# An Integrated Model for the Harvest, Storage, and Distribution of Perishable Crops
## 3 Problem Description and Model Formulation
### Model for Horizontal Collaboration between Suppliers  
We refer to the possibility of horizontal collaboration between multiple suppliers, who have in common some or all customers. In this context, the main hypothesis is that all suppliers are heterogeneous, that is, not competing with each other. Collaboration is a strategic decision that, in many cases, can lead to a significant reduction in operational costs. Given a set \(\mathcal{C}=\{1,2,\ldots,C\}\) of supplier companies, collaboration, as intended in this study, implies that one of them makes available its own fleet, and the depot as hub for deliveries. Basically, the remaining \((C-1)\) companies, called spoke-suppliers, send their goods to the hub, where the transshipment of goods from their vehicles to those of the hub is carried out, with the aim of optimizing the routing. The problem can be defined on a complete graph \(G^{\prime}=\{\mathcal{N}^{\prime},\mathcal{A}^{\prime}\}\). We denote by \(\mathcal{V}^{\prime}=\{1,2,\ldots,V^{\prime}\}\) the set of all customers (i.e., DCs), and by 0 the node-hub. Therefore, \(\mathcal{N}^{\prime}=\mathcal{V}^{\prime}\cup\{0\}\) is the set of nodes, while \(\mathcal{A}^{\prime}=\{(i,j):i,j\in\mathcal{N}^{\prime},i\
eq j\}\) is the set of arcs. We refer to \(\mathcal{V}^{\prime}{}_{c}\subset\mathcal{V}^{\prime}\) as the set of customers of company \(c\). \(d_{ij}\) and \(t_{ij}^{\prime}\) are the kilometric length of the arc \((i,j)\) and the time to travel it, respectively. \(K^{\prime}\) vehicles, each vehicle \(k\) having the same load capacity \(L^{\prime}\), are made available by the hub-supplier for the distribution of goods. We denote, respectively, by \(\sigma_{c}\) and \(\psi_{c}\) the fixed fuel and driver cost that the spoke-supplier \(c\) must bear for the depot-to-hub round trip. \(\tau_{c}\) is the maximum storage time according to the inventory policy of company \(c\), while \(K_{c}\) is the number of vehicles owned by company \(c\). Other company-related parameters are known from the solution of model \(M_{SS}^{c}\) for each supplier \(c\), namely \(\overline{y}_{jtc}\) and \(\overline{K}_{tc}\). \(\overline{y}_{jtc}\) is the optimal amount of product to be shipped to DC \(j\) at period \(t\) by supplier \(C\), and can be defined as follows:  
\[\overline{y}_{jtc}=\sum_{s=1}^{\tau_{c}}\sum_{k=1}^{K_{c}}\overline{y}_{jkt} \qquad\qquad\qquad\qquad j\in\mathcal{V}^{\prime}{}_{c},\;\;t=1,\ldots,T \tag{30}\]where \(\vec{y}_{jkt}^{\prime}\) is the optimal value of \(y_{jkt}^{\prime}\). \(\overline{K}_{lc}\) is the minimum number of vehicles, according to the load capacity, to be used by company \(c\) at period \(t\) to carry the amount of product \(\sum_{j\in\mathcal{V}_{c}^{\prime}}\overline{y}_{jtc}\). Table 3 summarizes the problem data. The goal is to minimize the overall routing cost. Decision variables \(x_{ijkt}\) and \(f_{lt}\) preserve the same meaning of \(M_{SS}\), while variables \(y_{jkt}^{\prime}\) are re-defined to take into account the multi-supplier nature of the model. Then, we denote by \(y_{jktc}\), the amount of product of supplier \(c\) shipped to DC \(j\) by vehicle \(k\) at period \(t\). Next, we introduce the model formulation.