Base-stock Level Research Articles

Dynamic pricing and inventory management of a dual-channel supply chain under different power structures

This paper studies a periodic review, joint dynamic pricing and inventory problem for a dual-channel supply chain with one manufacturer and one retailer, where the demand is stochastic and price sensitive. Three possible power structures between manufacturer and retailer in the dual-channel supply chain, i.e., Manufacturer Stackelberg, Retailer Stackelberg and Vertical Nash, are considered. Under each of the three power structures, decision models are developed with stochastic dynamic programming to determine how to adjust the pricing and inventory decisions in every period so that each member’s total expected discounted profit over the planning horizon is maximized. The effects of the power structure are analyzed by comparing the optimal dynamic pricing and inventory policies under different power structures. Our main findings include: (i) the optimal dynamic pricing and inventory policy of a dual-channel supply chain is an inventory-dependent base-stock-list-price policy; (ii) base stock levels and reduced prices are affected by starting inventory levels; (iii) the influence rules on reduced prices under different power structures are the same, while the influence rules on base stock levels vary according to the power structure; (iv) optimal pricing and inventory decisions are affected by the power structure, although the structural properties of the optimal policies under different power structures are the same. Results from numerical examples show that, for the two-period dual-channel supply chain, manufacturer and retailer prefer Vertical Nash when the wholesale price is low, while they prefer Manufacturer Stackelberg when the wholesale price is high.

On the estimation of the fill rate for the continuous (s, S) inventory system for the lost sales context.

In the continuous review reorder point, base-stock (s, S) policy, the replenishment order is launched when the inventory position reaches the reorder point, s. It is commonly assumed that the inventory position is exactly equal to the reorder point at the moment the order is launched, when actually it could be lower at that moment. This implies neglecting the possible undershoots at the reorder point, which has a direct impact on the calculation of the expected shortages per replenishment cycle. This article presents a method for an exact calculation of the fill rate (fraction of demand that is immediately satisfied from shelf) which takes explicit account of the existence of undershoots and is applicable to any discrete demand distribution function in a context of lost sales. This method is based on the determination of the stock probability vector at the moment the replenishment order is launched. Furthermore, neglecting the undershoots is shown to lead to an overestimation of the fill rate, particularly when we move farther away from the unitary demand assumption. From a practical point of view, this behaviour involves underestimating the base-stock level, S, when a target fill rate is set for its determination. The method proposed in this paper overcomes these shortcomings.

Evaluation of Unreliable Flow Lines with Limited Buffer Capacities and Spare Part Provisioning

We consider a model of an N-stage flow line with stochastic processing times and interstage buffers that decouple adjacent production stages. Machine downtimes are induced by failures of critical machine components. Each machine is assumed to have exactly one of these failure-prone components. To achieve high machine availability, it is assumed that spare parts for those failure-prone critical components are kept in stock. Failed components are immediately replaced with new, functioning components, and a one-for-one replenishment policy is applied for the restocking of those spare parts. We present a novel decomposition approach to approximate the average throughput and inventory for a system with an arbitrary number of machines, buffers, and spare parts. With a detailed numerical study, we analyze the impact of different parameter constellations on the approximation quality. We demonstrate the remarkable accuracy of our method by comparing our results with both exact and simulated values. Using our method, we further study the complex interaction and partial substitution effects between buffer sizes and spare part base-stock levels on the logistical performance of the flow line.

Echelon Base Stock Policy for Symbiotic and Reverse Logistic Flows

The main objective of this paper is to determine the base stock levels within the framework of an industrial symbiosis. At the operational level, we define the replenishment policies for finished and used products. The system is composed by two main warehouses and three enterprises located in the same industrial region, an eco-industrial park. For the first enterprise, we have two plants (or simply two manufacturing lines): the first plant manufactures finished products and the second one recovers used (wasted) finished products after collection and restores them. For the other enterprises, one enterprise consumes only finished products made by the first enterprise and the second one consumes by-products which could be used finished products or part of these used finished products coming from the second enterprise. This means that one enterprise consumes only finished products and generates wasted finished products and the second one could use these wasted finished products as inputs for its production. To simplify the study, we present only one kind of finished product but a generalization to several products and several customer and producer companies can simply be defined. The demands are uncertain such as the number of collected used finished products. The objective is then to define the base stock levels that minimize the long-run average expected total function, which is the sum of all economic, environmental and social costs. Numerical results are proposed to highlight our results.

Joint inventory and pricing control with lagged price responses

This paper studies the single stage, multiple period joint inventory and pricing planning problem with lagged price response. We prove that the optimal joint inventory and pricing policy takes the form of a state dependent list price base stock policy for the additive linear demand function with a first order auto-regressive distributed lag. Both the base stock level and the list price decisions are determined as functions of the effective price, which captures the current and past pricing impacts. Numerical studies are conducted to examine the impact of the lagged price response on the optimal decisions and the total profit.

Inventory Control involving Unknown Censored Demand and Convex Inventory Costs

We study inventory control involving lost sales and hence censored demand. In a long-run average framework, the demand distribution is largely unknown. As long as the stationary inventory costs are truly convex to the extent that the second lost item costs strictly more than the first lost item, the regret would be $\Omega(T^{2/3})$. We propose a policy that deliberately orders up to very high levels in designated learning periods and in the remaining doing periods, uses base-stock levels tailored to empirical distributions formed over the learning periods. A matching $O(T^{2/3})$ upper bound can be achieved by the policy. When inventory costs assume the special $V$-shaped form in which all holding cost rates are the same and so are all the lost cost rates, the bound could be extended to the case where items are nonperishable. Our numerical study indicates the $V$-shape restriction to be potentially relaxable. It also confirms the advantage of our policy over a slew of others from the literature.

Two-Echelon Inventory Model With Service Consideration and Lateral Transshipment

This paper considers a two-echelon inventory system with service consideration and lateral transshipment. So far, researchers have not extensively considered the use of lateral transshipment for such systems. Demand arrivals at both echelons follow the Poisson process. We introduce a continuous review base stock policy for the system in steady state, which determined the expected level for on-hand inventory, expected lateral transshipment level and expected backorder level. We showed that the model satisfied convexity with respect to base stock level. Computational experiments showed that the model with lateral transshipment performed better that the model without lateral transshipment.

Dynamic scheduling in make-to-stock production systems with setup times and random breakdowns: performance analysis and improved policies

We study the scheduling of production in a multi-item, failure-prone machine with setup times, with the objective of minimising long-run average inventory and backlog costs. We make two contributions to this problem, which has received significantly less attention compared to the standard stochastic economic lot scheduling problem (with random demand and production times). First, we adapt and implement a variety of existing scheduling policies and compare their performance through a set of numerical experiments spanning a wide range of operating conditions. This analysis allows us to identify important shortcomings of the policies, including their poor performance when items have widely different priorities (as dictated by the product of their cost and production rate parameters). Second, we leverage these insights to develop and analyse a new, easy-to-implement policy. This policy tightly controls the surplus of the highest-priority items using fixed base-stock levels and, for all other items, it determines their run lengths dynamically in such a way that long production runs become decreasingly likely with decreasing item priority. Our experiments show that our new policy greatly outperforms the benchmarking policies over a large and important set of operating regimes, comprising high-utilisation systems where the machine cycles through all items at a frequency comparable to or greater than the frequency of failure events.

How to Change a Running System—Controlling the Transition to Optimized Spare Parts Inventory Policies

Inventory optimization approaches typically optimize steady‐state performance, but do not consider the transition of an initial state to the optimized state. In this study, we address this transition. Our research is motivated by a company that implemented an improved inventory policy for its spare parts division. The improved policy suggested new base stock levels for the majority of the parts. For parts with increased base stock levels, inventory increases were realized after the part lead times, but for low‐demand parts with decreased base stock levels, inventory reductions were slow. As a result, inventory cost increased over the first months after the new inventory policy had been introduced and exceeded the inventory budget substantially. To avoid such undesirable effects, base stock level changes must be phased in. We consider a multi‐item spare parts inventory system, initially operating under an item approach inventory policy that achieves identical fill rates for all parts. Our approach addresses the transition to a superior system approach inventory policy that maximizes the system fill rate. We model the inventory transition as a finite‐horizon optimization problem and apply column generation and a marginal analysis heuristic to determine transient base stock levels for all parts. Using data from the company that motivated our research, we illustrate how the transition can be controlled to quickly improve fill rates without exceeding the initial inventory budget.

