Soft ordered approximations and incomplete information system

Abstract

The present paper aims to initiate the study of multi attribute group decision making in the presence of incomplete multi attribute and incomplete multi decision while making a decision with preferences in an incomplete information system. The concept of soft preference relation and soft dominance relation corresponding to decision attribute in incomplete multi criteria and incomplete multi decision information system is also presented. The discernibility matrix and discernibility function approach is proposed for decision attribute reduction to find all the reducts in an incomplete information system. A new technique of ranking the feasible alternative is proposed. Using the idea of soft dominance based rough sets, several uncertainty measures, such as approximate precision, rough degree, approximate quality and their mutual relationships are discussed in an incomplete information system. Finally the validity of these concepts are proved by applying these in solving multi-agent conflict analysis problems.

Keywords

Rough set soft set preference relation incomplete information

1. Introduction

The rough set theory, introduced by Pawlak [29] has been conceived as a useful tool to conceptualize and analyze various types of data. It has diverse applications in various fields such as artificial intelligence, cognitive sciences, medical, engineering, management sciences and economics etc. 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 [28]. 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.

Applications of rough set theory in case of conflict analysis is introduced by Pawlak in [30] 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 [7] is an extension of the model proposed by Pawlak by adding to the model some local aspects of conflicts. Subsequently Deja put forward three basic questions which are related to conflict analysis model: “What are the intrinsic reasons for the conflict?”, “How can a feasible consensus strategy be found?” and “Is it possible to satisfy all the agents?”

The concept of dominance based rough sets which are initiated by Greco et al. [11–13 , 15] extended the indiscernibility relation by a dominance relation by taking into account the preference order in value sets of criteria. Many extended models of the dominance based rough set approach have been proposed (see [4 , 36]). The hybridization of dominance based rough set approach and other mathematical tools have been created and applied to multi criteria decision analysis (see [3 , 36]).

Molodtsov’s soft set theory [27] 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. Maji et al. introduced several algebraic operations in soft set theory and examined their basic properties [25]. Ali et al. [1] proposed some new operations in soft set theory to further consolidate the algebraic basis of soft sets. In recent years, some hybrid uncertain models occur, e.g. fuzzy rough sets, fuzzy soft sets, rough soft sets, soft rough sets, soft rough fuzzy sets and soft fuzzy rough sets, to handle the uncertainty [9 , 37]. It is noted that all these hybrid systems have their own benefits and drawbacks.

Classical rough set theory can only deal with complete data sets. However, in many real world data sets, it may happen that some attribute values of an object are missing. For example, some measurement data of a patient in the clinical data set are missing, since the expense or difficulty of obtaining certain results. An information system with missing values is called an incomplete information system [21, 22]. A simple way of handling such systems is to transform the incomplete information system into a complete one, of which missing values can be deleted or filled with some values. However, the main disadvantage of this way is that some main information may be lost. Recently, extended rough set models have been developed to handle incomplete information systems directly without any preprocessing. Kryszkiewicz [21] defined the tolerance relation in incomplete information systems, and proposed the concepts of generalized decision and relative reduct.

The decision problem we consider, like the example of the Middle East conflict [30], is a problem of multi-criteria and multi-decision with preference group decision making. Classically, it is extension of the multiple criteria decision analysis problems such as multiple criteria classification with imprecise evaluations and assignments [8], multi attribute dominance fuzzy decision problems, multiple attribute dominance stochastic evaluation problems, multiple criteria group decision and so on [38]. Sun and Ma in [35], transform the multi-agent conflict decision making problem into a multi decision with preference group decision making problem and then gives a new approach to solve it by establishing and using a multi-decision preference dominance based rough set model.

Inspired by the idea of dominance based rough set approach to group decision making theory given by Greco et al. [14], as well as the study of the work by Chakhar and Saad [5], our objective is to consider a kind of multiple attribute (criteria) choice or ranking problem with multiple decisions (or the multiple criteria group decision problem), in which the decision attribute has a different preference description for each action, as in the Middle East example. For this kind of multiple attribute choice problem with multiple decision agents, the original definition of Pawlak rough set, which involves the relation of indiscernibility to identify granules of objects used to build lower and upper approximations, could not handle the evaluation of every action according to the multiple decision preference. Based on the dominance-based rough set approach [33], with two dominance relations, Sun and Ma defined one on the conditional attribute and the other on the decision attribute. Then, they present the lower and upper approximations of the dominance classes determined by the multiple decision attribute of each action using the dominance classes determined by the multiple attribute. Thus they tried to answer Deja’s questions related to conflict analysis problems. But still there are many areas for critics, for example “What are the intrinsic reasons for the conflict?” means what issues are focused by every agent and have different attitudes in a conflict, i.e., the core causes of the conflict. Another infirmity in Sun’s technique is that, it does not answer the question “Is it possible to satisfy all the agents?” One can neutrally asks the following interesting question: Is it possible to address/answer the aforesaid questions in the incomplete information systems?

In the present paper: (i) we present the idea of soft preference and soft dominance relation in an incomplete information system to solve a multi agent conflict analysis problem and tried to answer Deja’s first question in a best manner which is related to conflict analysis; (ii) another worth mentioning contribution of the present paper is that to define two types of approximations and applied these to discussed various properties of the approximations in an incomplete information system; (iii) another worth mentioning contribution of this paper is to disclose the ideas of two kinds of approximate precision, rough degree, approximate quality and their mutual relationship.

