Labor-management negotiation conflict analysis based on soft preference relation

Abstract

Conflict analysis plays a prominent role in negotiation during contract-management process in government and industry. The main problem to be solved is how to model conflict situation when there is uncertainty about agreement, disagreement and neutrality among agents in a conflict situation. This paper aims to introduce the novel concepts of the hybridized structures called soft preference relation and soft dominance relation. Further we initiate the approach to handle the labor-management negotiation conflict situation using soft preference and soft dominance relations. Another novelty of the proposed techniques is to classify exactly the agreement, disagreement and neutrality among all the agents in a conflict situation. In addition the proposed techniques can be applied to find the character of all the agents in the conflict situation when compared with other existing techniques.

Keywords

Conflict analysis soft set preference relation soft preference relation soft dominance relation

1 Introduction

Every one encounters conflict in everyday 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 [19]. 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 or more 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 [9].

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 [25], rough set theory [22], vague set theory [11], soft set theory [17] and hesitant fuzzy sets [21]. Decision making is one of the key components to achieve objectives in many areas, particularly in a field which obligates analyzing the conflict.

Applications of rough set theory in case of conflict analysis are introduced by Pawlak [23]. 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 Daja [9] 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 cause(s) 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.

Molodtsov’s soft set theory [17] was proposed as a general mathematical tool to deal with uncertainty the rationale 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 parameterization 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 [16]. Ali et al. [2] 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 [1]. Applications of soft sets in decision making problems have been studied by many authors in different contexts [6 , 13]. 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 [5 , 26–32]).

1.1 Motivation and significance

Sun et al. [24] developed a conflict analysis model based on rough set theory over two universes to overcome the above mentioned shortcomings in Pawlak’s conflict analysis. But in their proposals, still there are many areas for critics, for example, they did not answer the second question posed by Deja which is related to the development of proper and feasible consensus strategy.

The present paper aims to initiate the novel concepts of the hybridized structures called soft preference relation and soft dominance relation. Further we put forward the approach to handle the labor-management negotiation conflict situation using soft preference and soft dominance relations. Similarly the proposed technique addresses all the issues of the Sun’s model and answers the first and second questions of Deja in a more better way. Another novelty of the proposed techniques is to classify exactly the agreement, disagreement and neutrality among all the agents in a conflict situation. In addition the proposed techniques can be applied to find the character of all the agents in the conflict situation when compared with other existing techniques. The general algorithms for conflict problem are proposed and examined that our newly developed model is more efficient than the existing techniques/models.

The rest of the paper is organized as follows: In Section 2, labor-management negotiation problem is discussed and some existing conflict models have been perused. Section 3, consists on the novel idea of conflict analysis based on soft preference relation and soft dominance relation. Sections 4 and 5, mainly focus on the algorithms for labor-management negotiation problem.

2 Labor-management negotiation conflict analysis problem

In this section, we discuss an example of conflict analysis problem for labor-management negotiation. In this problem, there are five disputes or issues such as the, Incomes (C₁) , Working Conditions (C₂), Factory Profits (C₃) and Training or Promotion of the Personnel (C₄) . At the same time, there are ten agents {u₁, u₂, . . . u₁₀} who are included by the employees, the shareholders, the officers and other related personnel. The values of the integers are defined as follows: 0-against, 1-neutral and 2-favorable. The relationships of each agent to the considered issues are represented in Table 1.

Table 1
The relationships of each agent to the considered issues

C ₁ C ₂ C ₃ C ₄

u ₁ 2 0 1 0

u ₂ 1 2 0 1

u ₃ 0 1 0 0

u ₄ 1 1 0 2

u ₅ 0 2 0 1

u ₆ 1 2 1 0

u ₇ 2 2 1 2

u ₈ 0 1 0 2

u ₉ 2 2 1 0

u ₁₀ 0 0 0 1

	C ₁	C ₂	C ₃	C ₄
u ₁	2	0	1	0
u ₂	1	2	0	1
u ₃	0	1	0	0
u ₄	1	1	0	2
u ₅	0	2	0	1
u ₆	1	2	1	0
u ₇	2	2	1	2
u ₈	0	1	0	2
u ₉	2	2	1	0
u ₁₀	0	0	0	1

2.1 Deja’s conflict analysis model

Pawlak discussed the Middle East conflict analysis model based on rough set theory in [23]. He introduced the discernibility. matrix for the Middle East conflict situation. The Pawlak discernibility matrix of the conflict situation gives only the difference of attitudes between any two agents but gives no information about the preference(s) of agents. Analysis of conflict described in [23] 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 can not be determined, there is no way to specify the situation to avoid the conflict. Moreover, it can not be sure that the issues the agents vote represent the issues each agent takes care of. Though the Pawlak’s conflict analysis model has proven to be an effective method in practice, yet Deja in [8], 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?

2.2 Sun et al. conflict analysis model

Sun et al. in [24] tried to answer Deja’s questions by focusing the answer to first and second questions raised by Deja in [8], and tried to answer with the help of rough set theory over two universes. Their model can be summarized as:

Let U be the universe of agents and V be the universe of issues. The intrinsic causes in a conflict situation means that what issue are focussed by every agent and have different attitudes in a conflict. Also for any subset $Y \in P (V)$ (where $P (V)$ denotes the all subsets of 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 : ={ C₁, . . . , C_n } be the universe of issues in the conflict situation, f : ={ f², f⁰ } be the set valued mapping from U to $P (V),$ where $f^{2} : U \to P (V),$ $f^{2} (u) = {C \in V ∣ f (u, C) = 2 all u \in U},$ and $f^{0} : U \to P (V),$ $f^{0} (u) = {C \in V ∣ f (u, C) = 0 for all u \in U} .$ The mapping f² means the subset of issues of the universe V with which agent u of universe U agrees. The mapping f⁰ means the subset of issues of the universe V with which agent u of universe U disagrees. Similarly if g ={ g², g⁰ } is a set valued mapping from V to U, then

$g^{2} : V \to P (U),$ $g^{2} (C) : = {u \in U ∣ g (C, u) : = 2 all C \in V},$

$g^{0} : V \to P (U),$ $g^{0} (C) = {u \in U ∣ g (C, u) = 0 all C \in V} .$

Definition 1. [24] Let CS : = (U, V, f, g) be a conflict situation over two universes. Then $\begin{matrix} d_{v}^{0} (u_{i}, u_{j}) = {C_{k} \in V ∣ f^{0} (u_{i}) \cap f^{0} (u_{j}), \\ for all u_{i}, u_{j} \in U} \end{matrix}$ the consistent disagreement set of the agent u_i with u_j . Similarly $D_{V}^{0} : = {d_{v}^{0} (u_{i}, u_{j}) ∣ u_{i}, u_{j} \in U}$ is the consistent disagreement matrix and $h_{CS}^{0} : = \land {\lor d_{v}^{0} (u_{i}, u_{j}) ∣ d_{v}^{0} (u_{i}, u_{j}) \in D_{V}^{0}}$ is the consistent disagreement function of the CS : = (U, V, f, g).

Example 2. Consider the labor-managment negotiation conflict analysis as in Table 1. The disagreement matrix in the conflict situation is given in Table 2 (See Appendix 1).

The consistent disagreement function is defined by: $h_{CS}^{0} (u_{1}, u_{2}, . . ., u_{10}) : = {C_{2}, C_{3}} + {C_{3}, C_{4}} .$

This shows that all the agents in the labor-management negotiation conflict situation are not agree on the strategy {C₂, C₃} or the strategy {C₃, C₄ } . This means that all the agents are not agree on the issues C₂ and C₃ or on the issues C₃ and C₄ . So the agents may be against or neutral on these issues. Therefore the technique of [24] does not provide the exact information that on which issue all the agents in the labor-management negotiation conflict situation are against and on which issue they are neutral.

Similarly by aforesaid technique, the feasible consensus strategy for the labor-managment negotiation conflict situation is as follows: $\begin{matrix} feasible consensus strategy \\ = V - h_{CS}^{0} (u_{1}, u_{2}, ..., u_{10}) \\ = {C_{1}, C_{4}} (i) \end{matrix}$ and $\begin{matrix} feasible consensus strategy \\ = V - h_{CS}^{0} (u_{1}, u_{2}, ..., u_{10}) \\ = {C_{1}, C_{2}} (ii) \end{matrix}$ From (i) and (ii) all agents in the labor-management negotiation conflict situation are not against the strategy {C₁, C₄} or the strategy {C₁, C₂ } . This reveals that all the agents are not against the issues C₁ and C₄ or the issues C₁ and C₂ . Thus the behavior of all agents in the labor-management negotiation conflict situation are either neutral or favorable to these issues. But the technique of [24] does not clarify exactly on which issues all agents in the labor management negotiations conflict situation are neutral or favorable.

