New types of dominance based multi-granulation rough sets and their applications in conflict analysis problems

Abstract

In the present paper, we initiate a multi attribute group decision making problems in the presence of multi attribute and multi decision in decision making with preferences. Then resolving the problem, using two different approximation strategies, that is, seeking common reserving difference and seeking common rejecting difference, four kinds of dominance based multi-granulation rough sets are presented, namely, dominance based optimistic multi-granulation rough sets and dominance based pessimistic multi-granulation rough sets and their applications in solving a multi agent conflict analysis decision problem. The proposed method addresses the limitations of the Pawlak model and Sun’s conflict analysis model and thus improve these models. Finally, the results on labor management negotiation problems show that the proposed algorithms are more effective and efficient for feasible consensus strategy when compared with Sun’s technique.

Keywords

Rough sets preference relation multi granulation

1 Introduction

The rough set approach proposed by Pawlak [27] provides useful tools for reasoning from data. It has diverse applications in various fields such as 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. The concept of dominance based rough sets which is initiated by Greco et al. [11–13 , 15] deal with the disparities in miscellaneous kinds of multi criteria decision making problems. The idea abaft the rough set based on dominance relation is to alter the equivalence relation in Pawlak rough set [27] with the dominance relation that authorize to consider the preference order in the value set of the criteria. Many extended models of the dominance based rough set approach have been proposed (see [3 , 41]). The hybridization of dominance based rough set approach and other mathematical tools have been created and applied to multi criteria decision analysis (see [1 , 39–42]). 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 [26]. 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 [6].

Applications of rough set theory in case of conflict analysis is introduced by Pawlak in [28] 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 [6] is an extension and generalization 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?”

In [38] Sun and Ma developed a new multi-agent conflict analysis (labor management negotiations conflict analysis) problem based on preference relation with dominance and tried to answer Deja’s questions related to conflict analysis problems. But still there are many areas for critics, for example no answer to the third question in a good manner, and development of proper feasible consensus strategy is missing. Qian et al. in [32], 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 universe. In the present paper: (i) we define two types of optimistic approximations with the help of dominance (dominating/dominated) classes and applied these to discuss various properties of the approximations. The results on labor management negotiation problem show that the proposed algorithm based on optimistic approximations is more effective and efficient for feasible consensus strategy when compared with Sun’s technique; (iii) one the main contribution of this paper is to put forward the idea of two new kinds of pessimistic approximations with the help of dominance (dominating/dominated) classes and applied these to discuss related properties of the approximations. (iv) one another main contribution of this paper is to disclose the ideas of two kinds of approximate precision, rough degree, approximate quality and their mutual relationship.

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 focus our attention on the development of feasible consensus strategy for labor-management negotiation conflict analysis problem based on optimistic approximations. Further in section, we present another idea for the development of feasible consensus strategy in labor-management negotiations conflict analysis problem based on dominance multi-granulation rough sets called pessimistic rough sets. In addition, several uncertainty measures, such as approximate precision, rough degree, approximate quality and their mutual relationships are discussed.

2 Problem statement

The multiple decision problems with preference relations have been studied in this paper. In general these may be the multiple criteria group decision problems or multiple criteria and multiple decision with preference choice problems. It comprises of two parts, the first is the multiple criteria with predefined evaluations for every action and the other is the multiple decision with a predefined preference for every action. A decision problem may be considered as a system $S : = (A, C, D, E)$ , where A is a finite set of actions a_i, i = 1, 2, 3,…, n, 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). In this study, the values of both information functions f (a_i, C_j) and g (a_i, D_k) are integers. We call $S : = (A, C, D, E)$ the multiple attribute and multiple decision information system.

In order to show the 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, where 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
Multi-criteria and multi-decision information system

A	C ₁	C ₂	C ₃	C ₄	C ₅	D ₁	D ₂	D ₃	D ₄
a ₁	1	0	1	0	2	2	0	1	0
a ₂	1	0	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	0	2
a ₅	1	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	0	2	0	1	0	2
a ₉	1	2	1	1	2	2	2	1	0
a ₁₀	1	2	0	0	1	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 2

Dominating classes ${[a_{i}]}_{P_{C}}^{+} and Dominated classes {[a_{i}]}_{P_{C}}^{-}$

A	${[a_{i}]}_{P_{C}}^{+}$	${[a_{i}]}_{P_{C}}^{-}$
a ₁	{a₁, a₂, a₃, a₅, a₆, a₇, a₉}	{a₁}
a ₂	{a₂, a₆, a₇, a₉}	{a₁, a₂}
a ₃	{a₃, a₆, a₇}	{a₁, a₃}
a ₄	{a₄, a₅, a₆, a₇, a₈, a₉}	{a₄}
a ₅	{a₅, a₆, a₇, a₉}	{a₁, a₄, a₅}
a ₆	{a₆, a₇}	{a₁, a₂, a₃, a₄, a₅, a₆}
a ₇	{a₇}	{a₁, a₂, a₃, a₄, a₅, a₆, a₇}
a ₈	{a₈, a₉}	{a₄, a₈}
a ₉	{a₉}	${\begin{matrix} a_{1}, a_{2}, a_{4}, a_{5}, a_{8}, a_{9}, \\ a_{10}, a_{11}, a_{12} \end{matrix}}$
a ₁₀	{a₈, a₉, a₁₀, a₁₁, a₁₂}	{a₁₀}
a ₁₁	{a₉, a₁₁, a₁₂}	{a₁₀, a₁₁}
a ₁₂	{a₉, a₁₂}	{a₁₀, a₁₁, a₁₂}

