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

Matrix Representations and Extension of the Graph Model for Conflict Resolution

The graph model for conflict resolution (GMCR) 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, two critical components of which are status quo analysis and coalition analysis. In stability analysis, an equilibrium is a state that is stable for all decision makers (DMs) under appropriate stability definitions or solution concepts. Status quo analysis aims to determine whether a particular equilibrium is reachable from a status quo (or an initial state) and, if so, how to reach it. A coalition is any subset of a set of DMs. The coalition stability analysis within the graph model is focused on the status quo states that are equilibria and assesses whether states that are stable from individual viewpoints may be unstable for coalitions. Stability analysis began within a simple preference structure which includes a relative preference relationship and an indifference relation. Subsequently, preference uncertainty and strength of preference were introduced into GMCR but not formally integrated. In this thesis, two new preference frameworks, hybrid preference and multiple-level preference, and an integrated algebraic approach are developed for GMCR. Hybrid preference extends existing preference structures to combine preference uncertainty and strength of preference into GMCR. A multiple-level preference framework expands GMCR to handle a more general and flexible structure than any existing system representing strength of preference. An integrated algebraic approach reveals a link among traditional stability analysis, status quo analysis, and coalition stability analysis by using matrix representation of the graph model for conflict resolution. To integrate the three existing preference structures into a hybrid system, a new preference framework is proposed for graph models using a quadruple relation to express strong or mild preference of one state or scenario over another, equal preference, and an uncertain preference. In addition, a multiple-level preference framework is introduced into the graph model methodology to handle multiple-level preference information, which lies between relative and cardinal preferences in information content. The existing structure with strength of preference takes into account that if a state is stable, it may be either strongly stable or weakly stable in the context of three levels of strength. However, the three-level structure is limited in its ability to depict the intensity of relative preference. In this research, four basic solution concepts consisting of Nash stability, general metarationality, symmetric metarationality, and sequential stability, are defined at each level of preference for the graph model with the extended multiple-level preference. The development of the two new preference frameworks expands the realm of applicability of the graph model and provides new insights into strategic conflicts so that more practical and complicated problems can be analyzed at greater depth. Because a graph model of a conflict consists of several interrelated graphs, it is natural to ask whether well-known results of Algebraic Graph Theory can help analyze a graph model. Analysis of a graph model involves searching paths in a graph but an important restriction of a graph model is that no DM can move twice in succession along any path. (If a DM can move consecutively, then this DM's graph is effectively transitive. Prohibiting consecutive moves thus allows for graph models with intransitive graphs, which are sometimes useful in practice.) Therefore, a graph model must be treated as an edge-weighted, colored multidigraph in which each arc represents a legal unilateral move and distinct colors refer to different DMs. The weight of an arc could represent some preference attribute. Tracing the evolution of a conflict in status quo analysis is converted to searching all colored paths from a status quo to a particular outcome in an edge-weighted, colored multidigraph. Generally, an adjacency matrix can determine a simple digraph and all state-by-state paths between any two vertices. However, if a graph model contains multiple arcs between the same two states controlled by different DMs, the adjacency matrix would be unable to track all aspects of conflict evolution from the status quo. To bridge the gap, a conversion function using the matrix representation is designed to transform the original problem of searching edge-weighted, colored paths in a colored multidigraph to a standard problem of finding paths in a simple digraph with no color constraints. As well, several unexpected and useful links among status quo analysis, stability analysis, and coalition analysis are revealed using the conversion function. The key input of stability analysis is the reachable list of a DM, or a coalition, by a legal move (in one step) or by a legal sequence of unilateral moves, from a status quo in 2-DM or $n$-DM ($n > 2$) models. A weighted reachability matrix for a DM or a coalition along weighted colored paths is designed to construct the reachable list using the aforementioned conversion function. The weight of each edge in a graph model is defined according to the preference structure, for example, simple preference, preference with uncertainty, or preference with strength. Furthermore, a graph model and the four basic graph model solution concepts are formulated explicitly using the weighted reachability matrix for the three preference structures. The explicit matrix representation for conflict resolution (MRCR) that facilitates stability calculations in both 2-DM and $n$-DM ($n > 2$) models for three existing preference structures. In addition, the weighted reachability matrix by a coalition is used to produce matrix representation of coalition stabilities in multiple-decision-maker conflicts for the three preference frameworks. Previously, solution concepts in the graph model were traditionally defined logically, in terms of the underlying graphs and preference relations. When status quo analysis algorithms were developed, this line of thinking was retained and pseudo-codes were developed following a similar logical structure. However, as was noted in the development of the decision support system (DSS) GMCR II, the nature of logical representations makes coding difficult. The DSS GMCR II, is available for basic stability analysis and status quo analysis within simple preference, but is difficult to modify or adapt to other preference structures. Compared with existing graphical or logical representation, matrix representation for conflict resolution (MRCR) is more effective and convenient for computer implementation and for adapting to new analysis techniques. Moreover, due to an inherent link between stability analysis and post-stability analysis presented, the proposed algebraic approach establishes an integrated paradigm of matrix representation for the graph model for conflict resolution

