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In many production/distribution systems, materials flow in fixed lot sizes (e.g., in full truckloads or full containers) and under regular schedules (e.g., delivery every week). In this paper, we study a multiechelon serial system with batch ordering and fixed replenishment intervals. We derive the optimal inventory control policy, provide a distribution-function solution for its optimal control parameters, and present an efficient algorithm for computing those parameters. Further, we show that the optimal expected system cost is minimized when the ordering times for all stages are synchronized. In contrast to the known approach in the literature that develops a lower bound for the average cost of a given period for the classical serial system, we develop a lower bound for the average total cost over an appropriately defined cycle and then construct a policy that reaches the lower bound. We also discuss its extension to the nonlinear shortage cost case (i.e., the nonlinear cost case). This paper generalizes several recent results on the analysis of multiechelon systems.

We study inventory control of serial supply chains with continuous, Markov-modulated demand (MMD). Our goal is to simplify the computational complexity by resorting to certain approximation techniques, and, in doing so, to gain a deeper understanding of the problem. First, we perform a derivative analysis of the problem’s optimality equations and develop general, analytical solution bounds for the optimal policy. This leads to simple-to-compute near-optimal heuristic solutions, which also reveal an intuitive relationship with the primitive model parameters. Second, we establish an MMD central limit theorem under long replenishment lead time through asymptotic analysis. We show that the relative errors between our heuristic and the optimal solutions converge to zero as the lead time becomes sufficiently long, with the rate of convergence being the square root of the lead time. Third, we show that, by leveraging the Laplace transform, the computational complexity of our heuristic is superior to the existing methods. Finally, we provide the first set of numerical study for serial systems under MMD. The numerical results demonstrate that our heuristic achieves near-optimal performance even under short lead times and outperforms alternative heuristics in the literature. In addition, we observe that, in an optimally run supply chain under MMD, the internal fill rate can be high and the demand variability propagating upstream can be dampened, both different from the system behaviors under stationary demand. The online appendix is available at https://doi.org/10.1287/opre.2017.1615 .

We address how to configure the supply chain for a new product for which the design has already been decided. The central question is to determine what suppliers, parts, processes, and transportation modes to select at each stage in the supply chain. There might be multiple options to supply a raw material, to manufacture or assemble the product, and to transport the product to the customer. Each of these options is differentiated by its lead time and direct cost added. Given these various choices along the supply chain, the configuration problem is to select the options that minimize the total supply chain cost. We develop a dynamic program with two state variables to solve the supply chain configuration problem for supply chains that are modeled as spanning trees. We illustrate the problem and its solution with an industrial example. We use the example to show the benefit from optimization relative to heuristics and to form hypotheses concerning the structure of optimal supply chain configurations. We conduct a computational experiment to test these hypotheses.

Two critical decisions must be made daily when managing multiechelon repair and distribution systems for service parts: (1) allocating available repair capacity among different items and (2) allocating available inventories to field stocking locations to support service operations. In many such systems, procurement lead times for service parts are lengthy and variable, repair capacity is limited, and operational requirements change frequently—resulting in demand processes that are highly uncertain and nonstationary. As a consequence, it is common to have many items in short supply while others are abundant. In such environments, integrated real-time decision-support tools can provide significant value by reducing the impact of inventory imbalances and responding appropriately to the volatile nature of the demand processes. By \"integrated\" and \"real-time,\" we mean (respectively) tools that simultaneously consider key aspects of the current state of the operating environment in deciding what items to repair, where to ship available units, and by what mode to ship them. In this paper, we develop an integrated real-time model for making repair and inventory allocation decisions in a two-echelon reparable service parts system. We formulate the decision problem as a finite-horizon, periodic-review mathematical program, we show it can be formulated as a large-scale linear program, and we develop a practical heuristic method for solving the problem approximately. By simulating the operation of a service parts supply chain, we demonstrate the value of employing integrated decision models over using separate repair and inventory allocation rules for a range of environments where inventory imbalances exist. We also show that our heuristic approach is highly effective and that its inventory allocation subroutine, used as a stand-alone tool for making distribution decisions, outperforms a commonly used inventory allocation rule in most circumstances tested.

The life cycle of new products is becoming shorter and shorter in all markets. For electronic products, life cycles are measured in units of months, with 6- to 12-month life cycles being common. Given these short product life cycles, product demand is increasingly difficult to forecast. Furthermore, demand is never really stationary because the demand rate evolves over the life of the product. In this paper, we consider the problem of where in a supply chain to place strategic safety stocks to provide a high level of service to the final customer with minimum cost. We extend our model for stationary demand to the case of nonstationary demand, as might occur for products with short life cycles. We assume that we can model the supply chain as a network, that each stage in the supply chain operates with a periodic review base-stock policy, that demand is bounded, and that there is a guaranteed service time between every stage and its customers. We consider a constant service time (CST) policy for which the safety stock locations are stationary; the actual safety stock levels change as the demand process changes. We show that the optimization algorithm for the case of stationary demand extends directly to determining the safety stocks when demand is nonstationary for a CST policy. We then examine with an illustrative example how well the CST policy performs relative to a dynamic policy that dynamically modifies the service times. In addition, we report on numerical tests that demonstrate the efficacy of the proposed solution and how it would be deployed.

