Multi-location transshipment problem with capacitated transportation

We consider coordination among stocking locations through replenishment strategies that take explicitly into consideration transshipments, transfer of a product among locations at the same echelon level. We incorporate transportation capacity such that transshipment quantities between stocking locations are bounded due to transportation media or the location’s transshipment policy. We model different cases of transshipment capacity as a capacitated network flow problem embedded in a stochastic optimization problem. Under the assumption of instantaneous transshipments, we develop a solution procedure based on infinitesimal perturbation analysis to solve the stochastic optimization problem, where the objective is to find the policy that minimizes the expected total cost of inventory, shortage, and transshipments. Such a numerical approach provides the flexibility to solve complex problems. Investigating two problem settings, we show the impact of transshipment capacity between stocking locations on system behavior. We observe that transportation capacity constraints not only increase total cost, they also modify the inventory distribution throughout the network.     

Enhanced lateral transshipments in a multi-location inventory system

In managing an inventory network, two approaches to the pooling of stock have been proposed. Reactive transshipments respond to shortages at a location by moving inventory from elsewhere within the network, while proactive stock redistribution seeks to minimize the chance of future stockouts. This paper is the first to propose an enhanced reactive approach in which individual transshipments are viewed as an opportunity for proactive stock redistribution. We adopt a quasi-myopic approach to the development of a strongly performing enhanced reactive transshipment policy. In comparison to a purely reactive approach to transshipment, service levels are improved while a reduction in safety stock levels is achieved. The aggregate costs incurred in managing the system are significantly reduced, especially so for large networks. Moreover, an optimal policy is determined for small networks and it is shown that the enhanced reactive policy substantially closes the gap to optimality.     

An index heuristic for transshipment decisions in multi-location inventory systems based on a pairwise decomposition

In multi-location inventory systems, transshipments are often used to improve customer service and reduce cost. Determining optimal transshipment policies for such systems involves a complex optimisation problem that is only tractable for systems with few locations. Consequently simple heuristic transshipment policies are often applied in practice. This paper develops an approximate solution method which applies decomposition to reduce a Markov decision process model of a multi-location inventory system into a number of models involving only two locations. The value functions from the subproblems are used to estimate the fair charge for the inventory provided in a transshipment. This estimate of the fair charge is used as the decision criterion in a heuristic transshipment policy for the multi-location system. A numerical study shows that the proposed heuristic can deliver considerable cost savings compared to the simple heuristics often used in practice.     

Optimal Solutions in the Multi-location Inventory System with Transshipments

We consider a single-period multi-location inventory system where inventory choices at each location are centrally coordinated. Transshipments are allowed as recourse actions in order to reduce the cost of shortage or surplus inventory after demands are realized. This problem has not been solved to optimality before for more than two locations with general cost parameters. In this paper we present a simple and intuitive model that enables us to characterize optimal inventory and transshipment policies for three and four locations as well. The insight gained from these analytical results leads us to examine the optimality conditions of a greedy transshipment policy. We show that this policy will be optimal for two and three locations. For the n location model we characterize the necessary and sufficient conditions on the cost structure for which the greedy transshipment policy will be optimal.      

Managing transshipments in a multi-retailer system with approximate policies

Using the right transshipment policy is important when transshipments are exercised under demand uncertainty. Optimal transshipment policy can be quite complex in a multi-firm system as optimal actions depend on all system variables. Moreover, both how to select requested retailer and how to respond to requests are in question. We introduce simple, close-to-optimal heuristic transshipment policies for multiple retailers. We first show that heuristic policies may perform even better than self-optimal policy, which is explained by Braess’s paradox. Then we test the performances of various heuristics with respect to centrally optimal policy. When retailers can observe others’ inventory levels, more effective transshipments can be made. Otherwise, a random selection performs quite well. We also observe that although always-accept respond policy is quite close to centrally optimal in small systems, the performance of pairwise-optimal holdback levels to respond requests is more clear and consistent for larger systems.     

A Semi-Markov decision problem for proactive and reactive transshipments between multiple warehouses

Lateral transshipments are an effective strategy to pool inventories. We present a Semi-Markov decision problem formulation for proactive and reactive transshipments in a multi-location continuous review distribution inventory system with Poisson demand and one-for-one replenishment policy. For a two-location model we state the monotonicity of an optimal policy. In a numerical study, we compare the benefits of proactive and different reactive transshipment rules. The benefits of proactive transshipments are the largest for networks with intermediate opportunities of demand pooling and the difference between alternative reactive transshipment rules is negligible.     

