# An Integrated Model for the Harvest, Storage, and Distribution of Perishable Crops
## 3 Problem Description and Model Formulation
### Single-Supplier Model
The decisions to be made at each period \(t\) concern the amount \(Q_{t}\) of product harvested, the amount \(y_{jkt}^{s}\) of product of age \(s\) shipped to distribution center \(j\) by vehicle \(k\), the amount \(z_{t}^{s}\) of product of age \(s\) sold to spot customers. Binary variables \(r_{t}\) represent the possibility to choose whether or not to carry out the harvesting at each period. The inventory management is guaranteed by the variables \(I_{t}^{s}\), which define the inventory level of product of age \(s\) at the end of period \(t\). The inventory level at the beginning of the time horizon is supposed null. Continuous variables \(f_{jt}\) determine the time to serve DC-customer \(j\) at period \(t\), while binary variables \(x_{ijkt}\) are active in case arc \((i,j)\) is traveled by vehicle \(k\) at period \(t\). In Table 2, we show the notation for the decision variables of our optimization model. Next, we introduce the model formulation  
\[\max \sum_{j\in\mathcal{V}}\sum_{k=1}^{K}\sum_{t=1}^{T}\sum_{s=1}^{\tau}p_ {i}^{s}y_{jkt}^{s}+\sum_{t=1}^{T}\sum_{s=1}^{\tau}p_{t}^{s}z_{t}^{s}-\sum_{t=1}^ {T}\sum_{s=1}^{\tau}h_{t}I_{t}^{s}\] \[-\alpha\sum_{k=1}^{K}\sum_{t=1}^{T}\sum_{(i,j)\in\mathcal{A}}d_{ij }x_{ijkt}-\lambda\sum_{k=1}^{K}\sum_{t=1}^{T}\sum_{(i,j)\in\mathcal{A}}t_{ij}x_ {ijkt}\] \[-\rho\sum_{t=1}^{T}Q_{t}-\eta\sum_{t=1}^{T}r_{t} \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \(\mathcal{T}\) & harvesting/distribution time horizon, with \(\mathcal{T}=\{1,\ldots,T\}\); \\ \(\mathcal{W}\) & set of weeks of the harvesting/distribution time horizon, with \(\mathcal{W}=\{1,\ldots,W\}\); \\ \(\overline{T}_{w}\) & set of days of week \(w\); \\ \(\mathcal{V}\) & set of DCs of the main customer, with \(\mathcal{V}=\{1,2,\ldots,V\}\); \\ \(\mathcal{N}\) & set of nodes, including the company 0, with \(\mathcal{N}=\mathcal{V}\cup\{0\}\); \\ \(\mathcal{A}\) & set of arcs, with \(\mathcal{A}=\{(i,j):i,j\in\mathcal{N},i\
eq j\}\); \\ \(\mathcal{K}\) & set of vehicles, with \(\mathcal{K}=\{1,2,\ldots,K\}\); \\ \(L\) & load capacity of each vehicle; \\ \(d_{ij}\) & kilometric distance between node \(i\) and node \(j\), \((i,j)\in A\); \\ \(t_{ij}\) & time distance in minutes between node \(i\) and node \(j\), \((i,j)\in A\); \\ \(\alpha\) & fuel price per kilometer; \\ \(\lambda\) & wage rate for the drivers per minute; \\ \(\overline{Q}_{w}\) & amount of ripe product at week \(w\); \\ \(\overline{p}_{t}^{s}\) & unit selling price of product of age \(s\) at period \(t\) to the main customer; \\ \(p_{t}^{s}\) & unit revenue of product of age \(s\) at period \(t\) to the main customer; \\ \(D_{ij}\) & demand of product at period \(t\) by DC \(j\); \\ \(h_{t}\) & unit storage cost at period \(t\); \\ \(\tau\) & maximum storage time; \\ \(\delta\) & fraction of demand of the main customer to be satisfied at each period (i.e., service level); \\ \(M_{1}\), \(M_{2}\) & sufficiently high constants; \\ \(b_{t}^{s}\) & unit selling price of product of age \(s\) at period \(t\) to the spot customer; \\ \(G_{t}\) & demand of product at period \(t\) by spot customers; \\ \(I_{max}\) & storage capacity; \\ \(\theta\) & daily shipping time limit referring to the DCs of the main customer (i.e., service level); \\ \(\beta\) & unit reward related to the quality of service offered to the main customers; \\ \(\rho\) & unit production cost; \\ \(\eta\) & fixed daily harvesting cost (i.e., rental of specialized equipment); \\ \(\gamma\) & number of harvesting days per week. \\ \hline \end{tabular}
\end{table} 
Table 1: Notation: problem data.