傳統設限方法中，型一設限與型二設限常在壽命試驗中被使用，透過僅觀察設限時間前的故障元件來達到縮短實驗時間與降低實驗成本的目的。許多時候我們無法避免從實驗中提早移除部分的存活元件，這種允許元件在實驗尚未結束前即被移除的實驗方式，我們稱之為逐步設限。本篇論文我們以逐步設限方法收集受測元件壽命服從韋伯分配的壽命資料，並根據此類型的設限資料討論以下兩個在實務上重要的問題：(1)保固時間長度與(2)可靠度抽樣計畫。

在競爭激烈的市場中，廠商必須透過提供產品保固服務來吸引消費者。消費者會願意購買價格較高的產品，但前提是必須具有產品可靠度的保證。具有較長的保固時間通常意味著產品具有較高的可靠度，然而，提供一個沒有限制的保固是一種不切實際的做法，因為維持這樣的一個保固將導致生產者必須負擔高額的成本。我們首先對韋伯壽命分配的參數做最大概似估計與貝氏估計，並推論單樣本與雙樣本的貝氏預測區間。在保固政策的最佳設計問題中，我們考慮免費置換保固與比例負擔保固結合而成的混合型保固政策，提出一效用函數來決定使得生產者整體報酬最大的最佳保固時間，並舉兩個例子做分析討論。

對於第二個最佳設計的問題，我們結合逐步設限與第一失敗設限，提出以逐步第一失敗設限方法所收集的設限資料來設計可靠度抽樣計畫。在給定生產者風險、消費者風險與實驗成本預算限制下，提出三個不同的最佳化準則來得到最適的實驗配置與允收臨界值，並做數值研究、蒙地卡羅模擬與敏感度分析之討論。

In traditional censoring schemes, type-I and type-II censoring are often used in life-testing. To shorten experiment time and reduce experiment cost, failures are collected only before the censoring time. There are many scenarios that we cannot avoid removing some surviving units early from the life test. Such a life test that allows units removed before the termination of the experiment is called progressive censoring. In this dissertation, we consider the progressive censoring and assume the lifetime data are from a Weibull distribution. Based on this type of censored data, we discuss two important optimal design problems in practice: length of warranty and reliability sampling plan.

In an intensely competitive market, one way by which manufacturers attract consumers to their products is to provide warranties on the products. Consumers are willing to purchase a high-priced product only if they can be assured about the product's reliability. A longer warranty period
usually indicates better reliability. However, offering an unlimited warranty is unrealistic because maintaining such a policy needs very high cost. We first derive the maximum likelihood estimator and Bayes estimator for the parameters of the Weibull distribution and then obtain the one-sample and two-sample prediction interval.  For the optimal design problem, we consider a combined warranty which is a combination of free-replacement and pro-rata policies. We propose a utility function to determine the optimal warranty length which maximizes the expected value of the utility function. Two examples are discussed to illustrate the
application of the proposed method.

For the second problem, we combine the progressive censoring and first-failure censoring to develop a progressive first-failure censoring. Under the progressive first-failure censoring, we propose an approach to establish reliability sampling plans which minimize three different objective
functions under the constraint of total cost of experiment and given consumer's and producer's risks. Some numerical examples, Monte Carlo simulation and the sensitivity analysis are performed to demonstrate the proposed approach.

目錄

第一章 緒論                                                                  1
1.1 研究動機與目的...........................................................1
1.2 文獻回顧.................................................................4
1.3 本文架構.................................................................7
第二章 設限計畫與壽命分配                                                    9
2.1 一般傳統設限............................................................10
2.2 逐步設限................................................................12
2.3 逐步第一失敗設限........................................................16
2.4 壽命分配................................................................19
2.5 壽命試驗之期望實驗時間..................................................21
第三章 逐步型二設限計畫下的最適保固時間                                     34
3.1 參數估計................................................................35
3.1.1 最大概似估計量........................................................35
3.1.2 貝氏估計量............................................................37
3.1.3 貝氏可信區間..........................................................43
3.2 貝氏預測................................................................44
3.2.1 單樣本預測區間........................................................44
3.2.2 雙樣本預測區間........................................................46
3.3 估計和預測之數值分析和模擬研究..........................................48
3.3.1 數值範例..............................................................49
3.3.2 模擬研究..............................................................51
3.4 最佳化保固時間之決策....................................................54
3.4.1 保固政策..............................................................56
3.4.2 效用函數..............................................................58
3.4.3 最佳化保固時間........................................................62
3.5 最佳保固時間之數值分析..................................................63
3.5.1 數值範例 1............................................................64
3.5.2 數值範例 2............................................................68
第四章 逐步第一失敗設限下的可靠度抽樣計畫                                   73
4.1 參數估計................................................................74
4.2 成本限制下的可靠度抽樣計畫..............................................77
4.2.1 可靠度抽樣計畫........................................................77
4.2.2 成本限制下的最佳抽樣計畫..............................................78
4.2.3 演算法................................................................82
4.3 數值研究................................................................84
4.3.1 抽樣計畫..............................................................85
4.3.2 數值範例..............................................................89
4.3.3 參數估計之敏感度分析..................................................90
4.3.4 大樣本近似之準確性....................................................91
第五章 結論                                                                 96
附 錄                                                                       99
A 單樣本順序統計量預測分配期望值之證明......................................99
B 雙樣本順序統計量預測分配期望值之證明.....................................101
C 預測分配期望值之證明.....................................................102
參考文獻                                                                   104



表格目錄

表 3.1  自 Nelson (1982, Table 6.1) 所生成的逐步型二設限樣本................49
表 3.2  單樣本與雙樣本之 95% 貝氏預測區間...................................50
表 3.3  最大概似估計、貝氏估計和 Lindley's 近似值之平均估計值與估計風險.....53
表 3.4  單樣本預測區間的覆蓋率與預測區間之平均長度..........................54
表 3.5  雙樣本預測區間的覆蓋率與預測區間之平均長度..........................55
表 3.6  數值範例 1 在 FRW 保固政策下的最佳化保固時間........................65
表 3.7  數值範例 1 在 PRW 保固政策下的最佳化保固時間........................66
表 3.8  數值範例 1 在混合型保固政策下之最佳化保固時間.......................67
表 3.9  自 Wu et al. (2006b) 所生成的逐步型二設限樣本.......................68
表 3.10 數值範例 2 在 FRW 保固政策下的最佳化保固時間........................70
表 3.11 數值範例 2 在 PRW 保固政策下的最佳化保固時間........................71
表 3.12 數值範例 2 在混合型保固政策下之最佳化保固時間.......................72
表 4.1  最小化期望實驗時間的最適抽樣計畫 (n,k,d)............................86
表 4.2  最小化變異數共變異數矩陣行列式值的最適抽樣計畫(n,k,d)...............87
表 4.3  最小化分位數之平均漸近變異數的最適抽樣計畫 (n,k,d)..................88
表 4.4  逐步第一失敗設限之可靠度抽樣計畫....................................90
表 4.5  逐步第一失敗設限在不同壽命分配參數組合下的最佳實驗配置 (n,k)........92
表 4.6  逐步第一失敗設限於表 4.1 的允收機率之模擬值.........................93
表 4.7  逐步第一失敗設限於表 4.2 的允收機率之模擬值.........................94
表 4.8  逐步第一失敗設限於表 4.3 的允收機率之模擬值.........................95


圖形目錄

圖2.1 型一設限..............................................................10
圖2.2 型二設限..............................................................12
圖2.3 逐步型一設限..........................................................14
圖2.4 逐步型二設限..........................................................15
圖2.5 分群個數n = 8 的期望實驗時間E(Xm)/eμ..................................24
圖2.6 分群個數n = 10 的期望實驗時間E(Xm)/eμ.................................25
圖2.7 分群個數n = 12 的期望實驗時間E(Xm)/eμ.................................26
圖2.8 分群個數n = 15 的期望實驗時間E(Xm)/eμ.................................27
圖2.9 分群個數n = 8 時，逐步第一失敗設限與(a) 逐步型二設限；(b) 型二設限；(c) 完整樣本；(d) 第一失敗設限的期望實驗時間之比值.....................30
圖2.10 分群個數n = 10 時，逐步第一失敗設限與(a) 逐步型二設限；(b) 型二設限；(c) 完整樣本；(d) 第一失敗設限的期望實驗時間之比值...................31
圖2.11 分群個數n = 12 時，逐步第一失敗設限與(a) 逐步型二設限；(b) 型二設限；(c) 完整樣本；(d) 第一失敗設限的期望實驗時間之比值...................32
圖2.12 分群個數n = 15 時，逐步第一失敗設限與(a) 逐步型二設限；(b) 型二設限；(c) 完整樣本；(d) 第一失敗設限的期望實驗時間之比值...................33
圖3.1 單樣本預測............................................................44
圖3.2 雙樣本預測............................................................47
圖3.3 保固成本與不滿意成本..................................................60

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