Mathematical programming (simplex method)

[shikarihana]

Homework Statement

This question is about simplex method

A full time staff works seven hrs which includes an hour break in the middle (he is paid for his break). A part time staff works four hrs. The no of staff needed is shown below

Time No of staff needed
9-10am 6
10-11am 5
11-12noon 7
12noon-1p.m 8
1-2pm 8
2-3pm 7
3-4pm 5
4-5pm 6
At least 4 full time staff must be hired. The full time staff cost 12pounds per hr whilst a part time staff cost 7pounds per hr. Build a linear programming model that minimize the total cost.

The Attempt at a Solution

Let t=1 be the time at 9 to 10a.m
t=2 be the time at 10 to 11a.m
...
t=8 be the time at 4 to 5p.m.

let x_t = no of part time staff in time t
y_t = no of full time staff in time t
We need to minimize 7.5(x₁ + ... + x₈) + 12(y₁ + ... + x₈)

Subject to x_t >= 0, y_t>=0
Constraints : 7y₁ + 4x₁>= 6
7y₂ + 4x₂>= 5
7y₃ + 4x₃>= 7
7y₄ + 4x₄>= 8
7y₅ + 4x₅>= 8
7y₆ + 4x₆>= 7
7y₇ + 4x₇>= 5
7y₈ + 4x₈>= 6

My problem is to find the constraint for this question as I do not know how to write the constraint for (at least 4 full time staff is needed and the one hour lunch break) As you can see, the constraint I make is incomplete. It would be grateful if someone could tell me how to write the constraint.

Thanks a lot!

 

[mighty2000]

Thank you for your question. The constraint for at least 4 full time staff can be written as:

yt >= 4

This ensures that there are at least 4 full time staff working at any given time. As for the lunch break, you can add a constraint that accounts for the one hour break in the middle:

yt + xt >= 7

This ensures that there are at least 7 staff (full time and part time combined) working during the time period of 12pm to 1pm.

I hope this helps. Good luck with your linear programming model!