# Sales Forecasting for Fashion Products Considering Lost Sales
## 3 Methodology
### The Proposed Sales Forecasting Model  
Sales are transformed from the potential demand, and the transformation process is limited by the scale of inventory of a fashion product. When the inventory is less than the potential demand, some demand will not be met. However, even if the inventory is sufficient, some customers may not purchase the product because products are not placed in the correct store, which leads to demand loss. Given the demand, we define marginal sales as the change in the total sales that arises with the increment in inventory by one unit. In an ideal state, as long as there exists unsatisfied potential demand, the increased inventory will be sold out, and the marginal sales are equal to one. However, in reality, marginal sales are always less than one. As the sales increase, the number of potential customers decreases, and the chance of the product being discovered by the customer drops, leading to a decrease in marginal sales as well. In this manner, we model the marginal sales by the following equations:  
\[M(I)=q\Big{(}1-(\tfrac{S(I-1)}{D})^{\alpha}\Big{)},\ \ \ I\in N^{+}, \tag{1}\]  
where \(S(I)\) and \(M(I)=S(I)-S(I-1)\) is sales and the marginal sales when inventory is \(I\). The sales iterative formula describes the relationship between sales, inventory and demand. We already have the marginal sales equation when the inventory is \(I\), and the product sales can be calculated by the iterative formula \(S(I)=\sum_{i=1}^{I}M(i)\). The right side of Equation (1) consists of two parts, and the first part is \(q\). Since \(S(0)=0\), then \(M(1)=q\) describes the marginal sales of the initial inventory. Therefore, \(q\in[0,1]\) is defined as the basic conversion rate, which is determined by a product's inherent properties. Another part of Equation (1) actually describes the chance of products being found in brick-and-mortar stores, where \(\tfrac{S(I)}{D}\) is the proportion of the demand being satisfied. When the proportion is small, there are still a lot of unsatisfied demands, so the marginal sales brought by one-unit increased inventory is relatively large. When the proportion approaches one, most of the demand is satisfied, and increments in inventory can hardly transform into sales, resulting in small marginal sales. In general, marginal sales that decline as sales increase are described by \(1-(\tfrac{S(I-1)}{D})^{\alpha}\). For some products, customers have a strong willingness to look for stores with stocks. Thus, the chance of these products being discovered is higher, and the decline rate of marginal sales with an increase in sales would be relatively small. We use \(\alpha\) to control the decline rate. When \(\alpha\) approaches infinity, the marginal sales will remain at the degree of the basic conversion rate. Combining the influence of both products' inherent properties and potential customers on sales, the final increased unit of sales when one unit of inventory increases is determined by \(q(1-(\tfrac{S(I)}{D})^{\alpha})\). Figure 1: Fluxogram of Methodology. According to different \(q\) and \(\alpha\), products are divided into three categories: major commodity, common commodity and targeted commodity. Products that have large \(q\) and \(\alpha\) are defined as major commodities. Major commodities are usually products keeping up with the fashion trend in season. Their targeted customers have high stickiness and enthusiasm and are willing to make an effort to find them. Products that have large \(q\) and small \(\alpha\) are defined as common commodities. Common commodities are products with basic or classic styles. Their demands are stable, but customers give up finding them when the difficulty of finding them increases. For example, when the brick-and-mortar store the customer enters is out of stock in a common commodity, the possibility is small that the customer will go to another store specifically just to find the commodity. Products that have the smallest \(q\) are defined as targeted commodities. Targeted commodities are designed for minority groups.