A more efficient conflict analysis based on soft preference relation

Abstract

Conflicts and disagreements are associated with issues related to living beings in particular with humans. Certain steps towards resolving these issues help to minimize conflict. Very few researchers paid their attention towards this important topic to develop some techniques which are based on mathematical methods. Rough set theory as a new and powerful mathematical tool to handle uncertainty in decision making problems was used to study conflict analysis and decision making. Afterwards the Pawlak conflict analysis model was established. Subsequently Deja put forward some questions which are not answered by Pawlak conflict analysis model. In the present paper firstly, we introduce the notions of soft preference relation and soft dominance relation and analyzed the Middle East conflict. Secondly, we answered the questions posed by Deja. Thirdly, two new techniques of reduction of parameters/attributes are introduced and applied to the Middle East conflict.

Keywords

Conflict analysis prefernce relation soft preference relation

1 Introduction

Decision making is the most important part of selection when we have to choose one from two or more alternatives on the basis of information related to them. Generally, it is very difficult to attain complete knowledge about all the alternatives. This lack of knowledge is the major cause of uncertainty about them. There are multifarious approaches in decision making problems which have ability to handle uncertainty. These are based on fuzzy set theory [33], rough set theory [25], vague set theory [15], soft set theory [21] and hesitant fuzzy sets [29]. Decision making is one of the key components to achieve objectives in many areas, particularly in a field which obligates analyzing the conflict.

Every one encounters conflict in every day life. These are, no doubt, one of the most characteristic attributes of human nature. Their study is of utmost importance both practically and theoretically. Its 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 [23]. 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 two people have a difference of opinions. In 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 in conflict situations [13].

Applications of rough set theory in case of conflict analysis is introduced by Pawlak in [27]. He introduced 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 [13] 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?”

Pawlak conflict analysis model can be criticized for the following two reasons, one is that, it cannot point out the intrinsic causes of the conflict. That is, what issues are focused by every agent and have different attitudes in a conflict. Another flaw is that, it cannot find a feasible consensus strategy for solving a conflict, that is, the optimal strategy which satisfies all agents or a possible sub-optimal feasible consensus strategy which satisfies the agents as much as possible.

In [31] Sun et al. developed a conflict analysis model based on rough set theory over two universes to overcome the above mentioned shortcomings in Pawlak conflict analysis. But in their proposals, still there are many areas for critics, for example, development of proper and feasible consensus strategy is missing.

Molodtsov’s soft set theory [21] 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 limitation 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. introduced several algebraic operations in soft set theory and examined their basic properties [18]. Ali et al. [1] proposed several new operations in soft set theory to further consolidate the algebraic basis of soft sets. Ali initiated soft equivalence relation over a universe of discourse which gives rise to a classical information system and generates an approximation space in Pawlak sense [4]. Applications of soft sets (hybrid soft set) in decision making problems have been studied by many authors in different contexts (see [5 , 34–39]). Many decision making problems are characterized by the ranking of objects according to a set of criteria with predefined preference ordered decision classes, such as credit approval, stock risk estimation, university ranking, teaching evaluation, etc. (see [8 , 22]). In the present paper, we developed a new conflict analysis model which is based on soft preference relation and soft dominance relation to study the Middle East conflict situation. Further we answered different questions raised by various authors in an affirmative way. A general algorithm for conflict problem is proposed and examined that our newly developed model is more efficient than the existing techniques. We also introduced two new techniques of reduction of parameters/attributes to study subtilizely the Middle East conflict.

The remainder of this paper is organized as follows: Section 2 briefly reviews the existing conflict analysis models. Section 3 present the concepts of soft preference relation and conflict analysis based on this relation. Section 4 discusses the algorithm for classification of agents. Section 5 mainly focusses on the algorithm for selection of issues. In the last section core cause of the conflict analysis is presented.

2 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.

2.1 Pawlak conflict analysis model

Pawlak introduced the conflict analysis model based on rough set theory in [27]. In Pawlak’s conflict analysis model, relation of each agent to the issue is shown in the form of a table in which the agents are represented by rows and the issues by columns. The value assigned to each agent corresponding to an issue is from the set { - , 0, + } , where - means that the agent is against, 0 shows the neutrality of the agent and + depicts the favorablity of the agent towards a certain issue. Further, Pawlak in [27] discussed the Middle East conflict analysis problem. This conflict analysis problem consists of six agents and issues. The relationship of each agent to an issue is shown in Table 1.

Table 1
Middle East conflict table

U∖V a ₁ a ₂ a ₃ a ₄ a ₅

u ₁ - + + + +

u ₂ + 0 - - -

u ₃ + - - - 0

u ₄ 0 - - 0 -

u ₅ + - - - -

u ₆ 0 + - 0 +

U∖V	a ₁	a ₂	a ₃	a ₄	a ₅
u ₁	-	+	+	+	+
u ₂	+	0	-	-	-
u ₃	+	-	-	-	0
u ₄	0	-	-	0	-
u ₅	+	-	-	-	-
u ₆	0	+	-	0	+

Let U : ={ u₁, u₂, u₃, u₄, u₅, u₆ } be the universe of six agents, where $\begin{matrix} u_{1} : Israel u_{2} : Egypt u_{3} : Palestine \\ u_{4} : Jordan u_{5} : Syria u_{6} : Saudi Arabia \end{matrix}$

Let V : ={ a₁, a₂, a₃, a₄, a₅ } be the universe of five issues in the conflict situation, where $\begin{matrix} a_{1} : Autonomous Palestinian state on the West \\ Bank and Gaza, \\ a_{2} : Israeli military outpost along the Jordan River, \\ a_{3} : Israeli retains East Jerusalem, \\ a_{4} : Israeli military outposts on the Golan Heights, \\ a_{5} : Arab countries grant citizenship to Palestinians \\ who choose to remain within their boarders . \end{matrix}$

In information system, the attitude of six nations of the Middle East region towards the above issues is presented. Pawlak introduced the discernibility matrix for the conflict situation in [27].

