$)$ Tosla needs to schedule its workforce for the upcoming week in order

to maintain efficient operation across all its departments, including battery, body, assembly,

paint, and quality control. Each department has specific minimum and maximum staffing needs

for three shifts: morning, afternoon, and night for each day of the week $($Mon$-$Sun$).$ Tosla$$s

workforce is flexible, but workers have specific availabilities, preferences for working certain

shifts, and different effectiveness scores that measure their productivity in certain roles $($on a

scale of $1-10,$ with $10$ being the most preferred or effective$).$

The goal is to maximize the total preference$-$adjusted effectiveness of the workers scheduled

for each shift, subject to the following constraints:

$$ Each worker can only be assigned to work a maximum of one shift per day.

$$ Each worker can only work a maximum of $5$ days a week.

$$ Each worker can only be assigned to shifts that they are available for.

$$ The total number of workers assigned to a shift in a department must meet the department$$s

specific minimum and maximum staffing requirements.

Your task is to formulate this scheduling problem as a linear$/$integer programming problem

and solve it using an appropriate optimization software $($preferably Gurobi$).$

Use the randomly generated data given below to solve the model where we assume there

are $100$ skilled workers waiting to be assigned to a shift and department. Keep in mind that,

since this is a randomly generated dataset, some of you might be working with infeasible problems.

$($a$)(10$ points$)$Define the parameters, sets, decision variables, constraints, and the objective

you are using to model.

$($b$)(20$ points$)$Formulate the problem algebraically