Table 3

$Dominating classes {[a_{i}]}_{P_{D}}^{+} and Dominated classes {[a_{i}]}_{P_{D}}^{-}$

A	${[a_{i}]}_{P_{D}}^{+}$	${[a_{i}]}_{P_{D}}^{-}$
a ₁	{a₁, a₇, a₉, a₁₂}	{a₁}
a ₂	{a₂}	{a₂, a₅}
a ₃	{a₃}	{a₃, a₄, a₈}
a ₄	{a₃, a₄, a₇}	{a₄, a₈}
a ₅	{a₂, a₅, a₇, a₁₀}	{a₅}
a ₆	{a₆, a₇}	{a₆}
a ₇	{a₇}	${\begin{matrix} a_{1}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}, a_{9}, \\ a_{11}, a_{12}, \end{matrix}}$
a ₈	{a₃, a₄, a₇, a₈}	{a₈}
a ₉	{a₇, a₉}	{a₁, a₉, a₁₀}
a ₁₀	{a₁₀}	{a₅, a₁₀}
a ₁₁	{a₇, a₁₁}	{a₁₁}
a ₁₂	{a₇, a₁₂}	{a₁₂}

Table 4

Table for lower approximations

X _k	${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{o}$	${\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{o}$
X ₁	{a₁, a₂, a₃, a₄, a₆, a₇, a₉, a₁₂}	{a₁, a₂, a₃, a₄, a₆, a₁₂}
X ₂	{a₂, a₃, a₄, a₅, a₇, a₈, a₉, a₁₀, a₁₁, a₁₂}	{a₂, a₃, a₄, a₅, a₈, a₁₀, a₁₁, a₁₂}
X ₃	{a₁, a₆, a₇, a₉, a₁₁, a₁₂}	{a₁, a₆, a₁₁, a₁₂}
X ₄	{a₂, a₃, a₄, a₅, a₇, a₈, a₁₀, a₁₂}	{a₂, a₃, a₄, a₅, a₈, a₁₀, a₁₂}

Table 5

Decision table

δ _k	${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{o} - {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{o}$
δ ₁	{ a₇, a₉ }
δ ₂	{ a₇, a₉ }
δ ₃	{ a₇ }
δ ₄	{ a₇ }

Table 6

The lower approximations table

X _k	${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{p}$	${\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{p}$
X ₁	{a₂, a₃, a₆, a₇, a₉, a₁₂}	{a₁}
X ₂	{a₇, a₈, a₉, a₁₀, a₁₁, a₁₂}	{a₄, a₈, a₁₀, a₁₁, a₁₂}
X ₃	{a₆, a₇, a₉, a₁₁, a₁₂}	{a₁}
X ₄	{a₇}	{a₄, a₈, a₁₀}

Table 7

Decision table

δ _k	${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{p} - {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{p}$
δ ₁	{ a₂, a₃, a₆, a₇, a₉, a₁₂ }
δ ₂	{ a₇, a₉ }
δ ₃	{ a₆, a₇, a₉, a₁₁, a₁₂ }
δ ₄	{ a₇ }

Table 8

Result of the Algorithms

No	Data type	Sun et al.	Our proposed
1	Decision makers agreed on a₇	Two decision makers	All decision makers

The Table 1 describes a multi attribute and multi decision makers with preference for making a decision about the labor-management negotiations problem.

3 Literature Review

A simple conflict occurs when two persons have different points of view about a thing or event. In the following different conflict models have been presented. Some related definition are:

Given a non-empty set U, a relation R in U is called;

(i) Reflexive when (x, x) ∈ R for all x ∈ U.

(ii) Irreflexive when (x, x) ∉ R for some x ∈ U.

(iii) Symmetric when (x, y) ∈ R implies (y, x) ∈ R for all x, y ∈ U.

(iv) Antisymmetric when (x, y) ∈ R and (y, x) ∈ R implies x = y for all x, y ∈ U.

(v) Asymmetric when (x, y) ∈ R implies (y, x) ∉ R for all x, y ∈ U.

(vi) Transitive when (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R for all x, y ∈ U.

(vii) Complete when x ≠ y, (x, y) ∈ R or (y, x) ∈ R for all x, y ∈ U.

(viii) Strong complete when (x, y) ∈ R or (y, x) ∈ R for all x, y ∈ U.

(ix) Preorder relation when it is reflexive and transitive.

(x) Equivalence relation when it is reflexive, symmetric and transitive.

3.1 Deja conflict analysis model

Analysis of conflict described in [28] 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 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, Deja in [5], put forward three basic (below given) questions which are not answered by the Pawlak’s conflict analysis model:

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

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

3.2 Sun and Ma conflict analysis model

Now we briefly review Sun and Ma’s multi-criteria and multi-decision method [38] based on preference relation with dominance.

