# An Integrated Model for the Harvest, Storage, and Distribution of Perishable Crops
## 3 Problem Description and Model Formulation
### Model for Horizontal Collaboration between Suppliers
\[\min \alpha\sum_{k=1}^{K^{\prime}}\sum_{t=1}^{T}\sum_{(i,j)\in\mathcal{ A}^{\prime}}d_{ij}x_{ijkt}+\sum_{t=1}^{T}\sum_{c=1}^{C}\sigma_{c}\overline{K}_{lc}\] \[+\lambda\sum_{k=1}^{K^{\prime}}\sum_{t=1}^{T}\sum_{(i,j)\in \mathcal{A}^{\prime}}t_{ijk}x_{ijkt}+\sum_{t=1}^{T}\sum_{c=1}^{C}\psi_{c} \overline{K}_{lc} \tag{31}\]  
\[\sum_{k=1}^{K^{\prime}}y_{jktc}=\overline{y}_{jtc} c=1,\ldots,C,\ t=1,\ldots,T,\ j\in\mathcal{V}^{\prime}{}_{c} \tag{32}\] \[\sum_{c=1}^{C}\sum_{j\in\mathcal{V}_{c}^{\prime}}y_{jktc}\leq L^{\prime} t=1,\ldots,T,\ k=1,\ldots,K^{\prime}\] (33) \[y_{jktc}\leq L^{\prime}\sum_{i\in\mathcal{N}^{\prime}j\
eq j}x_{ ijkt} c=1,\ldots,C,\ j\in\mathcal{V}^{\prime}{}_{c}t=1,\ldots,T,\ k=1,\ldots,K^{\prime}\] (34) \[x_{ijkt}\leq\sum_{c\in C:j\in\mathcal{V}^{\prime}{}_{c}}y_{jktc} i\in\mathcal{N}^{\prime},j\in\mathcal{V}^{\prime},t=1,\ldots,T,\ k=1,\ldots,K^{\prime}\] (35) \[y_{jktc}\geq 0 c=1,\ldots,C,j\in\mathcal{V}^{\prime}{}_{c},\ t=1,\ldots,T,\ k=1, \ldots,K^{\prime} \tag{36}\]  
and the following constraints, introduced in \(M_{SS}\) and now referred to the graph \(G^{\prime}\): (13)-(15), (19)-(22), (27) and (28). The objective function (31) aims to minimize the routing costs, represented by four components: variable fuel cost, fixed fuel cost, variable driver cost, fixed driver cost. Constraints (32) guarantee that the demand from the distribution centers is met at each period. Constraints (33) ensure that the load capacity of each vehicle is not exceeded at each period. Constraints (34) and (35) ensure consistency between variables \(x_{ijkt}\) and \(y_{jktc}\), whose non negativity is established by constraints (36). In the remainder of this paper, we refer to this collaborative routing model as \(M_{CR}\). \begin{table}
\begin{tabular}{l l} \hline \hline \(\mathcal{C}\) & set of companies, with \(\mathcal{C}=\{1,2,\ldots,C\}\) \\ \(\mathcal{V}^{\prime}\) & set of DCs to be served, with \(\mathcal{V}^{\prime}=\{1,2,\ldots,V^{\prime}\}\) \\ \(\mathcal{N}^{\prime}\) & set of nodes, including the hub 0, with \(\mathcal{N}^{\prime}=\mathcal{V}^{\prime}\cup\{0\}\); \\ \(\mathcal{A}^{\prime}\) & set of arcs, with \(\mathcal{A}^{\prime}=\{(i,j):i,j\in\mathcal{N}^{\prime},i\
eq j\}\); \\ \(\mathcal{V}^{\prime}{}_{c}\) & subset of DCs to be served by company \(c\); \\ \(\mathcal{K}^{\prime}\) & set of vehicles, with \(\mathcal{K}^{\prime}=\{1,2,\ldots,K^{\prime}\}\); \\ \(d_{ij}^{\prime}\) & kilometric distance between node \(i\) and node \(j\), \((i,j)\in\mathcal{A}^{\prime}\); \\ \(t_{ij}^{\prime}\) & time distance in minutes between node \(i\) and node \(j\), \((i,j)\in\mathcal{A}^{\prime}\); \\ \(\sigma_{c}\) & fixed fuel cost for the depot-to-hub round trip by spoke-supplier \(c_{i}\) \\ \(\psi_{c}\) & fixed driver cost for the depot-to-hub round trip by spoke-supplier \(c_{i}\) \\ \(\overline{y}_{jtc}\) & amount of product to be shipped by company \(c\) to DC \(j\in\mathcal{V}^{\prime}{}_{c}\) at period \(t\); \\ \(\overline{K}_{lc}\) & number of vehicles to be used for the depot-to-hub round trip by spoke-supplier \(c\) at period \(t\). \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Notation: problem data.