I am currently trying to solve a problem where I need to minimise transport cost through the choice of vehicle (and how many of each choice) subject to a given demand.

The problem:

There are currently 3 vehicle sizes corresponding to their haulage capacity, an associated daily cost, and a daily demand.

I need an objective function that minimises the cost through the choice of vehicle whilst satisfying the sales demand, however, I do not know how to define an expression that is based on the number of vehicles used * the cost per day, as it depends on the weight.


# Vehicle capacity
truck_capacity_dict = {
    '7.5T': {'capacity': 7500},
    '12T': {'capacity': 12000},
    '44T': {'capacity': 44000}

# Vehicle daily costs
truck_capacity_dict = {
    '7.5T': {'rate': 350},
    '12T': {'rate': 660},
    '44T': {'rate': 2000}

# Daily demand 
sales_demand_tonnes = {
'2020-01-01': 300,
'2020-01-02': 293,
'2020-01-03': 176

Mathematically, this is similar to the below expression where the OF is to minimise the costs based on the choice of vehicle:

$$\min \sum V_{t, v} * C_{t, v} \forall t \subset T, v \subset V$$

However, I do not know how to formulate an expression in python that determines HOW MANY vehicles are chosen, as this depends on the weight.

  • $\begingroup$ Are you trying to model a VRP variant or it is about the cargo optimization problem? $\endgroup$ – A.Omidi Dec 16 '20 at 9:19

I suggest introducing $3n$ variables. Let $ c_{i} $ be the daily rate of each vehicle $i=1,2,3$.

$ x_{i,j} $ designates the quantity of vehicles which will be used every day: the subscript $j$ indicates the day in interest. Supposing to consider $n$ days, we have $j=1, 2, \ldots, n$ days. Clearly, $ x_{i,j} $ is a non-negative integer numbe and $ x_{i,j}=0$ means that no vehicle of kind $i-th$ is used in $j-th$ day.

The total cost based on the number of vehicles used is $ \min \sum_{j=1}^n \sum_{j=1}^3 c_i \cdot x_{ij}$


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