Date of Award

August 2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Engineering

First Advisor

Anoop Dhingra

Committee Members

Wilkistar Otieno, Matthew McGinty, Ilya Avdeev, Ron Perez

Keywords

Decentralized Problems, Design Optimization, Game Theory, Hierarchical Problems, Multi Level Programming, Optimization

Abstract

This dissertation presents a game theoretic approach to solve bi and multi-level optimization problems arising in mechanical design. Toward this end, Stackelberg (leader-follower), Nash, as well as cooperative game formulations are considered. To solve these problems numerically, a sensitivity based approach is developed in this dissertation. Although game theoretic methods have been used by several authors for solving multi-objective problems, numerical methods and the applications of extensive games to engineering design problems are very limited. This dissertation tries to fill this gap by developing the possible scenarios for multi-objective problems and develops new numerical approaches for solving them.

This dissertation addresses three main problems. The first problem addresses the formulation and solution of an optimization problem with two objective functions using the Stackelberg approach. A computational procedure utilizing sensitivity of follower's solution to leader's choices is presented to solve the bi-level optimization problem numerically. Two mechanical design problems including flywheel design and design of high speed four-bar mechanism are modeled based on Stackelberg game. The partitioning of variables between the leader and follower problem is discussed, and a variable partitioning metric is introduced to compare various variable partitions.

The second problem this dissertation focuses on is modeling the multi-objective optimization problem (MOP) as a Nash game. A computational procedure utilizing sensitivity based approach is also presented to find Nash solution of the MOP numerically. Some test problems including mathematical problems and mechanical design problems are discussed to validate the results. In a Nash game, the players of the game are at the same level unlike the Stackelberg formulation in which the players are at different levels of importance.

The third problem this dissertation addresses deals with hierarchical modeling of multi-level optimization problems and modeling of decentralized bi-level multi-objective problems. Generalizations of the basic Stackelberg model to consider cases with multiple leaders and/or multiple followers are missing from the literature. Three mathematical problems are solved to show the application of the algorithm developed in this research for solving hierarchical as well as decentralized problems.

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