3 Conflict analysis based on Soft preference relation

Inspired by the existing studies on conflict analysis based on Pawlak rough set theory and Sun et al. conflict analysis based on rough set theory over two universes and by matrix approach to conflict analysis. We stipulate a new conflict analysis model based on soft preference relation and soft dominance relation which 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 some 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 .

Strong 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-dominates 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. Throughout the paper, denote x ≽ y by the pair (x, y) . The other representation of the labor-management negotiation conflict information is given in Table 1(i) (See Appendix 2).

Definition 3. [17] A pair $R : = (F, E)$ is called a soft set over U, where E is a subset of parameters V and $F : E \to P (U)$ is a set valued mapping.

Definition 4. Let S : = (U, V, 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 $G (u) = {a_{i} ≽ a_{j} ∣ g (a_{i}, u) \geq g (a_{j}, u)}$ where a_i, a_j ∈ V . In information system (Table 1(i)) favorability (2) is preferred by neutrality (1) and neutrality (1) is preferred by against (0). In fact the 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.

Definition 5. If (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 an information system. In the square table if (C_i, C_j) ∈ Dom (G, I) , then (C_i, C_j) : =1, otherwise (C_i, C_j) : =0 where C_i, C_j ∈ V . We develop a new and more efficient algorithm for labor-management negotiation conflict analysis in the following sections.

4 Algorithm-I for Labor-management conflict analysis

This algorithm is used to find the feasible consensus strategy among the agents. The main steps of this algorithm are:

Input: Information System;

Construct soft preference relation (G, I) by using Definition 4;

Construct soft dominance relation Dom(G,I) by using to Definition 5;

Construct square table for Dom(G,I) by using 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 strategy respectively;

Output: Agreement, Disagreement and Neutral strategy.

Example 6. Consider the labor-managment negotiation conflict analysis as in Table 1(i).

Let $G : U \to P (V \times V)$ and (G, U) be a soft preference relation such that: $G (u_{1}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{2}), (C_{1}, C_{3}) \\ (C_{1}, C_{4}), (C_{2}, C_{4}), (C_{3}, C_{2}) \\ , (C_{3}, C_{4}), (C_{4}, C_{2}) \end{matrix}},$ $G (u_{2}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{3}), (C_{1}, C_{4}), \\ (C_{2}, C_{1}), (C_{2}, C_{3}), (C_{2}, C_{4}), \\ (C_{4}, C_{1}), (C_{4}, C_{3}) \end{matrix}},$ $G (u_{3}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{3}), (C_{1}, C_{4}), \\ (C_{2}, C_{1}), (C_{2}, C_{3}), (C_{2}, C_{4}), \\ (C_{3}, C_{1}), (C_{3}, C_{4}), (C_{4}, C_{1}), \\ (C_{4}, C_{3}) \end{matrix}},$ $G (u_{4}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{2}), (C_{1}, C_{3}), \\ (C_{2}, C_{1}), (C_{2}, C_{3}), (C_{4}, C_{1}), \\ (C_{4}, C_{2}), (C_{4}, C_{3}) \end{matrix}},$ $G (u_{5}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{3}), (C_{2}, C_{1}), \\ (C_{2}, C_{3}), (C_{2}, C_{4}), (C_{3}, C_{1}), \\ (C_{4}, C_{1}), (C_{4}, C_{3}) \end{matrix}},$ $G (u_{6}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{3}), (C_{1}, C_{4}), \\ (C_{2}, C_{1}), (C_{2}, C_{3}), (C_{2}, C_{4}), \\ (C_{3}, C_{1}), (C_{3}, C_{4}) \end{matrix}},$ $G (u_{7}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{2}), (C_{1}, C_{3}), \\ (C_{1}, C_{4}), (C_{2}, C_{1}), (C_{2}, C_{3}), \\ (C_{2}, C_{4}), (C_{4}, C_{1}), (C_{4}, C_{2}), \\ (C_{4}, C_{3}) \end{matrix}},$ $G (u_{8}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{3}), (C_{2}, C_{1}) \\ (C_{2}, C_{3}), (C_{3}, C_{1}), (C_{4}, C_{1}), \\ (C_{4}, C_{2}), (C_{4}, C_{3}) \end{matrix}},$ $G (u_{9}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{2}), (C_{1}, C_{3}), \\ (C_{1}, C_{4}), (C_{2}, C_{1}), (C_{2}, C_{3}), \\ (C_{2}, C_{4}), (C_{3}, C_{4}) \end{matrix}},$ $G (u_{10}) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), (C_{3}, C_{3}), \\ (C_{4}, C_{4}), (C_{1}, C_{2}), (C_{1}, C_{3}), \\ (C_{2}, C_{1}), (C_{2}, C_{3}), (C_{3}, C_{1}), \\ (C_{3}, C_{2}), (C_{4}, C_{1}), (C_{4}, C_{2}), \\ (C_{4}, C_{3}), \end{matrix}} .$ The soft dominance relation is given by $Dom (G, U) = {\begin{matrix} (C_{1}, C_{1}), (C_{2}, C_{2}), \\ (C_{3}, C_{2}), (C_{4}, C_{4}), \\ (C_{1}, C_{3}), \end{matrix}} .$ Construct the square table for Dom(G,U) as follows:

