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     

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     

Inventory control under substitutable demand: A stochastic game application

Substitutable product inventory problem is analyzed using the concepts of stochastic game theory. It is assumed that there are two substitutable products that are sold by different retailers and the demand for each product is random. Game theoretic nature of this problem is the result of substitution between products. Since retailers compete for the substitutable demand, ordering decision of each retailer depends on the ordering decision of the other retailer. Under the discounted payoff criterion, this problem is formulated as a two‐person nonzero‐sum stochastic game. In the case of linear ordering cost, it is shown that there exists a Nash equilibrium characterized by a pair of stationary base stock strategies for the infinite horizon problem. This is the unique Nash equilibrium within the class of stationary base stock strategies. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 359–375, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10018     

Multi-item service constrained (s,S) policies for spare parts logistics systems

Many organizations providing service support for products or families of products must allocate inventory investment among the parts (or, identically, items) that make up those products or families. The allocation decision is crucial in today's competitive environment in which rapid response and low levels of inventory are both required for providing competitive levels of customer service in marketing a firm's products. This is particularly important in high-tech industries, such as computers, military equipment, and consumer appliances. Such rapid response typically implies regional and local distribution points for final products and for spare parts for repairs. In this article we fix attention on a given product or product family at a single location. This single-location problem is the basic building block of multi-echelon inventory systems based on level-by-level decomposition, and our modeling approach is developed with this application in mind. The product consists of field-replaceable units (i.e., parts), which are to be stocked as spares for field service repair. We assume that each part will be stocked at each location according to an (s, S) stocking policy. Moreover, we distinguish two classes of demand at each location: customer (or emergency) demand and normal replenishment demand from lower levels in the multiechelon system. The basic problem of interest is to determine the appropriate policies (s_i S_i) for each part i in the product under consideration. We formulate an approximate cost function and service level constraint, and we present a greedy heuristic algorithm for solving the resulting approximate constrained optimization problem. We present experimental results showing that the heuristics developed have good cost performance relative to optimal. We also discuss extensions to the multiproduct component commonality problem.     

Priority and dynamic scheduling in a make‐to‐stock queue with hyperexponential demand

We consider the scheduling problem in a make‐to‐stock queue with two demand classes that can be differentiated based on their variability. One class experiences Poisson arrivals and the other class experiences hyperexponential renewal arrivals. We provide an exact analysis of the case where the demand class with higher variability is given non‐preemptive priority. The results are then used to compare the inventory cost performance of three scheduling disciplines, first‐come first‐serve and priority to either class. We then build on an existing dynamic scheduling heuristic to propose a modification that works well for our system. Extensions of the heuristic to more than two classes and to the case where demand state is known are also discussed. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006.     

Competition under time‐varying demands and dynamic lot sizing costs

We develop a competitive pricing model which combines the complexity of time‐varying demand and cost functions and that of scale economies arising from dynamic lot sizing costs. Each firm can replenish inventory in each of the T periods into which the planning horizon is partitioned. Fixed as well as variable procurement costs are incurred for each procurement order, along with inventory carrying costs. Each firm adopts, at the beginning of the planning horizon, a (single) price to be employed throughout the horizon. On the basis of each period's system of demand equations, these prices determine a time series of demands for each firm, which needs to service them with an optimal corresponding dynamic lot sizing plan. We establish the existence of a price equilibrium and associated optimal dynamic lotsizing plans, under mild conditions. We also design efficient procedures to compute the equilibrium prices and dynamic lotsizing plans.© 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Optimal control of a capacitated inventory system with multiple demand classes

This article studies the optimal control of a periodic‐review make‐to‐stock system with limited production capacity and multiple demand classes. In this system, a single product is produced to fulfill several classes of demands. The manager has to make the production and inventory allocation decisions. His objective is to minimize the expected total discounted cost. The production decision is made at the beginning of each period and determines the amount of products to be produced. The inventory allocation decision is made after receiving the random demands and determines the amount of demands to be satisfied. A modified base stock policy is shown to be optimal for production, and a multi‐level rationing policy is shown to be optimal for inventory allocation. Then a heuristic algorithm is proposed to approximate the optimal policy. The numerical studies show that the heuristic algorithm is very effective. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 43–58, 2011     

