# Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times
## 2 Literature Review
However, in many cases, there is a demand for products all year round, but with an increase in some in certain months. Such a case of seasonal demand, which is the focus of this paper, is closer to the one mentioned in the study by [PERSON] and [PERSON] [32]. They developed for the first time a seasonal demand inventory model with variable lead-time and resource constraints suited to the just in time (JIT) philosophy. They used a MIP model to solve the problem. However, in the current paper, the control on lead time is limited, based on practical settings. JIT settings are usually suitable for repetitive production. However, when there are so many product models, JIT is not appropriate as the same production line can be used to produce several different products, or the same product with different sizes, which can be produced every day. Another paper that dealt with lead time as a decision variable is by [PERSON] et al. [2], which concentrated on a single-level assembly system. [PERSON] [33] investigated the case when the demand of the goods follows the Sine function, such as when an item undergoes physical decay or deterioration over time. [PERSON] et al. [34] investigated the case when demand follows a ramp-type time-dependent function, with the assumption of zero lead time. Time variable demand was considered in a study by [PERSON] and [PERSON] [35] when only one type of raw material is required to fabricate the finished product. A distribution system with power demand pattern and backorders was investigated in a study by [PERSON] et al. [36]. Additionally, the power demand pattern case with zero lead time was studied by [PERSON] et al. [37], [PERSON] et al. [38], and [PERSON] et al. [39]. Imperfect economic manufacturing models were developed for power demand patterns in a study by [PERSON] et al. [40]. [PERSON] [41] investigated the case of seasonal demand using a simulation with a constant lead time. A simulation was also used in a study by [PERSON] et al. [42], where the integration of simulation modeling and the response surface methodology was performed to solve an order planning problem in the construction supply chain. [PERSON] and [PERSON] [43] considered the case of price and time-dependent seasonal demand rate. They assumed that inventory, once ordered, can be used for more than one season. [PERSON] et al. [44] considered fuzzy seasonal demand in a production inventory model but neglected the effect of lead time. An attempt for many practical settings was made in a study by [PERSON] et al. [45], where variable demand was assumed under the constraints of shelf life, budget, and storage capacity, but with a constant lead time. [PERSON] et al. [46] addressed the joint determination of pricing and ordering decisions, where a retailer sells seasonal products. However, they neglected the effect of lead time. [PERSON] and [PERSON] [3] investigated the periodic-review policy with seasonal demand, with known lead time, where there is one warehouse and \(N\) retailers. [PERSON] and [PERSON] [47] investigated an EOQ model under monsoon type fuzzy demand rate where cycle time is a decision variable, without any consideration for lead time, and shortages are not allowed. [PERSON] and [PERSON] [48] considered the case when the system has a seasonal demand within a cycle of one week, and generated heuristics for a periodic-review policy. They considered a one-day demand phase and a one-day lead time. Meta-heuristics were used by [PERSON] et al. [49] for the lot-sizing and scheduling problem. [PERSON] et al. [50] proposed a stock dynamic sizing optimization, where the safety stock is considered to fill up the demand variability, where the lead time is constant. The objective of this study is to propose a new novel way to plan the replenishment process between the supplier and the warehouse of any company that has a seasonal demand with a reasonable forecasting accuracy of the end product's demand.