Two‐dimensional failure modeling with minimal repair

In this paper, we discuss two‐dimensional failure modeling for a system where degradation is due to age and usage. We extend the concept of minimal repair for the one‐dimensional case to the two‐dimensional case and characterize the failures over a two‐dimensional region under minimal repair. An application of this important result to a manufacturer's servicing costs for a two‐dimensional warranty policy is given and we compare the minimal repair strategy with the strategy of replacement of failure. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Securing a border under asymmetric information

We study a stochastic interdiction model of Morton et al. IIE Transactions, 39 (2007):3–14 that locates radiation sensors at border crossings to detect and prevent the smuggling of nuclear material. In this model, an interdictor places sensors at customs checkpoints to minimize a potential smuggler's maximum probability of crossing a border undetected. We focus on a model variant in which the interdictor has different, and likely more accurate, perceptions of the system's parameters than the smuggler does. We introduce a model that is tighter and uses fewer constraints than that of Morton et al. We also develop a class of valid inequalities along with a corresponding separation procedure that can be used within a cutting‐plane approach to reduce computational effort. Computational results demonstrate the effectiveness of our approach.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 91–100, 2014     

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     

Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations

A Markovian arrival process of order n, MAP(n), is typically described by two n × n transition rate matrices in terms of rate parameters. While it is straightforward and intuitive, the Markovian representation is redundant since the minimal number of parameters is n² for non‐redundant MAP(n). It is well known that the redundancy complicates exact moment fittings. In this article, we present a minimal and unique Laplace‐Stieltjes transform (LST) representations for MAP(n)s. Even though the LST coefficients vector itself is not a minimal representation, we show that the joint LST of stationary intervals can be represented with the minimum number of parameters. We also propose another minimal representation for MAP(3)s based on coefficients of the characteristic polynomial equations of the two transition rate matrices. An exact moment fitting procedure is presented for MAP(3)s based on two proposed minimal representations. We also discuss how MAP(3)/G/1 departure process can be approximated as a MAP(3). A simple tandem queueing network example is presented to show that the MAP(3) performs better than the MAP(2) in queueing approximations especially under moderate traffic intensities. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 549–561, 2016     

Resource allocation with lumpy demand: To speed or not to speed?

In the classical EPQ model with continuous and constant demand, holding and setup costs are minimized when the production rate is no larger than the demand rate. However, the situation may change when demand is lumpy. We consider a firm that produces multiple products, each having a unique lumpy demand pattern. The decision involves determining both the lot size for each product and the allocation of resources for production rate improvements among the products. We find that each product's optimal production policy will take on only one of two forms: either continuous production or lot‐for‐lot production. The problem is then formulated as a nonlinear nonsmooth knapsack problem among products determined to be candidates for resource allocation. A heuristic procedure is developed to determine allocation amounts. The procedure decomposes the problem into a mixed integer program and a nonlinear convex resource allocation problem. Numerical tests suggest that the heuristic performs very well on average compared to the optimal solution. Both the model and the heuristic procedure can be extended to allow the company to simultaneously alter both the production rates and the incoming demand lot sizes through quantity discounts. Extensions can also be made to address the case where a single investment increases the production rate of multiple products. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

The proper balancing of repair and preventive maintenance activities in the determination of coordinated shipboard allowance lists

The determination as to the cost-effective number of spares for given types of items or equipment to be carried on board various types of ships is studied. This spare pool, known as the ship's COSAL, must provide prespecified levels of protection against stockouts for all uses of that item on board the given ship. This article derives and illustrates the methodology for optimally trading off the reduction in the ship's COSAL that can be gained by improving repair/resupply capabilities or by lowering the failure rate of the equipment through more or different types of preventive maintenance. A flexible class of preventive maintenance/repair response functions to cost is studied which are nonlinear and exhibit realistic diminishing returns. Two different types of assumptions are possible regarding the interdependencies of the resupply times across different ship types. A tractable budget allocation method is presented which can be used in a multiitem, multiship, multiechelon repair environment where there is one budget to cover all spares, all repair/resupply, and preventive maintenance activities. The technique incorporates different criticalities of shortages by type of ship and item. It can be used either in a budget building mode or a budget execution mode.     

