Inventory Theory - No Shortages Allowed - Deterministic Model
Book: "Introduction to Operations Research" 6th Ed. Hillier & Lieberman
Section 17.3 (Pg 762)
On page two how are they solving for $Q*$. When I take first and second derivative I get:
$$f^\prime(Q) = \frac{2ak}{x^3}$$
$$f^{\prime\prime}(Q) = -\frac{6ak}{x^4}$$
How is $Q* = \left(\frac{2ak}{h}\right)^{1/2}$?
Am I using the hint of $f^\prime$ is equal to zero incorrectly? Thanks for the help.
Not quite sure what you're asking here but on page 1, they show: $$ T = \frac{aK}{Q}+ac+\frac{hQ}{2} $$ So $$ \frac{\mathrm{d}T}{\mathrm{d}Q} = aK\frac{\mathrm{d}}{\mathrm{d}Q}Q^{-1} + \frac{h}{2}\frac{\mathrm{d}}{\mathrm{d}Q}Q = -aKQ^{-2} + \frac{h}{2} $$ You now set this to zero: $$\begin{aligned} -\frac{aK}{Q^{2}} + \frac{h}{2} &= 0 \\ \frac{aK}{Q^{2}} &= \frac{h}{2} \\ 2aK &= hQ^{2} \\ Q &= \sqrt{\frac{2aK}{h}} \end{aligned}$$