On-hand Inventory Research Articles

Economic Order Quantity (EOQ) Optimal Control Considering Selling Price and Salesman Initiative Cost

Retailers usually offer several types of similar products. A larger number of available stock products in display space will lead consumer to buy more, as well as giving a negative impression on other types of less available products. However, the amount of display space is limited so capacity of carrying the products is limited. Competition among products to increase demand rate is influenced by stock levels available in display space, price and salesmen’s initiative in promoting the products. The Economic Order Quantity (EOQ) to replenish the stock of the product is dependent on the on-hand inventory. Salesman’s initiative also affects maximum profit obtained by the seller. In this paper, Potryagin’s Maximal Principle is used to determine the state of the inventory levels response to control prices of products. Sensitivity analysis of capacity allocation display space is also presented numerically.

Trade-credit modeling for deteriorating item inventory system with preservation technology under random planning horizon

This paper deals with a stochastic deteriorating item inventory model with preservation technology and trade-credit finance. The planning horizon of many seasonal or fashionable items stochastically varies to some extent. On the other side, the demand of such items increases in initial phase followed by a constant. This type of demand pattern can be modeled as ramp-type demand. Deterioration of such products is a characteristic, which can be reduced by making the investment on latest equipment and technology. Consumption of such items within shelf life prevents to deterioration, which can be achieved by bulk sale. In order to stimulate the selling, trade-credit policy is also considered here. In these regards, this study examines the joint effect of preservation technology investment and trade-credit policy, wherein shortage is allowed and mixture of partial backlog and lost sales. Six cases may arise depending upon three parameters of time \(\mu \) (demand increases up to \(\mu \)), T (when on-hand inventory reaches to zero) and M (trade-credit period). The mathematical models are mainly categorized in two cases: (i) \(~\mu \le M\) and (ii) \(~\mu > M\). The model is illustrated through numerical experiments, sensitivity analysis, and graphical representation.

May reorder point help under disruptions?

We study a continuous-review inventory problem of a retailer observing constant deterministic demand and ordering from a supplier. This is an extension of Economic Order Quantity model (EOQ), in which both supplier and retailer are subject to random disruptions. The supplier is assumed to be at two states, which are available and unavailable. When the retailer is disrupted, all on-hand inventory is destroyed, but the retailer recovers immediately to serve customers. All unsatisfied demand at retailer is assumed to be lost. We create a mathematical model to determine optimal parameters of an order-up-to type policy for the retailer and investigate the importance of a non-zero reorder point for the retailer. With computational experiments, we identify when a non-zero reorder point is cost saving, and compare our solution with classical EOQ.

Optimal pricing, production, and inventory for deteriorating items under demand uncertainty: The finite horizon case

This paper aims to investigate the joint dynamic pricing and production decisions of deteriorating items with uncertain demand over a finite selling season, where the demand is price sensitive and the potential demand is characterized by a stochastic process. The stocks deteriorate physically at a constant fraction of the on-hand inventory. A joint dynamic pricing and production problem to maximize the total expected profit is modeled as a stochastic optimal control problem. We derive the closed-form solutions, which are in time-dependent linear feedback form of the inventory level when it is either positive or negative. It is shown that the manufacturer always benefits from a reduction in the volatility of potential market demand. In addition, to highlight the effectiveness of the joint dynamic strategy, we also consider the case of optimal production with a static price. A numerical example is presented to illustrate the validity of the optimal control policy, and sensitivity analysis on major parameters is performed to provide more managerial insights into deteriorating items.

An M/M/1 Queueing-Inventory System with Geometric Batch Demands and Lost Sales

This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples.

Implementation of an integrated pharmacy supply management strategy.

Implementation of an integrated pharmacy supply management strategy is described. In 2011, the formulary approval process and supply management for oncology medications were independent of each other at an oncology infusion center. Numerous nonformulary medications were kept on hand and reordered based on inventory levels that were established with inadequate usage information, while some formulary agents did not have on-hand inventory levels and had to be reordered on a patient-specific basis, which required paperwork and then a review by drug information staff per institutional policy. Because there was no true distinction in the ordering of formulary versus nonformulary oncology agents, the medical staff prescribed both in the same manner, leaving the pharmacy staff responsible for ensuring that enough quantities were on hand for many drugs, regardless of formulary status. Using supply chain management principles, a formal analysis of the on-hand inventory was performed. In addition, the formulary process for oncology drugs was restructured to align with how oncology drugs are managed for on-hand inventory levels. The alignment of these processes allowed the operation to have 1 supply strategy for the ambulatory oncology infusion center. As a result, inventory exhaustion rates were reduced by 70% and inventory turn rates improved by 78%. There was also significant time savings in the operational process streamlining, eliminating the rework and inefficiencies caused by an unclear process that was not fully captured in this assessment. Alignment of the formulary review process with inventory analyses that support supply management principles reduced inventory exhaustion while improving inventory turn rates.

