Modelling replenishment and transshipment decisions in periodic review multilocation inventory systems

Effective models of key operational decisions in multilocation inventory systems are important for a successful retail sector. This paper argues that much of the existing research in this area is not applicable to a highly competitive retail environment, particularly if periodic review replenishment policies are used. The paper develops a model of a periodic review multilocation inventory system that is suitable for this environment and investigates the characteristics of optimal replenishment and transshipment decisions. This motivates the development of three simple heuristic transshipment policies that are practical for systems with many locations. The results of a numerical study involving systems with five locations suggest that the performance of these heuristic policies is often close to optimal and can be considerably better than the performance of commonly used policies.     

A two-warehouse production model for deteriorating inventory items with time-dependent demands

A two-warehouse inventory model for deteriorating items with time-dependent demand has been developed. Compared with previous models, the model involves a free form time-dependent demand and a finite replenishment rate within a finite planning horizon. Rather than the heuristic approach of equal production cycle times adopted by Lee and Ma, an approach which permits variation in production cycle times is adopted to determine the number of production cycles and the times for replenishment during a finite planning horizon. Numerical examples are provided to illustrate the application of the model and the results indicate that the performance of the proposed approach is superior to that of the heuristic approach of Lee and Ma.     

A periodic review inventory system with two supply modes

We describe a periodic review inventory system in which there are two modes of resupply, namely a regular mode and an emergency mode. Orders placed through the emergency channel have a shorter supply lead time but are subject to higher ordering costs compared to orders placed through the regular channel. We analyze this problem within the framework of an order-up-to-R inventory control policy. At each epoch, the inventory manager must decide which of the two supply modes to use and then order enough units to raise the inventory position to a level R. We show that given any non-negative order-up-to level, either only the regular supply mode is used, or there exists an indifference inventory level such that if the inventory position at the review epoch is below the indifference inventory level, the emergency supply mode is used. We also develop procedures for solving for the two policy parameters, i.e., the order-up-to level and the indifference inventory level.     

A two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment

For the capacity of any warehouse is limited, it has to rent warehouse (RW) for storing the excess units over the fixed capacity W of the own warehouse (OW) in practice. The RW is assumed to offer better preserving facilities than the OW resulting in a lower rate of deterioration and is assumed to charge higher holding cost than the OW. In this paper, a two-warehouse inventory model for deteriorating items is considered with constant demand under conditionally permissible delay in payment. The purpose of this study is to find the optimal replenishment policies for minimizing the total relevant inventory costs. Useful theorems to characterize the optimal solutions have been derived. Furthermore, numerical examples are provided to illustrate the proposed model, sensitivity analysis of the optimal solutions with respect to major parameters is carried out and some managerial inferences are obtained.     

A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation

In the business transactions, the supplier usually offers a permissible delay in payment to his retailer to attract more sales. In addition, a permissible delay in payment may be applied as an alternative to price discount. Based on the above phenomena, we incorporate a permissible delay in payment into the model of Yang [1] and develop a two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation. The objective of this study is to derive the retailer’s optimal replenishment policy that maximizes the net present value of the profit per unit time. The necessary and sufficient conditions for an optimal solution are characterized. An algorithm is developed to find the optimal solution. Finally, numerical examples are provided to illustrate the proposed model. Sensitivity analysis is made and some managerial implications are presented.     

Managing decentralized inventory and transshipment

Any decentralized retail or wholesale system of competing entities requires a benefit sharing arrangement when competing entities collaborate after their demands are realized. For instance, consider a distribution system similar to the observed behavior of independent car dealerships. If a dealership does not have in stock the car requested by a customer, it might consider acquiring it from a competing dealer. Such behavior raises questions about procurement strategies that achieve system optimal (first-best) outcomes. In this paper, we examine the existence and uniqueness of pure strategy Nash equilibrium (PSNE) for a decentralized system that adopts a transfer payment approach proposed by Anupindi et al. (Manuf. Serv. Oper. Manag. 4(3):349–368, 2001). In particular, we state a set of conditions on cost parameters and distributions that guarantee uniqueness of PSNE and discuss its consequences. We also examine a situation with incomplete information and expand the scope of the earlier models by relaxing the assumption of satisfying local demand first. That is, we allow the retailers to transship their inventory regardless of the local demand status if such transshipment increases retailer’s profit, and observe that this model extension does not affect our results relative to the more restrictive case. In short, our results provide important insights, clarifications, and strategic limitations regarding collaborations in decentralized distribution system.     

