# Mathematical programming (simplex method)

1. Apr 7, 2010

### shikarihana

1. The problem statement, all variables and given/known data
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.

3. 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 xt = no of part time staff in time t
yt = no of full time staff in time t
We need to minimize 7.5(x1 + ... + x8) + 12(y1 + ... + x8)

Subject to xt >= 0, yt>=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!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution