Optimal Base-stock Level Research Articles

Control of a continuous production inventory system with production quantity restrictions

Motivated by a real-world application in the chemical industry, we consider the problem of controlling a production asset that has to run continuously around the clock. This requirement causes minimum and maximum production quantities per period, resulting from the minimally and maximally feasible production rates. Our goal in this paper is to find a good control rule for this production inventory setting. To this end, we first show that the setting is analytically equivalent to a periodic-review inventory system with an order band, for which a modified base-stock policy is optimal. We then develop an iterative solution algorithm to approximate the optimal base-stock level under a predefined service-level target. This algorithm is easily implementable in a spreadsheet software tool with additional Visual Basic for Applications coding. In a numerical study, we find that the approximation works very well for a broad set of parameters. In contrast, simple rules-of-thumb such as the usage of a “standard” base-stock level that does not consider the order band may result in severe errors. Either the chosen base-stock level is too low, causing the service-level target to be undershot (by up to 40% in our analyzed settings), or it is too high, resulting in an overshoot of the service-level target and thus in excessive inventory. In the real-world application example, the new solution approach identifies an inventory saving potential of about 17%.

Solution procedures for lost sales base-stock inventory systems with compound Poisson demand

The continuous review base-stock policy under the lost sales assumption has been extensively studied in the inventory literature. However, most of the research that deals with the optimal policy parameter determination under compound Poisson demand focuses on the stuttering Poisson case. In this paper, we extend earlier research by providing a method that calculates the best cost-driven base-stock policy under more general compound Poisson demand processes and for both the complete and the partial rejection case. This is done by developing a simple recursive formula that computes the steady-state probabilities of the number of outstanding orders for general demand and lead-time distributions. Based on this, we are able to obtain the optimal base-stock level under the complete rejection case and an approximate base-stock level under the partial rejection policy. Through a numerical investigation, we show that our approach outperforms existing methods with regard to computational requirements.

Spare Parts Inventory Control under a Fixed-Term Contract with a Long-Down Constraint

We are interested in service contracts for spare parts. We introduce a new performance measure, XLD (extreme long down), that limits the number of deliveries that are later than an agreed threshold during the contract period. We consider a single item, single location stockpoint serving multiple systems where demand is satisfied in an alternative way if the stockpoint is out of stock. Using a finite horizon Markov decision process, we derive the optimal spare parts inventory policy for meeting the contract at minimum costs. We prove that a state-dependent and time-dependent base stock policy is optimal. We also prove that the optimal base stock level is non-increasing in the number of allowed extreme long downs in the remaining contract period and that the optimal base stock level decreases at most by one unit per step. We formulate three heuristics for our finite horizon problem. These heuristics are a translation of heuristics commonly used for other service measures than the XLD measure. We assess their performance in comparison to the optimal policy. The results of our numerical study show that Heuristic 3, a myopic heuristic, is the best performing with an average optimality gap of 4.5% and an optimality gap lower than 5% in 81% of the instances. The maximum optimality gap is very high for all three heuristics, showing that important savings can be made by taking into account the actual contract performance and the remaining contract duration in stocking decisions.

Optimal Dynamic Order Scheduling under Capacity Constraints Given Demand‐Forecast Evolution

We consider a manufacturer without any frozen periods in production schedules so that it can dynamically update the schedules as the demand forecast evolves over time until the realization of actual demand. The manufacturer has a fixed production capacity in each production period, which impacts the time to start production as well as the production schedules. We develop a dynamic optimization model to analyze the optimal production schedules under capacity constraint and demand‐forecast updating. To model the evolution of demand forecasts, we use both additive and multiplicative versions of the martingale model of forecast evolution. We first derive expressions for the optimal base stock levels for a single‐product model. We find that manufacturers located near their market bases can realize most of their potential profits (i.e., profit made when the capacity is unlimited) by building a very limited amount of capacity. For moderate demand uncertainty, we also show that it is almost impossible for manufacturers to compensate for the increase in supply–demand mismatches resulting from long delivery lead times by increasing capacity, making lead‐time reduction a better alternative than capacity expansion. We then extend the model to a multi‐product case and derive expressions for the optimal production quantities for each product given a shared capacity constraint. Using a two‐product model, we show that the manufacturer should utilize its capacity more in earlier periods when the demand for both products is more positively correlated.

