Replacement models under additive damage

A production system which generates income is subject to random failure. Upon failure, the system is replaced by a new identical one and the replacement cycles are repeated indefinitely. In our breakdown model, shocks occur to the system in a Poisson stream. Each shock causes a random amount of damage, and these damages accumulate additively. The failure time depends on the accumulated damage in the system. The income from the system and the cost associated with a planned replacement depend on the accumulated damage in the system. An additional cost is incurred at each failure in service. We allow a controller to replace the system at any stopping time T before failure time. We will consider the problem of specifying a replacement rule that is optimal under the following criteria: maximum total long-run average net income per unit time, and maximum total long-run expected discounted net income. Our primary goal is to introduce conditions under which an optimal policy is a control limit policy and to investigate how the optimal policy can be obtained. Examples will be presented to illustrate computational procedures.     

Dynamic routing of prioritized warranty repairs

Recent years have seen a strong trend toward outsourcing warranty repair services to outside vendors. In this article we consider the problem of dynamically routing warranty repairs to service vendors when warranties have priority levels. Each time an item under warranty fails, it is sent to one of the vendors for repair. Items covered by higher priority warranty receive higher priority in repair service. The manufacturer pays a fixed fee per repair and incurs a linear holding cost while an item is undergoing or waiting for repair. The objective is to minimize the manufacturer's long‐run average cost. Because of the complexity of the problem, it is very unlikely that there exist tractable ways to find the optimal routing strategies. Therefore, we propose five heuristic routing procedures that are applicable to real‐life problems. We evaluate the heuristics using simulation. The simulation results show that the index‐based “generalized join the shortest queue” policy, which applies a single policy improvement step to an initial state‐independent policy, performs the best among all five heuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Termination policies for a two-state stochastic process

The purpose of this paper is to investigate and optimize policies which can be used to terminate a two-state stochastic process with a random lifetime. Such a policy consists of a schedule of times at which termination attempts should be made. Conditions are given which reduce the difficulty of finding the optimal policy by eliminating constraints and some boundary points from consideration. Finally, a bound for the optimal policy is derived for a case where some restrictions are imposed on the model.     

Optimal empty vehicle redistribution for hub‐and‐spoke transportation systems

This article considers the empty vehicle redistribution problem in a hub‐and‐spoke transportation system, with random demands and stochastic transportation times. An event‐driven model is formulated, which yields the implicit optimal control policy. Based on the analytical results for two‐depot systems, a dynamic decomposition procedure is presented which produces a near‐optimal policy with linear computational complexity in terms of the number of spokes. The resulting policy has the same asymptotic behavior as that of the optimal policy. It is found that the threshold‐type control policy is not usually optimal in such systems. The results are illustrated through small‐scale numerical examples. Through simulation the robustness of the dynamic decomposition policy is tested using a variety of scenarios: more spokes, more vehicles, different combinations of distribution types for the empty vehicle travel times and loaded vehicle arrivals. This shows that the dynamic decomposition policy is significantly better than a heuristics policy in all scenarios and appears to be robust to the assumptions of the distribution types. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal ordering policies when anticipating parameter changes in EOQ systems

The classical Economic Order Quantity Model requires the parameters of the model to be constant. Some EOQ models allow a single parameter to change with time. We consider EOQ systems in which one or more of the cost or demand parameters will change at some time in the future. The system we examine has two distinct advantages over previous models. One obvious advantage is that a change in any of the costs is likely to affect the demand rate and we allow for this. The second advantage is that often, the times that prices will rise are fairly well known by announcement or previous experience. We present the optimal ordering policy for these inventory systems with anticipated changes and a simple method for computing the optimal policy. For cases where the changes are in the distant future we present a myopic policy that yields costs which are near-optimal. In cases where the changes will occur in the relatively near future the optimal policy is significantly better than the myopic policy.     

