# Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times
## 3 Methodology
The supplier cannot produce every item type and size every day. Moreover, the demand for a certain type and size might not be big. Therefore, some suppliers might prefer to send a big truck every week (if needed). That means that there is a constraint not only on the capacity of the truck (and hence on the lot size), but also a constraint on the delivery times. Each supplier does not need to send each different type and size every week. However, at least some of them might be in a certain week. Such items are consolidated and sent to the warehouse in one truck to reduce transportation costs. Because the drinks' company needs to send the order in the previous week before the actual delivery and because the delivery date might be fixed on a certain day (for example, Tuesday), the actual lead time might be more than one week. However, there is a minimum lead time (MLT). For example, if the order by the company is sent to the supplier on Wednesday, it might not be delivered on the direct until next Tuesday. There is, however, a source of uncertainty, where sometimes the delivery is one or two days late and it might not arrive until Thursday. In this case, the lot size should cover the demand for 9 days instead of 7 days. In this paper, however, the assumption is that such a delay is only a small percent and the safety stock is enough to handle it. The lot will cover the demand for the period after the arrival of the shipment. Because of the variability of the demand, accurate forecasting must be considered, and a dynamic approach is needed to recalculate the replenishment times and quantities. Therefore, three questions arise:  
1. Should delivery be made in the next week? 2. If yes, how many weeks of demand should it cover (cycle time), and what about the next cycle times? 3. Based on the optimal cycle time, what are the optimal lot size and safety stocks? These questions should be answered the previous week before the next delivery date. The planning horizon is, however, larger than one week (it can be one year). That means that there is a primary plan based on the integer programming model, and this plan is re-evaluated every week based on the actual size of the inventory we have now. Therefore,  
Figure 2: Drinks company supply chain. the plan might indicate no delivery in a certain week, and then the new plan indicates delivery in that week. The currently available inventory size (P\({}_{0}\)) is the size of the inventory at the end of the current week. That means that the current week's demand is known or predicted with a high level of accuracy. The planning horizon starts the next week. That means that week 1 is the next week. If P\({}_{0}\) is expected not to cover the demand for at least the next week, an order must be triggered immediately. The following assumptions are made:  
* The demand rate from one week to another can be different. * Delivery times are discrete based on suppliers' constraints on them. * Every different product type can be studied independently. This assumption is reasonable because there are so many different item types, and the size of each one compared to the whole demand is not so high. Another factor is that delivery is carried out usually on a certain day in the week, and it is expected that many other items are needed in that week. Therefore, there is no need to assign a vehicle only for one item type. * Lot size can be any size lower than the maximum allowable one
* The initial inventory size at the end of the current week can be estimated with a good degree of accuracy
* The MLT should be less or equal to the minimum cycle time (MCT). For example, if MCT is one week, MLT can be one week or less. The constraint that delivery must be carried out at the beginning of the planning horizon is released in this paper. Another difference is that the decision about next week's delivery depends on a plan for the next several weeks (planning horizon), and not only the next week.