Fuel burn $()$ and weight $()$ are two important factors in aircraft design, and they

can be calculated as follows

$=2!$

$"+8"$

$"6!"+4!$

$=4!$

$"2"$

$"+2!"2!$

Here $!$ and $"$ are the relative spans and chords of the wing $($design variables$),$ respectively.

Conduct a multi$-$objective optimization to consider both fuel burn and weight. Requirements:

a$)$ Construct a composite objective function $$ using the weighted sum method with

weights $!="=0\mathrm{.}5($i$.$e$.,$ the fuel burn and weight are equally weighted$).$ Derive

the analytical expression of $$ as a function of $!$ and $"(20\%)$

b$)$ Find the optimal poin