| Summary: | 碩士 === 國立成功大學 === 工業與資訊管理學系 === 102 === SUMMARY
The subject of this study is optimization of multiple objective experimental designs. It is very difficult to optimize multiple responses simultaneously through input variables because of the trade-off between responses. However, many industries face this problem nowadays. This study combines Pareto frontier, the desirability function and manufacturing costs to deal with multiple objective experimental designs. We consider the situation that the total number of experiments is limited by the resource. Therefore, we adopt the near D-optimal criterion for the multiple objective experimental designs and use an algorithm to generate a near D-optimal experimental design. Using this near D-optimal design to do the experiments, we can get the experimental results. Then, we find all the design points on the Pareto frontier and calculate the response predictive values of design points based on the experimental results. Next, we use these predictive values to update the Pareto frontier. Finally, we add extra information of the manufacturing costs to find the design point which has the maximum ratio of utility and cost. We use a case to illustrate the process and we discuss the influence of variance-covariance matrix on the Pareto frontier. At the end of the example, sensitivity analysis is performed to demonstrate the effect of the maximum value of the cost differences. This study expects to provide an instrument for decision-makers.
Key words: Multiple objective experiments, Pareto frontier, desirability function
Introduction
Many quality characteristics are usually considered at the time when we measure the quality of products. These quality characteristics can be assessed by measuring the response variables. Response variables can be affected by one or several factors. When we deal with multiple response variables, we usually face the problem of trade-off. Trade-off means that one response variable can get the best result when the factors are set to specific levels, but the other response variables are not necessarily the best. That is, optimal factor settings for each response variable may be contradictory to each other. The experimental design that considers multiple responses is called multiple objective experimental designs. Ko et al. (2005), and Wu (2005) pointed out that the need for optimizing multiple objective experimental designs is increasing. Optimization of multiple objective experimental designs also has the trade-off problem. The common method is to convert response variables into a single value to access.
Generally, the more experiments are executed, the more accurate the result should be. In fact, we need to consider the cost when executing an experiment. The cost factors can be divided into two categories: the first category, the cost impacts on the total number of experiment. At this time, we need to find a feasible design under limited resource. The second category, the manufacturing costs of the different factor levels need to be considered. So, we need to consider the total manufacturing costs to produce one product.
In addition to the cost, limitation which exists between the factors also affects the design of experiment. So, we need to find an optimal experimental design. Berger and Wong (2005) pointed out that because of the existence of restrictions in the industrial, medical pharmaceutical, biomedical, epidemiology and other industries, the need to find the optimal experimental design problem is fairly common. Considering the total number of experiments is limited by cost and the restrictions between the factors. According to the description above, the purpose of this study is to find an optimal design of experiment under above limitation and then suggest a best design point when manufacturing costs are considered.
MATERIAL AND METHODS
In this study, we use a near D-optimal criterion to find a feasible design of experiment. The near D-optimal criterion does not depend on the variance-covariance matrix. In addition to the near D-optimal criterion, this study combines Pareto frontier, the desirability function and manufacturing costs to deal with the multiple objective experimental designs. The approach consists of the following steps: first, according to the response variables, we can get the Pareto frontier which is the set of non-dominated design points. We cannot tell which design points is better because they do not dominate each other. Second, we use the linear multiple responses model to predict the response variables. So, we can get the predictive values of non-executed experiments. Then, we update the Pareto frontier. Third, we set up an individual desirability function for each response variable according to the requirement of each response. The requirements can be divided into three categories: objective to maximize response variable, objective to minimize response variable and objective to be as close to the target as possible. After we calculate all individual desirability and the overall desirability of the design points on the Pareto frontier. Fourth, divide overall desirability by the total manufacturing cost of a design as the ratio of utility and cost. Finally, we choose the maximum ratio of utility and cost as the best design point.
RESULTS AND DISCUSSION
According to the case in this study, we find that the value of parameters is important when we calculate the overall desirability. The overall desirability is one of the important indices to choose the best design point. From the sensitivity analysis, we find the maximum value of cost differences has an impact on the best design point. If the factor with maximum value of the cost differences is set at high cost level, reducing its cost will not change the best design point. On the contrast, if the factor with maximum value of the cost differences is set at low cost level, reducing its cost may change the best design point.
For future research, the continuous settings of factors can be considered. The value of cost differences are fixed in this study. In the future, cost differences can be a function of factors.
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