A two‐echelon inventory‐location problem with service considerations

We study the problem of designing a two‐echelon spare parts inventory system consisting of a central plant and a number of service centers each serving a set of customers with stochastic demand. Processing and storage capacities at both levels of facilities are limited. The manufacturing process is modeled as a queuing system at the plant. The goal is to optimize the base‐stock levels at both echelons, the location of service centers, and the allocation of customers to centers simultaneously, subject to service constraints. A mixed integer nonlinear programming model (MINLP) is formulated to minimize the total expected cost of the system. The problem is NP‐hard and a Lagrangian heuristic is proposed. We present computational results and discuss the trade‐off between cost and service. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems

We consider a two‐echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base‐stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base‐stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base‐stock level at each DC and an upper bound for the optimal base‐stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base‐stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

When is a base stock policy optimal in recovering disrupted cyclic schedules?

The extended economic lot scheduling problem (EELSP) is concerned with scheduling the production of a set of items in a single facility to minimize the long-run average holding, backlogging, and setup costs. Given an efficient cyclic production schedule for the EELSP, called the target schedule, we consider the problem of how to schedule production after a single schedule disruption. We propose a base stock policy, characterized by a base stock vector, that prescribes producing an item until its inventory level reaches the peak inventory of the target schedule corresponding to the item's position in the production sequence. We show that the base stock policy is always successful in recovering the target schedule. Moreover, the base stock policy recovers the target schedule at minimal excess over average cost whenever the backorder costs are proportional to the processing times. This condition holds, for example, when the value of the items is proportional to their processing times, and a common inventory carrying cost and a common service level is used for all the items. Alternatively, the proportionality condition holds if the inventory manager is willing to select the service levels from a certain set that is large enough to guarantee any minimal level of service, and then uses the imputed values for the backorder costs. When the proportionality condition holds we provide a closed-form expression for the total relevant excess over average cost of recovering the target schedule. We assess the performance of the base stock policy when the proportionality condition does not hold through a numerical study, and suggest some heuristic uses of the base stock policy. © 1994 John Wiley & Sons, Inc.     

Centralized inventory control in a two‐level distribution system with Poisson demand

This paper introduces a new replenishment policy for inventory control in a two‐level distribution system consisting of one central warehouse and an arbitrary number of nonidentical retailers. The new policy is designed to control the replenishment process at the central warehouse, using centralized information regarding the inventory positions and demand processes of all installations in the system. The retailers on the other hand are assumed to use continuous review (R, Q) policies. A technique for exact evaluation of the expected inventory holding and backorder costs for the system is presented. Numerical results indicate that there are cases when considerable savings can be made by using the new (α₀, Q₀) policy instead of a traditional echelon‐ or installation‐stock (R, Q) policy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 798–822, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10040     

Managing waiting times of backordered demands in single‐stage (Q,r) inventory systems

We present a service constrained (Q, r) model that minimizes expected holding and ordering costs subject to an upper bound on the expected waiting time of demands that are actually backordered. We show that, after optimizing over r, the average cost is quasiconvex in Q for logconcave continuous lead time demand distributions. For logconcave discrete lead time demand distributions we find a single‐pass efficient algorithm based on a novel search stopping criterion. The algorithm also allows for bounds on the variability of the service measure. A brief numerical study indicates how the bounds on service impact the optimal average cost and the optimal (Q, r) choice. The discrete case algorithm can be readily adapted to provide a single pass algorithm for the traditional model that bounds the expected waiting time of all demands (backordered or not). © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 557–573, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10028     

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     

Optimal service rates of a service facility with perishable inventory items

In this paper we optimally control service rates for an inventory system of service facilities with perishable products. We consider a finite capacity system where arrivals are Poisson‐distributed, lifetime of items have exponential distribution, and replenishment is instantaneous. We determine the service rates to be employed at each instant of time so that the long‐run expected cost rate is minimized for fixed maximum inventory level and capacity. The problem is modelled as a semi‐Markov decision problem. We establish the existence of a stationary optimal policy and we solve it by employing linear programming. Several numerical examples which provide insight to the behavior of the system are presented. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 464–482, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10021     

