Question

A school is conducting optimization studies of the resources it has. One of the principal concerns of the Director is that of the staff. The problem he is currently facing is with the number of guards in the "Emergencies" section. To this end, he ordered a study to be carried out that yielded the following results:

Time Minimum number of guards required
O to 4 40
4 to 8 80
8 to 12 100
12 to 16 70
16 to 20 120
20 to 24 50

Each guard, according to Federal labor law, must work eight consecutive hours per day. Formulate the problem of hiring the minimum number of guards that meet the above requirements, as a Linear programing model. 

Answer 1

We have the following scheduling requirements

Shift	Time(24hour system)	Minimum number of guards
1	0-4	40
2	4-8	80
3	8-12	100
4	12-16	70
5	16-20	120
6	20-0	50

Now we are given that each guard must work 8 consecutive hours. So, every guard works for two consecutive shifts. Now, be the number of guards joining duty at the beginning of the 1st,2nd,3rd,4th,5th and 6th shifts, respectively. Then, as every guard works for two consecutive shifts, so the total number of guards in a shift is the number of guards in the previous shift and the number of guards that joined in the current shift. Using this, we can develop all the required constraints. And the objective function would be to minimize the total number of guards. So, the complete linear programming problem for the given situation is:-

Subject to

   (as the number of guards cannot be negative)

This is the required linear programming problem.

We can solve this using Excel Solver, which gives us the solution, the minimum number of guards required is 260, and the shiftwise allocation is given by the table(yellow cells)

Shift	1	2	3	4	5	6
Number of guards	40	40	60	10	110	0
Total guards	260						total	minimum
Min shift 1	1					1	40	40
Min shift 2	1	1					80	80
Min shift 3		1	1				100	100
Min shift 4			1	1			70	70
Min shift 5				1	1		120	120
Min shift 6					1	1	110	50