Projected Inventory Level Policies for Lost Sales Inventory Systems: Asymptotic Optimality in Two Regimes

We consider the canonical periodic review lost sales inventory system with positive lead-times and stochastic i.i.d. demand under the average cost criterion. We introduce a new policy that places orders such that the expected inventory level at the time of arrival of an order is at a fixed level and call it the Projected Inventory Level (PIL) policy. We prove that this policy has a cost-rate superior to the equivalent system where excess demand is back-ordered instead of lost and is therefore asymptotically optimal as the cost of losing a sale approaches infinity under mild distributional assumptions. We further show that this policy dominates the constant order policy for any finite lead-time and is therefore asymptotically optimal as the lead-time approaches infinity for the case of exponentially distributed demand per period. Numerical results show this policy also performs superior relative to other policies

Efficient Emission Reduction Through Dynamic Supply Mode Selection

We study the inbound supply mode and inventory management decision making for a company that sells an assortment of products. Stochastic demand for each product arrives periodically and unmet demand is backlogged. Each product has two distinct supply modes that may be different suppliers or different transport modes from the same supplier. These supply modes differ in terms of their carbon emissions, speed, and costs. The company needs to decide when to ship how much using which supply mode such that total holding, backlog, and procurement costs are minimized while the emissions associated with different supply modes across the assortment remains below a certain target level. Since the optimal policy for this inventory system is highly complex, we assume that shipment decisions for each product are governed by a dual-index policy. This policy dynamically prescribes shipment quantities with both supply modes based on the on-hand inventory, the backlog, and the products that are still in-transit. We formulate this decision problem as a mixed integer linear program that we solve through Dantzig-wolfe decomposition. We benchmark our decision model against two state-of-the-art approaches in a large test-bed based on real-life carbon emissions data. Relative to our decision model, the first benchmark lacks the flexibility to dynamically ship products with two supply modes while the second benchmark makes supply mode decisions for each product individually rather than holistically for the entire assortment. Our computational experiment shows that our decision model can outperform the first and second benchmark by up to 15 and 40 percent, respectively, for realistic targets for carbon emission reduction

Condition-Based Production for Stochastically Deteriorating Systems: Optimal Policies and Learning

Production systems deteriorate stochastically due to usage and may eventually break down, resulting in high maintenance costs at scheduled maintenance moments. This deterioration behavior is affected by the system's production rate. While producing at a higher rate generates more revenue, the system may also deteriorate faster. Production should thus be controlled dynamically to trade-off deterioration and revenue accumulation in between maintenance moments. We study systems for which the relation between production and deterioration is known and the same for each system as well as systems for which this relation differs from system to system and needs to be learned on-the-fly. The decision problem is to find the optimal production policy given planned maintenance moments (operational) and the optimal interval length between such maintenance moments (tactical). For systems with a known production-deterioration relation, we cast the operational decision problem as a continuous-time Markov decision process and prove that the optimal policy has intuitive monotonic properties. We also present sufficient conditions for the optimality of bang-bang policies and we partially characterize the structure of the optimal interval length, thereby enabling efficient joint optimization of the operational and tactical decision problem. For systems that exhibit variability in their production-deterioration relations, we propose a Bayesian procedure to learn the unknown deterioration rate under any production policy. Our extensive numerical study indicates significant profit increases of our approaches compared to the state-of-the-art

A multi-item approach to repairable stocking and expediting in a fluctuating demand environment

We consider a single inventory location where multiple types of repairable spare parts are kept for service and maintenance of several different fleets of assets. Demand for each part is a Markov modulated Poisson process (MMPP). Each fleet has a target for the maximum expected number of assets down for lack of a spare part. The inventory manager can meet this target by stocking repairables and by expediting the repair of parts. Expedited repairs have a shorter lead time. There are multiple repair shops (or departments) that handle the repair of parts and the load imposed on repair shops by expedited repairs is constrained. A dual-index policy makes stocking and expediting decisions that depend on demand fluctuations for each spare part type. We formulate the above problem as a non-linear non-convex integer programing problem and provide an algorithm based on column generation to compute feasible near optimal solutions and tight lower bounds. We show how to use the MMPP to model demand fluctuations in maintenance and other settings, including a moment fitting algorithm. We quantify the value of lead time flexibility and show that effective use of this flexibility can yield cost reductions of around 25 percent

Fleet readiness: Stocking spare parts and high tech assets

We consider a maintenance shop that is responsible for the availability of a fleet of assets; e.g., trains. Unavailability of assets may be due to active maintenance time or unavailability of spare parts. Both spare assets and spare parts may be stocked in order to ensure a certain fleet readiness, which is the probability of having sufficient assets available for the primary process (e.g., running a train schedule) at any given moment. This is different from guaranteeing a certain average availability, as is typically done in the literature on spare parts inventories. We analyze the corresponding system, assuming continuous review and base stock control. We propose an algorithm, based on a marginal analysis approach, to solve the optimization problem of minimizing holding costs for spare assets and spare parts. Since the problem is not item separable, even marginal analysis is time-consuming, but we show how to efficiently solve this problem. Using a numerical experiment, we show that our algorithm generally leads to a solution that is close to optimal and that it is much faster than an existing algorithm for a closely related problem. We further show that the additional costs that are incurred when the problem of stocking spare assets and spare parts is not solved jointly can be significant. A key managerial insight is that typically the number of spare assets to be acquired is very close to a lower bound that is determined only by the active maintenance time on the assets. It is typically not cost-effective to acquire more spare assets to cover spare parts unavailability

Economies of scale in recoverable robust maintenance location routing for rolling stock

We consider the problem of locating maintenance facilities in a railway setting. Different facility sizes can be chosen for each candidate location and for each size there is an associated annual facility costs that can capture economies of scale in facility size. Because of the strategic nature of facility location, the opened facilities should be able to handle the current maintenance demand, but also the demand for any of the scenarios that can occur in the future. These scenarios capture changes such as changes to the line plan and the introduction of new rolling stock types. We allow recovery in the form of opening additional facilities, closing facilities, and increasing the facility size for each scenario. We provide a two-stage robust programming formulation. In the first-stage, we decide where to open what size of facility. In the second-stage, we solve a NP-hard maintenance location routing problem. We reformulate the problem as a mixed integer program that can be used to make an efficient column-and-constraint generation algorithm. To show that our algorithm works on practical sized instances, and to gain managerial insights, we perform a case study with instances from the Netherlands Railways. A counter intuitive insight is that economies of scale only play a limited role and that it is more important to reduce the transportation cost by building many small facilities, rather than a few large ones to profit from economies of scale

Design of multi-component periodic maintenance programs with single-component models

Capital assets, such as wind turbines and ships, require maintenance throughout their long lifetimes. Assets usually need to go offline to perform maintenance, and such downs can be either scheduled or unscheduled. Since different components in an asset have different maintenance policies, it is key to have a maintenance program in place that coordinates the maintenance policies of all components, to minimize costs associated with maintenance and downtime. Single-component maintenance policies have been developed for decades, but such policies do not usually allow coordination between different components within an asset. We study a periodic maintenance policy and a condition-based maintenance policy in which the scheduled downs can be coordinated between components. In both policies, we assume that at unscheduled downs, a minimal repair is performed to keep the unscheduled downtime as short as possible. Both policies can be evaluated exactly using renewal theory, and we show how these policies can be used as building blocks to design and optimize maintenance programs for multi-component assets

Expediting in Two-Echelon Spare Parts Inventory Systems

We consider a two-echelon spare parts inventory system consisting of one central warehouse and multiple local warehouses. Each warehouse keeps multiple types of repairable parts to maintain several types of capital goods. The local warehouses face Poisson demand and are replenished by the central warehouse. We assume that unsatisfied demand is backordered at all warehouses. Furthermore, we assume deterministic lead times for the replenishments of the local warehouses. The repair shop at the central warehouse has two repair options for each repairable part: a regular repair option and an expedited repair option. Both repair options have stochastic lead times. Irrespective of the repair option, each repairable part uses a certain resource for its repair. Assuming a dual-index policy at the central warehouse and base stock control at the local warehouses, an exact and efficient evaluation procedure for a given control policy is formulated. To find an optimal control policy, we look at the minimization of total investment costs under constraints on both the aggregate mean number of backorders per capital good type and the aggregate mean fraction of repairs that are expedited per repair resource. For this non-linear non-convex integer programming problem, we develop a greedy heuristic and an algorithm based on decomposition and column generation. Both solution approaches perform very well with average optimality gaps of 1.56 and 0.23 percent, respectively, across a large test bed of industrial size. Based on a case study at Netherlands Railways, we show how managers can significantly reduce the investment in repairable spare parts when dynamic repair policies are leveraged to prioritize repair of parts whose inventory is critically low

The Decay Vendor: Timing Medical Radioisotope Production to Meet a Fixed Delivery Schedule

Over 85% of nuclear medicine procedures are conducted with Technetium-99m (Tc-99m), a decay product of Molybdenum-99 (Mo-99). Mo-99 has a half-life of 66.7 hours and is primarily produced by irradiating uranium in a nuclear research reactor. Due to its continuous decay the alignment of the supply chain’s just-in-time processes are critical. We present a basic structure to model and analyze this unique type of perishable supply chain. We consider how to integrate a production timing decision with a fixed delivery schedule. The objective is to maximize the viable amount of Mo-99 that makes it to the delivery given a stochastic lead time and departure. We compare a naïve approach for determining a production time to a greedy heuristic and show significant improvement potential of 5% on average and up to 22% in our test bed. We also investigate how different delivery schedules allow more viable Mo-99 to reach marke

Base-stock policies for lost-sales models: Aggregation and asymptotics

This paper considers the optimization of the base-stock level for the classical periodic review lost-sales inventory system. The optimal policy for this system is not fully understood and computationally expensive to obtain. Base-stock policies for this system are asymptotically optimal as lost-sales costs approach infinity, easy to implement and prevalent in practice. Unfortunately, the state space needed to evaluate a base-stock policy exactly grows exponentially in both the lead time and the base-stock level. We show that the dynamics of this system can be aggregated into a one-dimensional state space description that grows linearly in the base-stock level only by taking a non-traditional view of the dynamics. We provide asymptotics for the transition probabilities within this single dimensional state space and show that these asymptotics have good convergence properties that are independent of the lead time under mild conditions on the demand distribution. Furthermore, we show that these asymptotics satisfy a certain ow conservation property. These results lead to a new and computationally efficient heuristic to set base-stock levels in lost-sales systems. In a numerical study we demonstrate that this approach performs better than existing heuristics with an average gap with the best base-stock policy of 0.01% across a large test-bed