## 2 Mathematical Formulation
### Assumptions  
A number of assumptions were considered for this problem's formulation. * Three types of storage location were considered: warehouse, cold storage, and tanks. Each type could be in different locations and had different holding costs. * The formulation considered a single warehouse and cold storage, but multiple tanks were allowed. * Lead time for transportation of stock between storage locations was considered to be negligible. * Holding cost was assumed to be linear with the amount of stock. * While there was only one cold storage location, it contained multiple freezer units. * For the calculation of the risk of shortage, a horizon of 60 days was considered. * Filling a storage location more than its maximum capacity was not allowed. * The problem was considered deterministic. \begin{table}
\begin{tabular}{l l} \hline \hline
**Variable** & **Description** \\ \hline \multicolumn{3}{l}{Cold Storage Variables} \\ \(ET_{cold}\) & Correspondence of which storage location \(k\) is a cold storage \([n_{k}\times 1]\) \\ \(S_{cold}\) & Total stock at the cold storage location, \([IU]\)\([1\times 1]\) \\ \(\alpha_{1}\) & Individual freezer unit capacity, \([IU]\) \\ \(\alpha_{2}\) & Ratio of energy consumption increase in freezer units, with degree of fullness \\ \(\alpha_{3}\) & Nominal daily energy consumption of each freezer unit, \([EU]\) \\ \(\alpha_{4}\) & Mid-point of the sigmoid energy consumption increase \\ \(\alpha_{5}\) & Width of the sigmoid energy consumption increase \\ Warehouse Variables \\ \(ET_{warehouse}\) & Correspondence of which storage location \(k\) is a warehouse \([n_{k}\times 1]\) \\ \(S_{warehouse}\) & Total stock in the warehouse, \([IU]\)\([1\times 1]\) \\ \(FC_{Queuehouse}\) & Fixed energy consumption in the warehouse, \([EU]\)\([1\times 1]\) \\ \(\beta_{1}\) & Energy consumption per unit of material in the warehouse, consumed by transportation of stock, \([EU/IU]\)\([1\times 1]\) \\ \(\beta_{2}\) & Base energy consumed by HVAC in the warehouse, considering an empty warehouse, \([EU]\)\([1\times 1]\) \\ \(\beta_{3}\) & Linear decay rate of HVAC energy consumption in the warehouse, per unit of material, \([EU/IU]\)\([1\times 1]\) \\ Tank Variables \\ \(ET_{tank}\) & Correspondence of which storage location \(k\) is a tank, with multiple tanks allowed \([n_{k}\times n_{kt}]\) \\ \(S_{tank}\) & Total stock in each tank, \([IU]\)\([1\times n_{kt}]\) \\ \(\gamma\) & Energy consumption per tank, considering the tank at full capacity, \([EU]\)\([n_{kt}\times 1]\) \\ Stock Shortage Calculation Variables \\ \(L_{i}\) & Average lead time of purchasing product i, \([day]\). \\ \(\sigma_{i}\) & Standard deviation of the lead time of purchasing product i, \([day]\). \\ \(N_{i,t}\) & Requirements of material i for the t-th day after the problem, \([IU]\). \\ \(B_{i,t}\) & Stock of material \(i\) on day \(t\) after consumption has been subtracted and without stock replenishment. The values only contain the sign of the quantities, i.e., whether they are negative, null, or positive quantities. \\ \(ShD_{i}\) & Day when the base stock is exhausted. Takes the value \(n_{k}+1\) for materials whose base stock is never exhausted. \\ \(Sh1_{i}\) & Shortage value of type 1 for each material. \\ \(Sh2_{i}\) & Shortage value of type 2 for each material. \\ \(Sh3_{i}\) & Shortage value of type 3 for each material. \\ \(SW1\) & Weight of shortage type 1. \\ \(SW2\) & Weight of shortage type 2. \\ \(SW3\) & Weight of shortage type 3. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._All quantities were considered to be in generic units. Materials were in inventory units, \([IU]\), costs were in cost units, \([CU]\), and energy were in energy units, \([EU]\). * The total energy consumption of a tank was linearly proportional to the rate of fullness of the tank. Additionally, a few assumptions related to the proposed theorems can be presented.