A sequential scheduling problem with impatient jobs

There are n customers that need to be served. Customer i will only wait in queue for an exponentially distributed time with rate λ_i before departing the system. The service time of customer i has distribution F_i, and on completion of service of customer i a positive reward r_i is earned. There is a single server and the problem is to choose, after each service completion, which currently in queue customer to serve next so as to maximize the expected total return. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 659–663, 2015     

Scheduling multiple products on parallel machines with setup costs

We consider a class of production scheduling models with m identical machines in parallel and k different product types. It takes a time p_i to produce one unit of product type i on any one of the machines. There is a demand stream for product type i consisting of n_i units with each unit having a given due date. Before a machine starts with the production of a batch of products of type i a setup cost c is incurred. We consider several different objective functions. Each one of the objective functions has three components, namely a total setup cost, a total earliness cost, and a total tardiness cost. In our class of problems we find a relatively large number of problems that can be solved either in polynomial time or in pseudo‐polynomial time. The polynomiality or pseudo‐polynomiality is achieved under certain special conditions that may be of practical interest; for example, a regularity pattern in the string of due dates combined with earliness and tardiness costs that are similar for different types of products. The class of models we consider includes as special cases discrete counterparts of a number of inventory models that have been considered in the literature before, e.g., Wagner and Whitin (Manage Sci 5 (1958), 89–96) and Zangwill (Oper Res 14 (1966), 486–507; Manage Sci 15 (1969), 506–527). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

New heuristic methods for the capacitated multi‐facility Weber problem

In this paper we consider the capacitated multi‐facility Weber problem with the Euclidean, squared Euclidean, and ?_p‐distances. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the distance between them. We first present a mixed integer linear programming approximation of the problem. We then propose new heuristic solution methods based on this approximation. Computational results on benchmark instances indicate that the new methods are both accurate and efficient. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 2006     

Capacity games for partially complementary products under multivariate random demands

In this study, we consider n firms, each of which produces and sells a different product. The n firms face a common demand stream which requests all their products as a complete set. In addition to the common demand stream, each firm also faces a dedicated demand stream which requires only its own product. The common and dedicated demands are uncertain and follow a general, joint, continuous distribution. Before the demands are realized, each firm needs to determine its capacity or production quantity to maximize its own expected profit. We formulate the problem as a noncooperative game. The sales price per unit for the common demand could be higher or lower than the unit price for the dedicated demand, which affects the firm's inventory rationing policy. Hence, the outcome of the game varies. All of the prices are first assumed to be exogenous. We characterize Nash equilibrium(s) of the game. At the end of the article, we also provide some results for the endogenous pricing. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 59: 146–159, 2012     

Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

A note on equity across groups in facility location

An equity model between groups of demand points is proposed. The set of demand points is divided into two or more groups. For example, rich and poor neighborhoods and urban and rural neighborhoods. We wish to provide equal service to the different groups by minimizing the deviation from equality among groups. The distance to the closest facility is a measure of the quality of service. Once the facilities are located, each demand point has a service distance. The objective function, to be minimized, is the sum of squares of differences between all pairs of service distances between demand points in different groups. The problem is analyzed and solution techniques are proposed for the location of a single facility in the plane. Computational experiments for problems with up to 10,000 demand points and rectilinear, Euclidean, or general ?_p distances illustrate the efficiency of the proposed algorithm. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Optimization methods for large-scale multiechelon stockage problems

I examine the problem of determining inventory stockage levels and locations of different parts in a multiechelon system. This stockage problem is complicated by parts commonality—each part may be used by several different end items. Stockage levels and locations of each part affect the availability of end items that use the part, since an end item will be out of service if it requires a part that is not available. Of course, if the part is available at another nearby location, then the end item will be out of service for a shorter period of time. An essential feature of any model for this problem is constraints on operational availability of the end items. Because these constraints would involve nonconvex functions if the stockage levels were allowed to vary continuously, I formulate a 0–1 linear optimization model of the stockage problem. In this model, each part can be stocked at any of a number of prespecified levels at each echelon. The model is to minimize stockage cost of the selected items subject to the end-item availability constraints and limits on the total weight, volume, and number of different parts stocked at each echelon. Advantages and disadvantages of different Lagrangian relaxations and the simplex method with generalized upper-bounding capability are discussed for solving this stockage model.     

