Poisson Demand Research Articles

Exact evaluation and optimization of integer-ratio policies for stochastic one-warehouse multiple-retailer inventory systems

This paper considers one-warehouse multiple-retailer inventory systems. The retailers are non-identical and face Poisson demand processes. We consider an integer-ratio policy in which each retailer orders in a fixed interval that is either an integer or integer-ratio multiple of the warehouse's reorder interval. Base-stock policy and virtual allocation are assumed. We show that cost evaluations at the retailers are completely separable, leading to a simple approach to develop an optimal integer-ratio policy for a two-level distribution system with heterogeneous retailers.

Adaptive network traffic control with approximate dynamic programming based on a non-homogeneous Poisson demand model

ABSTRACT In this study, we develop a stochastic dynamic traffic-flow model subject to practical restrictions under the non-homogeneous Poisson vehicle arrival process. Using the cell transmission strategy, we establish traffic dynamics between two intersections. We also discuss simulating the random demand of source links from an estimated intensity function. Additionally, we propose an algorithm to optimize time interval division for aggregated data, aiming to enhance estimation performance. We explore applying our traffic flow model to the adaptive traffic network management problem, which is formulated as a Markov decision process. Leveraging approximate dynamic programming with recursive least squares-temporal difference learning, we achieve adaptive optimal policies. To validate our approach, we conduct a series of numerical experiments with random demands. The results of non-homogeneous Poisson demand conducted using random numbers and a real-word dataset indicate high efficiency with the piecewise constant, I-SMOOTH, and MNO-PQRS estimators. Compared to the Webster and Max-pressure control systems, our proposed approximate dynamic programming-based model exhibits superior stability and applicability.

Economic production with Poisson demand, lost sales, fixed‐rate discrete replenishment, and a constant setup time

We address a production/inventory problem for a single product and machine where demand is Poisson distributed, and the times for unit production and setup are constant. Demand not in stock is lost. We derive a solution for a produce‐up‐to policy that minimizes average cost per unit time, including costs of setup, inventory carrying, and lost sales. The machine is stopped periodically, possibly rendered idle, set up for a fixed period, and then restarted. The average cost function, which we derive explicitly, is quasi‐convex sparately in the produce‐up‐to level Q, the low‐level R that prompts a setup, and jointly in R equals Q. We start by finding the minimizing value of Q where R equals 0, and then extend the search over larger R values. The discrete search may end with R less than Q, or on the matrix diagonal where R equals Q, depending on the problem parameters. Idle time disappears in the cycle when R equals Q, and the two‐parameter system folds into one. This hybrid policy is novel in make‐to‐stock problems with a setup time. The number of arithmetic operations to calculate costs in the ( Q, R) matrix depends on a vector search over Q. The computation of the algorithm is bounded by a quadratic function of the minimizing value of Q. The storage requirements and number of cells visited are proportional to it.

A continuous approximation location-inventory model with exact inventory costs and nonlinear delivery lead time penalties

Most existing location-inventory models use exact locations (coordinates) for warehouses and customers but use simple approximations for inventory costs. A different approach is presented in this paper. While the locations are loosely modeled using a continuous approximation approach, a reorder point, order quantity inventory model with exact costs is used. Exact expressions for the transport time distribution and backorder time distribution (for Poisson demand) are combined with an exogenously given warehouse fulfillment time and form the delivery lead time. Customers perceive a long delivery lead time negatively. This creates a dynamic between the number of stocking locations used (which affects the transport time and cost) and the inventory policy (which affects the backorder time and inventory costs) because both influence the delivery lead time. Furthermore, we are able to model a nonlinear customer perception of the delivery lead time. Based on the presented model, the general characteristics and mechanisms of such a location-inventory problem are investigated. Among other things, we show for many probability distributions that the average backorder time per backorder is convex and that our model is quasi-convex, which allows for a wide range of optimization methods.

