, E)$ respectively. ${[a]}_{D om (F_{k}^{*}, C)}^{+}$ describes the set of objects that dominate a and ${[a]}_{D om (F_{k}^{*}, C)}^{-}$ describes the set of objects that dominated by a in terms of $D om (F_{k}^{*}, C)$ .

We now discuss some properties and applications of $(F_{k}^{*}, C)$ -dominating and $(F_{k}^{*}, C)$ -dominated sets, where $D om {(F_{k}^{*}, C)}^{+} = {a \in A : {[a]}_{D om (F_{k}^{*}, C)}^{+}}$ and $D om {(F_{k}^{*}, C)}^{-} = {a \in A : {[a]}_{D om (F_{k}^{*}, C)}^{-}} .$

Theorem 1. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and $D om (F_{k}^{*}, C)$ be a soft dominance relation corresponding to D_k, k : =1, 2, ... , |D| over A. Then the following hold:

(1) If C₁ ⊆ C, then $D om (F_{k}^{*}, C) \subseteq D om (F_{k}^{*}, C_{1});$

(2) If $a_{n} \in {[a_{m}]}_{D om (F_{k}^{*}, C)}^{+},$ then ${[a_{n}]}_{D om (F_{k}^{*}, C)}^{+} \subseteq {[a_{m}]}_{D om (F_{k}^{*}, C)}^{+};$

(3) ${[a_{m}]}_{D om (F_{k}^{*}, C)}^{+} : = \cup {{[a_{n}]}_{D om (F_{k}^{*}, C)}^{+} : a_{n} \in {[a_{m}]}_{D om (F_{k}^{*}, C)}^{+}};$

(4) ${[a_{m}]}_{D om (F_{k}^{*}, C)}^{+} : = {[a_{n}]}_{D om (F_{k}^{*}, C)}^{+}$ iff f (a_m, C_j) = f (a_n, C_j) or f (a_m, C_j) =∗ or f (a_n, C_j) =∗ and g (a_m, D_k) = g (a_n, D_k) or g (a_m, D_k) =∗ or g (a_n, D_k) =∗ for all C_j ∈ C and D_k ∈ D.

Proof. The proof is straightforward.□

Definition 5. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then for any X ⊆ A the lower and upper approximations with respect to $D om {(F_{k}^{*}, C)}^{+}$ and $D om {(F_{k}^{*}, C)}^{-}$ are: ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}} : = {a \in A : {[a]}_{D om (F_{k}^{*}, C)}^{+} \subseteq X}$ ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}} : = {a \in A : {[a]}_{D om (F_{k}^{*}, C)}^{+} \cap X : \neq \emptyset} .$

The lower approximation ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}}$ , is composed of all objects a from A such that all objects a_i, having at least the same evaluations on all of the considered criteria from C, also belong to X. Thus one can say that if an object a_i has at least as good in evaluation on the criteria from C as the object a belonging to ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}},$ then certainly, a_i belongs to X. The upper approximation ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}}$ is made up of all objects a from A whose evaluation on the criteria from C are not worse than the evaluation of at least one other object a_i belonging to X. If ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}} : = {(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}}$ then it is called soft dominating definable, otherwise soft dominating rough set, where ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{-}} : = {a \in A : {[a]}_{D om (F_{k}^{*}, C)}^{-} \subseteq X}$ ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{-}} : = {a \in A : {[a]}_{D om (F_{k}^{*}, C)}^{-} \cap X : \neq \emptyset} .$ If ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{-}} : = {(\underline{X})}_{D om {(F_{k}^{*}, C)}^{-}}$ then it is called soft dominated definable, otherwise soft dominated rough set.

Theorem 2. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then for any X, Y ⊆ A, the following assertions hold:

(1) ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}} \subseteq X \subseteq {(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}} .$

(2) ${(\underline{\emptyset})}_{D om {(F_{k}^{*}, C)}^{+}} : = {(\bar{\emptyset})}_{D om {(F_{k}^{*}, C)}^{+}} : = \emptyset .$

(3) ${(\underline{A})}_{D om {(F_{k}^{*}, C)}^{+}} :$ = ${(\bar{A})}_{D om {(F_{k}^{*}, C)}^{+}} : = A .$

(4) X ⊆ Y implies ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}} \subseteq {(\underline{Y})}_{D om {(F_{k}^{*}, C)}^{+}} .$

(5) X ⊆ Y implies ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}} \subseteq {(\bar{Y})}_{D om {(F_{k}^{*}, C)}^{+}} .$

(6) ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}} \cup {(\underline{Y})}_{D om {(F_{k}^{*}, C)}^{+}} \subseteq {(\underline{X \cup Y})}_{D om {(F_{k}^{*}, C)}^{+}} .$

(7) ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}} \cup {(\bar{Y})}_{D om {(F_{k}^{*}, C)}^{+}} = {(\bar{X \cup Y})}_{D om {(F_{k}^{*}, C)}^{+}} .$

(8) ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{+}} \cap {(\underline{Y})}_{D om {(F_{k}^{*}, C)}^{+}} = {(\underline{X \cap Y})}_{D om {(F_{k}^{*}, C)}^{+}} .$

(9) ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{+}} \cap {(\bar{Y})}_{D om {(F_{k}^{*}, C)}^{+}} \supseteq {(\bar{X \cap Y})}_{D om {(F_{k}^{*}, C)}^{+}} .$

Proof. The proof follows from the respective definitions.□

Theorem 3. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then for any X, Y ⊆ A, the following assertions hold:

(1) ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{-}} \subseteq X \subseteq {(\bar{X})}_{D om {(F_{k}^{*}, C)}^{-}} .$

(2) ${(\underline{\emptyset})}_{D om {(F_{k}^{*}, C)}^{-}} = {(\bar{\emptyset})}_{D om {(F_{k}^{*}, C)}^{-}} = \emptyset .$

(3) ${(\underline{A})}_{D om {(F_{k}^{*}, C)}^{-}}$ = ${(\bar{A})}_{D om {(F_{k}^{*}, C)}^{-}} = A .$

(4) X ⊆ Y implies ${(\underline{X})}_{D {om}^{k} {(H, C)}^{-}} \subseteq {(\underline{Y})}_{D om {(F_{k}^{*}, C)}^{-}} .$

(5) X ⊆ Y implies ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{-}} \subseteq {(\bar{Y})}_{D om {(F_{k}^{*}, C)}^{-}} .$

(6) ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{-}} \cup {(\underline{Y})}_{D om {(F_{k}^{*}, C)}^{-}} \subseteq {(\underline{X \cup Y})}_{D om {(F_{k}^{*}, C)}^{-}} .$

