Advances in Cone-Based Preference Modeling for Decision Making with Multiple Criteria

Decision making with multiple criteria requires preferences elicited from the decision maker to determine a solution set. Models of preferences, that follow upon the concept of nondominated solutions introduced by Yu (1974), are presented and compared within a unified framework of cones. Polyhedral and nonpolyhedral, convex and nonconvex, translated, and variable cones are used to model different types of preferences. Common mathematical properties of the preferences are discussed. The impact of using these preferences in decision making is emphasized

Numerical procedures in multiobjective optimization with variable ordering structures

Multiobjective optimization problems with a variable ordering structure instead of a partial ordering have recently gained interest due to several applications. In the last years a basic theory has been developed for such problems. The difficulty in their study arises from the fact that the binary relations of the variable ordering structure, which are defined by a cone-valued map which associates to each element of the image space a pointed convex cone of dominated or preferred directions, are in general not transitive. In this paper we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn-Graef-Younes method known from multiobjective optimization with a partial ordering is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison

Capturing preferences for inequality aversion in decision support

We investigate the situation where there is interest in ranking distributions (of income, of wealth, of health, of service levels) across a population, in which individuals are considered preferentially indistinguishable and where there is some limited information about social preferences. We use a natural dominance relation, generalized Lorenz dominance, used in welfare comparisons in economic theory. In some settings there may be additional information about preferences (for example, if there is policy statement that one distribution is preferred to another) and any dominance relation should respect such preferences. However, characterising this sort of conditional dominance relation (specifically, dominance with respect to the set of all symmetric increasing quasiconcave functions in line with given preference information) turns out to be computationally challenging. This challenge comes about because, through the assumption of symmetry, any one preference statement (“I prefer giving $100 to Jane and$110 to John over giving $150 to Jane and$90 to John”) implies a large number of other preference statements (“I prefer giving $110 to Jane and$100 to John over giving $150 to Jane and$90 to John”; “I prefer giving $100 to Jane and$110 to John over giving $90 to Jane and$150 to John”). We present theoretical results that help deal with these challenges and present tractable linear programming formulations for testing whether dominance holds between any given pair of distributions. We also propose an interactive decision support procedure for ranking a given set of distributions and demonstrate its performance through computational testing

Equitable Efficiency in Multiple Criteria Optimization

Equitable efficiency in multiple criteria optimization was introduced mathematically in the middle of nineteen-nineties. The concept tends to strengthen the notion of Pareto efficiency by imposing additional conditions on the preference structure defining the Pareto preference. It is especially designed to solve multiple criteria problems having commensurate criteria where different criteria values can be compared directly. In this dissertation we study some theoretical and practical aspects of equitably efficient solutions. The literature on equitable efficiency is not very extensive and provides very limited number of ways of generating such solutions. After introducing some relevant notations, we develop some scalarization based methods of generating equitably efficient solutions. The scalarizations developed do not assume any special structure of the problem. We prove an existence result for linear multiple criteria problems. Next, we show how equitably efficient solutions arise in the context of a particular type of linear complementarity problem and matrix games. The set of equitably efficient solutions, in general, is a subset of efficient solutions. The multiple criteria alternative of the linear complementarity problem dealt in our dissertation has identical efficient and equitably efficient solution sets. Finally, we demonstrate the relevance of equitable efficiency by applying it to the problem of regression analysis and asset allocation

Capturing preferences for inequality aversion in decision support

We investigate the situation where there is interest in ranking distributions (of income, of wealth, of health, of service levels) across a population, in which individuals are considered preferentially indistinguishable and where there is some limited information about social pref- erences. We use a natural dominance relation, generalized Lorenz dominance, used in welfare comparisons in economic theory. In some settings there may be additional information about preferences (for example, if there is policy statement that one distribution is preferred to an- other) and any dominance relation should respect such preferences. However, characterising this sort of conditional dominance relation (specifically, dominance with respect to the set of all symmetric increasing quasiconcave functions in line with given preference information) turns out to be computationally challenging. This challenge comes about because, through the as- sumption of symmetry, any one preference statement (“I prefer giving $100 to Jane and$110 to John over giving $150 to Jane and$90 to John”) implies a large number of other preference statements (“I prefer giving $110 to Jane and$100 to John over giving $150 to Jane and$90 to John”; “I prefer giving $100 to Jane and$110 to John over giving $90 to Jane and$150 to John”). We present theoretical results that help deal with these challenges and present tractable linear programming formulations for testing whether dominance holds between any given pair of distributions. We also propose an interactive decision support procedure for ranking a given set of distributions and demonstrate its performance through computational testing

Domination and Decomposition in Multiobjective Programming

During the last few decades, multiobjective programming has received much attention for both its numerous theoretical advances as well as its continued success in modeling and solving real-life decision problems in business and engineering. In extension of the traditionally adopted concept of Pareto optimality, this research investigates the more general notion of domination and establishes various theoretical results that lead to new optimization methods and support decision making. After a preparatory discussion of some preliminaries and a review of the relevant literature, several new findings are presented that characterize the nondominated set of a general vector optimization problem for which the underlying domination structure is defined in terms of different cones. Using concepts from linear algebra and convex analysis, a well known result relating nondominated points for polyhedral cones with Pareto solutions is generalized to nonpolyhedral cones that are induced by positively homogeneous functions, and to translated polyhedral cones that are used to describe a notion of approximate nondominance. Pareto-oriented scalarization methods are modified and several new solution approaches are proposed for these two classes of cones. In addition, necessary and sufficient conditions for nondominance with respect to a variable domination cone are developed, and some more specific results for the case of Bishop-Phelps cones are derived. Based on the above findings, a decomposition framework is proposed for the solution of multi-scenario and large-scale multiobjective programs and analyzed in terms of the efficiency relationships between the original and the decomposed subproblems. Using the concept of approximate nondominance, an interactive decision making procedure is formulated to coordinate tradeoffs between these subproblems and applied to selected problems from portfolio optimization and engineering design. Some introductory remarks and concluding comments together with ideas and research directions for possible future work complete this dissertation

Inequity-averse decisions in operational research

This thesis is on inequity-averse decisions in operational research, and draws on concepts from economics and operational research such as multi-criteria decision making (MCDM) and mathematical modelling. The main focus of the study is developing systematic methods and modelling to help decision makers (DMs) in situations where equity concerns are important. We draw on insights from the economics literature and base our methods on some of the widely accepted principles in this area. We discuss two equity related concerns, namely equitability and balance, which are distinguished based on whether anonymity holds or not. We review applications involving these concerns and discuss alternative ways to incorporate such concerns into operational research (OR) models. We point out some future research directions especially in using MCDM concepts in this context. Specifically, we observe that research is needed to design interactive decision support systems. Motivated by this observation, we study an MCDM approach to equitability. Our interactive approach uses holistic judgements of the DM to refine the ranking of an explicitly given (discrete) set of alternatives. The DM is assumed to have a rational preference relation with two additional equity-related axioms, namely anonymity and the Pigou-Dalton principle of transfers. We provide theoretical results that help us handle the computational difficulties due to the anonymity property. We illustrate our approach by designing an interactive ranking algorithm and provide computational results to show computational feasibility. We then consider balance concerns in resource allocation settings. Balance concerns arise when the DM wants to ensure justice over entities, the identities of which might affect the decision. We propose a bi-criteria modelling approach that has efficiency (quantified by the total output) and balance (quantified by the imbalance indicators) related criteria. We solve the models using optimization and heuristic algorithms. Our extensive computational experiments show the satisfactory behaviour of our algorithms