Let $P_{q}$ be an outranking relation on a universe A with reference to criteria q ∈ C such that $a_{i} P_{q} a_{j}$ means “a_i is at least as good as a_j with respect to criteria q.” Suppose that $P_{q}$ is a complete preorder, Sun and Ma employed dominance relation for the study of the problem as follows: a_i ≽ a_j by f (a_i, q) ≥ f (a_j, q) according to increasing preference, where q ∈ C and a_i, a_j ∈ A. For any subset of the conditional attributes Q ⊆ C, a_i ≽ _Qa_j means that a_i ≽ _qa_j for all q ∈ Q, that is, a_i dominates a_j with respect to all attributes in Q. The intersection of complete preorders is a partial preorder and $P_{Q} : = ⋂_{q \in Q} P_{q},$ and thus the dominance relation $P_{Q}$ is a partial preorder.

Definition 1. Let $S : = (A, C, D, E)$ be an information system. Then the lower and upper approximations for the k-grade D-dominating set $P_{D}^{k +} (A)$ of the action set A about the dominance relation $P_{C}^{+}$ is as follows: $\begin{matrix} \underline{R} (P_{D}^{k +} (A)) & : = & ⋂_{a_{i} \in A} \underline{R} (P_{D}^{k +} (a_{i})) \\ : = & ⋂_{a_{j} \in A} {a_{i} ∣ P_{C}^{+} (a_{i}) \subseteq P_{D}^{k +} (a_{j})}; \end{matrix}$ $\begin{matrix} \bar{R} (P_{D}^{k +} (A)) : = \underset{a_{i} \in A}{\cap} \bar{R} (P_{D}^{k} (a_{i})) \\ : = \underset{a_{j} \in A}{\cap} {a_{i} | p_{C}^{+} (a_{i}) \cap P_{D}^{k +} (a_{j}) : \neq Ø}, \end{matrix}$ where $\begin{matrix} P_{C}^{+} (a_{i}) = {a_{j} \in A : f (a_{j}, C) \geq f (a_{i}, C)} \end{matrix}$ and $\begin{matrix} P_{D}^{k +} (a_{i}) = {a_{j} \in A : g (a_{j}, d_{l}) \geq f (a_{i}, d_{l}) +, \\ + 0 \leq k \leq | K |, | K | \leq | D |} \end{matrix}$ with d_l ∈ K ⊆ D.

3.3 Sun and Ma’s Algorithm

Input:Information system $S : = (A, C, D, E);$

Step 1: Initialize the value of K: = |D|;

Step 2: Calculate $\begin{matrix} \underline{R} (P_{D}^{k +} (A)) : = \underset{a_{i} \in A}{\cap} \underline{R} (P_{D}^{k} (a_{i})) \\ : = \underset{a_{j} \in A}{\cap} {a_{i} | P_{C}^{+} (a_{i}) \subseteq P_{D}^{k +} (a_{j})}; \end{matrix}$

Step 3: If $\underline{R} (P_{D}^{k +} (A)) : \neq \emptyset;$

go to step 4.

otherwise,

k: = k - 1;

go back to step 3.

Step 4: Return maximum value k.

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

Example 1. In the labor-management negotiations conflict situation, there are five issues and four agents. Suppose there are twelve feasible alternatives 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. According to Sun and Ma’s algorithm

If k: = |D|: =4, then we find the alternative which satisfies all the agents. Then calculate the lower approximation $P_{D}^{4 +}$ about $S : = (A, C, D, E)$ as follows: $\begin{matrix} \underline{R} (P_{D}^{4 +} (A)) : = ⋂_{a_{i} \in A} \underline{R} (P_{D}^{4 +} (a_{i})) \\ : = ⋂_{a_{j} \in A} {a_{i} ∣ P_{C}^{+} (a_{i}) \subseteq P_{D}^{4 +} (a_{j})} : = \emptyset . \end{matrix}$ Thus there is no optimal alternative which satisfies all the agents for the conflict situation.

Let k: = k - 1: =3. Then calculate the lower approximation $P_{D}^{3 +}$ about $S : = (A, C, D, E)$ as follows: $\begin{matrix} \underline{R} (P_{D}^{3 +} (A)) : = ⋂_{a_{i} \in A} \underline{R} (P_{D}^{3 +} (a_{i})) \\ : = ⋂_{a_{j} \in A} {a_{i} ∣ P_{C}^{+} (a_{i}) \subseteq P_{D}^{3 +} (a_{j})} : = \emptyset . \end{matrix}$ As seen above, for k: =3 the $\underline{R} (P_{D}^{3 +} (A)) : = \emptyset .$

If k: = k - 1: =2, then $\begin{matrix} \underline{R} (P_{D}^{2 +} (A)) : = ⋂_{a_{i} \in A} \underline{R} (P_{D}^{2 +} (a_{i})) \\ : = ⋂_{a_{j} \in A} {a_{i} ∣ P_{C}^{+} (a_{i}) \subseteq P_{D}^{2 +} (a_{j})} : = {a_{7}} . \end{matrix}$ Since $\underline{R} (P_{D}^{2 +} (A)) : = {a_{7}}$ , showing that the two decision makers agreed on a₇ and the other two are not agreed for the conflict situation $S : = (A, C, D, E)$ .