(7) ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{-}} \cup {(\bar{Y})}_{D om {(F_{k}^{*}, C)}^{-}} : = {(\bar{X \cup Y})}_{D om {(F_{k}^{*}, C)}^{-}} .$

(8) ${(\underline{X})}_{D om {(F_{k}^{*}, C)}^{-}} \cap {(\underline{Y})}_{D om {(F_{k}^{*}, C)}^{-}} : = {(\underline{X \cap Y})}_{D om {(F_{k}^{*}, C)}^{-}} .$

(9) ${(\bar{X})}_{D om {(F_{k}^{*}, C)}^{-}} \cap {(\bar{Y})}_{D om {(F_{k}^{*}, C)}^{-}} \supseteq {(\bar{X \cap Y})}_{D om {(F_{k}^{*}, C)}^{-}} .$

Proof. The proof follows from the respective definitions.□

4 Multi-granulation rough sets based on soft dominance relation

According to two different approximations, Qian et al. [51, 52] developed two different multi-granulation rough sets including optimistic and pessimistic ones. In this section we present two kinds of soft dominance based multi-granulation rough sets, namely, soft dominance (softdominating/softdominated) based optimistic multi-granulation rough sets and soft dominance (soft dominating/soft dominated) based pessimistic multi-granulation rough sets.

Definition 6. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system, D₁, D₂, D₃, ... , D_k ∈ D and X ⊆ A. The soft dominance optimistic multi-granulation lower and upper approximations of X with respect to D₁, D₂, D₃, ... , D_k are defined by

$\begin{matrix} {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} & = {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{+} \\ \subseteq X \lor . . . \lor {[a]}_{D om (F_{| D |}^{*}, C)}^{+} \subseteq X}, \end{matrix}$ $\begin{matrix} {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{+} \cap X \neq \emptyset \\ \land . . . \land {[a]}_{D om (F_{| D |}^{*}, C)}^{+} \cap X \neq \emptyset} . \end{matrix}$

Then $({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}}, {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}})$ is called the soft dominating optimistic multi-granulation rough set if ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \neq {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}}$ . The optimistic multi-granulation boundary region of X is

${BN}^{o} (X) = {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} - {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} .$

Similarly we define

$\begin{matrix} {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{-} \subseteq \\ X \lor . . . \lor {[a]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X}, \end{matrix}$ $\begin{matrix} {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{-} \cap X \neq \emptyset \\ \land . . . \land {[a]}_{D om (F_{| D |}^{*}, C)}^{-} \cap X \neq \emptyset} . \end{matrix}$

Then $({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}, {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})$ is called the soft dominated optimistic multi-granulation rough set if ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \neq {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$ .

Theorem 4. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system with D₁, D₂, D₃, ... , D_k ∈ D. Then for all X₁, X₂, ... , X_n ⊆ A, the soft dominating optimistic multi-granulation rough sets have the following properties:

${\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \subseteq X_{j} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

${\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} = \emptyset = {\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

${\underline{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} = A = {\bar{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

X₁ ⊆ X₂ implies ${\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

X₁ ⊆ X₂ implies ${\bar{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

${\underline{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \subseteq ⋂_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}})$

${\bar{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \subseteq ⋂_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}});$

${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \supseteq ⋃_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}});$

${\bar{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \supseteq ⋃_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}});$

${\underline{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} = {({\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}})}^{c};$

${\bar{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}}$ = ${({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}})}^{c} .$

Proof. Straightforward.□

Theorem 5. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system with D₁, D₂, D₃, ... , D_k ∈ D. Then for all X₁, X₂, ... , X_n ⊆ A, the soft dominated optimistic multi-granulation rough sets have the following properties:

(1) ${\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq X_{j} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}};$

(2) ${\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = \emptyset = {\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}};$

(3) X₁ ⊆ X₂ implies ${\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}};$

(4) X₁ ⊆ X₂ implies ${\bar{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}};$

(5) ${\underline{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq ⋂_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}});$

(6) ${\bar{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq ⋂_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}});$

(7) ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \supseteq ⋃_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}});$

(8) ${\bar{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \supseteq ⋃_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}});$

(9) ${\underline{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = {({\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})}^{c};$

(10) ${\bar{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = {({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})}^{c};$

(11) ${\underline{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = A = {\bar{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$

Proof. (1) For all $x \in {\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}},$ we have ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X$ or ... or ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X .$ In addition, the soft dominated relation satisfies reflexivity, for as much $x \in {[x]}_{D om (F_{1}^{*}, C)}^{-}$ and $x \in {[x]}_{D om (F_{1}^{*}, C)}^{-}, . . .$ and $x \in {[x]}_{D om (F_{| D |}^{*}, C)}^{-} .$ It can be acquired that x ∈ X. Hence ${\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq X .$

For all x ∈ X, apparently, $x \in {[x]}_{D om (F_{1}^{*}, C)}^{-}$ and $x \in {[x]}_{D om (F_{2}^{*}, C)}^{-}$ ... and $x \in {[x]}_{D om (F_{| D |}^{*}, C)}^{-} .$ So ${[x]}_{D om (F_{1}^{*}, C)}^{-} \cap X \neq \emptyset$ and ... and ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \cap X \neq \emptyset .$ This implies that $x \in {\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ Therefore $X \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$

(2) It can be got from property (1) of this theorem that ${\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq \emptyset,$ moreover, it is clear that $\emptyset \subseteq {\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ Thus ${\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = \emptyset .$

Suppose that ${\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \neq \emptyset .$ Then there must be exists $x \in {\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$ such that ${[x]}_{D om (F_{1}^{*}, C)}^{-} \cap \emptyset \neq \emptyset$ and ... and ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \cap \emptyset \neq \emptyset,$ which is a contradiction. Hence ${\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = \emptyset .$

(3) Let $x \in {\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ Then by definition ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X_{1}$ or ... or ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X_{1} .$ But X₁ ⊆ X₂, so ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X_{2}$ or ... or ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X_{2} .$ This implies that $x \in {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ Therefore ${\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$

(4) The proof of (4) is similar to the proof of (3).

(5) Let $x \in {\underline{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ Then by definition ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq ⋂_{j = 1}^{n} X_{j}$ or ... or ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq ⋂_{j = 1}^{n} X_{j} .$ This implies that ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X_{2}$ and ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq ⋂_{j = 1}^{n} X_{2}$ and ... and ${[x]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X_{n}$ or ... or ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X_{2}$ and ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq ⋂_{j = 1}^{n} X_{2}$ and ... and ${[x]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X_{n} .$ Thus $x \in {\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$ and $x \in {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$ and ... $x \in {\underline{\sum_{k = 1}^{| D |} (X_{n})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ Hence $x \in ⋂_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}) .$ Therefore ${\underline{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \subseteq ⋂_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}) .$

(6) The proof of (6) is similar to the proof of (5).

