Lost Sales Case Research Articles

Continuous Review Model - Lost Sales Case: Risk Perspective

Continuous Review Model - Lost Sales Case: Risk Perspective

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.

Joint optimal pricing and inventory management policy and its sensitivity analysis for perishable products: Lost sale case

<p style='text-indent:20px;'>In the real world, the demand cannot be depicted exactly because of customer behavior cannot be forecasted without error. In this paper, we study the effect of the error of the estimated price-demand parameters by analyzing the sensitivity of the optimal joint pricing and ordering policy on the price-demand parameters based on a periodic-review, multi-period and lost sale inventory model for perishable products with constant quantity decay rate and price-sensitive demand. Firstly, we formulate the joint pricing and inventory control problem and find the optimal ordering quantity and the optimal price for deterministic price-demand function. The optimal solutions show that the retailer tends to set a lower price in early periods of each ordering cycle in order to reduce the inventory holding costs. Furthermore, the sensitivity of the optimal joint pricing and inventory control system with respect to the price-demand parameters is examined analytically and evaluated numerically. The sensitivity analysis reveals that compared to the optimal ordering quantity, the optimal prices are less sensitive in the demand-price parameters. Finally, according to the findings of the sensitivity analysis, a heuristic method of regulating the estimated demand-price parameters is employed to improve the average profit. <b>185</b> words.</p>

Rationing policies in a spare parts inventory system with customers differentiation

We consider a spare parts inventory system controlled by a base-stock policy. We assume several demand classes, where the demands of each class follow a compound Poisson process, and a stochastic replenishment lead-time common to all classes. For this inventory system, we investigate two rationing policies that can be employed when the inventory level is critically low. The first policy is the reservation policy, where a part of the stock is reserved for high priority demand classes, whereas the second one is new, which we call a threshold policy, and accepts demands of a given class until the pipeline stock caused by the particular class (plus the amount of the incoming order) reaches a critical level. On average, for 8% (12%) of all instances, the threshold policy performs better than the reservation policy under the full (partial) lost sales case and when the objective is to minimise the average on-hand inventory subject to fill rate service requirements. Furthermore, in many cases where the reservation policy outperforms the threshold policy, the gap between the two policies is small under both service requirements and total cost minimisation criteria. This suggests that a trade-off between ‘fairness’ and ‘cost’ is required, and that when the performance of two policies are similar, the threshold policy is generally favoured.

Fleet sizing of reusable articles under uncertain demand and turnaround times

Reusable Articles (RAs) are vital for supply chain logistics as they provide for the safe, economical and environmentally friendly movement of products and materials. The existing literature related to RAs has focused on the design aspects, fleet sizing, allocation of RAs, repositioning of empty RAs, and their end-of-life stages. This paper focuses on the fleet-sizing problem of RAs under uncertain demand and turnaround times. Probability theory-based analytical models are developed to determine the optimal fleet size of RAs in both lost-sales and back-order cases. The proposed analytical procedure can be used to develop models to find the optimal fleet sizes for any demand scenario with a known distribution. In the lost-sales case, analytical models are developed for uniform and normally distributed demand scenarios. A simulation model is developed to test the effectiveness of the average on-hand inventory in service-level estimation. Sensitivity analysis of the average on-hand inventory for the changes in demand and turnaround times have also been conducted and the key managerial insights are summarized. Analytical models are also developed for the back-order case and some specific managerial insights are presented.

Managing Perishable Inventory Systems with Multiple Demand Classes

In this paper, we study a multi-period stochastic perishable inventory system with multiple demand classes that require products of different ages. The firm orders the product with a positive leadtime and sells it to multiple demand classes, each only accepting products with remaining lifetime longer than a threshold. In each period, after demand realization, the firm decides how to allocate the on-hand inventory to different demand classes with different backorder or lost-sale cost. At the end of each period, the firm can dispose inventory of any age. We formulate this problem as a Markov decision process and characterize the optimal ordering, allocation, and disposal policies. When unfulfilled demand is backlogged, we show that the optimal order quantity is decreasing in the inventory levels and is more sensitive to the inventory level of fresher products, the optimal allocation policy is a sequential rationing policy, and the optimal disposal policy is characterized by a set of thresholds. For the lost-sale case, we show that the optimal allocation and disposal policies have the same structure but the optimal ordering policy may be different. Based on the structure of the optimal policy, we develop an efficient heuristic that is at most 4% away from the optimal cost in our numerical examples. Using numerical studies, we show that the ordering and allocation policies are close to optimal even if the firm cannot intentionally dispose products. Moreover, ignoring the difference between demand classes and using a simple allocation policy (e.g., FIFO) can significantly increase the total cost. We examine how the firm can improve the control of perishable items and show that the benefit of decreasing the leadtime is more significant than that of increasing the lifetime of the products or that of decreasing the acceptance threshold of the demand. The analysis is extended to systems with age dependent disposal cost and stochastic supply.

