Degeneracy in inventory models

In order‐quantity reorder‐point formulations for inventory items where backordering is allowed, some of the more common ways to prevent excessive stockouts in an optimal solution are to impose either a cost per unit short, a cost per stockout occasion, or a target fill rate. We show that these popular formulations, both exact and approximate, can become “degenerate” even with quite plausible parameters. By degeneracy we mean any situation in which the formulation either cannot be solved, leads to nonsensical “optimal” solutions, or becomes equivalent to something substantially simpler. We explain the reasons for the degeneracies, yielding new insight into these models, and we provide practical advice for inventory managers. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 686–705, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10037     

Inventory systems with imperfect demand information

An inventory system is described in which demand information may be incorrectly transmitted from the field to the stocking point. The stocking point employs a forwarding policy which attempts to send out to the field a quantity which, in general, is some function of the observed demand. The optimal ordering rules for the general n-period problem and the steady state case are derived. In addition orderings of the actual reorder points as functions of the errors are presented, as well as some useful economic interpretations and numerical illustrations.     

Use of dimensionless ratios to determine minimum-cost inventory quantities

The iteration usually necessary for simultaneous determination of minimum-cost order quantity and reorder point in (Q, r) inventory systems may be eliminated by a graphical technique employing dimensionless ratios. This technique is illustrated for three different types of stock-out penalty.     

Exact analysis of (R,s, S) inventory control systems with lost sales and zero lead time

We study an (R, s, S) inventory control policy with stochastic demand, lost sales, zero lead‐time and a target service level to be satisfied. The system is modeled as a discrete time Markov chain for which we present a novel approach to derive exact closed‐form solutions for the limiting distribution of the on‐hand inventory level at the end of a review period, given the reorder level (s) and order‐up‐to level (S). We then establish a relationship between the limiting distributions for adjacent values of the reorder point that is used in an efficient recursive algorithm to determine the optimal parameter values of the (R, s, S) replenishment policy. The algorithm is easy to implement and entails less effort than solving the steady‐state equations for the corresponding Markov model. Point‐of‐use hospital inventory systems share the essential characteristics of the inventory system we model, and a case study using real data from such a system shows that with our approach, optimal policies with significant savings in inventory management effort are easily obtained for a large family of items.     

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     

On the use of a structural equation for determining inventory order quantities

This article considers a structural equation useful for characterizing the order quantity of several inventory models. A correct interpretation of this equation is provided and it is stressed that the equation should be used in conjunction with another equation for the reorder point. Failure to do so may give rise to improper interpretations and invalid conclusions. A specific case like this is cited for the sake of illustration.     

The relationship between decision variables and penalty cost parameter in (Q,R) inventory models

This paper is concerned with the optimum decision variables found using order quantity, reorder point (Q, R) inventory models. It examines whether the optimum variables (Q* and R*) are necessarily monotonic functions of the backorder cost parameter (or equivalently of the performance objective). For a general class of models it is proved that R* must increase as the performance objective is raised, and an inequality condition is derived which governs how Q* will change. Probability distributions of lead time demand are cited or found for which Q* increases, Q* decreases, and Q* is independent of increases in performance objectives or backorder cost parameter.     

Synchronizing pricing and replenishment to serve forward-looking customers

We incorporate strategic customer waiting behavior in the classical economic order quantity (EOQ) setting. The seller determines not only the timing and quantities of the inventory replenishment, but also the selling prices over time. While similar ideas of market segmentation and intertemporal price discrimination can be carried over from the travel industries to other industries, inventory replenishment considerations common to retail outlets and supermarkets introduce additional features to the optimal pricing scheme. Specifically, our study provides concrete managerial recommendations that are against the conventional wisdom on “everyday low price” (EDLP) versus “high-low pricing” (Hi-Lo). We show that in the presence of inventory costs and strategic customers, Hi-Lo instead of EDLP is optimal when customers have homogeneous valuations. This result suggests that because of strategic customer behavior, the seller obtains a new source of flexibility—the ability to induce customers to wait—which always leads to a strictly positive increase of the seller's profit. Moreover, the optimal inventory policy may feature a dry period with zero inventory, but this period does not necessarily result in a loss of sales as customers strategically wait for the upcoming promotion. Furthermore, we derive the solution approach for the optimal policy under heterogeneous customer valuation setting. Under the optimal policy, the replenishments and price promotions are synchronized, and the seller adopts high selling prices when the inventory level is low and plans a discontinuous price discount at the replenishment point when inventory is the highest.     

Computation of constrained optimum quantities and reorder points for time-weighted backorder penalties

The purpose of this paper and the accompanying tables is to facilitate the calculation of constrained optimum order quantities and reorder points for an inventory control system where the criterion of optimality is the minimization of expected inventory holding, ordering, and time-weighted backorder costs. The tables provided in the paper allow the identification of the optimal solution when order quantities and/or reorder points are restricted to a set of values which do not include the unconstrained optimal solution.     

New algorithms for (Q,r) systems with complete backordering using a fill-rate criterion

We present an algorithm which determines optimal parameter values for order quantity-reorder point systems with complete backordering. The service level is measured as fraction of demand satisfied directly from shelf, also known as “fill-rate.” This algorithm differs from existing algorithms because an exact cost function is used rather than an approximation. We also present a new heuristic algorithm, which is more efficient computationally than the optimal procedure and provides excellent results. Results of extensive computational experience also are reported.     

More ado about economic order quantities (EOQ)

This paper is concerned with the determination of explicit expressions for economic order quantities and reorder levels, such that the cost of ordering and holding inventory is minimized for specific backorder constraints. Holding costs are applied either to inventory position or on-hand inventory, and the backorder constraint is considered in terms of the total number of backorders per year or the average number of backorders at any point in time. Through the substitution of a new probability density function in place of the normal p.d.f., explicit expressions are determined for the economic order quantities and the reorder points. The resulting economic order quantities are independent of all backorder constraints. It is also concluded that under certain conditions, the minimization of ordering costs and inventory holding costs (applied to inventory position), subject to a backorder constraint, is equivalent in terms of reorder levels to minimization of the safety level dollar investment subject to the same backorder constraint.     

The stochastic joint replenishment problem: A new policy,analysis, and insights

In this study, we propose a new parsimonious policy for the stochastic joint replenishment problem in a single‐location, N‐item setting. The replenishment decisions are based on both group reorder point‐group order quantity and the time since the last decision epoch. We derive the expressions for the key operating characteristics of the inventory system for both unit and compound Poisson demands. In a comprehensive numerical study, we compare the performance of the proposed policy with that of existing ones over a standard test bed. Our numerical results indicate that the proposed policy dominates the existing ones in 100 of 139 instances with comparably significant savings for unit demands. With batch demands, the savings increase as the stochasticity of demand size gets larger. We also observe that it performs well in environments with low demand diversity across items. The inventory system herein also models a two‐echelon setting with a single item, multiple retailers, and cross docking at the upper echelon. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

Transportation type problems with quantity discounts

It is known to be real that the per unit transportation cost from a specific supply source to a given demand sink is dependent on the quantity shipped, so that there exist finite intervals for quantities where price breaks are offered to customers. Thus, such a quantity discount results in a nonconvex, piecewise linear functional. In this paper, an algorithm is provided to solve this problem. This algorithm, with minor modifications, is shown to encompass the “incremental” quantity discount and the “fixed charge” transportation problems as well. It is based upon a branch-and-bound solution procedure. The branches lead to ordinary transportation problems, the results of which are obtained by utilizing the “cost operator” for one branch and “rim operator” for another branch. Suitable illustrations and extensions are also provided.     