A decomposition and coordination method for optimising (Q, S ) policies in a two-echelon distribution system with joint replenishment

A continuous-review, two-echelon distribution system with one central distribution centre (CDC) and multiple regional distribution centres (RDCs) is studied. The CDC jointly replenishes its inventories of multiple items from an external supplier, while each RDC replenishes its inventories of the items from the CDC. Each RDC faces a Poisson demand for each item, and the inventories of each stock in the system are controlled by a (Q, S ) policy. Under this policy, an order is placed by a stock whenever its aggregate demand since the last order reaches a given quantity, and the inventory position of each item is raised up to its order-up-to level after the order placement. The objective is to optimise these (Q, S ) policies so that the expected total cost of this system is minimised. We propose a decomposition and coordination method for this optimisation after deriving analytically the cost function of the system. Our extensive numerical experiments demonstrate the effectiveness of the proposed method. Furthermore, a parameter sensitivity analysis is conducted to analyse the impacts of some key system parameters on the performance of the method, and managerial insights are provided for optimising distribution systems with joint replenishment and real applications of the method.

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.

Exact analysis of One-Warehouse-Multiple-Retailer inventory systems with quantity restricted deliveries

Joint consideration and coordination of inventory and shipment decisions is an important challenge in the quest for sustainable distribution systems. This paper addresses this issue by presenting a method for exact analysis of the inventory level distributions in a centralized One-Warehouse-Multiple-Retailer (OWMR) inventory system with quantity restricted deliveries from the warehouse. The system is characterized by continuous review (R,nQ) policies, complete backordering, and compound Poisson demand. The class of quantity restricted delivery policies we consider is motivated by a common desire in practice to ship full load carriers. It generalizes the assumption of unrestricted partial deliveries often used in the literature, and allows the warehouse to only ship retailer specific delivery batch quantities to the different retailers. The delivery batches typically represent package or pallet sizes, containers, or full truckloads. An attractive feature of our approach is that it does not add to the computational effort of analyzing the special case of unrestricted partial deliveries available in the existing literature. A numerical study shows that accounting for quantitative delivery restrictions used in transportation and handling operations when optimizing the reorder points in multi-echelon systems can be very important. Failure to do so can lead to high backorder costs and inadequate customer service.

Inventory dispensation and restocking in multiclass inventory systems: A decoupling approach

We consider an inventory system with Poisson demand arrivals arising from multiple classes. Restocking epochs are specified by a given sequence. At each epoch, we can place an order for any number of items for immediate delivery. We have to decide the inventory dispensation policy, which determines whether we should satisfy a demand, or reject it, or backlog it until the next restocking epoch. We allow class‐dependent rejection and backlogging costs. We also incorporate holding costs and procurement costs. We obtain the optimal restocking and inventory dispensation policies under the total discounted cost criterion as well as the long‐run average cost criterion. In particular, we develop efficient algorithms to compute the optimal policies (OPs) by decoupling the inventory dispensation policy from the restocking policy. We show that the OP operates as follows: At the restocking epochs, we order up to a fixed base‐stock level. We satisfy the demands from a class if the inventory is above a critical level that depends on the class and time to the next restocking epoch; else we reject or backlog it depending on whether the next restocking epoch is later or earlier than a class‐dependent critical number. The numerical experiments for the stochastic and deterministic interstocking periods are conducted to compare the OPs and costs.

Evaluation and control of inventory distribution systems with quantity based shipment consolidation

AbstractJoint consideration of inventory and shipment decisions is an important aspect of obtaining economically and environmentally sustainable distribution systems. We consider this issue in the context of a one‐warehouse‐multiple‐retailer inventory system with quantity‐based shipment consolidation to groups of nonidentical retailers facing Poisson demand. The system is centralized, with free information sharing and access to real‐time point of sale data. Thus, demand information at the retailers is immediately conveyed upstream to the warehouse without any fixed ordering costs. However, fixed costs associated with handling and transporting goods from the warehouse to the retailers are reflected in the shipment consolidation policy. Stock replenishments at the warehouse are made from an outside supplier/manufacturer according to an (R, Q) policy. For this system, we derive an exact recursive method for determining the inventory level distributions at the retailers. This allows us to evaluate the expected inventory and shipment costs, fill‐rates, and transport emissions for the entire system. We also show how to optimize the system by providing bounds on the optimal shipment quantities and the warehouse reorder level.

