Multi-objective black-box optimization and optimal design of CUSUM

This thesis is composed of two topics: a method for the black-box multi-objective problem and an optimal design of the CUSUM control chart to detect an unknown mean shift. The first topic considers the multi-objective optimization problem, which has attracted interest from decision makers when a decision should be made by considering multiple performance measures simultaneously. If one performance measure can be constructed by aggregating all the measures, it becomes a single-objective optimization problem. In design problems in engineering, science, and management, however, decisions often need to be taken in the presence of many trade-offs between two or more conflicting objectives. The set of all the best trade-off solutions is called the Pareto set and its image in the objective space is called the Pareto front. Since each solution in the Pareto set cannot be better than the others for all the objectives, any rational final decision comes from the Pareto set. We consider continuous black-box bi-objective optimization problems with deterministic responses. An iterative algorithm called P AWS is developed to approximate the Pareto front with a relatively small number of evaluations. PAWS combines four well-known optimization and statistics procedures: design of experiments, response surface modeling, trust region method, and weighted sum method. At each iteration, PAWS uses a designed experiment to construct a quadratic metamodel to approximate each objective function. The trust region concept is applied to determine the local region that the metamodels are assumed to be appropriate. The weighted sum method is implemented with the metamodels to pursue Pareto optimal solutions. We compared PAWS with NSGA-II and B IMADS, which are multi-objective optimization algorithms based on a genetic algorithm and a direct-search method, respectively. Simulation results show that PAWS gives better quality of solutions than NSGA-II and BIMADS with a small number of evaluations. The second topic concerns a statistical method for quality control. A Cumulative Sum (CUSUM) control chart is one of the most popular methods used to detect a process mean shift. When one specific size of the mean shift is assumed, the CUSUM chart can be optimally designed in terms of average run length (ARL). In practice, however, the size of the mean shift is usually unknown, and the CUSUM chart can perform poorly when the actual size of the mean shift is significantly different from the assumed size. In this work, we assign a probability distribution to the size of the mean shift to represent its lack of knowledge. We use an ARL-based performance measure, called Expected Weighted Run Length (EWRL), and propose a method to optimally design a CUSUM chart based on EWRL. This method can be easily extended to other CUSUM-based control charts, such as weighted CUSUM and multi-CUSUM charts proposed in the literature. The numerical results show that the CUSUM or the CUSUM-based chart is improved by our proposed method in terms of EWRL