# Thread: Help with Optimization of functions of n-variables!

1. ## Help with Optimization of functions of n-variables!

Hi, I'm not sure what to use for a formula for this question. Any help is greatly appreciated

An accounting firm uses partners and staff to produce an audit. The quality of
the audit (as measured by reduction in litigation liability and the likelihood of
audit errors) is a function of the composition of the audit team. In particular

$\displaystyle r = P^1^/^2 S^1^/^4$

where r is the audit quality, P is the partner-hours devoted to the audit, and S
is the sta ff hours devoted to the audit. Notice that both partners and staff are
essential for the audit quality, and that audit quality increases in the amount
of either, but at a decreasing rate. The cost of a partner hour is 100 while
the cost of a sta ff hour is 20. The budget for this audit is 3000. How many
partner and sta ff hours will the accounting firm choose to maximize quality
subject to this budget constraint?

2. Note that this is actually a one dimensional problem.

We must maximize $\displaystyle r(p,s)=p^{1/2}s^{1/4}$ with $\displaystyle 100p+20s = 3000$ (assuming all of the budget is used)

Then $\displaystyle s = (3000-100p)/20 = 150-5p$. Thus we need to maximize $\displaystyle r(p,s)=\tilde{r}(p)= p^{1/2}(150-5p)^{1/4}$ (is only a function of p)

Thus find $\displaystyle 0<p<30$ such that $\displaystyle \tilde{r}'(p)=0$. (why?)

Good luck.

3. ## Re: Help with Optimization of functions of n-variables!

That is somehow an amazing solution Dinky, I think that formula will surely work to solve Sean's problem.

"I hate litigation support problems."