Production, Manufacturing and Logistics
Age-based inventory control in a multi-echelon system with emergency replenishments

https://doi.org/10.1016/j.ejor.2017.08.057Get rights and content

Highlights

  • A new multi-echelon inventory model with emergency supply is analyzed.

  • The inventory decision policy is based on the ages of the units in the system.

  • A heuristic solution procedure is developed which seems to work very well.

  • The environmental impact of making emergency transshipments is evaluated.

Abstract

This paper deals with a two-echelon spare parts inventory system with one central warehouse and a number of local warehouses facing Poisson demand. Normally, the local warehouses replenish stock from the central warehouse. However, if a demand cannot be satisfied within a pre-determined time window, an emergency shipment is requested from an outside supplier. Moreover, as in practice, this outside emergency supplier may not have ample capacity at the moment when the order was requested. This means that it is not always possible to realize such an emergency shipment within the stipulated time window. In many cases there are service agreements between the spare parts provider and the customer, that state the penalty cost if a customer demand is not met within an acceptable time window. The cost structure of such penalties is typically non-linear, and in this setting we consider piecewise constant penalty costs. That is, in cases where it is not possible to realize an emergency shipment from the outside supplier, a significant fixed cost is incurred. From a sustainability perspective, we also consider the production waste that arises from production stops when the time window is exceeded, and unplanned production stops lead to waste of perishable raw materials. We quantify the expected total CO2 emissions related to production waste and transportations connected to emergency shipments. In particular we show that it is important to take a more holistic view on CO2 emissions, and not only consider emissions related to transportation (as in most of the related literature).

Introduction

Today, spare part management plays a key role in many technical systems (Basten & van Houtum, 2014). Examples of such systems include, e.g., the aircraft and the oil rig systems where spare parts must be replaced in a timely fashion in order to avoid costly downtimes at customer sites. What is considered a “timely fashion” is often stipulated through service agreements between the supplier and the customer. A typical service agreement (or contract) of this type often involves a pre-specified acceptable customer waiting time, where the supplier is obliged to pay a fixed penalty if the spare part is not delivered within the acceptable time limit.

The motivation of the research presented in this paper stems from collaboration with Tetra Pak TS, a global spare parts provider for packaging machines. The spare parts are distributed through a central warehouse, which in turn distributes to local sites near their customers. When a spare part for a packaging machine fails at a customer site it is crucial to replace the broken spare part with a new one within a pre-specified time frame. Otherwise, a whole production batch of the product must be discarded due to the perishable nature of the product. Of course, a wasted production batch has a negative impact on the costs, but also on the global environment in the form of emissions during the production of the batch. Hence, an important question in design and control of sustainable supply chain systems is how to enable reduction in total CO2 emissions (related to production waste and transportation) without compromising customer service or increasing total costs.

This paper extends the literature concerning multi-echelon inventory control in several directions. We consider a continuous review base-stock inventory system with one central warehouse and a number of local sites. A vast majority of such inventory control models focus on linear backorder costs, see e.g. Axsäter (2015). However, it is well known that in many practical cases the backorder cost (as a function of the customer waiting time) is typically not a linear function. As discussed above, as a consequence of the service agreement between the customer and the service provider, the backorder cost structure is often a non-linear function of the waiting time. In our model, we consider the scenario where a large fixed cost is incurred if the customer waiting time for a spare part exceeds a certain target level. Moreover, in addition to regular replenishments, we use real-time inventory pipeline information and consider the option of making an emergency replenishment from an outside supplier if the time to replace a demanded spare part, via the regular supply channel, is too long. However, in real life situations, it is not always possible to orchestrate an emergency shipment that will arrive within the committed service time. The possibility of not being able to initiate an emergency shipment when requested may occur in situations where the outside supplier has insufficient inventory or transport (such as air freight) capacity. In our model, we take this scenario into account by assuming that, with a certain fixed probability, an emergency shipment can satisfy a demand within the committed service time. Finally, we investigate how the total expected CO2 emissions related to production waste due to perishable products when exceeding the acceptable waiting time and transportation are affected by the inclusion of emergency shipments. One notable and important (and perhaps counter intuitive) insight from this study is that using CO2 intensive emergency shipments may indeed lower the total expected CO2 emissions in a supply chain system.

The remainder of this paper is organized as follows. In Section 2, we present a literature review and in Section 3, we formulate our model. Section 4 describes the solution procedure and presents the system dynamics and performance metrics, together with a cost optimization procedure and how to quantify emissions. In Section 5, a numerical study follows and in the final section we conclude.

Section snippets

Literature review

The model presented in this paper is, of course, closely related to the literature on time window constraints, emergency shipments and to some extent lateral transshipments. However, in general, emergency shipments are from a modeling perspective equivalent to lost sales. Therefore, our work is also related to the literature on inventory models with partial backordering. In this literature review we will mainly focus on multi-echelon inventory models.

There is a well-established literature on

Model formulation

We consider a two-echelon continuous review inventory system with one central warehouse and N local sites. Customer demands occur only at the local sites and are assumed to follow independent Poisson processes with rate λi. Since we focus on spare part products, we assume that replenishments are made according to base-stock ordering policies. The transportation times from the warehouse to the local sites and from the outside supplier to the warehouse are constant. All unfulfilled demands are

Solution procedure

We start by noting that demands satisfied by emergency replenishments can be modeled as lost sales, since such demands are satisfied from a supplier outside the system. Hence, the emergency replenishment cost per item, ri, plus the cost of waiting for the emergency order is equivalent to a lost sales cost. Moreover, since some customers are satisfied from emergency replenishments instead of normal replenishments, the demand process at the central warehouse is no longer Poisson. The true demand

Numerical results

We have studied a number of test problems to optimize the base-stock levels and evaluate the emissions for the system. We consider an inventory system where N=2, i.e. the system has two local sites. We also assume, for simplicity, that the local sites are identical, although this is not necessary in the model.

For our set of test problems, we consider typical parameter values in the context of multi-echelon inventory control for spare parts. In this context L0 is typically substantially longer

Conclusions

In this paper, we have studied a two-echelon inventory system with non-linear backorder costs where it is possible to request an emergency transshipment from an outside supplier. The focus has been on a spare parts inventory system where a customer demand is satisfied if the spare part is delivered within a pre-defined time limit. The backorder cost structure is piecewise constant, where a significant fixed backorder cost is incurred if a demand is not satisfied within the pre-defined time

Acknowledgment

The authors would like to thank the two anonymous referees for their helpful comments.

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