Optimal allocation policy for a multi-location inventory system with a quick response warehouse

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Abstract

We study a multi-location inventory problem with a so-called quick response warehouse and base stock inventory control. In case of a stock-out at a local warehouse, a demand can be satisfied by a stock transfer from the quick response warehouse. We derive the optimal policy for when to apply such a stock transfer, as well as conditions under which that is always optimal. In a numerical experiment we compare the optimal allocation policy to simpler policies.

Introduction

In this paper we study a multi-location inventory model, with the special feature of a so-called Quick Response (QR) warehouse. When a local warehouse is out-of-stock, a demand can be satisfied by a stock transfer from this QR warehouse. This QR warehouse is situated at close distance to the local warehouses. Hence, by a stock transfer the demand is satisfied much faster than by an emergency shipment from a central warehouse (or from outside the network).

The main application of such a model is found in spare parts inventory networks, where ready-for-use parts are kept on stock for critical components of advanced technical systems. Examples of these include the key machines in a production line, trucks for a transportation company, medical equipment in hospitals, and printing systems in professional environments. Many break-downs of such systems can be solved by replacing a failed critical component by a spare part. Downtime of these systems is often very expensive because of loss of production/revenue, and therefore a timely delivery of the spare parts is required. If the requested spare part is on stock at a nearby local warehouse, then the part can be delivered within a short time. If this is not the case, alternative fast options have to be considered. By installing a quick response warehouse near multiple local warehouses, one obtains such an alternative fast option against a reasonable price.

We know of two companies where such quick response warehouses are used. Axsäter et al. [3] and Howard et al. [10] describe the use of quick response warehouses (referred to as ‘support warehouses’) at Volvo Parts Corporation, a global spare parts service provider. Rijk [17] describes the use of quick response stocks in the spare parts network of Océ, a manufacturer of professional printing systems. They have service contracts with many customers. These contracts imply that a high percentage (95%, say) of failures of printing systems at the customers has to be solved on the same business day. When a printing system fails, a service engineer is sent to that system and in many cases the failure can be solved by a replacement by a spare part of the engineers’ care stock. Because systems consist of many components that are subject to failures, it is impossible to carry all parts in the car stocks. Hence, in a significant percentage of cases, one uses spare parts supplied from a quick response stock by a courier (within two hours, say) and then the service engineer will still be able to solve the problem on the same business day.

Another application of the model with a QR warehouse is the combination of physical stores and an on-line shop, e.g. for books or fashion. Next to the physical stores, the on-line shop keeps items in inventory as well, located centrally. The demands of customers visiting the physical stores are satisfied immediately when there is stock on-hand, but their demands can be routed to the inventory of the on-line shop in case of a stock-out at the store. Hence, this on-line shop, which also has its own demand stream, acts as a QR warehouse.

In the above spare parts situations, one has multiple local warehouses and one QR warehouse, where many Stock-Keeping Units (SKU’s) are kept on stock. Demand rates are often low for (almost) all SKU’s. Further, the local warehouses and QR warehouse are supplied from a central warehouse by consolidated orders/shipments for all SKU’s together, and this is done once or multiple times per week, say. Because of the consolidated shipments, one has no or low fixed ordering costs per SKU and there is no reason to limit the number of orders per SKU. Hence, for the inventory control, often base stock policies are used; i.e., per warehouse and per SKU, the inventory position is increased up to a given base stock level at each ordering moment. The low demand rates and high ordering frequency imply that most order sizes are 0 or 1 (see also [3], [10], [12], [14], [17], [19], [21]). Further, we distinguish two planning levels (cf.[12], [21]):

  • 1.

    The tactical planning level: At this level, once per month or quarter, one generates forecasts for demand parameters and determines the base stock levels for all warehouses and SKU’s. Here, in order to keep decision support models tractable, it may be necessary to make simplifying assumptions for what happens at the operational planning level.

  • 2.

    The operational planning level: At this level, the base stock levels of the tactical planning level are taken as given and one decides per demand for a spare part on which option is used to satisfy the demand. A demand is preferably satisfied by the local warehouse where it occurs. If this is not possible, then a demand can be satisfied by a part of the QR warehouse (only if the QR warehouse has stock on hand) or via an emergency procedure that satisfies the demand from an external source. The policy for when the QR warehouse is used is denoted as the allocation policy.

