D. N. Prabhakar Murthy

Warranty Cost Analysis

For products sold with warranty, the manufacturer incurs costs resulting from the servicing of failures occurring under warranty. These costs depend on several factors, the most important being the reliability of the product. For planning purposes, the manufacturer must estimate (a) the expected cost of warranty per unit sold and (b) the expected cost per unit time over the product life cycle. This must be done in the product development stage. This article deals with models that may be used to obtain these cost estimates for products sold with free-replacement and pro rata warranties. Keywords: warranty; reliability; cost analysis; cost models; free-replacement warranty; pro rata warranty

Case Studies in Reliability and Maintenance

Contributors. Preface. Introduction and Overview (Wallace R. Blischke and D. N. P. Murthy). PART A: CASES WITH EMPHASIS ON PRODUCT DESIGN. Space Interferometer Reliability-Based Design Evaluation (Donald H. Ebbeler, et al.). Confidence Intervals for Hardware Reliability Predictions (Chun Kin Chan and Michael Tortorella). Allocation of Dependability Requirements in Power Plant Design (Seppo Virtanen and Per-Erik Hagmark). PART B: CASES WITH EMPHASIS ON DEVELOPMENT AND TESTING. The Determination of the Design Strength of Granite Used as External Cladding for Buildings (Malcolm J. Faddy, et al.). Use of Sensitivity Analysis to Assess the Effect of Model Uncertainty in Analyzing Accelerated Life Test Data (William Q. Meeker, et al.). Virtual Qualification of Electronic Hardware (Michael Osterman, et al.). Development of a Moisture Soak Model for Surface-Mounted Devices (Loon Ching Tang and Soon Huat Ong). Construction of Reliable Software in Resource-Constrained Environments (Mladen A. Vouk and Anthony T . Rivers). Modeling and Analysis of Software System Reliability (Min Xie, et al.). Information Fusion for Damage Prediction (Nozer D. Singpurwalla, et al.). PART C: CASES WITH EMPHASIS ON DEFECT PREDICTION AND FAILURE ANALYSIS. Use of Truncated Regression Methods to Estimate the Shelf Life of a Product from Incomplete Historical Data (William Q. Meeker and Luis A. Escobar). Determining Software Quality Using COQUALMO (Sunita Chulani, et al.). Use of Extreme Values in Reliability Assessment of Composite Materials (Linda C. Wolstenholme). Expert Judgment in the Uncertainty Analysis of Dike Ring Failure Frequency (Roger Cooke and Karen Slijkhuis). PART D: CASES WITH EMPHASIS ON MAINTENANCE AND MAINTAINABILITY. Component Reliability, Replacement, and Cost Analysis with Incomplete Failure Data (Nicholas A. J. Hastings). Maintainability and Maintenance-A Case Study on Mission Critical Aircraft and Engine Components (U. Dinesh Kumar and John Crocker). Photocopier Reliability Modeling Using Evolutionary Algorithms (Michael Bulmer and John Eccleston). Reliability Model for Underground Gas Pipelines (Roger M. Cooke, et al.). RCM Approach to Maintaining a Nuclear Power Plant (Gilles C. Zwingelstein). Case Experience Comparing the RCM Approach to Plant Maintenance with Modeling Approach (Xisheng Jia and Anthony H. Christer). PART E: CASES WITH EMPHASIS ON OPERATIONS OPTIMIZATION AND REENGINEERING. Mean Residual Life and Optimal Operating Conditions for Industrial Furnace Tubes (Elsayed A. Elsayed). Optimization of Dragline Load (Peter G. A. Townson, et al.). Fords Reliability Improvement Process-A Case Study on Automotive Wheel Bearings (Karl D. Majeske, et al.). Reliability of Oil Seal for Transaxle-A Science SQC Approach at Toyota (Kakuro Amasaka and Shunji Osaki) PART F: CASES WITH EMPHASIS ON PRODUCT WARRANTY. Warranty Data Analysis for Assessing Product Reliability (Peter C. Sander, et al.). Reliability and Warranty Analysis of a Motorcycle Based on Claims Data (Bermawi P. Iskandar and Wallace R. Blischke). Index.

Weibull model selection for reliability modelling

A large number of models have been derived from the two-parameter Weibull distribution and are referred to as Weibull models. They exhibit a wide range of shapes for the density and hazard functions, which makes them suitable for modelling complex failure data sets. The WPP and IWPP plot allows one to determine in a systematic manner if one or more of these models are suitable for modelling a given data set. This paper deals with this topic.