Dynamic Warranty Reserve Planning Over a Finite Horizon

This article studies the warranty reserve planning problem for a firm that makes electronic products and maintains a reserve fund for warranty claims. For accounting purposes, the reserve fund is adjusted periodically. We model this problem in a multiperiod single-product setting.We propose a model that incorporates the randomness of product sales and failures to explicitly characterize the time-varying behavior of the warranty demand, which allows the firm to dynamically manage the reserve fund. Since the firm will eventually discontinue the product, we investigate a multiperiod dynamic reserve planning model to optimize warranty reserve levels over a finite horizon. We derive a closed-form expression of time-dependent warranty demand with general product sales and failure distributions. From the perspective of after-sales service, this result can be used to explain the dynamics involved in demand for warranty service. With the objective of minimizing the discounted total loss, we obtain the optimal reserve-setting policy and prove that the firm should set the reserve fund at either zero or a warranty-demand-dependent base stock level. To show the applicability and flexibility of our model, numerical experiments are conducted for some typical sales patterns and the Weibull lifetime distribution.

Inventory decision in a periodic review inventory model with two complementary products

Interaction effect across complementary products plays an important role in characterizing the optimal inventory policy. The inventory levels of complementary products are interrelated due to interaction between demand streams. In this paper, we consider a periodic review base-stock policy in the presence of two complementary products with interrelated demands and joint replenishment. Demands are modeled by a Poisson process and any unmet demand is lost. Demands can be in sets of one unit of each or jointly. If an arrival demand requests two products jointly and one of the products is not in stock, then the whole demand is lost. We aim to investigate how this interrelated demand phenomenon influences the optimal base-stock levels and the period length of a periodic review policy. We utilize the renewal reward theorem to derive the explicit expression of the expected profit rate in the system. The goal is to determine the optimal period length and the base-stock levels such that the expected profit rate is maximized. Enumeration and approximation algorithms are employed to find the optimal and near-optimal solutions, respectively. The approximation algorithm is based on a scenario with independent demand processes which results in an explicit expression for the long-run profit per time unit and leads to analytical solutions for optimal policies. Our numerical results reveal that the solutions obtained by the approximation algorithm are close to optimal solutions. Numerical experiences show that the maximum profit in the system is achieved if the proportion of customers with jointly demand increases. Moreover, the interaction effect between demand processes has a significant impact on the control policy performance when the units lost sales and unit holding costs are high, and the demand rare is low.

Volume Flexibility at Responsive Suppliers in Reshoring Decisions

We investigate how volume flexibility, defined by a sourcing cost premium beyond a base capacity, at a local responsive supplier impacts the decision to reshore supply. The buyer also has access to a remote supplier that is cheaper with no restrictions to volume flexibility. We show that with unit lead time difference between both suppliers, the optimal dual sourcing policy is a modified dual base-stock policy with three base-stock levels. Fast orders follow a modified base-stock policy with two base-stock levels: the inventory position is raised to the higher base-stock level by using the base capacity; no overtime is used if this order increases the inventory position above the lower base-stock level. Else, the base capacity and overtime are used to raise the inventory position up to the low base-stock level. After ordering from the fast supplier, an order to the slow supplier is placed to raise the inventory position to a third base-stock level. Surprisingly, in contrast to single sourcing with limited volume flexibility, a more complex dual sourcing model results in a “simpler policy that replaces demand in each period. The latter allows analytical insights into the sourcing split between the responsive and the remote supplier. Our analysis shows how increased volume flexibility at responsive suppliers promotes the decision to reshore operations and effectively serves as a (cost) benefit. It also shows how investing in base capacity or additional volume flexibility act as strategic substitutes.