For the tactical planning level, one may assume that always a stock transfer is applied when the local warehouse is out of stock and the QR warehouse has positive stock. Under this straightforward allocation policy, the decision on the base stock levels at the tactical level may be supported by inventory models with lateral transshipments (a stock transfer from the QR warehouse to fulfill a demand at a local warehouse is a lateral transshipment). There is rich literature on these models; see [15] for an overview. Of particular interest are models with a partial pooling structure; see e.g. [13].

In this paper, we focus on the allocation policy that is needed at the operational planning level. Our contribution is as follows. We formulate the problem as a Markov decision problem (MDP), with the objective to minimize the costs for the stock transfers from the QR warehouse and the use of the emergency procedure. Both cost factors may differ per local warehouse. Next, we use event-based dynamic programming (cf. [11]) to derive the optimal structure of the allocation policy. We derive also simple, sufficient conditions under which it is always optimal to satisfy a demand by a part from the QR warehouse if it has a part available. Furthermore, by means of a numerical study, we provide insights into the question when especially costs reductions can be achieved by the use of the dynamic optimal allocation policy, compared to a static allocation policy.

To the best of our knowledge, in the literature no results are available on the optimal allocation policy, i.e., the optimal use of the stock at the QR warehouse. Basically, our model is a combination of a so-called overflow model and a stock rationing model. Overflow models, in which unsatisfied demands are routed to another source, typically arise in telecommunication models, e.g. in call centers (see [8]). However, in these models, no costs are incorporated for the routing or blocking of demands, whereas we show these costs to play an important role in the optimal policy when incorporated. In stock rationing models (see [20] and the references therein), multiple demand classes are served from a single stock point. If demand processes are Poisson, the optimal policy for satisfying demands is a so-called critical level policy (see e.g. [22], [9], [5]), which prescribes a (net) stock level (the critical level) for each demand class from which on their demands are satisfied. However, the overflow demand streams in our model are not Poisson processes, and thus such a critical level policy fails to be optimal. In fact, an overflow demand stream is a special case of a Markov modulated Poisson process (MMPP, cf. [7]). The optimal policy depends on the states of each of these processes, i.e. on the stock levels at all local warehouses.

As stated above, our model also relates to spare parts models with lateral transshipments. In this stream, either an optimal policy (for the combined inventory and allocation problem) is derived for a two-location model [2], [24], or for a symmetric multi-location setting [18]. The results for two-location models cannot be extended by the techniques used to general multi-location models. We, however, do so for our model in which we only allow lateral transshipments from the QR warehouse to the local warehouses.

In our model, the stock transfers from the QR warehouse are applied in a reactive manner. One could also consider to apply them proactively, which may lead to a further performance improvement. This topic is related to proactive lateral transshipments (see [15] for an overview) and would be a good topic for future research.

The organization of the rest of the paper is as follows. In Section 2, we present our model and the MDP formulation. The structural results are derived in Section 3, where in the last part extensions for three model variations are discussed. The numerical results are presented in Section 4.

Section snippets

Model

The model that we consider is mainly motivated by spare parts applications as described in Section 1. Below, in Section 2.1, we first give the pure mathematical description of the allocation problem. Next, in Section 2.2, we discuss the main assumptions. Finally, the MDP formulation is given in Section 2.3.

Structural results

In this section we prove our main result: the structure of the optimal policy of the QR warehouse. For this, we first introduce the properties convexity and supermodularity. Each of the operators in the value function preserves these properties, hence the value function satisfies them. From this, the optimal policy structure is derived, as well as conditions under which it simplifies.

Numerical results

In a numerical study we show how much is to be gained by executing the optimal policy, compared to two simpler policies. For two examples, we vary the arrival rates and cost parameters, and compare the average costs per time unit of executing three possible policies: (i) the optimal policy, (ii) a naive policy always satisfying all demands, and (iii) a state-independent threshold policy with optimal thresholds, the so-called optimal critical level policy.

Under a critical level policy, for each

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