Manufacturing Management Case $($Show all your works: including all computational procedures and details$)$

General Precision Machinery Company $($GPM$)$ produces and sells Reagent Measurement Devices $($RMD$).$ The weekly demand for RMD is $1,000$ units. Each RMD unit is assembled using two units of Type $1$ components and one unit of Type $2$ components. Manufacturing a Type $2$ component typically takes around $0\mathrm{.}6$ weeks, while a Type $1$ component is completed in approximately $0\mathrm{.}4$ weeks. The selling price of each RMD unit is $$600,$ with the cost per unit of the Type $1$ component set at $$100$ and the Type $2$ component at $$200.$ However, maintaining inventory $($In Components Process Stock$)$ of the Type $1$ component in production incurs a weekly holding cost of $$10$ per unit $($For example, if GPM hold $2$ units of inventory for one week, $2,$ while for the Type $2$ component, this cost is $$12$ per unit per week. Upon completion, the finished components undergo rigorous scrutiny by the quality control department. Only those components that meet the department's stringent quality standards are forwarded to the assembly plant for further processing and integration into the final assembly. While Type $2$ components consistently achieve a $100\%$ success rate in production, Type $1$ components are less reliable, with approximately $5\%$ of units rendered defective due to imperfections in the manufacturing process. These defective units are discarded entirely. Considering that the weekly demand for RMD is $1,000$ units and $5\%$ of Type $1$ components are deemed defective, it becomes clear that Type $1$ components must be produced in quantities exceeding immediate demand $(2000$ units$)$ to account for potential defects. The quality inspection period is negligible and can be disregarded. Once the components pass inspection, they are sent to the assembly plant for final product assembly, a process estimated to take around $0\mathrm{.}2$ weeks$/$per component $($The assembly time per component is $0\mathrm{.}2$ weeks. Therefore, for example, assembling $2$ Type $1$ components and $1$ Type $2$ component into a single RMD would take $0\mathrm{.}6$ weeks.$).$ Components held in inventory during this stage $($In Assembly Process Stock$)-$incur holding costs. For inventory management, the weekly holding cost is $$10$ per unit for Type $1$ components, while for Type $2$ components, it is higher at $$20$ per unit per week. Once manufacturing is complete, the products are transported to retail stores operated by GPM in major cities. The transportation process takes approximately $0\mathrm{.}5$ weeks. During transport, products $($In Transportation Process Stock$)$ incur transportation management costs, amounting to about $$10$ per unit per week. Upon arrival at GPM$-$operated stores, RMD units are typically sold to customers after an average of $0\mathrm{.}2$ weeks in stock $($Finished Goods Process Stock$).$ For inventory management, the weekly inventory holding cost for a RMD unit amounts to $$30.$ Some RMD units may not be properly assembled during the assembly process, resulting in malfunctioning products. $10\%$ of RMD units assembled during the assembly process are malfunctioning products. Customers who purchase these malfunctioning units return them, and the returned units are sent back to GPM$\text{'}$s assembly plant for proper reassembly. The transportation time for malfunctioning products to the assembly plant is $0\mathrm{.}5$ weeks, during which the units $($In Transportation Process Stock$)$ incur transportation management costs of approximately $$10$ per unit per week. Upon arrival at the assembly plant, the malfunctioning units are disassembled and reintroduced into the assembly process along with other components. Since the disassembly time is very short, it can be disregarded. Once the assembly is complete $($taking approximately $0\mathrm{.}2$ weeks$/$per component$),$ the customers who returned the products collect them directly. To compensate for the inconvenience, GPM offers a $$50$ return refund to affected customers.

$($a$)$ Evaluate the$$

Weekly Profit$=$Weekly Sales Revenue$$Component Costs$$Inventory$/$Transportation Management Costs$$Scrap Costs$$Return Refund

$($b$)$ Evaluate the following two options and choose the optimal option $($assuming their investment costs are identical$)$

OPTION $1$: Improve the component manufacturing process to reduce the defect rate of Component $1$ to zero.

OPTION $2$: Enhance the assembly process to eliminate the occurrence of malfunctioning RMD units.