One can naturally ask the interesting question. Is it possible to develop a technique which make consensus among all the agents in above conflict problem?

4 Multi-granulation rough sets based on dominance relation

According to two different approximations, Qian et al. [32, 33] developed two different multi-granulation rough sets including optimistic and pessimistic ones. In this section we present two kinds of dominance based multi-granulation rough sets namely, dominance (dominating/dominated) based optimistic multi-granulation rough sets and dominance (dominating/dominated) based pessimistic multi-granulation rough sets.

Definition 2. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system, and X ⊆ A. The dominance relation defined by conditional attribute C is denoted by $P_{C}$ and dominance relation defined by decision attribute D is denoted by $P_{D} .$ Let ${[a_{i}]}_{P_{C}}^{+} = {a_{j} \in A : f (a_{j}, C) \geq f (a_{i}, C)}$ ${[a_{i}]}_{P_{C}}^{-} = {a_{j} \in A : f (a_{j}, C) \leq f (a_{i}, C)},$ represent the C-dominating and C-dominated set with respect to a_i over the conditional attribute C of $S$ . Similarly, we can also define the dominance relation over the decision attribute for the multi attributes with multi decisions information system. For any a_i, a_j ∈ A ${[a_{i}]}_{P_{D}}^{+} = {a_{j} \in A : g (a_{j}, D) \geq f (a_{i}, D)}$ ${[a_{i}]}_{P_{D}}^{-} = {a_{j} \in A : g (a_{j}, D) \leq f (a_{i}, D)}$

represent the D-dominating and D-dominated set with respect to a_i over the decision attribute D of $S$ . The dominance optimistic multi-granulation lower and upper approximations of X are defined by ${\underline{{(X)}^{o}}}_{P_{C}^{+} + P_{D}^{+}} = {a \in A : {[a]}_{P_{C}}^{+} \subseteq X \lor {[a]}_{P_{D}}^{+} \subseteq X},$ $\begin{matrix} {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o} = {a \in A : {[a]}_{P_{C}}^{-} \cap X \neq \emptyset \land {[a]}_{P_{D}}^{-} + \\ + \cap X \neq \emptyset} . \end{matrix}$

Then $({\underline{{(X)}^{o}}}_{P_{C}^{+} + P_{D}^{+}}, {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o})$ is called the dominating optimistic multi-granulation rough set if ${\underline{{(X)}^{o}}}_{P_{C}^{+} + P_{D}^{+}} \neq {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o}$ . The optimistic multi-granulation boundary region of X is ${BN}^{o} (X) = {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o} - {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o} .$ Similarly we define ${\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o} = {a \in A : {[a]}_{P_{C}}^{-} \subseteq X \lor {[a]}_{P_{D}}^{-} \subseteq X},$ $\begin{matrix} {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o} = {a \in A : {[a]}_{P_{C}}^{+} \cap X \neq \emptyset \land . . . \land {[a]}_{P_{D}}^{+} + \\ + \cap X \neq \emptyset} . \end{matrix}$ Then $({\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o}, {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o})$ is called the dominated optimistic multi-granulation rough set if ${\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o} \neq {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o}$ .

Theorem 1. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Then for all X₁, X₂,…, X_n ⊆ A, the dominating optimistic multi-granulation rough sets has the following properties:

${\underline{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o} \subseteq X_{j} \subseteq {\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o};$

${\underline{(\emptyset)}}_{P_{C}^{+} + P_{D}^{+}}^{o} = \emptyset = {\bar{(\emptyset)}}_{P_{C}^{+} + P_{D}^{+}}^{o};$

${\underline{(A)}}_{P_{C}^{+} + P_{D}^{+}}^{o} = A = {\bar{(A)}}_{P_{C}^{+} + P_{D}^{+}}^{o};$

X₁ ⊆ X₂ implies ${\underline{(X_{1})}}_{P_{C}^{+} + P_{D}^{+}}^{o} \subseteq {\underline{(X_{2})}}_{P_{C}^{+} + P_{D}^{+}}^{o};$

X₁ ⊆ X₂ implies ${\bar{(X_{1})}}_{P_{C}^{+} + P_{D}^{+}}^{o} \subseteq {\bar{(X_{2})}}_{P_{C}^{+} + P_{D}^{+}}^{o};$

${\underline{{(⋂_{j = 1}^{n} X_{j})}^{o}}}_{P_{C}^{+} + P_{D}^{+}} \subseteq ⋂_{j = 1}^{n} ({\underline{{(X_{j})}^{o}}}_{P_{C}^{+} + P_{D}^{+}})$

${\bar{(⋂_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o} \subseteq ⋂_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o});$

${\underline{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o} \supseteq ⋃_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o});$

${\bar{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o} \supseteq ⋃_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{o});$

${\underline{(X^{c})}}_{P_{C}^{+} + P_{D}^{+}}^{o} = {({\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o})}^{c};$

${\bar{(X^{c})}}_{P_{C}^{+} + P_{D}^{+}}^{o}$ = ${({\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o})}^{c} .$

Proof. Straightforward.