Scholarship at UWindsor Annual Report: Nov 2012-2013

This is the first annual report for the Scholarship at UWindsor institutional Repository

Strategic management for large engineering projects : the stakeholder value network approach

Thesis (Ph. D. in Technology, Management, and Policy)--Massachusetts Institute of Technology, Engineering Systems Division, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 280-302).A critical element of the challenges and opportunities for today's large engineering projects are associated with the multi-type and networked relationships between these projects and their various stakeholders. This dissertation advances a multidisciplinary approach-Stakeholder Value Network (SVN) analysis-as a unique lens to examine, understand, model, and manage these stakeholder relationships. The SVN approach, based on the Social Exchange Theory (SET), unifies both social and economic relationships into a common framework, under which all the stakeholder relationships are formed by the use of subjective utility analysis and the comparison of alternatives. Next, restricted and generalized exchanges are identified as two basic patterns for stakeholders to exchange both tangible and intangible value, and from this, the missing links between relationship types and exchange patterns are also discovered. In the end, the network implications, such as stakeholder importance or salience, are inferred as the outcome of both value exchanges and the structural properties of the network consisting of stakeholders and their exchange relationships. According to the above theoretically grounded assumptions, a four-step methodological framework (viz., Mapping, Quantifying, Searching, and Analyzing) is developed for the SVN analysis. As part of this development, a network utility model is built to quantify the value delivered to the focal organization (viz., the large engineering projects) through the channel of generalized exchanges. Meanwhile, the benefits from as well as a feasible way for the integration of stakeholders and strategic issues are explored under the SVN framework. In addition, for the purpose of reducing the egocentric bias associated with the pre-selection of a focal organization, the four-step framework is further developed to interpret the implications of the SVN from the perspective of the whole network. The computational challenges arising from this new development are met by the construction of a dedicated mathematical tool for the SVN analysis, namely, the Dependency Structure Matrix (DSM) modeling platform. Corresponding to the two-stage development of the methodological framework, two large real-world engineering projects are studied respectively: The first one, Project Phoenix, is a retrospective case and applies the SVN analysis from the focal organization perspective. Based on this case study, the descriptive accuracy of the SVN analysis is validated, through a comparison of important stakeholders derived from Mangers' Mental Model, the "Hub-and-Spoke" Model, and the SVN Model. Specifically, it is found that Managers' Mental Model is similar to the "Hub-and-Spoke" Model, and both models miss the Public Media and the Local Governments as important stakeholders at the beginning of the project. On the contrary, even with only prior information, the SVN Model identifies the importance of these two stakeholders by capturing the impacts of indirect stakeholder relationships as generalized exchanges. The reasons why generalized exchanges matter for today's large engineering projects are further examined from psychological, sociological, economic, and managerial aspects. The second one, China's Energy Conservation Campaign, is a prospective case and applies the SVN analysis from the whole network perspective. In this case study, five basic principles are first proposed for modeling the intraorganizational hierarchies of large and important stakeholders, and then these principles are tested as an effective means to manage the structural complexity of the SVN in the modeling process. During this process, the instrumental power of the SVN analysis is demonstrated. The SVN approach becomes complete with the above theory, methodology, tool, and meaningful findings from two representative case studies. At the end of this dissertation, two conceptual innovations are conceived to bridge the gap between the SVN analysis and systems architecting, and the theoretical, methodological, as well as empirical directions of future research on the SVN approach are also discussed.by Wen Feng.Ph.D.in Technology, Management, and Polic

Recent advances in petri nets and concurrency

CEUR Workshop Proceeding