Cooperation and game-theoretic cost allocation in stochastic inventory models with continuous review

We study cooperation strategies for companies that continuously review their inventories and face Poisson demand. Our main goal is to analyze stable cost allocations of the joint costs. These are such that any group of companies has lower costs than the individual companies. If such allocations exist they provide an incentive for the companies to cooperate.     

Joint pricing and inventory replenishment decisions with returns and expediting

We study a single-item periodic-review model for the joint pricing and inventory replenishment problem with returns and expediting. Demand in consecutive periods are independent random variables and their distributions are price sensitive. At the end of each period, after the demand is realized, a buyer can return excess stocks to a supplier. Or, if there are stockouts, the buyer can place an expediting order at the supplier to reduce the amount of shortage. Unfilled demands are fully backlogged. We characterize the optimal dynamic policy that determines the pricing, inventory replenishment, and adjustment decisions in each period so that the total expected discounted profit is maximized. For a very general stochastic demand function, we can show that the optimal replenishment policy is a modified base-stock policy, the optimal pricing policy is a modified base-stock-list-price policy, and the optimal policy for inventory adjustment follows a dual-threshold policy. We further study the operational effect of returns and expediting. Analytical and numerical results demonstrate that returns and expediting lead to a significant profit increase in a number of situations, including limited supply capacity, sufficient flexibility of the expediting order, high demand uncertainty, and a price-sensitive market.     

Dynamic pricing inventory control under fixed cost and lost sales

This paper studies a periodic review pricing and inventory replenishment problem which encounters stochastic demands in multiple periods. In many inventory control problems, the unsatisfied demand is traditionally assumed to be backlogged but in this paper is assumed to be lost. In many practical problems, a consumer who could not buy what he/she wants in one store is not willing to wait until that store restocks it but tries to buy alternatives in other stores. Also, in this paper, the random variable for the demand function is assumed to be general, which means that any probability function for the random variable can be applied to our result. Cost terms consist of the holding cost by the leftover, the shortage cost by lost sales, and the strictly positive fixed ordering cost. The objective of this paper is to dynamically and simultaneously decide the optimal selling price and replenishment in each period by maximizing the expected profit over the finite selling horizon. We show that, under the general assumption on the random variable for the demand, the objective function is K$K$-concave, an (s,S)$(s\text{,}S)$ policy is optimal for the replenishment and the optimal price is determined based on the inventory level after the replenishment in each period.     

Coordinating pricing and inventory control in a fluctuating environment

This paper addresses the simultaneous determination of pricing and inventory replenishment strate- gies under a fluctuating environment. Specifically, we analyze the single item, periodic review model. The demand consists of two parts： the deterministic component, which is influenced by the price, and the stochastic component （perturbation）. The distribution of the stochastic component is determined by the current state of an exogenous Markov chain. The price that is charged in any given period can be specified dynamically. A replenishment order may be placed at the beginning of some or all of the periods, and stockouts are fully backlogged. Ordering costs that are lower semicontinuous, and inventory/backlog （or surplus） costs that are continuous with polynomial growth. Finite-horizon and infinite-horizon problems are addressed. Existence of optimal policies is established. Furthermore, optimality of （s,S,p）-type policies is proved when the ordering cost consists of fixed and proportional cost components and the surplus cost （these costs are all state-dependent） is convex.     

Bayesian solution to pricing and inventory control under unknown demand distribution

This paper addresses the simultaneous determination of pricing and inventory control with learning. The Bayesian formulation of this model results in a dynamic program with a multi-dimension state-space. We show that the state-space of the Bayesian model can be reduced under some conditions and characterize the structure of the optimal policy.     

A model for heuristic coordination of real life distribution inventory systems with lumpy demand

This paper presents an approximation model for optimizing reorder points in one-warehouse N-retailer inventory systems subject to highly variable lumpy demand. The motivation for this work stems from close cooperation with a supply chain management software company, Syncron International, and one of their customers, a global spare parts provider. The model heuristically coordinates the inventory system using a near optimal induced backorder cost at the central warehouse. This induced backorder cost captures the impact that a reorder point decision at the warehouse has on the retailers’ costs, and decomposes the multi-echelon problem into solving N + 1 single-echelon problems. The decomposition framework renders a flexible model that is computationally and conceptually simple enough to be implemented in practice.     

An inventory control model with stochastic review interval and special sale offer

Periodic review inventory models are widely used in practice, especially for inventory systems in which many different items are purchased from the same supplier. However, most of periodic review models have assumed a fixed length of the review periods. In practice, it is possible that the review periods are of a random (stochastic) length. This paper presents an inventory control model in the case of random review intervals and special sale offer from the supplier. The replenishment interval is assumed to obey from two different distributions, namely, exponential and uniform distributions. Also, shortages are allowed in the term of partial backordering. For this model, its convexity condition is discussed and closed form solutions are proposed.     