Disposal versus rework – Inventory control in a production system with random yield

In a production environment where random yield plays a fairly significant role, a decision has to be made on how to handle products that do not satisfy given quality requirements. We consider a single-stage production system with a positive production time and random yield. To ensure that only high quality items are sold to the customer, a post-production quality control system has been put in place. We compare two different strategies for defective items: disposal or rework. Disposal is possible without any time delay whereas the rework process requires a positive rework time. While disposed-of items leave the production process, reworked products stay in the process and are assumed to be as good as products that are perfect when they are initially produced. The end products are stored in a warehouse to satisfy stochastic demand. We show how to determine the optimal base-stock level, which is very difficult because of unknown covariances between orders. Subsequently, an optimization model is proposed to support the planner’s decision on which strategy to choose when it comes to whether to dispose of or rework defective items. By means of a sensitivity analysis we show which parameters directly affect this decision and give managerial insights. The analysis indicates that significant cost reductions can be obtained by choosing the best strategy for defective products.

Exact analysis of the [formula omitted] inventory policy in a two-echelon production–inventory system

We consider a two-echelon production–inventory system with a single manufacturer operating from a warehouse and a distribution center (DC) using the continuous review (R,Q) inventory policy where transportation times between the two echelons are generally distributed. We first derive the optimal reorder point at the DC, and then, propose a heuristic to approximate the optimal base-stock level in the warehouse as well as the batch order size at the DC.

Inventory management of reusable surgical supplies.

We investigate the inventory management practices for reusable surgical instruments that must be sterilized between uses. We study a hospital that outsources their sterilization services and model the inventory process as a discrete-time Markov chain. We present two base-stock inventory models, one that considers stockout-based substitution and one that does not. We derive the optimal base-stock level for the number of reusable instruments to hold in inventory, the expected service level, and investigate the implied cost of a stockout. We apply our theoretical results to a dataset collected from a surgical unit at a large tertiary care hospital specializing in colorectal operations. We demonstrate how to implement our model when determining base-stock levels for future capacity expansion and when considering alternative stockout protocols. Our analysis suggests that the hospital can reduce the number of reusable instrument sets held in inventory if on-site sterilization techniques (e.g., flash sterilization) are employed. Our results will guide future procurement decisions for surgical units based on costs and desired service levels.

Optimal production, pricing, and substitution policies in continuous review production-inventory systems

We consider the optimal production, pricing, and substitution policies of a continuous-review production-inventory system with two products: a high-end product and a low-end product. Each product has its associated customer stream; however, the demands for the low-end product may be satisfied by the high-end product. We formulate the problem as a Markov decision process and characterize the structure of the optimal control policy, which specifies when to produce for each product, when to use the high-end product as a substitute, and how to set the optimal prices. We show that a base-stock production policy is optimal; however, the optimal base-stock level for each product depends on the inventory level of the other product and it features a monotonic property. We also demonstrate that the optimal substitution policy is a rationing policy with the rationing level depending on the total inventory amount. We find that the optimal prices can be either decreasing or increasing in the inventory levels, depending on the forms of demand functions. Furthermore, we utilize numerical experiments to investigate the impact of different system characteristics on the benefits of using substitution and dynamic pricing. Finally, we investigate when the dynamic pricing strategy and the substitution strategy are complements or substitutes.

Optimal Control Policy for Stochastic Inventory Models with Two Substitutable Products

In this paper, we consider a stochastic inventory system of two products where a fi xed proportion of the unsatisfied ed demand for one product purchases the other product as a substitute. We formulate the optimal inventory control problem as a dynamic program; and by developing two key technical results, we derive several important structural properties (including joint concavity and submodularity) of the value functions. Based on these properties, we show that the optimal inventory policy for each product in each period is a base-stock policy where the base-stock level is a decreasing function of the maximum of the initial inventory level of the other product and a critical control number. We further prove that, as the initial inventory level of the other product increases, the base-stock level of each product converges to the optimal base-stock level of an auxiliary single-product lost-sales inventory system. We also provide intuitive interpretations and a numerical example to facilitate the understanding of our analytical results.

Dynamic Stochastic Inventory Management with Reference Price Effects

We analyze the joint inventory and pricing decisions of a firm when demand depends on not only the current selling price but also a memory-based reference price and customers are loss averse. The presence of reference price effect leads to a nonconcave one-period expected revenue in price and reference price. We introduce a transformation technique that allows us to prove under some mild assumptions the optimality of a reference-price-dependent base-stock list-price policy, which is characterized by a base-stock level and a target reference price. In addition, we show that the target reference price is increasing in the reference price, but except in the loss-neutral case, the base-stock level is not monotone in the reference price. We also show that in the steady state of the model with the reference price effect, the optimal price is lower while the optimal base-stock level is higher than their counterparts in the model without the reference price effect.