(7) For $X_{1} \subseteq ⋃_{j = 1}^{n} X_{j},$ $X_{2} \subseteq ⋃_{j = 1}^{n} X_{j}, . . ., X_{n} \subseteq ⋃_{j = 1}^{n} X_{j}$ for j = 1, 2, ... , n. On the grounds of the property (3) of this theorem, we can get ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \supseteq {\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}},$ ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \supseteq {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}, . . .,$ ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \supseteq {\underline{\sum_{k = 1}^{| D |} (X_{n})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$ So ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} \supseteq ⋃_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}) .$

(8) The proof of (8) is similar to the proof of (7).

(9) For all $a \in {\underline{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}},$ we have ${[a]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X^{c}$ or ... or ${[a]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X^{c}$ if and only if ${[a]}_{D om (F_{1}^{*}, C)}^{-} \cap X = \emptyset$ or ... or ${[a]}_{D om (F_{| D |}^{*}, C)}^{-} \cap X = \emptyset$ if and only if $a \notin ({\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})$ if and only if $a \in {({\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})}^{c} .$

(10) The proof of (10) is similar to the proof of (9).

(11) ${\underline{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} = {({\bar{\sum_{k = 1}^{| D |} (A^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})}^{c} = {({\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})}^{c} = {(\emptyset)}^{c} = A .$

$({\bar{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}) = {({\underline{\sum_{k = 1}^{| D |} (A^{c})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}})}^{c}$ = (∅) ^c = A.□

To develop the feasible consensus strategy among the agents on feasible alternative(s) and to respond the questions (ii) and (iii) of Deja, we develop the following algorithm utilizing the notion of soft dominance multi-granulation approximations.

4.1 Proposed algorithm for conflict analysis model

Input: Information system $S^{*} : = (A, C, D, E);$

Step 1 : Construct $(F_{k}^{*}, C)$ corresponding to D_k for all k : =1, 2, ... |D| // according to Definition 2;

Step 2 : Construct $D om (F_{k}^{*}, C),$ for all k : =1, 2, ... |D| // according to Definition 3;

Step 3 : Construct ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}}$ and ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$ // according to Definition 6, where X_k ={ a ∈ A : g (a, D_k) ≠0 } ;

Step 4 : If ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$ then

$δ_{k} : = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} or δ_{k} : = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}}$

otherwise $δ_{k} = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} - {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}};$

Step 5 : $δ = ⋂_{k : = 1}^{| D |} δ_{k} \neq \emptyset .$

go to output.

Otherwise,

|D| = |D| - 1 ;

go back to step 5.

Output: Feasible alternative(s) for feasible consensus strategy.

Example 1. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For the labor-mangement negotiations conflict situation, with five issues C_j where j = 1, ... , |C| = 5 and four agents D_k, k = 1, 2, ... , |D| = 4. The system comprises of twelve feasible alternatives a_i, i = 1, 2, ... , |A| = 12 pre-established by experts or obtained by other approaches and that every agent has given its preference in advance for every feasible action. The results are presented in Table 1. Construct soft preference relations for the conditional attributes corresponding to decision attributes, $(F_{1}^{*}, C)$ corresponding to D₁, similarly $(F_{2}^{*}, C)$ corresponding to D₂ and so on from the given incomplete information system $S^{*} : = (A, C, D, E) .$ Applying Definition 3 and Definition 4 to construct soft dominance relations. $D om (F_{1}^{*}, C) = {\begin{matrix} (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), (a_{6}, a_{6}), \\ (a_{7}, a_{7}), (a_{8}, a_{8}), (a_{9}, a_{9}), (a_{10}, a_{10}), (a_{11}, a_{11}), (a_{12}, a_{12}), \\ (a_{1}, a_{2}), (a_{2}, a_{4}), (a_{2}, a_{8}), (a_{2}, a_{10}), (a_{3}, a_{1}), (a_{3}, a_{4}), \\ (a_{3}, a_{5}), (a_{3}, a_{6}), (a_{3}, a_{8}), (a_{3}, a_{10}), (a_{3}, a_{11}), (a_{6}, a_{2}), \\ (a_{6}, a_{4}), (a_{7}, a_{1}), (a_{7}, a_{2}), (a_{7}, a_{3}), (a_{7}, a_{4}), (a_{7}, a_{5}), \\ (a_{7}, a_{6}), (a_{8}, a_{5}), (a_{8}, a_{10}), (a_{8}, a_{11}), (a_{9}, a_{1}), (a_{9}, a_{2}), \\ (a_{9}, a_{4}), (a_{9}, a_{5}), (a_{9}, a_{8}), (a_{9}, a_{10}), (a_{9}, a_{11}), (a_{9}, a_{12}), \\ (a_{10}, a_{4}), (a_{10}, a_{8}), (a_{11}, a_{10}), (a_{12}, a_{10}), (a_{12}, a_{11}) \end{matrix}} .$

The soft dominance classes from $D om (F_{1}^{*}, C)$ are given in Table 2.

Table 2
Dominating classes ${[a_{i}]}_{D om (F_{1}^{}, C)}^{+} and Dominated classes {[a_{i}]}_{D om (F_{1}^{}, C)}^{-}$