Evaluation of stock allocation policies in a divergent inventory system with shipment consolidation

In this paper we consider a one-warehouse N-retailer inventory system characterized by access to real-time point-of-sale data, and a time based dispatching and shipment consolidation policy at the warehouse. More precisely, inventory is reviewed continuously, while a consolidated shipment (for example, a truck) to all retailers is dispatched from the warehouse at regular time intervals. The focus is on investigating the cost benefits of using state-dependent myopic allocation policies instead of a simple FCFS (First-Come-First-Serve) rule to allocate shipped goods to the retailers. The analysis aims to shed some light on when, if ever, FCFS is a reasonable policy to use in this type of system? The FCFS allocations of items to retailers are determined by the sequence in which retailer orders (or equivalently customer demands) arrive to the warehouse. Applying the myopic policy enables the warehouse to postpone the allocation decision to the moment of shipment (from the warehouse) or the moments of delivery (to the different retailers), and to base it on the inventory information available at those times. The myopic allocation method we study is often used in the literature on periodic review systems.     

Integrated inventory models considering permissible delay in payment and variant pricing strategy

The allocation of cost savings is very important for the success of the joint relationship between the buyer and vendor in supply chain management. This paper develops integrated models with permissible delay in payments for determining the optimal replenishment time interval and replenishment frequency. In addition, the variant pricing strategy is employed to obtain both sides’ cost savings in order to entice buyers to join long-term cooperative relationships. A simple solution algorithm is presented to allocate the cost savings in the integration model, and a numerical example is used to demonstrate the feasibility of the proposed integration models.     

On the value of customer information for an independent supplier in a continuous review inventory system

We consider the inventory control problem of an independent supplier in a continuous review system. The supplier faces demand from a single customer who in turn faces Poisson demand and follows a continuous review (R, Q) policy. If no information about the inventory levels at the customer is available, reviews and ordering are usually carried out by the supplier only at points in time when a customer demand occurs. It is common to apply an installation stock reorder point policy. However, as the demand faced by the supplier is not Markovian, this policy can be improved by allowing placement of orders at any point in time. We develop a time delay policy for the supplier, wherein the supplier waits until time t after occurrence of the customer demand to place his next order. If the next customer demand occurs before this time delay, then the supplier places an order immediately. We develop an algorithm to determine the optimal time delay policy. We then evaluate the value of information about the customer’s inventory level. Our numerical study shows that if the supplier were to use the optimal time delay policy instead of the installation stock policy then the value of the customer’s inventory information is not very significant.     

An inventory model with unidirectional lateral transshipments

This paper deals with a continuous review inventory system with Poisson demand, in which lateral transshipments are allowed. In case of a shortage at a location, another location acts as a supplier, if it is possible. A common assumption in earlier papers is that transshipments are allowed between all locations. This network configuration may, however, not be the best choice for many reasons. One such reason is that it may be difficult to establish contracts between locations regarding the design of the transshipment policy. Another reason is that a system with many transshipment links is much more complex than a system with few transshipment links. In this paper, we study a system where transshipments are allowed only in one direction. This may be a reasonable policy if the locations have very different backorder/lost sales costs. Our approach is relatively simple and fast, and works well in most cases.     

An inventory control model for modal split transport: A tailored base-surge approach

Firms are increasingly interested in transport policies that enable a shift in cargo volumes from road (truck) transport to less expensive, more sustainable, but slower and less flexible transport modes like railway or inland waterway transport. The lack of flexibility in terms of shipment quantity and delivery frequency may cause unnecessary inventories and lost sales, which may outweigh the savings in transportation costs. To guide the strategic volume allocation, we examine a modal split transport (MST) policy of two modes that integrates inventory controls.We develop a single-product–single-corridor stochastic MST model with two transport modes considering a hybrid push–pull inventory control policy. The objective is to minimize the long-run expected total costs of transport, inventory holding, and backlogging. The MST model is a generalization of the classical tailored base-surge (TBS) policy known from the dual sourcing literature with non-identical delivery frequencies of the two transport modes. We analytically solve approximate problems and provide closed-form solutions of the modal split. The solution provides an easy-to-implement solution tool for practitioners. The results provide structural insights regarding the tradeoff between transport cost savings and holding cost spending and reveal a high utilization of the slow mode. A numerical performance study shows that our approximation is reasonably accurate, with an error of less than 3% compared to the optimal results. The results also indicate that as much as 85% of the expected volume should be split into the slow mode.     

Joint replenishment and pricing decisions in inventory systems with stochastically dependent supply capacity

In this paper, we study a joint optimization problem of replenishment and pricing for a periodic-review inventory system with random supply capacity. When making replenishment and pricing decisions at the beginning of each period, the firm only knows the supplier’s available capacity in the current period, but does not know what will be the available capacity in future periods. The salient feature of our model is that the random supply capacities for different periods are dependent. Several stochastic dependency structures are considered for the supply capacity sequence, including the one-lag and the multi-lag dependency.     

