Dual-index Policy Research Articles

Online Learning for Dual-Index Policies in Dual-Sourcing Systems

Problem definition: We consider a periodic-review dual-sourcing inventory system with a regular source (lower unit cost but longer lead time) and an expedited source (shorter lead time but higher unit cost) under carried-over supply and backlogged demand. Unlike existing literature, we assume that the firm does not have access to the demand distribution a priori and relies solely on past demand realizations. Even with complete information on the demand distribution, it is well known in the literature that the optimal inventory replenishment policy is complex and state dependent. Therefore, we focus our attention on a class of popular, easy-to-implement, and near-optimal heuristic policies called the dual-index policy. Methodology/results: The performance measure is the regret, defined as the cost difference of any feasible learning algorithm against the full-information optimal dual-index policy. We develop a nonparametric online learning algorithm that admits a regret upper bound of [Formula: see text], which matches the regret lower bound for any feasible learning algorithms up to a logarithmic factor. Our algorithm integrates stochastic bandits and sample average approximation techniques in an innovative way. As part of our regret analysis, we explicitly prove that the underlying Markov chain is ergodic and converges to its steady state exponentially fast via coupling arguments, which could be of independent interest. Managerial implications: Our work provides practitioners with an easy-to-implement, robust, and provably good online decision support system for managing a dual-sourcing inventory system. Funding: This work was supported by the Amazon Research Award. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0323 .

Efficient emission reduction through dynamic supply mode selection

Reducing the carbon footprint of global supply chains is a challenge for many companies. Governmental emission regulations are increasingly stringent, and consumers are increasingly environmentally conscious. Companies should therefore integrate carbon emissions in their supply chain decision making. In this paper, 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 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 target level. We assume that shipment decisions for each product are governed by a dual-index policy for which we optimize the parameters. 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. Our computational experiment shows that our decision model can outperform the first and second benchmark by up to 15 and 40 percent, respectively, for moderate carbon emission reduction targets.

Dual sourcing inventory management with nonconsecutive lead times from a supply chain perspective: a numerical study.

We study a stochastic multi-period two-echelon dual sourcing inventory system where the buyer can source a product from two different suppliers: a regular and an expedited supplier. The regular supplier is a low-cost offshore supplier, whereas the expedited supplier is a responsive nearshore supplier. Such dual sourcing inventory systems have been well studied in the literature, mostly being solely evaluated from the buyer's perspective. Since the buyer's decisions have an impact on the supply chain profit, we adopt the perspective of the entire supply chain, i.e., by taking the suppliers explicitly into consideration. In addition, we study this system for general (nonconsecutive) lead times for which the optimal policy is unknown or very complex. We numerically compare the performance of two different policies in a two-echelon setting: the Dual-Index Policy (DIP) and the Tailored Base-Surge Policy (TBS). From earlier studies we know that when the lead time difference is one period, DIP is optimal from the buyer's perspective, but not necessarily from the supply chain perspective. On the other hand, when the lead time difference grows to infinity, TBS becomes optimal for the buyer. In this paper, we evaluate the policies numerically (under various conditions) and we show that from a supply chain perspective, TBS typically outperforms DIP at a limited lead time difference of a few time periods. Based on data collected from 51 manufacturing firms, the results of our paper imply for many supply chains with a dual sourcing setting that TBS quickly becomes a beneficial policy alternative, especially given its simple and appealing structure.

Dual sourcing models with stock-out dependent substitution

Companies use different criteria such as lead time, cost and quality to evaluate suppliers; often using multiple suppliers with the aim of reducing stockout risk. But in many industries there may be significant differences between the quality levels of different suppliers. Thus quality-sensitive companies may prefer an item from a primary supplier, but be forced to accept substitute products of lesser quality in case of a stock-out. Motivated by an example in the aviation industry, we introduce a Dual Sourcing problem With Stock-out dependent substitution (DSWS) which includes quality differences. Due to nonconvexity of the multi-period model, analytical characterization of the optimal policy appears intractable. To overcome this problem, we prove a relation between the optimal cost of DSWS and costs of three other problems - dual sourcing without substitution and single sourcing problems with and without backlogging. This leads us to propose the use of the dual index policy (and a variant) as heuristics for DSWS, and to develop an algorithm for parameter optimization of our heuristics. Extensive numerical experiments show that the dual index policy outperforms all other candidate solutions from the literature by at least 8%. Our experiments show that the utilization of the back-up supplier leads to substantial cost savings and service rate increase, especially in case of high differences between holding cost rates of different quality items.

1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing systems. Recent numerical results suggest that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation. In “1.79-Approximation Algorithms for Continuous Review Single-Sourcing Lost-Sales and Dual-Sourcing Inventory Models,” the author provides a theoretical foundation for this phenomenon in two classical inventory models. First, in a continuous review lost-sales model with lead times and Poisson demand, he proves that a capped base-stock policy has a worst-case performance guarantee of 1.79 by conducting an asymptotic analysis under a large penalty cost and lead time. Second, in a more complex continuous review dual-sourcing model with general lead times and Poisson demand, he proves that a similar capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. The results provide a deeper understanding of the superior numerical performance of capped policies and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

Technical Note—Understanding the Performance of Capped Base-Stock Policies in Lost-Sales Inventory Models

Single-sourcing lost-sales inventory systems with lead times are notoriously difficult to optimize. In this paper, we propose a new family of capped base-stock policies and provide a new perspective on constructing a practical hybrid policy combining two well-known heuristics: base-stock and constant-order policies. Each capped base-stock policy is associated with two parameters: a base-stock level and an order cap. We prove that for any fixed order cap, the capped base-stock policy converges exponentially fast in the base-stock level to a constant-order policy, providing a theoretical foundation for a phenomenon by which a capped dual-index policy converges numerically to a tailored base-surge policy recently observed in other work in a different but related dual-sourcing inventory model. As a consequence, there exists a sequence of capped base-stock policies that are asymptotically optimal as the lead time grows. We also numerically demonstrate its superior performance in general (including small lead times) by comparing it with otherwell-known heuristics.

Comparing policies for the stochastic multi-period dual sourcing problem from a supply chain perspective

We study a supply chain that consists of a buyer and two suppliers. The buyer faces stochastic demand and has two different supply sources for the same product: a slower regular supplier and a more expensive expedited supplier. Such dual sourcing inventory systems have been widely studied in the literature, evaluating what is best for the buyer. As the decisions of the buyer may adversely affect the costs of the suppliers, and also, potentially, the supply chain profit, we adopt the perspective of the entire supply chain. We compare the performance of two different policies in a multi-period and two-echelon setting: the Dynamic Order Policy (DOP) and the Standing Order Policy (SOP). In the long run, the DOP policy converges to the Dual-Index Policy (DIP), which is optimal for the buyer in the case of a lead time difference of one period. On the other hand, the SOP policy is appealing from a practical perspective as a fixed (standing) quantity is ordered from the regular supplier in each period. We show that from our supply chain perspective, the DOP policy is not necessarily the better performing policy, and we define the conditions for the preferred policy. We evaluate the policies numerically under various conditions to obtain relevant managerial insights. We find that the preferred policy from a supply chain perspective is mainly the result of a trade-off between responsiveness, flexibility, and cost. Flexibility appears to be valuable if there is a substantial cost difference between the suppliers.

A 1.79-approximation Algorithm for a Continuous Review Lost-Sales Inventory Model

Stochastic inventory systems with lead times are often challenging to optimize, including single-sourcing lost-sales and dual-sourcing inventory systems. Recent numerical results have suggested that capped policies demonstrate superior performance over existing heuristics. However, the superior performance lacks a theoretical foundation, and why such policies generally perform so well remains a major open question. In this paper, we provide a theoretical foundation for this phenomenon in two classical inventory models with lead times. First, in a continuous review lost-sales inventory model with lead times and Poisson demand, we prove that a so-called capped base-stock policy has a worst-case performance guarantee of 1.79, by conducting an asymptotic analysis under a large penalty cost and lead time. Second, we extend the analysis to a more complex continuous review dual-sourcing inventory model with general lead times and Poisson demand, and also prove that a so-called capped dual-index policy has a worst-case performance guarantee of 1.79 under large lead time and ordering cost differences. Our results provide a deeper understanding of the superior numerical performance of capped policies, and present a new approach to proving worst-case performance guarantees of simple policies in hard inventory problems.