The paper is organized as follows: Section 2, deals mainly with the problem statement. In Section 3, we present the idea of soft preference and soft dominance relations in an incomplete information system to solve a multi agent conflict analysis labor-management negotiations problem. In Section 4, highlights the study of core cause of conflict analysis. In Section 5, our attention is concentrated on the development of feasible consensus strategy for labor-management negotiation conflict analysis problem based on soft dominance rough sets in an incomplete information system. In addition, several uncertainty measures, such as approximate precision, rough degree, approximate quality, and their mutual relationships are discussed in an incomplete information system.

2. Problem statement

The incomplete multiple decision problems with preference relations have been studied in this paper. In general these may be the incomplete multi criteria group decision problems or incomplete multi criteria and incomplete multi decision with preference choice problem. It comprises of two parts, the first is the incomplete multi criteria with predefined evaluations for every action and the other is the incomplete multi decision with a predefined preference for every action. An incomplete decision problem may be considered as an S^∗ = (A, C, D, E), where A is a finite set of actions a_i, C is a finite set of conditional attributes C_j, D is a finite set of decision attributes D_k, 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 article, the values of both information functions f (a_i, C_j) and g (a_i, D_k) are integers. The value of f (a_i, C_j) describes the evaluation of action a_i based on criterion C_j and the value of g (a_i, D_k) describes the evaluation of action a_i by the decision maker D_k . That is, it describes the preference of decision maker D_k for action a_i . Predominantly speaking, the greater the value of g (a_i, D_k) , the greater the preference for action a_i by decision maker D_k . It may happen that some attribute values of an action are missing. To indicate such a situation, a distinguished value is often assigned. We denote special symbol “∗” to indicate the missing attribute value. For an information system, if there exist a_i ∈ A, C_j ∈ C and D_k ∈ D such that f (a_i, C_j) =∗ or g (a_i, D_k) = ∗ .

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

Table 1
Incomplete multi criteria and multi decision information table

A C ₁ C ₂ C ₃ C ₄ C ₅ D ₁ D ₂ D ₃ D ₄

a ₁ 1 0 1 ∗ 2 2 ∗ 1 ∗

a ₂ 1 ∗ 1 1 2 1 3 0 1

a ₃ 1 0 2 0 2 3 1 0 3

a ₄ 1 1 0 0 2 1 1 ∗ 2

a ₅ ∗ 1 1 0 2 0 2 0 1

a ₆ 1 1 2 1 2 1 0 2 0

a ₇ 1 1 2 2 2 3 2 3 2

a ₈ 1 2 ∗ 0 2 0 ∗ 0 2

a ₉ 1 2 1 1 2 2 2 1 ∗

a ₁₀ 1 2 0 0 ∗ ∗ 3 0 3

a ₁₁ 1 2 1 0 1 0 1 2 0

a ₁₂ 1 2 1 1 1 2 1 1 1

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

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

3. Proposed conflict analysis model

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

According to Deja [6], the conflict analysis decision task is proposed into three problems. In the present paper, we initiate the notion of soft preference and soft dominance relation and applied these to solve the problems/questions posed by Deja.

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

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

Definition 2.Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with multi decisions information system and $F_{k}^{*} : C \to P (A \times A)$ be a set valued mapping from C to $P (A \times A),$ where C denotes the set of parameters or attributes and $P (A \times A)$ the set of all subsets of A × A . If $F_{k}^{*} (C_{j})$ is a preference relation for all C_j ∈ C then $(F_{k}^{*}, C)$ is called a soft preference relation corresponding to the decision attributes where $F_{k}^{*} (C_{j}) = {\begin{matrix} a_{m} ≽ a_{n} ∣ f (a_{m}, C_{j}) \geq f (a_{n}, C_{j}) or f (a_{m}, C_{j}) = * or f (a_{n}, C_{j}) = * \\ and g (a_{m}, D_{k}) \geq g (a_{n}, D_{k}) or g (a_{m}, D_{k}) = * or g (a_{n}, D_{k}) = * \end{matrix}} .$ Denote a_m ≽ a_n by (a_m, a_n). In an incomplete multi attributes with incomplete multi decisions information system $S^{*} = (A, C, D, E),$ we answer to the first question of Deja’s which is related to conflict analysis model, that is, “What are the intrinsic reasons for the conflict?” This means what issues are focussed by every agent or feasible action and have different attitudes in a conflict corresponding to a specific decision maker. For any issue C_j ∈ C, we obtain attitude information for every agent with respect to issue C_j ∈ C by using $F_{k}^{*} (C_{j})$ which is a preference relation corresponding to decision maker D_k, where and Thus we have answered the first question of Deja’s which is related to conflict analysis model.

Definition 3. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and $(F_{k}^{*}, C)$ be a soft preference relation corresponding to the decision attributes over A . Then there is dominance relation associated with which can be denoted by $D om (F_{k}^{*}, C)$ and is defined by: $D om (F_{k}^{*}, C) = ⋂_{C_{j} \in C} F_{k}^{*} (C_{j})$ where and It is worth mentioning that in an incomplete information system $S^{*},$ the $D om (F_{k}^{*}, C)$ depicts that which agent(s) is/are preferred by decision maker (D_k) on all criteria by other agents.

Definition 4. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and {} be a family of soft dominance relations over A . Then there exists a soft dominance relation $D {om}^{k} (H^{*}, C)$ such that $D o m^{k} (ℋ^{*}, C) = \cap_{k = 1}^{| D |} D o m (ℱ %_{k}^{*}, C) .$ In a similar fashion, family of dominance relations $D {om}^{k} (H^{*}, C)$ shows which of the agent(s) preferred on all criteria by all decision maker(s). But still we do not have sufficient information about the agent(s) on which all decision maker(s) are agreed. So by Deja’s second question which is related to conflict analysis, “How can a feasible consensus strategy be found”? This means that to find feasible alternative(s) on which all the optimal decision makers agreed or at least suboptimal decision makers. If all the decision makers don’t agree on any feasible alternative(s), then in order to develop feasible consensus strategy some authors remove certain decision maker(s) from the information system (e.g. see [35]).