Sampling-Based Approximation for Serial Multi-Echelon Inventory System

We consider inventory management of an infinite-horizon, serial system with unknown demand distribution. Each stage orders from its upstream stage and incurs inventory holding cost while the most downstream stage faces random demand and incurs inventory holding and demand backlogging cost. Lead times between stages are constants. The objective is to minimize the expected total discounted cost over the planning horizon. We apply the sample average approximation (SAA) method and study the performance of the resulting solution under the empirical distribution function constructed from a demand sample. We derive the sample size (i.e., distribution-free sample size) required to guarantee the performance of the SAA solution be close to the optimum with a given probability. This result is obtained by first deriving a decomposable and tight cost upper bound of the whole system that depends on the (given) echelon base-stock levels and then showing the cost difference between the SAA and optimal solutions can be measured by the distance between empirical and real demand distribution functions. Furthermore, when the one-period demand distribution function is absolutely continuous and has an increasing failure rate, we derive a smaller sample size (i.e., distribution-dependent sample size) that guarantees the same performance. A special case of this result for the newsvendor problem generalizes the existing result by Levi et al. (2015). In addition, both the distribution-free and the distribution-dependent sample sizes increase polynomially as the number of stages increases. Finally, our numerical study corroborates the theoretical results.}

Technical Note—Understanding the Performance of Capped Base-Stock Policies in Lost-Sales Inventory Models

Single-sourcing lost-sales inventory systems with lead times are notoriously difficult to optimize. In this paper, we propose a new family of capped base-stock policies and provide a new perspective on constructing a practical hybrid policy combining two well-known heuristics: base-stock and constant-order policies. Each capped base-stock policy is associated with two parameters: a base-stock level and an order cap. We prove that for any fixed order cap, the capped base-stock policy converges exponentially fast in the base-stock level to a constant-order policy, providing a theoretical foundation for a phenomenon by which a capped dual-index policy converges numerically to a tailored base-surge policy recently observed in other work in a different but related dual-sourcing inventory model. As a consequence, there exists a sequence of capped base-stock policies that are asymptotically optimal as the lead time grows. We also numerically demonstrate its superior performance in general (including small lead times) by comparing it with otherwell-known heuristics.

‘Buy online and pick up in-store’: Implications for the store inventory

‘Buy online and pick up in-store’ (BOPS), wherein a customer buys the product online and picks it up from the physical store at her convenience, is fast gaining popularity. However, BOPS service complicates the store's inventory policy as BOPS orders, while waiting to be picked up, block inventory and thus delay replenishment requests. The store now has to maintain a base-stock level that is optimal considering both walk-in and BOPS orders. We discuss an inventory control policy that suits this situation. The problem is accentuated when the online channel and the physical store are under different ownerships, since the revenue from BOPS orders will now get shared between the two. In such a scenario, the store may opt for an inventory policy that favours walk-in orders over BOPS requests, as every BOPS order fetches less revenue than every walk-in customer. The extent of restriction on BOPS orders will depend on the revenue-sharing policy and also on the relative frequencies of walk-in and BOPS demands. We discuss two inventory policies that can help the store prioritize walk-in demands over BOPS demands. We compare the performances of the policies proposed, and show that each policy has its claim to supremacy over others in different operating environments, characterized by the input parameters.

A comparison of inventory policies for perishable goods

Past research on inventory management of perishables introduced models in which demand is sensitive to the age of the product. For such models, we prove that a fixed-order quantity policy is optimal under certain conditions and show that its expected cost is closer to optimal than that of the base-stock level policy when there is demand for units of all ages. We also show numerically when substituting older products to fulfill the demand for new (or vice versa) is beneficial.

Fuzzy modeling approach to on-hand stock levels estimation in (R, S) inventory system with lost sales