Let T : = (U, I), B ⊆ I. By a discernibility matrix of B in T, if T is understood, we will mean a n × n (n := |U|) matrix which is defined by: $γ (x, y) : = {a \in B ∣ a (x) \neg = a (y)} .$

So γ (x, y) is the set of all issues that discern agents x and y. The discernibility matrix for the conflict situation presented in Table 2 (a & b), where the entries of the table are the issues about which any two agents have different attitudes.

Table 2(a)

Discernibility matrix for Middle East conflict

	u ₁	u ₂
u ₁
u ₂	a_1,a₂, a₃, a₄, a₅
u ₃	a_1,a₂, a₃, a₄, a₅	a₂, a₅
u ₄	a_1,a₂, a₃, a₄, a₅	a_1,a₂, a₄
u ₅	a_1,a₂, a₃, a₄, a₅	a ₂
u ₆	a_1,a₃, a₄	a_1,a₂, a₄, a₅

Table 2(b)

Discernibility matrix for the Middle East conflict

	u ₃	u ₄	u ₅	u ₆
u ₁
u ₂
u ₃
u ₄	a_1,a₄, a₅
u ₅	a ₅	a₂, a₄
u ₆	a_1,a₂, a₄, a₅	a₂, a₅	a_1,a₂, a₄, a₅

From the above discernibility matrix of the conflict situation only the difference of attitudes between any two agents is obtained but no information is obtained about the preference of agents.

2.2 Deja conflict analysis model

Analysis of conflict described in [27] 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 [12], put forward three basic (below given) questions which are not answered by the Pawlak’s conflict analysis model:

What are the intrinsic reasons for the conflict?

How can a feasible consensus strategy be found?

Is it possible to satisfy all the agents?

2.3 Sun et al. conflict analysis model

Sun et al. in [31] tried to answer Deja’s questions and focussed to answer the first and second questions raised by Deja in [12]. They concentrated on two problems of “What are the intrinsic conflict reasons?” and “How can a feasible consensus strategy be found?” and tried to answer these with the help of rough set theory over two universes. Their model can be summarized as the following:

Let U be the universe of agents and V be the universe of issues. The intrinsic cause in a conflict situation means that what are focused by every agent and have difference attitudes in a conflict. Also for any subset $Y \in P (V)$ (where $P (V)$ denotes the set of all subsets on V), we call Y the feasible strategy for a conflict situation. That is, optimal strategy which satisfies all agents or a possible sub-optimal feasible consensus strategy which satisfies the agents as much as possible. Let U : ={ u₁, ... , u_m } be the universe of agents and V : ={ a₁, ... , a_n } be the universe of issues in the conflict situation, f : ={ f⁺, f^- } be the set valued mappings from U to $P (V),$ where $\begin{matrix} f^{+} : U \to P (V), \\ f^{+} (u) : = {a \in V ∣ f (u, a) : = +}, for all u \in U \end{matrix}$ and $\begin{matrix} f^{-} : U \to P (V), \\ f^{-} (u) : = {a \in V ∣ f (u, a) : = -}, for all u \in U . \end{matrix}$

The mapping f⁺ means the subset of issues of the universe V with which agent u of universe U agrees. The mapping f^- means the subset of issues of the universe V with which agent u of universe U disagrees. For any feasible strategy $Y \in P (V)$ on an issue of universe V, the approximations are: $\begin{matrix} {\underline{apr}}_{f}^{+} (Y) & : = & {u \in U ∣ f^{+} (u) \subseteq Y}, \\ {\underline{apr}}_{f}^{-} (Y) & : = & {u \in U ∣ f^{-} (u) \subseteq Y}, \\ {\bar{apr}}_{f}^{+} (Y) & : = & {u \in U ∣ f^{+} (u) \cap Y \neg = \emptyset}, \\ {\bar{apr}}_{f}^{-} (Y) & : = & {u \in U ∣ f^{-} (u) \cap Y \neg = \emptyset} . \end{matrix}$

Example 1. Consider the Middle East conflict as in Table 1. Let Y : = { a₂, a₃, a₅ } ⊆ V. Then ${\underline{apr}}_{f}^{+} (Y) : = {u_{6}}, {\bar{apr}}_{f}^{+} (Y) : = {u_{1}, u_{6}}, {\underline{apr}}_{f}^{-} (Y) : = {u_{4}, u_{6}}$ and ${\bar{apr}}_{f}^{-} (Y) : = {u_{2}, u_{3}, u_{4}, u_{5}, u_{6}} .$ According to Sun et al. [31]: $\begin{matrix} Agreement subset : = R_{f}^{+} (Y) : \\ = {\underline{apr}}_{f}^{+} (Y) - {\underline{apr}}_{f}^{-} (Y) : = \emptyset, \\ Disagreement subset : = R_{f}^{-} (Y) : \\ = {\underline{apr}}_{f}^{-} (Y) - {\underline{apr}}_{f}^{+} (Y) : = {u_{4}}, \\ Neutral subset : = U - (R_{f}^{+} (Y) \cup R_{f}^{-} (Y)) : \\ = {u_{1}, u_{2}, u_{3}, u_{5}, u_{6}} . \end{matrix}$