Understanding the Performance of Capped Base-Stock Policies in Lost-Sales Inventory Models

Single-sourcing lost-sales inventory systems with lead times are notoriously difficult to optimize. In this work, we propose a new family of capped base-stock policies and provide a new perspective of constructing a practical hybrid policy combining two well-known heuristics: base-stock and constant-order policies. Each capped base-stock policy is associated with two parameters: a base-stock level and an order cap. We prove that for any fixed order cap, the capped base-stock policy converges exponentially fast in the base-stock level to a constant-order policy, providing a theoretical foundation for a phenomenon that a capped dual-index policy converges numerically to a Tailored Base-Surge policy recently observed in Sun and Van Mieghem (2019) in a different but related dual-sourcing inventory model. As a consequence, there exists a sequence of capped base-stock policies that are asymptotically optimal as the lead time grows. We also numerically demonstrate its superior performance in general (including small lead times) by comparing with other well-known heuristics.

Dual-index Policies for Serial Systems with Dual Delivery Modes and Batch Orders

In this paper, we consider a stochastic serial inventory system with dual delivery/supply modes for each echelon stage. The two modes in each stage take different lead times, incur different ordering costs and require different base order batch sizes. We study a class of dual-index echelon-(R,nQ) policies which is a nature extension of the well-studied dual-index policies (Veeraraghavan and Scheller-Wolf 2008, Zhou and Yang 2016) and echelon-(R,nQ) policy (Chen 2000) that have been proved to be near-optimal/optimal for the single-echelon dual-sourcing system and the batch-ordering serial system, respectively. Based on the concept of strong Q-jump-convexity, we prove that this policy is indeed optimal for the case in which the lead time difference between the two delivery modes in each stage is one period and the order batch sizes satisfy an integer-ratio constraint. For the general case, we analyze the system dynamics under this policy and show separability results for the policy parameters. Based on these properties, we provide a simulation-based optimization procedure for policy parameters. Three simple heuristics are then provided to find the near-optimal policy parameters, which are shown to be very efficient by the numerical studies.We also numerically investigate the value of the dual delivery modes and observe that as long as the regular delivery mode of the upstream echelon is only marginally less favorable than the regular delivery mode of the downstream echelon, the downstream echelon should be given priority by firms thinking of adding an expedited mode to the system.

Managing a Hybrid RDC-DC Inventory System

In this paper, we study a hybrid RDC-DC serial inventory system where the RDC replenishes its stock from an outside supplier, while the DC faces random demand and replenishes its stock from the RDC. However, unlike in the traditional serial system, the DC itself can replenish its inventory from the outside supplier as well. We propose two simple and easy-to-implement policies for the system. The first policy, which we call the three-index policy, combines the characteristics of the echelon-base-stock policy for the serial system (Clark and Scarf 1960) and the dual-index policy for the dual-sourcing system (Veeraraghavan and Scheller 2008). We show that the order-up-to level of the DC from the RDC can be computed by a newsboy fractile. A simulation-based optimization procedure for the policy is provided. We then develop another policy (the ALP policy) based on the three-index policy and the multimodularity of the problem proved by Sapra (2017). The policy applies the linear programming approach to approximately solve the value function of the dynamic programming formulation of the problem. Our numerical results show that both the three-index policy and the ALP policy are comparable to the policy computed via dynamic programming, and that the former performs slightly better. In addition, the outside supply option of the DC can draw considerable cost savings under both policies. We also conduct a numerical study with problem parameters calibrated by the actual data from a consumer goods company in China to glen insights on the management of the system.

Robust Dual Sourcing Inventory Management: Optimality of Capped Dual Index Policies and Smoothing

Problem definition: How to dynamically replenish inventory from two supply sources or shipping modes with general lead times. The fast source is more expensive than the slow source. Academic/practical relevance: Dual sourcing provides supply chain flexibility to mitigate demand and supply risk. Despite its relevance in practice, characterizing the optimal dual sourcing policy is extremely challenging, and the optimal stochastic policy for nonconsecutive lead times has been unknown for over 50 years. Methodology: We present and solve a robust rolling horizon model for periodically reviewed dual sourcing inventory systems that minimizes the total cost of purchasing, inventory holding, and backlogging. Results: We prove that the optimal robust policy is a dual-index, dual-base-stock policy that constrains or “caps” the slow order and provide closed-form expressions for the three control parameters. The optimality result is established for general lead times and can accommodate nonstationary demand. Our numerical study shows that, as the lead time difference grows, the capped dual index policy increasingly smooths slow orders and for stationary demand, converges to the tailored base-surge policy, which places a constant slow order and has been shown to be asymptotically optimal. Managerial implications: Our capped dual index policy is easy to understand, explain, and implement in practice. In an extensive simulation study, the capped dual index policy performs as well as and can even outperform the best heuristics presented in the stochastic inventory literature.

Robust Dual Sourcing Inventory Management: Optimality of Capped Dual Index Policies

(1) Problem Definition: How to dynamically replenish inventory from two supply sources or shipping modes with general lead times. The fast source is more expensive than the slow source. (2) Academic/Practical Relevance: Dual sourcing provides supply chain flexibility to mitigate demand and supply risk. Despite its relevance in practice, characterizing the optimal dual sourcing policy is extremely challenging, and the optimal stochastic policy for non-consecutive lead times has been unknown for over 50 years. (3) Methodology: We present and solve a robust rolling horizon model for periodically reviewed dual sourcing inventory systems that minimizes the total cost of purchasing, inventory holding and backlogging. (4) Results: We prove that the optimal robust policy is a dual index, dual base-stock policy that constrains or “caps” the slow order and provide closed-form expressions for the three control parameters. The optimality result is established for general lead times and can accommodate non-stationary demand. Our numerical study shows that as the lead time difference grows, the capped dual index policy increasingly smooths slow orders and, for stationary demand, converges to the tailored base surge policy, which places a constant slow order and has been shown to be asymptotically optimal. (5) Managerial Implications: Our capped dual index policy is easy to understand, explain and implement in practice. In an extensive simulation study, the capped dual index policy performs as well as, and can even outperform, the best heuristics presented in the stochastic inventory literature.

Continuous-Review (R, nQ) Policies for Inventory Systems with Dual Delivery Modes

We study a continuous-review, infinite-horizon inventory system with compound Poisson demand and dual sourcing/delivery modes. Ordering from either source/mode incurs a fixed cost and the expedited mode provides a shorter lead time than the regular mode. As the optimal ordering policy is unknown, while expected to be very complicated, we propose a class of simple policies called single-index (R, nQ) policies—when ordering from each mode, based on the inventory position, the system follows an (R, nQ)-type of policy. We provide an exact procedure to compute the expected long-run average cost. Specifically, we first analyze the steady-state distribution of the inventory position, which is found to be no longer uniform in general as in the classic (R, nQ) inventory system where only one delivery mode is available. We then develop a recursive procedure, which overcomes the order-crossing effect, to determine the steady-state distribution of the inventory level. Two simple heuristics for computing near-optimal policy parameters are provided. For a special case where ordering from the regular mode incurs no fixed cost and follows a base-stock policy, we derive closed-form solution bounds by applying normal approximation. To assess the performance of the single-index (R, nQ) policy, we further study a more complicated class of policies called dual-index (R, nQ) policies and numerically illustrate that the simpler single-index policy performs close to the dual-index policy. Finally, the performance of the single-index policy is also shown comparable to the policy computed via dynamic programming.

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.

Dual index production and subcontracting policies for assemble-to-order systems

ABSTRACTWe analyze tradeoffs related to production and subcontracting decisions in an assemble-to-order system with capacity constraints and stochastic lead times. We assume that component replenishment is carried out by orders to a subcontractor and component stock levels at the manufacturer are determined by dual index-based policies. Furthermore, customer demands for the final product are immediately satisfied if all of the required components are in stock; otherwise, they are back-ordered. In order to maintain high service levels, the manufacturer reserves the option to produce components internally. Using queuing models, we analyze the tradeoffs related to internal manufacturing versus subcontracting under different types of dual index policies. We use Matrix-Geometric methods to conduct an exact analysis for an assemble-to-order system with two components and develop a decomposition-based algorithm to analyze the performance of systems with more than two products. Numerical studies provide useful insights on the performance of the various dual index policies under study.

Analysis of a two-echelon inventory system with two supply modes

In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We extend results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and illustrate that dual-sourcing can lead to significant cost savings in cases with high demand uncertainty, high backlogging cost or long lead times.

Efficient optimization of the dual-index policy using Markov chains

This article considers the inventory control of a single product in one location with two supply sources facing stochastic demand. A premium is paid for each product ordered from the faster “emergency” supply source. Unsatisfied demand is backordered and ordering decisions are made periodically. The optimal control policy for this system is known to be complex. For this reason a type of base-stock policy known as the Dual-Index Policy (DIP) is used as the as control mechanism for this inventory system. Under this policy ordering decisions are based on a regular and an emergency inventory position and their corresponding order-up-to levels. Previous work on this policy assumes deterministic lead times and uses simulation to find the optimal order-up-to levels. This article provides an alternate proof for the result that separates the optimization of the DIP in two one-dimensional problems. An insight from this proof allows the model to be generalized to accommodate stochastic regular lead times and provide an approximate evaluation method based on limiting results so that optimization can be done without simulation. An extensive numerical study shows that this approach yields excellent results for deterministic lead times and good results for stochastic lead times.

New Policies for the Stochastic Inventory Control Problem with Two Supply Sources

We study an inventory system under periodic review in the presence of two suppliers (or delivery modes). The emergency supplier has a shorter lead-time than the regular supplier, but the unit price he offers is higher. Excess demand is backlogged. We generalize the recently studied class of dual index policies [Veeraraghavan, S., A. Scheller-Wolf. 2008. Now or later: Dual index policies for capacitated dual sourcing systems. Oper. Res. 56(4) 850–864] by proposing two classes of policies. The first class consists of policies that have an order-up-to structure for the emergency supplier. We provide analytical results that are useful for determining optimal or near-optimal policies within this class. This analysis and the policies we propose leverage our observation that the classical “lost sales inventory problem” is a special case of this problem. The second class consists of policies that have an order-up-to structure for the regular supplier. Here, we derive bounds on the optimal order quantity from the emergency supplier, in any period, and use these bounds for finding effective policies within this class. Finally, we undertake an elaborate computational investigation to compare the performance of the policies we propose with that of dual index policies. One of our policies provides an average cost-saving of 1.1% over the best dual index policy and has the same computational requirements. Another policy that we propose has a cost performance similar to the best dual index policy, but its computational requirements are lower.

A comparison of the constant-order and dual-index policy for dual sourcing

We analyze a single-product, periodic-review, stochastic demand inventory model with backorders and two supply options, a regular and a more expensive expedited one, with deterministic, offsetting lead times. Since the optimal policy for such a problem is generally unknown, several simpler policies have been proposed in the literature, e.g., the single-index (SIP), dual-index (DIP), or constant-order policy (COP). In previous research the DIP has been found to perform closely to the optimal policy in specific numerical settings. The COP, on the other hand, is very appealing from a practical point of view. We explore the relative cost performance of these two policies in a numerical study to gain insights into appropriate policy choices. We find that in settings, where dual sourcing is most valuable, the simpler COP delivers satisfactory results, sometimes even better ones than the DIP, and therefore represents a legitimate policy alternative for practical situations.