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.      

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.     

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.     

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.     

Managing Dynamic Inventory Systems with Product Returns: A Markov Decision Process

This paper presents a Markov decision process for managing inventory systems with Markovian customer demand and Markovian product returns. Employing functional analysis, we prove the existence of the optimal replenishment policies for the discounted-cost and average-cost problems when demand, returns, and cost functions are of polynomial growth. Our model generalizes literature results by integrating Markovian demand, Markovian returns, and positive replenishment lead times. In particular, the optimality of the reorder point, order-up-to policies is proved when the order cost consists of fixed setup and proportional cost components and the inventory surplus cost is convex. We then make model extensions to include different cost components and to differentiate returned products from new ones. Finally, we derive managerial insights for running integrated closed-loop supply chains. At the aggregate level, returns reduce effective demand while many structural characteristics of inventory models are intact. A simple heuristic for managing systems with returns is to still utilize literature results without returns, but effective demand is lower than customer demand.     

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.     

Optimal transshipments and reassignments under periodic or cyclic holding cost accounting

In a centrally managed system, inventory at a retailer can be transshipped to a stocked-out retailer to meet demand. As the inventory at the former retailer may be demanded by future customers of that retailer and transshipment time/cost is non-negligible, it can be more profitable to not transship in some situations. When unsatisfied demand is backordered, reassignment of inventory to a previously backordered demand can perhaps become profitable as demand uncertainty resolves over time. Despite this intuition, we prove that no reassignments are necessary for cost optimality under periodic holding cost accounting in a two-retailer system. This remains valid for multi-retailer systems according to numerical analyses. When holding costs are accounted for only at the end of each replenishment cycle, reassignments are necessary for optimality but insignificant in reducing the total cost. In most instances tested, the decrease in total cost from reassignments is below 2% for end of cycle holding cost accounting. These results simplify transshipment policies and facilitate finding good policies in both implementation and future studies, as reassignments can be omitted from consideration in optimization models under periodic holding cost accounting and in approximation models under cyclical cost accounting.     

A new partial pooling structure for spare parts networks

Motivated by real-life spare parts networks, we introduce a new spare parts inventory model with lateral transshipment. We consider a multi-item, multi-location, single-echelon system with base stock control and aggregate mean waiting time constraints. The local warehouses are divided into two types: main and regular local warehouses. Lateral transshipment is allowed from main local warehouses only. A practical advantage of this structure is that only a limited number of local warehouses has to be equipped to provide lateral transshipment. This structure represents a new form of partial pooling, with no pooling (zero main locals) and full pooling (zero regular locals) as special cases. We develop an accurate and fast approximate evaluation method, and exploit this method in a heuristic procedure for the base stock level determination. We show that only a small number of main locals is sufficient to obtain most of the full pooling benefits. We also apply our methods to case data of ASML, an original equipment manufacturer in the semiconductor supplier industry. As a result of our work ASML was able to improve spare parts planning.     

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.     

The use of simple calibrations of individual locations in making transshipment decisions in a multi-location inventory network

Demands occur at each location in a network of stock-holding retail outlets. Should a location run out of stock between successive replenishments, then subsequent demands may be met either by transshipping from another location in the network or by an emergency supply from a central depot. We deploy an approximate stochastic dynamic programming approach to develop a class of interpretable and implementable heuristics for making transshipment decisions (whether and from where to transship) which make use of simple calibrations of the candidate locations. The calibration for a location depends upon its current stock, the time to its next replenishment and the identity of the location needing stock. A numerical investigation shows strong performance of the proposed policies in comparison with standard industry practice (complete pooling, no pooling) and a recently proposed heuristic. It points to the possibility of substantial cost savings over current practice.     

A semi-Markov model with holdout transshipment policy and phase-type exponential lead time

In this paper, a semi-Markov decision model of a two-location inventory system with holdout transshipment policy is reviewed under the assumption of phase-type exponential replenishment lead time rather than exponential lead time. The phase-type exponential lead time more closely approximates fixed lead time as the number of phases increases. Unlike past research in this area which has concentrated on the simple transshipment policies of complete pooling or no pooling, the research presented in this paper endeavors to develop an understanding of a more general class of transshipment policy. In addition, we propose an effective method to approximate the dynamic holdout transshipment policy.     