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

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

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

4. Core cause of conflict analysis

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 the multi-attributes with multi-decisions incomplete information system $S^{*}$ based on soft dominance relation. For more clarification we present:

4.1. Discernibility matrix and discernibility function for multi-attributes with multi-decisions in incomplete information system

In this subsection, a method based on the discernibility matrix and discernibility function is proposed for decision attribute reduction to find all the reducts in the multi-attributes with multi-decisions in an incomplete information system $S^{*} = (A, C,$ D, E).

Definition 5. Let $S^{*} = (A, C, D, E)$ be a multi-attributes with multi-decisions incomplete information system. We present the discernibility matrix based approach to compute all the decision reducts in $S^{*}$ , for a_i, a_j ∈ A and $D_{D}^{*} (a_{i}, a_{j}) = {\begin{matrix} {D_{1} \in D : (a_{i}, a_{j}) \notin D om (F_{1}^{*}, C)} & (a_{i}, a_{j}) \notin D {om}^{k} (H^{*}, C); \\ \emptyset & otherwise . \end{matrix}$

Definition 6. Let $S^{*} = (A, C, D, E)$ be a multi-attributes with multi-decisions incomplete information system, a discernibility function f of the system can be defined by $f = \underset{(a_{i}, a_{j}) \in A \times A}{\land} {\lor {D_{D}^{*} (a_{i}, a_{j}) : D_{D}^{*} (a_{i}, a_{j}) \neq \emptyset}},$ where $\lor D_{D}^{*} (a_{i}, a_{j})$ is the disjunction of all attribute in $D_{D}^{*} (a_{i}, a_{j}),$ indicating that the object pair (a_i, a_j) can be distinguished by any attribute in $D_{D}^{*} (a_{i}, a_{j}),$ $\land {\lor D_{D}^{*} (a_{i}, a_{j})}$ is the conjunction of all $D_{D}^{*} (a_{i}, a_{j}),$ indicating that the family of discernible objects pair can be distinguished by a set of attributes satisfying $\land {\lor D_{D}^{*} (a_{i}, a_{j})} .$

Example 2. (Continued from Example 1) For the reduction of decision attributes in multi-criteria and multi-decision incomplete information system $S^{*} = (A, C, D, E),$ applying the Definitions 5 and 6. Thus we can get the following Table 2 .

Table 2
The discernibility matrix for reduction of decision attributes

a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈ a ₉ a ₁₀ a ₁₁ a ₁₂

a ₁ ∅ ∅ D D D D D D D D D D

a ₂ $\begin{matrix} D_{1}, \\ D_{3} \end{matrix}$ ∅ D D ₄ $\begin{matrix} D_{1}, \\ D_{4} \end{matrix}$ D D D ₄ D $\begin{matrix} D_{3}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{1}, \\ D_{3}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{1}, \\ D_{3}, \\ D_{4} \end{matrix}$

a ₃ D ₃ D ∅ $\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$ D $\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$ D $\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$ D

a ₄ D D D ∅ D D D D D D D D

a ₅ $\begin{matrix} D_{1} \\ D_{3} \end{matrix}$ D D $\begin{matrix} D_{1}, \\ D_{4} \end{matrix}$ ∅ D D D D D D D

a ₆ D ₁ $\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{1}, \\ D_{2}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$ $\begin{matrix} D_{1}, \\ D_{2}, \\ D_{4} \end{matrix}$ ∅ D D D D D D

a ₇ ∅ D ₂ D ₄ ∅ ∅ ∅ ∅ D D D D D

a ₈ $\begin{matrix} D_{1}, \\ D_{3} \end{matrix}$ D $\begin{matrix} D_{1}, \\ D_{4} \end{matrix}$ D ₁ ∅ D D ∅ D $\begin{matrix} D_{3}, \\ D_{4} \end{matrix}$ D ₃ D

a ₉ ∅ D ₂ D ∅ ∅ D D ∅ ∅ D ₂ $\begin{matrix} D_{3}, \\ D_{4} \end{matrix}$ D ₄

a ₁₀ D D D ∅ D D D ∅ D ∅ D D

a ₁₁ D D D D D D D D D $\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$ ∅ D

a ₁₂ D D D D D D D D D $\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$ D ₃ ∅

	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈	a ₉	a ₁₀	a ₁₁	a ₁₂
a ₁	∅	∅	D	D	D	D	D	D	D	D	D	D
a ₂	$\begin{matrix} D_{1}, \\ D_{3} \end{matrix}$	∅	D	D ₄	$\begin{matrix} D_{1}, \\ D_{4} \end{matrix}$	D	D	D ₄	D	$\begin{matrix} D_{3}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{1}, \\ D_{3}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{1}, \\ D_{3}, \\ D_{4} \end{matrix}$
a ₃	D ₃	D	∅	$\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$	D	$\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$	D	$\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{2}, \\ D_{3}, \\ D_{4} \end{matrix}$	D
a ₄	D	D	D	∅	D	D	D	D	D	D	D	D
a ₅	$\begin{matrix} D_{1} \\ D_{3} \end{matrix}$	D	D	$\begin{matrix} D_{1}, \\ D_{4} \end{matrix}$	∅	D	D	D	D	D	D	D
a ₆	D ₁	$\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{1}, \\ D_{2}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$	$\begin{matrix} D_{1}, \\ D_{2}, \\ D_{4} \end{matrix}$	∅	D	D	D	D	D	D
a ₇	∅	D ₂	D ₄	∅	∅	∅	∅	D	D	D	D	D
a ₈	$\begin{matrix} D_{1}, \\ D_{3} \end{matrix}$	D	$\begin{matrix} D_{1}, \\ D_{4} \end{matrix}$	D ₁	∅	D	D	∅	D	$\begin{matrix} D_{3}, \\ D_{4} \end{matrix}$	D ₃	D
a ₉	∅	D ₂	D	∅	∅	D	D	∅	∅	D ₂	$\begin{matrix} D_{3}, \\ D_{4} \end{matrix}$	D ₄
a ₁₀	D	D	D	∅	D	D	D	∅	D	∅	D	D
a ₁₁	D	D	D	D	D	D	D	D	D	$\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$	∅	D
a ₁₂	D	D	D	D	D	D	D	D	D	$\begin{matrix} D_{2}, \\ D_{4} \end{matrix}$	D ₃	∅

