$1.$ For the problem, please provide mathematical formulations for the optimizations in negative null form specifying the following:

i$.$ Objective function

ii$.$ Decision variables

iii. Constraints

iv$.$ Typology of variables $($continuous$,$ integer, or mixed$)$

v$.$ Typology of functions $($linear$,$ convex, non$-$linear$)$

please solve the optimization

a$)$ Consider a $1000$ MW coal plant that can sell its generated electricity for $$0\mathrm{.}05/$kWh$.$ At what capacity factor should the plant operate over the course of a year in order to maximize its profits? The fuel cost of the plant is $$32/$MWh and its operating cost is a function of the capacity factor xcf as follows: