Cost analysis of pro rata and free-replacement warranties

This article examines the short run total costs and long run average costs of products under warranty. Formulae for both consumer cost under warranty and producer profit are derived. The results in the case of the pro rata warranty correct a mistake appearing in Blischke and Scheuer [5]. We also show that expected average cost to both the producer and the consumer of a product under warranty depends on both the mean of the product lifetime distribution and on its failure rate.     

Warranty reserves for nonstationary sales processes

Estimation of warranty costs, in the event of product failure within the warranty period, is of importance to the manufacturer. Costs associated with replacement or repair of the product are usually drawn from a warranty reserve fund created by the manufacturer. Considering a stochastic sales process, first and second moments (and thereby the variance) are derived for the manufacturer's total discounted warranty cost of a single sale for single‐component items under four different warranty policies from a manufacturer's point of view. These servicing strategies represent a renewable free‐replacement, nonrenewable free‐replacement, renewable pro‐rata, and a nonrenewable minimal‐repair warranty plans. The results are extended to determine the mean and variance of total discounted warranty costs for the total sales over the life cycle of the product. Furthermore, using a normal approximation, warranty reserves necessary for a certain protection level, so that reserves are not completely depleted, are found. Results and their managerial implications are studied through an extensive example. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 499–513, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10023     

Stochastic modeling for computational warranty analysis

We examine two key stochastic processes of interest for warranty modeling: (1) remaining total warranty coverage time exposure and (2) warranty load (total items under warranty at time t). Integral equations suitable for numerical computation are developed to yield probability law for these warranty measures. These two warranty measures permit warranty managers to better understand time‐dependent warranty behavior, and thus better manage warranty cash reserves. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Comparisons of renewable warranties

Two types of warranties are commonly applied. There are replacement‐free warranty and pro‐rata warranty. The third warranty is the mixing of the above two. That is it has a replacement‐free stage followed by a pro‐rata stage. In addition to the above classification, warranties can also be a renewable or nonrenewable. No matter which warranty is used, it always incurs certain cost to the manufacturer. Moreover, often manufacturers want to improve the quality of products by burn‐in before delivering them to consumers, and this incurs an extra cost. This article considers the random costs for both burn‐in and renewable warranty, and derives the mean warranty cost. Then the mean costs corresponding to pure replacement‐free warranty, mixing renewable warranty, and pure pro‐rata warranty are compared. Furthermore, the mean costs under the same warranty policy associated with different lifetime distributions of products are also compared. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 91–106, 1999     

Periodic inspection policy with preventive maintenance

This article considers a modified inspection policy with periodic check intervals, where the unit after check has the same age as before with probability p and is as good as new with probability q. The mean time to failure and the expected number of checks before failure are derived, forming renewal-type equations. The total expected cost and the expected cost per unit of time until detection of failure are obtained. Optimum inspection policies which minimize the expected costs are given as a numerical example.     

Warranty claims modeling

A company wishes to estimate or predict its financial exposure in a reporting period of length T (typically one quarter) because of warranty claims. We propose a fairly general random measure model which allows computation of the Laplace transform of the total claim made against the company in the reporting interval due to warranty claims. When specialized to a Poisson process of both sales and warranty claims, statistical estimation of relevant quantities is possible. The methodology is illustrated by analyzing automobile sales and warranty claims data from a large car manufacturer for a single car model and model year. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

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.     

On the value of information in dynamic production/inventory problems under forecast evolution

In this article we explore how total system costs and inventory positions are affected when forecasts are incorporated explicitly in production/inventory systems. We assume that forecasts for demand of a certain item are available in each period, and they evolve from one period to the next in accordance with an additive evolution model. In order to analyze the effects of the forecasts on the production/inventory system we compare the optimal ordering policy and the expected costs of the model that keeps forecasts with that of a comparable standard inventory model. We show that under mild assumptions the former yields lower expected costs and inventory levels than the latter. © 1996 John Wiley & Sons, Inc.     

Optimal ordering policies for a product that perishes in two periods subject to stochastic demand

This paper considers the problem of computing optimal ordering policies for a product that has a life of exactly two periods when demand is random. Initially costs are charged against runouts (stockouts) and outdating (perishing). By charging outdating costs according to the expected amount of outdating one period into the future, a feasible one period model is constructed. The central theorem deals with the n-stage dynamic problem and demonstrates the appropriate cost functions are convex in the decision variable and also provides bounds on certain derivatives. The model is then generalized to include ordering and holding costs. The paper is concluded with a discussion of the infinite horizon problem.     

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.     