From the discernibility matrix, we can construct the discernibility function as follows: $\begin{matrix} f & = & D_{1} \land D_{2} \land D_{3} \land D_{4} \land {D_{1} \lor D_{3}} \\ \land {D_{2} \lor D_{4}} \land {D_{1} \lor D_{4}} \land {D_{3} \lor D_{4}} \\ \land {D_{1} \lor D_{2} \lor D_{4}} \land {D_{1} \lor D_{3} \lor D_{4}} \\ \land {D_{2} \lor D_{3} \lor D_{4}} \\ = & D_{1} \land D_{2} \land D_{3} \land D_{4} . \end{matrix}$

By applying the absorption laws, we find that {D₁, D₂, D₃, D₄} is the reduct of the incomplete information system $S^{*}$ .

Definition 7. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and $D {om}^{k} (H^{*}, C)$ be a soft dominance relation. For a ∈ A, define the soft dominance classes by ${[a]}_{D {om}^{k} (H^{*}, C)}^{+} = {a_{i} \in A : a_{i} ≽_{D {om}^{k} (H^{*}, C)} a}$ and ${[a]}_{D {om}^{k} (H^{*}, C)}^{-} = {a_{i} \in A : a_{i} ⪯_{D {om}^{k} (H^{*}, C)} a}$ which represent $(H^{*}, C)$ -k-dominating set and $(H^{*}, C)$ -k-dominated set with respect to a, over the set of conditional attributes C corresponding to the decision attributes D_k, of $S^{*} = (A, C, D, E)$ respectively. ${[a]}_{D {om}^{k} (H^{*}, C)}^{+}$ describes the set of objects that dominate a and ${[a]}_{D {om}^{k} (H^{*}, C)}^{-}$ describes the set of objects that dominated by a in terms of $D {om}^{k} (H^{*}, C) .$

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

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

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

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

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

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

Proof. The proof is straightforward.□

Definition 8. Let $S^{*} = (A, C, D, E)$ be an incomplete information system. For any X ⊆ A, the approximate accuracy $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X)$ and $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{-}} (X)$ of X about ${[a]}_{D {om}^{k} (H^{*}, C)}^{+}$ and ${[a]}_{D {om}^{k} (H^{*}, C)}^{-}$ are defined by $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) = \frac{| {[a]}_{D {om}^{k} (H^{*}, C)}^{+} \cap X |}{| {[a]}_{D {om}^{k} (H^{*}, C)}^{+} |}$ and $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{-}} (X) = \frac{| {[a]}_{D {om}^{k} (H^{*}, C)}^{-} \cap X |}{| {[a]}_{D {om}^{k} (H^{*}, C)}^{-} |} .$ It is routine to verify that $0 \leq ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) \leq 1$ and $0 \leq ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{-}} (X) \leq 1$ . For any X, Y ⊆ A, the following properties hold:

(1) $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) = 1$ if and only if $a \in {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}},$

(2) $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) = 0$ if and only if $a \in {({\underline{X}}^{c})}_{D {om}^{k} {(H^{*}, C)}^{+}},$

(3) $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X \cup Y) = ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) + ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (Y) - ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X \cap Y),$

(4) $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) = 1 - ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X^{c}),$

(5) If X ⊆ Y, then $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) \leq ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (Y),$

(6) $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (\emptyset) = 0$ and $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (A) = 1,$ for every a ∈ A,

(7) $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X) + ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{+}} (X^{c}) = 1 .$

Analogous properties also hold in the case of $ς_{{[a]}_{D {om}^{k} (H^{*}, C)}^{-}} (X) .$

5. Method for finding rank of feasible alternatives

In the present section we propose a new technique of ranking the feasible alternative in an incomplete information system $S^{*}$ .

Definition 9. Let $S^{*} = (A, C, D, E)$ be an incomplete information system. The soft dominance degree between a_i and a_j with respect to $D {om}^{k} {(H^{*}, C)}^{+}$ and $D {om}^{k} {(H^{*}, C)}^{-}$ is defined by $D_{D o m^{k} {(ℋ^{*}, C)}^{+}} (a_{i}, a_{j}) = 1 - \frac{| {[a_{i}]}_{% D o m^{k} (ℋ^{*}, C)}^{+} - {[a_{j} %]}_{D o m^{k} (ℋ^{*}, C)}^{+} |}{| A |}$ and $D_{D o m^{k} {(ℋ^{*}, C)}^{-}} (a_{i}, a_{j}) = 1 - \frac{| {[a_{i}]}_{% D o m^{k} (ℋ^{*}, C)}^{-} - {[a_{j} %]}_{D o m^{k} (ℋ^{*}, C)}^{-} |}{| A |} .$