Fill rate estimation in periodic review policies with lost sales using simple methods

Purpose: The exact estimation of the fill rate in the lost sales case is complex and time consuming. However, simple and suitable methods are needed for its estimation so that inventory managers could use them.Design/methodology/approach: Instead of trying to compute the fill rate in one step, this paper focuses first on estimating the probabilities of different on-hand stock levels so that the fill rate is computed later.Findings: As a result, the performance of a novel proposed method overcomes the other methods and is relatively simple to compute.Originality/value: Existing methods for estimating stock levels are examined, new procedures are proposed and their performance is assessed.

Optimal production scheduling with customer-driven demand substitution

This paper deals with the production scheduling problem with customer-driven demand substitution. We consider a manufacturing system in a make-to-stock environment which is potentially able to produce a large variety of product options (the so-called long-term product assortment) but, for reasons of capacity and operative limitations, only a subset of those options can be available in stock at the same time (the so-called short-term product assortment). In such a context, typical of fields where high-variety strategies are applied, the first-choice option of the customer could be unavailable at a certain instant of time. In that case, if production is planned by taking demand substitution issues into consideration, other options which are good substitutes will be available, thus increasing the probability that the customer chooses to substitute. The paper proposes two mixed-integer linear programming models (for both the lost sale case and the backorder case) for optimising the production schedule by jointly considering (i) capacity and production constraints, and costs on one hand, (ii) and demand substitution issues on the other hand. An extensive experimental analysis has allowed us to evaluate the models’ behaviour in a variety of operative scenarios and to draw some concluding remarks.

Analyzing a lost-sale stochastic inventory model with Markov-modulated demands: A simulation-based optimization study

A consumer demand that presents auto-correlated components is a class of demand commonly found in competitive markets in which consumers may develop preferences for certain products which influence their willingness to purchase them again. This behavior may be observed in inventory systems whose products are subject to promotion plans in which mechanisms that incentivize the demand are implemented. Inventory systems that ignore these dependency components may severely impair their performance. This paper analyzes a stochastic inventory model where the control review system is periodic, is categorized as a lost-sale case, and is exposed to this class of auto-correlated demand pattern. The demand for products is characterized as a discrete Markov-modulated demand in which product quantities of the same item may relate to one another according to an empirical probability distribution. A simulation-based optimization that combines simulated annealing, pattern search, and ranking and selection (SAPS&RS) methods to approximate near-optimal solutions to this problem is employed. Lower and upper bounds for a range of near-optimal solutions are determined by the pattern search step enhanced by ranking and selection—indifferent zone. Results indicate that inventory performance significantly declines as the autocorrelation increases and is disregarded.

Analysing the effects of serially-correlated demands on costs and service levels: a lost-sale case inventory system

This paper studies a stochastic inventory system exposed to demands that exhibit serial-correlation components. The relationship between different levels of serial-correlation and inventory cost components as well as reorder points and order quantities are analysed in this study. Understanding these relationships provides opportunities for managers to leverage logistic and cross-functional supply chain levers which may be used to inform managerial risk mitigation plans. We consider a lost-sale inventory system with continuous monitoring while applying an enhanced version of simulated annealing that incorporates pattern search, ranking and selection to generate sub-optimal solutions to this problem. We employ an experimental design methodology to quantify the magnitude and significance of main effects and interactions. The analysis of the variance demonstrates that holding costs and the order quantity become substantial and significant as the auto-correlation increases in the studied setting. Total costs and service levels are compared between solutions that ignore autocorrelation factors and those that account for these dependencies.

Inventory policies for a partially observed supply capacity model

This paper considers a multi-period inventory problem with partially observed supply capacity in the lost sales case. Partially observed supply means that exact available supply in a period is observed only when the order quantity is not less than the supply capacity. Then, these observations are used to update the supply capacity distribution from one period to the next. For this inventory problem with partially observed supply information and random demand, we establish the inventory model according to a known Markov decision process(MDP) space. The existence of an optimal policy for this inventory problem is proved. Finally, some numerical examples considering Poisson distributed demand are given to verify the ability to find an optimal order quantity.