Outsourcing warranty repairs: Dynamic allocation

In this paper we consider the problem of minimizing the costs of outsourcing warranty repairs when failed items are dynamically routed to one of several service vendors. In our model, the manufacturer incurs a repair cost each time an item needs repair and also incurs a goodwill cost while an item is awaiting and undergoing repair. For a large manufacturer with annual warranty costs in the tens of millions of dollars, even a small relative cost reduction from the use of dynamic (rather than static) allocation may be practically significant. However, due to the size of the state space, the resulting dynamic programming problem is not exactly solvable in practice. Furthermore, standard routing heuristics, such as join‐the‐shortest‐queue, are simply not good enough to identify potential cost savings of any significance. We use two different approaches to develop effective, simply structured index policies for the dynamic allocation problem. The first uses dynamic programming policy improvement while the second deploys Whittle's proposal for restless bandits. The closed form indices concerned are new and the policies sufficiently close to optimal to provide cost savings over static allocation. All results of this paper are demonstrated using a simulation study. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Some aspects in estimating warranty and post‐warranty repair demands

When customers buy a product, they are often eligible for free repairs for a certain warranty period. In this article, we study some important aspects, which are often overlooked in the literature but are of interest to the manufacturer, in estimating both warranty and post‐warranty repair demands. We consider that the installed base of the product (i.e., the number of units of the product actually in use) varies with time due to both new sales and units being taken out of service. When estimating warranty and post‐warranty repair demands, we explicitly address the fact that customers may not always request repairs for failed units. For the case where the product failure time distribution is exponential, we derive the closed‐form expressions for both types of repair demands of a single unit and of the time‐varying installed base. The insights into some risk‐related quantities are also presented. Furthermore, the proposed model is extended by considering delayed warranty claims that are frequently seen in practice. Numerical examples illustrate that understanding both types of repair demands and the related decision variables is important for managing the obligatory and profitable repair services. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 499–511, 2013     

Optimizing usage and maintenance decisions for k‐out‐of‐n systems of moving assets

We consider an integrated usage and maintenance optimization problem for a k‐out‐of‐n system pertaining to a moving asset. The k‐out‐of‐n systems are commonly utilized in practice to increase availability, where n denotes the total number of parallel and identical units and k the number of units required to be active for a functional system. Moving assets such as aircraft, ships, and submarines are subject to different operating modes. Operating modes can dictate not only the number of system units that are needed to be active, but also where the moving asset physically is, and under which environmental conditions it operates. We use the intrinsic age concept to model the degradation process. The intrinsic age is analogous to an intrinsic clock which ticks on a different pace in different operating modes. In our problem setting, the number of active units, degradation rates of active and standby units, maintenance costs, and type of economic dependencies are functions of operating modes. In each operating mode, the decision maker should decide on the set of units to activate (usage decision) and the set of units to maintain (maintenance decision). Since the degradation rate differs for active and standby units, the units to be maintained depend on the units that have been activated, and vice versa. In order to minimize maintenance costs, usage and maintenance decisions should be jointly optimized. We formulate this problem as a Markov decision process and provide some structural properties of the optimal policy. Moreover, we assess the performance of usage policies that are commonly implemented for maritime systems. We show that the cost increase resulting from these policies is up to 27% for realistic settings. Our numerical experiments demonstrate the cases in which joint usage and maintenance optimization is more valuable. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 418–434, 2017     

Limiting behavior of the stochastic sequential assignment problem

The stochastic sequential assignment problem (SSAP) considers how to allocate available distinct workers to sequentially arriving tasks with stochastic parameters such that the expected total reward obtained from the sequential assignments is maximized. Implementing the optimal assignment policy for the SSAP involves calculating a new set of breakpoints upon the arrival of each task (i.e., for every time period), which is impractical for large‐scale problems. This article studies two problems that are concerned with obtaining stationary policies, which achieve the optimal expected reward per task as the number of tasks approaches infinity. The first problem considers independent and identically distributed (IID) tasks with a known distribution function, whereas in the second problem tasks are derived from r different unobservable distributions governed by an ergodic Markov chain. The convergence rate of the expected reward per task to the optimal value is also obtained for both problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

The single and multi‐item transshipment problem with fixed transshipment costs

This article deals with supply chain systems in which lateral transshipments are allowed. For a system with two retailers facing stochastic demand, we relax the assumption of negligible fixed transshipment costs, thus, extending existing results for the single‐item case and introducing a new model with multiple items. The goal is to determine optimal transshipment and replenishment policies, such that the total centralized expected profit of both retailers is maximized. For the single‐item problem with fixed transshipment costs, we develop optimality conditions, analyze the expected profit function, and identify the optimal solution. We extend our analysis to multiple items with joint fixed transshipment costs, a problem that has not been investigated previously in the literature, and show how the optimality conditions may be extended for any number of items. Due to the complexity involved in solving these conditions, we suggest a simple heuristic based on the single‐item results. Finally, we conduct a numerical study that provides managerial insights on the solutions obtained in various settings and demonstrates that the suggested heuristic performs very well. © 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 637–664, 2014     

A stochastic constrained optimal replacement model

In this paper a stochastically constrained replacement model is formulated. This model determines a sequence of replacement dates such that the total “current account” cost of all future costs and capital expenditures over an infinite time horizon for the n initial incumbent machines is minimized subject to the constraints that an expected number of machines are in a chosen utility class at any point in time. We then indicate one possible solution method for the model.     

