Production, Manufacturing and Logistics
Determining near optimal base-stock levels in two-stage general inventory systems

https://doi.org/10.1016/j.ejor.2013.07.025Get rights and content

Highlights

  • We consider a two-level general inventory system.

  • We propose an easily implementable, scalable decomposition heuristic for determining base-stock levels.

  • The base-stock levels identified by the heuristic are near optimal.

  • Considerable improvements can be obtained compared to common- sense heuristics.

Abstract

This paper proposes an easily implementable, scalable decomposition heuristic for determining near optimal base stocks in two-level general inventory systems. In this heuristic, the general system is decomposed into assembly systems—one for each end product. For these assembly systems, the base-stock levels are calculated separately, taking into account risk-pooling effects for the common components. Our numerical analyses yield two main insights: First, the base-stock levels determined by the heuristic are close-to-optimal. Second, considerable improvements can be obtained compared to common-sense heuristics.

Introduction

This paper considers a two-level general inventory system. Outside suppliers deliver components to a raw material warehouse. Customers’ demands are fulfilled from stocks of finished products. For replenishing these stocks, components are taken out of the warehouse and assembled. We have product-specific components as well as common components, which are included in the bill-of-materials of multiple end products. The assembly capacity is unlimited. The lead times for producing the finished product and for supply of components are deterministic and may be significant. There are no fixed costs for supply and assembly. Demands are stochastic. In case of shortages of finished products, demands are backordered. Both components and finished products can be held in stock at the expense of item-specific inventory holding costs. All problem parameters are stationary. The objective is to minimize the long-run average expected inventory and backorder costs.

This setting has a high practical relevance. Examples are the production of circuit boards (e.g., Grotzinger, Srinivasan, Akella, & Bollapragada, 1993) and of surgery sets (e.g., customer procedure trays at Paul Hartmann AG). Surgery sets consist of up to 60 different components. Customers can configure their sets on their own, thereby choosing between several thousands of different components. Often customers order their sets continuously during some years. Before a set can be sold, it has to be sterilized. Since sterilization and transport after assembly take at least 3 weeks (including quarantine time), stocks of finished products have to be built up. Capacities can often be regarded as unlimited in this context. Assembly personnel is extendable rather easily at short-term (e.g., through flexible working hours), and sterilization activities can be outsourced. As the (deterministic) sterilization and transport times are much larger than the assembly time, the assumption of deterministic production lead times seems appropriate in this context.

In practice, inventory management in such systems is often driven by simple heuristics without focus on cost-effectiveness. We believe that a major reason for this is the lack of scalable approaches which yield near optimal solutions and do not require heavy computation. Even the potential benefits from more sophisticated methods are largely unknown. Probably the most important topics in this context are the choice of the safety stocks and the strategy for component allocation. For component allocation, we assume a fixed and—in our opinion—favorable procedure. The main contribution of this paper is a new, simple heuristic for determining the safety stocks. We assume that the inventory levels of products and components are controlled using base-stock policies. Our heuristic decomposes the general system into separate assembly systems. For these assembly systems, the optimal base stocks can be determined using standard approaches. Among those is the procedure of Shang and Song (2003), which is the base for the heuristic developed by us. The main idea of our heuristic is to take the risk-pooling effects due to component commonality into account. When calculating the base stocks of the common components, the cumulated secondary demand for all their successors is considered. Apart from our new heuristic, we evaluate two “common-sense” heuristics in computational tests: level-by-level optimization and holding stocks of finished products only. Our heuristic performs well. In all of its variants, the resulting costs increased by less than 0.9–1.2% on the average, compared to the optimal solution from enumerating all combinations of base stocks between wide upper and lower limits. Compared to the common-sense heuristics, the improvement is considerable. The solutions have been more than 7.9% cheaper.

In the literature, effective heuristics for determining safety stocks in general inventory systems are rare. Despite the significant advances during the last years (e.g., Song and Yao, 2002, Benjaafar and ElHafsi, 2006, Song and Zhao, 2009, Lu and Song, 2005, Lu et al., 2010, Dogru et al., 2010, Nadar et al., 2012), this is also true for assemble-to-order (ATO) systems (see also Song & Zipkin, 2003 for this argument). ATO systems represent an important special case of the setting investigated in this paper: In ATO systems we have negligible lead times for assembly, such that finished products are not held in inventory. (A further related problem class is the repair kit problem (e.g., Smith, Chambers, & Shlifer, 1980). This problems share with ATO systems the assumption of zero finished product inventory since repair jobs cannot be done in advance. A main difference is that for component replenishment, the repairman undertakes a tour to a central depot. Thus, the repair kit models usually consider a single period or a periodic review policy with negligible component replenishment lead times.)