In Table 3, C . Sum represent the column wise sum and R . Sum denote the row wise sum.

Table 3
Square Table 3. For Dom(G,U)

C ₁ C ₂ C ₃ C ₄ R . Sum

C ₁ 1 0 1 0 2

C ₂ 0 1 0 0 1

C ₃ 0 0 1 0 1

C ₄ 0 0 0 1 1

C . sum 1 1 2 1

	C ₁	C ₂	C ₃	C ₄	R . Sum
C ₁	1	0	1	0	2
C ₂	0	1	0	0	1
C ₃	0	0	1	0	1
C ₄	0	0	0	1	1
C . sum	1	1	2	1

By using the aforesaid algorithm, from Table 4, the agreement strategy: ={ C₁ }, disagreement strategy: ={ C₃ } and neutral strategy: ={ C₂, C₄ }. This shows that all agents in the labor-managment negotiation conflict situation have agreed on the strategy {C₁}, disagreed on the strategy {C₃} and all agents in the labor-managment negotiation conflict situation are neutral on the strategy {C₂, C₄}. Also we see that the feasible consensus strategy {C₁ } ⊆ 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, which is helpful to build the consensus and solve the conflict situation for the decision makers.

Table 4

For strategy

V	C ₁	C ₂	C ₃	C ₄
R . Sum- C . Sum	1	0	-1	0

Definition 7. Let CS : = (U, V, f, g) be a conflict situation over two universes. Then $\begin{matrix} d_{U}^{0} (C_{i}, C_{j}) = {u_{k} \in U ∣ g^{0} (C_{i}) \cap g^{0} (C_{j}), \\ \forall C_{i}, C_{j} \in V} \end{matrix}$ the consistent disagreement set of the agent u_i with u_j and $\begin{matrix} d_{U}^{2} (C_{i}, C_{j}) = {u_{k} \in U ∣ g^{2} (C_{i}) \cap g^{2} (C_{j}), \\ \forall C_{i}, C_{j} \in V} \end{matrix}$ the consistent agreement set of the agent u_i with u_j, where $D_{U}^{0} : = {d_{U}^{0} (C_{i}, C_{j}) ∣ C_{i}, C_{j} \in V}$ the consistent disagreement matrix and $D_{U}^{2} : = {d_{U}^{2} (C_{i}, C_{j}) ∣ C_{i}, C_{j} \in V}$ the consistent agreement matrix. Similarly $h_{CS}^{0} : = \land {\lor d_{U}^{0} (C_{i}, C_{j}) ∣ d_{U}^{0} (C_{i}, C_{j}) \in D_{U}^{0}}$ and $h_{CS}^{2} : = \land {\lor d_{U}^{2} (C_{i}, C_{j}) ∣ d_{U}^{2} (C_{i}, C_{j}) \in D_{U}^{2}}$ is the negative alliance function and positive alliance function, respectively, of the CS : = (U, V, f, g) .

