Base-stock Level Research Articles

Augmenting Monte Carlo Tree Search for managing service level agreements

Monte Carlo Tree Search (MCTS) is an algorithmic technique utilized in reinforcement learning, a subfield of artificial intelligence, that combines tree-based search and random sampling for decision-making in uncertain environments. Although MCTS has been successfully used for playing complex games such as Chess and Go, without customizing the original algorithm using domain knowledge, it struggles to effectively solve complex supply chain problems. This study proposes several augmenting mechanisms for MCTS, tailored for managing service level agreements. Furthermore, we enhance the proposed solution for products/services where adjusting the base-stock level is feasible. The results demonstrate that even with non-stationary demand, where most optimization methods reach their limits, employing these augmentation mechanisms significantly improves MCTS performance.

Optimal ordering based on shortage scenarios under presence of a low-quality substitute and supply uncertainty

Global trade has shifted from short-distance supply chains within countries to long-distance supply chains between countries. As a result, delivery lead time has become a key operational factor, and uncertainty causes stock-outs, resulting in massive losses. An important solution for shortages is product substitutes, but they can affect the quality of the end products. Moreover, even when demand is expected and planned for, unexpected last-minute demand makes operational conditions challenging. This paper considers a single-period problem of replenishing a high-quality product under both lead time and demand-related uncertainty. Stock-outs during and at the end of the replenishment period can be handled only by providing a low-quality substitute product that requires repeated replacements, therefore affecting pending demand. We provide detailed modeling of possible stock-out scenarios and show that, in presence of these two types of uncertainty, the optimal order quantity of a high-quality product can be determined using an upper and lower threshold and a base stock level equal to anticipated (planned) demand over the replenishment period. In particular, we derive a closed-form solution for the optimal order quantity under general probability distributions of lead time and demand and explicit solutions for uniform distributions. We also show the conditions under which optimal order quantities can be insensitive to the cost of the high-quality product.

Numerical Analysis of A Spare Parts Supply Chain With Additive Manufacturing

With recent successes, the use of additive manufacturing (AM) has shown to be a viable alternative for the production of high quality spare parts in remote military or humanitarian missions. However, effective methods for finding inventory policies in such multi-echelon networks with dual sourcing have so far been missing. In a numerical analysis, we confirm that the use of AM in downstream echelons leads to significant cost reduction, as well as reducing inventory levels. A key insight we obtain is that AM reduces the dependency of upstream stock outs, opening the door for efficient heuristics to optimize base stock levels, like the method presented.

Deep Reinforcement Learning for One-Warehouse Multi-Retailer inventory management

The One-Warehouse Multi-Retailer (OWMR) system is the prototypical distribution and inventory system. Many OWMR variants exist, e.g. demand in excess of supply may be completely back-ordered, partially back-ordered, or lost. Prior research has focused on the study of heuristic reordering policies such as echelon base-stock levels coupled with heuristic allocation policies. Constructing well-performing policies is time-consuming and must be redone for every problem variant. By contrast, Deep Reinforcement Learning (DRL) is a general purpose technique for sequential decision making that has yielded good results for various challenging inventory systems. However, applying DRL to OWMR problems is nontrivial, since allocation involves setting a quantity for each retailer: The number of possible allocations grows exponentially in the number of retailers. Since each action is typically associated with a neural network output node, this renders standard DRL techniques intractable. Our proposed DRL algorithm instead inferences a multi-discrete action distribution which has output nodes that grow linearly in the number of retailers. Moreover, when total retailer orders exceed the available warehouse inventory, we propose a random rationing policy that substantially improves the ability of standard DRL algorithms to train good policies because it promotes the learning of feasible retailer order quantities. The resulting algorithm outperforms general-purpose benchmark policies by ∼1−3% for the lost sales case and by ∼12−20% for the partial back-ordering case. For complete back-ordering, the algorithm cannot consistently outperform the benchmark.

