P. L. Yu

Multiple-criteria decision making : concepts, techniques, and extensions

1. Introduction.- 1.1. The Needs and Basic Elements.- 1.2. An Overview of the Book.- 1.3. Notation.- 2. Binary Relations.- 2.1. Preference as a Binary Relation.- 2.2. Characteristics of Preferences.- 2.3. Optimality Condition.- 2.4. Further Comments.- Exercises.- 3. Pareto Optimal or Efficient Solutions.- 3.1. Introduction.- 3.2. General Properties of Pareto Optimal Solutions.- 3.3. Conditions for Pareto Optimality in the Outcome Space.- 3.3.1. Conditions for a General Y.- 3.3.2. Conditions when Y Is ??-Convex.- 3.3.3. Boundedness of Tradeoff and Proper Efficiency.- 3.4. Conditions for Pareto Optimality in the Decision Space.- 3.4.1. Conditions in Terms of Single Criterion Maximization.- 3.4.2. Conditions in Terms of Differentiability.- 3.4.3. Decomposition Theorems of X0(??) and X0(?>).- 3.4.4. An Example.- 3.5. Further Comments.- 3.6. Appendix: Generalized Gordon Theorem.- 3.7. Appendix: Optimality Conditions.- Exercises.- 4. Goal Setting and Compromise Solutions.- 4.1. Introduction.- 4.2. Satisficing Solutions.- 4.2.1. Goal Setting.- 4.2.2. Preference Ordering and Optimality in Satisficing Solutions.- 4.2.3. Mathematical Programs and Interactive Methods.- 4.3. Compromise Solutions.- 4.3.1. Basic Concepts.- 4.3.2. General Properties of Compromise Solutions.- 4.3.3. Properties Related to p.- 4.3.4. Computing Compromise Solutions.- 4.3.5. Interactive Methods.- 4.4. Further Comments.- Exercises.- 5. Value Function.- 5.1. Revealed Preference from a Value Function.- 5.2. Conditions for Value Functions to Exist.- 5.3. Additive and Monotonic Value Functions and Preference Separability.- 5.3.1. Additive and Monotonic Value Functions and Implied Preference Separability.- 5.3.2. Conditions for Additive and Monotonic Value Functions.- 5.3.3. Structures of Preference Separability and Value Functions.- 5.4. Further Comments.- Exercises.- 6. Some Basic Techniques for Constructing Value Functions.- 6.1. Constructing General Value Functions.- 6.1.1. Constructing Indifference Curves (Surfaces).- 6.1.2. Constructing the Tangent Planes and the Gradients of Value Functions.- 6.1.3. Constructing the Value Function.- 6.2. Constructing Additive Value Functions.- 6.2.1. A First Method for Constructing Additive Value Functions.- 6.2.2. A Second Method for Constructing Additive Value Functions.- 6.3. Approximation Method.- 6.3.1. A General Concept.- 6.3.2. Approximation for Additive Value Functions.- 6.3.3. Eigenweight Vectors for Additive Value Functions.- 6.3.4. Least-Distance Approximation Methods.- 6.4. Further Comments.- 6.5. Appendix: Perron-Frobenius Theorem.- Exercises.- 7. Domination Structures and Nondominated Solutions.- 7.1. Introduction.- 7.2. Domination Structures.- 7.3. Constant Dominated Cone Structures.- 7.3.1. Cones and their Polars.- 7.3.2. General Properties of N-Points.- 7.3.3. A Characterization of N-Points.- 7.3.4. Cone-Convexity and N-Points.- 7.3.5. N-Points in the Decision Space.- 7.3.6. Existence, Properness, and Duality Questions.- 7.4. Local and Global N-Points in Domination Structures.- 7.5. Interactive Approximations for N-Points with Information from Domination Structures.- 7.6. Further Comments.- 7.7. Appendix: A Constructive Proof of Theorem 7.3.- Exercises.- 8. Linear Cases, MC- and MC2-Simplex Methods.- 8.1. N-Points in the Linear Case.- 8.2. MC-Simplex Method and Nex-Points.- 8.2.1. MC-Simplex Method and Set of Optimal Weights.- 8.2.2. Decomposition of the Weight Space.- 8.2.3. Connectedness and Adjacency of Nex-Points, and a Method for Locating Nex Set.- 8.3. Generating the Set N from Nex-Points.- 8.3.1. The Need for the Entire Set N.- 8.3.2. Decomposition of the Set N into Nondominated Faces.- 8.3.3. Method to Locate All N-Faces and Examples.- 8.4. MC2-Simplex Method and Potential Solutions in Linear Systems.- 8.4.1. Introduction.- 8.4.2. Potential Solutions of Linear Systems.- 8.4.3. The MC2-Simplex Method.- 8.4.4. Separation, Adjacency, and Connectedness.- 8.4.5. Duality of MC2 Programs.- 8.4.6. An Example.- 8.5. Further Comments.- 8.6. Appendix: Proof of Lemma 8.2.- Exercises.- 9. Behavioral Bases and Habitual Domains of Decision Making.- 9.1. Introduction.- 9.2. Behavioral Bases for Decision Making.- 9.2.1. A Model for Decision/Behavior Processes-Overview.- 9.2.2. Internal Information Processing Center-The Brain.- 9.2.3. Goal Setting, Self-Suggestion, and State Valuation.- 9.2.4. Charge Structures and Significance Ordering of Events.- 9.2.5. Least Resistance Principle, Discharge, and Problem Solving.- 9.2.6. External Information Inputs.- 9.3. Habitual Domains.- 9.3.1. Definition and Formation of Stable Habitual Domains.- 9.3.2. The Expansion of Habitual Domains.- 9.3.3. Interaction of Different Habitual Domains.- 9.3.4. Implications of Studying Habitual Domains.- 9.4. Some Observations in Social Psychology.- 9.4.1. Social Comparison Theory.- 9.4.2. Halo Effect.- 9.4.3. Projection Effect (Assumed Similarity).- 9.4.4. Proximity Theory.- 9.4.5. Reciprocation Behaviors.- 9.4.6. Similarity Effect.- 9.4.7. Scapegoating Behavior (Displacement of Aggression).- 9.4.8. Responsibility Diffusion or Deindividuation in Group Behavior.- 9.5. Some Applications.- 9.5.1. Self-Awareness, Happiness, and Success.- 9.5.2. Decision Making.- 9.5.3. Persuasion, Negotiation, and Gaming.- 9.5.4. Career Management.- 9.6. Further Comments.- 9.7. Appendix: Existence of Stable Habitual Domains.- Exercises.- 10. Further Topics.- 10.1. Interactive Methods for Maximizing Preference Value Functions.- 10.1.1. Adapted Gradient Search Method.- 10.1.2. Surrogate Worth Tradeoff Method.- 10.1.3. Zionts-Wallenius Method.- 10.2. Preference over Uncertain Outcomes.- 10.2.1. Stochastic Dominance (Concepts Based on CDF).- 10.2.2. Mean-Variance Dominance (Concepts Based on Moments).- 10.2.3. Probability Dominance (Concept Based on Outperforming Probability).- 10.2.4. Utility Dominance (Concept Based on Utility Functions).- 10.2.5. Some Interesting Results.- 10.3. Multicriteria Dynamic Optimization Problems.- 10.3.1. Finite Stage Dynamic Programs with Multicriteria.- 10.3.2. Optimal Control with Multicriteria.- 10.4. Second-Order Games.- 10.4.1. Decision Elements and Decision Dynamics.- 10.4.2. Second-Order Games.- 10.4.3. Second-Order Games and Habitual Domains.- Exercises.

Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives

Although there is no universally accepted solution concept for decision problems with multiple noncommensurable objectives, one would agree that agood solution must not be dominated by the other feasible alternatives. Here, we propose a structure of domination over the objective space and explore the geometry of the set of all nondominated solutions. Two methods for locating the set of all nondominated solutions through ordinary mathematical programming are introduced. In order to achieve our main results, we have introduced the new concepts of cone convexity and cone extreme point, and we have explored their main properties. Some relevant results on polar cones and polyhedral cones are also derived. Throughout the paper, we also pay attention to an important special case of nondominated solutions, that is, Pareto-optimal solutions. The geometry of the set of all Pareto solutions and methods for locating it are also studied. At the end, we provide an example to show how we can locate the set of all nondominated solutions through a derived decomposition theorem.

The set of all nondominated solutions in linear cases and a multicriteria simplex method

Abstract In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.

Compromise Solutions, Domination Structures, and Salukvadze’s Solution

We outline the concepts of compromise solutions and domination structures in such a way that the underlying assumptions and their implications concerning the solution concept suggested by Salukvadze may be clearer. An example is solved to illustrate our discussion.

Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios

A new eigenweight vector is derived for the data of pairwise weight ratios. The well-known eigenweight vector derived by Saaty is then compared and contrasted in the light of least-distance approximation models. It is shown that the new eigenweight vector commands advantages over Saatys, including less rigid assumptions on the error terms, robustness of solution, in addition to the fact that the new eigenweight vector can be computed very easily. The reader can construct other types of eigenweight vectors and least-distance approximation models using the framework of this article.

Domination Structures and Multicriteria Problems in N-Person Games.

Multiple criteria decision problems with one decision maker have been recognized and discussed in the recent literature in optimization theory, operations research and management science. The corresponding concept with n-decision makers, namely multicriteria n-person games, has not yet been extensively explored.In this paper we first demonstrate that existing solution concepts for single criterion n-person games in both normal form and characteristic function form induce domination structures (similar to those defined and studied by Yu [39] for multicriteria single decision maker problems) in various spaces, including the payoff space, the imputation space and the coalition space. This discussion provides an understanding of some underlying assumptions of the solution concepts and provides a basis for generalizing and generating new solution concepts not yet defined. Also we illustrate that domination structures may be regarded as a measure of power held by the players.We then illustrate that a multicriteria problem can naturally arise in decision situations involving (partial) conflict among n-persons. Using our discussion of solution concepts for single criterion games as a basis, various approaches for resolving both normal form and characteristic function form multicriteria n-person games are proposed. For multicriteria games in characteristic function form, we define a multicriteria core and show that there exists a single ‘game point’ whose core is equal to the multicriteria core. If we reduce a multicriteria game to a single criterion game, domination structures which are more general than ‘classical’ ones must be considered, otherwise some crucial information in the game may be lost. Finally, we discuss a parametrization process which, for a given multicriteria game, associates a single criterion game to each point in a parametric space. This parametrization provides a basis for the discussion of solution concepts in multicriteria n-person games.

Estimating criterion weights using eigenvectors: a comparative study

Abstract In addition to Saatys eigenvector, there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios. As the judgment of the ratios are dependent on personal experience, learning, situations and state of mind, inconsistencies and degree of easiness or judgment on these individual ratios can be different, we study the properties of the different eigen weight vectors, including that of Saaty and that recently proposed by Cogger and Yu. A general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed.

The market speed of adjustment to new information

Abstract A definition of market adjustment is proposed in terms of the time it takes market attributes to reflect new information. Properties of the proposed definition are discussed. In order to operationalize the concept, a statistical method is introduced to estimate the adjustment times. Empirical examples are used to illustrate the proposed method. Some possible economic interpretations are given. The properties of the estimator are also investigated by simulation and analytical methods.

Generalization of domination structures and nondominated solutions in multicriteria decision making

The concepts of domination structures and nondominated solutions are important in tackling multicriteria decision problems. We relax Yus requirement that the domination structure at each point of the criteria space be a convex cone (Ref. 1) and give results concerning the set of nondominated solutions for the case where the domination structure at each point is a convex set. A practical necessity for such a generalization is discussed. We also present conditions under which a locally nondominated solution is also a globally nondominated solution.

Optimal competence set expansion using deduction graphs

A competence set is a collection of skills used to solve a problem. Based on deduction graph concepts, this paper proposes a method of finding an optimal process so as to expand a decision makers competence set to enable him to solve his problem confidently. Using the concept of minimum spanning tree, Yu and Zhang addressed the problem of the optimal expansion of competence sets. In contrast, the method proposed here enjoys the following advantages: it can deal with more general problems involving intermediate skills and compound skills; it can find the optimal solution by utilizing a 0–1 integer program; and it can be directly extended to treat multilevel competence set problems, and thus is more practically useful.