Warranty reserving

This paper considers a problem of warranty reserving, namely, the current practice of setting aside part of a product revenue to meet future claims arising from the warranty. We define a Compound Poisson stochastic model for warranty claims and reserve and obtain, using a sample paths technique, the long-run probability distribution of a warranty reserves, managed under alternative warranties and reserve policies.     

Warranty policies and burn-in

Warranty is an important factor for consumer durable products in the marketplace. However, the warranty cost may drastically reduce profitability. Burn in is a common procedure to improve the quality of products after they have been produced, but it is also costly. By taking both the burn-in procedure and warranty policy into consideration, several cost functions can be formulated and optimized. Assuming that the failure-rate function of the product has a bathtub shape, it is shown that the optimal burn-in times that minimize the considered cost functions never exceed the first change point of the failure-rate function. The continuous dependence of the optimal burn-in times on the model parameters and the underlying distribution is also established. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 199–209, 1997     

Warranty analysis and renewal function estimation

Estimation of the expected cost of a warranty for a stochastically failing unit is closely tied to estimation of the renewal function. The renewal function is a basic tool also used in probabilistic models arising in other areas such as reliability theory, inventory theory, and continuous sampling plans. In these other areas, estimation of a straight line approximation of the renewal function instead of direct estimation of the renewal function has proved successful. This approximation is based on a limit expression for large values of the argument, say t, of the renewal function. However, in warranty analusis, typically t is small compared to the mean failure time of the unit. Hence, alternative methods for renewal function estimation, both parametric and nonparametric, are presented and discussed. An important aspect of this paper is to discuss the performance of the renewal function estimators when only a small number of failed units is available. A Monte Carlo study is given which suggests guidelines for choosing an estimator under various circumstances.     

Warranty design under buyer and seller risk aversion

This paper provides a framework in which warranty policies for non-repairable items can be evaluated according to risk preferences of both buyers and sellers. In particular, a warranty price schedule is established such that sellers are indifferent among the policies. Given this schedule, a buyer's response is expressed by selecting the price-warranty combination that minimizes disutility. Within this framework, a warranty can be viewed as an instrumet of risk management that can induce more sales and greater profitability. For given utility functions, analytical results for the development of a price schedule are developed. Numerical results illustrate the substitution effects between warranty terms, prices, and risk parameters.     

不完全维修策略的产品保修期研究

以保修期内的保修费用和可用度2个目标作为优化决策依据,权衡厂家和顾客的利益及需求,建立基于改善因子的不完全预防性维修模型,并通过实例确定厂家和顾客共同达到最大满意度的最佳维修间隔期及保修期,验证了模型的实用性。     

军民融合条件下新型军械装备保修模式研究

针对如何引入承制方优质技术力量支援部队维修保障能力建设这一问题,分析我军现役军械装备维修保障力量、保障模式存在的不足和地方装备维修保障力量的能力特点,明确承制方在军民融合条件下装备维修保障力量中的重要地位.在借鉴国外研究成果的基础上,深入分析和系统设计了我军军械装备的保修模式.结合部队巡检机制,建立了一维免费-按比例收费组合保修的费效模型来验证采购合同中保修条款的合理性.最后进行了案例分析,证明了模型的有效性和实用性.     

Approximating nonstationary two-priority non-preemptive queueing systems

An approximation for analyzing transient and nonstationary two-priority non-preemptive queueing systems is presented. This system has a three-dimensional state space, and through use of state-space partitioning in conjunction with use of conditional surrogate distributions with constant parameters an approximation is designed. Regardless of system capacity K, the approximation requires the numerical solution of only ten differential equations, compared to the K² + K+1 Kolmogorov-forward equations required for the classic solution. Time-dependent approximations of the mean number of entities of type i and of the probability of a type-i entity being in service are obtained. Empirical test results over a wide range of systems indicate the approximation is quite accurate.     

