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Math Help - Help with Optimization of functions of n-variables!

  1. #1
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    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

    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?
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Note that this is actually a one dimensional problem.

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

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

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

    Good luck.
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  3. #3
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    Midland, TX
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    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."
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