## Optimization and Decision Theory

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This book includes how to solve solutions of multi-objective problems considered through genetic algorithms. The problem of multi-objective optimization is described, as well as a primer on classic techniques of optimization and meta techniques, which include using heuristics to solve such problems. The book will be an important element of the library of graduate and postgraduate students in the fields of operations research, decision theory, optimization theory, linear algebra, interval analysis and fuzzy sets among others.

The theory of decision deals with analyzing how a person chooses an action that, among a set of possible actions, leads to the best result given the preferences. If a person should invest or not in capital goods, what career a person is going to choose, what car is foing to buy, are very common problems that affect us in our daily life and to the which - in formal terms - is faced with the theory of decision. On the other hand, in recent years its influence in disciplines such as psychology and economics has been so great, along with applied mathematics, sociology, political science and philosophy - which have that it is very difficult to address some of these specialties without having a knowledge of theory of the decision.

Decision theory has become an indispensable working tool in disciplines as varied as economics, psychology, science politics, sociology or philosophy. Nevertheless, he remains a great stranger for many social scientists despite its great influence. In this work the basic elements of decision theory are presented first then focus on decision theory in situations of uncertainty. Thus, after explaining some classic decision criteria under uncertainty, we discuss the normative model of expected subjective utility (SEU). The limitations of this theory lead us to the more recent descriptive models that are Support of Herbert Simón’s limited rationality theory as the model the adaptive decision maker or the theory of ecological rationality.

Life abounds in situations where it is necessary to make decisions. In some cases, the consequences of decisions depend only on one side, which makes the decision. For example, a programmer makes a decision in any programming language will encrypt algorithm that solves a given problem. However, the consequences of decisions often depend not only on one side but also on the interaction with the decisions taken by the other side, so that the outcome of the decision on the one hand depends on the decisions of others or other parties. It is often the case that such a situation characterized by conflicting-antagonistic interests of the participants in decision-making, ie. we say that the parties to make decisions in the conflict. In the game of chess, the result of the game depends not only moves one player more than another move and their interests are conflicting, because each side wants to win another.

This situation of uncertainty in decision-making in mathematical games is the field of operations research that deals with the analysis of these problems and finding optimal solutions, and is called game theory.

Game theory means the mathematical theory of decision-making process by the opponent (the participants, players) that are in conflict (conflict) or are involved in competitive conditions. The term game means a model of real conflict situations. The game can be added to the relevant rules, which define the rules of conduct of participants in the game, and the goal of game theory is that the exact mathematical algorithm analyzes the conflict situation and determine the reasonable behavior of the players and the course of the conflict, ie. to determine the optimal strategy for each of the participants in the game.

Game theory has the task of finding solutions in situations of competition, which is partially or completely conflicting interests are at least two opponents (ie. According to this theory among the participants in the game). The solution of the conflict is determined by the actions of all parties involved in the conflict. Game theory deals with situations that have the following characteristics: a) there must be at least two players; b) The game starts by having one or more players choose between defined alternatives; c) after the selection is associated with the first move, the result is determined by the situation that determines who makes the next selection and what are his options open; d) the rules of the game are certain rules for determining which specifies the mode of behavior of players; e) any move in the game ends the situation that determines the payout of each bonificiranog player (extra nine player is the one who makes choices and receive payments). There are many examples in different areas of life that can be observed and studied as a conflict situation. A good number of economic problems in the field of market - competitive relations contain conflicts of different interests, so they can be analyzed and solved using game theory.

In this book the solutions of multiobjective problems are considered through genetic algorithms, which consist of random searches in the search space marked by the restrictions, obtaining solutions increasingly efficient. In order to achieve this, two new methodologies are proposed, the first one (MOEGA), which elitism as an interesting concept not to lose the good results that are have achieved and obtain a Pareto border close to the real and the second (MOEGA-P) which considers the preferences of the decision maker in an interactive way, such that the decision maker can direct the search of the algorithm towards the area of its interest.

MOEGA get better solutions in the problem of the backpack comparing it with algorithms like the SPEA2 and NSGAII and MOEGA-P allows the decision- maker to obtain only a portion of the pareto according to your preferences acquiring knowledge of the problem, such that for it will be much easier for him to decide between this small group of alternatives. In addition to in the case of the backpack, MOEGA-P offers the decision maker alternative solutions that does not consider an algorithm without preferences and much closer to the Pareto frontier real, because restricting the search area with preferences, takes advantage of the cost to find more efficient solutions instead of looking at areas that are no longer are of interest to the decision maker.

In this book, the problem of multiobjective optimization is described, mentioning some classic techniques of optimization as well as some meta techniques - heuristics that are used to solve such problems. Since the interest of this research are the genetic algorithms, will be deepened in this type of methodology. The following chapter will describe the main methodologies of multiobjective genetic algorithms and will mention how the preferences of the decision maker. Taking into account the weaknesses and strengths of these methodologies, the next chapter describes an interactive methodological proposal where the decision-maker intervenes in the search process for better results and according to your preferences; But not before proposing another new methodology of algorithms genetic algorithm that obtains better results comparing it with the already existing ones. Then, the results obtained with the two methodological proposals are described along with the conclusions and some ideas of possible future work.