I'm working on a project involving quadratic equations and I'm supposed to use them to determine the optimal selling price and quantity for a product.
I've tried using a linear demand equation and then modeling profit and price with a quadratic, but the results I have are pretty inaccurate.
Here's a brief explanation of the steps I took:
Linear equation modeled using real world data:
$q = -8.8p + 6320$
q - quantity
p - price
$P = pq - cq$
P - profit, c - production cost
$c = $190 + $\frac{$7000}{q}$ (Here the 7000 USD cost includes tools, marketing and storage price and it is supposed to be distributed over q, the qty of the desks to be sold. The 190 USD cost is a wooden board, desk legs and wood treatment, so this is per desk)
If I replace q with the demand equation in the Profit equation:
$$P = p(-8.8p + 6320) - \frac{190+7000}{-8.8p+6320})(-8.8p + 6320)$$
$$P = -8.8p^2 + 6320p -190(-8.8p + 6320) - 7000$$ $$P = -8.8p^2 + 7992p - 1207800$$
454 selling price, 606747 profit
Quantity of desks to be manufactured and sold: $$q = -8.8 \times 454 + 6320 q = 6320 - 3995 q = 2325 \ desks$$
I do know the $7000 budget to produce 2000 desks isn't nearly enough, but that's easily fixable.
How do I make a more realistic, non-linear estimation? Any help is appreciated?
you may be looking for a solver (Gurobi, CPLEX) to optimize your profit. Math formulation would be
Assuming $q = 6320 -8.8p $ is already derived
Max $P = q(p-c)$
s.t.
$ 190 + cq \le B$ where B = 7000
$ q +8.8p = 6320$
$ 0 \le q,p$