Resource Allocation Problem using Statistics

#1
My client is a children's charity that provides entertainment (or experiences) for sick children in 10 different hospitals. I have the following business problem that am unsure how to go about it:

Problem: How many staffing hours (resources) to allocate at each hospital based on the number of experiences to children. We want to increase the amount of staffing hours only if it results in a significant increase in experiences to children. So for example if we were to increase hours by 20% then ideally we would like the number of experiences to increase by the same amount, if not more. Therefore, we need to work out the optimal hours that maximises the experiences.

My initial idea is that we look at say the past 2 years, for each day we know the number of staffing hours and the number of experiences. We can work out the relationship between those two variables, with a GLM, and see if the number of experiences plateau's after a certain number of staffing hours. The other big dependant variable here is the number of hospital admissions because we know that higher admissions means more experiences provided to children. Unfortunley we don't have that info for each day, but only as an annual figure, so unsure how we account for this.

I just want to know if there are other ways to go about this or if I'm on the right track?
 

hlsmith

Not a robit
#2
If you think there may be a nonlinear relationship then examining a general additive model may be beneficial. Also if you are thinking about increasing hours to a values larger than in your datasets, watch out for possible issues related to extrapolation (out of sample generalizations).

Secondarily, if you have a define relationship - linear programing is a good option for examining optimization problems. It is the technique behind saying you have so much wood and want to build chairs and tables, which each have their own ROI, which should you build more of and why.
 
#3
Thanks for your response. I will look into doing a general additive model and see if the relationship is non-linear. Am hoping it would be because it would allow us to choose an optimal staffing hours which would be a stationary point in the curve.

I might consider a linear program after doing a GLM first.