What is wrong with the one step result?
Q = Qr + n*q;
Costqu = 1/2*c*q^2 + F;
MCostqu = c*q;
Profitqu = p*q - Costqu;
Demand = A - B*p; polys = {p - MCostqu, Demand - Q,
ExpectedValue[Profitqu, WeibullDistribution[2, 2], Qr] - Inv}
(* Out[1695]= {p - c q,
A - B p - n q - Qr, -Inv + 1/2 (-2 F + 2 p q - c q^2)} *)
We can solve as below. Note that this is equivalent to the original formulation.
sol = Solve[polys == 0, {n, p, q}]
(* Out[1696]= {{n -> (-2 B c F - 2 B c Inv -
Sqrt[2] A Sqrt[c] Sqrt[F + Inv] +
Sqrt[2] Sqrt[c] Sqrt[F + Inv] Qr)/(2 (F + Inv)),
p -> -Sqrt[2] Sqrt[c] Sqrt[F + Inv],
q -> -((Sqrt[2] Sqrt[F + Inv])/Sqrt[c])}, {n -> (-2 B c F -
2 B c Inv + Sqrt[2] A Sqrt[c] Sqrt[F + Inv] -
Sqrt[2] Sqrt[c] Sqrt[F + Inv] Qr)/(2 (F + Inv)),
p -> Sqrt[2] Sqrt[c] Sqrt[F + Inv],
q -> (Sqrt[2] Sqrt[F + Inv])/Sqrt[c]}} *)
Now it's straightforward to check the result.
Simplify[polys /. sol]
(* Out[1697]= {{0, 0, 0}, {0, 0, 0}} *)