Optimal service‐capacity allocation in a loss system

We consider a loss system with a fixed budget for servers. The system owner's problem is choosing the price, and selecting the number and quality of the servers, in order to maximize profits, subject to a budget constraint. We solve the problem with identical and different service rates as well as with preemptive and nonpreemptive policies. In addition, when the policy is preemptive, we prove the following conservation law: the distribution of the total service time for a customer entering the slowest server is hyperexponential with expectation equal to the average service rate independent of the allocation of the capacity. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 81–97, 2015     

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     

Allocation of quality improvement targets based on investments in learning

Purchased materials often account for more than 50% of a manufacturer's product nonconformance cost. A common strategy for reducing such costs is to allocate periodic quality improvement targets to suppliers of such materials. Improvement target allocations are often accomplished via ad hoc methods such as prescribing a fixed, across‐the‐board percentage improvement for all suppliers, which, however, may not be the most effective or efficient approach for allocating improvement targets. We propose a formal modeling and optimization approach for assessing quality improvement targets for suppliers, based on process variance reduction. In our models, a manufacturer has multiple product performance measures that are linear functions of a common set of design variables (factors), each of which is an output from an independent supplier's process. We assume that a manufacturer's quality improvement is a result of reductions in supplier process variances, obtained through learning and experience, which require appropriate investments by both the manufacturer and suppliers. Three learning investment (cost) models for achieving a given learning rate are used to determine the allocations that minimize expected costs for both the supplier and manufacturer and to assess the sensitivity of investment in learning on the allocation of quality improvement targets. Solutions for determining optimal learning rates, and concomitant quality improvement targets are derived for each learning investment function. We also account for the risk that a supplier may not achieve a targeted learning rate for quality improvements. An extensive computational study is conducted to investigate the differences between optimal variance allocations and a fixed percentage allocation. These differences are examined with respect to (i) variance improvement targets and (ii) total expected cost. For certain types of learning investment models, the results suggest that orders of magnitude differences in variance allocations and expected total costs occur between optimal allocations and those arrived at via the commonly used rule of fixed percentage allocations. However, for learning investments characterized by a quadratic function, there is surprisingly close agreement with an “across‐the‐board” allocation of 20% quality improvement targets. © John Wiley & Sons, Inc. Naval Research Logistics 48: 684–709, 2001     

Asymptotically optimal component assembly plans in repairable systems and server allocation in parallel multiserver queues

T identical exponential lifetime components out of which G are initially functioning (and B are not) are to be allocated to N subsystems, which are connected either in parallel or in series. Subsystem i, i = 1,…, N, functions when at least K_i of its components function and the whole system is maintained by a single repairman. Component repair times are identical independent exponentials and repaired components are as good as new. The problem of the determination of the assembly plan that will maximize the system reliability at any (arbitrary) time instant t is solved when the component failure rate is sufficiently small. For the parallel configuration, the optimal assembly plan allocates as many components as possible to the subsystem with the smallest K_i and allocates functioning components to subsystems in increasing order of the K_i's. For the series configuration, the optimal assembly plan allocates both the surplus and the functioning components equally to all subsystems whenever possible, and when not possible it favors subsystems in decreasing order of the K_i's. The solution is interpreted in the context of the optimal allocation of processors and an initial number of jobs in a problem of routing time consuming jobs to parallel multiprocessor queues. © John Wiley & Sons, Inc. Naval Research Logistics 48: 732–746, 2001     

Asymptotic properties of imperfect repair models and estimation of repair efficiency

The aim of this articles is to study the asymptotic behavior of two imperfect repair models, called Arithmetic Reduction of Intensity and Arithmetic Reduction of Age models. These models have been proposed by Doyen and Gaudoin (Reliab Eng Syst Safe 84 (2004) 45–56) and include many usual virtual age models. First, it is proved that the failure intensity of these models is asymptotically almost surely equivalent to a deterministic increasing function with a cumulative error proportional to a logarithm. Second, the almost sure convergence and asymptotic normality of several estimators of repair efficiency are derived, when the wear‐out process without repair is known. Finally, the coverage rate of the asymptotic confidence intervals issued from those estimators is empirically studied. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

“Government contracting options: A model and application“∗

This paper models and simulates a government‐contractor principal‐agent weapon system repair model. Insights are derived as to how government repair contracts should be constructed so as to induce optimal contractor behavior. The paper's general conclusion is that the best contracting approach combines a lump‐sum payment that does not vary with the number of units repaired, expensive item cost‐sharing, and a contractor‐provided availability guarantee. Provided there is intercontractor competition, this type of contract performs well even if the government is poorly informed about weapon system break patterns or repair costs.     

Warranty costs: An age‐dependent failure/repair model

An age‐dependent repair model is proposed. The notion of the “calendar age” of the product and the degree of repair are used to define the virtual age of the product. The virtual failure rate function and the virtual hazard function related to the lifetime of the product are discussed. Under a nonhomogeneous Poisson process scenario the expected warranty costs for repairable products associated with linear pro‐rata, nonrenewing free replacement and renewing free replacement warranties are evaluated. Illustration of the results is given by numerical and graphical examples. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Tail hazard rate ordering properties of order statistics and coherent systems

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k‐out‐of‐n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Allocation of two redundancies in two‐component series systems