Some Observations on: Improving Production Policy for a Deteriorating Item Under Permissible Delay in Payments with Stock-Dependent Demand Rate

Das et al. (Comput Math Appl 60(7):1973–1985, 2010) proposed a production inventory model for a deteriorating item under permissible delay in payments assuming that the demand is stock dependent. In the production inventory model, the production rate is partially constant and dependent upon on both on-hand inventory and demand. The production inventory model assumes that the supplier gives a price discount and permissible delay in payment. In this paper, some shortcomings in the solutions of the numerical example given in Das et al. (2010) are identified, discussed and corrected. Moreover, this paper presents the optimal solutions to the numerical example as well as the correct sensitivity analysis.

Iterative Approach to Calculating the Order Fill Rate Under Purchase Dependence

For the business environment under purchase dependence that characterizes customer purchase patterns observed in such areas as marketing, manufacturing systems, and distribution systems, this paper proposes a new approximate approach to calculate the order fill rate, probability of filling an entire customer order immediately from the shelf. The new approximate approach divides customer orders into item orders and calculates fill rates of all order types to approximate the order fill rate. In order to avoid the curse of dimensionality and prevent the solution from diverging for the large instances, we develop the greed iterative search algorithm (GISA) based on the Gauss-Seidel method. Through the computational analysis that compares the GISA with the simulation, we demonstrate that the GISA is a dependable algorithm to derive the stationary joint distribution of on-hand inventories in the type-K pure system. In addition, we present some managerial insights.

Multisourcing Supply Network Design: Two-Stage Chance-Constrained Model, Tractable Approximations, and Computational Results

In this paper, we study a multisourcing supply network design problem, in which each retailer faces uncertain demand and can source products from more than one distribution center (DC). The decisions to be simultaneously optimized include DC locations and inventory levels, which set of DCs serves each retailer, and the amount of shipments from DCs to retailers. We propose a nonlinear mixed integer programming model with a joint chance constraint describing a certain service level. Two approaches—set-wise approximation and linear decision rule-based approximation—are constructed to robustly approximate the service level chance constraint with incomplete demand information. Both approaches yield sparse multisourcing distribution networks that effectively match uncertain demand using on-hand inventory, and hence successfully reach a high service level. We show through extensive numerical experiments that our approaches outperform other commonly adopted approximations of the chance constraint.

Dynamic inventory rationing: How to allocate stock according to managerial priorities. An empirical study

In spare part distribution, contracts characterized by different unit prices, lead times, minimum order quantities are usually offered to different types of customers. In this context, one of the main problems is the decision on the allocation of the on–hand inventory to the different customers in order to minimize the total back–order cost. In this paper, we propose two allocation policies that use a dynamic critical level to reserve on-hand inventory for high priority customers. The first policy sets the critical level according to a target service level for high priority demand, while the other sets the critical level according to the back-order costs for the different classes of customers. The effectiveness of the two policies with respect to no rationing or stock piling polices is shown using the data of a European spare part company. Moreover, the relationship between the two policies is evaluated from a managerial perspective.

A global simulation-optimisation approach for inventory management in a decentralised supply chain

ABSTRACTThe optimal calibration of inventory management policies in a multi-echelon linear supply chain (SC) constitutes a complex problem. The difficulty to provide a clear and unique answer leads to frustration in using serious games for educational purposes. In this work, a simulation-based optimisation approach, using a dynamic tabu search metaheuristic, is developed and applied to a specific case of a four-echelon linear SC. A stochastic demand with constant mean is used and different service levels to the market are considered. The results indicate significant differences of the required on-hand inventory, cost contributions and cost element importance between the different echelons. The non-linear influence of the service level on the various parameters and costs is also observed. The obtained results are consistent and provide a clear answer to the stated problem. The methodology developed can be easily generalised to other cases.

Closing the Gap: A Learning Algorithm for Lost-Sales Inventory Systems with Lead Times

We consider a periodic-review, single-product inventory system with lost sales and positive lead times under censored demand. In contrast to the classical inventory literature, we assume the firm does not know the demand distribution a priori and makes an adaptive inventory-ordering decision in each period based only on the past sales (censored demand) data. The standard performance measure is regret, which is the cost difference between a learning algorithm and the clairvoyant (full-information) benchmark. When the benchmark is chosen to be the (full-information) optimal base-stock policy, Huh et al. [Huh WT, Janakiraman G, Muckstadt JA, Rusmevichientong P (2009a) An adaptive algorithm for finding the optimal base-stock policy in lost sales inventory systems with censored demand. Math. Oper. Res. 34(2):397–416.] developed a nonparametric learning algorithm with a cubic-root convergence rate on regret. An important open question is whether there exists a nonparametric learning algorithm whose regret rate matches the theoretical lower bound of any learning algorithms. In this work, we provide an affirmative answer to this question. More precisely, we propose a new nonparametric algorithm termed the simulated cycle-update policy and establish a square-root convergence rate on regret, which is proven to be the lower bound of any learning algorithm. Our algorithm uses a random cycle-updating rule based on an auxiliary simulated system running in parallel and also involves two new concepts, namely the withheld on-hand inventory and the double-phase cycle gradient estimation. The techniques developed are effective for learning a stochastic system with complex system dynamics and lasting impact of decisions. This paper was accepted by Yinyu Ye, optimization.

