Theory and application of conflict resolution with hybrid preference in colored graphs

An algebraic approach is proposed to calculate stabilities in a colored graph with hybrid preference. The algebraic approach establishes a hybrid framework for stability analysis by combining strength of preference and unknown preference. The hybrid system is more general than existing models, which consider preference strength and preference uncertainty separately. Within the hybrid preference structure, matrix representations of four basic stabilities in a colored graph are extended to include mild, strong, and uncertain preference and algorithms are developed to calculate efficiently the inputs essential to the stability definitions. A specific case study, including multiple decision makers and hybrid preference, is used to illustrate how the proposed method can be applied in practice.     

A matrix approach to status quo analysis in the graph model for conflict resolution

An algebraic method is developed to carry out status quo analysis within the framework of the graph model for conflict resolution. As a form of post-stability analysis, status quo analysis aims at confirming that possible equilibria, or states stable for all decision-makers, are in fact reachable from the status quo or any other initial state. Although pseudo-codes for status quo analysis have been developed, they have never been implemented within a practical decision support system. The novel matrix approach to status quo analysis designed here is convenient for computer implementation and easy to employ, as is illustrated by an application to a real-world conflict case. Moveover, the proposed explicit matrix approach reveals an inherent link between status quo analysis and the traditional stability analysis and, hence, provides the possibility of establishing an integrated paradigm for stability and status quo analyses.     

Using matrices to link conflict evolution and resolution in a graph model

The graph model for conflict resolution provides a convenient and effective means to model and analyze a strategic conflict. Standard practice is to carry out a stability analysis of a graph model, and then to follow up with a post-stability analysis, an important component of which is status quo analysis. A graph model can be viewed as an edge-colored graph, but the fundamental problem of status quo analysis – to find a shortest colored path from the status quo node to a desired equilibrium – is different from the well-known network analysis problem of finding the shortest path between two nodes. The only matrix method that has been proposed cannot track all aspects of the evolution of a conflict from the status quo state. Our explicit algebraic approach is convenient for computer implementation and, as demonstrated with a real world case study, easy to use. It provides new insights into a graph model, not only identifying all equilibria reachable from the status quo, but also how to reach them. Moreover, this approach bridges the gap between stability analysis and status quo analysis in the graph model for conflict resolution.     

A matrix-based approach to searching colored paths in a weighted colored multidigraph

An algebraic approach to finding all edge-weighted-colored paths within a weighted colored multidigraph is developed. Generally, the adjacency matrix represents a simple digraph and determines all paths between any two vertices, and is not readily extendable to colored multidigraphs. To bridge the gap, a conversion function is proposed to transform the original problem of searching edge-colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph. Moreover, edge weights can be used to represent some preference attribute. Its potentially wide realm of applicability is illustrated by a case study: status quo analysis in the graph model for conflict resolution. The explicit matrix function is more convenient than other graphical representations for computer implementation and for adapting to other applications. Additionally, the algebraic approach reveals the relationship between a colored multidigraph and a simple digraph, thereby providing new insights into algebraic graph theory.     

Status quo analysis in the graph model for conflict resolution

The graph model for conflict resolution, an analysis paradigm for strategic conflicts, has mainly relied on stability analysis for its conclusions. This paper proposes algorithms to apply another analysis technique, status quo analysis, to a graph model. Status quo analysis is dynamic and forward-looking, in contrast to Stability Analysis, which is static and contingent. Status quo analysis is carried out by means of a directed graph that tracks moves and countermoves from a status quo state, and a table that records the reachability status of states from the status quo. Different algorithms are proposed to produce status quo diagrams with and without restrictions on moves; more efficient versions of the algorithm for the case of transitive graphs are also put forward. Properties of diagrams generated by different algorithms are investigated. A case study illustrates how status quo analysis can be applied, and how it interacts with other analysis methodologies.     

Interval fuzzy preferences in the graph model for conflict resolution

A new analysis technique, appropriate to situations of high preference uncertainty, is added to the graph model for conflict resolution methodology. Interval fuzzy stabilities are now formulated, based on decision makers’ (DMs’) interval fuzzy preferences over feasible scenarios or states in a conflict. Interval fuzzy stability notions enhance the applicability of the graph model, and generalize its crisp and fuzzy preference-based stability ideas. A graph model is both a formal representation and an analysis procedure for multiple participant-multiple objective decisions that employs stability concepts representing various forms of human behavior under conflict. Defined based on a type-2 fuzzy logic, an interval fuzzy preference for one state over another is represented by a subinterval of [0, 1] indicating an interval-valued preference degree for the first state over the second. The interval fuzzy stabilities put forward in this research are interval fuzzy Nash stability, interval fuzzy general metarational stability, interval fuzzy symmetric metarational stability, and interval fuzzy sequential stability. A state is interval fuzzy stable for a DM if moving to any other state is not adequately desirable to the DM; where adequacy is measured by the interval fuzzy satisficing threshold of the DM and farsightedness, involving possible moves and countermoves by DMs, is determined by the interval fuzzy stability notion selected. Note that infinitely many degrees in an interval-valued preference are preserved in characterizing the desirability of a move. A state from which no DM can move to any sufficiently desirable scenario is an interval fuzzy equilibrium, and is interpreted as a possible resolution of the strategic conflict under study. The new stability concept is illustrated through its application to an environmental conflict that took place in Elmira, Ontario, Canada. Insightful results are identified and discussed.     

Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations

In a multi-attribute decision-making (MADM) context, the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives. This paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations. This proposed approach includes three stages. First, pairwise efficiency scores are computed using two DEA models: the CCR model and the proposed cross-evaluation DEA model. Second, the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation. Third, by use of the row wise summation technique, we yield a priority vector, which is used for ranking decision-making units (DMUs). For the case of a single output and a single input, the preference relation can be directly obtained from the original sample data. The proposed approach is validated by two numerical examples.     

Synchronization of networks with time-varying couplings

In this paper, we present a review of our recent works on complete synchro-nization analyses of networks of the coupled dynamical systems with time-varying cou-plings. The main approach is composed of algebraic graph theory and dynamic system method. More precisely, the Hajnal diameter of matrix sequence plays a key role in in-vestigating synchronization dynamics and the joint graph across time periods possessing spanning tree is a doorsill for time-varying topologies to reach synchronization. These techniques with proper modification count for diverse models of networks of the cou-pled systems, including discrete-time and continuous-time models, linear and nonlinear models, deterministic and stochastic time-variations. Alternatively, transverse stability analysis of general time-varying dynamic systems can be employed for synchronization study as a special case and proved to be equivalent to Hajnal diameter.     

基于图模型理论有序偏好下的冲突反问题研究

在冲突谈判中,能获知对手偏好是掌握谈判主动性的重要条件。本文基于冲突分析图模型理论构建了一种获取对手偏好的方法。该方法通过深入分析冲突分析图模型中Nash、GMR和SEQ三种稳定性定义,利用反向思维,建立求解对手偏好最少约束条件的数学模型。该方法能让决策者在预知冲突结局的前提下,得到对手的全部偏好信息。以“云南曲靖陆良县铬污染”冲突事件为例,通过对该事件引发的冲突进行建模和偏好分析,在已知冲突最终结局的前提下,运用数学模型,省环保厅可以得到陆良化工企业的所有偏好序,使其在冲突谈判中做到知己知彼,同时也验证了该方法的可行性和有效性。案例分析过程可以从战略层面为谈判中的一方提供参考。     

基于分段策略评分的冲突主体偏好排序方法及其应用

冲突中各利益主体的偏好信息对冲突局势的演变和纠纷调解具有重要影响。现有的冲突偏好排序方法主要基于决策者对冲突局势或状态、策略权重和声明信息的主观判断和理解,缺乏科学的数据来源支撑。为准确获取冲突主体的偏好信息,本文提出了一种基于调查法的分段策略冲突偏好排序方法。首先,根据决策者类别将冲突策略集合进行分段,并通过问卷、调研等方法获取每个冲突主体对所有分段策略的重要度评分信息。在此基础上,计算决策者对各个冲突状态的综合偏好评分,进而得到状态偏好的排序结果。最后以医患纠纷为例,对比分析了传统策略权重法和分段策略评分法的偏好排序和稳定性分析结果,进一步验证了所提方法的有效性。     

Fuzzy graphs modelling for HazMat telegeomonitoring

Telegeomonitoring system development combines two heterogeneous technologies: the geographical information systems technology (GIS) and telecommunications technology. In this paper, we give the system components for telegeomonitoring transportation of hazardous materials. The telegeomonitoring system uses GIS to capture civil infrastructure (urban network, land use, industries, etc.) and decision support systems technology to allow risks analysis and evaluate routing strategies that minimize transportation risk. Routing algorithms are to this effect adapted to graphs of the fuzzy risk. A new algebraic structure is proposed to solve a path-finding problem in a fuzzy graph. This algebraic structure is adapted precisely to solve the problem of the K-best fuzzy shortest paths. The approach that we proposed, consists of defining generic structures of operator’s traversal problem in fuzzy graphs. The principal contribution of our approach is to build adequate structures of path algebra to solve the problem of graph traversal in a fuzzy graph without negative circuits. Foundations of the system studied in this work will be able to be transposed to other fields of transportation.     

Generalized metarationalities in the graph model for conflict resolution

A metarational tree is defined within the graph model for conflict resolution paradigm, providing a general framework within which rational behavior in models with two decision makers (DMs) can be described more comprehensively. A new definition of stability for a DM that depends on the total number, h, of moves and counter-moves allowed is proposed. Moreover, the metarational tree can be refined so that all moves must be unilateral improvements, resulting in a new set of stability definitions for each level of the tree. Relationships among stabilities at various levels of the basic and refined trees are explored, and connections are established to existing stability definitions including Nash stability, general metarationality, symmetric metarationality, sequential and limited-move stability, and policy equilibria.     

An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers’ incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.     