It is observed that:

(i) For the feasible strategy Y : = { a₂, a₃, a₅ } ∈ P (V) using Sun et al. technique, Agreement subset: = ∅ , Disagreement subset : ={ u₄ } and Neutral subset: = { u₁, u₂, u₃, u₅, u₆ }. Since in the information system (Table 1) the values for u₄ in the columns of a₂, a₃ and a₅ are in disagreement, that is “-”. Also the case of u₅ is not different from u₄, but in [31] it is treated as neutral, so the agents of the same behavior have been put in different coalitions which is causing a confusion. (ii) For the feasible strategy Y : = { a₃ } ∈ P (V), Agreement subset: = ∅ , Disagreement subset: ={ u₆ } and Neutral subset: = { u₁, u₂, u₃, u₄, u₅ }. Since in the information system (Table 1) the attitude of u₆ in the column of a₃ is in disagreement that is “-”. Also the cases of u₂, u₃, u₄ and u₅ are not different from u₆, but according to [31], they should be in neutral, so the agents of the same character grouped in different coalitions which is an infirmity of the Sun et al technique. (iii) For another strategy Y : = { a₁, a₃ } ∈ P (V), the agreement subset: = { u₂, u₃, u₅ } , Disagreement subset: ={ u₁, u₆ } and Neutral subset: = { u₄ }. Now in the information system (Table 1) the behavior of u₄ and u₆ in the columns of a₁ and a₃ are same. But by technique of [31], these agents of the same character can be shown in different coalitions which is an other infirmity of the Sun et al. technique. (iv) A snage noted in [31] is that it does not show how much an agent is in favor or against the issue(s). (v) The technique of [31] declared the conflict problem as undecided one, which is another flaw of this technique as seen below: Let V : ={ a₁, a₂, a₃, a₄, a₅ } the set of all issues. Then $\begin{matrix} {\underline{apr}}_{f}^{+} (V) & : = & {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}}, \\ {\underline{apr}}_{f}^{-} (V) & : = & {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}} . \end{matrix}$

Agreement subset: = ∅ , Disagreement subset : = ∅ , Neutral subset: = { u₁, u₂, u₃, u₄, u₅, u₆ }. So for the feasible consensus strategy V : ={ a₁, a₂, a₃, a₄, a₅ }, the set of all issues of the conflict situation, there is no agreement subset and disagreement subset for V, the attitudes of all agents u₁, u₂, u₃, u₄, u₅ and u₆ are neutral for the feasible consensus strategy V.

3 Conflict analysis based on soft preference relation

Inspired by the existing studies on conflict analysis based on Pawlak rough set theory as well as Sun et al. conflict analysis based on rough set theory over two universes, we stipulate a new conflict analysis model based on soft preference relation that will help to ameliorate the above limitations in the existing approaches in the literature.

In order to overcome the above mentioned shortcomings, a new conflict analysis model with the help of soft set theory is developed which is free from all such weaknesses and work more efficaciously.

3.1 Basic properties of binary relations

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

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

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

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

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

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

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

Complete when x ¬ = y, (x, y) ∈ R or (y, x) ∈ R for all x, y ∈ U.

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

Preorder relation when it is reflexive and transitive.

Equivalence relation when it is reflexive, symmetric and transitive.

Let U be a set of agents and V be a set of issues. Let ⪰_a be an out ranking relation on U with respect to issue a ∈ V such that x ⪰ _ay means “x is at least good as y with respect to criterion a”. Suppose that ⪰_a is a complete preorder, it is strongly complete (which means that for each x, y ∈ U at least one of x ⪰ _ay and y ⪰ _ax exists) and transitive binary relation defined on U. Thus x and y are always comparable with respect to criterion a ∈ V. We say that object x E-dominate object y with respect to E ⊆ V (denoted by xD_Ey) if x ⪰ _ay for all a ∈ E. Since the intersection of complete preorders is a partial preorder, D_E : = ⋂ _a∈E ⪰ _a, the dominance relation D_E is a partial preorder.

Definition 2. [21] A pair $R : = (F, E)$ is called a soft set over U, where E is a subset of parameters of V and $F : E \to P (U)$ is a set valued mapping.

Definition 3. [4] If $F : E \to P (U \times U)$ is a mapping from a subset of parameters E to the power set of U × U, then the soft set (F, E) over U × U is called a soft binary relation over U. A soft binary relation (F, E) over a set U is called a soft equivalence relation over U if ∅ ¬ = F (a) is an equivalence relation over U if for all a ∈ E.

Definition 4. Let $S : = (U, V, \tilde{f})$ be a conflict situation and $F : E \to P (U \times U)$ be a set valued mapping from E to $P (U \times U),$ where E ⊆ V denotes the set of parameters or attributes and $P (U \times U)$ the set of all subsets of U × U. If ∅ ¬ = F (a) is a preference relation for all a ∈ E, then (F, E) will be called a soft preference relation, where $F (a) : = {u_{i} ⪰ u_{j} ∣ \tilde{f} (u_{i}, a) \geq \tilde{f} (u_{j}, a), u_{i}, u_{j} \in U}$ and $\tilde{f} : U \times V \to {-, 0, +}$ such that $\tilde{f} (u, a) : = s,$ where s∈ { - , 0, + } and u ∈ U, a ∈ V. In fact $\tilde{f}$ is more general than Sun et al. f in [31]. Throughout the paper we will use (u_i, u_j) for u_i ⪰ u_j. Preference relations are efficient tools to address decision making problems that only require the decision makers to compare one pair of objects at a time. In information system (Table 1) favorability (+) is preferred by neutrality (0) and neutrality (0) is preferred by against (-).