On the effect of centralisation on expected profits in a multi-location Newsboy problem

This paper is a multi-location, single-period, single-product inventory problem with an opportunity for centralisation. The decentralised model in which a separate inventory is kept at every location is compared to the centralised system in which the demands are satisfied from one central warehouse. The two systems are then compared when excess demands are penalised and lost, and in the more general case where a portion or all of the excess demand at a location may be reallocated among other locations with the remaining inventory. The revenue, salvage, cost, and penalty functions at each location are assumed to be identical. Expected profits are evaluated and conditions under which the centralised model outperforms the decentralised one are derived.     

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.     

Evaluation of cycle-count policies for supply chains with inventory inaccuracy and implications on RFID investments

Inventory record inaccuracy leads to ineffective replenishment decisions and deteriorates supply chain performance. Conducting cycle counts (i.e., periodic inventory auditing) is a common approach to correcting inventory records. It is not clear, however, how inaccuracy at different locations affects supply chain performance and how an effective cycle-count program for a multi-stage supply chain should be designed. This paper aims to answer these questions by considering a serial supply chain that has inventory record inaccuracy and operates under local base-stock policies. A random error, representing a stock loss, such as shrinkage or spoilage, reduces the physical inventory at each location in each period. The errors are cumulative and are not observed until a location performs a cycle count. We provide a simple recursion to evaluate the system cost and propose a heuristic to obtain effective base-stock levels. For a two-stage system with identical error distributions and counting costs, we prove that it is more effective to conduct more frequent cycle counts at the downstream stage. In a numerical study for more general systems, we find that location (proximity to the customer), error rates, and counting costs are primary factors that determine which stages should get a higher priority when allocating cycle counts. However, it is in general not effective to allocate all cycle counts to the priority stages only. One should balance cycle counts between priority stages and non-priority stages by considering secondary factors such as lead times, holding costs, and the supply chain length. In particular, more cycle counts should be allocated to a stage when the ratio of its lead time to the total system lead time is small and the ratio of its holding cost to the total system holding cost is large. In addition, more cycle counts should be allocated to downstream stages when the number of stages in the supply chain is large. The analysis and insights generated from our study can be used to design guidelines or scorecard systems that help managers design better cycle-count policies. Finally, we discuss implications of our study on RFID investments in a supply chain.     

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.     

Optimal ordering policies for periodic-review systems with replenishment cycles

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.     

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.     

A comparison of different inventory policies of the one-period multi-location inventory system

In this paper we compare different policies that must meet some demand requirement in a multi-center one-period inventory system. More specifically, this paper concerns a multi-location, one-period inventory problem with linear holding costs and a probability of out-of-stock constraint at each location. Such systems are common in the public sector. Three different policies are considered: (a) A decentralized policy, (b) a centralized policy without priorities, and (c) a centralized priority policy. The superiority of policy (c) over policies (a) and (b) is discussed and demonstrated for some specific examples. It is shown that this superiority is very sensitive to the size of the system.     

Optimal control of a dual service rate M/M/1 production-inventory model

We analyse a dual-source, production-inventory model in which the processing times at a primary manufacturing resource and a second, contingent resource are exponentially distributed. We interpret the contingent source to be a subcontractor, although it could also be overtime production. We treat the inventory and contingent sourcing policies as decision variables in an analytical study and, additionally, allow the primary manufacturing capacity to be a decision variable in a subsequent numerical study. Our goal is to gain insight into the use of subcontracting as a contingent source of goods and whether it can fulfill real-world managers' expectations for improved performance. We prove that a stationary, non-randomised inventory and subcontracting policy is optimal for our M/M/1 dual-source model and, moreover, that a dual base-stock policy is optimal. We then derive an exact closed-form expression for one of the optimal base stocks, which to our knowledge is the first closed-form solution for a dual-source model. We use that closed-form result to advantage in a numerical study from which we gain insight into how optimal capacity, subcontracting, and inventory policies are set, and how effectively a contingent source can reduce total cost, capacity cost, and inventory cost. We find that (i) the contingent source can reduce total cost effectively even when contingent sourcing is expensive and (ii) contingent sourcing reduces capacity cost more effectively than it does inventory cost.