Tesla is producing four models in its facilities in cities around the world: Model $3,$ Model S$,$ Model X$,$ and Model Y$.$ It is estimated that demand for each car model in $2025$ will be $400,000,500,000$ or $600,000$ units; with probabilities $0\mathrm{.}4,0\mathrm{.}2,$ and $0\mathrm{.}4,$ respectively. The company is focusing on $4$ assembly plants in this problem: Fremont $($Model $3),$ Austin $($Model S$),$ Gru $$nheide $($Model X$),$ and Shanghai $($Model Y$)$ each dedicated to one particular model.

a$)$ The company decides to establish $500,000$ units of capacity in each plant prior to $2025$ and has no chance to change that within the year. What is the expected total sales for Tesla in $2025?$b$)$ If the company can adopt a complete flexibility configuration and still establishes $500,000$ units of capacity in each assembly line prior to $2025,$ what is the expected total sales for Tesla in $2025?$

$1$

c$)$ It turns out that a complete flexibility configuration can be very costly$.$ Tesla is interested in a limited flexibility configuration such that each model can be produced in one other line. Tesla decided to invest in plants such that Fremont can also produce Model S$,$ Austin can also produce Model $3,$ Gru $$nheide can also produce Model Y$,$ Shanghai can also produce Model X$.$ What is the expected total sales for Tesla in $2025,$ if it uses this configuration? $($Assume again that the capacity of each plant is $500,000).$

d$)$ Recommend a better limited flexibility configuration to Tesla. Construct a demand scenario in which the total demand is partially satisfied in the previous configuration in part c$)$ but fully satisfied in the configuration you suggested.