The initial provisioning decision for insurance type items

Initial provisioning decisions (inventory stocking requirements) for low demand items often have to be made without much knowledge of what future demand rates will be. When the nature of an item is such that little demand for it is expected, the problem of whether to stock initially or risk not stocking the item is most critical. This report discusses this problem and presents decision procedures which can be used to handle this aspect of initial provisioning. The procedures relate an item's provisioning desirability to its provisioning characteristics, such as expected cost, expected resupply time, current information on its likely demand rate, and to an overall operating policy or criterion. The criterion function measures the total system degredation as a function of the events of having items out of stock when demand occurs. Several different policy functions are discussed and the provisioning decision rules which apply to each are presented. Demand rate information is handled through a Bayesian type approach. The decision rules presented in this report can be utilized to either determine stocking requirements within a budgetary constraint, or determine the relative stocking desirability on an item-by-item basis.     

Inventory rotation policies for slow moving items

In this article we present a stochastic model for determining inventory rotation policies for a retail firm which must stock many hundreds of distinctive items having uncertain heterogeneous sales patterns. The model develops explicit decision rules for determining (1) the length of time that an item should remain in inventory before the decision is made on whether or not to rotate the item out of inventory and (2) the minimum sales level necessary for retaining the item in inventory. Two inventory rotation policies are developed, the first of which maximizes cumulative expected sales over a finite planning horizon and the second of which maximizes cumulative expected profit. We also consider the statistical behavior of items having uncertain, discrete, and heterogeneous sales patterns using a two-period prediction methodology where period 1 is used to accumulate information on individual sales rates and this knowledge is then used, in a Bayesian context, to make sales predictions for period 2. This methodology assumes that over an arbitrary time interval sales for each item are Poisson with unknown but stationary mean sales rates and the mean sales rates are distributed gamma across all items. We also report the application of the model to a retail firm which stocks many hundreds of distinctive unframed poster art titles. The application provides some useful insights into the behavior of the model as well as some interesting aspects pertaining to the implementation of the results in a “real-world” situation.     

Models for multi-item continuous review inventory policies subject to constraints

Models are formulated for determining continuous review (Q, r) policies for a multiitem inventory subject to constraints. The objective function is the minimization of total time-weighted shortages. The constraints apply to inventory investment and reorder workload. The formulations are thus independent of the normal ordering, holding, and shortage costs. Two models are presented, each representing a convex programming problem. Lagrangian techniques are employed with the first, simplified model in which only the reorder points are optimized. In the second model both the reorder points and the reorder quantities are optimized utilizing penalty function methods. An example problem is solved for each model. The final section deals with the implementation of these models in very large inventory systems.     

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.     

Cost allocation rules for elastic single‐attribute situations

Many cooperative games, especially ones stemming from resource pooling in queueing or inventory systems, are based on situations in which each player is associated with a single attribute (a real number representing, say, a demand) and in which the cost to optimally serve any sum of attributes is described by an elastic function (which means that the per‐demand cost is non‐increasing in the total demand served). For this class of situations, we introduce and analyze several cost allocation rules: the proportional rule, the serial cost sharing rule, the benefit‐proportional rule, and various Shapley‐esque rules. We study their appeal with regard to fairness criteria such as coalitional rationality, benefit ordering, and relaxations thereof. After showing the impossibility of combining coalitional rationality and benefit ordering, we show for each of the cost allocation rules which fairness criteria it satisfies. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 271–286, 2017     

The distribution of recoverable inventory items from a repair center when the number of consumption centers is large

This paper considers real-time decision rules for an inventory system where items are repaired than “used up.” The problem is to decide which user in the system has the greatest need for the newly available inventory items coming out of repair. The main result shows that two published approahes, the Transportation Time Look Ahead policy and METRIC, are optimal when the number of users gets large. A useful byproduct of the proof is a lower bound on the average backorder rate for a repair-inventory system of any size.     

Spares stocking policies for repairable items with dependent repair times

This paper analyzes the problem of determining desirable spares inventory levels for repairable items with dependent repair times. The problem is important for repairable products such as aircraft engines which can have very large investment in spares inventory levels. While existing models can be used to determine optimal inventory spares levels when repair times are independent, the practical considerations of limited repair shop capacity and prioritized shop dispatching rules combine to make repair times not independent of one another. In this research a simulation model of a limited capacity repair facility with prioritized scheduling is used to explore a variety of heuristic approaches to the spares stocking decision. The heuristics are also compared with use of a model requiring independent repair times (even though that assumption is not valid here). The results show that even when repair time dependencies are present, the performance of a model which assumes independent repair times is quite good.     

