Ignacy Kaliszewski - Böcker

"Daprima importa sappere di che cosa si tratta." ("The ?rst thing is to know what the talk is about".) Attributed to Vilfredo Pareto. This book results from my continuous and deep interest in multiple criteria decision making (MCDM). Eleven years ago I wrote in my p- vious monograph: "This work results from my interest in the ?eld of vector optimization. I stumbled ?rst upon this subject in 1982 [ ... ]. I was attracted then by a gap between vector optimization used to serve as a formal model for multiple objective decision problems and the - cision problems themselves, the gap nonexistent in scalar optimization. Roughly speaking, vector optimization provides methods for ranking - cisions according to a partial order whereas decision making requires a linear ordering of decisions." This declaration is still valid and nothing needs to be changed. To be more speci?c, this book is a fruit of my dissatisfaction with the current state-of-the-art of MCDM. MCDM is a branch of science, whose declared ultimate goal is to provide practical tools. However, we cannot say, and this is regrettable, that all present MCDM methods and algorithms are in popular use by those who make complex decisions and for that purpose are in need of methodological or computational support.

This monograph presents interactive mathematics within a self-contained framework, the cone separation technique. This framework affords a unified approach to multiple objective decision problems and their most common model, the vector optimization problem, regardless of their linear, discrete, continuous, convex, or nonconvex nature. With this approach, the analysis of relations between efficient decisions is significantly broadened, new facts are identified and proved. The cone separation technique gives the decision maker or researcher better methods for analyzing potential efficiency solutions. Specifically, the framework can add any of the following to a solution: a simple way of creating hierarchical structures over sets of efficient decisions; a way of visualizing the decision making process by a method of graphic approximation; a method to calculate trade-offs and gain-to-loss ratios; and sensitivity analysis of efficient solutions with respect to perturbation analysis.

"Daprima importa sappere di che cosa si tratta." ("The ?rst thing is to know what the talk is about".) Attributed to Vilfredo Pareto. This book results from my continuous and deep interest in multiple criteria decision making (MCDM). Eleven years ago I wrote in my p- vious monograph: "This work results from my interest in the ?eld of vector optimization. I stumbled ?rst upon this subject in 1982 [ ... ]. I was attracted then by a gap between vector optimization used to serve as a formal model for multiple objective decision problems and the - cision problems themselves, the gap nonexistent in scalar optimization. Roughly speaking, vector optimization provides methods for ranking - cisions according to a partial order whereas decision making requires a linear ordering of decisions." This declaration is still valid and nothing needs to be changed. To be more speci?c, this book is a fruit of my dissatisfaction with the current state-of-the-art of MCDM. MCDM is a branch of science, whose declared ultimate goal is to provide practical tools. However, we cannot say, and this is regrettable, that all present MCDM methods and algorithms are in popular use by those who make complex decisions and for that purpose are in need of methodological or computational support.

This work results from my interest in the field of vector optimiza­ tion. I stumbled first upon this subject in 1982 during my six months visit to the Istituto di Elaborazione della Informazione in Pisa, Italy, supported by a fellowship of the (Italian) Consiglio Nationale delle Richerche. I was attracted then by a gap between vector optimiza­ tion used to serve as a formal model for multiple objective decision problems and the decision problems themselves, the gap nonexis­ tent in scalar optimization. Roughly speaking, vector optimization provides methods for ranking decisions according to a partial order whereas decision making requires a linear ordering of decisions. The book deals with vector optimization. However, vector opti­ mization is considered here not only as a topic of research in itself but also as a basic tool for decision making. In consequence, all results presented here are aimed at exploiting and understanding the structure of elements (decisions) framed by a vector optimiza­ tion problem with the underlying assumption that the results should be interpretable in terms and applicable in the context of decision making. Computational tractability of results is therefore of special concern throughout this book. A unified framework for presentation is offered by the Cone Sep­ aration Technique (CST) founded on the notion of cone separation.

This electroniccompanion contains models of problems to be solved built in Excel spreadsheetfiles.Optimizationmodels are too often oversimplifications of decision problems met in practice.