Write a Python function that, given a list of 3D component dimensions and material properties, computes the optimal extrusion profile to minimize weight while meeting a target stiffness. Return the profile dimensions and weight reduction percentage versus a standard profile.

MECE framework: 1) Validate inputs (dimensions, material density, target stiffness). 2) Compute baseline weight and stiffness of standard profile using beam theory. 3) Generate candidate cross‑sectional profiles (e.g., I‑beam, T‑beam, rectangular). 4) For each candidate, calculate weight = density × volume and stiffness = (E × I)/L using simplified beam equations. 5) Filter candidates that satisfy stiffness ≥ target. 6) Select candidate with minimum weight. 7) Return dimensions and weight‑reduction % relative to baseline.

The solution begins by validating the input list of dimensions and material properties, ensuring no missing or negative values. Next, I compute the baseline weight and stiffness of a standard extrusion profile using the formula weight = density × volume and stiffness = (E × I)/L, where I is the second moment of area. I then generate a set of candidate cross‑sections (I‑beam, T‑beam, rectangular) and iterate through them. For each candidate, I calculate its weight and stiffness using the same simplified beam equations. Candidates that meet or exceed the target stiffness are retained, and I select the one with the lowest weight. Finally, I return the chosen profile dimensions and the weight‑reduction percentage compared to the baseline. This approach balances computational efficiency with design rigor, ensuring the solution is both practical and scalable.