Techniques for establishing ergodic and recurrence properties of continuous-valued markov chains

We present techniques for classifying Markov chains with a continuous state space as either ergodic or recurrent. These methods are analogous to those of Foster for countable space chains. The theory is presented in the first half of the paper, while the second half consists of examples illustrating these techniques. The technique for proving ergodicity involves, in practice, three steps: showing that the chain is irreducible in a suitable sense; verifying that the mean hitting times on certain (usually bounded) sets are bounded, by using a “mean drift” criterion analogous to that of Foster; and finally, checking that the chain is such that bounded mean hitting times for these sets does actually imply ergodicity. The examples comprise a number of known and new results: using our techniques we investigate random walks, queues with waiting-time-dependent service times, dams with general and random-release rules, the s-S inventory model, and feedback models.     

On the value of information in dynamic production/inventory problems under forecast evolution

In this article we explore how total system costs and inventory positions are affected when forecasts are incorporated explicitly in production/inventory systems. We assume that forecasts for demand of a certain item are available in each period, and they evolve from one period to the next in accordance with an additive evolution model. In order to analyze the effects of the forecasts on the production/inventory system we compare the optimal ordering policy and the expected costs of the model that keeps forecasts with that of a comparable standard inventory model. We show that under mild assumptions the former yields lower expected costs and inventory levels than the latter. © 1996 John Wiley & Sons, Inc.     

Proactive release procedures for just‐in‐time job shop environments,subject to machine failures

In this paper, we consider just‐in‐time job shop environments (job shop problems with an objective of minimizing the sum of tardiness and inventory costs), subject to uncertainty due to machine failures. We present techniques for proactive uncertainty management that exploit prior knowledge of uncertainty to build competitive release dates, whose execution improves performance. These techniques determine the release dates of different jobs based on measures of shop load, statistical data of machine failures, and repairs with a tradeoff between inventory and tardiness costs. Empirical results show that our methodology is very promising in comparison with simulated annealing and the best of 39 combinations of dispatch rules & release policies, under different frequencies of breakdowns. We observe that the performance of the proactive technique compared to the other two approaches improves in schedule quality (maximizing delivery performance while minimizing costs) with increase in frequency of breakdowns. The proactive technique presented here is also computationally less expensive than the other two approaches. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

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 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     

On the (S − 1, S) lost sales inventory model with priority demand classes

In this paper an inventory model with several demand classes, prioritised according to importance, is analysed. We consider a lot‐for‐lot or (S ? 1, S) inventory model with lost sales. For each demand class there is a critical stock level at and below which demand from that class is not satisfied from stock on hand. In this way stock is retained to meet demand from higher priority demand classes. A set of such critical levels determines the stocking policy. For Poisson demand and a generally distributed lead time, we derive expressions for the service levels for each demand class and the average total cost per unit time. Efficient solution methods for obtaining optimal policies, with and without service level constraints, are presented. Numerical experiments in which the solution methods are tested demonstrate that significant cost reductions can be achieved by distinguishing between demand classes. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 593–610, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10032     

Inventory control in divergent supply chains with time‐based dispatching and shipment consolidation

This article analyses a divergent supply chain consisting of a central warehouse and N nonidentical retailers. The focus is on joint evaluation of inventory replenishment and shipment consolidation effects. A time‐based dispatching and shipment consolidation policy is used at the warehouse in conjunction with real‐time point‐of‐sale data and centralized inventory information. This represents a common situation, for example, in various types of vendor managed inventory systems. The main contribution is the derivation of an exact recursive procedure for determining the expected inventory holding and backorder costs for the system, under the assumption of Poisson demand. Two heuristics for determining near optimal shipment intervals are also presented. The results are applicable both for single‐item and multiitem systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 59–71, 2011     

Inventory models with a mixture of backorders and lost sales

This article presents several single-echelon, single-item, static demand inventory models for situations in which, during the stockout period, a fraction b of the demand is backordered and the remaining fraction 1 - b is lost forever. Both deterministic and stochastic demand are considered. although the case of stochastic demand is treated heuristically. In each situation, a mathematical model representing the average annual cost of operating the inventory system is developed. and an optimum operating policy derived. At the extremes b=1 and b=0 the models presented reduce to the usual backorders and lost sales cases, respectively.     