Example 8. Consider the labor-managment negotiation conflict analysis as in Table 1. According to Definition 7, the positive alliance matrix is given in Table 5 (See Appendix 3). From Table 5, the positive alliance function is given by $\begin{matrix} h_{CS}^{2} (C_{1}, C_{2}, C_{3}, C_{4}) \\ = {u_{6}, u_{7},} + {u_{4}, u_{6}, u_{9}} \\ + {u_{4}, u_{6}, u_{9}} (iii) \end{matrix}$ From (iii) , there are three alliances which are not against on the set of issues {C₁, C₂, C₃, C₄ } . This means that the behavior of these alliances on the set of issues {C₁, C₂, C₃, C₄} may be favorable or neutral. In a similar way, the negative alliance matrix is given in Table 6 (See Appendix 4). From Table 6, the negative alliance function of the labor-management negotiation conflict situation is $\begin{matrix} h_{CS}^{0} (C_{1}, C_{2}, C_{3}, C_{4}) \\ = {u_{1}, u_{3}, u_{10},} + {u_{3}, u_{6}, u_{10}} (iv) . \end{matrix}$ From (iv) , there are three alliances which are not favorable on the set of issues {C₁, C₂, C₃, C₄ } . This reveals that the behavior of these alliances on the set of issues {C₁, C₂, C₃, C₄} may be against or neutral. But the Sun et al. technique do not provide information about the agents u₂ and u₅ on the set of issues {C₁, C₂, C₃, C₄} whether they are against, neutral or favorable.

Definition 9. Let S : = (U, V, 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 $F (a) : = {u_{i} ≽ u_{j} ∣ f (u_{i}, a) \geq f (u_{j}, a)}$ where u_i, u_j ∈ U .

Remark 10. For any issue C_i ∈ V, we can easily obtain attitude information for every agent with respect to issue C_i ∈ V by using F (C_i) which is a preference relation, since F (C_i) is more informative than those of [24], the two set valued mappings. Thus we have answered the question “What are the intrinsic reasons for the conflict?” by using F (C_i) where C_i ∈ V .

Definition 11. If (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: $Dom (F, E) : = ⋂_{C \in E} F (C) .$ 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 .

Now we present methodology for the feasible consensus strategy of the conflict situation. We develop another algorithm for labor-management negotiation conflict analysis in the following section.

5 Algorithm-II for labor-management conflict analysis

This algorithm is used to find the intrinsic reasons for the conflict and another feasible consensus strategy among the agents. The main steps of this algorithm are as follows.

Input: Information System:

Construct soft preference relation (F, V) by using Definition 9;

Construct soft dominance relation Dom(F, V) by using Definition 11;

Construct square table for Dom(F, V) by using Definition 11;

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 number(s) including zero and negative number(s) including zero showing positive and negative alliances respectively;

Output: Positive alliance and Negative alliance.

Remark 12. Using the above algorithm, we may have two sets such as positive alliance and negative alliance of the labor-managment negotiation conflict situation. Their relations are given below:

(Positive alliance set)∩(negative alliance set):≠∅.

(Positive alliance set)∪(negative alliance set):= U.

(Positive alliance set)∩(negative alliance set):=Neutral alliance set.

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

Example 13. Consider the labor-managment negotiation conflict analysis as in Table 1.

Using the proposed algorithm,

Positive alliance: = { u₂, u₄, u₅, u₇, u₈, u₉ } , Negative alliance: ={ u₁, u₃, u₅, u₆, u₈, u₁₀ } then Neutral alliance: = { u₅, u₈ } .

Remark 14.

If all the agents in positive alliance of the labor-managment negotiation conflict situation are in agreement with all criteria, then every subset of this alliance also have the same nature as that of the parent alliance.

Similarly if all the agents in negative alliance of the labor-managment negotiations conflict situation are in disagreement with all criteria, then every subset of this alliance also have the same nature as that of the parent alliance.

In [24] information about some agents are missing, for example, the nature (classification) of u₂ and u₅, but our proposed technique discussed all agents, as seen in Example 13 that u₂ and u₅ have been put rightly in positive alliance.

6 Conclusion

In daily life, we always face 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 the labor-management negotiation conflict analysis. This model was studied initially by Sun et al. in [24] with the help of discernibility matrix and tried to answer the Deja’s questions. But their technique has also some shortcomings which make the study of conflict analysis more ambiguous.

In this paper, a new conflict analysis model to study the labor-management negotiation conflict situation 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. Two general algorithms have been proposed in this paper. Thus the proposed conflict analysis model depict the real spirit of the conflict situation. The proposed model can be extended to study other conflicts around the world.

Footnotes

Acknowledgments

C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).