a _i ${[a_{i}]}_{D om (F_{1}^{}, C)}^{+}$ ${[a_{i}]}_{D om (F_{1}^{}, C)}^{-}$

a ₁ {a₁, a₃, a₇, a₉} {a₁, a₂}

a ₂ {a_1,a₂, a₆, a₇, a₉} {a₂, a₄, a₈, a₁₀, a₁₁}

a ₃ {a₃, a₇} {a₁, a₃, a₄, a₅, a₆, a₈, a₁₀, a₁₁}

a ₄ {a₂, a₃, a₄, a₆, a₇, a₉, a₁₀} {a₄}

a ₅ {a₃, a₅, a₇, a₈, a₉} {a₅}

a ₆ {a₆, a₇} {a₂, a₄, a₆}

a ₇ {a₇} {a₁, a₂, a₃, a₄, a_5,a₆, a₇}

a ₈ {a₃, a₈, a₁₀} {a₅, a₈, a₁₀, a₁₁}

a ₉ {a₉} {a₁, a₂, a₄, a₅, a₈, a₉, a₁₀, a₁₁, a₁₂}

a ₁₀ {a₂, a₃, a₈, a₉, a₁₀, a₁₁, a₁₂} {a₄, a₈, a₁₀}

a ₁₁ {a₂, a₃, a₈, a₉, a₁₁, a₁₂} {a₁₀, a₁₁}

a ₁₂ {a₉, a₁₂} {a₁₀, a₁₁, a₁₂}

Next $D om (F_{2}^{*}, C) = {\begin{matrix} (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), (a_{6}, a_{6}), \\ (a_{7}, a_{7}), (a_{9}, a_{9}), (a_{10}, a_{10}), (a_{11}, a_{11}), (a_{12}, a_{12}), (a_{1}, a_{2}) \\ (a_{2}, a_{1}), (a_{2}, a_{4}), (a_{2}, a_{5}), (a_{2}, a_{8}), (a_{2}, a_{10}), (a_{2}, a_{11}), \\ (a_{2}, a_{12}), (a_{3}, a_{1}), (a_{5}, a_{1}), (a_{5}, a_{4}), (a_{6}, a_{1}), (a_{7}, a_{1}), \\ (a_{7}, a_{3}), (a_{7}, a_{4}), (a_{7}, a_{5}), (a_{7}, a_{6}), (a_{5}, a_{4}), (a_{8}, a_{1}), \\ (a_{8}, a_{3}), (a_{8}, a_{4}), (a_{8}, a_{5}), (a_{8}, a_{10}), (a_{8}, a_{11}), (a_{9}, a_{1}), \\ (a_{9}, a_{4}), (a_{9}, a_{5}), (a_{9}, a_{8}), (a_{9}, a_{11}), (a_{9}, a_{12}), \\ (a_{8}, a_{8}), (a_{10}, a_{4}), (a_{10}, a_{8}), (a_{12}, a_{11}) \end{matrix}} .$

The soft dominance classes from $D om (F_{2}^{*}, C)$ are given in Table 3.

Table 3

Dominating classes ${[a_{i}]}_{D om (F_{2}^{*}, C)}^{+} and Dominated classes {[a_{i}]}_{D om (F_{2}^{*}, C)}^{-}$

a _i	${[a_{i}]}_{D om (F_{2}^{*}, C)}^{+}$	${[a_{i}]}_{D om (F_{2}^{*}, C)}^{-}$
a ₁	{a₁, a₂, a₃, a₅, a₆, a₇, a₈, a₉}	{a₁}
a ₂	{a₁, a₂}	{a₁, a₂, a₄, a₅, a₈, a₁₀, a₁₁, a₁₂}
a ₃	{a₃, a₇, a₈}	{a₁, a₃}
a ₄	{a₂, a₄, a₅, a₇, a₈, a₉, a₁₀}	{a₄}
a ₅	{a₅, a₇, a₈, a₉}	{a₁, a₄, a₅}
a ₆	{a₆, a₇}	{a₁, a₆}
a ₇	{a₇}	{a₁, a₃, a₄, a_5,a₆, a₇}
a ₈	{a₂, a₈, a₉, a₁₀}	{a₁, a₃, a₄, a_5,a₈, a₁₀, a₁₁}
a ₉	{a₉}	{a₁, a₄, a₅, a₈, a₉, a₁₁, a₁₂}
a ₁₀	{a₂, a₁₀}	{a₄, a₈, a₁₀}
a ₁₁	{a₂, a₈, a₉, a₁₁, a₁₂}	{a₁₁}
a ₁₂	{a₂, a₉, a₁₂}	{a₁₁, a₁₂}

$D om (F_{3}^{*}, C) = {\begin{matrix} (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), (a_{6}, a_{6}), \\ (a_{7}, a_{7}), (a_{9}, a_{9}), (a_{10}, a_{10}), (a_{11}, a_{11}), (a_{12}, a_{12}), (a_{1}, a_{2}), \\ (a_{2}, a_{4}), (a_{2}, a_{5}), (a_{2}, a_{8}), (a_{5}, a_{4}), (a_{6}, a_{1}), (a_{6}, a_{2}), \\ (a_{6}, a_{3}), (a_{6}, a_{4}), (a_{6}, a_{5}), (a_{7}, a_{1}), (a_{7}, a_{2}), (a_{7}, a_{3}), \\ (a_{7}, a_{4}), (a_{7}, a_{5}), (a_{7}, a_{6}), (a_{8}, a_{3}), (a_{8}, a_{4}), (a_{8}, a_{5}), \\ (a_{9}, a_{1}), (a_{9}, a_{2}), (a_{9}, a_{4}), (a_{9}, a_{5}), (a_{9}, a_{8}), (a_{9}, a_{10}), \\ (a_{9}, a_{12}), (a_{10}, a_{4}), (a_{10}, a_{8}), (a_{11}, a_{10}), (a_{12}, a_{10}), (a_{8}, a_{8}) \end{matrix}} .$

The soft dominance classes from $D om (F_{3}^{*}, C)$ are given inTable 4.

Table 4

Dominating classes ${[a_{i}]}_{D om (F_{3}^{*}, C)}^{+} and Dominated classes {[a_{i}]}_{D om (F_{3}^{*}, C)}^{-}$

a _i	${[a_{i}]}_{D om (F_{3}^{*}, C)}^{+}$	${[a_{i}]}_{D om (F_{3}^{*}, C)}^{-}$
a ₁	{a₁, a₆, a₇, a₉}	{a₁, a₂}
a ₂	{a₁, a₂, a₆, a₇, a₉}	{a₂, a₄, a₅, a₈}
a ₃	{a₃, a₆, a₇, a₈}	{a₃}
a ₄	{a₂, a₄, a₅, a₆, a₇, a₈, a₉, a₁₀}	{a₄}
a ₅	{a₂, a₅, a₆, a₇, a₈, a₉}	{a₄, a₅}
a ₆	{a₆, a₇}	{a₁, a₂, a₃, a₄, a_5,a₆}
a ₇	{a₇}	{a₁, a₂, a₃, a₄, a_5,a₆, a₇}
a ₈	{a₂, a₈, a₉, a₁₀}	{a₄, a₈, a₁₀}
a ₉	{a₉}	{a₁, a₂, a₄, a₅, a₈, a₉, a₁₀, a₁₂}
a ₁₀	{a₉, a₁₀, a₁₁, a₁₂}	{a₄, a₈, a₁₀}
a ₁₁	{a₁₁}	{a₁₀, a₁₁}
a ₁₂	{a₉, a₁₂}	{a₁₀, a₁₂}

$D om (F_{4}^{*}, C) = {\begin{matrix} (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), (a_{6}, a_{6}), \\ (a_{7}, a_{7}), (a_{9}, a_{9}), (a_{10}, a_{10}), (a_{11}, a_{11}), (a_{12}, a_{12}), (a_{1}, a_{2}), \\ (a_{2}, a_{1}), (a_{3}, a_{1}), (a_{5}, a_{1}), (a_{6}, a_{1}), (a_{7}, a_{1}), (a_{7}, a_{2}), \\ (a_{7}, a_{4}), (a_{7}, a_{5}), (a_{7}, a_{6}), (a_{8}, a_{1}), (a_{8}, a_{4}), (a_{8}, a_{5}), \\ (a_{8}, a_{11}), (a_{9}, a_{1}), (a_{9}, a_{2}), (a_{9}, a_{4}), (a_{9}, a_{5}), (a_{9}, a_{8}), \\ (a_{9}, a_{10}), (a_{10}, a_{4}), (a_{10}, a_{8}), (a_{12}, a_{11}) \end{matrix}} .$

The soft dominance classes from $D om (F_{4}^{*}, C)$ are given in the following Table 5.

