Consider convex facility cost function and compare to current linear one

[loladekhuijzen]

We currently use the overprovision costs.
Cost(Lⱼ) = w × (s_min - Lⱼ) if Lⱼ < s_min Cost(Lⱼ) = 0 if Lⱼ ≥ s_min

What if we use the total cost per student instead?
Cost(Lⱼ) = (C_0 + α × max(0, Lⱼ - s_min)) / Lⱼ
α = C_0/s_min? (the cost doubles when load doubles past s_min)

[jipclaassens]

@cjacobscrisioni deelt de kosten functie

[cjacobscrisioni]

Quick analysis of costs: I have taken the total primary school costs from Jacobs-Crisioni et al, 2023, and divided them by nr of pupils. According to Excel, the result is best approximated with a power law function y=51712*x^-0.535 , r^2 of ~0.91

My proposal is to use this as the convect cost per student function. @loladekhuijzen , OK?

[loladekhuijzen]

yes! @cjacobscrisioni, we have implemented the power law facility function for the Greedy algorithm and performed a grid search comparing it against the overprovision cost function.

power law using greedy.pdf

to be able to implement the power law function in the bounded scenario, we likely need to approximate it using a piecewise linear approximation, as we are using a Linear Programming model.

[cjacobscrisioni]

@cjacobscrisioni suggereert lineaire functie met 2 kinks

[cjacobscrisioni]

Linear approximation of convex costs (x = number of students):

0 < x < 10: -11597x + 124656.48
10 <= x < 150: -39.866x + 9044.9
x >= 150: -0.326x + 3717.5

Methodology: I have taken linear approximations of cost for small / medium / large schools in Excel, and then adjusted the constant cost bit to ensure that the three subfunctions have smooth transitions at the intended inflection points (ie, they lead to the same cost per student at 10 students / 150 students, respectively).

[cjacobscrisioni]

@loladekhuijzen , @MaartenHilferink , can you work with the above approximation ^?

[loladekhuijzen]

Thanks @cjacobscrisioni! Sorry for the oversight, but since our models use the total facility cost function, we first apply the cost per student with the load, and so (if I'm not mistaken) this approximation would no longer be linear.

During today's meeting, we concluded that the power law function may close schools too aggressively, and we want to aim for something that more closely resembles the cost function we had before. What that function should look like isn't clear yet, Lewis suggested we share our current ideas with Bernard and ask for his input.