Replenishment and transshipment in periodic-review systems with a fixed order cost

Small businesses with many stocking locations face high ordering costs and imbalanced inventories. To address those challenges, we consider an inventory system of multiple retailers: each has Poisson demand and backorders are permitted. They order together periodically and there is an opportunity to transship available inventory between retailers in each order cycle. Each order incurs a fixed cost, and there is a variable transshipping cost, a holding cost for inventory, and a penalty cost for backorders. The system aims to determine a feasible ordering and transshipping plan which minimizes the long-run average total cost over time. We provide an analytical approach to evaluating costs, construct bounds to search for the optimal order interval, order-up-to level, and timing of transshipment, and determine the amounts to transship based on a marginal cost. Our numerical studies find that, first, the optimal order interval is between that in the no-transshipment case and an EOQ-like calculation. Second, the optimal timing of transshipments is in the middle of an order cycle or a little later. Third, the optimal order interval and timing of transshipments are insensitive to the introduction of differences between retailers. Fourth, transshipment effectively reduces costs, especially when transshipment costs are low or different between retailers. These findings can simplify the task of cost optimization in complex inventory systems and develop practical measures to further reduce costs.

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.

A serial inventory system with lead-time-dependent backordering: A reduced-state approximation

We study a serial inventory system where the external customers may have a maximum time that they would be willing to wait for delivery in cases of stock-out and the demand would be lost if the remaining delivery lead time of the next available item is longer. This lead-time-dependent backordering behavior subsumes the models of partial backordering regardless of the wait that a customer would experience. In the inventory literature, this behavior has only been analyzed in single-location settings. We study this behavior in a multi-stage setting. We consider continuous review policies at all stages facing external Poisson demands. Using the method of supplementary variables, we define the stochastic process representing the inventory system and obtain the expressions for the operating characteristics of the inventory system. Based on the solution structures for the special cases, we propose an approximate solution which rests on replacing the state-dependent purchasing decision of the customer with an averaged-out purchase probability computed using only the age of the oldest item. An extensive numerical study indicates that the proposed approximation performs very well. Our numerical study provides additional insights about the sensitivity and allocation of stock levels across stages.

The joint replenishment problem: Optimal policy and exact evaluation method

We propose a new method to evaluate any stationary joint replenishment policy under compound Poisson demand. The method makes use of an embedded Markov chain that only considers the state of the system after an order is placed. The resulting state space reduction allows exact analysis of instances that until now could only be evaluated using approximation procedures. In addition, the size of the state space is not affected if we include nonzero lead times, backlog, and lost sales. We characterize the optimal joint replenishment policy, and use these characteristics to develop a greedy-optimal algorithm that generalizes the can-order policy, a well-known family in the class of joint replenishment policies. We numerically show that this generalized can-order policy only marginally improves the best conventional can-order policy. For sizeable systems with multiple items, the latter can now be found using our exact embedded Markov-chain method. Finally, we use our method to improve and extend the well-known decomposition approach.

The newsvendor problem with a non-stationary demand process and exact accounting of holding costs

This work presents an extension of the classical newsvendor model that considers the inventory costs more accurately based on the actual stock-level within the selling period, and not on the stock-level at the end of it. The new feature of this model is that the selling period, which is relatively long, is comprised of n epochs, where the demand distribution in each epoch is known but is not stationary, and holding costs are considered only for the epochs in which the item was stored. A mathematical model is developed to calculate the expected profit, and an optimality equation is provided from which the optimal order quantity can be derived. Using a numerical analysis in a factorial experimental design of a non-homogeneous Poisson demand process, we evaluate the performance of the suggested model in comparison to two approximated standard newsvendor models that disregard the exact inventory costs.

혼성제조시스템을 위한 생산용량할당정책 수립과 지연재고정책의 이익 성과 효과성 분석

We consider a hybrid manufacturing system that produces components based on both long-term contract with higher priority and short-term contract with lower priority. The manufacturer operates production on a make-to-stock mode for the long-term contract and on a make-to-order mode for the short-term contract. Recently, hybrid manufacturing has received attention from academia and industry because it can contribute to profit enhancement through revenue channel diversification. In this paper, using a Markov decision process model, we study the structure of capacity rationing and admission control policies under the assumption of non-preemptive batch production and compound Poisson demand process. We also investigate the effectiveness of backlogging short-term contract orders on profit and examine the impact of long-term contract demand sizes on profit with varying probabilities of each demand size and variance of demand.