Table 5

Dominating classes ${[a_{i}]}_{D om (F_{4}^{*}, C)}^{+} and Dominated classes {[a_{i}]}_{D om (F_{4}^{*}, C)}^{-}$

a _i	${[a_{i}]}_{D om (F_{4}^{*}, C)}^{+}$	${[a_{i}]}_{D om (F_{4}^{*}, C)}^{-}$
a ₁	{a₁, a₂, a₃, a₅, a₆, a₇, a₈, a₉}	{a₁, a₂}
a ₂	{a₁, a₂, a₇}	{a₁, a₂}
a ₃	{a₃}	{a₁, a₃}
a ₄	{a₄, a₇, a₈, a₉, a₁₀}	{a₄}
a ₅	{a₅, a₇, a₈, a₉}	{a₁, a₅}
a ₆	{a₆, a₇}	{a₁, a₆}
a ₇	{a₇}	{a₁, a₂, a₄, a_5,a₆, a₇}
a ₈	{a₈, a₉}	{a₁, a₄, a₅, a₈, a₁₁}
a ₉	{a₉}	{a₁, a₂, a₄, a_5,a₈, a₉, a₁₀}
a ₁₀	{a₉, a₁₀}	{a₄, a₈, a₁₀}
a ₁₁	{a₈, a₁₁, a₁₂}	{a₁₁}
a ₁₂	{a₁₂}	{a₁₁, a₁₂}

Thus from Tables 6 and 7, we get δ : = { a₇ }. Hence action a₇ is the feasible alternative for solving this conflict analysis problem, on which all agents have agreed in the conflict situation $S^{*} .$ That is, we find a feasible consensus strategy which satisfies all the agents. So we not only show whether there exists an optimal alternative (action a₇) that satisfies all agents, but also present a method to find the consensus for a given conflict situation. Therefore we answered the second and third questions posed by Deja [10] for the classical Pawlak conflict analysis decision making model [47].

Table 6

Table for lower approximations

X _k	${\underline{\sum_{k = 1}^{4} (X_{k})}}_{D o m^{o} {(ℱ_{k}^{*}, C)}^{+}}$	${\underline{\sum_{k = 1}^{4} (X_{k})}}_{D o m^{o} {(ℱ_{k}^{*}, C)}^{-}}$
X ₁	{a₁, a₂, a₃, a₄, a₆, a₇, a₉, a₁₂}	{a₁, a₂, a₃, a₄, a₆, a₁₂}
X ₂	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}, \\ a_{9}, a_{10}, a_{11}, a_{12} \end{matrix}}$	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{8}, a_{9}, \\ a_{10}, a_{11}, a_{12} \end{matrix}}$
X ₃	{a₁, a₆, a₇, a₉, a₁₁, a₁₂}	{a₄, a₆, a₉, a₁₁, a₁₂}
X ₄	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}, \\ a_{9}, a_{10}, a_{12} \end{matrix}}$	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{8}, a_{9}, \\ a_{10}, a_{12} \end{matrix}}$

Table 7

Decision table

δ _k	${\underline{\sum_{k = 1}^{4} (X_{k})}}_{D {om}^{o} {(F_{k}^{}, C)}^{+}} - {\underline{\sum_{k = 1}^{4} (X_{k})}}_{D {om}^{o} {(F_{k}^{}, C)}^{-}}$
δ ₁	{ a₇, a₉, a₁₀ }
δ ₂	{ a₇ }
δ ₃	{ a₁, a₇, a₉ }
δ ₄	{ a₇ }

Now we present another new technique for the development of feasible consensus strategy based on soft dominance multi-granulation pessimistic rough sets.

Definition 7. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system, D₁, D₂, D₃, ... , D_k ∈ D and X ⊆ A. The pessimistic multi-granulation soft dominance lower and upper approximations of X with respect to D₁, D₂, D₃, ... , D_k are defined by

$\begin{matrix} {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{+} \subseteq \\ X \land . . . \land {[a]}_{D om (F_{| D |}^{*}, C)}^{+} \subseteq X}, \\ {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{-} \cap X \neq \emptyset \\ \lor . . . \lor {[a]}_{D om (F_{| D |}^{*}, C)}^{-} \cap X \neq \emptyset} . \end{matrix}$

Then $({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}}, {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}})$ is called the soft dominating pessimistic multi-granulation rough set if ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \neq {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}}$ . The pessimistic multi-granulation boundary region of X is

${BN}^{p} (X) = {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} - {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} .$

Similarly we define

$\begin{matrix} {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{-} \subseteq X \\ \land . . . \land {[a]}_{D om (F_{| D |}^{*}, C)}^{-} \subseteq X}, \\ {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = & {a \in A : {[a]}_{D om (F_{1}^{*}, C)}^{+} \cap X \neq \emptyset \\ \lor . . . \lor {[a]}_{D om (F_{| D |}^{*}, C)}^{+} \cap X \neq \emptyset} . \end{matrix}$ Then $({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}}, {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}})$ is called the soft dominated pessimistic multi-granulation rough set if ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \neq {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}}$ .