Optimal Base-Stock Inventory-Management Policies of Cement Retailers under Supply-Side Disruptions

The current study aims to identify some optimal base-stock inventory-management policies that maximize the expected long-run profitability of cement retailers under potential supply-side disruptions. Unlike existing articles, the proposed economic order-quantity model considers periodically varying random demand rates of deteriorating items together with partially back-ordered shortages in the face of those random disruptions. This study computes the global concavity to execute the exemplary aspect for the optimal base-stock level under a slew of cost components and a fixed cycle length. Regarding the optimal pricing-related policies, this study proposes that cement retailers should stock from nearby supplier points. Unlike existing articles, we find that changes to either the unit-holding cost or the unit-lost sales cost have hardly any determining effect on the long-run profitability of retailers. When supply-side disruptions are more likely to occur during peak seasons, this study advocates for a planned capacity addition and higher base-stock levels of cement bags.

Admission control for a capacitated supply system with real-time replenishment information

Control towers can provide real-time information on logistic processes to support decision making. The question however, is how to make use of it and how much it may save. We consider this issue for a company supplying expensive spare parts and which has limited production capacity. Besides deciding on base stock levels, it can accept or reject customers. The real-time status information is captured by a k-Erlang distributed replenishment lead time. First we model the problem with patient customers as an infinite-horizon Markov decision process and minimize the total expected discounted cost. We prove that the optimal policy can be characterized using two thresholds: a base work storage level that determines when ordering takes place and an acceptance work storage level that determines when demand of customers should be accepted. In a numerical study, we show that using real-time status information on the replenishment item and adopting admission control can lead to significant cost savings. The cost savings are highest when the optimal admission threshold is a work storage level with a replenishment item halfway in process. This finding is different from the literature, where it is stated that the cost increase of ignoring real-time information is negligible under either the lost sales or the backordering case. Next we study the problem where customers are of limited patience. We find that the optimal admission policy is not always of threshold type. This is different from the literature which assumes an exponential production lead time.

Fixed-interval order-up-to policies and myopic optimal warehouse stock allocation for one-warehouse multiple-retailer systems

We consider a centralized two-level distribution system that consists of one warehouse and multiple retailers. The retailers are non-identical and face independent Poisson demand. The system adopts a fixed-interval order-up-to policy, in which the warehouse and retailers each order in a fixed interval to raise the echelon inventory position to a fixed base stock level. Shortages in the system are fully backlogged but transshipments between retailers are not allowed. The fixed-interval order-up-to policy has been studied in the literature under virtual allocation that reserves a unit of warehouse stock immediately after a unit of demand occurs. We consider an inventory control system that is different in two aspects. First, the warehouse implements myopic optimal allocation that allocates stock to retailers only at points of delivery to minimize system cost. Second, the warehouse adopts a non-stationary base stock policy to order stock for retailers according to their replenishment schedules. We develop methods to exactly evaluate and fully optimize this inventory control system. In a numerical study, we demonstrate that myopic optimal allocation reduces system cost by an average of 5.89% with a range from 1.45% to 10.13% as compared to virtual allocation. In addition, we develop an iterative procedure to alleviate the problem of dimensionality for myopic optimal warehouse stock allocation and a close-to-optimal heuristic solution that can significantly reduce computation.

Inventory control with modulated demand and a partially observed modulation process

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. We present an attainability condition, AC, that guarantees the existence of an optimal myopic base stock policy if the reorder cost $$K=0$$ and the existence of an optimal (s, S) policy if $$K>0$$ , where both policies depend on the belief function of the modulation process. Assuming AC holds, we show that (i) when $$K=0$$ , the value of the optimal base stock level is constant within regions of the belief space and that each region can be described by two linear inequalities and (ii) when $$K>0$$ , the values of s and S and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. A heuristic and bounds for the $$K=0$$ case are presented when AC does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.