Preliminary Data Analysis

The objectives of preliminary data analysis are to edit the data to prepare it for further analysis, describe the key features of the data, and summarize the results. This chapter deals with quantitative and qualitative approaches to achieving these objectives. Topics covered include scales of measurement, types of data, graphical methods of analysisᾢincluding histograms, probability plots, and other graphical representations of data, and basic descriptive statisticsᾢmean, median, fractiles, standard deviation, and so forth. The chapter concludes with a discussion of the use of probability plots in preliminary model selection.

Cost Models for Two-Dimensional Warranties

Multi-dimensional warranties usually involve a time dimension (similar to that in one-dimensional warranties) and one or more usage dimensions. There are several notions of usage. We confine our discussion to two-dimensional warranties where the warranty is characterized by a rectangular region under which the warranty expires when the item reaches an age W or the usage reaches a level U, whichever comes first. The cost analysis is a more difficult than that of one-dimensional warranties. For 2-D warranties, failures are random points scattered over the two-dimensional warranty region as opposed to being random points along the time axis in the case of one-dimensional warranties. We discuss the three different approaches that have been proposed for modeling failures and warranty claims. Costs depend not only on failures but also on several other factors. We discuss these and derive cost models for some simplified cases.

Cost Models for One-Dimensional Warranties

When a manufacturer offers a warranty, all legitimate claims under warranty must be serviced. The number of claims that might be expected depends on the reliability of the product. Servicing of the claims results in additional costs to the manufacturer. There are several notions of warranty costs, each of which leads to a distinct cost model. In this chapter we focus on models for prediction of warranty costs as a function of product reliability for various one-dimensional warranties. These models play a critical role in warranty management.

Basic Statistical Inference

Once data are collected, edited, summarized and otherwise prepared for detailed analysis, basic methods of statistical inference are applied to address stated experimental and management objectives. In this chapter, we look at several key statistical techniques that are used in inference, particularly in the context of reliability and warranty analysis. These include (1) estimation, including maximum likelihood, several other methods of point estimation, and confidence intervals; (2) hypothesis testing, including comparison of two population means; (3) nonparametric methods for comparing populations; (4) tolerance intervals for estimating population fractiles; and (5) rank correlation for measuring data relationships.

Optimal Control of Unreliable Dynamic Systems with Discrete Time Inspections

The optimal control of unreliable dynamic systems where the mode (working or failed) is continuously inspected or monitored has received attention in the past. We consider the problem where instead of continuous inspection we have either no inspection or inspection at discrete time instants. The optimal feedback control is derived for the case where the system has no renewable capabilities. The case where a fixed number of renewals are allowed is briefly discussed.

Sub-optimal control of sparsely coupled systems

In this paper we consider the sub-optimal control of sparsely coupled systems, characterized by linear differential equations with quadratic cost functional. The method proposed here converts the problem to a canonical form and identifies the variables from each sub-system which are strongly interacting. This is done by defining a threshold level matrix based on the eigenvalues of the sub-systems and their cost matrices. The elements of the coupling matrix in the canonical form is compared with the corresponding elements in the threshold level matrix to decide whether the coupling is significant or not. Sub-optimal controls are derived for each sub-system incorporating complete state of the sub-system and only those states from other sub-systems which are strongly interacting. A considerable saving in computation, as well as reduced hardware coat in terms of information transmission is achieved. The sub-optimal controller is derived for both constrained and unconstrained information structures. A few examples are given to illustrate the method.

Supplementary Warranty Data

Warranty claims data alone are not adequate for purposes of statistical inference in many cases, including estimation of product reliability, prediction of future claims, costs, and so forth. To address many of these problems, additional data, called “supplementary warranty data,” are required. In this chapter, we focus on the various forms of supplementary warranty data that may provide the additional information needed for inference. The most important additional data needed for estimation and other inferences about product reliability are service times of item that did not fail. These data are censored data. For other purposes, a broader class of supplementary data is needed. This may include data from several different sourcesᾢsome internal (from different units or sections of the manufacturer) and others external. For maximum benefit, the collection and analysis of data must be done in the context of the product life cycle. The chapter discusses the various sources of data, the characterization of the data and the role of data in management of the warranty process.