Mathematical Model of (R,Q) Inventory Policy under Limited Storage Space for Continuous and Periodic Review Policies with Backlog and Lost Sales

This paper involves developing new mathematical expressions to find reorder point and order quantity for inventory management policies that explicitly consider storage space capacity. Both continuous and periodic reviews, as well as backlogged and lost demand during stockout, are considered. With storage space capacity, when on-hand inventory exceeds the capacity, the over-ordering cost of storage at an external warehouse is charged on a per-unit-period basis. The objective is to minimize the total cost, consisting of ordering, shortage, holding, and over-ordering costs. Demand and lead time are stochastic and discrete in nature. Demand during varying lead time is modeled using an empirical distribution so that the findings are not subject to assumptions of demand and lead time probability distributions. Due to the complexity of the developed mathematical expressions, the problems are solved using an iterative method. The method is tested with problem instances that use real data from industry. Optimal solutions of the problem instance are determined by performing exhaustive search. The proposed method can effectively find optimal solutions for continuous review policies and near optimal solutions for periodic review policies. Fundamental insights about the inventory policies are reported from a comparison between continuous review and periodic review solutions, as well as a comparison between backlog and lost sales cases.

Closing the Gap: A Learning Algorithm for the Lost-Sales Inventory System with Lead Times

We consider a periodic-review single-product inventory system with lost-sales and positive lead times under censored demand. In contrast to the classical inventory literature, we assume the firm does not know the demand distribution a priori, and makes adaptive inventory ordering decision in each period based only on the past sales (censored demand) data. The standard performance measure is regret, which is the cost difference between a feasible learning algorithm and the clairvoyant (full-information) benchmark. When the benchmark is chosen to be the (full-information) optimal base-stock policy, Huh et al. [Mathematics of Operations Research 34(2): 397-416 (2009)] developed a nonparametric learning algorithm with a cubic-root convergence rate on regret. An important open question is whether there exists a nonparametric learning algorithm whose regret rate matches the theoretical lower bound of any learning algorithms. In this work, we provide an affirmative answer to the above question. More precisely, we propose a new nonparametric algorithm termed the simulated cycle-update policy, and establish a square-root convergence rate on regret, which is proven to be the lower bound of any learning algorithms. Our algorithm uses a random cycle-updating rule based on an auxiliary simulated system running in parallel, and also involves two new concepts, namely, the withheld on-hand inventory and the double-phase cycle gradient estimation. The techniques developed are effective for learning a stochastic system with complex systems dynamics and lasting impact of decisions.

Inventory control with and without deliveries in several pieces

An inventory system is operated as a base stock system under a compound Poisson demand process. Besides having inventory and backorder costs, also a cost is incurred for each order that is delivered in several pieces to the customer (irrespective of whether some items in the order are delivered on time, or whether they are all delivered late but at different times). The possible split deliveries are assumed to happen automatically, meaning that there cannot at the same time be a positive on-hand inventory and a positive backlog. We develop a mathematical cost expression for this system to be optimized, denoted Model 1. We compare Model 1 to another model, denoted Model 2, where it is forbidden to make any of these split deliveries; thus at the same time, there can be a positive on-hand inventory and a positive backlog. We compare Model 2 to the standard textbook model, denoted Model 0, where split deliveries occur automatically and are costless. Thereby, we address a claim made by Zipkin (Foundations of inventory management. McGraw-Hill, Boston, 2000). We also examine the threshold value for the delivery split cost, which makes Model 1 and 2 perform equally well, and we examine how this threshold value depends on the other common parameters of the models.

A Simple Heuristic for Joint Inventory and Pricing Models with Lead Time and Backorders

We study a joint inventory and pricing problem in a single-stage system with a positive lead time. We consider both additive and multiplicative demand forms. This problem is, in general, intractable due to its computational complexity. We develop a simple heuristic that resolves this issue. The heuristic involves a myopic pricing policy that generates each period’s price as a function of the initial inventory level and a base-stock policy for inventory replenishment. In each period, the firm monitors its so-called price-deflated inventory position and places an order to reach a target base-stock level. The price-deflated inventory position weights the on-hand and pipeline inventory according to a factor that reflects the sensitivity of price to the net inventory level. To assess the effectiveness of our heuristic, we construct an upper bound to the exact system. The upper bound is based on an information-relaxation approach and involves a penalty function derived from the proposed heuristic. A numerical study suggests that the heuristic is near-optimal. The heuristic approach can be applied to a wide variety of inventory systems, such as systems with fixed ordering costs or fixed batch sizes. The heuristic enables us to explore the use of price as a lever to balance supply and demand. Our findings indicate that a responsive strategy (that effectively reduces the replenishment lead time) leads to a more stable pricing policy and that the value of dynamic pricing increases with lead time. This paper was accepted by Martin Lariviere, operations management.