A Simulation-Based Optimization Approach to a Lost Sale Stochastic Inventory Model

This paper describes a stochastic inventory model where the control review system is periodic; demand contains auto-correlated components; and categorized as a lost sale case. The authors propose a simulation-based optimization based on using a combination of simulated annealing, pattern search, and ranking and selection methods to search and approximate solutions to this problem. Simulated annealing is employed to stochastically nominate and pre-select solutions in a decision space. Pattern search is used to systematically define a grid of competitive neighbors around pre-selected solutions. Ranking and selection is used to evaluate the performance of such competing pre-selected alternatives. On one hand, results show that service level in terms of filling rates deteriorate as the autocorrelation grows and is ignored. In contrast, service levels were kept almost invariable to the effects of the serially correlated components for solutions suggested using the proposed algorithm.

Bounds and Heuristics for Optimal Bayesian Inventory Control with Unobserved Lost Sales

In most retail environments, when inventory runs out, the unmet demand is lost and not observed. The sales data are effectively censored by the inventory level. Factoring this censored data effect into demand estimation and inventory control decision makes the problem difficult to solve. In this paper, we focus on developing bounds and heuristics for this problem. Specifically, we consider a finite-horizon inventory control problem for a nonperishable product with unobserved lost sales and a demand distribution having an unknown parameter. The parameter is estimated sequentially by the Bayesian updating method. We first derive a set of solution upper bounds that work for all prior and demand distributions. For a fairly general monotone likelihood-ratio distribution family, we derive relaxed but easily computable lower and upper bounds along an arbitrary sample path. We then propose two heuristics. The first heuristic is derived from the solution bound results. Computing this heuristic solution only requires the evaluation of the objective function in the observed lost-sales case. The second heuristic is based on the approximation of the first-order condition. We combine the first-order derivatives of the simpler observed lost-sales and perishable-inventory models to obtain the approximation. For the latter case, we obtain a recursive formula that simplifies the computation. Finally, we conduct an extensive numerical study to evaluate and compare the bounds and heuristics. The numerical results indicate that both heuristics perform very well. They outperform the myopic policies by a wide margin.

A price-setting newsvendor model under CVaR decision criterion with emergency procurement

In this paper, we consider a newsvendor model in which a risk-averse manager faces a stochastic price-dependent demand in either an additive or a multiplicative form. An emergency purchase option is allowed after the realization of demand to satisfy the units that are short. By adopting conditional value-at-risk (CVaR) as the decision criterion, we aim to investigate the optimal pricing and ordering decisions, and the effects of parameter changes in such a setting. We provide sufficient conditions for the uniqueness of the optimal policy for both demand models. We perform comparative statics analysis to show how the optimal pricing and ordering decision behaves when changing parameters. We also compare our results with those of the newsvendor with a general utility function and with CVaR criterion under lost sales assumption. Our key results include: (i) For both demand models, the optimal selling price is decreasing in risk aversion. Hence, the optimal price of a risk-averse newsvendor is not greater than the optimal price of a risk-neutral newsvendor. (ii) In contrary to the lost sales case, for the multiplicative demand model, the optimal order quantity may not be monotonic in risk aversion. Consequently, the optimal risk-averse order quantity may be lower or higher than the optimal risk-neutral counterpart. (iii) For the additive model, the optimal order quantity is strictly increasing in the emergency purchase price, while for the multiplicative model the optimal order quantity has no such a monotonic property. Some numerical examples are conducted to verify our claims and gain more insights about the risk-averse decision-making behaviors.

Optimality of State-Dependent (s, S) Policies in Inventory Models with Markov-Modulated Demand and Lost Sales

Markov-modulated processes have been used in stochastic inventory models with setup costs for modeling demand under the influence of uncertain environmental factors, such as fluctuating economic and market conditions. The analyses of these models have been carried out in the literature only under the assumption that unsatisfied demand is fully backlogged. The lost sales situation occurs in many retail establishments such as department stores and supermarkets. We use the analysis of the Markovian demand model with backlogging to analyze the lost sales case; in particular, we establish the optimality of an (s, S)-type policy under fairly general conditions.

Advance demand information and a restricted production capacity: on the optimality of order base-stock policies

This paper considers the optimality of order aggregation in a single-item production/inventory problem with advance demand information and a restricted production capacity. The advance demand information is modeled by introducing a positive customer order lead time. The paper proves, when customer order lead times are less than a threshold value, it is allowed to aggregate the orders over time when establishing the optimal production decision. This implies the optimality of an order base-stock policy. It shows also that in case of linear inventory cost, the positive effect of advance demand information is equal to a cost reduction that is proportional to idle time and foreknowledge horizon. The results hold for the backlogging case as well as for the lost-sales case.

