Multi-Echelon Inventory Control with General Backorder Cost Structures and Emissions

Abstract: The contribution of this licentiate thesis lies in the modelling of a multi-echelon inventory system with more general backorder cost structures. Backorder costs are traditionally assumed to be linear, whereas we consider structures that are more general functions of the customers’ time in backorder. The system studied is a two-echelon system with one central warehouse and a number of local sites, all applying continuous review base stock policies, where demand is Poisson. We add to the literature by extending previous results for this type of system where we present closed form expressions of the probability distributions for the inventory levels and the probabilities of a customer’s waiting time exceeding a certain time level.We also take a first step towards quantifying the expected emissions that are consequences of not fulfilling customer demands in time. We include the possibility of fast but more polluting emergency replenishments to prevent this type of emissions. From our numerical study we can conclude that, in some cases, the expected total emission for the system can be reduced by using emergency replenishments although they are more emission intense than the normal replenishments. Furthermore, we take into account that it may not always be possible to arrange an emergency replenishment within a stipulated time.The study of this specific inventory system is inspired by a special type of service contract that exists in several industries. These contracts stipulate that the service providing company are obliged to pay a large fixed penalty fee if a requested item cannot be delivered within a certain time limit. These contracts imply a non-linear backorder cost and introduce the concept of an acceptable waiting time limit for which a customer is willing to wait. If the customer receives the requested item within this time limit, the demand is considered to be met, but if the limit is exceeded there are both economic and environmental impacts.Our motivating case considers a spare parts provider handling spare parts for machines used in for example the dairy industry. Let us say that the production process is down due to a halt in a critical machine because a crucial spare part is needed. If the spare part can be delivered within a certain time limit, the machine can be repaired and the production process resumed. If the machine cannot be up and running again within a certain amount of time, the whole production batch must be discarded due to the perishable nature of the product and the production equipment must be cleaned. Apart from the economic consequence, and hence the service contract, there is an environmental impact of products going to waste.

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