Theorem 6. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system in which D₁, D₂, D₃, ... , D_k ∈ D. Then for all X₁, X₂, ... , X_n ⊆ A, the pessimistic multi-granulation soft dominated rough sets have the following properties:

(1) ${\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq X_{j} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}};$

(2) ${\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = \emptyset = {\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}};$

(3) ${\underline{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = A = {\bar{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}};$

(4) X₁⊆ X₂ implies ${\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}};$

(5) $X_{1} \subseteq X_{2} implies {\bar{\sum_{| D | = 1}^{| D |} (X_{1})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}};$

(6) ${\underline{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = ⋂_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}});$

(7) ${\bar{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq ⋂_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}});$

(8) ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \supseteq ⋃_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}});$

(9) ${\bar{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = ⋃_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}});$

(10) ${\underline{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = {({\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}})}^{c};$

(11) ${\bar{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}}$ = ${({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}})}^{c} .$

Proof. The proof follows from the respective definitions.□

Theorem 7. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system in which D₁, D₂, D₃, ... , D_k ∈ D. Then for all X₁, X₂, ... , X_n ⊆ A, the pessimistic multi-granulation soft dominated rough sets have the following properties:

(1) ${\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \subseteq X_{j} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

(2) ${\underline{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = \emptyset = {\bar{\sum_{k = 1}^{| D |} (\emptyset)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

(3) ${\underline{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = A = {\bar{\sum_{k = 1}^{| D |} (A)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

(4) X₁⊆ X₂ implies ${\underline{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

(5) $X_{1} \subseteq X_{2} implies {\bar{\sum_{k = 1}^{| D |} (X_{1})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \subseteq {\bar{\sum_{k = 1}^{| D |} (X_{2})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

(6) ${\underline{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = ⋂_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}});$

(7) ${\bar{\sum_{k = 1}^{| D |} (⋂_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \subseteq ⋂_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}});$

(8) ${\underline{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \supseteq ⋃_{j = 1}^{n} ({\underline{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}});$

(9) ${\bar{\sum_{k = 1}^{| D |} (⋃_{j = 1}^{n} X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = ⋃_{j = 1}^{n} ({\bar{\sum_{k = 1}^{| D |} (X_{j})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}});$

(10) ${\underline{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} = {({\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}})}^{c};$

(11) ${\bar{\sum_{k = 1}^{| D |} (X^{c})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}}$ = ${({\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}})}^{c} .$

Proof. The proof is straightforward.□

Theorem 8. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Suppose $D om (F_{1}^{*}, C) \subseteq D om (F_{2}^{*}, C) \subseteq . . . . \subseteq D om (F_{| D |}^{*}, C) .$ Then

(1) ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = {\underline{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}}$

(2) ${\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = {\bar{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}} .$

Proof. (1) Let $a \in {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} .$ Then ${[a]}_{D om (F_{1}^{*}, C)}^{+} \subseteq X \land . . . \land {[a]}_{D om (F_{| D |}^{*}, C)}^{+} \subseteq X .$ Which implies that ${[a]}_{D om (F_{| D |}^{*}, C)}^{+} \subseteq X .$ This implies that $a \in {\underline{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}} .$ Thus ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq {\underline{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}} .$ Now for any $a \in {\underline{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}}$ implies ${[a]}_{D om (F_{| D |}^{*}, C)}^{+} \subseteq X .$ But $D om (F_{1}^{*}, C) \subseteq D om (F_{2}^{*}, C) \subseteq . . . . \subseteq D om (F_{| D |}^{*}, C)$ so ${[a]}_{D om (F_{1}^{*}, C)}^{+} \subseteq X \land . . . \land {[a]}_{D om (F_{| D |}^{*}, C)}^{+} \subseteq X .$ This implies that $a \in {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} .$ Hence ${\underline{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} .$ Therefore ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}}$ $= {\underline{(X)}}_{D om {(F_{| D |}^{*}, C)}^{+}} .$ The proof of (2) is analogous to that of (1).□

Definition 8. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For any X ⊆ A, the approximate precisions $ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)$ and $ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X)$ of X about $D {om}^{p} {(F_{k}^{*}, C)}^{+}$ and $D {om}^{p} {(F_{k}^{*}, C)}^{-}$ are defined by: $\begin{matrix} ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) = \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} and \\ ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) = \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} |}{| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} |} \end{matrix}$ where X ≠ ∅ , and | · | denotes the cardinality of a set.

Let $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) = 1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)$ and $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) = 1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) .$ Then $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)$ and $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X)$ are called the rough degrees of X about $D {om}^{p} {(F_{k}^{*}, C)}^{+}$ and $D {om}^{p} {(F_{k}^{*}, C)}^{-}$ respectively.

By definition, $0 \leq μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) \leq 1$ and $0 \leq ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) \leq 1 .$ It can be seen that $ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) = 1$ if and only if ${\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = X = {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}}$ .

The following theorems describe the relationship of the precisions $ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)$ and $ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X)$ also the rough degrees $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)$ and $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X)$ for the intersection and union of any subsets X and Y of the universe A.

Theorem 9. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A. Then the rough degree and precision of the subsets X, Y, X ∪ Y and X ∩ Y satisfy the following relations. $\begin{array}{l} (1) μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cup Y) | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ \leq μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | + μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ - μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | . \end{array}$ $\begin{array}{l} (2) ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ \geq ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | + ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ - ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | . \end{array}$

Proof. (1) By definition of rough degree,

$\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cup Y) \\ = 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X \cup Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X \cup Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} \\ = 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X \cup Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} \\ \leq 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} . \end{matrix}$

This implies that $\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cup Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$

Similarly

$\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ = 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X \cap Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X \cap Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} \\ = 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X \cap Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} \\ \leq 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} . \end{matrix}$

Hence $\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$ for any sets A and B |A ∪ B| = |A| + |B| - |A ∩ B|. It follows that $\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cup Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | + | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ + | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | + | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ + | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | + | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{i = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$ Now by definition of rough degree $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) = 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}$ and $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y) = 1 - \frac{| {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| {\bar{\sum_{i = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} .$ We obtain $\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \end{matrix}$ and $\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$ Therefore

$\begin{matrix} μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cup Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq {| {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} \\ + {| {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - | {\underline{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |} \\ - μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ + μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$

(2) By definition of rough degree $μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) : = 1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) .$ So $\begin{array}{l} μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cup Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ \leq μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ + μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ - μ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | . \end{array}$

Therefore $\begin{matrix} {1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cup Y)} \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq {1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)} | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ + {1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y)} | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - {1 - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y)} \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = | {(\bar{X})}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | + | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ + ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ = | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ + ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$ By routine simplifications, we get $\begin{array}{l} ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cup Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ \geq ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ + ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ - ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cap Y) \\ | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | . \end{array}$

□

Theorem 10. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A. Then the rough degree and precision of the subsets X, Y, X ∪ Y and X ∩ Y of A satisfy the following relations. $\begin{matrix} (1) & μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | \\ \leq & μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | + μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | \\ - μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | . \end{matrix}$ $\begin{matrix} (2) ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | \\ \geq & ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | + ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (Y) | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | \\ - ρ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | . \end{matrix}$

Proof. It is analogous to the proof of Theorem 9.□

Definition 9. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For any subset X of A, the approximate qualities $γ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X)$ and $γ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X)$ of X about $D {om}^{p} {(F_{k}^{*}, C)}^{+}$ and $D {om}^{p} {(F_{k}^{*}, C)}^{-}$ are defined by: $\begin{matrix} γ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) : = \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} |}{| A |} and \\ γ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) : = \frac{| {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} |}{| A |} . \end{matrix}$ Clearly $0 \leq γ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) \leq 1$ and $0 \leq γ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) \leq 1 .$

The following theorem illustrates the relationship between rough degree and approximate quality of the intersection and union of subsets X and Y of the universe A.