Expected time delay in multi‐item inventory systems with correlated demands

This paper considers multi‐item inventory systems where a customer order may require several different items (i.e., demands are correlated across items) and customer satisfaction is measured by the time delays seen by the customers. Most inventory models on time delay in the literature assume each demand only requires one item (i.e., demands are not correlated across items or are independent). In this paper, we derive an exact expression for the expected total time delay. We show that when items are actually correlated, assuming items are independent leads to an overestimate of the total time delay. However, (1) it is extremely difficult in practice to obtain the demand information for all demand types (especially in a system with tens of thousands of part numbers), and (2) the problem becomes too complicated to be of practical interest when the correlation is considered. We then explore the possibility of including the demand information partially and develop bounds for the time delays. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 671–688, 1999     

Optimal pricing and inventory control policy in periodic‐review systems with fixed ordering cost and lost sales

This paper studies a periodic‐review pricing and inventory control problem for a retailer, which faces stochastic price‐sensitive demand, under quite general modeling assumptions. Any unsatisfied demand is lost, and any leftover inventory at the end of the finite selling horizon has a salvage value. The cost component for the retailer includes holding, shortage, and both variable and fixed ordering costs. The retailer's objective is to maximize its discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We show that, under a mild assumption on the additive demand function, at the beginning of each period an (s,S) policy is optimal for replenishment, and the value of the optimal price depends on the inventory level after the replenishment decision has been done. Our numerical study also suggests that for a sufficiently long selling horizon, the optimal policy is almost stationary. Furthermore, the fixed ordering cost (K) plays a significant role in our modeling framework. Specifically, any increase in K results in lower s and higher S. On the other hand, the profit impact of dynamically changing the retail price, contrasted with a single fixed price throughout the selling horizon, also increases with K. We demonstrate that using the optimal policy values from a model with backordering of unmet demands as approximations in our model might result in significant profit penalty. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Optimal inventory control policy for periodic‐review inventory systems with inventory‐level‐dependent demand

We consider a setting in which inventory plays both promotional and service roles; that is, higher inventories not only improve service levels but also stimulate demand by serving as a promotional tool (e.g., as the result of advertising effect by the enhanced product visibility). Specifically, we study the periodic‐review inventory systems in which the demand in each period is uncertain but increases with the inventory level. We investigate the multiperiod model with normal and expediting orders in each period, that is, any shortage will be met through emergency replenishment. Such a model takes the lost sales model as a special case. For the cases without and with fixed order costs, the optimal inventory replenishment policy is shown to be of the base‐stock type and of the (s,S) type, respectively. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

A heterogeneous arrival and service queueing loss model

Consider a single-server exponential queueing loss system in which the arrival and service rates alternate between the paris (γ₁, γ₁), and (γ₂, μ₂), spending an exponential amount of time with rate cμ_i in (γ_i, μ_i), i = 1.2. It is shown that if all arrivals finding the server busy are lost, then the percentage of arrivals lost is a decreasing function of c. This is in line with a general conjecture of Ross to the effect that the “more nonstationary” a Poisson arrival process is, the greater the average customer delay (in infinite capacity models) or the greater the precentage of lost customers (in finite capacity models). We also study the limiting cases when c approaches 0 or infinity.     

A single‐period inventory placement problem for a serial supply chain

This article addresses the inventory placement problem in a serial supply chain facing a stochastic demand for a single planning period. All customer demand is served from stage 1, where the product is stored in its final form. If the demand exceeds the supply at stage 1, then stage 1 is resupplied from stocks held at the upstream stages 2 through N, where the product may be stored in finished form or as raw materials or subassemblies. All stocking decisions are made before the demand occurs. The demand is nonnegative and continuous with a known probability distribution, and the purchasing, holding, shipping, processing, and shortage costs are proportional. There are no fixed costs. All unsatisfied demand is lost. The objective is to select the stock quantities that should be placed different stages so as to maximize the expected profit. Under reasonable cost assumptions, this leads to a convex constrained optimization problem. We characterize the properties of the optimal solution and propose an effective algorithm for its computation. For the case of normal demands, the calculations can be done on a spreadsheet. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:506–517, 2001     

A diffusion model for the control of a multipurpose reservoir system

This paper develops a methodology for optimizing operation of a multipurpose reservoir with a finite capacity V. The input of water into the reservoir is a Wiener process with positive drift. There are n purposes for which water is demanded. Water may be released from the reservoir at any rate, and the release rate can be increased or decreased instantaneously with zero cost. In addition to the reservoir, a supplementary source of water can supply an unlimited amount of water demanded during any period of time. There is a cost of C_i dollars per unit of demand supplied by the supplementary source to the i^th purpose (i = 1, 2, …, n). At any time, the demand rate R_i associated with the i^th purpose (i = 1, 2, …, n) must be supplied. A controller must continually decide the amount of water to be supplied by the reservoir for each purpose, while the remaining demand will be supplied through the supplementary source with the appropriate costs. We consider the problem of specifying an output policy which minimizes the long run average cost per unit time.     