Analysis of second order properties of production–inventory systems with lost sales

We consider a single-item production–inventory system under a base stock policy for inventory control. We model the system as a closed Gordon–Newell network. The population size of the network is equal to the base stock level, which is the sum of the finished goods and work-in-process inventory. Each exogenous demand, which follows a Poisson process, releases a production order for a new unit and increases the amount of the work-in-process inventory. When there are no items in the finished goods inventory available, arriving demand is lost. The replenishment network operates with state dependent service rates, which we assume to be increasing and concave. First, we analyze the queue length behavior of a two node system and provide conditions under which the mean queue length at the production server is convex in the number of customers in the system. We prove that this leads to convexity of a standard cost function. Using Norton’s theorem, we are able to generalize our results for arbitrarily large production–inventory systems.

Dynamic supplier competition and cooperation for buyer loyalty on service

A firm’s service-driven always-a-share behavior may have a significant effect on the competitive and cooperative inventory policies of its suppliers. To explore this effect, we consider a model of a repeat buyer (she) sharing her patronage among two heterogeneous newsvendor-type suppliers over an infinite horizon. To enjoy the best service advantage, the buyer plays one supplier (him) against the other by rewarding product availability with repurchase (loyalty) and punishing stockouts with switching (disloyalty) in the next period. Faced with this behavior, the optimal ordering policy of each supplier is a basestock policy with a non-negative “active” basestock level when the supplier has the buyer’s loyalty and a zero basestock level when he does not. Under competition, the optimal active basestock level of each supplier is greater than his myopic basestock level and increases in the other supplier’s active basestock level. Under a mild condition, the active basestock levels of both suppliers have at least one pure-strategy Nash equilibrium solution. If the suppliers cooperate, the optimal active basestock level of the supplier with the highest/lowest myopic profit is greater/smaller than his myopic basestock level. To better comprehend these results, we apply them to the case where the buyer’s demand is exponentially distributed. This allows us to obtain exact expressions for the optimal active basestock levels and payoff functions, which we then use in a numerical sensitivity analysis. We conclude with a discussion of the extension of the results to more than two suppliers.

Simulation based model for component replenishment in multi-product ATO systems with shared resources

ABSTRACT With increasing product complexity, sophistication and customization, Assemble-to-Order (ATO) systems have gained a lot of popularity in recent years. ATO systems have the advantage of delivering customer orders at shorter leadtimes by manufacturing components to stock. However, for an on-time delivery of the final assembled product, the corresponding components must be replenished and be available when needed for assembly in a timely yet cost-effective manner. This research investigates the production and subcontracting decisions in the multi-product ATO systems. We also provide insights on resource allocation decisions among various components and how does randomness in the service times impact these decisions. Using, Monte Carlo simulation approach, we identified that when the manufacturer is cheaper the direction towards optimality is by having the threshold values kept close to the base stock level for all components.

Configuring information updating schemes of partial imperfect advance demand information in production-inventory systems

We consider a make-to-stock system, where a supplier produces a single product to meet a single customer’s order. The order quantity is fixed, but the order time is random. Before the order becomes due, it goes through multiple stages, and the transition time from one stage to the next is random. The customer provides advance demand information (ADI) by updating the order status information to the supplier when the order progresses to a key stage. We study the supplier’s optimal production policy. By formulating this problem as a Markov decision process, we show that the optimal production policy is a state-dependent base-stock policy, and the optimal base-stock level after receiving the update is higher than before. We extend the model to the case with multiple updates. We numerically examine the configuration of information updating schemes, i.e. the timing and number of updates. We find that partial ADI with one update can capture most of the benefits of full ADI in most cases. However, the timing of update is crucial for the benefit of ADI. For a few cases where only one update is less effective, increasing the number of updates slightly can capture most of the benefits of full ADI.