Theorem 11. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A. Then the rough degree and approximate quality of the subsets X, Y, X ∪ Y and X ∩ Y of A satisfy the following relations: $\begin{matrix} (1) μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ \leq & | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | + | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | \\ - | A | {γ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X) + γ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (Y)} \\ - μ_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} | . \end{matrix}$ $\begin{matrix} (2) μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | \\ \leq & | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | + | {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | \\ - | A | {γ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X) + γ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (Y)} \\ - μ_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}} | . \end{matrix}$

Proof. It is analogous to the proof of Theorem 9.□

The following theorem describes the relationship between approximate precision and approximate quality of the intersection and union of two sets.

Theorem 12. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y are subsets of A. Then the approximate quality and precision of the subsets X, Y, X ∪ Y and X ∩ Y of A satisfy the following relations: $\begin{array}{l} (1) ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | \\ \geq | A | {γ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X) + γ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (Y)} \\ - ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{+}} | . \end{array}$

$\begin{array}{l} (2) ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} (X \cup Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} \cup {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} | \\ \geq | A | {γ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} (X) + γ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} (Y)} \\ - ρ_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} (X \cap Y) | {\bar{\sum_{k = 1}^{| D |} (X)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} \cap {\bar{\sum_{k = 1}^{| D |} (Y)}}_{D o m^{p} {(ℱ_{k}^{*}, C)}^{-}} | . \end{array}$

Proof. It is analogous to the proof of Theorem 9.□

Definition 10. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system in which D₁, D₂, D₃, ... , D_k ∈ D. Then for all X ⊆ A, the rough membership of a in X is defined by $ϑ_{X}^{D om {(F_{k}^{*}, C)}^{+}} (a) = \frac{| {[a]}_{D om (F_{k}^{*}, C)}^{+} \cap X |}{| {[a]}_{D om (F_{k}^{*}, C)}^{+} |}$ and $ϑ_{X}^{D om {(F_{k}^{*}, C)}^{-}} (a) = \frac{| {[a]}_{D om (F_{k}^{*}, C)}^{-} \cap X |}{| {[a]}_{D om (F_{k}^{*}, C)}^{-} |} .$ The maximal rough membership and minimal rough membership of a in X are defined by $Ω_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) = ⋁_{k = 1}^{| D |} ϑ_{X}^{D om {(F_{k}^{*}, C)}^{+}} (a);$ $Ψ_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) = ⋀_{k = 1}^{| D |} ϑ_{X}^{D om {(F_{k}^{*}, C)}^{+}} (a) .$ Similarly in the case of $D om {(F_{k}^{*}, C)}^{-}$ we have $Ω_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{-}} (a) = ⋁_{k = 1}^{| D |} ϑ_{X}^{D om {(F_{k}^{*}, C)}^{-}} (a);$ $Ψ_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{-}} (a) = ⋀_{k = 1}^{| D |} ϑ_{X}^{D om {(F_{k}^{*}, C)}^{-}} (a) .$

Theorem 13. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system in which D₁, D₂, D₃, ... , D_k ∈ D. Then for all X ⊆ A, the following hold.

(1) $Ω_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) = 1 \Leftrightarrow a \in {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

(2) $Ψ_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) = 1 \Leftrightarrow a \in {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}};$

(3) $0 < Ω_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) \leq 1 \Leftrightarrow a \in {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

(4) $0 < Ψ_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) \leq 1 \Leftrightarrow a \in {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{-}} .$

Proof. We prove only part (1). The proofs of the remaining parts are analogous to the proof of (1). $\begin{matrix} (1) Ω_{X}^{\sum_{k = 1}^{| D |} D om {(F_{k}^{*}, C)}^{+}} (a) \\ = 1 \Leftrightarrow ⋁_{k = 1}^{| D |} ϑ_{X}^{D om {(F_{k}^{*}, C)}^{+}} (a) = 1 \\ \Leftrightarrow \exists k \in {1, 2, . . ., | D |} such that {[a]}_{D om (F_{k}^{*}, C)}^{+} \subseteq X \\ \Leftrightarrow a \in {\underline{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} . \end{matrix}$ □

Example 2. (Continued from Example 1) Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Let X_k ⊆ A where

X_k ={ a ∈ A : g (a, D_k) ≠0 } and k = 1, 2, ... , |D|. Then we have Table 8 and Table 9.

Table 8

Maximal rough member ships of a_i in X₁

a _i	$Ω_{X}^{\sum_{k = 1}^{\| D \|} D om {(F_{k}^{*}, C)}^{+}} (a_{i})$	$Ω_{X}^{\sum_{k = 1}^{\| D \|} D om {(F_{k}^{*}, C)}^{-}} (a_{i})$
a ₁	1	1
a ₂	1	1
a ₃	1	1
a ₄	1	1
a ₅	0.66	0.66
a ₆	1	1
a ₇	1	0.85
a ₈	0.75	0.57
a ₉	1	75
a ₁₀	1	0.66
a ₁₁	0.66	0.5
a ₁₂	1	1

Table 9

Minimal rough member ships of a_i in X₁

a _i	$Ψ_{X}^{\sum_{k = 1}^{\| D \|} D om {(F_{k}^{*}, C)}^{+}} (a_{i})$	$Ψ_{X}^{\sum_{k = 1}^{\| D \|} D om {(F_{k}^{*}, C)}^{-}} (a_{i})$
a ₁	0.75	1
a ₂	1	0.5
a ₃	0.66	0.62
a ₄	0.71	1
a ₅	0.5	0
a ₆	1	0.83
a ₇	1	0.83
a ₈	0.5	0.25
a ₉	1	0.57
a ₁₀	0.71	0.66
a ₁₁	0	0
a ₁₂	1	0.5

To develop the feasible consensus strategy among the agents on feasible alternative(s) and to respond the questions (ii) and (iii) of Deja, we develop another novel algorithm utilizing the notion of soft dominance multi-granulation approximations.