Clearly $0 < D_{D {om}^{k} {(H^{*}, C)}^{+}} (a_{i}, a_{j}) \leq 1$ and $0 < D_{D {om}^{k} {(H^{*}, C)}^{-}} (a_{i}, a_{j}) \leq 1 .$ Furthermore following properties hold:

(1) If ${[a_{i}]}_{D {om}^{k} (H^{*}, C)}^{+} \cap {[a_{j}]}_{D {om}^{k} (H^{*}, C)}^{+} = \emptyset,$ then

(2) If ${[a_{i}]}_{D {om}^{k} (H^{*}, C)}^{+} \subseteq {[a_{j}]}_{D {om}^{k} (H^{*}, C)}^{+},$ then $D_{D {om}^{k} {(H^{*}, C)}^{+}} (a_{I}, a_{j}) = 1;$

(3) If ${[a_{i}]}_{D {om}^{k} (H^{*}, C)}^{+} = {[a_{j}]}_{D {om}^{k} (H^{*}, C)}^{+},$ then $D_{D {om}^{k} {(H^{*}, C)}^{+}} (a_{I}, a_{z}) = D_{D {om}^{k} {(H^{*}, C)}^{+}} (a_{j}, a_{z})$ for all z ∈ A .

Obviously, one can construct soft dominance relation matrices with respect to $D {om}^{k} {(H^{*}, C)}^{+}$ and $D {om}^{k} {(H^{*}, C)}^{-} .$ From these matrices, the whole soft dominance degree of each object is defined by $D^{* +} (a_{i}) = \frac{1}{| A |} % \sum D_{D o m^{k} {(ℋ^{*}, C)}^{+}} (a_{i}, a_{j}) for all a_{j} \in A$ and $D^{* -} (a_{i}) = \frac{1}{| A |} % \sum D_{D o m^{k} {(ℋ^{*}, C)}^{-}} (a_{i}, a_{j}) for all a_{j} \in A .$

Example 3. (Continued from Example 1) For the ranking of feasible alternative in a multi-criteria and multi-decision incomplete information system $S^{*} = (A, C, D, E),$ the soft dominance classes from $D {om}^{k} (H^{*}, C)$ are given in the following Table 3:

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

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

It is seen that the dominance classes in $A / D {om}^{k} (H^{*}, C)$ do not constitute a partition of A, rather a covering of A . The soft dominance degree is given by $D_{D o m^{k} {(ℋ^{*}, C)}^{+}} (a_{1}, a_{2}) = 1 - \frac{| {[a_{1}]}_{% D o m^{k} (ℋ^{*}, C)}^{+} - {[a_{2} %]}_{D o m^{k} (ℋ^{*}, C)}^{+} |}{| A |} = 1 - 0.1666 = 0.8334.$ In a similar manner we can find all $D_{D {om}^{k} {(H^{*}, C)}^{+}} (a_{I}, a_{j}),$ i, j = 1, 2,. . . , 12 . The soft dominance relation matrix with respect to $D {om}^{k} {(H^{*}, C)}^{+}$ is $(\begin{matrix} 1 & 0.833 & 0.750 & 0.916 & 0.916 & 0.833 & 0.833 & 0.833 & 0.833 & 0.750 & 0.750 & 0.750 \\ 0.916 & 1 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 \\ 0.916 & 0.916 & 1 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 \\ 0.833 & 0.666 & 0.666 & 1 & 0.833 & 0.750 & 0.750 & 0.833 & 0.750 & 0.750 & 0.666 & 0.666 \\ 0.833 & 0.666 & 0.666 & 0.833 & 1 & 0.750 & 0.750 & 0.833 & 0.750 & 0.666 & 0.666 & 0.666 \\ 0.916 & 0.833 & 0.833 & 0.916 & 0.916 & 1 & 0.916 & 0.833 & 0.833 & 0.833 & 0.833 & 0.833 \\ 1 & 0.916 & 0.916 & 1 & 1 & 1 & 1 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 \\ 0.833 & 0.750 & 0.750 & 0.916 & 0.916 & 0.750 & 0.750 & 1 & 0.833 & 0.833 & 0.750 & 0.750 \\ 1 & 0.916 & 0.916 & 1 & 1 & 0.916 & 0.916 & 1 & 1 & 0.916 & 0.916 & 0.916 \\ 0.916 & 0.916 & 0.916 & 1 & 0.916 & 0.916 & 0.916 & 1 & 0.916 & 1 & 0.916 & 0.916 \\ 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 1 & 0.916 \\ 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 0.916 & 1 \end{matrix}) .$

Furthermore, $D^{* +} (a_{1}) = 0.8330,$ $D^{* +} (a_{2}) = 0.8538,$ $D^{* +} (a_{3}) = 0.923,$ $D^{* +} (a_{4}) = 0.7635,$ $D^{* +} (a_{5}) = 0.7565,$ $D^{* +} (a_{6}) = 0.8746,$ $D^{* +} (a_{7})$ = 0.951, $D^{* +} (a_{8}) = 0.8192,$ $D^{* +} (a_{9}) = 0.951,$ $D^{* +} (a_{10}) = 0.9370,$ $D^{* +} (a_{11}) = 0.923$ and $D^{* +}$ (a₁₂) = 0.923 . Thus we achieve the following soft dominating ranking: $\begin{matrix} a_{7} & = & a_{9} ≽ a_{10} ≽ a_{3} = a_{11} \\ = & a_{12} ≽ a_{6} ≽ a_{2} ≽ a_{1} ≽ a_{8} ≽ a_{4} ≽ a_{5} . \end{matrix}$

The feasible consensus strategy for multi-criteria and multi-decision in incomplete information system is developed with the help of raking (softdominatingranking) which is a novel idea. Thus from the soft dominating ranking, actions a₇ and a₉ are feasible alternatives for solving this conflict analysis problem, on which all agents have agreed in the incomplete conflict situation $S^{*} .$ The result shows that there is an optimal alternative (actions a₇ and a₉) which satisfy all agents in the incomplete conflict situation $S^{*} .$ Thus a feasible consensus strategy is developed.

