Graduation Year

2021

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Industrial and Management Systems Engineering

Major Professor

Hadi Charkhgard, Ph.D.

Committee Member

Changhyun Kwon, Ph.D.

Committee Member

Susana Lai-Yuen, Ph.D.

Committee Member

Xiaopeng (Shaw) Li, Ph.D.

Committee Member

He Zhang, Ph.D.

Keywords

Geometric Mean Optimization, Multi-objective Optimization, Nash Bargaining Solution, Nash Social Welfare, Optimization Over the Efficient Set

Abstract

This dissertation presents three different contributions to an important class of optimization problems known as Multiplicative Programs (MPs). The first group of contributions contains the development and analysis of several multi-objective optimization-based based algorithms designed to find the optimal solution of Mixed Integer Linear MPs. As for the second group, the application of a special class of MPs in radiotherapy planning is presented. Finally, in the last group, a new technique for conducting the multiplication process in the objectives of MPs is presented. Using this technique, we introduce a family of novel solution methods that are capable of solving both maximum and minimum MPs.

Regarding the first group of contributions, we show that MPs can be viewed as special cases of the problem of optimization over the efficient set in Multi-objective optimization. Based on this observation, we develop several solution approaches with their own unique properties. In the first solution approach, we embed an existing algorithm capable of solving continuous MPs with two multiplying terms in a branch-and-bound framework to solve mixed integer instances. In our second approach, we develop a criterion space search algorithm for solving any mixed integer MPs with any number of multiplying terms. In our last solution approach, we develop three different algorithms that, in addition to solving any mixed integer MPs with any number of multiplying terms, are capable of handling the multiplying terms with different powers. Specifically, the advantage of our last solution approach is that it handles the powers with the minimal impact on the computational complexity in practice.

In the second group of contributions, we study the fluency map optimization problem in Intensity Modulated Radiation Therapy (IMRT) from a cooperative game theory point of view. We consider the cancerous and healthy organs in a patient's body as players of a game, where cancerous organs seek to eliminate the cancerous cells and healthy organs seek to receive no harm. We balance these trade-off by transforming the fluency map optimization problem into a bargaining game. Then, using the concept of Nash Social Welfare, we find a solution for our bargaining game by creating and optimizing an MP. The importance of our solution is that it simultaneously satisfies efficiency and fairness in the trade-offs. Finally, we demonstrate the advantages of our proposed approach by implementing it on two different cancer cases.

In the third and last group of contributions, we present the novel idea of binary-encoding the multiplication operation analogously to how a computer conducts it internally. Based on this idea, we develop a new family of solution methods for MPs. One of our methods is to solve the multiplicative programs bit-by-bit, i.e., iteratively computing the optimal value of each bit of the objective function. In an extensive computational study, we explore a number of solution methods that solve MPs faster and more accurately.

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