Theorem 2. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Then for all X₁, X₂,…, X_n ⊆ A, the dominated optimistic multi-granulation rough sets satisfy the following properties:

${\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o} \subseteq X_{j} \subseteq {\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o};$

${\underline{(\emptyset)}}_{P_{C}^{-} + P_{D}^{-}}^{o} = \emptyset = {\bar{(\emptyset)}}_{P_{C}^{-} + P_{D}^{-}}^{o};$

${\underline{(A)}}_{P_{C}^{-} + P_{D}^{-}}^{o} = A = {\bar{(A)}}_{P_{C}^{-} + P_{D}^{-}}^{o};$

X₁ ⊆ X₂ implies ${\underline{(X_{1})}}_{P_{C}^{-} + P_{D}^{-}}^{o} \subseteq {\underline{(X_{2})}}_{P_{C}^{-} + P_{D}^{-}}^{o};$

X₁ ⊆ X₂ implies ${\bar{(X_{1})}}_{P_{C}^{-} + P_{D}^{-}}^{o} \subseteq {\bar{(X_{2})}}_{P_{C}^{-} + P_{D}^{-}}^{o};$

${\underline{{(⋂_{j = 1}^{n} X_{j})}^{o}}}_{P_{C}^{-} + P_{D}^{-}} \subseteq ⋂_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o})$

${\bar{(⋂_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o} \subseteq ⋂_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o});$

${\underline{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o} \supseteq ⋃_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o});$

${\bar{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o} \supseteq ⋃_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{o});$

${\underline{(X^{c})}}_{P_{C}^{-} + P_{D}^{-}}^{o} = {({\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o})}^{c};$

${\bar{(X^{c})}}_{P_{C}^{-} + P_{D}^{-}}^{o}$ = ${({\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{o})}^{c} .$

Proof. Straightforward.

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

4.1 Proposed algorithm for conflict analysis model

Input:Information system $S : = (A, C, D, E);$

Step 1: Construct ${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{o}$ and ${\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{o}$ // according to Definition 2, where X_k ={ a ∈ A: g (a, D_k) ≠0 };

Step 2: If ${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{o} = {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{o}$ then $δ_{k} = {\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{o} or δ_{k} = {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{o}$ otherwise $δ_{k} = {\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{o} - {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{o};$

Step 3: If $δ : = ⋂_{k : = 1}^{| D |} δ_{k} \neq \emptyset;$

go to output.

Otherwise,

|D| = |D| - 1;

go back to step 3.

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

Time complexity of the algorithm

The time complexity of the algorithm is O (4n + i (k - i)).

Example 2. (Continued from Example 1). Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Construct dominance relations. $P_{C} = {\begin{matrix} (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), (a_{6}, a_{6}), \\ (a_{7}, a_{7}), (a_{8}, a_{8}), (a_{9}, a_{9}), (a_{10}, a_{10}), (a_{11}, a_{11}), (a_{12}, a_{12}), \\ (a_{2}, a_{1}), (a_{5}, a_{1}), (a_{5}, a_{4}), (a_{6}, a_{1}), (a_{6}, a_{3}), (a_{6}, a_{4}), \\ (a_{6}, a_{5}), (a_{8}, a_{4}), (a_{9}, a_{4}), (a_{9}, a_{8}), (a_{3}, a_{1}), (a_{6}, a_{2}), \\ (a_{7}, a_{1}), (a_{7}, a_{2}), (a_{7}, a_{3}), (a_{7}, a_{4}), (a_{7}, a_{5}), (a_{7}, a_{6}), \\ (a_{8}, a_{10}), (a_{9}, a_{1}), (a_{9}, a_{2}), (a_{9}, a_{5}), (a_{9}, a_{10}), \\ (a_{9}, a_{11}), (a_{9}, a_{12}), (a_{11}, a_{10}), (a_{12}, a_{10}), (a_{12}, a_{11}) \end{matrix}} .$

The dominance classes from $P_{C}$ are given in the following table Next $P_{D} = {\begin{matrix} (a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), (a_{6}, a_{6}), \\ (a_{7}, a_{7}), (a_{9}, a_{9}), (a_{10}, a_{10}), (a_{11}, a_{11}), (a_{12}, a_{12}), \\ (a_{7}, a_{1}), (a_{7}, a_{4}), (a_{7}, a_{5}), (a_{7}, a_{6}), (a_{7}, a_{8}), (a_{7}, a_{9}), \\ (a_{7}, a_{11}) (a_{7}, a_{12}) (a_{9}, a_{1}), (a_{8}, a_{8}), (a_{10}, a_{5}), (a_{12}, a_{1}) \\ (a_{2}, a_{5}), (a_{3}, a_{4}), (a_{3}, a_{8}), (a_{4}, a_{8}) \end{matrix}} .$

The dominance classes from $P_{D}$ are given in Table 2 and Table 3.

Using Table 4 and Table 5, δ: = { a₇ }. Thus 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 satisfy all agents. So we not only show whether there exists an optimal alternative (action a₇) that satisfy all agents, but also present a method to find the consensus for a given conflict situation. Therefore we answer the second and third questions posed by Deja [5] for the classical Pawlak conflict analysis decision making model [28].