Warehouse sizing to minimize inventory and storage costs

This paper considers a warehouse sizing problem whose objective is to minimize the total cost of ordering, holding, and warehousing of inventory. Unlike typical economic lot sizing models, the warehousing cost structure examined here is not the simple unit rate type, but rather a more realistic step function of the warehouse space to be acquired. In the cases when only one type of stock‐keeping unit (SKU) is warehoused, or when multiple SKUs are warehoused, but, with separable inventory costs, closed form solutions are obtained for the optimal warehouse size. For the case of multi‐SKUs with joint inventory replenishment cost, a heuristic with a provable performance bound of 94% is provided. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 299–312, 2001     

A computationally efficient approach for determining inventory levels in a capacitated multiechelon production‐distribution system

The system under study is a single item, two‐echelon production‐inventory system consisting of a capacitated production facility, a central warehouse, and M regional distribution centers that satisfy stochastic demand. Our objective is to determine a system base‐stock level which minimizes the long run average system cost per period. Central to the approach are (1) an inventory allocation model and associated convex cost function designed to allocate a given amount of system inventory across locations, and (2) a characterization of the amount of available system inventory using the inventory shortfall random variable. An exact model must consider the possibility that inventories may be imbalanced in a given period. By assuming inventory imbalances cannot occur, we develop an approximation model from which we obtain a lower bound on the per period expected cost. Through an extensive simulation study, we analyze the quality of our approximation, which on average performed within 0.50% of the lower bound. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 377–398, 2000     

Base‐stock policies in capacitated assembly systems: Convexity properties

We study an assembly system with a single finished product managed using an echelon base‐stock or order‐up‐to policy. Some or all operations have capacity constraints. Excess demand is either backordered in every period or lost in every period. We show that the shortage penalty cost over any horizon is jointly convex with respect to the base‐stock levels and capacity levels. When the holding costs are also included in the objective function, we show that the cost function can be written as a sum of a convex function and a concave function. Throughout the article, we discuss algorithmic implications of our results for making optimal inventory and capacity decisions in such systems.© 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

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     

Integrated capacity,demand, and production planning with subcontracting and overtime options

Models for integrated production and demand planning decisions can serve to improve a producer's ability to effectively match demand requirements with production capabilities. In contexts with price‐sensitive demands, economies of scale in production, and multiple capacity options, such integrated planning problems can quickly become complex. To address these complexities, this paper provides profit‐maximizing production planning models for determining optimal demand and internal production capacity levels under price‐sensitive deterministic demands, with subcontracting and overtime options. The models determine a producer's optimal price, production, inventory, subcontracting, overtime, and internal capacity levels, while accounting for production economies of scale and capacity costs through concave cost functions. We use polyhedral properties and dynamic programming techniques to provide polynomial‐time solution approaches for obtaining an optimal solution for this class of problems when the internal capacity level is time‐invariant. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Performance evaluation of a production/inventory system with periodic review and endogenous lead times

This paper considers a discrete time, single item production/inventory system with random period demands. Inventory levels are reviewed periodically and managed using a base‐stock policy. Replenishment orders are placed with the production system which is capacitated in the sense that there is a single server that sequentially processes the items one at a time with stochastic unit processing times. In this setting the variability in demand determines the arrival pattern of production orders at the queue, influencing supply lead times. In addition, the inventory behavior is impacted by the correlation between demand and lead times: a large demand size corresponds to a long lead time, depleting the inventory longer. The contribution of this paper is threefold. First, we present an exact procedure based on matrix‐analytic techniques for computing the replenishment lead time distribution given an arbitrary discrete demand distribution. Second, we numerically characterize the distribution of inventory levels, and various other performance measures such as fill rate, base‐stock levels and optimal safety stocks, taking the correlation between demand and lead times into account. Third, we develop an algorithm to fit the first two moments of the demand and service time distribution to a discrete phase‐type distribution with a minimal number of phases. This provides a practical tool to analyze the effect of demand variability, as measured by its coefficient of variation, on system performance. We also show that our model is more appropriate than some existing models of capacitated systems in discrete time. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007