(R.Q) inventory problem with unknown mean demand and learning

We address a single product, continuous review model with stationary Poisson demand. Such a model has been effectively studied when mean demand is known. However, we are concerned with managing new items for which only a Bayesian prior distribution on the mean is available. As demand occurs, the prior is updated and our control parameters are revised. These include the reorder point (R) and reorder quantity (Q). Deemer, taking a clue from some earlier RAND work, suggested using a model appropriate for known mean, but using a Compound Poisson distribution for demand rather than Poisson to reflect uncertainty about the mean. Brown and Rogers also used this approach but within a periodic review context. In this paper we show how to compute optimum reorder points for a special problem closely related to the problem of real interest. In terms of the real problem, subject to a qualification to be discussed, the reorder points found are upper bounds for the optimum. At the same time, the reorder points found can never exceed those found by the Compound Poisson (Deemer) approach. And they can be smaller than those found when there is no uncertainty about the mean. As a check, the Compound Poisson and proposed approach are compared by simulation.     

Sensitivity of the (Q,R) inventory model to penalty cost parameter

This paper treats an approximate continuous review inventory model with backlogging of excess demand and stochastic leadtime. The major result derived is that the behavior of the optimal order size with respect to the shortage cost parameter is determined solely by the “conditional mean residual life” function corresponding to the leadtime demand distribution. Some minor results and illustrative examples are also included.     

An (s,S) model for inventory with exponential lifetimes and renewal demands

Inventory control of products with finite lifetimes is important in many modern business organizations. It has been an important and difficult research subject. Here, we study the (s, S) continuous review model for items with an exponential random lifetime and a general renewal demand process through a Markov process. We derive a fundamental rate conservation theorem and show that all the other system performance measures can be obtained easily through the expected reorder cycle length. This leads to a simple expression for the total expected long run cost rate function in terms of the expected reorder cycle length. Subsequently, we derive formulas for computing the expected cycle lengths for the general renewal demand as well as for a large class of demands characterized by the phase type interdemand time distribution. We show analytically when the cost as a function of the reorder level is monotone, concave, or convex. We also show analytically that, depending on the behavior of the expected reorder cycle, the cost as a function of the order‐up level is either monotone increasing or unimodal. These analytical properties enable us to understand the problem and make the subsequent numerical optimization much easier. Numerical studies confirm and illustrate some of the analytical properties. The results also demonstrate the impact of various parameters on the optimal policy and the cost. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 39–56, 1999     

Production-location problems with demand considerations

The production-location problem of a profit maximizing firm is considered. A model is developed for a single firm, facing the joint problems of determining the optimal plant location, the optimal input mix, and the optimal plant size. A homothetic production function is used as the model of the production technologies, and the existence of a sequential “separability” between the production, or input mix, problem and the location problem is demonstrated.     

Interval estimation of a global optimum for large combinatorial problems

Consider an “intractable” optimization problem for which no efficient solution technique exists. Given a systematic procedure for generating independent heuristic solutions, we seek to obtain interval estimates for the globally optimal solution using statistical inference. In previous work, accurate point estimates have been derived. Determining interval estimates, however, is a considerably more difficult task. In this paper, we develop straightforward procedures which compute confidence intervals efficiently in order to evaluate heuristic solutions and assess deviations from optimality. The strategy presented is applicable to a host of combinatorial optimization problems. The assumptions of our model, along with computational experience, are discussed.     

Systems defense games: Colonel blotto,command and control

The classical “Colonel Blotto” games of force allocation are generalized to include situations in which there are complementarities among the targets being defended. The complementarities are represented by means of a system “characteristic function,” and a valuation technique from the theory of cooperative games is seen to indicate the optimal allocations of defense and attack forces. Cost trade-offs between systems defense and alternative measures, such as the hardening of targets, are discussed, and a simple example is analyzed in order to indicate the potential of this approach.     

Optimal reject allowance with constant marginal production efficiency

A job shop must fulfill an order for N good items. Production is conducted in “lots,” and the number of good items in a lot can be accurately determined only after production of that lot is completed. If the number of good items falls short of the outstanding order, the shop must produce further lots, as necessary. Processes with “constant marginal production efficiency” are investigated. The revealed structure allows efficient exact computation of optimal policy. The resulting minimal cost exhibits a consistent (but not universal) pattern whereby higher quality of production is advantageous even at proportionately higher marginal cost.