# An Integrated Model for the Harvest, Storage, and Distribution of Perishable Crops
## 3 Problem Description and Model Formulation
### Single-Supplier Model
\begin{table}
\begin{tabular}{l l} \hline \(Q_{t}\geq 0\) & amount of product harvested at period \(t\); \\ \(r_{t}\in\{0,1\}\) & binary variable equal to 1 if harvesting is made at period \(t\); \\ \(y_{jkt}^{s}\geq 0\) & amount of product of age \(s\) shipped to DC \(j\) by vehicle \(k\) at period \(t\); \\ \(I_{t}^{s}\geq 0\) & inventory level of product of age \(s\) at the end of period \(t\); \\ \(x_{ijkt}\in\{0,1\}\) & binary variable equal to 1 if vehicle \(k\) travels arc \((i,j)\) at period \(t\); \\ \(f_{ft}\geq 0\) & time to serve DC \(j\) at period \(t\); \\ \(z_{t}^{s}\geq 0\) & amount of product of age \(s\) sold at period \(t\) to spot customers. \\ \hline \end{tabular}
\end{table} 
Table 2: Notation: decision variables. \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\] \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\] (4) \[Q_{t}\leq M_{1}r_{t} t=1,\ldots,T\] (5) \[\sum_{k=1}^{K}\sum_{s=1}^{\tau}y_{jkt}^{s}\leq D_{tj} t=1,\ldots,T,\ j\in\mathcal{V}\] (6) \[\sum_{k=1}^{K}\sum_{s=1}^{\tau}y_{jkt}^{s}\geq\delta D_{tj} t=1,\ldots,T,\ j\in\mathcal{V}\] (7) \[\sum_{s=1}^{\tau}z_{t}^{s}\leq G_{t} t=1,\ldots,T\] (8) \[I_{0}^{s}=0 s=1,\ldots,\tau\] (9) \[I_{t}^{1}=Q_{t}-\sum_{j\in\mathcal{V}}\sum_{k=1}^{K}y_{jkt}^{1} -z_{t}^{1} t=1,\ldots,T\] (10) \[I_{t}^{s}=I_{t-1}^{s-1}-\sum_{j\in\mathcal{V}}\sum_{k=1}^{K}y_{ jkt}^{s}-z_{t}^{s} t=1,\ldots,T,\ s=2,\ldots,\tau\] (11) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]0 (12) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]1 (13) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]2 (14) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]3 (15) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]4 (16) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]5 \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]6 (17) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]7 \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]8 (18) \[\sum_{t\in\overline{T}_{w}}Q_{t}=\overline{Q}_{w} w=1,\ldots,W \tag{3}\]9 (19) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]0 (20) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]1 (21) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]2 (22) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]3 (23) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]4 (24) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]5 (25) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]6 (26) \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]7\[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]8 \[\sum_{t\in\overline{T}_{w}}r_{t}=\gamma w=1,\ldots,W\]9 (27) \[Q_{t}\leq M_{1}r_{t} t=1,\ldots,T\]0 (28) \[Q_{t}\leq M_{1}r_{t} t=1,\ldots,T\]1  
The objective function (2) maximizes the profit and comprises seven parts: revenue from the main customer, revenue from spot customers, inventory cost, fuel and driver cost (i.e., routing cost), production cost, harvesting cost. Constraints (3) and (4) regulate the amount of daily harvested product. The consistency between harvesting variables (\(Q_{t}\)) and harvesting frequency variables (\(r_{t}\)) is ensured by constraints (5). Constraints (6) and (7) state that the amount of product shipped to the DCs must not exceed their demand and must be consistent with the contractualized service level at each period, respectively. Constraints (8) ensure that the demand of spot customers is not exceeded at each period. Constraints (9)-(12) regulate the inbound/outbound mechanism of product to/from the inventory, considering its limited capacity and the assumed initial null level. Constraints (13)-(15) guarantee the flow balancing on the depot of the company and on each node-customer, respectively. Constraints (16) ensure that the load capacity of each vehicle is not exceeded at each period.