
系統識別號 
U00262606201415243700 
論文名稱(中文) 
考慮效用成本比最大化之限制條件下多目標實驗設計 
論文名稱(英文) 
Constrained multiobjective experimental designs for the maximum ratio of utility and cost 
校院名稱 
成功大學 
系所名稱(中) 
工業與資訊管理學系 
系所名稱(英) 
Department of Industrial and Information Management 
學年度 
102 
學期 
2 
出版年 
103 
研究生(中文) 
簡仲廷 
研究生(英文) 
ChungTing 
學號 
R36011034 
學位類別 
碩士 
語文別 
中文 
論文頁數 
58頁 
口試委員 
指導教授張裕清 口試委員王泰裕 口試委員蔡青志

中文關鍵字 
多目標實驗設計
柏拉圖前緣
渴望函數

英文關鍵字 
Multiple objective experiments
Pareto frontier
desirability function

學科別分類 

中文摘要 
本研究處理的對象為多目標實驗設計，而欲同時最佳化多個反應變數時，往往都會面臨到取捨的問題，因此在處理多個反應變數的最佳化問題是相當困難的，但在現今許多產業都會面臨到這個問題。本研究結合了柏拉圖最佳解的概念和渴望函數來處理多目標實驗設計的問題，在考量到成本因素而使得實驗總次數有限制之條件下，本研究利用Chang(1997)所提出的準則並將演算法改寫成符合本研究所要處理的問題，找出一個近似Doptimal的可行實驗設計，並根據此近似Doptimal的實驗設計進行實驗並且分析，接著根據實驗結果來找出位在柏拉圖前緣上的設計點，再計算出未執行實驗的設計點之反應變數預測值，來更新柏拉圖前緣上的設計點，並透過渴望函數將反應變數進行轉換，最後考量製造成本後，找出效用成本比最大的設計點做為最佳的設計。之後透過實例驗證來說明本研究的研究方法流程，並且討論變異數共變異數矩陣對前緣的影響。最後，透過進行敏感度分析來說明最大成本差額的影響。預期在考量製造成本的多目標實驗設計下，本研究可以提供給決策者做為一個判斷的標準。

英文摘要 
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 tradeoff 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 Doptimal criterion for the multiple objective experimental designs and use an algorithm to generate a near Doptimal experimental design. Using this near Doptimal 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 variancecovariance 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 decisionmakers.
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 tradeoff. Tradeoff 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 tradeoff 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 Doptimal criterion to find a feasible design of experiment. The near Doptimal criterion does not depend on the variancecovariance matrix. In addition to the near Doptimal 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 nondominated 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 nonexecuted 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.

論文目次 
目錄
摘要 I
致謝 V
目錄 VI
表目錄 IX
圖目錄 XI
第一章 緒論 1
1.1研究背景與動機 1
1.2研究目的 2
1.3研究假設 2
1.4研究架構 3
第二章 文獻回顧 4
2.1單目標實驗設計 4
2.2多目標實驗設計 5
2.2.1因子篩選 6
2.2.2多目標實驗設計分析 7
2.2.3渴望函數(Desirability Function) 10
2.3多目標問題 12
2.3.1多目標最佳化問題定義 12
2.3.2柏拉圖最佳解 13
2.3.3多目標最佳解方法 14
2.4最佳化設計 16
2.4.1單目標最佳化設計準則 16
2.4.2多目標最佳化設計 17
2.5資源利用成本 17
第三章 模型建構 19
3.1研究模型 19
3.1.1問題描述與基本假設 19
3.1.2迴歸模型的參數估計 21
3.1.3反應變數預測值 21
3.2研究方法 22
3.2.1尋找最佳的設計矩陣 22
3.2.2尋找柏拉圖前緣 25
3.2.3更新柏拉圖前緣 26
3.2.4渴望函數的應用 26
3.2.5考量成本之效用最大化 28
3.2.6執行驗證實驗 29
3.3研究方法流程 30
3.4敏感度分析 31
3.4.1成本差額的凌駕產生 31
3.4.2成本差額敏感度分析 31
3.4.3額敏感度分析的可能情形 32
第四章 實證分析 34
4.1資料來源與說明 34
4.2實例驗證 34
4.2.1研究假設與因子設定 34
4.2.2尋找最佳設計矩陣 36
4.2.3尋找柏拉圖前緣 37
4.2.4更新柏拉圖前緣 38
4.2.5最佳設計點之選擇 41
4.3成本差額之敏感度分析 45
4.3.1成本差額凌駕產生 45
4.3.2敏感度分析 46
4.4執行驗證實驗 47
4.4.1驗證實驗之結果 47
4.4.2最終前緣之敏感度分析 49
4.5小結 50
第五章 結論與未來研究建議 52
5.1結論 52
5.2貢獻與重要性 52
5.3未來研究建議 53
參考文獻 54

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