Transshipment incentive contracts in a multi-level supply chain

Transshipments within a supply chain can be difficult to implement as the costs and benefits are often incurred by different parties. This difficulty becomes even more problematic when the costs and benefits are not completely known by all parties. The primary purpose of this paper is to introduce the role of asymmetric information into the design of supply chain transshipment contracts. Using a representative supply chain from within the soft drink industry as an example, a multi-level contracting framework is developed that aligns incentives to encourage transshipments and improve performance in the absence of all parties having full information. Analysis of the proposed framework suggests that, even if a transshipment is likely to be unprofitable to the transshipping dyad, it may still be best for the entire supply chain. Moreover, overall supply chain inventories with transshipments do not necessarily increase relative to the no-transshipment case.     

Heuristics for assessing emergency transshipments

It is not uncommon for organizations with multiple stock-keeping facilities to have some locations facing shortages while others have excess inventory. This paper examines emergency transshipments of product as an approach to solve this problem. Specifically, two heuristics are developed to assist managers in determining when stock transfers should be made. The heuristics produce critical values for on-hand inventory levels above which transshipments minimize overall expected costs. They also provide insight into the factors affecting transshipment usage.     

Location and allocation for distribution systems with transshipments and transportion economies of scale

Locating transshipment facilities and allocating origins and destinations to transshipment facilities are important decisions for many distribution and logistic systems. Models that treat demand as a continuous density over the service region often assume certain facility locations or a certain allocation of demand. It may be assumed that facility locations lie on a rectangular grid or that demand is allocated to the nearest facility or allocated such that each facility serves an equal amount of demand. These assumptions result in suboptimal distribution systems. This paper compares the transportation cost for suboptimal location and allocation schemes to the optimal cost to determine if suboptimal location and allocation schemes can produce nearly optimal transportation costs. Analytical results for distribution to a continuous demand show that nearly optimal costs can be achieved with suboptimal locations. An example of distribution to discrete demand points indicates the difficulties in applying these results to discrete demand problems.     

Transshipment and rebalancing policies for library books

Library customers can soon order books online and specify a location to collect them from. Libraries exchange books between locations to meet these requests. Two types of exchanges take place: transshipments from library to library to fulfill the requests and rebalancing to redistribute books between libraries. This research determines optimal decisions for transshipments and rebalancing, so that logistic costs in the library system are minimized. In current practice, libraries typically send the book back to the original library after return. We consider a more general policy, in which we rebalance books in anticipation of demand. Moreover, we determine the optimal location from which to transship a book when it is unavailable at the location of demand. By means of stochastic dynamic programming, we derive the optimal policy for small instances. For larger instances we present two heuristics: the cluster and the expected shortage reduction (ESR) heuristic. The ESR heuristic proves to be near-optimal and significantly outperforms current practice.     

Multi-location transshipment problem with capacitated production

We consider coordination among stocking locations through replenishment strategies that explicitly take into account lateral transshipments, i.e., transfer of a product among locations at the same echelon level. Our basic contribution is the incorporation of supply capacity into the traditional transshipment model. Our goal is to analyze the impact on system behavior and on stocking locations’ performance of the fact that the supplier may fail to fulfill all the replenishment orders. We therefore formulate the capacitated supply scenario as a network flow problem embedded in a stochastic optimization problem, which is solved through a sample average approximation method. We find that, depending on the production capacity, system behavior can vary drastically. Moreover, in a production-inventory system, we find evidence that either capacity flexibility (i.e., extra production) or transshipment flexibility (i.e., pooling) is required to maintain a desired level of service.     

Inventory models with lateral transshipments: A review

Lateral transshipments within an inventory system are stock movements between locations of the same echelon. These transshipments can be conducted periodically at predetermined points in time to proactively redistribute stock, or they can be used reactively as a method of meeting demand which cannot be satisfied from stock on hand. The elements of an inventory system considered, e.g. size, cost structures and service level definition, all influence the best method of transshipping. Models of many different systems have been considered. This paper provides a literature review which categorizes the research to date on lateral transshipments, so that these differences can be understood and gaps within the literature can be identified.     

Optimal lateral transshipment policies for a two location inventory problem with multiple demand classes

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are expensive, ready–for–use spare parts are kept in stock to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival. Each stockpoint faces demands from multiple demand classes. We are interested in the optimal lateral transshipment policy. There are three ways in which a demand can by satisfied: from own stock, via a lateral transshipment, or via an emergency procedure. Using stochastic dynamic programming, we characterize and prove the structure of the optimal policy, that is, the policy for satisfying the demands which minimizes the average operating costs of the system. This optimal policy is a threshold type policy, with state-dependent thresholds at each stockpoint for every demand class. We show a partial ordering in these thresholds in the demand classes. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, two policies that are often assumed in the literature. Furthermore, we study several model extensions which fit in the same modeling framework.     

Inventory pooling of repairable spare parts with non-zero lateral transshipment time and delayed lateral transshipments

In this paper, we develop an analytical model to estimate several performance measures in a single-item, multi-company, repairable inventory system where complete pooling of stock is permitted among the companies. Compared to previous research, this paper is different in that both non-zero lateral transshipment time and delayed lateral transshipments are considered in the model. We model the system as a multi-dimensional Markovian problem and solve it using a two-stage solution method. The proposed method is very accurate and computationally efficient.     