4.2 Proposed algorithm for conflict analysis model

Input: Information system $S^{*} : = (A, C, D, E);$

Step 1 : Construct $(F_{k}^{*}, C)$ corresponding to the D_k for all k : =1, 2, ... |D| // according to Definition 2;

Step 2 : Construct $D om (F_{k}^{*}, C),$ for all k : =1, 2, ... , |D| // according to Definition 3;

Step 3 : Construct ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}}$ and ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}}$ // according to Definition 7, where X_k ={ a ∈ A : g (a, D_k) ≠0 } ;

Step 4 : If ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}}$ then

$δ_{k} = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} or δ_{k} = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}}$

otherwise $δ_{k} = {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} - {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{-}};$

Step 5 : $δ = ⋂_{k : = 1}^{| D |} δ_{k} \neq \emptyset .$

go to output.

Otherwise,

|D| = |D| - 1 ;

go back to step 5.

Output: Feasible alternative(s) for feasible consensus strategy. Feasible alternative(s) for feasible consensus strategy.

Example 3. (Continued from Example 1) Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Applying the above algorithm to select feasible alternative(s) for feasible consensus strategy, we have the following tables:

Thus from Table 10 Table 11 we have δ : = { a₇ }. Hence action a₇ is the feasible alternative for solving the conflict analysis problem, on which all agents have agreed in the conflict situation $S^{*} .$ The result shows that there is an optimal alternative (action a₇) which satisfy all the agents in the conflict situation $S^{*} .$ Consequently we answered second and third questions of Deja in an affirmative way.

Table 10
The lower approximations table

X _k ${\underline{\sum_{k = 1}^{4} (X_{k})}}_{D {om}^{p} {(F_{k}^{}, C)}^{+}}$ ${\underline{\sum_{k = 1}^{4} (X_{k})}}_{D {om}^{p} {(F_{k}^{}, C)}^{-}}$

X ₁ {a₂, a₆, a₇, a₉, a₁₂} {a₁, a₄}

X ₂ {a₇, a₈, a₉, a₁₀, a₁₁, a₁₂} {a₁, a₂, a₄, a₅, a₈, a₉, a₁₀, a₁₁, a₁₂}

X ₃ {a₆, a₇, a₉} {a₄}

X ₄ {a₇, a₈, a₉, a₁₂} {a₁, a₄, a₅, a₁₀}

Table 11

Decision table

δ _k	${\underline{\sum_{k = 1}^{4} (X_{k})}}_{D {om}^{p} {(F_{k}^{}, C)}^{+}} - {\underline{\sum_{k = 1}^{4} (X_{k})}}_{D {om}^{p} {(F_{k}^{}, C)}^{-}}$
δ ₁	{a₂, a₆, a₇, a₉, a₁₂}
δ ₂	{a₇}
δ ₃	{a₆, a₇, a₉}
δ ₄	{a₇, a₈, a₉, a₁₂}

In the last, it is necessary to discuss the relationship between optimistic and pessimistic approximations.

Theorem 14. Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system in which D₁, D₂, D₃, ... , D_k ∈ D. Then for any subset X of A, the following hold:

(1) ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}};$

(2) ${\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} \subseteq {\bar{\sum_{k = 1}^{| D |} (X)}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} .$

Proof. Follows from the respective definitions.□

Example 4. (Continued from Example 1) Let $S^{*} : = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system.

Let X_k ⊆ A where X_k ={ a ∈ A : g (a, D_k) ≠0 } and k = 1, 2, ... , |D|. From Tables 6 and 10, we have ${\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{p} {(F_{k}^{*}, C)}^{+}} \subseteq {\underline{\sum_{k = 1}^{| D |} (X_{k})}}_{D {om}^{o} {(F_{k}^{*}, C)}^{+}} .$

4.3 Advantages and Significance

(i) The proposed model answer questions posed by Deja.

(ii) Another advantage of the proposed model is that, it can be used for ranking the feasible alternatives simpler than other existing technique/models.

(iii) The proposed model can be used for reduction of attributes.

(iv) Using the proposed model, one can find two types of rough degree, precision and approximate quality and their relationship.

(v) As a real world application, the proposed model can be applied to solve the Middle East conflict analysis problem, determining the governor election results in Indonesia, selection process at university level, for selection of appropriate medicine for a disease.

(vi) The proposed model can be applied to Sun’s problem.

5 Conclusion

The classical rough set theory presented by Pawlak has been applied to deal with uncertain knowledge in information system while studying the conflict. Deja further enriched the literature of conflict analysis. In this research article, we have presented the idea of soft preference and soft dominance relation in an incomplete information system to solve a multi agent conflict analysis for labor-management negotiations problem. We have developed a feasible consensus strategy for labor-management negotiations conflict analysis problem based on optimistic approximations in an incomplete information system. Further we have described another idea for the development of feasible consensus strategy in labor-management negotiations conflict analysis problem based on soft dominance multi-granulation rough sets called pessimistic rough sets. In addition, this paper presented several uncertainty measures, such as approximate precision, rough degree, approximate quality, maximal and minimal rough memberships and their mutual relationships in an incomplete information system.

In our future study of soft ordered based multi-granulation rough sets may be the following topics to be considered:

(1) To apply the proposed technique to solve the Middle East Conflict analysis problem.

(2) To initiate the study of variable precision soft ordered based multi-granulation rough sets with applications.

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a _i	${[a_{i}]}_{D om (F_{1}^{*}, C)}^{+}$	${[a_{i}]}_{D om (F_{1}^{*}, C)}^{-}$
a ₁	{a₁, a₃, a₇, a₉}	{a₁, a₂}
a ₂	{a_1,a₂, a₆, a₇, a₉}	{a₂, a₄, a₈, a₁₀, a₁₁}
a ₃	{a₃, a₇}	{a₁, a₃, a₄, a₅, a₆, a₈, a₁₀, a₁₁}
a ₄	{a₂, a₃, a₄, a₆, a₇, a₉, a₁₀}	{a₄}
a ₅	{a₃, a₅, a₇, a₈, a₉}	{a₅}
a ₆	{a₆, a₇}	{a₂, a₄, a₆}
a ₇	{a₇}	{a₁, a₂, a₃, a₄, a_5,a₆, a₇}
a ₈	{a₃, a₈, a₁₀}	{a₅, a₈, a₁₀, a₁₁}
a ₉	{a₉}	{a₁, a₂, a₄, a₅, a₈, a₉, a₁₀, a₁₁, a₁₂}
a ₁₀	{a₂, a₃, a₈, a₉, a₁₀, a₁₁, a₁₂}	{a₄, a₈, a₁₀}
a ₁₁	{a₂, a₃, a₈, a₉, a₁₁, a₁₂}	{a₁₀, a₁₁}
a ₁₂	{a₉, a₁₂}	{a₁₀, a₁₁, a₁₂}