## 3 Materials and Methods
### Cost Structure  
As noted by [61], cost parameters in inventory models are approximations of actual conditions and do not need to match real-world costs exactly. Instead, as performance evaluations indicate, these parameters should be seen as control variables that can be adjusted to ensure effective management.  
Total inventory-related cost in the observed period (C\({}_{\text{TOT}}\)) considered in this study consists of ordering (C\({}_{\text{O}}\)), transportation (C\({}_{\text{T}}\)), holding (C\({}_{\text{H}}\)), and penalty costs (C\({}_{\text{P}}\)), similar to the works of [37; 38; 45; 62; 63; 64; 65; 66; 67]. Total costs are calculated according to Equation (1):  
\[\text{C}_{\text{TOT}}=\text{C}_{\text{O}}+\text{C}_{\text{T}}+\text{C}_{\text {H}}+\text{C}_{\text{P}} \tag{1}\]  
Ordering costs (C\({}_{\text{O}}\)) are calculated with fixed ordering costs charged for each launched order (C\({}_{\text{o,f}}\)), and the number of orders in the observed period (N\({}_{\text{o}}\)) is defined by Equation (2). Fixed ordering costs are charged for processing an order, regardless of its size.  
\[\text{C}_{\text{o}}=\text{C}_{\text{o,f}}\text{N}_{\text{o}} \tag{2}\]  
Transportation costs (C\({}_{\text{T}}\)) related to inventory replenishments are charged per delivery and depend on the total number of deliveries in the observed period (N\({}_{\text{d}}\)). Transportation costs arising from inventory replenishments consist of depreciation cost (C\({}_{\text{d}}\)) defined per vehicle type as specified in Table 1, multiplied by the LF as a result of inventory replenishments increased with the cost of diesel fuel calculated as a product of fuel consumption (F) and the unit fuel price (C\({}_{\text{f}}\)) as per [68]. Depreciation costs are calculated assuming a standard yearly depreciation rate of 25%. The load factor represents the ratio of transported inventory replenishment and payload capacity of the vehicle used. Empty trip factor (ETF) is a percentage of distance vehicles running empty and loaded. According to our SC model, vehicles pick up empty, so the ETF is 100%. We assume six different vehicles are available to DC for inventory replenishments. Vehicles transporting inventory replenishments are used daily, i.e., 1461 days in a four-year period.  
Transportation costs related to inventory replenishments are defined by Equation (3).  
\[\mathrm{C_{T}=N_{d}(C_{d}\cdot LF+F\cdot C_{f})} \tag{3}\]  
Inventory holding costs (\(\mathrm{C_{H}}\)) are calculated according to Equation (4), with fixed inventory holding cost (\(\mathrm{C_{h,f}}\)) multiplied by the average inventory level within the period (N).  
\[\mathrm{C_{H}=C_{h,f}\cdot AIL\cdot N} \tag{4}\]  
Penalty costs (\(\mathrm{C_{P}}\)) are charged for a product shortage and are calculated according to Equation (5), where \(C_{p,f}\) denotes representative fixed penalty costs. In practice, penalty costs cannot be determined with absolute certainty since they represent, besides the lost sales, effects that are challenging to quantify, such as the loss of customer goodwill. Therefore, this paper considers a representative fixed penalty cost per missing unit, which, in real-world conditions, depends on the type and value of the product itself, the market, and the managerial implications of missing items. The lost sales in the observed period (LS) are directly related to FRs of 89.9% and 99.9% used in this research.  
\[\mathrm{C_{P}=C_{p,f}\cdot LS} \tag{5}\]