Consider a logistics system consisting of $n$ production sites and $m$ warehouses. For a given product, the monthly production capacity of the production sites is $p_i$ units, with $i = 1,\dots, n$. The monthly demand of the warehouses is $d_j$ units, with $j = 1,\dots, m$. The cost of transporting one unit of goods from the production site to the warehouse is $c_{ij}$ monetary units. Provide the linear programming model for planning the minimum cost of the monthly transport of goods from production sites to warehouses in compliance with production capacity and demand. Let us now consider the fact that the transport is carried out by vans which can carry at most $C$ units of the product in question. The cost of using a van to travel once any route from $i$ to $j$, regardless of the quantity of goods transported and the distance traveled, is $L$ monetary units. The cost of use per van and per route must now be added to the monthly transport cost. Modify the first model to take into account the additional costs, generated an integer linear programming model. Finally, consider the fact that vans can be rented from different companies. The capacities and costs per section are indicated as $C_k$ and $L_k$, with $k = 1,\dots, s$. Extend the model to also decide which company to rent the vans from, taking into account the fact that only one must be chosen.
This is my attempt:
First model. Let be $x_{ij}$ the quantity of product transported from the production site $i$ to the warehouse $j$. \begin{align} \text{minimize } \sum_{i=1}^n\sum_{j=1}^m c_{ij}x_{ij}\\ \sum_{j=1}^m x_{ij} &\leq p_i && \text{for $i=1,\dots,n$}\\ \sum_{i=1}^n x_{ij}&\geq d_j && \text{for $j=1,\dots,m$}\\ x_{ij}&\geq 0&&\text{for $i=1,\dots,n$ and $j=1,\dots,m$} \end{align}
Second model. Let $y_{ij}=1$ if the van travels from $i$ to $j$ and $0$ otherwise. \begin{align} \text{minimize } \sum_{i=1}^n\sum_{j=1}^m c_{ij}x_{ij}+L\sum_{i=1}^n\sum_{j=1}^m y_{ij}\\ \sum_{j=1}^m x_{ij} &\leq p_i&&\text{for $i=1,\dots,n$}\\ \sum_{i=1}^n x_{ij}&\geq d_j&&\text{for $j=1,\dots,n$}\\ x_{ij} &\leq C y_{ij}&&\text{for $i=1,\dots,n$ and $j=1,\dots,m$}\\ x_{ij}&\geq 0&&\text{for $i=1,\dots,n$ and $j=1,\dots,m$}\\ y_{ij}&\in \{0,1\}&&\text{for $i=1,\dots,n$ and $j=1,\dots,m$} \end{align}
Third model. Let $z_k=1$ if I rent from the company $k$ and $0$ otherwise. \begin{align} \text{minimize } \sum_{i=1}^n\sum_{j=1}^m c_{ij}x_{ij}+L\sum_{i=1}^n\sum_{j=1}^m y_{ij}+\sum_{k=1}^s L_k z_k\\ \sum_{j=1}^m x_{ij} &\leq p_i&&\text{for $i=1,\dots,n$}\\ \sum_{i=1}^n x_{ij}&\geq d_j&&\text{for $j=1\dots,m$}\\ \sum_{k=1}^s z_k&=1\\ x_{ij} &\leq C y_{ij}&&\text{for $i=1,\dots,n$ and $j=1,\dots,m$}\\ x_{ij} &\leq C_k z_{k}&&\text{for $k=1,\dots,s$}\\ x_{ij} &\geq 0&&\text{for $i=1,\dots,n$ and $j=1,\dots,m$}\\ z_k&\in \{0,1\}&&\text{for $k=1,\dots,s$} \end{align}
What do you think about it?
The first two look good. For the third one, you should not use $C$ and $L$. Instead, the fixed cost is $\sum_{i,j,k} L_k y_{i,j,k}$, and replace your fourth and fifth constraints with \begin{align} x_{i,j} &\le \sum_k C_k y_{i,j,k} &&\text{for all $i,j$}\\ y_{i,j,k} &\le z_k &&\text{for all $i,j,k$}\\ \end{align}
You should also specify that $y_{i,j,k} \in \{0,1\}$.