Control of closed-loop systems

We considered a closed loop system with demands that occur according to a Poisson process and where the market sojourn time of the items are Gamma distributed. There is a certain probability that an item will return. If they will return, then they will be send to the repair centre, where the items will be checked, cleaned and repaired if necessary. The repair time is exponential. We simulated the demand, return and repair times with a simulation model. We kept these times fixed for the four models that we consider: model I where we have the exponential repair times with the inventory position updated when an item returns to the repair centre, model II where we have exponential repair times with the inventory position updated when an item returns to inventory, model III has a deterministic repair time that is equal to the mean of the stochastic repair times and the inventory position is updated when an item returns to the repair centre and as last we have model IV that is equal to model III except that the inventory is not updated when an item returns to the inventory. We use two methods to optimize the models. The first one is to meet a certain fill rate of % and the second one is based on minimizing the total average costs per item. We determine the reorder point that is associated with these optimums and the amount of necessary items to fulfill % of the demand. We have simulated for a time horizon of 337 days. The results do not differ much between the stochastic and deterministic models. The necessary amount of items that are needed are around 2716 for models I and III, and 2658 for models II and IV. Using the first method to obtain a fill rate of 80%, we found associated costs of about € 173 for models I and III, and € 177 for models II and III with reorder points 63 and 43 respectively. When we optimize the total average costs, we find costs of more or less € 138 for models I and III and € 133.50 for models II and III with a reorder points 110 and 88 respectively. All the four models have a fill rate of 100% each. If we compare the two methods with each other, we see that using the method to optimize the costs are giving lower costs with higher fill rates. The model with the lowest costs are given by model IV, that is the deterministic model where the inventory position is updated when an item returns to the inventory, which is € 133.61 with a fill rate of 100%.