Revenue and Distributional Consequences of Alternative Outdoor Recreation Pricing Mechanisms: Evidence from a Micropanel Data Set

This article uses a system of Poisson demand equations to examine the revenue potential associated with uniform, site-differentiated, and income-differentiated recreational access fees for more than 130 lakes in the state of Iowa. We also consider optimal fees in the spirit of Ramsey (1927) and demonstrate how the new insights from Banzhaf and Smith (2022) can empirically inform discussions of user fees. We find that user fees could be used to raise revenue for the maintenance of recreation infrastructure, but that they are generally regressive. Fees differentiated by income groups can attenuate (but not eliminate) this regressivity.

1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing systems. Recent numerical results suggest that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation. In “1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models,” the author provides a theoretical foundation for this phenomenon in two classical inventory models. First, in a continuous review lost-sales model with lead times and Poisson demand, he proves that a capped base-stock policy has a worst-case performance guarantee of 1.79 by conducting an asymptotic analysis under a large penalty cost and lead time. Second, in a more complex continuous review dual-sourcing model with general lead times and Poisson demand, he proves that a similar capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. The results provide a deeper understanding of the superior numerical performance of capped policies and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

A Replenishment Inventory Model with a Stock-Dependent Demand and Age–Stock-Dependent Cost Functions in a Random Environment

This paper investigates an [Formula: see text] continuous-review perishable inventory model with a stock-dependent Poisson demand process, full backordering (with an extension for lost sales) and uncertainty in lead time and shelf life. Four types of costs are considered: a fixed cost of an order and each outdated item; age-dependent costs of an item (i.e., holding and salvage costs), given by a function of its remaining shelf life; and a delay cost of a backlogged demand unit, which is a function of its delay duration. Applying the supplementary variable technique, we obtain the joint probability-density function of the number of items in the system and the remaining time and thereby obtain the optimal parameters minimizing the long-run average total cost. Numerical experiments show that supply chain profits are enhanced by integrating the age components into replenishment decisions, and ignoring the shelf age- and delay-dependent costs may result in a substantial loss (up to 25%). It further appeared that estimating the lead-time distribution by an exponential one is significantly more costly, in particular as the c.v. differs from 1. In contrast, an exponential shelf life may provide a good heuristic for other shelf-life distributions.

Robust newsvendor problems with compound Poisson demands

We consider the robust newsvendor problem where the demand follows a compound Poisson distribution, but its distribution is only partly known. This assumption means that customers arrive according to a Poisson process with a given intensity, while the size of customer demand is another random variable. Yet, the newsvendor only knows the expectation and variance of the demand sizes. Given limited information, a plausible approach, put forth in prior work, is to evaluate the moments of the aggregate demand (from all customers) and then determine the respective order quantity. Instead, this paper suggests to make the best use of all the information contained in the first few moments of demand sizes as well as the structural properties of a compound demand distribution to compute a better ordering quantity. To achieve this goal, we propose a new decision model employing Panjer’s recursion as constraints. The attendant optimization problem is then solved via convex relaxation using McCormick envelopes. Numerical results confirm that the newsvendor can gain a significant increase in expected profit using the new modeling approach to make his/her ordering decision.

A fluid approximation for the single-leg fare allocation problem with nonhomogeneous poisson demand

Fare allocation for legs and O&D pairs plays a crucial role in airline revenue management. Despite a large number of dynamic pricing studies, there are only a few widely adopted studies in which assumptions affect most tactical decisions with potentially large impacts on airline profitability. These decisions involve approximating future pricing schemes, allocation of fare classes, and setting booking limits. We propose a fare allocation model for a single leg in the presence of a realistic nonhomogeneous Poisson demand with an increasing rate. We aim to compute when and how to markup the price for an airfare product (switch to an upper fare class) to maximize the expected revenue. We study a fluid approximation of the underlying stochastic problem considering independent demand from each customer segment and examine different properties that lead to several important insights. Finally, we propose a dynamic look-ahead pricing scheme to compare our fluid approximation results against the well-known EMSRb heuristic and a dynamic programming solution on randomly generated booking request data. Numerical examples illustrate the effectiveness of our proposed approach.