Appendices

Appendix 1

Table 2. Disagreement matrix $D_{V}^{0}$ for labor-managment negotiations conflict situation

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀
u ₁	${\begin{matrix} C_{2}, \\ C_{4} \end{matrix}}$
u ₂	∅	{C₃}
u ₃	{C₄}	{C₃}	${\begin{matrix} C_{1}, \\ C_{3}, \\ C_{4} \end{matrix}}$
u ₄	∅	{C₃}	{C₃}	{C₃}
u ₅	∅	{C₃}	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$	{C₃}	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$
u ₆	${\begin{matrix} C_{2}, \\ C_{4} \end{matrix}}$	∅	{C₄}	∅	∅	${\begin{matrix} C_{2}, \\ C_{4} \end{matrix}}$
u ₇	∅	∅	∅	∅	∅	∅	∅
u ₈	∅	∅	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$	{C₃}	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$	∅	∅	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$
u ₉	{C₄}	∅	{C₄}	∅	{C₄}	{C₄}	∅	∅	{C₄}
u ₁₀	{C₂}	{C₃}	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$	{C₃}	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$	{C₂}	∅	${\begin{matrix} C_{1}, \\ C_{3} \end{matrix}}$	∅	${\begin{matrix} C_{1}, \\ C_{2}, \\ C_{3} \end{matrix}}$

Appendix 2

Information Table 1(i) for labor-management negotiation

	u ₁	u ₂	u ₃	u ₄	u ₅	u ₆	u ₇	u ₈	u ₉	u ₁₀
C ₁	2	1	0	1	0	1	2	0	2	0
C ₂	0	2	1	1	2	2	2	1	2	0
C ₃	1	0	0	0	0	1	1	0	1	0
C ₄	0	1	0	2	1	0	2	2	0	1

Appendix 3

Table 5. Positive alliance matrix

	C ₁	C ₂	C ₃	C ₄
C ₁	{u₁, u₇, u₉}
C ₂	{u₇, u₉}	{u₂, u₅, u₆, u₇, u₉}
C ₃	∅	{u₆}	{u₆}
C ₄	{u₇}	{u₇}	∅	{u₄, u₇, u₈}

Appendix 4

Table 6. Negative alliance matrix

	C ₁	C ₂	C ₃	C ₄
C ₁	{u₃, u₅, u₈, u₁₀}
C ₂	{u₁₀}	{u₁, u₆, u₁₀}
C ₃	{u₃, u₅, u₈, u₁₀}	{u₁₀}	{u₂, u₃, u₄, u₅, u₈, u₁₀}
C ₄	{u₃}	{u₁, u₆}	∅	{u₁, u₃, u₆, u₉}

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Labor-management negotiation conflict analysis based on soft preference relation

Abstract

Keywords

1 Introduction

1.1 Motivation and significance

2 Labor-management negotiation conflict analysis problem

Table 1 The relationships of each agent to the considered issues C 1 C 2 C 3 C 4 u 1 2 0 1 0 u 2 1 2 0 1 u 3 0 1 0 0 u 4 1 1 0 2 u 5 0 2 0 1 u 6 1 2 1 0 u 7 2 2 1 2 u 8 0 1 0 2 u 9 2 2 1 0 u 10 0 0 0 1

2.2 Sun et al. conflict analysis model

3 Conflict analysis based on Soft preference relation

3.1 Basic properties of binary relations

4 Algorithm-I for Labor-management conflict analysis

Table 3 Square Table 3. For Dom(G,U) C 1 C 2 C 3 C 4 R . Sum C 1 1 0 1 0 2 C 2 0 1 0 0 1 C 3 0 0 1 0 1 C 4 0 0 0 1 1 C . sum 1 1 2 1

6 Conclusion

Footnotes

Acknowledgments

Appendices

References

Table 1
The relationships of each agent to the considered issues

C ₁ C ₂ C ₃ C ₄

u ₁ 2 0 1 0

u ₂ 1 2 0 1

u ₃ 0 1 0 0

u ₄ 1 1 0 2

u ₅ 0 2 0 1

u ₆ 1 2 1 0

u ₇ 2 2 1 2

u ₈ 0 1 0 2

u ₉ 2 2 1 0

u ₁₀ 0 0 0 1

Table 3
Square Table 3. For Dom(G,U)

C ₁ C ₂ C ₃ C ₄ R . Sum

C ₁ 1 0 1 0 2

C ₂ 0 1 0 0 1

C ₃ 0 0 1 0 1

C ₄ 0 0 0 1 1

C . sum 1 1 2 1