Inventory control under corporate income tax and accrual accounting

Corporate income tax constitutes a significant portion of financial costs in business operations. A typical tax system is characterized by the tax structure, accounting method, and loss carryover rule. In this article, we study the optimal inventory policies under taxation and accrual accounting. We consider a lost-sales periodic-review inventory system under a proportional tax with loss carryforward. We formulate the problem as a cyclic stochastic dynamic program with multiple accounting periods, each of which consists of multiple review periods. We show that an income-dependent basestock policy is optimal. We find that tax function convexity and loss carryforward may introduce conflicting incentives into inventory decisions: the former drives risk-averse decisions whereas the latter induces risk-seeking behavior. We identify two intertemporal effects of taxation, the intra-accounting-period effect and the inter-accounting-period effect, both inducing smaller basestock levels. We further extend the analysis to the progressive tax, backordering system, and cash accounting method. Our numerical study examines the effects of various characteristics of taxation on the optimal policies. Our results demonstrate that ignoring taxation in inventory decisions may lead to significant losses especially when the tax rate and demand uncertainty are sufficiently high, which reveals the value of tax considerations in inventory management.

Safety stock placement in supply chains with expediting

Numerous documented implementations demonstrate the guaranteed-service (GS) model of safety stock optimization has applicability to practice. The standard GS (S-GS) model employs safety stock as the only countermeasure to satisfy a specified demand bound, assuming any demand in excess of the bound will be addressed by countermeasures which are not modeled. We extend the S-GS model to explicitly include expediting as a second countermeasure. Thus, our two-countermeasure GS (2C-GS) formulation addresses more demand variability explicitly than the S-GS model. We characterize a stage’s optimal base-stock level as the maximum of two base-stock calculations, and show how the resulting optimization problem can be efficiently solved using existing GS solution techniques. For normally distributed demand, we analytically characterize when the S- and 2C-GS approaches deploy different configurations of safety stock. For a series of numerical experiments, we find that the 2C-GS model places safety stocks at more stages than the S-GS model. When compared to the 2C-GS solution, the suggested safety stock positioning of the S-GS approach results in an average cost increase of 7%, and a maximum of 42%, in real-world supply chain settings. Furthermore, the benefits from expediting increase when downstream stages have greater expediting capability.

Comparison of dynamic and static pricing strategies in a dual-channel supply chain with inventory control

This paper studies the dynamic versus static pricing problem of a manufacturer–retailer dual-channel supply chain where demand is stochastic and price sensitive, and inventory can be replenished periodically. Four different pricing strategies, i.e., both members take dynamic pricing (DD strategy), retailer takes dynamic pricing while manufacturer takes static pricing (DS strategy), retailer takes static pricing while manufacturer takes dynamic pricing (SD strategy), and both members take static pricing (SS strategy), are considered. Under each of the pricing strategies, stochastic dynamic programming models are developed to determine the optimal pricing and inventory policy so that each member’s total expected discounted profit over the planning horizon is maximized. Numerical studies are carried out to compare the performance of different pricing strategies. Results show that: the optimal inventory policy under different pricing strategies is a base-stock type; under DD strategy, both members should reduce prices as long as one member’s initial inventory level is above its base stock level; under DS and SD strategies, the member adopting dynamic pricing should reduce price if its initial inventory level is above its base stock level; there exists a Nash pricing strategy which is affected by market parameters including demand variety, market size, channel preference, and price sensitivities.

Evaluation of a joint replenishment policy with sales-based threshold and stochastic demand

In this study, we consider a stochastic joint replenishment problem with Poisson demand, where multiple items are replenished jointly through a common supplier. We study a novel type of joint replenishment policy controlled by a sales threshold, under which new orders are placed to restore every item’s inventory position to a base-stock level when the system’s accumulated sales generated since the last order reach a certain threshold. The sales threshold enables our model to capture multiple items’ value heterogeneity that has been ignored by the quantity-based threshold in traditional joint replenishment control policy. We first characterize the inventory dynamics of the joint replenishment model by the semi-Markov chain method and then provide an approach to evaluate the exact long-run average system inventory-related costs. We further provide an approach to optimize the joint replenishment policy. Numerical experiments are conducted to compare the sales-based threshold policy with the traditional quantity-based threshold policy. Our results show that the sales-based threshold policy can significantly outperform the quantity-based threshold policy, especially for a system with a high holding cost rate, short lead time, low fixed ordering cost, and large heterogeneity among products. This is because the sales-based threshold adds an extra dimension to differentiate the products in addition to the use of different stock levels for multiple items. Our results help practitioners understand the underlying factors that drive the effectiveness of the sales-based threshold policy.

