# Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times
## 2 Literature Review  
Supply chain planning in a dynamic environment assuming dynamic demand using optimization models has been widely investigated in the literature. An example of this is shown in the study by [PERSON] et al. [7] which proposed a dynamic supply chain design, and the study by [PERSON] et al. [8], which investigated the production-planning problem for production-time-dependent products. Both suggested a MIP model and a heuristic algorithm. Mathematical programming in warehouses was also investigated in a study by [PERSON] et al. [9]. Some studies concentrated on the literature review of economic order quantity such as [PERSON] and [PERSON] [4] and [PERSON] et al. [10]. The first attempt to include the trend in demand was by [PERSON] [11], who used substantial calculations. At first, he calculated the cycle times, which are different based on time-dependent demand, and then based on these delivery times, the best lot size was found. The optimal delivery times were found for a different total number of deliveries on a specified horizon. To simplify the solution, [PERSON] [12] developed an approximation \"Silver-Meal\" heuristic. Later, [PERSON] [13] extended the time horizon of [PERSON] so that it no longer influenced the replenishment times to simplify the calculations. More practical settings started to appear. For example, [PERSON] and [PERSON] [14] developed an inventory model involving lead time as a decision variable. Later, some studies included the shortage costs besides the trend in demand in the investigation, such as the study by [PERSON] [15]. However, there were still some limiting assumptions; both the lead time and the initial and final inventory levels were zero. Moreover, a study by [PERSON] et al. [16] investigated only the replenishment policy with linear decreasing demand. [PERSON] et al. [17] investigated the inventory replenishment policy for a linear trend in demand, where two steps are needed to examine the classical no-shortage inventory replenishment policy. [PERSON] et al. [18] relaxed the linear trend when they investigated replenishment with non-linear decreasing demand. Other studies that came later investigated the increasing demand pattern, such as [PERSON] and [PERSON] [19]  
Factors other than demand were also investigated. For example, [PERSON] and [PERSON] [20] considered the case of imperfect quality items. The effect of rework on production order quantity was considered by [PERSON] et al. [21] and [PERSON] et al. [22]. Moreover, [PERSON] et al. [23] investigated the replenishment policy in the case of fuzzy stock cost of each unit quantity and the order cost of each cycle. [PERSON] et al. [24] investigated the replenishment system when the supplier's warehouse has limited capacity and there is an upper bound on the number of orders in the Vendor Managed Inventory (VMI) System. A JIT system was considered in a study by [PERSON] et al. [25], but assuming constant demand. Restrictions on delivery due dates were considered in a study by [PERSON] et al. [26] in the JIT environment, but the demand rate was assumed to be constant. [PERSON] et al. [27] investigated finding the optimal production and inventory policy in a multiproduct bakery unit, using mixed-integer programming (MIP). The seasonality effect was investigated at first for products that were in demand for only a short period. [PERSON] and [PERSON] [28] concentrated on products such as winter sporting goods. The same direction was followed by [PERSON] et al. [29] to study an EQQ model for deteriorating items. Some later studies investigated seasonal demand such as the study by [PERSON] et al. [30], where they concentrated on items with fixed selling seasons. The effect of fluctuations in demand and unit price was investigated in a study by [PERSON] et al. [31], especially for high-tech products where the unit cost declines significantly over a short product life cycle while the demand increases. Again, there are some limiting assumptions such as the zero-lead time and the zero initial inventory level.