$($b$)$ NovaTech $($Nova$)$ manufactures high$-$end gaming laptops, and they're concerned about the recent negative publicity surrounding a competitor's product failure. Historically, Nova has encountered occasional issues with overheating in their laptops, which can typically be detected and repaired during an internal inspection. The inspection process costs $$6$ per laptop, with repair costs amounting to $$1\mathrm{.}25$ per unit. Last year, all $150,000$ laptops were inspected internally, revealing that $8\%$ had overheating issues. Additionally, another $4\%$ of laptops had overheating problems that went undetected during internal inspections.

Defective units that were sold and shipped to customers had to be recalled for repairs. The cost of recalling and repairing these laptops amounts to $$10$ per unit. However, the direct costs of external failures don't fully capture the consequences. For $25\%$ of the externally failed units, negative reviews and word$-$of$-$mouth damage led to a loss of future sales, resulting in a $$400$ reduction in profits for each affected unit.

Required:

$($i$)$ Calculate the cost of the appraisal process.

$($ii$)$ Determine the cost of internal failures.

$($iii$)$ Calculate the direct costs associated with external failures.

$($iv$)$ Identify the opportunity cost linked to external failures.

$($v$)$ What are the total quality costs?

$($vi$)$ Nova is considering an alternative inspection plan to reduce upfront costs. This new technique costs $$2$ per laptop inspected and detects only $6\%$ of laptops with overheating issues during inspection, with the remaining $6\%$ identified after sale and shipping. What are the total quality costs associated with this alternative approach?