Formulating a Mixed Integer Programming Problem

[I Like Pi]

Homework Statement

Not sure if this type of math goes in this section, but I have a quick question regarding a MIP problem. It's simple to grasp, but I'm not sure whether or not I am formulating this problem correctly.

A city needs to hire workers to clean the snow. The city is divided into j areas, which require a certain number of trucks, n_j, within area j. The workers are required to place a tender at the beginning of the ear, which includes the price per truck and the number of trucks they can supply. We must formulate a model where i workers are tendered, t_i is the trucks available from worker i, c_ij is the price per truck per area j, and m_j is the minimum number of workers for area j. And the workers can't supply more trucks than are necessary for their areas

Homework Equations

The Attempt at a Solution

Let xij = number of trucks supplied by contractor i to area j
Let yij = 1 if worker i supplies in trucks in area j, 0 if worker i does not supply trucks in area j

I want to minimize Z = $\sum$c_ijx_ij

My constraints are:

$\sum$x_ij ≤ t_iy_i (i'm not sure about the y_i part)

$\sum$x_ij = n_j

$\sum$y_ij ≥ m_j

 

[mighty2000]

I would suggest the following approach to help you formulate your problem correctly:

1. Clearly define your objective: In this case, it seems like your objective is to minimize the cost of hiring workers to clean the snow.

2. Identify all relevant variables: From the problem statement, it seems like the variables you will need are the number of workers, the number of trucks supplied by each worker, the price per truck, and the minimum number of workers required for each area.

3. Define the relationships between the variables: Based on the problem statement, it seems like the number of workers and the number of trucks supplied by each worker are directly related. The price per truck may also be a factor in determining the number of workers to hire. Additionally, there is a minimum number of workers required for each area.

4. Formulate the objective function: Based on your objective of minimizing cost, your objective function could be something like Z = \sumcijxij, where cij is the price per truck in area j supplied by contractor i.

5. Formulate the constraints: Your constraints should reflect the relationships between the variables you identified in step 2. For example, you could have a constraint that limits the total number of trucks supplied by all workers to be less than or equal to the total number of trucks needed in each area. You could also have a constraint that ensures each area has at least the minimum number of workers required.

6. Test your model: Once you have formulated your model, it is important to test it with different scenarios to ensure it is working as intended.

Overall, it seems like you are on the right track with your formulation. However, I would suggest including the price per truck in your constraints to ensure that the total cost is minimized. Additionally, you may want to consider adding a constraint that ensures each worker can only supply trucks to areas they are assigned to.