USA Mathematically show importance of an incorrect COGS

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I work at a small manufacturing business and we currently write off Inventory into COGS as soon as we purchase it - we have no inventory account. We also do not track hours per job, so payroll/labor is certainly also incorrectly incorporated into COGS. I am not an accountant, but I have done enough research to setup a cost accounting system at our company to fix these problems. I need additional help convincing my boss the importance and value that this system adds to our company.

1. Any suggestions on how to show the value of this? Obviously I can list off that our balance sheets are incorrect, our inventory isnt under control, cant calculate margins, etc. which this fixes.

2. Is there any way to mathematically graph COGS vs some important dependent variable to show how an incorrect COGS can be dangerous?
 

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