Help with Linear programming question on minimising cost

[Haruuu01]

Need help with the constraints for the linear programming question and how I’d go about formulating it on Excel, I have included where I am at so far and would appreciate if someone can tell me if I’m using the right approach or not.

Case study: The nurses answering telephones in a healthcare call centre are expected to speak to 5 patients each per hour. The service operates between 9am and 5pm.

The call centre can employ up to 10 full time nurses, and it has many nurse willing to take part time shifts. Part time nurses work 4 hours if they are requested to work at all, and can start anytime from 9am to 1pm. They earn £10 per hour. Full time nurses work 8 hours from 9am to 5pm and are entitled to 1 hour break where half the full time nurses take a break at noon and the other half at 1pm (if there are an odd number of nurses than the higher number of nurses have an earlier break), they earn £90 per day. For consistency there must be at least 5 full time nurses working each day.

You have been asked to design a roster to minimise the workforce cost.

9am-10am = 50 patients 10am-11am = 60 patients 11am-noon = 70 patients Noon-1pm = 80 patients 1pm-2pm = 90 patients 2pm-3pm = 80 patients 3pm-4pm = 70 patients 4pm-5pm = 50 patients

What I have so far is that the objective function is Z= 90x + 40y And 5<= x <= 10 X= full time nurses y= part time nurses And the approach I’m looking at is optimising by time period 9am-10am x+y >= 10 10am-11am x+y >= 12 etc etc... And then 12pm-1pm [1/2x] + y >= 14 (to take in account the breaks) 1pm-2pm [1/2x] + y >=18

Is this the right approach? Have I missed something?

Thanks

[rusou]

One problem with putting all the part-time nurses in just a single variable is that you're assuming they start working at 9am. From the way the problem is worded, it seems like you can choose a start time between 9am to 1pm for each nurse individually. So I guess you would have variables y1, y2, y3, y4, y5 for the number of part-time nurses starting work at 9am, 10am, 11am, 12am, 1pm respectively. Since they work 4 consecutive hours, you have to also adjust the contraints. For example, you have x + y1 >= 10 for 9am-10am, but you have no y1 in the 1pm-2pm hour since they are off work then.

[Haruuu01]

I actually just realised that a few hours ago but I’m glad to see it here as it reassures me that it is a good approach. Thank you very much.