PROBLEM $1(4$ pts$)$: XYZ Machine Inc. is planning for an upgrade project for one of its metal$-$forming process. The project consists of two parts: i$)$ upgrade of the equipment with better yield $($i$.$e$.,$ lower number of defective parts produced$)$ and ii$)$ upgrade the inspection equipment$/$devices used to sort out the defective parts. For the equipment upgrade part, there are three machine models being considered:

Machine Model A with capital cost of $$350,000,$ with yield of $90\%($provided by OEM$)$

Machine Model B with capital cost of $$400,000,$ with yield of $93\%($provided by OEM$)$

Machine Model C with capital cost of $$450,000,$ with yield of $99\%($provided by OEM$)$ For the inspection equipment, there are two models being considered:

a$.$ Inspection machine $x$ with capital cost of $$50,000,$ with accuracy of $95\%($sorting $95\%$ of defectives$)$

b$.$ Inspection machine Y with capital cost of $$150,000,$ with accuracy of $99\%($sorting $99\%$ of defectives$)$ The management team has set the total budget for this upgrade project to be $$500,000.$ The target for the Outgoing Quality $($percentage of defective parts after the inspection$)$ to be less than $0\mathrm{.}0007(0\mathrm{.}07\%).$ Based on the budget of $$500,000,$ which Machine model and which inspection machine should be selected to achieve the target of outgoing quality $)(0\mathrm{.}0007?$