不同直觉偏好信息下的多属性决策方法

研究了属性值为实数且决策者对属性的偏好信息以直觉判断矩阵或残缺直觉判断矩阵给出的模糊多属性决策问题.首先介绍了直觉判断矩阵、一致性直觉判断矩阵、残缺直觉判断矩阵、一致性残缺直觉判断矩阵等概念,而后分别考虑关于直觉判断矩阵和残缺直觉判断矩阵的多属性决策问题,接着建立了基于直觉判断矩阵和残缺直觉判断矩阵的多属性群决策模型,通过求解这些模型获得属性的权重.进而给出了不同直觉偏好信息下的多属性决策方法.最后通过一个例子说明了该方法的可行性和实用性.     

On the statistical foundations of non-linear utility theory: The case of status quo-dependent preferences

An account of non-expected utility is given which resumes concepts of μ–σ-analysis from statistical decision theory and combines them with standard principles of preference theory such as weak order, continuity and stochastic dominance. A three-parameter family of probability-dependent utility functions is specified, which is governed by the decision maker's aspiration level, distribution of present wealth, or status quo, and discount parameter for future risks. The approach offers a simple resolution of the Allais Paradox and explains basic patterns of probability-dependent risk attitudes arising in theoretical and applied decision analysis.     

Cross-efficiency aggregation in DEA models using the evidential-reasoning approach

Cross-efficiency in data envelopment analysis (DEA) models is an effective way to rank decision-making units (DMUs). The common methods to aggregate cross-efficiency do not consider the preference structure of the decision maker (DM). When a DM’s preference structure does not satisfy the “additive independence” condition, a new aggregation method must be proposed. This paper uses the evidential-reasoning (ER) approach to aggregate the cross-efficiencies obtained from cross-evaluation through the transformation of the cross-efficiency matrix to pieces of evidence. This paper provides a new method for cross-efficiency aggregation and a new way for DEA models to reflect a DM’s preference or value judgments. Additionally, this paper presents examples that demonstrate the features of cross-efficiency aggregation using the ER approach, including an empirical example of the evaluation practice of 16 basic research institutes in Chinese Academy of Sciences (CAS) in 2010 that illustrates how the ER approach can be used to aggregate the cross-efficiency matrix produced from DEA models.     

Choquet-based optimisation in multiobjective shortest path and spanning tree problems

This paper is devoted to the search of Choquet-optimal solutions in finite graph problems with multiple objectives. The Choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives. We first present a condition on preferences (name hereafter preference for interior points) that characterizes preferences favouring compromise solutions, a natural attitude in various contexts such as multicriteria optimisation, robust optimisation and optimisation with multiple agents. Within Choquet expected utility theory, this condition amounts to using a submodular capacity and a convex utility function. Under these assumptions, we focus on the fast determination of Choquet-optimal paths and spanning trees. After investigating the complexity of these problems, we introduce a lower bound for the Choquet integral, computable in polynomial time. Then, we propose different algorithms using this bound, either based on a controlled enumeration of solutions (ranking approach) or an implicit enumeration scheme (branch and bound). Finally, we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes.     

Learning to rank: a ROC-based graph-theoretic approach

This note summarizes the main results presented in the author’s Ph.D. thesis, supervised by Luc Boullart and Bernard De Baets. The thesis was defended on 14th October 2008 at Universiteit Gent. It is written in English and available for download at . The work deals with preference learning, with emphasis on the ranking and ordinal regression machine learning settings and their connections to decision theory. Based on receiver operator characteristics analysis and graph theory, new performance measures are proposed to evaluate this type of models, and new algorithms are presented to compute and optimize these performance measures efficiently. Furthermore, the relationship with other settings like pairwise preference learning and multi-class classification is discussed.     

Overlapping Domain Decomposition Algorithms for General Sparse Matrices

Domain decomposition methods for finite element problems using a partition based on the underlying finite element mesh have been extensively studied. In this paper, we discuss algebraic extensions of the class of overlapping domain decomposition algorithms for general sparse matrices. The subproblems are created with an overlapping partition of the graph corresponding to the sparsity structure of the matrix. These algebraic domain decomposition methods are especially useful for unstructured mesh problems. We also discuss some difficulties encountered in the algebraic extension, particularly the issues related to the coarse solver.     

Multiple levels of preference in interactive strategic decisions

A multiple-level preference ranking structure is developed within the paradigm of the Graph Model for Conflict Resolution to study multi-objective decision making in conflict situations more realistically. In this structure, a decision maker may have multiple levels of preference for one state or scenario over another; for example, if state A is preferred to state B, it may be mildly preferred at level 1, more strongly preferred at level 2, …, or maximally preferred at level r, where r>0 is a fixed parameter. The number of levels, r, is unrestricted in this system, thereby extending earlier two-level (r=1) and three-level (r=2) structures. Multilevel versions of four stability definitions, Nash stability, general metarationality, symmetric metarationality, and sequential stability, are defined for the graph model with this extended preference structure, and the relationships among them are investigated. A specific case study, including multiple decision makers and multiple levels of preference, is carried out to illustrate how the new solution concepts can be applied in practice.