Definition 10. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then for any X ⊆ A the lower and upper approximations with respect to $D {om}^{k} {(H^{*}, C)}^{+}$ and $D {om}^{k} {(H^{*}, C)}^{-}$ are: ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {a \in A : {[a]}_{D {om}^{k} (H^{*}, C)}^{+} \subseteq X}$ ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {a \in A : {[a]}_{D {om}^{k} (H^{*}, C)}^{-} \cap X \neq \emptyset} .$ The lower approximation ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ , is composed of all objects a from A such that all objects a_i, having at least the same evaluations on all of the considered criteria from C, also belong to X . Thus one can say that if an object a_i has at least as good in evaluation on the criteria from C as the object a belonging to ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}},$ then certainly, a_i belongs to X . The upper approximation ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ is made up of all objects a from A whose evaluation on the criteria from C are not worse than the evaluation of at least one other object a_i belonging to X . If ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ then it is called soft dominating definable, otherwise soft dominating rough set, where ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {a \in A : {[a]}_{D {om}^{k} (H^{*}, C)}^{-} \subseteq X}$ ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {a \in A : {[a]}_{D {om}^{k} (H^{*}, C)}^{+} \cap X \neq \emptyset} .$ If ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{-}}$ then it is called soft dominated definable, otherwise soft dominated rough set.

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

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

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

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

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

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

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

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

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

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

Proof. Follows from the respective definitions.□

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

(1) ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = ⋃_{y \in X} {[y]}_{D {om}^{k} (H^{*}, C)}^{+};$

(2) ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = ⋃_{y \in X} {[y]}_{D {om}^{k} (H^{*}, C)}^{-} .$

Theorem 4. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then for any X ⊆ A :

(1) ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = A - {(\bar{X^{c}})}_{D {om}^{k} {(H^{*}, C)}^{-}};$

(2) ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = A - {(\underline{X^{c}})}_{D {om}^{k} {(H^{*}, C)}^{+}} .$

Theorem 5. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then for any X ⊆ A :

(1) ${(\underline{X})}_{D {om}^{k - 1} {(H^{*}, C)}^{+}} \supseteq {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} .$

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

Proof. (1) For any $x \in {(\underline{X})}_{D {om}^{k - 1} {(H^{*}, C)}^{+}}$ , it follows that ${[x]}_{D {om}^{k - 1} (H^{*}, C)}^{+} \subseteq X .$ Which implies that ${[x]}_{D {om}^{k} (H^{*}, C)}^{+} \subseteq {[x]}_{D {om}^{k - 1} (H^{*}, C)}^{+} .$ This implies that ${[x]}_{D {om}^{k} (H^{*}, C)}^{+} \subseteq X .$ Hence $x \in {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} .$ Therefore ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \supseteq {(\underline{X})}_{D {om}^{k - 1} {(H^{*}, C)}^{+}} .$

(2) For any $x \in {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}},$ it follows that ${[x]}_{D {om}^{k} (H^{*}, C)}^{-} \cap X \neq \emptyset .$ We have ${[x]}_{D {om}^{k} (H^{*}, C)}^{-} \subseteq {[x]}_{D {om}^{k - 1} (H^{*}, C)}^{-} .$ This implies that ${[x]}_{D {om}^{k - 1} (H^{*}, C)}^{-} \cap X \neq \emptyset .$ Hence $x \in {(\bar{X})}_{D {om}^{k - 1} {(H^{*}, C)}^{+}} .$ Therefore ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \subseteq {(\bar{X})}_{D {om}^{k - 1} {(H^{*}, C)}^{+}} .$ □

The following theorems give a connection between topology and soft dominating/dominated definable sets.

Theorem 6. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then the collection of soft dominating definable sets form a topology on A .

Theorem 7. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then the collection of soft dominated definable set form a topology on A .

Definition 11. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For any X ⊆ A, the approximate precisions $ρ_{D {om}^{k} {(H^{*}, C)}^{+}} (X)$ and $ρ_{D {om}^{k} {(H^{*}, C)}^{-}} (X)$ of X about $D {om}^{k} {(H^{*}, C)}^{+}$ and $D {om}^{k} {(H^{*}, C)}^{-}$ are defined as: $\begin{matrix} ρ_{D {om}^{k} {(H^{*}, C)}^{+}} (X) & = & \frac{| {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} |}{| {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} |} and \\ ρ_{D {om}^{k} {(H^{*}, C)}^{-}} (X) & = & \frac{| {(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} |}{| {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} |} \end{matrix}$ where X ≠ ∅ , and | · | denotes the cardinality of a set.

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

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

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

Theorem 8. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A . Then the rough degree and precision of the sets X, Y, X ∪ Y and X ∩ Y satisfy the following relations.

$\begin{matrix} (1) μ_{D {om}^{k} {(H^{*}, C)}^{+}} (X \cup Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \cup {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | \\ \leq & μ_{D {om}^{k} {(H^{*}, C)}^{+}} (X) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} | + μ_{D {om}^{k} {(H^{*}, C)}^{+}} (Y) | {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | \\ - μ_{D {om}^{k} {(H^{*}, C)}^{+}} (X \cap Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \cap {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | . \\ (2) ρ_{D {om}^{k} {(H^{*}, C)}^{+}} (X \cup Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \cup {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | \\ \geq & ρ_{D {om}^{k} {(H^{*}, C)}^{+}} (X) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} | + ρ_{D {om}^{k} {(H^{*}, C)}^{+}} (Y) | {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | \\ - ρ_{D {om}^{k} {(H^{*}, C)}^{+}} (X \cap Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \cap {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | . \end{matrix}$ Theorem 9. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A . Then the precision of the sets X, Y, X ∪ Y and X ∩ Y satisfy the following relations. $\begin{matrix} (1) μ_{D {om}^{k} {(H^{*}, C)}^{-}} (X \cup Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} \cup {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | \\ \leq & μ_{D {om}^{k} {(H^{*}, C)}^{-}} (X) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} | + μ_{D {om}^{k} {(H^{*}, C)}^{-}} (Y) | {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | \\ - μ_{D {om}^{k} {(H^{*}, C)}^{-}} (X \cap Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} \cap {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | . \\ (2) ρ_{D {om}^{k} {(H^{*}, C)}^{-}} (X \cup Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} \cup {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | \\ \geq & ρ_{D {om}^{k} {(H^{*}, C)}^{-}} (X) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} | + ρ_{D {om}^{k} {(H^{*}, C)}^{-}} (Y) | {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | \\ - ρ_{D {om}^{k} {(H^{*}, C)}^{-}} (X \cap Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} \cap {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | . \end{matrix}$

Definition 12. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For any X ⊆ A, the approximate quality $γ_{D {om}^{k} {(H^{*}, C)}^{+}} (X)$ and $γ_{D {om}^{k} {(H^{*}, C)}^{-}} (X)$ of X about $D {om}^{k} {(H^{*}, C)}^{+}$ and $D {om}^{k} {(H^{*}, C)}^{-}$ are defined as Follows: $γ_{D o m^{k} {(ℋ^{*}, C)}^{+}} (X) = \frac{| {(\underline{X})}_{D % o m^{k} {(ℋ^{*}, C)}^{+}} |}{| A |} and γ_{D o m^{k} {(ℋ^{*}, C)}^{-}} (X) = \frac{| {(\underline{X})}_{D % o m^{k} {(ℋ^{*}, C)}^{-}} |}{| A |} .$ Clearly $0 \leq γ_{D {om}^{k} {(H^{*}, C)}^{+}} (X) \leq 1$ and $0 \leq γ_{D {om}^{k} {(H^{*}, C)}^{-}} (X) \leq 1 .$

The following theorem illustrate the relationship between rough degree and approximate quality about the intersection and union of sets X and Y on the universe A .

Theorem 10. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A . Then the approximate quality of the sets X, Y, X ∪ Y and X ∩ Y satisfy the following relations. $\begin{matrix} (1) μ_{D {om}^{k} {(H^{*}, C)}^{+}} (X \cup Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \cup {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | \\ \leq & | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} | + | {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | - A {γ_{D {om}^{k} {(H^{*}, C)}^{+}} (X) + γ_{D {om}^{k} {(H^{*}, C)}^{+}} (Y)} \\ - μ_{D {om}^{k} {(H^{*}, C)}^{+}} (X \cap Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} \cap {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}} | . \end{matrix}$ $\begin{matrix} (2) μ_{D {om}^{k} {(H^{*}, C)}^{-}} (X \cup Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} \cup {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | \\ \leq & | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} | + | {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | - A {γ_{D {om}^{k} {(H^{*}, C)}^{-}} (X) + γ_{D {om}^{k} {(H^{*}, C)}^{-}} (Y)} \\ - μ_{D {om}^{k} {(H^{*}, C)}^{-}} (X \cap Y) | {(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} \cap {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} | . \end{matrix}$

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

Theorem 11. {Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A . Then the approximate quality and precision of the sets X, Y, X ∪ Y and X ∩ Y satisfy the following relations. $(1) ρ_{D o m^{k} {(ℋ^{*}, C)}^{+}} (X \cup Y) | {(\bar{X})}_{D o m^{k} {(ℋ^{*}, C)}^{+}} \cup {(\bar{Y})}_{D o m^{k} {(ℋ^{*}, C)}^{+}} | \geq A (γ_{D o m^{k} {(ℋ^{*}, C)}^{+}} (X) + γ_{D o m^{k} {(ℋ^{*}, C)}^{+}} (Y)) - ρ_{D o m^{k} {(ℋ^{*}, C)}^{+}} (X \cap Y) | {(\bar{X})}_{D % o m^{k} {(ℋ^{*}, C)}^{+}} \cap {(\bar{Y} %)}_{D o m^{k} {(ℋ^{*}, C)}^{+}} | .$ $(2) ρ_{D o m^{k} {(ℋ^{*}, C)}^{-}} (X \cup Y) | {(\bar{X})}_{D o m^{k} {(ℋ^{*}, C)}^{-}} \cup {(\bar{Y})}_{D o m^{k} {(ℋ^{*}, C)}^{-}} | \geq A (γ_{D o m^{k} {(ℋ^{*}, C)}^{-}} (X) + γ_{D o m^{k} {(ℋ^{*}, C)}^{-}} (Y)) - ρ_{D o m^{k} {(ℋ^{*}, C)}^{-}} (X \cap Y) | {(\bar{X})}_{D % o m^{k} {(ℋ^{*}, C)}^{-}} \cap {(\bar{Y} %)}_{D o m^{k} {(ℋ^{*}, C)}^{-}} | .$

Definition 13. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A . Then

X and Y are called lower soft dominating rough equal if ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {(\underline{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ , denoted as X≂Y .

X and Yare called upper soft dominating rough equal if ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}},$ denoted as X ≃ Y .

X and Yare called soft dominating rough equal

if ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {(\underline{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ and ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ , denoted as X ≈ Y .