Definition 5. (F, E) is a soft preference relation over U, then there is dominance relation associated with which can be denoted by Dom (F, E) and is defined as follows: $Dom (F, E) : = ⋂_{a \in E} F (a) .$ This type of relation can be represented in a square table which is very useful in information system. In the square table if (u_i, u_j) ∈ Dom (F, E), then (u_i, u_j) : =1, otherwise (u_i, u_j) : =0 where u_i, u_j ∈ U. We develop a more efficient algorithm for Middle East conflict analysis in the following section.

4 Algorithm for Middle East conflict analysis

Input information table $S : = (U, V, \tilde{f}),$ where number of elements in U and V are m and n respectively.

Construct soft preference relation (F, V)// according to Definition 4.

Construct soft dominance relation Dom (F, V) and its square table // according to Definition 5.

Sum up the scores of agents row wise and column wise in the square table.

Find the difference of row wise sum and column wise sum.

Based on Step 5, we may have positive, negative and non-negative numbers showing agreement, disagreement and neutral subsets respectively.

Using soft preference relation we give affirmative response to the aforesaid drawbacks of Sun et al. technique:

Example 6. Consider the Information Table 1 for the Middle East conflict analysis.

(i) If E : = { a₂, a₃, a₅ } ⊆ V, then a soft preference relation F : E → P (U × U) which is defined by $\begin{matrix} F (a_{2}) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), (u_{3}, u_{3}), \\ (u_{4}, u_{4}), (u_{5}, u_{5}), (u_{6}, u_{6}), \\ (u_{1}, u_{2}), (u_{1}, u_{3}), (u_{1}, u_{4}), \\ (u_{1}, u_{5}), (u_{1}, u_{6}) (u_{2}, u_{3}), \\ (u_{2}, u_{4}), (u_{2}, u_{5}), (u_{3}, u_{4}), \\ (u_{3}, u_{5}), (u_{4}, u_{3}), (u_{4}, u_{5}), \\ (u_{5}, u_{3}), (u_{5}, u_{4}), (u_{6}, u_{1}), \\ (u_{6}, u_{2}), (u_{6}, u_{3}), (u_{6}, u_{4}), \\ (u_{6}, u_{5}) \end{matrix}}, \\ F (a_{3}) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), (u_{3}, u_{3}), \\ (u_{4}, u_{4}), (u_{5}, u_{5}) (u_{6}, u_{6}), \\ (u_{1}, u_{2}), (u_{1}, u_{3}), (u_{1}, u_{4}), \\ (u_{1}, u_{5}), (u_{1}, u_{6}) (u_{2}, u_{3}), \\ (u_{2}, u_{4}), (u_{2}, u_{5}), (u_{2}, u_{6}) \\ , (u_{3}, u_{2}) . (u_{3}, u_{4}) (u_{3}, u_{5}), \\ (u_{3}, u_{6}), (u_{4}, u_{2}) (u_{4}, u_{3}), \\ (u_{4}, u_{5}), (u_{4}, u_{6}), (u_{5}, u_{2}), \\ (u_{5}, u_{3}), (u_{5}, u_{4}), (u_{5}, u_{6}), \\ (u_{6}, u_{2}), (u_{6}, u_{2}), (u_{6}, u_{4}), \\ (u_{6}, u_{5}) \end{matrix}}, \\ F (a_{5}) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), (u_{3}, u_{3}), \\ (u_{4}, u_{4}), (u_{5}, u_{5}) (u_{6}, u_{6}), \\ (u_{1}, u_{2}), (u_{1}, u_{3}), (u_{1}, u_{4}), \\ (u_{1}, u_{5}), (u_{1}, u_{6}) (u_{2}, u_{4}), \\ (u_{2}, u_{5}), (u_{3}, u_{2}), (u_{3}, u_{4}), \\ (u_{3}, u_{5}), (u_{4}, u_{2}), (u_{4}, u_{5}), \\ (u_{5}, u_{2}), (u_{5}, u_{4}) (u_{6}, u_{1}), \\ (u_{6}, u_{2}), (u_{6}, u_{3}), (u_{6}, u_{4}), \\ (u_{6}, u_{5}) \end{matrix}} . \end{matrix}$

The soft dominance relation is given by $Dom (F, E) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), (u_{3}, u_{3}), \\ (u_{4}, u_{4}), (u_{5}, u_{5}), (u_{6}, u_{6}), \\ (u_{1}, u_{2}), (u_{1}, u_{3}), (u_{1}, u_{4}), \\ (u_{1}, u_{5}), (u_{1}, u_{6}), (u_{2}, u_{4}), \\ (u_{2}, u_{5}), (u_{3}, u_{4}), (u_{3}, u_{5}), \\ (u_{4}, u_{5}), (u_{5}, u_{4}) (u_{6}, u_{2}), \\ (u_{6}, u_{3}), (u_{6}, u_{4}), (u_{6}, u_{5}) \end{matrix}} .$

Construct the square table for Dom (F, E) as follows:

In Table 3, where C. S represent the column wise sum and R. S denote the row wise sum respectively.

