Dynamic programming approach to the optimization of Naval aircraft rework and replacement policies

This paper describes a method for determining optimal repair and replacement policies for aireraft, with specific reference to the F–4. The objective of the analysis is to choose the set of policies from all possible alternatives over a finite planning horizon which minimizes the cost of operations. A dynamic program is presented which seeks an optimal path through a series of decision periods, when each period begins with the choice of keeping an aircraft, reworking it before further operation, or buying a new one. We do not consider changes in technology. Therefore, when a replacement does occur, it is made with a similar aircraft. Multivariate statistical techniques are used to estimate the relevant costs as a function of age, and time since last rework.     

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     

On parallel machine replacement problems with general replacement cost functions and stochastic deterioration

The parallel machine replacement problem consists of finding a minimum cost replacement policy for a finite population of economically interdependent machines. In this paper, we formulate a stochastic version of the problem and analyze the structure of optimal policies under general classes of replacement cost functions. We prove that for problems with arbitrary cost functions, there can be optimal policies where a machine is replaced only if all machines in worse states are replaced (Worse Cluster Replacement Rule). We then show that, for problems with replacement cost functions exhibiting nonincreasing marginal costs, there are optimal policies such that, in any stage, machines in the same state are either all kept or all replaced (No‐Splitting Rule). We also present an example that shows that economies of scale in replacement costs do not guarantee optimal policies that satisfy the No‐Splitting Rule. These results lead to the fundamental insight that replacement decisions are driven by marginal costs, and not by economies of scale as suggested in the literature. Finally, we describe how the optimal policy structure, i.e., the No‐Splitting and Worse Cluster Replacement Rules, can be used to reduce the computational effort required to obtain optimal replacement policies. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Heuristics for multicomponent joint replacement: Applications to aircraft engine maintenance

This paper considers the maintenance of aircraft engine components where economies exist for joint replacement because (a) the aircraft must be pulled from service for maintenance and (b) repair of some components requires removal and disassembly of the engine. It is well known that the joint replacement problem is difficult to solve exactly, because the optimal solution does not have a simple structured form. Therefore, we formulate three easy-to-implement heuristics and test their performance against a lower bound for various numerical examples. One of our heuristics, the base interval approach, in which replacement cycles for all components are restricted to be multiples of a specified interval, is shown to be robustly accurate. Moreover, this heuristic is consistent with maintenance policies used by commercial airlines in which periodic maintenance checks are made at regular intervals. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 435–458, 1998     

Block replacement policies for a two‐component system with failure dependence

Block replacement and modified block replacement policies for two‐component systems with failure dependence and economic dependence are considered in this paper. Opportunistic maintenance policies are also considered. Where tractable, long‐run costs per unit time are calculated using renewal theory based arguments; otherwise simulation studies are carried out. The management implications for the adoption of the various policies are discussed. The usefulness of the results in the paper is illustrated through application to a particular two‐component system. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Strong turnpike policies in the single-item capacitated lot-sizing problem with periodical dynamic parameter

This article considers optimization problems in a discrete capacitated lot sizing model for a single product with limited backlogging. The demand as well as the holding and backlogging costs are assumed to be periodical in time. Nothing is assumed about types of the cost functions. It is shown that there exists an optimal infinite inverse policy and a strong turnpike policy. A forward algorithm for computing optimal policies relative to the class of batch ordering type policies is derived. Some backward procedure is adopted to determine a strong turnpike policy. The algorithm is simple, and it terminates after the a number of steps equal to the turnpike horizon. Some remarks on the existence of rolling horizontal plans and forecast horizons are also given. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 775–790, 1997     

Maintenance and flight scheduling of low observable aircraft

In this article, we focus on relatively new maintenance and operational scheduling challenges that are faced by the United States Air Force concerning low‐observable (LO) or stealth aircraft. The LO capabilities of an aircraft degrade stochastically as it flies, making it difficult to make maintenance scheduling decisions. Maintainers can address these damages, but must decide, which aircraft should be put into maintenance, and for how long. Using data obtained from an active duty Air Force F‐22 wing and interviews with Air Force maintainers and program specialists, we model this problem as a generalization of the well‐known restless multiarmed bandit superprocess. Specifically, we use an extension of the traditional model to allow for actions that require varying lengths of time, and generate two separate index policies from a single model; one for maintenance actions and one for the flying action. These index policies allow maintenance schedulers to intuitively, quickly, and effectively rank a fleet of aircraft based on each aircraft's LO status and decide, which aircraft should enter into LO maintenance and for how long, and which aircraft should be used to satisfy daily sortie requirements. Finally, we present extensive data‐driven, detailed simulation results, where we compare the performance of the index policies against policies currently used by the Air Force, as well as some other possible more naive heuristics. The results indicate that the index policies significantly outperform existing policies in terms of fully mission capable (FMC) rates. In particular, the experiments highlight the importance of coordinated maintenance and flying decisions. © 2015 Wiley Periodicals, Inc. 62:60–80, 2015     

Note on “An Optimal Recursive Method for Various Inventory Replenishment Models with Increasing Demand and Shortages” by Teng et al.

