$\backslash (1-\backslash )$

Cat $\backslash (\backslash$dot$\{$C$\}\backslash )$ makes a product using three machines $\backslash (-$X$,$ Y $\backslash )$ and $\backslash ($ Z $\backslash ).$ The capacity of each machine is as follows:

The demand for the product is $1,000$ units per week. For every additional unit sold per week, net present value increases by $\backslash (\backslash$$ $50,000\backslash ).$ Cat Co is considering the following possible purchases $($they are not mutually exclusive$)$:

Purchase $1$ Replace machine $\backslash ($ X $\backslash )$ with a newer model. This will increase capacity to $1,100$ units per week and costs $\backslash (\backslash$$ $6\backslash$mathrm$\{$~m$\}\backslash ).$

Purchase $2$ Invest in a second machine Y$,$ increasing capacity by $550$ units per week. The cost of this machine would be $\backslash (\backslash$$ $6\mathrm{.}8\backslash$mathrm$\{$~m$\}\backslash ).$

Purchase $3$ Upgrade machine $\backslash ($ Z $\backslash )$ at a cost of $\backslash (\backslash$$ $7\mathrm{.}5\backslash$mathrm$\{$~m$\}\backslash ),$ thereby increasing capacity to $1,050$ units.

Required:

Which is Cat Co's best course of action?