Difficulty formulating Linear Programming Model Equation

I'm dealing with the following question and am not able to formulate a linear programming model for it: -
Suppose that the minimum number of security staff required at different hours are at the ith hour (i=1,2,…,24) are outlined as follows (see picture)
Each staff must work in 6 consecutive hours. The salary per hour is $35. Determine how many staff should start working at the beginning of ith hour to minimize the total cost.


So i converted the hours into 6 hour shifts
Time Period Shift Interval Min. No. of Staff Needed
1 Midnight - 6 AM 30
2 6 AM - 12 PM 44
3 12PM - 6 PM 36
4 6PM - Midnight 57

i. Let x1, x2, x3, x4 be the number of staff beginning their shift at the beginning of periods 1,2,3 and 4.
ii. Since the per hour salary is $35 per hour, each staff member will earn $210 (i.e. $35 * 6hours)
iii. Let Z be the total cost which needs to be minimised
iv. The objective function will look like this: -
Minimize Z = 210(x1 + x2 + x3 + x4)
v. Subject to the following Constraints: -
x1 + x2 ≥ 30
x2 + x3 ≥ 44
x3 + x4 ≥ 36
x4 + x1 ≥ 57
x1, x2, x3, x4 ≥ 0



Well-Known Member
Hi Austin,

Please try Let x0,x1, x2, x3, x4, ...,x23 be the number of staff beginning at hour 0,1,2,...23
Since the salary is the same you can ignore it and minimize (x0+x1+...x23)
Hint: please notice that a staff member that started at Xi will continue to work for 6 hours, how will you represent it?
Last edited: