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Article Excerpt
We extend the guaranteed service, supply chain modeling framework to allow for an arbitrary, integer review period or ordering frequency at each stage. We define a notation for the cyclic inventory dynamics that review periods introduce and generalize inventory-balance equations to accommodate three different periodic-review operating policies--constant base stock, constant safety stock, and adaptive base stock. As a form of validation, we apply the model to the Celanese acetic acid supply chain and show that inventory metrics of the new model differ by more than 30 percent from those derived through the simpler modeling approach of aggregating a review period into lead time.
Key words: multiechelon; inventory system; safety stock; optimization; dynamic programming application; review periods.
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Leading-edge companies have integrated lean manufacturing and Six Sigma processes deeply into their organizations to reduce the total length and cost of their supply chains while maintaining or increasing service to their customers. Against this backdrop, firms are also outsourcing significant portions of their operations and integrating their processes more tightly into the delivery networks of their customers. An integrated and long-lasting supply chain improvement process increases inventory turns, increases return on assets, and decreases cash-to-cash cycle times. These factors improve corporate performance. However, continued advances in lean initiatives and the increasing complexity of supply chains pose modeling challenges. In particular, these improvements increase the importance of correctly modeling operating times and policies across the supply chain; they raise the bar that a useful model must cross.
Periodic-review models make up a rich research stream within the field of inventory management; for a broad survey of such models, we refer the reader to Federgruen (1993). However, the complexity that review periods introduce seems to have limited results for chains with stage-dependent review periods to specialized systems. Papers illustrative of this work include Graves (1996) and van Houtum et al. (2003). Graves (1996) develops a computationally intensive, exact evaluation approach of inventory levels in a distribution system facing demands with independent increments. Van Houtum et al. (2003) prove that a base-stock policy is optimal for a serial-line network with nested review periods.
Developing a model to optimize inventory levels and locations across a supply chain in the presence of review periods entails trading off exactness and tractability. Our approach is similar in spirit to that of Lee and Billington (1993), who develop a decentralized model to set inventory levels across a multiechelon supply chain subject to demand and supply variability. To make their model tractable, the authors richly characterize a single-stage inventory model; however, they assume that the demand process each stage faces is an allocation of the end-item demands that the stage satisfies.
In this paper, we extend the discrete-time, supply chain modeling framework that Simpson (1958) originally described to allow for an arbitrary, integral review period at each stage of a chain as well as different inventory policies. Simpson (1958) defined the guaranteed service modeling (GSM) framework for a serial-line and distribution network. Graves and Willems (2000), which we hereafter refer to as GW, extended the framework to supply chains that are modeled as spanning trees; they formulated a deterministic, dynamic program to optimize the spanning tree models. Humair and Willems (2006) further generalized the network structure to so-called clusters of commonality. Application of this modeling approach at Hewlett-Packard was a 2003 Edelman Prize finalist (Billington et al. 2004). All previous GSM work assumes a single underlying review period that is common to all stages. We summarize the GSM framework in Appendix 1--Reviewing the GSM Framework.
Although single-stage models often readily accommodate review periods, stage interactions can greatly complicate multiechelon models. First, review periods complicate demand propagation. A stage that reviews periodically typically orders periodically and so generates intermittent demand. Intermittent or more general, nonstationary incoming demand, combined with periodic review, compounds the complication. Although nested review periods effectively negate intermittency and seem broadly appropriate, they do not always appear in practice, as the real-world example in the Application at Celanese section illustrates. Without nesting, one must account for not just review-period lengths but also for staggering. A stage that orders every two days must distinguish weekly demand originating on Mondays from weekly demand originating on Fridays. In addition, many different ordering policies exist. A stage might always order up to a fixed, precalculated base-stock target. This case seems most straightforward, and we consider it first. Alternatively, a stage might order to maintain a constant safety-stock level, choose a fully adaptive base-stock policy, or even smooth demand. Ordering behavior affects inventory dynamics at the stage in question and further complicates demand propagation. In the Extension for General Review Periods section, we describe the models and briefly generalize the dynamic GW program to accommodate review periods.
The Application at Celanese section demonstrates the importance of review periods by presenting the successful application of this model at Celanese, a $6 billion chemical company. Celanese and the chemicals industry in general encompass a host of review-period variations that are often too critical to ignore. Boats operating under fixed schedules transport many raw materials and finished goods. Customers are assigned specific days to order each week, and some distribution centers review at different frequencies. Sometimes requirements are simply transmitted monthly, and sometimes they are smoothed over the monthly review cycle. Finally, the capital intensity of the business makes cyclic schedules commonplace. Our modeling framework addresses each of these issues. Although we focus on the application at Celanese, we have integrated our review-periods research into the Optiant PowerChain software application. More than a dozen Fortune 500 companies, including Black and Decker, Boston Scientific, Hewlett-Packard, Honeywell, Intel, and Procter & Gamble, have applied it.
We offer some conclusions in the Conclusion section.
Extension for General Review Periods
Extending the GSM framework to include review periods involves two primary complications--characterizing internal demand streams and generalizing the inventory-balance equation. In a single-stage setting (Hadley and Whitin 1963), the time interval of interest is the order cycle that elapses between consecutive orders or consecutive replenishments. The multiechelon setting involves three cycles at each stage. The order cycle still exists and still equals the review period. However, this cycle operates in concert with two additional cycles. First, incoming demand might be intermittent or more generally cyclic, and the cycle length of inbound demand depends on downstream review periods. A third cycle governs inventory dynamics at the stage itself as well as outgoing demand transmission to its suppliers. We assume that the review period defines the frequency with which a stage places demands on its suppliers or, more generally, modifies its ordering behavior. In addition, the suppliers receive demand information only through these orders, although they know their customers' inventory policies.
The Demand Propagation Under a Constant Base-Stock Target and Single-Stage Model sections develop the notation and inventory-balance equation for a single stage that resets its inventory position to a constant base-stock target in the presence of arbitrary review periods. Using two examples, the Example section illustrates the inventory dynamics that the balance equation implies. The Demand Bounds and Service-Level Targets section connects the demand bounds to service-level targets, and the Optimization section generalizes the GW dynamic program to account for review periods. The Adaptive Base-Stock and Constant Safety-Stock Policies and Demand Smoothing sections extend the analysis to adaptive base-stock targets and a particular version of smoothing.
Demand Propagation Under a Constant Base-Stock Target
If a stage has a constant base-stock target, its ordering process under review periods remains simple; each review period, it resets its inventory position to the target by ordering the demand incurred since it last reviewed. We denote the length of the stage j review period by [R.sub.j] and a corresponding offset by [[omega].sub.k] [member of] {0, 1, 2, ..., [R.sub.j] -1}. That is, stage j places orders at times [[omega[.sub.j] + n x [R.sub.j] for n = 0, 1, 2, .... Offsets permit discrimination among stages that, for example, review weekly but on different days. Although the external demand processes remain stationary, stage-dependent review periods make the internal demands cyclic, and we denote the length of the demand cycle that stage j faces by [[lambda.sup.in.sub.j]. That is, for integers n and some fixed time t, the demands that stage j faces at times t + n x [[lambda].sup.in.sub.j] are independent and identically distributed. Because stage j might not order every period, the cycle length of the demand process that stage j generates might differ from that of the demand process that it faces. In particular, this outgoing cycle length is the least common multiple of the incoming cycle length and stage j's review period. We denote the length of stage j's outbound demand cycle by [[lambda].sup.out.sub.j] = LCM([[lambda].sup.in.sub.j], [R.sub.j]). Similarly, the inbound demand-cycle length at stage j is the least common multiple of the outbound demand-cycle lengths generated by the stages immediately downstream from stage j. That is, [[lambda].sup.in.sub.j] = LCM({[[lambda].sup.out.sub.j] | k: (j, k) [member of] A}). We can calculate...
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