Purpose: One challenge in inventory control models is to know the stock available at the beginning of the cycle to satisfy future demands, i.e. to know the on-hand stock levels at order delivery. For inventory managers, this knowledge is necessary to both determine service levels and establish the control parameters of the inventory policy. However, the calculation of on-hand stock levels when unfilled demand is lost is mathematically complex since on-hand stock cannot be negative by definition. The purpose of this paper is to propose a new approach to estimate on-hand stock levels when the inventory is periodically reviewed and unfilled demand is lost, through the use of fuzzy techniques.Design/methodology/approach: This paper applies fuzzy set techniques for the calculation of the on-hand stock levels at order delivery in the lost sales context, based on the uncertainty that real demand introduces. To this end, we propose a new approach based on modeling the on-hand stock as an imprecise Markov chain using possibility functions, which reduces significantly the computational effort required to obtain the on-hand stock levels.Findings: To illustrate the performance of the proposed method, two experiments are carried out. The first experiment shows that the proposed fuzzy method correctly calculates on-hand stock levels with insignificant deviation with respect the exact vector. Additionally, the results illustrate that the fuzzy method simplifies the calculation and highly reduces the computational efforts. The second experiment shows the performance of the fuzzy method when it is used to estimate service levels by means of the fill rate. The results show that the proposed method accurately estimates the fill rate with average deviations lower than 0.00015.Practical implications: Knowing the on-hand stock vector is important for inventory managers to establish the control parameters of the system, i.e. to determine the minimum base stock level, S, that guarantees the achievement of a target service level. The difficulty of this estimation is that to obtain the on-hand stock vector in a lost sales context requires a huge computational effort and it is difficult to implement in companies’ information systems. However, the proposed fuzzy method leads to a very accurate calculation of the on-hand stock vector significantly reducing the computational costs, which makes this method easily implementable in practical environments.Originality/value: Fuzzy set techniques have been widely used in inventory models to introduce the uncertainty of demand, costs or shortage. However, to the best of our knowledge, this is the first paper which deals directly with fuzzy estimation of on-hand levels.

Repairable spare parts provisioning for multiregional expanding fleets of equipment under performance-based contracting

Performance-based service is becoming a dominant business strategy, especially in the capital-intensive equipment industry. When new equipment is released into multiregional markets, managing repairable spare parts provisioning so as to maximize the service profit from an increasing number of installations becomes challenging since the demand for spare parts varies over time across multiregional markets. A two-echelon repairable spare parts service network is studied to support performance-based service under multiregional fleet expansions. This study proposes an adaptive multi-phase basestock policy that dynamically adjusts the basestock levels in the two-echelon warehouses and the repair capacity in the central depot to meet the nonstationary demands. A profit-centric model is formulated that considers linear-step revenue and availability bands. Using the structural properties of the optimization model, a greedy algorithm is developed based on marginal analysis to solve the problem. Numerical experiements are conducted using a case study of a real business problem and its variants to demonstrate the effectiveness and applicability of the proposed model and the greedy algorithm.

Analysis of base-stock perishable inventory systems with general lifetime and lead-time

A base-stock inventory system for perishables with Markovian demand and general lead-time and lifetime distributions is investigated. Using a queueing network model, we derive explicit expressions of the stationary distribution of the inventory state together with the total expected cost in a base-stock system with lost-sales. Next, we show some monotonicity properties of the cost function and propose a procedure to derive the optimal base-stock. Extensions to the backorders case and to a dual-sourcing system with multiple warehouses are also provided. Our results generalize existing ones from the literature where the lifetime and the lead-time follow either deterministic or exponential distributions. Finally, we investigate the effect of the lifetime and lead-time variability on the system cost and the optimal base-stock level. In particular, we show the substantial errors made when assuming deterministic or exponential distributions for the lifetime. This further shows the need to have results beyond exponential or deterministic assumptions.

Time-based service constraints for inventory systems with commitment lead time

We consider a firm which targets a certain level of customer responsiveness with a minimum operational cost. The firm employs a preorder strategy, in which customers place their orders before their actual need based on a predefined time window called commitment lead time and, in turn, they receive a bonus. The firm aims to determine the optimal control policy, which consists of the optimal commitment lead time and the optimal base-stock level, such that the total operational cost is minimized subject to a time-based service level target. We formulate two time-based service criteria to measure the firm’s responsiveness in terms of average customer waiting time and individual customer waiting time in two constraint optimization models called ACO and CCO, respectively. We derive the exact probability distribution of customer waiting time, and we propose exact and heuristic optimization algorithms to solve the ACO and CCO models. We show that the commitment lead time not only increases the firm’s responsiveness level but also decreases the customers’ maximum waiting time. We find that when the firm targets a high level of responsiveness, the value of commitment lead time is more highlighted. We study multiple interesting extensions with two customer classes, random replenishment lead times and a combined model with two time-based service level targets.