In a recent paper, Teng, Chern, and Yang consider four possible inventory replenishment models and determine the optimal replenishment policies for them. They compare these models to identify the best alternative on the basis of minimum total relevant inventory costs. The total cost functions for Model 1 and Model 4 as derived by them are not exact for the comparison. As a result, their conclusion on the least expensive replenishment policy is incorrect. The present article provides the actual total costs for Model 1 and Model 4 to make a correct comparison of the four models. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 602–606, 2000     

Two‐stage component test plans for testing the reliability of a series system

A system reliability is often evaluated by individual tests of components that constitute the system. These component test plans have advantages over complete system based tests in terms of time and cost. In this paper, we consider the series system with n components, where the lifetime of the i‐th component follows exponential distribution with parameter λ_i. Assuming test costs for the components are different, we develop an efficient algorithm to design a two‐stage component test plan that satisfies the usual probability requirements on the system reliability and in addition minimizes the maximum expected cost. For the case of prior information in the form of upper bounds on λ_i's, we use the genetic algorithm to solve the associated optimization problems which are otherwise difficult to solve using mathematical programming techniques. The two‐stage component test plans are cost effective compared to single‐stage plans developed by Rajgopal and Mazumdar. We demonstrate through several numerical examples that our approach has the potential to reduce the overall testing costs significantly. © 2002 John Wiley & Sons, Inc. Naval Research Logistics, 49: 95–116, 2002; DOI 10.1002/nav.1051     

Operating Costs of Aging Air Force Aircraft: Adjusting for Aggregate Budget Effects

The rate at which operating costs grow as aircraft age is important for setting operating budgets and for deciding when to replace aging systems. While studies using data from the 1990's typically found 1–3 percent real rates of growth in operating costs as aircraft age, studies using data from the 2000's found greater rates, for example in the 4–6 percent range. Growth in the total U.S. Air Force budget during the 2000s appears to explain much of the higher estimated annual growth rates in operating costs per flying hour beyond the growth rate intrinsic to the aging of the fleet.     

The LP/POMDP marriage: Optimization with imperfect information

A new technique for solving large‐scale allocation problems with partially observable states and constrained action and observation resources is introduced. The technique uses a master linear program (LP) to determine allocations among a set of control policies, and uses partially observable Markov decision processes (POMDPs) to determine improving policies using dual prices from the master LP. An application is made to a military problem where aircraft attack targets in a sequence of stages, with information acquired in one stage being used to plan attacks in the next. © 2000 John Wiley & Sons, Inc., Naval Research Logistics 47: 607–619, 2000     

Machine maintenance with workload considerations

Machine maintenance is modeled in the setting of a single‐server queue. Machine deterioration corresponds to slower service rates and failure. This leads to higher congestion and an increase in customer holding costs. The decision‐maker decides when to perform maintenance, which may be done pre‐emptively; before catastrophic failures. Similar to classic maintenance control models, the information available to the decision‐maker includes the state of the server. Unlike classic models, the information also includes the number of customers in queue. Considered are both a repair model and a replacement model. In the repair model, with random replacement times, fixed costs are assumed to be constant in the server state. In the replacement model, both constant and variable fixed costs are considered. It is shown in general that the optimal maintenance policies have switching curve structure that is monotone in the server state. However, the switching curve policies for the repair model are not always monotone in the number of customers in the queue. Numerical examples and two heuristics are also presented. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

The parallel replacement problem with demand and capital budgeting constraints

A generalized parallel replacement problem is considered with both fixed and variable replacement costs, capital budgeting, and demand constraints. The demand constraints specify that a number of assets, which may vary over time, are required each period over a finite horizon. A deterministic, integer programming formulation is presented as replacement decisions must be integer. However, the linear programming relaxation is shown to have integer extreme points if the economies of scale binary variables are fixed. This allows for the efficient computation of large parallel replacement problems as only a limited number of 0–1 variables are required. Examples are presented to provide insight into replacement rules, such as the “no‐splitting‐rule” from previous research, under various demand scenarios. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 40–56, 2000     

A linear programming model for minimum-cost procurement and operation of marine corps training aircraft

In military situations of sharply increasing combat activity the Marine Corps is faced with training problems in its expanding aviator corps. Additional training aircraft are required, and procurement decisions must be made. In view of the significant costs involved in procurement and operation of new high performance aircraft, it is very desirable to buy and operate an efficient mix of aircraft necessary for training the pilots to make the Marine Aircraft Wings essentially 100-percent tactically qualified. The mathematical model presented here enables computation of a least-cost mix of training aircraft which satisfies certain specified training requirements. The basic element allowing tradeoffs is the commonality of training available in the F4, RF4, A6, and EA6 types of aircraft. Both airframe oriented and mission oriented training are necessary, but the airframe oriented training can be conducted in either of the aircraft possessing the commonality. Training requirements over a five year period are considered, and the mix of training aircraft has the minimum five year procurement and operating cost.     

