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During i t s development t e s t i n g , a system may undergo modification t o remedy design weaknesses which t h e tests r e v e a l . The changes are made t o improve t h e r e l i a b i l i t y of t h e system, and i f such improvanent o c c u r s , we s a y t h a t r e l i a b i l i t y growth i s t a k i n g place.This RAND Memorandum d e r i v e s s t a t i s t i c a l methods f o r e s t i m a t i n g r e l i a b i l i t y under growth assumptions which merely r e q u i r e t h a t changes made t o t h e system which is being t e s t e d do not make i t worse. The methods d i f f e r from previous methods, which e i t h e r ignore growth o r assume t h a t growth i s t a k i n g place i n a n assigned f u n c t i o n a l manner. While t h i s Memorandum i s addressed p r i m a r i l y t o s t a t i s t i c i a n s , i t should a l s o be of i n t e r e s t t o test engineers and managers concernedw i t h a s s e s s i n g a system's r e l i a b i l i t y .taken as a p a r t of t h e Apollo Contingency Planning Study which RAND is conducting f o r Headquarters, NASA, under Contract NASr-21(09).of t h e a u t h o r s , Richard E. Barlow, i s a c o n s u l t a n t t o The RAND Corporation. The i n v e s t i g a t i o n w a s under- One -V- SUMMAKYThis study examines t h e problem of e s t i m a t i n g t h e r e l i a b i l i t y o f a system t h a t is undergoing development t e s t i n g . I n such a program, changes a r e made t o t h e system from time t o time i n o r d e r t o i n c r e a s e i t s r e l i a b i l i t y . n o t d e l e t e r i o u s , and, u n l i k e some previous work i n t h i s area, i t does n o t assume t h a t system modifications cause r e l i a b i l i t y growth according t o a p r e s c r i b e d f u n c t i o n a l form. The method described h e r e i n does, however, r e q u i r e t h a t each f a i l u r e be c l a s s i f i e d a s i n h e r e n t o r ref l e c t i n g a c o r r e c t a b l e cause. This study assumes t h a t t h e s e changes a r e a t l e a s tThe study proceeds on t h e supposition t h a t t h e test program is conducted i n K s t a g e s , with s i m i l a r items b e i n g t e s t e d w i t h i n each s t a g e . For each s t a g e , t h e number o f i n h e r e n t f a i l u r e s , o f a s s i g n a b l e cause f a i l u r e s , and o f successes is recorded.It is supposed t h a t t h e p r o b a b i l i t y of a n inherent f a i l u r e , t h e test program and t h a t t h e p r o b a b i l i t y of an a s s i g n a b l e cause f a i l u r e i n t h e i -t h s t a g e , qi, does n o t i n c r e a s e from s t a g e t o s t a g e of t e s t i n g . This Memorandum obtains maximum l i k e l i h o o d e s t i m a t e s o f q and a l s o o b t a i n s .a c o n s e r v a t i v e lower confidence i n t e r v a l f o r r t h e r e l i a b i l i t y o f t h e system i n i t s f i n a l c o n f i g u r a t i o n o f t h e t e s t progra remains t h e same throughout ,and o f t h e q 's s u b j e c t t o t h e c o n d i t i o n t h a t they be non-increasi...

Under a free-replacement warranty of duration W, the customer is provided. for an initial cost of C, as many replacement items as needed to provide service for a period W.

Two types of warranties are analyzed. These are the free-replacement warranty, under which failed items are replaced free of charge until a specified total operating time has been achieved, and the pro rata warranty, under which items that fail prior to a specified time are replaced at pro rata cost to the buyer. Both the buyer's and seller's points of view are considered. 'The basis of the analysis is a comparison of warrantied and unwarrantied (otherwise identical) items with regard to long-run cost to the buyer and long-run profit to the seller.Application of the results requires knowledge of certain characteristics of the life distribution of the items in question. Parametric and nonparametric methods of estimation of these characteristics from incomplete data are discussed. Single and multiple failure-mode situations are considered.Some solutions to the problem are illustrated using incomplete data on failure times of an aircraft component. INTRODUCTICINAs individual consumers we frequently purchase goods or services which include a warranty of one kind or another. Most of these purchases are relatively small and most of us rarely pay much attention to the exact nature of the warranty nor do we think about its actual cost to us. From the point of view of the manufacturer of the item or supplier of the service, however, such considerations are very important ones since they can rather drastically affect profitability. In situations involving large, complex, costly pieces of' equipment and dealings with other organizations (corporations, government agencies, and so forth) rather than individual consumers, the nature and cost of the warranty can become very important, indeled, both to the supplier and to the consumer. In this paper we shall analyze some common warranty policies from both points of view.In analyzing wairranty policies, the basic approach taken is a comparison of costs to the consumer and profits of the supplier, of warrantied versus unwarrantied items. From the point of view of the supplier, this comparison leads to the derivation of a differential price structure which will equate long-run profit in the two situations. From the point of view of the consumer, it involves a differential price structure such that, in the long run, the consumer is indifferent with regard to whether he purchases a warrantied or an unwarrantied item.