Optimization of two-stage production/inventory systems under order base stock policy with advance demand information

It is important to share demand information among the members in supply chains. In recent years, production and inventory systems with advance demand information (ADI) have been discussed, where advance demand information means the information of demand which the decision maker obtains before the corresponding actual demand arrives. Appropriate production and inventory control using demand information leads to the decrease of inventory and backlog costs. For a single stage system, the optimal base stock and release lead time have been discussed in the literature. In practical production systems the manufacturing system has multiple processes. The multiple stage production and inventory system with ADI, however, has been analyzed by simulation or assuming exponential processing time. That is, their theoretical analysis and optimization of release lead time and base stock level have little been obtained because of its difficulty. In this paper, theoretical analysis of a two-stage production inventory system with advance demand information is developed, where the processing time is assumed deterministic and identical; demand arrival process is Poisson, and an order base stock policy is adopted. Using the analytical results, optimal release lead time and optimal base stock levels for minimizing the average cost on the holding and backlog costs are explicitly derived.

Reservation policies of advance orders in the presence of multiple demand classes

We consider an inventory system where customers provide advance order information. Specifically each customer order contains two attributes: a request date when the order is received and a due date specifying when the customer wants the order delivered. We assume M different customer classes whose demands are generated by independent Poisson processes. All customers in the same class have the same constant demand lead-time (the difference between the due date and the request date), and all demand lead-times are less than the replenishment lead-time of the inventory system. The inventory system is operated as a base-stock system where a replenishment order is issued instantaneously upon the receipt of a customer order. A key characteristic is that customers with a long demand lead-time can be overtaken by customers with a shorter demand lead-time. Therefore, a potentially important issue is: should items in stock be reserved for specific order requests, and if so, when? Here it is understood that after the reservation, the item cannot be used to fill other customer requests. We describe a general reservation policy, meaning that to each customer class a given reservation delay arises between receipt of an order and the reservation. Assuming that the revenue of an order depends on the demand lead-time, we propose a profit optimization model where the expected profit is the difference between the expected revenue and the expected inventory holding costs. We derive the profit function for the general reservation policy and find the optimal base stock level and the reservation policy by exhaustive search. In our numerical results we find cases where, by using the general reservation policy, there are improvements by more than 5% compared to either making no reservations at all, or making a complete reservation (that is, reservation immediately upon receipt) of items to incoming orders. Our numerical results also show that instead of using a general reservation policy, one can, without much loss of optimality, resort to a simpler reservation policy, denoted Backward Delay, BD.

A characterization of optimal base-stock levels for a multistage serial supply chain

In this article, we present a multistage model to optimize inventory control decisions under stochastic demand and continuous review. We first formulate the general problem for continuous stages an ...

Backup Sourcing Decisions for Coping with Supply Disruptions under Long-Term Horizons

This paper studies a buyer’s inventory control problem under a long-term horizon. The buyer has one major supplier that is prone to disruption risks and one backup supplier with higher wholesale price. Two kinds of sourcing methods are available for the buyer: single sourcing with/without contingent supply and dual sourcing. In contingent sourcing, the backup supplier is capacitated and/or has yield uncertainty, whereas in dual sourcing the backup supplier has an incentive to offer output flexibility during disrupted periods. The buyer’s expected cost functions and the optimal base-stock levels using each sourcing method under long-term horizon are obtained, respectively. The effects of three risk parameters, disruption probability, contingent capacity or uncertainty, and backup flexibility, are examined using comparative studies and numerical computations. Four sourcing methods, namely, single sourcing with contingent supply, dual sourcing, and single sourcing from either of the two suppliers, are also compared. These findings can be used as a valuable guideline for companies to select an appropriate sourcing strategy under supply disruption risks.

Managing Finite-Horizon Down-Side Risks of Stochastic and Capacitated Supply Chains

We consider minimizing the down-side risk of a supply chain with random demand and capacity. A base-stock inventory to be determined buffers the demand-capacity interaction. The Conditional Value-at-Risk of the finite-horizon holding and back-ordering cost measures down-side risk. We demonstrate the problem is intractable and derive an approximation of the optimal base-stock level. A simulation experiment indicates the approximation yields optimal or near-optimal solutions. We study the manager’s behavior regarding the optimal base-stock level as the supply chain parameters change.