The allocation of redundancies in a system to optimize the reliability of system performance is an interesting problem in reliability engineering and system security. In this article, we focus on the optimal allocation of two exponentially distributed active (standby) redundancies in a two‐component series system using the tool of stochastic ordering. For the case of active redundancy, stochastic comparisons are carried out in terms of the likelihood ratio and reversed hazard rate orders. For the case of standby redundancy, a likelihood ratio ordering result is developed. The results established here generalize and strengthen corresponding results in the recent literature. In addition, several numerical examples are used to explicate the results. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Dual sourcing: Responsive hedging against correlated supply and demand uncertainty

This article analyzes dual sourcing decisions under stochastically dependent supply and demand uncertainty. A manufacturer faces the trade‐off between investing in unreliable but high‐margin offshore supply and in reliable but low‐margin local supply, where the latter allows for production that is responsively contingent on the actual demand and offshore supply conditions. Cost thresholds for both types of supply determine the optimal resource allocation: single offshore sourcing, single responsive sourcing, or dual sourcing. Relying on the concept of concordance orders, we study the effects of correlation between supply and demand uncertainty. Adding offshore supply to the sourcing portfolio becomes more favorable under positive correlation, since offshore supply is likely to satisfy demand when needed. Selecting responsive capacity under correlated supply and demand uncertainty is not as straightforward, yet we establish the managerially relevant conditions under which responsive capacity either gains or loses in importance. Our key results are extended to the broad class of endogenous supply uncertainty developed by Dada et al. [Manufact Serv Operat Mange 9 (2007), 9–32].© 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

The “signature” of a coherent system and its application to comparisons among systems

Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a system's lifetime distribution as a function of the system's “signature,” that is, as a function of the vector p= (p₁, … , p_n), where p_i is the probability that the system fails upon the occurrence of the ith component failure. Sufficient conditions are provided for the lifetime of one system to be larger than that of another system in three different senses: stochastic ordering, hazard rate ordering, and likelihood ratio ordering. Further, a new preservation theorem for hazard rate ordering is established. In the final section, the notion of system signature is used to examine a recently published conjecture regarding componentwise and systemwise redundancy. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 507–523, 1999     

Repairable consecutive‐k‐out‐of‐n: F system with Markov dependence

In this article, a model for a repairable consecutive‐k‐out‐of‐n: F system with Markov dependence is studied. A binary vector is used to represent the system state. The failure rate of a component in the system depends on the state of the preceding component. The failure risk of a system state is then introduced. On the basis of the failure risk, a priority repair rule is adopted. Then the transition density matrix can be determined, and the analysis of the system reliability can be conducted accordingly. One example each of a linear and a circular system is then studied in detail to explain the model and methodology developed in this paper. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 18–39, 2000     

Impact of reseller's and sales agent's forecasting accuracy in a multilayer supply chain

We consider a three‐layer supply chain with a manufacturer, a reseller, and a sales agent. The demand is stochastically determined by the random market condition and the sales agent's private effort level. Although the manufacturer is uninformed about the market condition, the reseller and the sales agent conduct demand forecasting and generate private demand signals. Under this framework with two levels of adverse selection intertwined with moral hazard, we study the impact of the reseller's and the sales agent's forecasting accuracy on the profitability of each member. We show that the manufacturer's profitability is convex on the reseller's forecasting accuracy. From the manufacturer's perspective, typically improving the reseller's accuracy is detrimental when the accuracy is low but is beneficial when it is high. We identify the concrete interrelation among the manufacturer‐optimal reseller's accuracy, the volatility of the market condition, and the sales agent's accuracy. Finally, the manufacturer's interest may be aligned with the reseller's when only the reseller can choose her accuracy; this alignment is never possible when both downstream players have the discretion to choose their accuracy. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 207–222, 2014     

Contracts and coordination: Supply chains with uncertain demand and supply

Considering a supply chain with a supplier subject to yield uncertainty selling to a retailer facing stochastic demand, we find that commonly studied classical coordination contracts fail to coordinate both the supplier's production and the retailer's procurement decisions and achieve efficient performance. First, we study the vendor managed inventory (VMI) partnership. We find that a consignment VMI partnership coupled with a production cost subsidy achieves perfect coordination and a win‐win outcome; it is simple to implement and arbitrarily allocates total channel profit. The production cost subsidy optimally chosen through Nash bargaining analysis depends on the bargaining power of the supplier and the retailer. Further, motivated by the practice that sometimes the retailer and the supplier can arrange a “late order,” we also analyze the behavior of an advance‐purchase discount (APD) contract. We find that an APD with a revenue sharing contract can efficiently coordinate the supply chain as well as achieve flexible profit allocation. Finally, we explore which coordination contract works better for the supplier vs. the retailer. It is interesting to observe that Nash bargaining solutions for the two coordination contracts are equivalent. We further provide recommendations on the applications of these contracts. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 305–319, 2016