Table 3
Square table for Dom (F,E)

u ₁ u ₂ u ₃ u ₄ u ₅ u ₆ R. S

u ₁ 1 1 1 1 1 1 6

u ₂ 0 1 0 1 1 0 3

u ₃ 0 0 1 1 1 0 3

u ₄ 0 0 0 1 1 0 2

u ₅ 0 0 0 1 1 0 2

u ₆ 0 1 1 1 1 1 5

C. S 1 3 3 6 6 2

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	R. S
u ₁	1	1	1	1	1	1	6
u ₂	0	1	0	1	1	0	3
u ₃	0	0	1	1	1	0	3
u ₄	0	0	0	1	1	0	2
u ₅	0	0	0	1	1	0	2
u ₆	0	1	1	1	1	1	5
C. S	1	3	3	6	6	2

Table 4

Table4

U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
R. S - C. S	5	0	0	-4	-4	3

By using the aforesaid algorithm, from Table 4, the agreement subset: = { u₁, u₆ } , disagreement subset: ={ u₄, u₅ } and neutral subset: = { u₂, u₃ }. So we say that for any feasible strategy E : = { a₂, a₃, a₅ } ⊆ V of the conflict situation, the agents u₄ and u₅ disagree with E : = { a₂, a₃, a₅ } , the agents u₁ and u₆ agree with E : ={ a₂, a₃, a₅ } and the attitude of the agents u₂ and u₃ are neutral for feasible strategy E : = { a₂, a₃, a₅ }. It is more pertinent to distinguish the agreement and disagreement subset of the agents than to distinguish neutral subset attitudes of the conflict situation, so that to provide some information for decision makers. Hence we disclose information about agreement and disagreement for any feasible strategy in the conflict situation by agreement subset and disagreement subset. Also we ascertain the feasible consensus strategy E : = { a₂, a₃, a₅ } ⊆ V for conflict situation by choosing the maximum cardinality of the agreement subset. Thus we have answered the question “How can a feasible consensus strategy be found?” by using the agreement subset. (ii) Let for the feasible strategy E : = { a₃ } ⊆ V, according to the proposed technique: agreement subset: = { u₁ } , disagreement subset: ={ u₂, u₃, u₄, u₅, u₆ } and neutral subset: = ∅. (iii) For feasible strategy E : = { a₁, a₃ } ⊆ V, according to soft preference technique: agreement subset: = { u₂, u₃, u₅ } , disagreement subset: ={ u₄, u₆ } and neutral subset: = { u₁ }. (iv) Proposed technique also gives/shows how much agent(s) is/are in favour/against of some issue(s). (v) Let for feasible strategy V : = {a₁, a₂, a₃, a₄, a₅} that is the set of all issues or criterion. Then by our technique, agreement subset: = { u₂, u₃, u₆ } , disagreement subset: ={ u₄, u₅ } and neutral subset: ={ u₁ } depicting the real spirit of the conflict situation, while in this case Sun et al. technique gives that the conflict table under consideration is neutral. Next for any issue a_i ∈ V, we can easily obtain attitude information for every agent with respect to issue a_i ∈ V by using F (a_i) which is a preference relation, since F (a_i) is more informative than the [31] two set valued mappings. Thus we answered the question “What are the intrinsic conflict reasons?” by using F (a_i) where a_i ∈ V. In conclusion we replied in a best way to Deja’s questions.

Now we present another methodology for the feasible consensus strategy of the conflict situation. The other representation of the Middle East conflict Information table is given below:

According to Sun et al. [31]:

Let g ={ g⁺, g^- } be a set valued mapping from V to U, where

$g^{+} : V \to P (U), g^{+} (a) : = {u \in U ∣ g (a, u) : = +}$ for all a ∈ V,

$g^{-} : V \to P (U),$ g^- (a) : = {u ∈ U ∣ g (a, u) : = -} for all a ∈ V.

Now for any combination of agents $X \in P (U)$ in a conflict situation, the approximations are defined as follows: $\begin{matrix} {\underline{apr}}_{g}^{+} (X) & : = & {a \in V ∣ g^{+} (a) \subseteq X}, \\ {\underline{apr}}_{g}^{-} (X) & : = & {a \in V ∣ g^{-} (a) \subseteq X}, \\ {\bar{apr}}_{g}^{+} (X) & : = & {a \in V ∣ g^{+} (a) \cap X \neg = \emptyset}, \\ {\bar{apr}}_{g}^{-} (X) & : = & {a \in V ∣ g^{-} (a) \cap X \neg = \emptyset} . \end{matrix}$

Example 7. Consider the Middle East conflict analysis as in Table 5. Let X : = { u₁, u₂, u₃, u₅ } ⊆ U. Then ${\underline{apr}}_{g}^{+} (X) : = {a_{1}, a_{3}, a_{4}}$ and ${\underline{apr}}_{g}^{-} (X) : = {a_{1}, a_{4}} .$ According to [31]: Agreement subset: = { a₃ } , Disagreement subset : = ∅ , Neutral subset: = { a₁, a₂, a₄, a₅ }. It is noted that for X : = { u₁, u₂, u₃, u₅ } ⊆ U in the conflict analysis of Table 5, a₃ and a₄ behaves alike, wherein they have been placed in different alliances. Now, we propose a new and general algorithm for conflict analysis model which is immaculate from above flaw.

Table 5

Middle East conflict Table

V∖U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
a ₁	-	+	+	0	+	0
a ₂	+	0	-	-	-	+
a ₃	+	-	-	-	-	-
a ₄	+	-	-	0	-	0
a ₅	+	-	0	-	-	+

Definition 8. Let $S : = (U, V, \tilde{g})$ be a conflict situation and $G : I \to P (V \times V)$ be a set valued mapping from I to $P (V \times V),$ where I ⊆ U denotes the set of parameters or agents and $P (V \times V)$ the set of all subsets of V × V. If ∅ ¬ = G (u) is a preference relation for all u ∈ I then (G, I) is called a soft preference relation, where

$G (u) : = {a_{i} ⪰ a_{j} ∣ \tilde{g} (a_{i}, u) \geq \tilde{g} (a_{j}, u), a_{i}, a_{j} \in V}$ and

$\tilde{g} : V \times U \to {-, 0, +}$ such that $\tilde{g} (a, u) : = w,$ where w∈ { - , 0, + } and u ∈ U, a ∈ V. In fact $\tilde{g}$ is more general than Sun et al. g in [31].

Definition 9. (G, I) is a soft preference relation over V, then there is dominance relation associated with which can be denoted by Dom (G, I) and is defined as:

$Dom (G, I) : = ⋂_{u \in I} G (u) .$ This type of relation can be represented in a square table which is very useful in information system. In the square table if (a_i, a_j) ∈ Dom (G, I), then (a_i, a_j) : =1, otherwise (a_i, a_j) : =0 where a_i, a_j ∈ V. We develop a new and more efficient algorithm for Middle East conflict analysis in the following section.

5 Algorithm for Middle East conflict analysis

Input information table $S : = (U, V, \tilde{g}),$ where number of elements in U and V are m and n respectively.

Construct soft preference relation (G, I)// according to Definition 8.

Construct soft dominance relation Dom (G, I) and its square table // according to Definition 9.

Sum up the scores of agents row wise and column wise in the square table.

Find the difference of row wise sum and column wise sum.

Based on Step 5, we may have positive, negative and non-negative numbers showing agreement, disagreement and neutral subsets respectively.

Example 10. Consider the Middle East conflict analysis as in Table 5. Let I : = { u₁, u₂, u₃, u₅ } ⊆ U. Then by our second algorithm Agreement subset: = { a₂, a₅ } , Disagreement subset : = { a₃, a₄ } , Neutral subset: = { a₁ }. Thus for I : = {u₁, u₂, u₃, u₅} ⊆ U in a conflict situation, all agents in I agree on the feasible strategy {a₂, a₅} of universe V. Consequently we found out the feasible consensus strategy {a₂, a₅ } ⊆ V for conflict situation by choosing the maximum cardinality of the agreement subset. Thus the example of the Middle East conflict analysis does not necessarily reflect the present day situation in this region but is used here only as an illustration of the basic ideas consider in this paper.

6 Core cause of conflict analysis

Selecting pre-defined subsets of attributes is a very useful and interesting idea [11] which has been discussed in numerous papers. In the classic rough set approach [26], where reduction of attributes remains one of the main issues, the reduced subsets of attributes, referred to as reducts, have been particularly intensively studied, in both classic [24] and extended forms [7, 17]. The reducts, introduced in classic rough set approach, remain an important and inspiring notion as they involve the idea of finding attribute subsets that are minimal with regard to inclusion and guarantee the same quality of approximation as the whole set of attributes [26]. In [2] it has been seen that there is a very close relationship between soft sets and rough sets. So it is natural to ask, “Can we develop a method of reduction of parameters for soft sets as we do in case of rough sets for attribute reduction without loosing important information?” The present paper aims to initiate another concept of reduction of parameters/attributes in Middle East conflict analysis based on soft dominance relation to categories agents in the classes of neutrality, agreement and disagreement. For more clarification we present:

Example 11. Consider the Information Table 1 for the Middle East conflict analysis. If E : = { a₁, a₂, a₃, a₄, a₅ } ⊆ V, then (F, E) where F : E → P (U × U) is a soft preference relation. The soft dominance relation is given by: $Dom (F, E) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), \\ (u_{3}, u_{3}), (u_{4}, u_{4}), \\ (u_{5}, u_{5}), (u_{6}, u_{6}), \\ (u_{2}, u_{5}), (u_{3}, u_{5}), \\ (u_{6}, u_{4}) \end{matrix}} .$

Construct the square table for Dom (F, E) as follows:

From Tables 6 and 7, the agreement subset: = {u₂, u₃, u₆}, disagreement subset: = {u₄, u₅} and neutral subset: = {u₁}. So for any feasible strategy. E : = { a₁, a₂, a₃, a₄, a₅ } ⊆ V of the conflict situation, the agents u₄ and u₅ disagree with E : = { a₁, a₂, a₃, a₄, a₅ } , the agents u₂, u₃ and u₆ agree with E : ={ a₁, a₂, a₃, a₄, a₅ } and the attitude of the agent u₁ is neutral for feasible strategy E : = { a₁, a₂, a₃, a₄, a₅ }. One of the basic question is that ‘which parameter(s)/attribute(s) plays dominant role in the (Middle East) conflict problem’? That is which is/are the core cause of the conflict. So far there is no such technique which can find the core cause of the conflict in a conflict problem. We try to give affirmative answer the aforesaid question by a new method of reduction of the parameter(s)/attribute(s) using soft dominance relation to classify the agents on the bases of their behavior as neutral, agreement and disagreement. If we eliminate a₁ from V and construct soft preference relation for E₁ : = { a₂, a₃, a₄, a₅ } ⊆ V which is (F, E₁) where F : E₁ → P (U × U). Then the soft dominance relation for (F, E₁) is defined by: $Dom (F, E_{1}) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), \\ (u_{3}, u_{3}), (u_{4}, u_{4}), \\ (u_{5}, u_{5}), (u_{6}, u_{6}), \\ (u_{1}, u_{2}), (u_{1}, u_{3}), \\ (u_{1}, u_{4}), (u_{1}, u_{5}), \\ (u_{1}, u_{6}), (u_{2}, u_{5}), \\ (u_{3}, u_{5}), (u_{4}, u_{5}), \\ (u_{6}, u_{2}), (u_{6}, u_{3}), \\ (u_{6}, u_{4}), (u_{6}, u_{5}) \end{matrix}} .$