A non-stationary periodic review production–inventory model with uncertain production capacity and uncertain demand

In this paper we consider a nonstationary periodic review dynamic production–inventory model with uncertain production capacity and uncertain demand. The maximum production capacity varies stochastically. It is known that order up-to (or base-stock, critical number) policies are optimal for both finite horizon problems and infinite horizon problems. We obtain upper and lower bounds of the optimal order up-to levels, and show that for an infinite horizon problem the upper and the lower bounds of the optimal order up-to levels for the finite horizon counterparts converge as the planning horizons considered get longer. Furthermore, under mild conditions the differences between the upper and the lower bounds converge exponentially to zero.     

A new algorithm for a multi-item periodic review inventory system

In this paper, we deal with a multi-item, stochastic, periodic review inventory system with general cost structure which permits partial or complete backlogging of unfilled demand. Since both the (, S) policy and the mixed reorder policy are not optimal, we derive several properties of an optimal ordering policy and propose a new algorithm for computing it. This algorithm is based on the policy iteration method (PIM), but reduces substantially computation times in the policy evaluation and improvement routines of the PIM.     

Fill rate of single-stage general periodic review inventory systems

This paper develops an exact formula for the fill rate of a single-stage inventory system that uses a general periodic-review base-stock policy. For normal demand, we present a fill-rate expression that uses the standard normal PDF and CDF, and develop two approximations for the fill rate.     

A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time

The purpose of this article is to investigate a stochastic integrated supplier-retailer inventory problem. The model analyzed in this article explores the problem of the protection interval, the backorder price discount, the lead time, and the numbers of shipments from the supplier to the retailer in one production run as control variables to widen applications for an integrated periodic review inventory model. We consider the situation in which the supplier and the retailer establish a long-term strategic partnership and contract to jointly determine the best strategy. We assume that the protection interval demand follows a normal distribution. Our objective is to determine the optimal review period, the optimal backorder price discount, the optimal lead time, and the optimal number of shipments from the supplier to the retailer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, an algorithm of finding the optimal solution is developed. Also, the sensitivity analysis included and a numerical example is given to illustrate the results of the proposed model.     

Lot sizing for products with finite demand horizon and periodic review inventory policy

This paper analyzes and develops an efficient algorithm for solving a simple generalization of the classical economic order quantity (EOQ) model where the demand horizon is finite and a periodic review inventory policy is followed. Such problems could arise in production shops where the product under consideration is going to be discontinued and/or replaced by some other product after a certain known number of periods. Such problems could also arise as subproblems in large prodution-inventory problems were demand rate is constant in some periods and varying in others.     

A semi-Markov model with holdout transshipment policy and phase-type exponential lead time

In this paper, a semi-Markov decision model of a two-location inventory system with holdout transshipment policy is reviewed under the assumption of phase-type exponential replenishment lead time rather than exponential lead time. The phase-type exponential lead time more closely approximates fixed lead time as the number of phases increases. Unlike past research in this area which has concentrated on the simple transshipment policies of complete pooling or no pooling, the research presented in this paper endeavors to develop an understanding of a more general class of transshipment policy. In addition, we propose an effective method to approximate the dynamic holdout transshipment policy.     

Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are expensive, ready–for–use spare parts are kept in stock to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival. Each stockpoint faces demands from multiple demand classes. We are interested in the optimal lateral transshipment policy. There are three ways in which a demand can by satisfied: from own stock, via a lateral transshipment, or via an emergency procedure. Using stochastic dynamic programming, we characterize and prove the structure of the optimal policy, that is, the policy for satisfying the demands which minimizes the average operating costs of the system. This optimal policy is a threshold type policy, with state-dependent thresholds at each stockpoint for every demand class. We show a partial ordering in these thresholds in the demand classes. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, two policies that are often assumed in the literature. Furthermore, we study several model extensions which fit in the same modeling framework.     

A two-echelon inventory model with lost sales

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R_i) control policy. The warehouse operates an (SQ, (S − 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L_i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.     

A stochastic inventory model

Summary In some cases arising in certain industries or military installations not only the demand for a particular commodity is a stochastic variable but its supply as well. In these cases it is convenient to consider the inventory level resulting from the interaction of supply and demand as a third stochastic variable. The variation of the inventory level in time can then be considered as a stochastic process. If this process is ergodic, the total inventory cost over a certain timeT may be represented as a function of the mean inventory level. This mean level can then be manipulated in such a way as to minimize the total inventory cost.

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Zusammenfassung Es kommt vor, daß in gewissen Industriezweigen sowohl der Verbrauch, als auch die Anlieferung eines bestimmten Gutes stochastische Variable sind. In solchen Fällen ist es zweckmäßig, wenn man die aus der Zusammenwirkung von Verbrauch und Anlieferung resultierende Vorratsmenge als eine dritte stochastische Variable einführt. Man kann dann die Oszillationen der Vorratsmenge in der Zeit als einen stochastischen Prozeß auffassen. Falls dieser Prozeß ergodisch ist, können die gesamten Vorratshaltungskosten für eine bestimmte ZeitT dargestellt werden als eine Funktion der mittleren Vorratsmenge. Diese mittlere Vorratsmenge kann dann so bestimmt werden, daß sie die gesamten Vorratshaltungskosten minimalisiert.

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SHAPE Air Defence Technical Centre. Formerly with Tidewater Oil Company, Los Angeles, California, where the present problem was originally investigated.

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Vorgel. v.:J. Nitsche.     

Pricing and lot-sizing for a deteriorating item in a periodic review inventory system with shortages

We consider a single product that is, subject to continuous decay, a multivariate demand function of price and time, shortages allowed and completely backlogged in a periodic review inventory system in which the selling price is allowed to adjust upward or downward periodically. The objective of this paper is to determine the periodic selling price and lot-size over multiperiod planning horizon so that the total discount profit is maximized. The proposed model can be used as an add-in optimizer like an advanced planning system in an enterprise resource planning system that coordinates distinct functions within a firm. Particular attention is placed on investigating the effect of periodic pricing jointly with shortages on the total discount profit. The problem is formulated as a bivariate optimization model solved by dynamic programming techniques coupled with an iterative search process. An intensive numerical study shows that the periodic pricing is superior to the fixed pricing in profit maximization. It also clarifies that shortages strategy can be an effective cost control mechanism for managing deterioration inventory.     

Note on the fill rate of single-stage general periodic review inventory systems

For base-stock policies, Zhang and Zhang [J. Zhang, J. Zhang, Fill rate of single-stage general periodic review inventory systems, Operations Research Letters 35 (2007) 503-509] derive the fill rate, defined as the long-run average fraction of demand satisfied immediately. We derive the same expression for the fill rate defined as the ratio of expected demand satisfied immediately to expected demand, and generalize to (R,Q) policies.     

Investigation of two-warehouse inventory problems in interval environment under inflation via particle swarm optimization

ABSTRACT

In this paper, a two-warehouse inventory problem has been investigated under inflation with different deterioration effects in two separate warehouses (rented warehouse, RW, and owned warehouse, OW). The objective of this investigation is to determine the lot-size of the cycle of the two-warehouse inventory system by minimizing the average cost of the system. Considering different inventory policies, the corresponding models have been formulated for linear trend in demand and interval valued cost parameters. In OW, shortages, if any, are allowed and partially backlogged with a variable rate dependent on the duration of the waiting time up to the arrival of the next lot. The corresponding optimization problems have been formulated as non-linear constrained optimization problems with interval parameters. These problems have been solved by an efficient soft computing method, viz. practical swarm optimization. To illustrate the model, a numerical example has been solved with different partially backlogging rates. Then to study the effect of changes of different system parameters on the optimal policy, sensitivity analyses have been carried out graphically by changing one parameter at a time and keeping the others at their original values. Finally, a fruitful conclusion has been reached regarding the selection of an appropriate inventory policy of the two-warehouse system.     

A comparison of different dispatching policies in two-warehouse inventory systems for deteriorating items over a finite time horizon

This paper considers a two-warehouse inventory problem for deteriorating items with a constant demand rate over a finite time horizon. A modified first-in-first-out dispatching policy is first proposed, and a new two-warehouse inventory model based on this dispatching policy is developed. The results of this model are then compared with those of other models based on classical dispatching policies, such as the last-in-first-out, modified last-in-first-out and first-in-first-out dispatching policies. We also prove the existence and uniqueness of the optimal solutions for the models considered. Finally, a numerical example is presented to illustrate the results, and several key conditions are derived for comparing the general cases of these four models.