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

Definition 3. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system, and X ⊆ A. The pessimistic multi-granulation dominance lower and upper approximations of X are defined by ${\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} = {a \in A : {[a]}_{P_{C}}^{+} \subseteq X \land {[a]}_{P_{D}}^{+} \subseteq X},$ $\begin{matrix} {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} = {a \in A : {[a]}_{P_{C}}^{-} \cap X \neq \emptyset \lor {[a]}_{P_{D}}^{-} + \\ + \cap X \neq \emptyset} . \end{matrix}$

Then $({\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p}, {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p})$ is called the dominating pessimistic multi-granulation rough set if ${\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \neq {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p}$ . The pessimistic multi-granulation boundary region of X is ${BN}^{p} (X) = {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} - {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} .$ Similarly we define ${\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} = {a \in A : {[a]}_{P_{C}}^{-} \subseteq X \land {[a]}_{P_{D}}^{-} \subseteq X},$ $\begin{matrix} {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} = {a \in A : {[a]}_{P_{C}}^{+} \cap X \neq \emptyset \lor {[a]}_{P_{D}}^{+} + \\ + \cap X \neq \emptyset} . \end{matrix}$ Then $({\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p}, {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p})$ is called the dominated pessimistic multi-granulation rough set if ${\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} \neq {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p}$ .

Theorem 3. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Then for all X₁, X₂,…, X_n ⊆ A, the pessimistic multi-granulation dominating rough sets has the following properties:

${\underline{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p} \subseteq X_{j} \subseteq {\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p};$

${\underline{(\emptyset)}}_{P_{C}^{+} + P_{D}^{+}}^{p} = \emptyset = {\bar{(\emptyset)}}_{P_{C}^{+} + P_{D}^{+}}^{p};$

${\underline{(A)}}_{P_{C}^{+} + P_{D}^{+}}^{p} = A = {\bar{(A)}}_{P_{C}^{+} + P_{D}^{+}}^{p};$

X₁⊆ X₂ ⇒ ${\underline{(X_{1})}}_{P_{C}^{+} + P_{D}^{+}}^{p} \subseteq {\underline{(X_{2})}}_{P_{C}^{+} + P_{D}^{+}}^{p};$

$X_{1} \subseteq X_{2} \Rightarrow {\bar{(X_{1})}}_{P_{C}^{+} + P_{D}^{+}}^{p} \subseteq {\bar{(X_{2})}}_{P_{C}^{+} + P_{D}^{+}}^{p};$

${\underline{(⋂_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p} = ⋂_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p});$

${\bar{(⋂_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p} \subseteq ⋂_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p});$

${\underline{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p} \supseteq ⋃_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p});$

${\bar{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p} = ⋃_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p});$

${\underline{(X^{c})}}_{P_{C}^{+} + P_{D}^{+}}^{p} = {({\bar{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p})}^{c};$

${\bar{(X^{c})}}_{P_{C}^{+} + P_{D}^{+}}^{p} = {({\underline{(X_{j})}}_{P_{C}^{+} + P_{D}^{+}}^{p})}^{c} .$

Proof. Follows from the respective definitions.

Theorem 4. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Then for all X₁, X₂,…, X_n ⊆ A, the pessimistic multi-granulation dominated rough sets satisfy the following properties:

${\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p} \subseteq X_{j} \subseteq {\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p};$

${\underline{(\emptyset)}}_{P_{C}^{-} + P_{D}^{-}}^{p} = \emptyset = {\bar{(\emptyset)}}_{P_{C}^{-} + P_{D}^{-}}^{p};$

${\underline{(A)}}_{P_{C}^{-} + P_{D}^{-}}^{p} = A = {\bar{(A)}}_{P_{C}^{-} + P_{D}^{-}}^{p};$

$X_{1} \subseteq X_{2} \Rightarrow {\underline{(X_{1})}}_{P_{C}^{-} + P_{D}^{-}}^{p} \subseteq {\underline{(X_{2})}}_{P_{C}^{-} + P_{D}^{-}}^{p};$

$X_{1} \subseteq X_{2} \Rightarrow {\bar{(X_{1})}}_{P_{C}^{-} + P_{D}^{-}}^{p} \subseteq {\bar{(X_{2})}}_{P_{C}^{-} + P_{D}^{-}}^{p};$

${\underline{(⋂_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p} = ⋂_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p});$

${\bar{(⋂_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p} \subseteq ⋂_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p});$

${\underline{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p} \supseteq ⋃_{j = 1}^{n} ({\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p});$

${\bar{(⋃_{j = 1}^{n} X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p} = ⋃_{j = 1}^{n} ({\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p});$

${\underline{(X^{c})}}_{P_{C}^{-} + P_{D}^{-}}^{p} = {({\bar{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p})}^{c};$

${\bar{(X^{c})}}_{P_{C}^{-} + P_{D}^{-}}^{p}$ = ${({\underline{(X_{j})}}_{P_{C}^{-} + P_{D}^{-}}^{p})}^{c} .$

Proof. Follows from the respective definitions.

Definition 4. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. For any X ⊆ A, the approximate precisions $ρ_{P_{C}^{+} + P_{D}^{+}} (X)$ and $ρ_{P_{C}^{-} + P_{D}^{-}} (X)$ of X are defined by: $\begin{matrix} ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) : = \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} and ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X) \\ : = \frac{| {\underline{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} |}{| {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} |} \end{matrix}$ where X: ≠ ∅, and | · | denotes the cardinality of a set.