We are aware of two bodies of work dealing with scalable heuristics for general inventory systems (see also Graves and Willems, 2003, de Kok and Fransoo, 2003 for overviews). The first is the so-called “guaranteed-service model approach” (e.g., Inderfurth and Minner, 1998, Graves et al., 2000, Humair and Willems, 2006). Here a fixed external service time and upper bounded demand are assumed. Then each stage is able to quote a guaranteed, deterministic service time to its successors, which simplifies the determination of the safety stocks. The scope differs significantly from ours because the trade-off between inventory holding and backorder costs for finished products is not optimized by these models—this trade-off is exogenously predetermined by the amount of the external service time and the demand bound.

The second body corresponds to the “stochastic-service model approach”. On the one hand, this includes large-scale, generic supply chain models where the stochastic lead times between stages are approximated by a random variable, while assuming that at most one component can be out of stock (e.g., Ettl, Feigin, Lin, & Yao, 2000). This assumption is especially critical in assembly structures with many components. There, we have a significant probability for stockouts of more than one component at one instance of time. Our heuristic does not rely on this assumption.

On the other hand, some researchers propose heuristics particularly designed for inventory systems with an assembly focus. Chew, Lee, and Lau (2010) determine safety stocks using infinitesimal perturbation analysis and a steepest descent algorithm. This requires significant problem-specific computing, in contrast to the heuristic proposed here. de Kok and Visschers (1999) propose a decomposition algorithm. For the special case where different components have the longest cumulative lead times, they outline a concrete solution procedure. More general settings, like the one investigated in this paper, are not considered. In this special case, the system is decomposed into serial systems, like our approach. The solution procedure of de Kok and Visschers (1999), however, cannot be compared with ours since its objective is different: the minimization of inventory subject to a service-level constraint. However, there is one important analogy: De Kok and Visschers solve the serial systems separately. Since they do not take risk-pooling effects into account, however, their solutions will often become suboptimal. We investigated an analogous scenario in our computational tests of Section 4 (αi = 0). There, the degree of suboptimality ranged between 1.8% and 5.1% compared to the preferred variant of our heuristic.

This paper is organized as follows. Section 2 presents the model and the allocation procedure. The decomposition heuristic is described in Section 3. The computational study in Section 4 provides insights into the solution quality of the heuristic and the potential savings compared to common-sense heuristics. Section 5 concludes the paper.

Section snippets

Model and allocation procedure

Consider a two-level general inventory system. There are i = 1,  , m finished products. The products are assembled on resources with unlimited capacities using components j = 1,  , n. Denote the number of units of component j required to assemble product i by rji, where rji  0. The components are procured from outside suppliers with unlimited capacities. The lead times Li and Lj for assembly and supply are deterministic. Inventory holding is allowed for all items. Denote by hi and hj the (local)

Decomposition heuristic

This section provides the main contribution of this paper, the new decomposition heuristic. The basic idea of the heuristic is simple: To decompose the general system into assembly systems and to calculate the optimal base-stock levels for them. For each product, the decomposition generates an assembly system, which consists of this product and its predecessors (see Fig. 1 for an illustration). Based on that, there is a straightforward approach for calculating the base stocks for the original,

Outline

Two major questions for determining the theoretical and practical value of the heuristic are:

  • What is the optimality gap of the heuristic?

  • How relevant is it to have near optimal base stocks? How large are the savings compared to common-sense strategies?

We have conducted a computational study to obtain insights into these questions. There, we have evaluated 1664 test instances using simulation. These instances are based on different combinations of parameters regarding the system structure and

Conclusion

This paper introduced a new heuristic for determining near optimal base stocks in general two-level inventory systems. The heuristic is simple and scalable. It can be implemented easily (e.g., using the standard functionalities of a spreadsheet) with low computational effort. An important limitation of the heuristic is that it does not account for fixed costs. In our opinion, this is a beneficial area for future research.

Acknowledgements

We would like to thank Stefan Schmid, Jaroslav Novak, and Sabine Stabenow for discussions on safety stock planning as well as two anonymous referees for their helpful remarks.

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