# Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times
## 1 Introduction  
Replenishment planning is necessary to balance service levels, inventory ordering and holding costs. The decision-maker must react to the customer's dynamic needs, and at the same time keep the costs as low as possible. Therefore, a trade-off must be planned. A clear picture of the situation on the ground is necessary to optimize the system. This planning is widely known in theory and practice. However, sometimes restrictions enforced by suppliers must be taken into consideration, such as the possible times of delivery, which has been overlooked in the literature. A simple economic order quantity formula (EOQ) was developed by [PERSON] in 1913 [1]. However, this formula was too simple to reflect many practical sititions such as changes in demand and suppliers' constraints. For example, a lot of research studies assume a constant demand rate or a deterministic trend in demand rate. A lot of studies have neglected the lead time or considered it to be constant [2]. In many studies about the replenishment system, the phrase \"lead time\" was not mentioned at all. Other assumptions are the zero-inventory point at the exact time of lot coming to the warehouse, therefore neglecting safety stock. The capacitated replenishment system assumes constraints on the minimum size of the lot or a maximum warehouse capacity, but it assumes no constraints on the delivery dates. In other words, studies assumed no constraints on the delivery times. Therefore, delivery time was traditionallyconsidered a continuous variable. Some of the limitations were on the design of the supply chain network [3]. Many attempts were made to include more practical situations such as the trend in demand. Still, however, most of the research depended heavily on many unpractical assumptions. Practitioners are struggling to find studies without non-practical assumptions and are easy to understand [4]. Most of the research used substantial calculations including calculus, nonlinear programming, and other mathematical methods [5]. Therefore, practitioners would use the theoretical formulas as a starting point, and then use their experience to modify the results. Another point to mention is that the demand here is between suppliers and the company. The ordering cost is usually large since the distances between different supply chain parties are long. Therefore, supplying the needed products according to the actual known demand for short periods is very expensive as it includes too many orders and therefore results in huge ordering costs. The situation escalates when there are hundreds or even thousands of different item types, where the demand size for each one of them is not so large. Research has favoured the JIT principle when firms have tried to meet high and consistent levels of demand [6]. This is the not case in this paper. Another point is that in the well-known periodic review system, period order quantity (POQ) is used. However, the assumption in this method is that there must be a delivery every period, no matter how much the demand size is (very high or very low). An optimal solution, however, might be reducing the number of deliveries in the low demand period, and increasing it in the high demand period. The model in this paper optimally combines the net demand for one or more periods. This study is motivated by a case study about a drinks company in Germany for the manufacturing and distribution of drinks. Several types of drinks are produced or bought from other companies and then sold to retailers. Unit prices are relatively stable, but demand is generally larger in the summer. Suppliers cannot produce every product type every day, and therefore, a certain product type can be supplied on a certain day in the week, or, for example, two days in the week. This is especially true when the distance between suppliers and retailers or consolidation warehouses is relatively long to reduce transportation costs. In other words, there is a constraint on the delivery times which are discrete and not continuous.