a performance index
A performance index is the clever move of squeezing several material properties into one number that you simply maximize, the way a fuel-economy rating blends engine and weight into a single score you can compare at a glance.
Here is the derivation for a light, stiff tie rod: you want minimum mass for a given stiffness. Mass is m = rho x A x L (rho is density, A is cross-section area, L is length). The rod's stiffness is S = E x A / L (E is Young's modulus), so A = S x L / E. Substitute back and m = S x L^2 x (rho / E). Length and stiffness are fixed, so to make m smallest you make E / rho largest — that is the index for this problem. Change the part to a beam loaded in bending and the index becomes E^(1/2)/rho instead.
This matters because materials with a high index cluster along one guideline on an Ashby chart, so a single line picks your winners. The honest caveat: the index depends on the loading mode and on what is held fixed. A tie, a beam, and a panel give the indices E/rho, E^(1/2)/rho, and E^(1/3)/rho respectively — quote the wrong one and you select the wrong material.
Two rods must carry the same tensile load at the same stiffness. Steel has E=200 GPa, rho=7.8 g/cm^3; aluminum has E=70 GPa, rho=2.7 g/cm^3. The tie index E/rho is 25.6 for steel and 25.9 for aluminum — nearly identical, so for a simple tie they weigh about the same. Switch to a bending beam, where the index is E^(1/2)/rho: steel gives 1.8, aluminum 3.1 — now aluminum is clearly the lighter choice.
Same two metals, different loading, different winner — because the right index depends on how the part is loaded.
A performance index only makes sense once you have fixed the geometry constraint (tie, beam, or panel); quoting 'E/rho' without saying which loading mode is a common mistake.