Optimal Inventory Policy with Two Suppliers

We analyze a periodic-review inventory model where the decision maker can buy from either of two suppliers. With the first supplier, the buyer incurs a high variable cost but negligible fixed cost; with the second supplier, the buyer incurs a lower variable cost but a substantial fixed cost. Consequently, ordering costs are piecewise linear and concave. We show that a reduced form of generalized (s, S) policy is optimal for both finite and (discounted) infinite-horizon problems, provided that the demand density is log-concave. This condition on the distribution is much less restrictive than in previous models. In particular, it applies to the normal, truncated normal, gamma, and beta distributions, which were previously excluded. We concentrate on the situation in which sales are lost, but explain how the policy, cost assumptions, and proofs can be altered for the case where excess demand is backordered. In the lost sales case, the optimal policy will have one of three possible forms: a base stock policy for purchasing exclusively at the high variable cost (HVC) supplier; an (sLVC, SLVC) policy for buying exclusively from the low variable cost (LVC) supplier; or a hybrid (s, SHVC, SLVC) policy for buying from both suppliers.

Newsvendor Networks: Inventory Management and Capacity Investment with Discretionary Activities

We introduce a class of models, called newsvendor networks, that allow for multiple products and multiple processing and storage points and investigate how their single-period properties extend to dynamic settings. Such models provide a parsimonious framework to study various problems of stochastic capacity investment and inventory management, including assembly, commonality, distribution, flexibility, substitution and transshipment. Newsvendor networks are stochastic models with recourse that are characterized by linear revenue and cost structures and a linear input-output transformation. While capacity and inventory decisions are locked in before uncertainty is resolved, some managerial discretion remains via ex-post input-output activity decisions. Ex-post decisions involve both the choice of activities and their levels and can result in subtle benefits. This discretion in choice is captured through alternate or “nonbasic” activities that can redeploy inputs and resources to best respond to resolved uncertain events. Nonbasic activities are never used in a deterministic environment; their value stems from discretionary flexibility to meet stochastic demand deviations from the operating point. The optimal capacity and inventory decisions balance overages with underages. Continuing the classic newsvendor analogy, the optimal balancing conditions can be interpreted as specifying multiple “critical fractiles” of the multivariate demand distribution; they also suggest appropriate measures for and trade-offs between product service levels. This paper shows that the properties of optimal newsvendor network solutions extend to a dynamic setting under plausible conditions. Indeed, we establish dynamic optimality of inventory and capacity policies for the lost sales case. Depending on the nonbasic activities, this also extends to the backordering case. Analytic- and simulation-based solution techniques and graphical interpretations are presented and illustrated by a comprehensive example that features discretionary input commonality and a flexible processing resource.

Policy mechanisms for supply chain coordination

The problem is to determine a review period and stocking policy that are mutually beneficial to a producer and a retailer. In our model, the retailer uses a periodic review, base stock policy for ordering the item from the producer's Distribution Center (DC). Excess customer demand is assumed to be lost. A make-to-order production system supplies to the DC. We show that given a review period, unless the manufacturer agrees to share the cost of carrying a fraction of the safety stocks at the retailer, the two will not agree upon the level of stocks to be carried in the store. We prove that there is an equilibrium value for this fraction, such that the retailer and the manufacturer are always in agreement with regard to the stocking level. We then show that complete coordination on the stocking level as well as the review period can be achieved solely through carrying out negotiations on credit terms. These theoretical results are used to construct an algorithm for calculating the optimal policy parameters for a supply chain. As part of the analysis we suggest a modification of the base stock policy for the positive lag lost sales case of periodic review inventory models that consistently outperforms the base stock policy in our numerical studies.

Policy mechanisms for supply chain coordination

The problem is to determine a review period and stocking policy that are mutually beneficial to a producer and a retailer. In our model, the retailer uses a periodic review, base stock policy for ordering the item from the producer's Distribution Center (DC). Excess customer demand is assumed to be lost. A make-to-order production system supplies to the DC. We show that given a review period, unless the manufacturer agrees to share the cost of carrying a fraction of the safety stocks at the retailer, the two will not agree upon the level of stocks to be carried in the store. We prove that there is an equilibrium value for this fraction, such that the retailer and the manufacturer are always in agreement with regard to the stocking level. We then show that complete coordination on the stocking level as well as the review period can be achieved solely through carrying out negotiations on credit terms. These theoretical results are used to construct an algorithm for calculating the optimal policy parameters for a supply chain. As part of the analysis we suggest a modification of the base stock policy for the positive lag lost sales case of periodic review inventory models that consistently outperforms the base stock policy in our numerical studies.