Thanks! There appear to be many different SQRT examples in optimal inventory management, but the SQRT's seem to come from two different sources: fixed cost amortizations and random walks from statistical deviations in Gaussian/normal distributions. Here's the simplest example I can think of: withdrawing cash from an ATM machine. s is constant spend rate per year; s = total $ spent in one year. r is simple interest rate per year. A is ATM fee per withdrawal; A is independent of amount of cash withdrawn. C is the amount of cash withdrawn on each trip to the ATM. We want to find the optimal amount of cash C* to withdraw on each trip to the ATM. C* minimizes the total ATM fees in one year + total interest foregone in one year. Trips to the ATM in one year = s/C Total ATM fees in one year = A*(s/C) Mean amount of cash on hand = C/2 (spend rate is constant) Simple interest foregone in one year = r*(C/2) Total costs in one year: T = A*(s/C) + r*(C/2) dT/dC = -s*A/C^2 + r/2 = 0 Solving for C, we get C* = C = sqrt(2*s*A/r) Suppose A=$5, r=3%, s=$30/year, C*=$100. i.e., even though I spend only $30 cash per year, I'm still best off withdrawing $100 every trip to the ATM. #trips to the ATM in one year = 30/100 = .3 trips/year ATM costs per year = $5*0.3 = $1.50 Mean cash on hand = $50 Simple interest foregone per year = $50*3/100 = $1.50 So, with optimum cash withdrawal amounts, the ATM fees equal the interest foregone. https://en.wikipedia.org/wiki/Economic_order_quantity At 06:37 AM 3/6/2018, Seb Perez-D wrote:

Square roots do appear in optimal inventory https://www.wiley.com/legacy/wileychi/waters/supp/Equations.pdf

In particular (S,s) models https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.24.1.183

Cheers,

Seb

On 5 March 2018 at 21:33, Henry Baker <hbaker1@pipeline.com> wrote:

...

I seem to recall a classical inventory management problem whose optimum solution involves a sqrt function.

Since I never took a Operations Research course, I think that this problem showed up in a calculus class & used Lagrange multipliers.

But after Googling a while, I couldn't find it.

Anyone here recall a similar problem?