Two Backorder Compensation Mechanisms in Inventory Systems with Impatient Customers

We study a compensation mechanism design problem with customer‐choice behavior in a continuous review setting where the production and demand processes are stochastic. When a stockout occurs, the firm controls backorders on the basis of certain compensation policies. Customers make decisions to maximize their utility, which is decreasing in the price, the waiting time, and the customer's impatience factor. We assume that the impatience factor is private information held by the customer only. Two compensation mechanisms are designed to control backorders, namely uniform compensation and priority auction with an admission price. Under uniform compensation, the firm offers the same discount to all customers, whereas under auction compensation, priority is granted according to the customers' bid prices. We obtain the optimal stockout price and base stock level under each mechanism, and analyze the properties of the respective optimal policies. Assuming linear waiting costs with uniformly distributed impatience factor, we find that the auction mechanism (1) maintains a lower base stock level and results in greater profit and (2) benefits customers with relatively lower or higher impatience factors, but customers with a medium impatience factor may be rendered worse off. We further show that both compensation mechanisms are suitable for products with a high unit profit, a high lost sales penalty cost, and a high holding cost.

Models for integrated production-inventory systems: steady state and cost analysis

We consider a two-echelon production-inventory system with a central supplier connected to production systems (servers) at several locations, each with a local inventory. Demand of customers arrives at each production system according to a Poisson process and is lost if the local inventory is depleted. To satisfy a customer’s demand, a server at the production system takes exactly one unit of raw material from the associated local inventory. The central supplier manufactures raw material to replenish the local inventories, which are controlled by a continuous review base stock policy. We derive stationary distributions of joint queue length and inventory processes in explicit product form. After performing a cost analysis, we find out that the global search for the vector of optimal base stock levels can be reduced to a set of independent optimisation problems. The explicit form of the stationary distribution enables us to get additional structural insights, e.g. about monotonicity properties and stability conditions. Obtaining the product form relies on some simplifying assumptions. The results are therefore compared with simulations of a more realistic system, which supports to use it as approximation.

Optimizing Reorder Intervals for Two-Echelon Distribution Systems with Stochastic Demand

We consider a periodic-review inventory system in which N non-identical retailers replenish from a warehouse, which further replenishes from an outside vendor with ample supply. Each facility faces Poisson demand and replenishes according to a base-stock policy in a fixed time interval. Fixed costs are incurred for placing an order. The warehouse fills the retailers’ orders in the same sequence as the occurrence of the demand at the retailers. The objective is to minimize the average system cost per period. This paper develops an evaluation scheme and provides a method to obtain the optimal base-stock levels and reorder intervals. Specifically, with fixed reorder intervals, we show that the optimal base-stock levels can be obtained by generalizing the result in the literature. To find the optimal reorder intervals, we first allocate the total system cost to each facility and then construct a lower bound to the allocated facility cost. These lower bound functions, which are separable functions of reorder intervals, can be used to derive bounds for the optimal reorder intervals. The key to tightening the bounds is to obtain a near-optimal total cost. Thus, we propose a simple heuristic that modifies the algorithm that solves the deterministic counterpart. The results of numerical studies suggest that the optimal reorder intervals tend to satisfy integer-ratio relationships and that the suggested heuristic can generate effective integer-ratio policies for large systems.

Inventory optimization for perishables subject to supply disruptions

We study a retailer selling perishable products with fixed lifetime. The retailer’s supply system is subject to random disruptions. Assuming a base-stock policy, deterministic demand and stationary cost parameters, we determine the expression for the optimal base-stock level. We show that if the retailer manages to make the system safer, it is possible to operate with the same expected cost even with products with shorter lifetimes. We conclude that the retailer should concentrate on reducing disruptions’ durations instead of their occurrence rate.

A modelling and simulation approach to assessment of a negative binomial approximation in a multi-echelon inventory system

Some studies in the multi-echelon inventory systems literature have used a negative binomial distribution to approximate a critical random variable arising in the model. Graves (1996) developed a multi-echelon inventory model with fixed replenishment intervals, where each site follows a base stock policy. He proposed in the one-warehouse, N-retailer case a negative binomial distribution to approximate a random variable he referred to as ‘uncovered demand’. Computational evidence was provided to demonstrate the effectiveness of the approximation. Graves then suggested search procedures for approximately optimal base stock levels at the warehouse and N identical retailers under two customer service criteria: 1) probability of no stockout; 2) fill rate. A separate analytical evaluation of the negative binomial approximation has been preliminarily reported elsewhere. In the current study, we apply a modelling and simulation approach to assess whether the approximation-based search procedures, in fact, lead to optimal or near-optimal stock levels.