An optimal critical level policy for inventory systems with two demand classes

Traditional inventory systems treat all demands of a given item equally. This approach is optimal if the penalty costs of all customers are the same, but it is not optimal if the penalty costs are different for different customer classes. Then, demands of customers with high penalty costs must be filled before demands of customers with low penalty costs. A commonly used inventory policy for dealing with demands with different penalty costs is the critical level inventory policy. Under this policy demands with low penalty costs are filled as long as inventory is above a certain critical level. If the inventory reaches the critical level, only demands with high penalty costs are filled and demands with low penalty costs are backordered. In this article, we consider a critical level policy for a periodic review inventory system with two demand classes. Because traditional approaches cannot be used to find the optimal parameters of the policy, we use a multidimensional Markov chain to model the inventory system. We use a sample path approach to prove several properties of this inventory system. Although the cost function is not convex, we can build on these properties to develop an optimization approach that finds the optimal solution. We also present some numerical results. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Revenue maximization in two-station tandem queueing systems

We study optimal pricing for tandem queueing systems with finite buffers. The service provider dynamically quotes prices to incoming price sensitive customers to maximize the long-run average revenue. We present a Markov decision process model for the optimization problem. For systems with two stations, general-sized buffers, and two or more prices, we describe the structure of the optimal dynamic pricing policy and develop tailored policy iteration algorithms to find an optimal pricing policy. For systems with two stations but no intermediate buffer, we characterize conditions under which quoting either a high or a low price to all customers is optimal and provide an easy-to-implement algorithm to solve the problem. Numerical experiments are conducted to compare the developed algorithms with the regular policy iteration algorithm. The work also discusses possible extensions of the obtained results to both three-station systems and two-station systems with price and congestion sensitive customers using numerical analysis.     

Optimal renewal policies for complex systems

Most maintenance and replacement models for industrial equipment have been developed for independent single-component machines. Most equipment, however, consists of multiple components. Also, when the maintenance crew services several machines, the maintenance policy for each machine is not independent of the states of the other machines. In this paper, two dynamic programming replacement models are presented. The first is used to determine the optimal replacement policy for multi-component equipment. The second is used to determine the optimal replacement policy for a multi-machine system which uses one replacement crew to service several machines. In addition, an approach is suggested for developing an efficient replacement policy for a multi-component, multi-machine system.     

Periodic review inventory management with contingent use of two freight modes with fixed costs

We study a stochastic inventory model of a firm that periodically orders a product from a make‐to‐order manufacturer. Orders can be shipped by a combination of two freight modes that differ in lead‐times and costs, although orders are not allowed to cross. Placing an order as well as each use of each freight mode has a fixed and a quantity proportional cost. The decision of how to allocate units between the two freight modes utilizes information about demand during the completion of manufacturing. We derive the optimal freight mode allocation policy, and show that the optimal policy for placing orders is not an (s,S) policy in general. We provide tight bounds for the optimal policy that can be calculated by solving single period problems. Our analysis enables insights into the structure of the optimal policy specifying the conditions under which it simplifies to an (s,S) policy. We characterize the best (s,S) policy for our model, and through extensive numerical investigation show that its performance is comparable with the optimal policy in most cases. Our numerical study also sheds light on the benefits of the dual freight model over the single freight models. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Optimal burn‐in procedure for periodically inspected systems

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we study burn‐in procedure for a system that is maintained under periodic inspection and perfect repair policy. Assuming that the underlying lifetime distribution of a system has an initially decreasing and/or eventually increasing failure rate function, we derive upper and lower bounds for the optimal burn‐in time, which maximizes the system availability. Furthermore, adopting an age replacement policy, we derive upper and lower bounds for the optimal age parameter of the replacement policy for each fixed burn‐in time and a uniform upper bound for the optimal burn‐in time given the age replacement policy. These results can be used to reduce the numerical work for determining both optimal burn‐in time and optimal replacement policy. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

A partial characterization of the optimal ordering/rationing policy for a periodic review system with two demand classes and backordering