Applications of renewal theory in analysis of the free-replacement warranty

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. Payments of C are not made at fixed intervals of length W, but in random cycles of length Y = W + γ(W), where γ(W) is the (random) remaining life-time of the item in service W time units after the beginning of a cycle. The expected number of payments over the life cycle, L, of the item is given by M_Y(L), the renewal function for the random variable Y. We investigate this renewal function analytically and numerically and compare the latter with known asymptotic results. The distribution of Y, and hence the renewal function, depends on the underlying failure distribution of the items. Several choices for this distribution, including the exponential, uniform, gamma and Weibull, are considered.     

A robust policy for serial agile production systems

One of the major problems in modeling production systems is how to treat the job arrival process. Restrictive assumptions such as Markovian arrivals do not represent real world systems, especially if the arrival process is generated by job departures from upstream workstations. Under these circumstances, cost‐effective policies that are robust with respect to the nature of the arrival process become of interest. In this paper, we focus on minimizing the expected total holding and setup costs in a two‐stage produce‐to‐order production system operated by a cross‐trained worker. We will show that if setup times are insignificant in comparison with processing times, then near‐optimal policies can be generated with very robust performances with respect to the arrival process. We also present conditions under which these near‐optimal policies can be obtained by using only the arrival and service rates. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Contractual agreements for coordination and vendor‐managed delivery under explicit transportation considerations

We consider the coordination problem between a vendor and a buyer operating under generalized replenishment costs that include fixed costs as well as stepwise freight costs. We study the stochastic demand, single‐period setting where the buyer must decide on the order quantity to satisfy random demand for a single item with a short product life cycle. The full order for the cycle is placed before the cycle begins and no additional orders are accepted by the vendor. Due to the nonrecurring nature of the problem, the vendor's replenishment quantity is determined by the buyer's order quantity. Consequently, by using an appropriate pricing schedule to influence the buyer's ordering behavior, there is an opportunity for the vendor to achieve substantial savings from transportation expenses, which are represented in the generalized replenishment cost function. For the problem of interest, we prove that the vendor's expected profit is not increasing in buyer's order quantity. Therefore, unlike the earlier work in the area, it is not necessarily profitable for the vendor to encourage larger order quantities. Using this nontraditional result, we demonstrate that the concept of economies of scale may or may not work by identifying the cases where the vendor can increase his/her profits either by increasing or decreasing the buyer's order quantity. We prove useful properties of the expected profit functions in the centralized and decentralized models of the problem, and we utilize these properties to develop alternative incentive schemes for win–win solutions. Our analysis allows us to quantify the value of coordination and, hence, to identify additional opportunities for the vendor to improve his/her profits by potentially turning a nonprofitable transaction into a profitable one through the use of an appropriate tariff schedule or a vendor‐managed delivery contract. We demonstrate that financial gain associated with these opportunities is truly tangible under a vendor‐managed delivery arrangement that potentially improves the centralized solution. Although we take the viewpoint of supply chain coordination and our goal is to provide insights about the effect of transportation considerations on the channel coordination objective and contractual agreements, the paper also contributes to the literature by analyzing and developing efficient approaches for solving the centralized problem with stepwise freight costs in the single‐period setting. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Optimal scheduling of reader—systems

We consider a reader—writer system consisting of a single server and a fixed number of jobs (or customers) belonging to two classes. Class one jobs are called readers and any number of them can be processed simultaneously. Class two jobs are called writers and they have to be processed one at a time. When a writer is being processed no other writer or readers can be processed. A fixed number of readers and writers are ready for processing at time 0. Their processing times are independent random variables. Each reader and writer has a fixed waiting cost rate. We find optimal scheduling rules that minimize the expected total waiting cost (expected total weighted flowtime). We consider both nonpreemptive and preemptive scheduling. The optimal nonpreemptive schedule is derived by a variation of the usual interchange argument, while the optimal schedule in the preemptive case is given by a Gittins index policy. These index policies continue to be optimal for systems in which new writers enter the system in a Poisson fashion. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 483–495, 1998