Table 6
Square table for Dom (F,E)

u ₁ u ₂ u ₃ u ₄ u ₅ u ₆ R. S

u ₁ 1 0 0 0 0 0 1

u ₂ 0 1 0 0 1 0 2

u ₃ 0 0 1 0 1 0 2

u ₄ 0 0 0 1 0 0 1

u ₅ 0 0 0 0 1 0 1

u ₆ 0 0 0 1 0 1 2

C. S 1 1 1 2 3 1

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	R. S
u ₁	1	0	0	0	0	0	1
u ₂	0	1	0	0	1	0	2
u ₃	0	0	1	0	1	0	2
u ₄	0	0	0	1	0	0	1
u ₅	0	0	0	0	1	0	1
u ₆	0	0	0	1	0	1	2
C. S	1	1	1	2	3	1

Table 7

U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
R. S - C. S	0	1	1	-1	-2	1

Construct the square table for Dom (F, E₁) as follows:

From Tables 8 and 9, the agreement subset: = {u₁, u₆}, disagreement subset: = {u₂, u₃, u₄, u₅} and neutral subset: = ∅. We see that the elimination of a₁, disturbs the classification (agreement, disagreement, neutral) of the agents in Table 7. Therefore a₁ is the core attribute of the Middle East conflict situation. If a₂ is reduced from V, then the soft preference relation for E₁ : = { a₁, a₃, a₄, a₅ } ⊆ V is (F, E₂) where F : E₂ → P (U × U), and the soft dominance relation for (F, E₂) is given by: $Dom (F, E_{2}) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), \\ (u_{3}, u_{3}), (u_{4}, u_{4}), \\ (u_{5}, u_{5}), (u_{6}, u_{6}), \\ (u_{2}, u_{5}), (u_{3}, u_{2}), \\ (u_{3}, u_{5}), (u_{5}, u_{2}), \\ (u_{6}, u_{4}) \end{matrix}} .$

Table 8

Square table for Dom (F, E₁)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	R. S
u ₁	1	1	1	1	1	1	6
u ₂	0	1	0	0	1	0	2
u ₃	0	0	1	0	1	0	2
u ₄	0	0	0	1	1	0	2
u ₅	0	0	0	0	1	0	1
u ₆	0	1	1	1	1	1	5
C. S	1	3	3	3	6	2

Table 9

U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
R. S - C. S	5	-1	-1	-1	-5	3

Construct the square table for Dom (F, E₂) as follows:

From Tables 10 and 11, the agreement subset:={u₃, u₆}, disagreement subset:={u₂, u₄, u₅} and neutral subset:={u₁}. We see that the elimination of a₂, makes the classification (agreement, disagreement, neutral) of the agents different from that in Table 7. Therefore a₂ is another core attribute of the Middle East conflict situation. Similarly if a₃ is reduced from V i.e E : = {a₁, a₂, a₄, a₅} ⊆ V, then (F, E₃) is a soft preference relation, where F : E₃ → P (U × U). The soft dominance relation is given by: $Dom (F, E_{3}) : = {\begin{matrix} (u_{1}, u_{1}), (u_{2}, u_{2}), \\ (u_{3}, u_{3}), (u_{4}, u_{4}), \\ (u_{5}, u_{5}), (u_{6}, u_{6}), \\ (u_{2}, u_{5}), (u_{3}, u_{5}), \\ (u_{6}, u_{4}) \end{matrix}} .$

Table 10

Square table for Dom (F, E₂)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	R. S
u ₁	1	0	0	0	0	0	1
u ₂	0	1	0	0	1	0	2
u ₃	0	1	1	0	1	0	3
u ₄	0	0	0	1	0	0	1
u ₅	0	1	0	0	1	0	2
u ₆	0	0	0	1	0	1	2
C. S	1	3	1	2	3	1

Table 11

U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
R. S - C. S	0	-1	2	-1	-1	1

Construct the square table for Dom (F, E₃) as follows:

From Tables 12 and 13, the agreement subset:={u₂, u₃, u₆}, disagreement subset: ={u₄, u₅} and neutral subset: ={u₁}. Here the reducing of a₃ does not affect the classification (agreement, disagreement, neutral) ability of the agents in Table 7. So a₃ is dispensable in Table 1. Continuing in a similar fashion we find a set of core parameters/attributes {a₁, a₂, a₅}. Thus the elimination of a₃ and a₄ do not disturb the classification ability of the agents in Table 1. The Table 14 provides us the same classification with minimum parameters/attributes.