Prediction of inventory sustainability during a drug shortage.

A methodology for predicting how long the on-hand inventory of a given medication will last during a supply shortage is described; a practical example of application of the methodology is provided. Single-site data on consumption of i.v. tobramycin over an eight-month evaluation period were collected using commercial software that tabulates barcode-assisted medication administration (BCMA) events; administered doses were standardized as 1200-mg "vial-equivalents" and summed over the review period. The total number of vial-equivalents consumed was divided by the number of "non-zero weeks of consumption" (i.e., weeks during which any tobramycin use occurred) to obtain a mean ± S.D. weekly consumption rate; this rate was multiplied by the total i.v. tobramycin on-hand supply (in vial-equivalents) to determine the mean number of potentially sustainable weeks of therapy in the event a shortage were to restrict the future supply of the drug. Overall, 99.6 vial-equivalents of i.v. tobramycin were used during the evaluation period. The mean ± S.D. number of vial-equivalents used per non-zero week of consumption was 3.11 ± 1.26. A manual count of pharmacy inventory revealed that 102.9 vial-equivalents were available at the time of analysis. The mean predicted duration of supply was 33 weeks (95% confidence interval, -126 to 192 weeks). Available BCMA data on tobramycin consumption over eight months were used to calculate the mean number of weeks the on-hand supply of the drug could be expected to last during a persistent drug shortage.

Late season low inventory assortment effects in the newsvendor problem

The assumption of the newsvendor being able to satisfy demand as long as on-hand inventory is positive does not hold for a non-homogenous product. Consumers who do not find a unit of the product which satisfies their secondary features preferences may not purchase the product even though the newsvendor has positive on-hand inventory. This is likely to occur late in the season as inventory level declines. We solve a newsvendor problem in which the probability of purchase by consumers is increasing in on-hand inventory for any inventory level below that which is needed to have a complete assortment. We identify the sufficient optimality condition for the order quantity. We show that, unlike the case of inventory-dependent demand models in the literature, the optimal order quantity may decrease due to the assortment effect. We investigate two types of pre-end of season discounts, immediate all-units and delayed, as ways to mitigate the late season assortment effect and show that in some cases, they can increase the newsvendor׳s profit and free up the shelf space for other products.

An inventory model with declining demand market for deteriorating items under a trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions for deteriorating items under a trade credit policy to reflect more realistic situations within an economic product quantity framework. In this paper, we analyse an inventory model when the supplier offers the retailer a credit period to settle the account, if the ret ailer orders a large quantity. The proposed study is meant for a declining demand market in which shortages are not allowed. The demand rate is a decreasing function of time and the deterioration rate is a constant fraction of the on-hand inventory. The mathematical formulation is explored by numerical examples. An analysis of the sensitivity of parameters on the optimal solution of our proposed study is carried out.

Optimal replenishment rate for inventory systems with compound Poisson demands and lost sales: a direct treatment of time-average cost

Supply contracts are designed to minimize inventory costs or to hedge against undesirable events (e.g., shortages) in the face of demand or supply uncertainty. In particular, replenishment terms stipulated by supply contracts need to be optimized with respect to overall costs, profits, service levels, etc. In this paper, we shall be primarily interested in minimizing an inventory cost function with respect to a constant replenishment rate. Consider a single-product inventory system under continuous review with constant replenishment and compound Poisson demands subject to lost-sales. The system incurs inventory carrying costs and lost-sales penalties, where the carrying cost is a linear function of on-hand inventory and a lost-sales penalty is incurred per lost sale occurrence as a function of lost-sale size. We first derive an integro-differential equation for the expected cumulative cost until and including the first lost-sale occurrence. From this equation, we obtain a closed form expression for the time-average inventory cost, and provide an algorithm for a numerical computation of the optimal replenishment rate that minimizes the aforementioned time-average cost function. In particular, we consider two special cases of lost-sales penalty functions: constant penalty and loss-proportional penalty. We further consider special demand size distributions, such as constant, uniform and Gamma, and take advantage of their functional form to further simplify the optimization algorithm. In particular, for the special case of exponential demand sizes, we exhibit a closed form expression for the optimal replenishment rate and its corresponding cost. Finally, a numerical study is carried out to illustrate the results.