We consider a finite horizon periodic review, single product inventory system with a fixed setup cost and two stochastic demand classes that differ in their backordering costs. In each period, one must decide whether and how much to order, and how much demand of the lower class should be satisfied. We show that the optimal ordering policy can be characterized as a state dependent (s,S) policy, and the rationing structure is partially obtained based on the subconvexity of the cost function. We then propose a simple heuristic rationing policy, which is easy to implement and close to optimal for intensive numerical examples. We further study the case when the first demand class is deterministic and must be satisfied immediately. We show the optimality of the state dependent (s,S) ordering policy, and obtain additional rationing structural properties. Based on these properties, the optimal ordering and rationing policy for any state can be generated by finding the optimal policy of only a finite set of states, and for each state in this set, the optimal policy is obtained simply by choosing a policy from at most two alternatives. An efficient algorithm is then proposed. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

A dynamic inventory system with recycling

This paper deals with a periodic review inventory system in which a constant proportion of stock issued to meet demand each period feeds back into the inventory after a fixed number of periods. Various applications of the model are discussed, including blood bank management and the control of reparable item inventories. We assume that on hand inventory is subject to proportional decay. Demands in successive periods are assumed to be independent identically distributed random variables. The functional equation defining an optimal policy is formulated and a myopic base stock approximation is developed. This myopic policy is shown to be optimal for the case where the feedback delay is equal to one period. Both cost and ordering decision comparisons for optimal and myopic policies are carried out numerically for a delay time of two periods over a wide range of input parameter values.     

Dynamic inventory management with cash flow constraints

In this article, we consider a classic dynamic inventory control problem of a self‐financing retailer who periodically replenishes its stock from a supplier and sells it to the market. The replenishment decisions of the retailer are constrained by cash flow, which is updated periodically following purchasing and sales in each period. Excess demand in each period is lost when insufficient inventory is in stock. The retailer's objective is to maximize its expected terminal wealth at the end of the planning horizon. We characterize the optimal inventory control policy and present a simple algorithm for computing the optimal policies for each period. Conditions are identified under which the optimal control policies are identical across periods. We also present comparative statics results on the optimal control policy. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections

An EMQ model with a production process subject to random deterioration is considered. The process can be monitored through inspections, and both the lot size and the inspection schedule are subject to control. The “in-control” periods are assumed to be generally distributed and the inspections are imperfect, i.e., the true state of the process is not necessarily revealed through an inspection. The objective is the joint determination of the lot size and the inspection schedule, minimizing the long-run expected average cost per unit time. Both discrete and continuous cases are examined. A dynamic programming formulation is considered in the case where the inspections can be performed only at discrete times, which is typical for the parts industry. In the continuous case, an optimum inspection schedule is obtained for a given production time and given number of inspections by solving a nonlinear programming problem. A two-dimensional search procedure can be used to find the optimal policy. In the exponential case, the structure of the optimal inspection policy is established using Lagrange's method, and it is shown that the optimal inspection times can be found by solving a nonlinear equation. Numerical studies indicate that the optimal policy performs much better than the optimal policy with periodic inspections considered previously in the literature. The case of perfect inspections is discussed, and an extension of the results obtained previously in the literature is presented. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 165–186, 1998     

Rolling‐horizon replenishment: Policies and performance analysis

We consider a rolling‐horizon (RH) replenishment modeling framework under which a buyer can update demand information and inventory status, modify order quantities committed previously, place an advanced order for a new period at the end of the RH, and move along in time seamlessly. We show that the optimal order policy for the two‐period RH problem is a dual‐threshold type for updating period(s) plus a base‐stock type for the advanced order. We provide analytical formulas and algorithms to compute the optimal thresholds and the optimal base‐stock level exactly. With our analytical results and numerical procedures, we demonstrate the significant value of RH replenishment in matching supplies to demands more closely. We also show that with RH updating (flexibility), the value of additional demand information beyond the RH diminishes quickly. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Optimal policies for M/M/m queue with two different kinds of (N,T)‐policies