An effective heuristic for the review period in an inventory model with staggered deliveries and normal demands

Consider a single-item, periodic review, infinite-horizon, undiscounted, inventory model with stochastic demands, proportional holding and shortage costs, and full backlogging. Orders can arrive in every period, and the cost of receiving them is negligible (as in JIT). Every T periods, one audits the stocks and chooses a delivery schedule for each of the next T periods, thus incurring a fixed audit cost and—when one schedules actual deliveries—a fixed order cost. The problem is to find a review period T and an ordering policy minimizing the average cost. An earlier article developed an algorithm for computing an optimal T, and undertook a numerical study to evaluate various approximations. Assuming normal demands, we characterize the asymptotic behavior (for large μ/σ) of the optimal T and establish the asymptotic optimality of a heuristic, calculable on a spreadsheet. A numerical study indicates that patterns established here for large μ/σ hold for σ/μ above 2.     

Optimal dynamic pricing and inventory control with stock deterioration and partial backordering

This paper studies the optimal dynamic pricing and inventory control policies in a periodic-review inventory system with fixed ordering cost and additive demand. The inventory may deteriorate over time and the unmet demand may be partially backlogged. We identify two sufficient conditions under which (s,S,p) policies are optimal.     

An order expediting policy for continuous review systems with manufacturing lead-time

Expedited shipments are often seen in practice. When the inventory level of an item gets dangerously low after an order has been placed, material managers are often willing to expedite the order at extra fixed and/or variable costs. This paper proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of ordinary (s,Q)$(s\text{,}Q)$ models. Besides the two usual operational parameters: reorder point s and order quantity Q, it consists of a third parameter called the expedite-up-to level R. If inventory falls below R at the end of the manufacturing lead-time, the buyer can request the upstream supplier to deliver part of an outstanding order via a fast transportation mode. The amount expedited will raise inventory to R, while the remaining order is delivered via a slow (regular) supply mode. Simple procedures are developed to obtain optimal operational parameters. Computational results show that the proposed policy can save large costs for a firm if service level is high, demand variability is large, the extra cost for expediting is small, or the manufacturing lead-time is long.     

Stochastic leadtimes in a one-warehouse,N-retailer inventory system with the warehouse not carrying stock

This study extends upon a multi-echelon inventory model developed by Graves, introducing in the one-warehouse, N-retailer case—as Graves suggested—stochastic leadtimes between the warehouse and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels at the retail sites in the case where the warehouse carries no stock (e.g., serves as a cross-dock point) were investigated analytically. Two alternative treatments of stochastic leadtime distributions were considered. Using as a baseline Graves’ computational study under deterministic leadtimes, results of the current study suggest that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime.     

Evolutionary Pareto optimizers for continuous review stochastic inventory systems

Multi-objective inventory control has been studied for a long time. The trade-off analysis of cycle stock investment and workload, so called the exchange curve concept, possibly dates back to several decades ago. A classical way to such trade-off analysis is to utilize the Lagrangian relaxation technique or interactive method to search for the optimum in a sequence of single objective optimization problems. However, the field of optimization has been changed over the last few decades since the concept of evolutionary computation was introduced. In this paper, a continuous review stochastic inventory system with three objectives about cost and shortage is resolved by evolutionary computation in order to plan for the control policies under backordering and lost sales. Two evolutionary optimizers, multi-objective electromagnetism-like optimization (MOEMO) and multi-objective particle swarm optimization (MOPSO), are employed to well and fast approximate the non-dominated policies in term of lot size and safety stock. Trade-offs are observed in a non-dominated set that no one excels the others in all objectives. Computational results show that the evolutionary Pareto optimizers could generate trade-off solutions potentially ignored by the well-known simultaneous method. Comparisons between the results of backordering and lost sales indicate that decision makers will make more deliberate choices about lot sizing and safety stocking when unsatisfied demand is completely lost.     

A continuous covering location model with risk consideration

This paper proposes a continuous covering location model with risk consideration. The investigated model is an extension of the discrete covering location models in continuous space. The objective function consists of installation and risk costs. Because of uncertain covering radius, customer satisfaction degree of covering radius is introduced by fuzzy concept. Since, the uncertainty may cause risk of uncovering customers; the risk cost is added to the objective function. The installation cost is assigned to a zone with a predetermined radius from its center. The model is solved by a fuzzy method named α$\alpha$-cut. After solving the model based on different α$\alpha$-values, the zones with the largest possibilities are determined for locating new facilities and the best locations are calculated based on the obtained possibilities. Then, the model is solved to determine the best covering values. This paper, also introduces a risk analysis method based on Response Surface Methodology (RSM) to consider risk management in the location models. Finally, a numerical example is expressed to illustrate the proposed model.