Proposition 1. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. Then the relations ≂, ≃ and ≈ are equivalence relation.

Table 4

The lower approximations table

X _k	${(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{+}}$	${(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{-}}$
X ₁	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{6}, a_{7}, a_{9}, \\ a_{10}, a_{12} \end{matrix}}$	{a₁, a₂, a₃, a₄, a₆, a₁₂}
X ₂	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}, a_{9}, \\ a_{10}, a_{11}, a_{12} \end{matrix}}$	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{8}, a_{9}, \\ a_{10}, a_{11}, a_{12} \end{matrix}}$
X ₃	{a₁, a₆, a₇, a₉, a₁₁, a₁₂}	{a₄, a₆, a₁₁, a₁₂}
X ₄	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}, \\ a_{9}, a_{10}, a_{12} \end{matrix}}$	${\begin{matrix} a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{8}, a_{9}, \\ a_{10}, a_{12} \end{matrix}}$

Theorem 12. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For any X, Y, X^∼, Y^∼ ⊆ A, the following properties hold:

(1) X≂Y if and only if (X ∩ Y) ≂X and (X ∩ Y) ≂Y,

(2) X ≃ Y if and only if (X ∪ Y) ≂X and (X ∪ Y) ≂Y,

(3) If X≂X^∼ and Y≂Y^∼, then (X ∩ Y) ≂ (X^∼ ∩ Y^∼) ,

(4) If X ≃ X^∼ and Y ≃ Y^∼, then (X ∪ Y) ≃ (X^∼ ∪ Y^∼) ,

(5) If X≂Y, then X ∩ (∼Y) ≂ ∅ ,

(6) If X ≃ Y, then X ∪ (∼Y) ≃ A,

(7) If X≂∅ or Y≂ ∅ , then X ∩ Y≂ ∅ ,

(8) If X ≃ A or Y ≃ A, then X ∪ Y ≃ A .

Definition 14. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system and X, Y ⊆ A . Then

X and Y are called lower soft dominated rough equal if ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {(\underline{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} .$ It is denoted by X≂Y .

X and Y are called upper soft dominated rough equal if ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}} .$ It is denoted by X ≃ Y .

Set X and Y are called soft dominated rough equal if ${(\underline{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {(\underline{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}}$ and ${(\bar{X})}_{D {om}^{k} {(H^{*}, C)}^{-}} = {(\bar{Y})}_{D {om}^{k} {(H^{*}, C)}^{-}}$ . It is denoted by X ≈ Y .

Theorem 13. Let $S^{*} = (A, C, D, E)$ be an incomplete multi attributes with incomplete multi decisions information system. For any X, Y, X^∼, Y^∼ ⊆ A, the following properties hold:

(1) X≂Y if and only if (X ∩ Y) ≂X and (X∩ Y) ≂Y ;

(2) X ≃ Y if and only if (X ∪ Y) ≂X and (X∪ Y) ≂Y ;

(3) If X≂X^∼ and Y≂Y^∼, then (X∩ Y) ≂ (X^∼ ∩ Y^∼) ;

(4) If X ≃ X^∼ and Y ≃ Y^∼, then (X∪ Y) ≃ (X^∼ ∪ Y^∼) ;

(5) If X≂Y, then X∩ (∼Y) ≂ ∅ ;

(6) If X ≃ Y, then X∪ (∼Y) ≃ A ;

(7) If X≂∅ or Y≂ ∅ , then X∩ Y≂ ∅ ;

(8) If X ≃ A or Y ≃ A, then X ∪ Y ≃ A .

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

5.1. Proposed algorithm for conflict analysis model

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

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

Step 2 : Construct $D om (F_{k}^{*}, C) = ⋂_{C_{j} \in C} F_{k}^{*} (C_{j}),$ for all according to Definition;

Step 3 : Construct $D om (F_{k}^{*}, C)$ // according to Definition 4;

Step 4 : Construct ${(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ and ${(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{-}}$ // according to Definition 10, where X_k ={ a ∈ A : g (a, D_k) ≠0 } ;

Step 5 : If ${(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{+}} = {(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{-}}$ then $δ_{k} = {(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{+}}$ or ${(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{-}}$ otherwise $δ_{k} = {(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{+}} - {(\underline{X_{k}})}_{D {om}^{k} {(H^{*}, C)}^{-}};$

Step 6 :

go to output.

Otherwise,

go back to step 6 .

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 4. (Continued from Example 1) For selecting the feasible alternative for feasible consensus strategy, utilizing the above algorithm we have:

Table 5
Decision table

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

From Table 4 and Table 5, we get 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 the 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 answered the second and third questions posed by Deja [6] for the classical Pawlak conflict analysis decision making model of [30].

5.2. Novelty and Advantages

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

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

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

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

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

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

6. Conclusion

This paper presents the idea of soft preference and soft dominance relation in an incomplete information system to solve a multi agent conflict analysis for labor management negotiations problem. We have developed a feasible consensus strategy for labor management negotiations conflict analysis problem based on soft dominance rough sets in an incomplete information system. Further, this paper, several uncertainty measures, such as approximate precision, rough degree, approximate quality, and their mutual relationships are presented and discussed in an incomplete information system.

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Soft ordered approximations and incomplete information system

Abstract

Keywords

1. Introduction

2. Problem statement

4. Core cause of conflict analysis

4.1. Discernibility matrix and discernibility function for multi-attributes with multi-decisions in incomplete information system

Table 3 Dominating classes [ a i ] D om k ( H ∗ , C ) + and Dominated classes [ a i ] D om k ( H ∗ , C ) -

Table 5 Decision table

6. Conclusion

References

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

Table 5
Decision table