In this paper, two different kinds of (N, T)‐policies for an M/M/m queueing system are studied. The system operates only intermittently and is shut down when no customers are present any more. A fixed setup cost of K > 0 is incurred each time the system is reopened. Also, a holding cost of h > 0 per unit time is incurred for each customer present. The two (N, T)‐policies studied for this queueing system with cost structures are as follows: (1) The system is reactivated as soon as N customers are present or the waiting time of the leading customer reaches a predefined time T, and (2) the system is reactivated as soon as N customers are present or the time units after the end of the last busy period reaches a predefined time T. The equations satisfied by the optimal policy (N*, T*) for minimizing the long‐run average cost per unit time in both cases are obtained. Particularly, we obtain the explicit optimal joint policy (N*, T*) and optimal objective value for the case of a single server, the explicit optimal policy N* and optimal objective value for the case of multiple servers when only predefined customers number N is measured, and the explicit optimal policy T* and optimal objective value for the case of multiple servers when only predefined time units T is measured, respectively. These results partly extend (1) the classic N or T policy to a more practical (N, T)‐policy and (2) the conclusions obtained for single server system to a system consisting of m (m ≥ 1) servers. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 240–258, 2000     

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     

An examination of the effects of the criterion functional on optimal fire-support policies

This paper examines the dependence of the structure of optimal time-sequential fire-support policies on the quantification of military objectives by considering four specific problems, each corresponding to a different quantification of objectives (i.e. criterion functional). We consider the optimal time-sequential allocation of supporting fires during the “approach to contact” of friendly infantry against enemy defensive positions. The combat dynamics are modelled by deterministic Lanchester-type equations of warfare, and the optimal fire-support policy for each one-sided combat optimization problem is developed via optimal control theory. The problems are all nonconvex, and local optima are a particular difficulty in one of them. For the same combat dynamics, the splitting of supporting fires between two enemy forces in any optimal policy (i.e. the optimality of singular subarcs) is shown to depend only on whether the terminal payoff reflects the objective of attaining an “overall” military advantage or a “local” one. Additionally, switching times for changes in the ranking of target priorities are shown to be different (sometimes significantly) when the decision criterion is the difference and the ratio of the military worths (computed according to linear utilities) of total infantry survivors and also the difference and the ratio of the military worths (computed according to linear utilities) of total infantry survivors and also the difference and the ratio of the military worths of the combatants' total infantry losses. Thus, the optimal fire-support policy for this attack scenario is shown to be significantly influenced by the quantification of military objectives.     

Social benefits of government investment in enterprises

A critical issue for many governments is boosting the adoption rates of products or technologies that enhance consumer surplus or total social welfare. Governments may, for example, pay subsidies to producers or to consumers to stimulate the manufacture or consumption of specific products, for example, energy-efficient appliances or more effective drugs. This research proposes a strategic government investment policy, namely, share acquisition, and demonstrates its effectiveness in reaching societal objectives. We consider a Cournot quantities-choice market comprised of homogeneous firms where the government intervenes to buy shares, and turning private firms into state-owned enterprises. We recognize that purchasing a single private firm is the optimal policy for the government to reach its societal objectives. Additionally, taking into consideration financial constraints, we find that the optimal stake increases with the budget. Compared with the optimal output-based subsidy policy, when the budget is low, the optimal government investment policy induces a higher consumer surplus. In addition, in differentiated Cournot competition, under which firms compete in selling substitutable products, we find that when the budget is sufficient, the optimal stake purchased first decreases and then increases according to the substitutability level among products.     

Inventory control of by-products

The system to be controlled produces n products simultaneously in fixed proportions every time it is activated. Demands for the products in any period are components of an n dimensional vector random variable with known distribution function. Cases of excess demands backlogged and excess demands lost are considered. In the former the notion of k convexity can be generalized to guarantee relatively simple form for the optimal policy in an n decision problem. In the latter, this generalization was not successful although when there is no setup cost, a convexity argument can be used to show that the optimal policy has a simple form.