Modelling replenishment and transshipment decisions in periodic review multilocation inventory systems

Effective models of key operational decisions in multilocation inventory systems are important for a successful retail sector. This paper argues that much of the existing research in this area is not applicable to a highly competitive retail environment, particularly if periodic review replenishment policies are used. The paper develops a model of a periodic review multilocation inventory system that is suitable for this environment and investigates the characteristics of optimal replenishment and transshipment decisions. This motivates the development of three simple heuristic transshipment policies that are practical for systems with many locations. The results of a numerical study involving systems with five locations suggest that the performance of these heuristic policies is often close to optimal and can be considerably better than the performance of commonly used policies.     

An analysis of p-median location problem: Effects of backup service level and demand assignment policy

Any solution to facility location problems will consider determining the best suitable locations with respect to certain criteria. Among different types of location problems, involving emergency service system (ESSs) are one of the most widely studied in the literature, and solutions to these problems will mostly aim to minimize the mean response time to demands. In practice, however, a demand may not be served from its nearest facility if that facility is engaged in serving other demands. This makes it a requirement to assign backup services so as to improve response time and service quality. The level of backup service is a key, strategic-level planning factor, and must be taken into consideration carefully. Moreover, in emergency service operations conducted in congested demand regions, demand assignment policy is another important factor that affects the system performance. Models failing to adopt sufficient levels of backup service and realistic demand assignment policies may significantly deteriorate the system performance.Considering the classic p-median problem (pMP) location model, this paper investigates the effects of backup service level, demand assignment policy, demand density, and number of facilities and their locations on the solution performance in terms of multiple metrics. For this purpose, we adopt a combined optimization and simulation approach. We will first modify the classic pMP to account for distances to backup services. Next, we employ a discrete event simulation to evaluate the performance of location schemes obtained from the deterministic mathematical model. Our results provide insights for decision-makers while planning ESS operations.     

An inventory model with returns and lateral transshipments

In this paper, we propose a single-item inventory model with returns. The model allows lateral transshipment of returns from one inventory system to another. Each inventory system is under continuous review and an (r, Q) policy is employed as the inventory control. An approximated closed-form solution of the system steady-state probability distribution is derived when Q is large. The approximated inventory cost and replenishment cost can be written in terms of this distribution. We show that the rejection rate of returns is reduced significantly when transshipment of returns is allowed between the inventory systems.     

Service differentiation through selective lateral transshipments

We consider a multi-item spare parts problem with multiple warehouses and two customer classes, where lateral transshipments are used as a differentiation tool. Specifically, premium requests that cannot be met from stock at their preferred warehouse may be satisfied from stock at other warehouses (so-called lateral transshipments). We first derive approximations for the mean waiting time per class in a single-item model with selective lateral transshipments. Next, we embed our method in a multi-item model minimizing the holding costs and costs of lateral and emergency shipments from upstream locations in the network. Compared to the option of using only selective emergency shipments for differentiation, the addition of selective lateral transshipments can lead to significant further cost savings (14% on average).     

Simple,efficient heuristics for multi-item multi-location spare parts systems with lateral transshipments and waiting time constraints

This paper deals with the analysis of a multi-item, continuous review model of a multi-location inventory system of repairable spare parts, in which lateral and emergency shipments occur in response of stock-outs. The objective is to determine close-to-optimal stocking policies minimizing the total cost for inventory holding, lateral transshipments, and emergency shipments subject to a target level for the average waiting times at all locations. We structure the optimization problem as a combinatorial problem and four different heuristics are developed and evaluated in terms of their total costs and computation times. It is shown that the greedy-type heuristic has the best performance. A numerical study is carried out to look at the relative cost savings obtained from the use of multi-item approach and lateral transshipments.     

Multi-item spare parts systems with lateral transshipments and waiting time constraints

This paper deals with the analysis of a multi-item, continuous review model of two-location inventory systems for repairable spare parts, used for expensive technical systems with high target availability levels. Lateral and emergency shipments occur in response to stockouts. A continuous review basestock policy is assumed for the inventory control of the spare parts. The objective is to minimize the total costs for inventory holding, lateral transshipments and emergency shipments subject to a target level for the average waiting time per demanded part at each of the two locations. A solution procedure based on Lagrangian relaxation is developed to obtain both a lower bound and an upper bound on the optimal total cost. The upper bound follows from a heuristic solution. An extensive numerical experiment shows an average gap of only 0.31% between the lower and upper bounds. The experiment also gives insights into the relative improvement achieved by applying lateral transshipments and or the system approach. We also apply the proposed model to actual data from an air carrier company.