Optimal resource extraction with uncertain reserves

We consider the problem of finding a plan that maximizes the expected discounted return when extracting a nonrenewable resource having uncertain reserves. An extraction plan specifies the rate at which the resource is extracted as a function of time until the resource is exhausted or the time horizon is reached. The return per unit of resource extracted may depend on the rate of extraction, time, and the amount of resource previously extracted. We apply a new method called the generalized search optimization technique to find qualitative features of optimal plans and to devise algorithms for the numerical calculation of optimal plans.     

Quality control and the sales process

Product quality is emerging as a major strategic instrument for competition. The purpose of this article is to assess the effects of quality control on sales, and, vice versa, the effects of the sales process on quality control. A model relating quality control and the sales process (advertising, repeat purchase, and word-of-mouth effects) is developed to evaluate the above relationships. Two special cases, with degenerate and beta distribution for defect items in the production lot, are analyzed in detail. In the former case, analytical results for the optimal quality control schemes are obtained, whereas in the latter, efficient bounds are derived to search for the optimal scheme. It is shown, analytically and numerically, that the sales parameters have significant impact on whether more “stringent” or “tighter” quality control is warranted. Future research directions are also discussed.     

Causative matrix technique for deriving interim period transition probabilities in nonstationary markov process

The Markov assumption that transition probabilities are assumed to be constant over entire periods has been applied in economic and social structures, for example, in the analysis of income and wage distributions. In many cases, however, nonstationary transition probabilities exist over different periods. Based on causative matrix technique, this study shows a binomial approximation for obtaining nonstationary interim transition probabilities under undisturbance when the first and the last transition matrices are known.     

Estimating negative binomial demand for retail inventory management with unobservable lost sales

The importance of effective inventory management has greatly increased for many major retailers because of more intense competition. Retail inventory management methods often use assumptions and demand distributions that were developed for application areas other than retailing. For example, it is often assumed that unmet demand is backordered and that demand is Poisson or normally distributed. In retailing, unmet demand is often lost and unobserved. Using sales data from a major retailing chain, our analysis found that the negative binomial fit significantly better than the Poisson or the normal distribution. A parameter estimation methodology that compensates for unobserved lost sales is developed for the negative binomial distribution. The method's effectiveness is demonstrated by comparing parameter estimates from the complete data set to estimates obtained by artificially truncating the data to simulate lost sales. © 1996 John Wiley & Sons, Inc.     

A dynamic,nonstationary inventory problem for a price/quantity setting firm

A dynamic and nonstationary model is formulated for a firm which attempts to minimize total expected costs over a finite planning horizon. The control variables are price and production. The price p and the demand ζ are linked through the relationship ζ = g(p) + η, where g(p) is the riskless demand curve and η is a random variable. The general model allows for proportional ordering costs, convex holding and stockout costs, downward sloping riskless demand curve, backlogging, partial backlogging, lost sales, partial spoilage of inventory, and two modes of collecting revenue. Sufficient conditions are developed for this problem to have an optimal policy which resembles the single critical number policy known from stochastic inventory theory. It is also shown what set of parameters will satisfy these sufficiency conditions.     

Control of entry to a nonstationary queuing system

In this article the control of entry of customers to a queuing system with s servers is considered. It is assumed that the arrivals form a nonstationary Poisson process with a periodic rate. The service times are assumed to be exponentially distributed with a parameter independent of time. The cost structure considered is the same as that of Naor. It is demonstrated numerically that, like the stationary cases, the average expected benefit of customers per unit of time is a unimodal function of the critical point. And, also, the social critical point is smaller than the individual critical point. These suggest the use of a search technique for finding the social critical point. The results show successful application of the discrete version of the Fibonacci search.     

新型光电装备延伸保修决策方法研究

延伸保修是开展军民融合式装备保障的重要技术手段之一,通过合理选择延伸保修策略,可以在有限的经济条件下较好地完成维修保障任务,提高装备维修保障效益.针对新型光电装备特点,在考虑安全性、任务性和经济性因素的背景下,提出一种由逻辑决断分析法和模糊层次分析法相结合的延伸保修决策方法.通过逻辑决断分析从定性角度选择合理的延伸保修策略,采用模糊层次分析法对逻辑决断得到的结果进行定量补充分析,为类似装备的延伸保修决策提供决策参考.     