Let $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) : = 1 - ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X)$ and $μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X) : = 1 - ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X) .$ Then $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X)$ and $μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X)$ are called the rough degrees of X.

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

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

Theorem 5. Let $S : = (A, C, D, E)$ be a multi attributes with 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. $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X {\cup^{}}^{} Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup^{} {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \leq \\ μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + μ_{P_{C}^{+} + P_{D}^{+}}^{p} (Y) | {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X {\cap^{}}^{} Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} {\cap^{}}^{} {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$ $\begin{array}{l} 2. ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup^{} Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \geq \\ ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (Y) | {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X {\cap^{}}^{} Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap^{} {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{array}$

1. By definition of rough degree, $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup Y) = 1 - \frac{| {(\underline{X \cup Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {(\bar{X \cup Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} |} \\ = 1 - \frac{| {(\underline{X \cup Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} |} \\ = 1 - \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} . \end{matrix}$

This implies that $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$

Similarly $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) = 1 - \frac{| {\underline{(X \cap Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\underline{(X \cap Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} \\ = 1 - \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\underline{(X \cap Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} \\ \leq 1 - \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} . \end{matrix}$

Hence $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$

for any sets A and B, |A ∪ B| = |A| + |B| - |A ∩ B|. It follows that $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ = & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | + | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + & | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ = & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | + | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & | {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + & | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | + | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - & μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$

Now by definition of rough degree $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) = 1 - \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}$ and $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (Y) = 1 - \frac{| {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} .$ We obtain $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | = | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |$ and $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (Y) | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | = | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | .$ Therefore $\begin{matrix} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq & {| {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} \\ + {| {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - | {\underline{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |} \\ - μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ = μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | + μ_{P_{C}^{+} + P_{D}^{+}}^{p} (Y) | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$

2. By definition of rough degree $μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) : = 1 - ρ_{P_{C}^{+} + P_{D}^{+}} (X) .$ Since $\begin{array}{l} μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{C}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{C}^{+}}^{p} | \\ \leq μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) | {\bar{(X)}}_{P_{C}^{+} + P_{C}^{+}}^{p} | + μ_{P_{C}^{+} + P_{C}^{+}}^{p} (Y) | {\bar{(Y)}}_{P_{C}^{+} + P_{C}^{+}}^{p} | \\ - μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{C}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{C}^{+}}^{p} | . \end{array}$

Therefore $\begin{matrix} {1 - ρ_{P_{C}^{+} + P_{D}^{+}} (X \cup Y)} | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq & {1 - ρ_{P_{C}^{+} + P_{D}^{+}} (X)} | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + {1 - ρ_{P_{C}^{+} + P_{D}^{+}} (Y)} | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - {1 - ρ_{P_{C}^{+} + P_{D}^{+}} (X \cap Y)} \\ | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ = & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | - ρ_{P_{C}^{+} + P_{D}^{+}} (X) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - ρ_{P_{C}^{+} + P_{D}^{+}} (Y) | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + ρ_{P_{C}^{+} + P_{D}^{+}} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ = & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - ρ_{P_{C}^{+} + P_{D}^{+}} (X) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - ρ_{P_{C}^{+} + P_{D}^{+}} (Y) | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ + ρ_{P_{C}^{+} + P_{D}^{+}} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$

By routine simplifications, we get $\begin{array}{l} ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup^{} Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \geq ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} | ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \\ - ρ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {(\bar{X})}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {(\bar{Y})}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{array}$

Theorem 6. Let $S : = (A, C, D, E)$ be a multi attributes with 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. $\begin{array}{l} 1. μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cup^{} Y) | {(\bar{X})}_{P_{C}^{-} + P_{D}^{-}}^{p} \cup^{} {(\bar{Y})}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ \leq μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X) | {(\bar{X})}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ + μ_{P_{C}^{-} + P_{D}^{-}}^{p} (Y) | {(\bar{Y})}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ - μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cap^{} Y) | {(\bar{X})}_{P_{C}^{-} + P_{D}^{-}}^{p} \cap^{} {(\bar{Y})}_{P_{C}^{-} + P_{D}^{-}}^{p} | . \end{array}$ $\begin{array}{l} 2. ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cup^{} Y) | {(\bar{X})}_{P_{C}^{-} + P_{D}^{-}}^{p} \cup^{} {(\bar{Y})}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ \leq ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X) | {(\bar{X})}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ + ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (Y) | {(\bar{Y})}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ - ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cap^{} Y) | {(\bar{X})}_{P_{C}^{-} + P_{D}^{-}}^{p} \cap^{} {(\bar{Y})}_{P_{C}^{-} + P_{D}^{-}}^{p} | . \end{array}$

Proof. It is analogous to Theorem 5.

