## 3 Conflict Analysis Based on Soft Preference Relation  
Inspired by the existing studies on conflict analysis based on [PERSON]'s rough set theory, as well as [PERSON] et al. 's conflict analysis based on rough set theory over two universes, we propose a new conflict analysis model based on a soft preference relation that will help to improve 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 works more efficaciously. Let \(U\) be a set of agents and \(V\) be a set of issues. Let \(\succeq_{a}\) be an outranking relation on \(U\) with respect to issue \(a\in V\) such that \(x_{1}\succeq_{a}x_{2}\) means \"\(x_{1}\) is at least as good as \(x_{2}\) with respect to criterion \(a.\) Suppose that \(\succeq_{a}\) is a complete preorder, it is strongly complete (which means that for each \(x_{1},x_{2}\in U\) at least one of \(x_{1}\succeq_{a}x_{2}\) or \(x_{2}\succeq_{a}x_{1}\) exists) and a transitive binary relation defined on \(U.\) Thus \(x_{1}\) and \(x_{2}\) are always comparable with respect to criterion \(a\in V.\) We say that an object \(x_{1}\)\(E\)-dominates an object \(x_{2}\) with respect to \(E\subseteq V\) (denoted by \(x_{1}D_{E}x_{2}\)), if \(x_{1}\succeq_{a}x_{2}\) for all \(a\in E.\) Since the intersection of complete preorders is a partial preorder, \(D_{E}=\bigcap\limits_{a\in E}\succeq_{a}\), so the dominance relation \(D_{E}\) is a partial preorder. Definition 1: [5] A pair \(\Re=(F,E)\) is called a soft set over \(U\), where \(E\) is a subset of parameters of \(V\) and \(F:E\rightarrow\mathcal{P}\left(U\right)\) is a set valued mapping. Definition 2: [8] If \(F:E\rightarrow\mathcal{P}\left(U\times U\right)\) is a mapping from a subset of parameters \(E\) to the power set of \(U\times U\), then the soft set \((F,E)\) over \(U\times U\) 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 \(\mathcal{O}\
eq F\left(a\right)\) is an equivalence relation over \(U\) for all \(a\in E\). Definition 3: Let \(\mathcal{IS}=\left(U,V,\widetilde{f}\right)\) be the Middle East conflict situation and \(F:E\rightarrow\mathcal{P}\left(U\times U\right)\) be a set valued mapping from \(E\) to \(\mathcal{P}\left(U\times U\right)\), where \(E\subseteq V\) denotes the set of parameters or attributes and \(\mathcal{P}\left(U\times U\right)\) the set of all subsets of \(U\times U\). If \(\mathcal{O}\
eq F\left(a\right)\) is a preference relation for all \(a\in E\), then \((F,E)\) is called a soft preference relation, where:  
\[F\left(a\right)=\left\{u_{i}{\succeq}u_{j}\mid\widetilde{f}\left(u_{i},a \right)\geq\widetilde{f}\left(u_{j},a\right)\,,\,u_{i},u_{j}\in U\right\}\]  
and:  
\[\widetilde{f}:U\times V\rightarrow\left\{-,0,+\right\},\]  
such that \(\widetilde{f}\left(u,a\right)=s\), where \(s\in\left\{-,0,+\right\}\) and \(u\in U,a\in V\). In fact \(\widetilde{f}\) is more general than [PERSON] et al. 's \(f\) of Reference [4]. Throughout the paper, we will use \(\left(u_{i},u_{j}\right)\) for \(u_{i}{\succeq}u_{j}\). 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 the information system (Table 1), favorability \((+)\) is preferred to neutrality \((0)\), while neutrality \((0)\) is preferred to antagonism \((-)\). Definition 4: Let \(\mathcal{IS}=\left(U,V,\widetilde{f}\right)\) be the Middle East conflict situation and \((F,E)\) be a soft preference relation over \(U\).