We present two significant enhancements to the guaranteed-service (GS) model for multiechelon safety stock placement. First, we let each stage's expected inventory cost be a generalized nonconcave non-closed-form function of its incoming and outgoing service time. This allows the GS model to incorporate important phenomena such as variable stage times and nonnested review periods, which previous GS literature has not allowed. Second, we optimize the generalized cost GS model for directed acyclic networks, rather than assembly/distribution networks or trees. For the resulting NP-hard optimization problem, we present a provably optimal algorithm that runs within minutes for 29 chains from a data set of 38 real-world supply chains ranging from 8 to 2,025 stages. We also present two significantly faster yet near-optimal heuristics. One heuristic is motivated by the structure of the formulation's dual space, whereas the other heuristic simply terminates the optimization algorithm after a fixed number of iterations. As a performance benchmark, on the 38 chains, the first heuristic has an average optimality gap of approximately 1.1% and average run time of 88 seconds, whereas the second heuristic has an average optimality gap of 2.8% and an average run time of 5.9 seconds.

We present two significant enhancements to the guaranteed-service (GS) model for multiechelon safety stock placement. First, we let each stage's expected inventory cost be a generalized nonconcave non-closed-form function of its incoming and outgoing service time. This allows the GS model to incorporate important phenomena such as variable stage times and nonnested review periods, which previous GS literature has not allowed. Second, we optimize the generalized cost GS model for directed acyclic networks, rather than assembly/distribution networks or trees. For the resulting NP-hard optimization problem, we present a provably optimal algorithm that runs within minutes for 29 chains from a data set of 38 real-world supply chains ranging from 8 to 2,025 stages. We also present two significantly faster yet near-optimal heuristics. One heuristic is motivated by the structure of the formulation's dual space, whereas the other heuristic simply terminates the optimization algorithm after a fixed number of iterations. As a performance benchmark, on the 38 chains, the first heuristic has an average optimality gap of approximately 1.1% and average run time of 88 seconds, whereas the second heuristic has an average optimality gap of 2.8% and an average run time of 5.9 seconds.

This paper studies a periodic-review, serial inventory system in which echelon ( r , nQ , T ) policies are implemented. Under such a policy, each stage reviews its inventory in every T period and orders according to an echelon ( r , nQ ) policy. Two types of fixed costs are considered: one is associated with each order batch Q , and the other is incurred for each inventory review. The objective is to find the policy parameters such that the average total cost per period is minimized. This paper provides a method for obtaining heuristic and optimal policy parameters. The heuristic is based on minimizing lower and upper bounds on the total cost function. These total cost bounds, which are separable functions of the policy parameters, are obtained in two steps: First, we decompose the total cost into costs associated with each stage, which include a penalty cost for holding inadequate stock. Second, we construct lower and upper bounds for the penalty cost by regulating downstream policy parameters. To find the optimal solution, we further construct cost bounds for each echelon (a subsystem that includes a stage and all of its downstream stages) by regulating holding and backorder cost parameters. The echelon lower-bound cost functions, as well as the stage cost bounds, generate bounds for the optimal solution. In a numerical study, we find that the heuristic is near optimal when the ratio of the fixed cost to the holding cost at the most downstream stage is large. We also find that changing the optimal batch sizes may not affect the optimal reorder intervals or, equivalently, the delivery schedules under some conditions.

In this paper, we study the impact of coordinated replenishment and shipment in inventory/distribution systems. We analyze a system with multiple retailers and one outside supplier. Random demand occurs at each retailer, and the supplier replenishes all the retailers. In traditional inventory models, each retailer orders directly from the supplier whenever the need arises. We present a new, centralized ordering policy that orders for all retailers simultaneously. The new policy is equivalent to the introduction of a warehouse with no inventory that is in charge of the ordering, allocation, and distribution of inventory to the retailers. Under such a policy, orders for some retailers may be postponed or expedited so that they can be batched with other retailers' orders, which results in savings in ordering and shipping costs. In addition to the policy we propose for supplying inventory to the retailers, we also consider three other policies that are based on these well-known policies in the literature: (a) can-order policy, (b) echelon inventory policy, and (c) fixed-replenishment interval policy. Furthermore, we create a framework for simultaneously making inventory and transportation decisions by incorporating the transportation costs (or limited truck capacities). We numerically compare the performance of our proposed policy with these policies to identify the settings in which each policy would perform well.