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     

Distribution free bounds for service constrained (Q,r) inventory systems

A classical and important problem in stochastic inventory theory is to determine the order quantity (Q) and the reorder level (r) to minimize inventory holding and backorder costs subject to a service constraint that the fill rate, i.e., the fraction of demand satisfied by inventory in stock, is at least equal to a desired value. This problem is often hard to solve because the fill rate constraint is not convex in (Q, r) unless additional assumptions are made about the distribution of demand during the lead‐time. As a consequence, there are no known algorithms, other than exhaustive search, that are available for solving this problem in its full generality. Our paper derives the first known bounds to the fill‐rate constrained (Q, r) inventory problem. We derive upper and lower bounds for the optimal values of the order quantity and the reorder level for this problem that are independent of the distribution of demand during the lead time and its variance. We show that the classical economic order quantity is a lower bound on the optimal ordering quantity. We present an efficient solution procedure that exploits these bounds and has a guaranteed bound on the error. When the Lagrangian of the fill rate constraint is convex or when the fill rate constraint does not exist, our bounds can be used to enhance the efficiency of existing algorithms. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 635–656, 2000     

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.     

Coordination of pricing and inventory control across products

We consider the joint pricing and inventory‐control problem for a retailer who orders, stocks, and sells two products. Cross‐price effects exist between the two products, which means that the demand of each product depends on the prices of both products. We derive the optimal pricing and inventory‐control policy and show that this policy differs from the base‐stock list‐price policy, which is optimal for the one‐product problem. We find that the retailer can significantly improve profits by managing the two products jointly as opposed to independently, especially when the cross‐price demand elasticity is high. We also find that the retailer can considerably improve profits by using dynamic pricing as opposed to static pricing, especially when the demand is nonstationary. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Order-statistic calculation,costs, and service in an (s,Q) inventory system

A retailer or distributor of finished goods, or the manager of a spare-parts inventory system, must generally forecast the major portion of demand. A specific customer-service level p (fraction of replenishment intervals with no stockout) implies two challenges: achieve the service within a small interval plus or minus, and do so with a minimum-cost investment in inventory. The pth fractile of lead-time demand (LTD) is the reorder point (ROP) for this service measure, and is often approximated by that fractile of a normal distribution. With this procedure, it is easy to set safety stocks for an (s, Q) inventory system. However, Bookbinder and Lordahl [2] and others have identified cases where the normal approximation yields excessive costs and/or lower service than desired. This article employs an order-statistic approach. Using available LTD data, the ROP is simply estimated from one or two of the larger values in the sample. This approach is sufficiently automatic and intuitive for routine implementation in industry, yet is distribution free. The order-statistic method requires only a small amount of LTD data, and makes no assumptions on the form of the underlying LTD distribution, nor even its parameters μ and ρ. We compare the order-statistic approach and the normal approximation, first in terms of customer service and then using a model of expected annual cost. Based upon characteristics of the available LTD data, we suggest a procedure to aid a practitioner in choosine between the normal and order-statistic method. © 1994 John Wiley & Sons, Inc.     

Exact algorithms for integrated facility location and production planning problems

We consider a class of facility location problems with a time dimension, which requires assigning every customer to a supply facility in each of a finite number of periods. Each facility must meet all assigned customer demand in every period at a minimum cost via its production and inventory decisions. We provide exact branch‐and‐price algorithms for this class of problems and several important variants. The corresponding pricing problem takes the form of an interesting class of production planning and order selection problems. This problem class requires selecting a set of orders that maximizes profit, defined as the revenue from selected orders minus production‐planning‐related costs incurred in fulfilling the selected orders. We provide polynomial‐time dynamic programming algorithms for this class of pricing problems, as well as for generalizations thereof. Computational testing indicates the advantage of our branch‐and‐price algorithm over various approaches that use commercial software packages. These tests also highlight the significant cost savings possible from integrating location with production and inventory decisions and demonstrate that the problem is rather insensitive to forecast errors associated with the demand streams. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011