Definition 5. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. For any subset X of A, the approximate qualities $γ_{P_{C}^{+} + P_{D}^{+}} (X)$ and $γ_{P_{C}^{-} + P_{D}^{-}} (X)$ of X are defined by: $\begin{matrix} γ_{P_{C}^{+} + P_{D}^{+}} (X) : = \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| A |} and γ_{P_{C}^{-} + P_{D}^{-}} (X) \\ : = \frac{| {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} |}{| A |} . \end{matrix}$ Clearly $0 \leq γ_{P_{C}^{+} + P_{D}^{+}} (X) \leq 1$ and $0 \leq γ_{P_{C}^{-} + P_{D}^{-}} (X) \leq 1 .$

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 7. Let $S : = (A, C, D, E)$ be a multi attributes with 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. $\begin{matrix} 1 . μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cup Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cup {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ \leq & | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | + | {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | \\ - | A | {γ_{P_{C}^{+} + P_{D}^{+}} (X) + γ_{P_{C}^{+} + P_{D}^{+}} (Y)} \\ - μ_{P_{C}^{+} + P_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \cap {\bar{(Y)}}_{P_{C}^{+} + P_{D}^{+}}^{p} | . \end{matrix}$ $\begin{matrix} 2 . μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cup Y) | {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} \cup {\bar{(Y)}}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ \leq & | {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} | + | {\bar{(Y)}}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ - | A | {γ_{P_{C}^{-} + P_{D}^{-}} (X) + γ_{P_{C}^{-} + P_{D}^{-}} (Y)} \\ - μ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} \cap {\bar{(Y)}}_{P_{C}^{-} + P_{D}^{-}}^{p} | . \end{matrix}$

Proof. It is analogous to Theorem 5.

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

Theorem 8. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system and X, Y are subset of A. Then the approximate quality and precision of the subsets X, Y, X ∪ Y and X ∩ Y of A satisfy the following relations:

$\begin{array}{l} ρ_{p_{C}^{+} + p_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{p_{C}^{+} + p_{D}^{+}}^{p} \cup {\bar{(Y)}}_{p_{C}^{+} + p_{C}^{+}}^{p} | \\ \geq | A | {γ_{p_{C}^{+} + p_{D}^{+}} (X) + γ_{p_{C}^{+} + p_{D}^{+}} (Y)} \\ - ρ_{p_{C}^{+} + p_{D}^{+}}^{p} (X \cap Y) | {\bar{(X)}}_{p_{C}^{+} + p_{D}^{+}}^{p} \cap {\bar{(Y)}}_{p_{C}^{+} + p_{D}^{+}}^{p} | . \end{array}$

$\begin{array}{l} ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} \cup {\bar{(Y)}}_{P_{C}^{-} + P_{D}^{-}}^{p} | \\ \geq | A | {γ_{P_{C}^{-} + P_{D}^{-}} (X) + γ_{P_{C}^{-} + P_{D}^{-}} (Y)} \\ - ρ_{P_{C}^{-} + P_{D}^{-}}^{p} (X \cap Y) | {\bar{(X)}}_{P_{C}^{-} + P_{D}^{-}}^{p} \cap {\bar{(Y)}}_{P_{C}^{-} + P_{D}^{-}}^{p} | . \end{array}$

Proof. It is analogous to Theorem 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 dominance multi-granulation approximations.

5 Proposed algorithm for conflict analysis model

Input:Information system $S : = (A, C, D, E);$

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

Step 2: If ${\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{p} = {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{p}$ then $δ_{k} : = {\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{p} or δ_{k} : = {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{p}$ otherwise $δ_{k} = {\underline{(X_{k})}}_{P_{C}^{+} + P_{D}^{+}}^{p} - {\underline{(X_{k})}}_{P_{C}^{-} + P_{D}^{-}}^{p};$

Step 3: If $δ : = ⋂_{k : = 1}^{| D |} δ_{k} \neq \emptyset;$

go to output.

Otherwise,

|D| = |D| - 1;

go back to step 3.

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

Order of complexity for above algorithm is O (4n + i (k - i)).

Example 3. (Continued from Example 1) Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Applying the above algorithm to select feasible alternative(s) for feasible consensus strategy, we have Table 6 and Table 7.

Hence δ: = { a₇ }. Thus 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.

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

Theorem 9. Let $S : = (A, C, D, E)$ be a multi attributes with multi decisions information system. Then for any subset X of A, we have

${\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} \subseteq {\underline{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o};$

${\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{o} \subseteq {\bar{(X)}}_{P_{C}^{+} + P_{D}^{+}}^{p} .$

Proof. Follows from the respective definitions.

6 Conclusion

In this paper we have developed a feasible consensus strategy for labor-management negotiations conflict analysis problem based on optimistic approximations. Further, we present another new idea for the development of feasible consensus strategy in labor-management negotiations conflict analysis problem based on dominance multi-granulation rough sets called pessimistic rough sets. In addition, this paper aims to present several uncertainty measures, such as approximate precision, rough degree, approximate quality and their mutual relationships are discussed. At the end, it is seen that the proposed technique works even much better when it is compared with Sun’s techniques (see Table 8).

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New types of dominance based multi-granulation rough sets and their applications in conflict analysis problems

Abstract

Keywords

1 Introduction

2 Problem statement

Table 1 Multi-criteria and multi-decision information system

3.1 Deja conflict analysis model

3.2 Sun and Ma conflict analysis model

3.3 Sun and Ma’s Algorithm

4 Multi-granulation rough sets based on dominance relation

4.1 Proposed algorithm for conflict analysis model

5 Proposed algorithm for conflict analysis model

6 Conclusion

References

Table 1
Multi-criteria and multi-decision information system