# An Integrated Model for the Harvest, Storage, and Distribution of Perishable Crops
## 3 Problem Description and Model Formulation
### Single-Supplier Model  
We refer to harvesting, storage, and distribution issues faced by an agricultural company, along a discete time horizon \(\mathcal{T}=\{1,\dots,T\}\), which belongs to the harvesting/distribution season, prior to which planting and growing of crop occur. We denote by \(t\in\mathcal{T}\) a generic period index. The company, which deals with a single perishable agri-product, has contractual obligations to a main customer, but can also exploit the favorable opportunities offered by spot customers. The main customer has a set of distribution centers (DCs) to be served along \(\mathcal{T}\). The company agrees with the main customer a planting plan, in order to guarantee the availability of the product to the DCs during \(\mathcal{T}\). We denote by \(\rho\) the unit production cost, which takes into account planting and growing activities. Based on the planting plan,we denote by \(\overline{Q}_{w}\) the amount of product, which is ripe to be harvested during the week \(w\) of \(\mathcal{T}\). The harvesting activities can be carried out \(\gamma\) days per week and each harvesting day implies a fixed cost \(\eta\), related to the rental of specialized equipment. Each product unit (kg), once harvested can be immediately shipped to the customers or stored for later deliveries. The depot has a capacity \(I_{max}\), each product unit has a storage cost per period \(h_{t}\) and is arranged according to its age \(s\) (\(s=1\) means fresh product, \(s=2\) means that the product was harvested yesterday, etc.). According to company policies, the product cannot be stored for more than \(\tau\) time periods, after which it must be discarded. Disposal costs are not taken into account explicitly. The freshness of the product determines its market value, then we denote by \(\overline{p}_{t}^{s}\) and \(b_{t}^{s}\) (\(b_{t}^{s}>\overline{p}_{t}^{s}\)) the unit selling price of product of age \(s\) at period \(t\), to the main customer and spot customers, respectively. A reward \(\beta\) can be earned by the company for each product unit shipped to the main customer, depending on the quality of service. This latter is related to the agreed daily shipping time limit \(\theta\), and/or to the agreed fraction \(\delta\) of the demand, which must be guaranteed to each DC at each period. Therefore, the unit revenue \(p_{t}^{s}\) of product of age \(s\) at period \(t\) can be computed as follows:  
\[p_{t}^{s}=\overline{p}_{t}^{s}+\beta(\theta,\delta)\hskip 72.27 ptt=1,\ldots,T,\ s=1, \ldots,\tau \tag{1}\]  
The amount of product marketable at each period depends on the market demand; we denote by \(D_{lj}\) the demand by the distribution center \(j\) at period \(t\), while \(G_{t}\) is the demand by spot customers at period \(t\). \(K\) vehicles, each vehicle \(k\) having the same load capacity \(L\), are available for the distribution to the DCs. We denote by \(\mathcal{V}=\{1,2,\ldots,V\}\) the set of customers (i.e., DCs), and by 0 the node-depot of the company. Then, the problem can be defined on a complete graph \(G=\{\mathcal{N},\mathcal{A}\}\), where \(\mathcal{N}=\mathcal{V}\cup\{0\}\) is the set of nodes, while \(\mathcal{A}=\{(i,j):i,j\in\mathcal{N},i\
eq j\}\) is the set of arcs. \(d_{ij}\) and \(t_{ij}\) are the kilometric distance of the arc \((i,j)\) and the time to travel it, respectively. Fuel and driver cost define the overall routing cost. We denote by \(\alpha\) the fuel price per kilometer, while \(\lambda\) is the wage rate per minute for the drivers. There are no routing or distribution costs for spot customers sales. In Table 1, the notation of all problem data is shown. The goal is to maximize profit. Revenue depends on sales to the main customer and spot customer. Cost depends on the inventory management, the routing of the vehicles, the production (planting/growing) and harvesting activities.