逐步設限資料下可靠度與保固之研究

Reliability and Warranty Based on Progressively Censored Data

黃炫融(Syuan-Rong Huang)

指導教授 ： 吳碩傑

淡江大學/商管學院/管理科學學系博士班/博士(2013年)

https://doi.org/10.6846/TKU.2013.01247

摘要

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。

關鍵字

期望實驗時間 ； 最大概似法 ； 後驗預測分配 ； 可靠度抽樣計畫 ； 效用函數 ； 韋伯分配

並列摘要

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.

並列關鍵字

Expected test time ； Maximum likelihood method ； Posterior predictive distribution ； Reliability sampling plan ； Utility function ； Weibull distribution

參考文獻

Agrawal, J., Richardson, P. S. and Grimm, P. E. (1996). The relationship between warranty and product reliability. The Journal of Consumer Affairs, 30, 421-443.

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Ali Mousa, M. A. M. and Jaheen, Z. F. (2002). Statistical inference for the Burr model based on progressively censored data. Computers & Mathematics with Applications, 43, 1441-1449.

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