Optimal maintenance policies for a safety‐critical system and its deteriorating sensor

We consider the integrated problem of optimally maintaining an imperfect, deteriorating sensor and the safety‐critical system it monitors. The sensor's costless observations of the binary state of the system become less informative over time. A costly full inspection may be conducted to perfectly discern the state of the system, after which the system is replaced if it is in the out‐of‐control state. In addition, a full inspection provides the opportunity to replace the sensor. We formulate the problem of adaptively scheduling full inspections and sensor replacements using a partially observable Markov decision process (POMDP) model. The objective is to minimize the total expected discounted costs associated with system operation, full inspection, system replacement, and sensor replacement. We show that the optimal policy has a threshold structure and demonstrate the value of coordinating system and sensor maintenance via numerical examples. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 399–417, 2017     

Return policy: Hassle‐free or your money‐back guarantee?

This article compares the profitability of two pervasively adopted return policies—money‐back guarantee and hassle‐free policies. In our model, a seller sells to consumers with heterogeneous valuations and hassle costs. Products are subject to quality risk, and product misfit can only be observed post‐purchase. While the hassle‐free policy is cost advantageous from the seller's viewpoint, a money‐back guarantee allows the seller to fine‐tune the consumer hassle on returning the product. Thus, when the two return policies lead to the same consumer behaviors, the hassle‐free policy dominates. Conversely, a money‐back guarantee can be more profitable even if on average, high‐valuation consumers experience a lower hassle cost than the low‐valuation ones. The optimal hassle cost can be higher when product quality gets improved; thus, it is not necessarily a perfect proxy or signal of the seller's quality. We further allow the seller to adopt a mixture of these policies, and identify the concrete operating regimes within which these return policies are optimal among more flexible policies. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 403–417, 2014     

Optimum step‐stress accelerated life test plans for log‐location‐scale distributions

This article presents new tools and methods for finding optimum step‐stress accelerated life test plans. First, we present an approach to calculate the large‐sample approximate variance of the maximum likelihood estimator of a quantile of the failure time distribution at use conditions from a step‐stress accelerated life test. The approach allows for multistep stress changes and censoring for general log‐location‐scale distributions based on a cumulative exposure model. As an application of this approach, the optimum variance is studied as a function of shape parameter for both Weibull and lognormal distributions. Graphical comparisons among test plans using step‐up, step‐down, and constant‐stress patterns are also presented. The results show that depending on the values of the model parameters and quantile of interest, each of the three test plans can be preferable in terms of optimum variance. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Lateral stock transshipments in a two‐location inventory system with fixed and joint replenishment costs

We address the problem of inventory management in a two‐location inventory system, in which the transshipments are carried out as means of emergency or alternative supply after demand has been realized. This model differs from previous ones as regards its replenishment costs structure, in which nonnegligible fixed replenishment costs and a joint replenishment cost are considered. The single period planning horizon is analyzed, with the form and several properties of the optimal replenishment and transshipment policies developed, discussed and illustrated. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 525–547, 1999     

An optimal recursive method for various inventory replenishment models with increasing demand and shortages

We establish various inventory replenishment policies to solve the problem of determining the timing and number of replenishments. We then analytically compare various models, and identify the best alternative among them based on minimizing total relevant costs. Furthermore, we propose a simple and computationally efficient optimal method in a recursive fashion, and provide two examples for illustration. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 791–806, 1997     

Approximation algorithms for capacitated stochastic inventory systems with setup costs

We develop the first approximation algorithm with worst‐case performance guarantee for capacitated stochastic periodic‐review inventory systems with setup costs. The structure of the optimal control policy for such systems is extremely complicated, and indeed, only some partial characterization is available. Thus, finding provably near‐optimal control policies has been an open challenge. In this article, we construct computationally efficient approximate optimal policies for these systems whose demands can be nonstationary and/or correlated over time, and show that these policies have a worst‐case performance guarantee of 4. We demonstrate through extensive numerical studies that the policies empirically perform well, and they are significantly better than the theoretical worst‐case guarantees. We also extend the analyses and results to the case with batch ordering constraints, where the order size has to be an integer multiple of a base load. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 304–319, 2014