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

can be calculated as follows

$=21^2+82^2612+41$

$=41^222^2+21221$

Here $1$ and $2$ 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 $1=2=0\mathrm{.}5($i$.$e$.,$ the fuel burn and weight are equally weighted$).$ Derive

the analytical expression of $$ as a function of $1$ and $2$

b$)$ Find the optimal point