Example $5$: A simple assembly process

A manufacturer produces two subparts, A and B $,$ and one end product C$.$ Subpart B also can be sold with price $20$ t$//$unit$,$ while selling price of $\backslash ($ C $\backslash )$ is $\backslash (30\backslash$mathrm$\{$tl$\}/\backslash$mathrm$\{$unit$\}\backslash ).$ Raw materials used in the production process and the assembly chart of the parts are shown in the following figure:

The manufacturer wants to find optimum production volumes of the subparts and the end product to maximize the profit