Demand estimation in lost sales inventory systems

This article considers the problem of estimating parameters of the demand distribution in lost sales inventory systems. In periods when lost sales occur demand is not observed; one knows only that demand is larger than sales. We assume that demands form a sequence of IID normal random variables, which could be a residual demand process after filtering out seasonality and promotional nonstationarities. We examine three estimators for the mean and standard deviation: maximum likelihood estimator, BLUE (best linear unbiased estimator), and a new estimator derived here. Extensive simulations are reported to compare the performance of the estimators for small and large samples and a variety of parameter settings. In addition, I show how all three estimators can be incorporated into sequential updating routines. © 1994 John Wiley & Sons, Inc.     

Continuous sampling plans for markov-dependent production processes

This article discusses the behavior of three continuous sampling plans: continuous sampling plan 1 (CSP 1) and continuous sampling plan 2 (CSP 2) developed by Dodge [5] and Dodge and Torrey [7], and multilevel continuous sampling plan 2 (MLP 2) developed by Lieberman and Solomon [11], when the quality of successive units in a continuous production process follows a two-state time-homogeneous Markov chain. We first derive the average outgoing quality (AOQ) expressions of these plans. Exact procedures for determining the average outgoing quality limit (AOQL) can be obtained only for CSP 1. For CSP 2 and MLP 2 plans, iterative procedures have been used to obtain the AOQL contours. For these plans, it is assumed that the serial correlation coefficient between the two consecutive random variables of the Markov chain is known. In addition, estimation procedures for the coefficient are given. We show that if the serial correlation coefficient of the Markov chain is positive (negative), the AOQL is increased (decreased) as compared to the case when the successive units in the production process follows a Bernoulli pattern. Let r denote the number of production units examined in succession which are found to be of good quality and k denote the inverse of the sampling fraction employed when quality is good. Then if r and k are sufficiently small, it is observed from the graph that, for small departures of the serial correlation coefficient from zero, the AOQL values do not differ significantly for each of the three plans; whereas for sufficiently large values of r and k, the AOQL values differ significantly. Various aspects of these plans, such as their operating characteristics 2 (OC 2) and the serial correlation coefficient, are discussed.     

Single-stage,multi-product production/inventory systems with lost sales

A production/inventory system consisting of a single processor producing three product types and a warehouse is considered. For each product type, the demand process is assumed to be Poisson and the processing time is phase-type. Excess demand is lost. Products have a priority structure and the processor's attention is shared by all the products according to a switching rule. Production of a product continues until its target level is reached. Then, a switch-over takes place if another product needs the processor's attention. A set-up process takes place every time a switch-over occurs. An (R, r) continuous-review inventory control policy is used to start and stop the production of each product. The underlying Markov chain is studied and its steady-state distribution is obtained recursively. Through the recursive procedure, the steady-state balance equations to be dealt with are significantly reduced to a manageable set. The procedure is implemented on a supercomputer and examples are provided to show its efficiency and stability for a range of model parameters. We analyzed the joint behaviors of the inventory levels of the three products as their demand rates increase. Finally we introduced a cost minimizing objective function to guide design efforts. © 1995 John Wiley & Sons, Inc.     

Inventory models with a mixture of backorders and lost sales

This article presents several single-echelon, single-item, static demand inventory models for situations in which, during the stockout period, a fraction b of the demand is backordered and the remaining fraction 1 - b is lost forever. Both deterministic and stochastic demand are considered. although the case of stochastic demand is treated heuristically. In each situation, a mathematical model representing the average annual cost of operating the inventory system is developed. and an optimum operating policy derived. At the extremes b=1 and b=0 the models presented reduce to the usual backorders and lost sales cases, respectively.