Inventory management of perishable products with fixed shelf life for a single echelon system

Perishable goods play an important role in our daily lives as well as in the retail industry. For products such as bakery goods, inventory optimization is very much important as it has a shorter shelf life and uncertain demand due to dynamic consumer behaviour. Baked products are as popular as they've ever been, however, consumer demand is shifting in fascinating ways. Consumers are more conscious about their health than ever before and their demand for fresh foods has skyrocketed. There lies the importance of requirement of an effective inventory management system in the bakery sector to provide fresh products reducing the wastage. Mathematical formulation is framed for a single echelon system considering only the retailer with bread as an example. Two methods of inventory management are being compared, in the first one, the entire inventory is available in rack on the beginning of the day, whereas in the second case a reservation policy is being introduced. The best base-stock level of bread to be kept by the retailer for the above two cases are tested with two ordering policies namely order-up-to-S and (s, S) policies. Demand streams are generated following uniform distribution, as it gives equal probability over a given range.

Two-echelon inventory location model with response time requirement and lateral transshipment

This study presents a new model for a two-echelon location-inventory system with response time constraints. This system controls inventory with a (S-1, S) policy and comprises of a finite collection of customers, a finite collection of service facilities and a single plant. This paper's main novelty is the incorporation of lateral transshipment into a two-echelon location-inventory system with response time requirement. By using a continuous-time Markov process approach, we determine expected on-hand inventory level in steady state, expected lateral transshipment level in steady state and expected backorder level in steady state. We utilize these steady state levels to formulate a mixed integer nonlinear programming model which incorporates lateral transshipment into an integrated location-inventory system with response time constraint. The model minimizes the total system cost and simultaneously determines: optimal location and number of service facilities, the optimal assignment of customers and base-stock level. We exploit the model's properties using Lagrange decomposition and we show that the model is convex. The model is tested on a real-world scenario using GAMS and our model returned lower costs following comparisons with a model without lateral transshipment. We also establish that lateral transshipment results to consistency of expected cost with varying response time requirement.

An asymptotic perspective on risk pooling: Limitations and relationship to transshipments

In this article we provide a novel perspective on risk pooling approaches by characterizing and comparing their asymptotic performance, highlighting the conditions under which one approach dominates the other. More specifically, we determine the inventory policy and the expected total costs of systems under physical and information pooling as the number of locations grows. We show that physical pooling dominates information pooling in settings with no additional per-location costs for operating the centralized system. In the presence of such costs, however, information pooling becomes a viable alternative to physical pooling. Through asymptotic analysis, we also address the grouping problem, the division of a given set of non-identical locations into an ordered collection of mutually exclusive and collectively exhaustive subsets of predetermined sizes and demonstrate that homogeneous groups, comprising locations with similar demand volatility, achieve a lower expected total cost. Finally, the convergence of the expected total costs and the base stock levels under the two pooling approaches is demonstrated through a simple numerical illustration. Our analysis supports the assertion that it is important to consider not only the individual characteristics of each location in isolation, but also the interactions among them, when designing pooling systems.

Quantity-based emergency shipment policies

Either structurally or in an ad hoc manner, emergency shipments are widely utilized for preventing the potentially negative impacts of stock-outs. In addition to having policies for their regular orders, companies need to decide when and for how many units to place emergency orders. In this article, we consider the periodic-review problem of a retailer who uses a quantity-based emergency shipment policy. Under this policy an emergency order is triggered if the inventory level within the review period falls below a certain level. We consider the base-stock policy for the regular replenishment orders. The goal is to determine the optimal base-stock level, the optimal period length as well as the optimal size and threshold value of emergency orders such that the total expected cost rate is minimized. We use renewal reward theory to derive the expressions of operating characteristics. The expected cost rate’s properties are analyzed to develop an optimization algorithm for finding the optimal policy parameters. We compare our policy with the time-based and hybrid emergency shipment policies and conclude that the quantity-based policy can bring substantial benefits compared to other policies.