Table 12

Square table for Dom (F, E)

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	R. S
u ₁	1	0	0	0	0	0	1
u ₂	0	1	0	0	1	0	2
u ₃	0	0	1	0	1	0	2
u ₄	0	0	0	1	0	0	1
u ₅	0	0	0	0	1	0	1
u ₆	0	0	0	1	0	1	2
C. S	1	1	1	2	3	1

Table 13

U	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆
R. S - C. S	0	1	1	-1	-2	1

Table 14

Middle East conflict

U∖V	a ₁	a ₂	a ₅
u ₁	-	+	+
u ₂	+	0	-
u ₃	+	-	0
u ₄	0	-	-
u ₅	+	-	-
u ₆	0	+	+

Further, it is natural to ask which agent plays a dominant role in the information system Table 5. We give affirmative response to this question by using the aforementioned technique and thus we see that {u₁, u₂, u₃, u₄} play leading role in the Middle East conflict analysis. In sum {u₁, u₂, u₃, u₄} give rise despotically to core cause of conflict and {u₅, u₆} play dispensable role in the Middle East conflict problem. Another representation of Table 5 with minimum number of agents is given in Table 15.

Table 15

Middle East conflict

V∖U	u ₁	u ₂	u ₃	u ₄
a ₁	-	+	+	0
a ₂	+	0	-	-
a ₃	+	-	-	-
a ₄	+	-	-	0
a ₅	+	-	0	-

6.1 Some advantages of proposed algorithms

Both the algorithms rightly classify the agents and criteria.

The proposed algorithms answer questions (i) and (ii) of Deja in a more better way than Sun et al.

These algorithms can be used in a convenient way in reduction of attributes (see Section 6).

Another advantage of the proposed algorithms is that they can be used for ranking of feasible alternative more simply than other existing technique (for example TOPSIS method). Since every algorithm/technique has its own benefits and drawbacks. So far our proposed algorithms/techniques solve all the relevant existing problems.

Conclusion. In daily life, we are always faced with various decision making pursuits. Some decision are simple and some are complex. When one faces a complex problem with the conflicting criteria, an effective decision making method becomes helpful to make an acceptable and reasonable decision. This paper aims to discuss Middle East conflict. This model was studied initially by Pawlak in [27] with the help of rough set theory on the single universe and their drawbacks which are highlighted by Deja, where he put some questions related to conflict analysis. Sun et al. built a new conflict analysis model and tried to answer the Deja’s questions with help of rough set theory over two universes. But they have also some shortcomings which build the study of conflict analysis more ambiguous. In this paper, a new conflict analysis model to study the Middle East conflict analysis problem has been analyzed with the help of soft preference relation and soft dominance relation. Further, it has been tried to answer Deja’s questions in a positive manner. The proposed model can be extended to study other conflicts around the world. A general algorithm has been proposed in this paper. Thus the proposed conflict analysis model depicting the real spirit of the conflict situation. Apart of these, two new techniques of reduction of parameters/attributes are introduced and applied to the Middle East conflict.

Footnotes

Acknowledgments

The authors would like to thank all the anonymous reviewers and associate editor Professor Jianming Zhan for thorough reading of this manuscript and for the thoughtful comments and constructive suggestions, which helped to improve the earlier version of this manuscript.

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A more efficient conflict analysis based on soft preference relation

Abstract

Keywords

1 Introduction

2 Literature review

2.1 Pawlak conflict analysis model

Table 1 Middle East conflict table U∖V a 1 a 2 a 3 a 4 a 5 u 1 - + + + + u 2 + 0 - - - u 3 + - - - 0 u 4 0 - - 0 - u 5 + - - - - u 6 0 + - 0 +

2.3 Sun et al. conflict analysis model

3 Conflict analysis based on soft preference relation

3.1 Basic properties of binary relations

4 Algorithm for Middle East conflict analysis

Table 3 Square table for Dom (F,E) u 1 u 2 u 3 u 4 u 5 u 6 R. S u 1 1 1 1 1 1 1 6 u 2 0 1 0 1 1 0 3 u 3 0 0 1 1 1 0 3 u 4 0 0 0 1 1 0 2 u 5 0 0 0 1 1 0 2 u 6 0 1 1 1 1 1 5 C. S 1 3 3 6 6 2

6 Core cause of conflict analysis

Table 6 Square table for Dom (F,E) u 1 u 2 u 3 u 4 u 5 u 6 R. S u 1 1 0 0 0 0 0 1 u 2 0 1 0 0 1 0 2 u 3 0 0 1 0 1 0 2 u 4 0 0 0 1 0 0 1 u 5 0 0 0 0 1 0 1 u 6 0 0 0 1 0 1 2 C. S 1 1 1 2 3 1

Footnotes

Acknowledgments

References

Table 1
Middle East conflict table

U∖V a ₁ a ₂ a ₃ a ₄ a ₅

u ₁ - + + + +

u ₂ + 0 - - -

u ₃ + - - - 0

u ₄ 0 - - 0 -

u ₅ + - - - -

u ₆ 0 + - 0 +

Table 3
Square table for Dom (F,E)

u ₁ u ₂ u ₃ u ₄ u ₅ u ₆ R. S

u ₁ 1 1 1 1 1 1 6

u ₂ 0 1 0 1 1 0 3

u ₃ 0 0 1 1 1 0 3

u ₄ 0 0 0 1 1 0 2

u ₅ 0 0 0 1 1 0 2

u ₆ 0 1 1 1 1 1 5

C. S 1 3 3 6 6 2

Table 6
Square table for Dom (F,E)

u ₁ u ₂ u ₃ u ₄ u ₅ u ₆ R. S

u ₁ 1 0 0 0 0 0 1

u ₂ 0 1 0 0 1 0 2

u ₃ 0 0 1 0 1 0 2

u ₄ 0 0 0 1 0 0 1

u ₅ 0 0 0 0 1 0 1

u ₆ 0 0 0 1 0 1 2

C. S 1 1 1 2 3 1