Dynamic inventory control with limited capital and short‐term financing

For most firms, especially the small‐ and medium‐sized ones, the operational decisions are affected by their internal capital and ability to obtain external capital. However, the majority of the literature on dynamic inventory control ignores the firm's financial status and financing issues. An important question that arises is: what are the optimal inventory and financing policies for firms with limited internal capital and limited access to external capital? In this article, we study a dynamic inventory control problem where a capital‐constrained firm periodically purchases a product from a supplier and sells it to a market with random demands. In each period, the firm can use its own capital and/or borrow a short‐term loan to purchase the product, with the interest rate being nondecreasing in the loan size. The objective is to maximize the firm's expected terminal wealth at the end of the planning horizon. We show that the optimal inventory policy in each period is an equity‐level‐dependent base‐stock policy, where the equity level is the sum of the firm's capital level and the value of its on‐hand inventory evaluated at the purchasing cost; and the structure of the optimal policy can be characterized by four intervals of the equity level. Our results shed light on the dynamic inventory control for firms with limited capital and short‐term financing capabilities.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 184–201, 2014     

Optimal policies for a multi-product inventory system with negotiable lead times

The objective of this paper is to determine the optimum inventory policy for a multi-product periodic review dynamic inventory system. At the beginning of each period two decisions are made for each product. How much to “normal order” with a lead time of λ_n periods and how much to “emergency order” with a lead time of λ_e periods, where λe = λ_n - 1. It is assumed that the emergency ordering costs are higher than the normal ordering costs. The demands for each product in successive periods are assumed to form a sequence of independent identically distributed random variables with known densities. Demands for individual products within a period are assumed to be non-negative, but they need not be independent. Whenever demand exceeds inventory their difference is backlogged rather than lost. The ordering decisions are based on certain costs and two revenue functions. Namely, the procurement costs which are assumed to be linear for both methods of ordering, convex holding and penalty costs, concave salvage gain functions, and linear credit functions. There is a restriction on the total amount that can be emergency ordered for all products. The optimal ordering policy is determined for the one and N-period models.     

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.     

Optimal outbound dispatch policies: Modeling inventory and cargo capacity

In this paper, we present an optimization model for coordinating inventory and transportation decisions at an outbound distribution warehouse that serves a group of customers located in a given market area. For the practical problems which motivated this paper, the warehouse is operated by a third party logistics provider. However, the models developed here may be applicable in a more general context where outbound distribution is managed by another supply chain member, e.g., a manufacturer. We consider the case where the aggregate demand of the market area is constant and known per period (e.g., per day). Under an immediate delivery policy, an outbound shipment is released each time a demand is realized (e.g., on a daily basis). On the other hand, if these shipments are consolidated over time, then larger (hence more economical) outbound freight quantities can be dispatched. In this case, the physical inventory requirements at the third party warehouse (TPW) are determined by the consolidated freight quantities. Thus, stock replenishment and outbound shipment release policies should be coordinated. By optimizing inventory and freight consolidation decisions simultaneously, we compute the parameters of an integrated inventory/outbound transportation policy. These parameters determine: (i) how often to dispatch a truck so that transportation scale economies are realized and timely delivery requirements are met, and (ii) how often, and in what quantities, the stock should be replenished at the TPW. We prove that the optimal shipment release timing policy is nonstationary, and we present algorithms for computing the policy parameters for both the uncapacitated and finite cargo capacity problems. The model presented in this study is considerably different from the existing inventory/transportation models in the literature. The classical inventory literature assumes that demands should be satisfied as they arrive so that outbound shipment costs are sunk costs, or else these costs are covered by the customer. Hence, the classical literature does not model outbound transportation costs. However, if a freight consolidation policy is in place then the outbound transportation costs can no longer be ignored in optimization. Relying on this observation, this paper models outbound transportation costs, freight consolidation decisions, and cargo capacity constraints explicitly. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 531–556, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10030     

Storage problems when demand is “all or nothing”

An inventory of physical goods or storage space (in a communications system buffer, for instance) often experiences “all or nothing” demand: if a demand of random size D can be immediately and entirely filled from stock it is satisfied, but otherwise it vanishes. Probabilistic properties of the resulting inventory level are discussed analytically, both for the single buffer and for multiple buffer problems. Numerical results are presented.