How to Formulate an Employee Scheduling Linear Program?

[sara_87]

Homework Statement

I have an employee scheduling table. The hours of observation and the required number of assistants are as follows:
shift: assistants(a)
12 mid - 2 am a=3
2 am - 4 am a=4
4 am - 6 am a=4
6 am - 8 am a=6
8 am - 10 am a=8
10 am - 12 noon a=10
12 noon - 2 pm a=11
2 pm - 4 pm a=10
4 pm - 6 pm a=8
6 pm - 8 pm a=6
8 pm - 10 pm a=5
10 pm - 12 am a=4

Each assistant is paid at the samr hourly rate. Each assistant works eight consecutive hours. I have to only formulate an employee-scheduling Linear Program (i.e write down the objective function and constraints saying what are the decision variable, verbal model and mathematical model)

Homework Equations

The Attempt at a Solution

I don't know how to start because all the other question i did on linear programming are simple ones and not about scheduling. please give me a few hints or lead me in the right direction.
Thank you

 

[mighty2000]

for your question. Scheduling can be a complex problem to solve using linear programming, but with the right approach, it can be broken down into manageable pieces. Here are some steps you can follow to formulate your employee-scheduling LP:

1. Identify the decision variables: In this case, the decision variable is the number of assistants assigned to each shift. Let's call this variable x.

2. Write down the objective function: The objective of this LP is to minimize the total cost of the assistants' wages. Since each assistant is paid at the same hourly rate, the total cost will be directly proportional to the number of assistants assigned to each shift. Therefore, the objective function can be written as:

Minimize Z = 3x1 + 4x2 + 4x3 + 6x4 + 8x5 + 10x6 + 11x7 + 10x8 + 8x9 + 6x10 + 5x11 + 4x12

3. Write down the constraints: The constraints in this LP will ensure that each shift is adequately staffed and that each assistant works eight consecutive hours. The constraints can be written as:

x1 + x2 + x3 >= 3 (for the 12 am - 2 am shift)
x2 + x3 + x4 >= 4 (for the 2 am - 4 am shift)
x3 + x4 + x5 >= 4 (for the 4 am - 6 am shift)
x4 + x5 + x6 >= 6 (for the 6 am - 8 am shift)
x5 + x6 + x7 >= 8 (for the 8 am - 10 am shift)
x6 + x7 + x8 >= 10 (for the 10 am - 12 pm shift)
x7 + x8 + x9 >= 11 (for the 12 pm - 2 pm shift)
x8 + x9 + x10 >= 10 (for the 2 pm - 4 pm shift)
x9 + x10 + x11 >= 8 (for the 4 pm - 6 pm shift)
x10 + x11 + x12 >= 6 (for the 6 pm - 8 pm shift)
x11 + x12 + x1 >= 5 (for the 8 pm - 10 pm shift)
x12 + x1