# Optimization problem with input variables

I've been trying to solve an optimization problem using Mathematica. I'm 100% familiar with the programme, I just practised a few tutorials available online, but they were very small and the problem is a bit larger. This is a screenshot of the actual problem

This is my attempt at using Mathematica to solve the problem.

I also copied the script from the screenshot above. The script is as follows:

NMaximize[{(1/T) (p (Q - E) + 10 x - 20 Q - 10 -
5 ((6000) 25^0.6 e^(-0.1 p) + (Subscript[t, 1]^3/(6 (0.04)) -
Subscript[t, 1]^2/2) + Subscript[Qt, 1]) - (5/2) (25 +
E) (T - Subscript[t, 1])),
Q == 25 + (((6000) 25^0.6 e^(-0.1 p))/(2 (0.04))) (0.04^2 - (0.04 -
T)^2 - (2 (0.04) (25^(1 - 0.6) - E^(1 - 0.6)))/(6000 (1 -
0.6) e^(-0.1 p))) && Q >= w && Subscript[t, 1] =
0.04 - Sqrt[(0.04 -
T)^2 - (2 (0.04) (25^(1 - 0.6) - E^(1 - 0.6)))/(6000 (1 -
0.6) e^(-0.1 p))] && Subscript[t, 1] >= 0 && E >= 0 &&
w >= E && Subscript[t, 1] >= 0 && T >= Subscript[t, 1] &&
m >= T}, {E, p, T, Q, Subscript[t, 1]}]

• Welcome to the Mathematica Stack Exchange. E represents the base of natural log. In general use lower case letters for user-defined symbols. Please update the post with this change.
– Syed
Nov 26 '21 at 13:09

In addition to correcting the equations, also include the constraint that p > 0
Clear["Global*"]
$Version (* "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)" *) (sys = {(1/t) (p (q - e) + s*e - c*q - o - h (α*w^β*E^(-λ*p) (t1^3/(6 m) - t1^2/2) + q*t1) - (h/2) (w + e) (t - t1)), q == w + ((α w^β E^(-λ*p))/(2 m))*(m^2 - (m - t)^2 - (2 m (w^(1 - β) - e^(1 - β)))/(α (1 - β) E^(-λ*p))), q >= w, t1 == m - Sqrt[(m - t)^2 + (2 m (w^(1 - β) - e^(1 - β)))/(α (1 - β) E^(-λ*p))], t1 >= 0, e >= 0, w >= e, t >= t1, m >= t, p > 0}); α = 6000; β = 0.6; λ = 0.1; c = 20; (* $$/unit *) h = 5; (*$$/unit/year *) m = 0.04; (* years *) o = 10; (* $$/order *) s = 10; (*$$/unit *) w = 25; (* units *)  Maximizing, NMaximize[sys, {e, p, t, q, t1}] (* {8366.52, {e -> 6.51561, p -> 28.1919, t -> 0.0231873, q -> 39.6589, t1 -> 0.00645804}} *)  • Don;t you think that$e$in the question stands for E in Wolfram Language (see the attached image))? Nov 26 '21 at 16:01 • @user64494 - Yes, that is why I entered e as E and converted the variable E to e. Nov 26 '21 at 16:12 • You should rewrite$e^{...}\$ as Exp[...], not as e^...  with a variable e. Hope I am clear. Nov 26 '21 at 16:18
• @user64494 - compare my equations with the image. Where the image showed e^x (exponentiation) I used E^x; where the image showed the variable E raised to a power (I.e., E^x) I changed this to e^x since the variable is now e Nov 26 '21 at 16:27
• @user64494 - in the image the only Exp terms are e^(-λ*p); in my equations these are all entered as E^(-λ*p). And in my equations, the image's E^(1-β) are entered as e^(1-β)`. I agree with your concept but not your reading of my equations. Nov 26 '21 at 16:43