Task $1$: A mobile phone manufacturer is exploring ways to lower costs in its assembly process. The company is considering the purchase and installation of automated systems to handle the phone battery installation, which could reduce labor, material, and energy costs. The cost of procuring and installing the automation equipment is $$500,000,$ with an estimated useful life of $3$ years. A full$-$time technician will be required for ongoing maintenance and programming at a cost of $$65,000$ per year. With the automated system in place, the company can reallocate $4$ full$-$time employees who currently handle battery installation. These employees each earn $$55,000$ per year. Additionally, there is a quality control inspector earning $$52,000$ per year who will still be required after automation. The current cost of battery adhesives is $$0\mathrm{.}6$ per unit, and the energy cost is $$0\mathrm{.}03$ per unit. With the new system, both costs will be reduced by $40\%.$ How many mobile phones should be produced and sold over the next three years to justify the investment in the automation system?

$[2\mathrm{.}5$ marks$]$

Task $2$: Ten samples of $75$ kg concrete blocks produced by a machine are collected. Each sample has three observations $($provided in the following table$)$:

Sample $12345678910$

Observations $74\mathrm{.}375\mathrm{.}674\mathrm{.}873\mathrm{.}975\mathrm{.}274\mathrm{.}475\mathrm{.}373\mathrm{.}875\mathrm{.}174\mathrm{.}7$

$74\mathrm{.}974\mathrm{.}575\mathrm{.}074\mathrm{.}274\mathrm{.}675\mathrm{.}774\mathrm{.}874\mathrm{.}575\mathrm{.}074\mathrm{.}2$

$75\mathrm{.}275\mathrm{.}175\mathrm{.}375\mathrm{.}075\mathrm{.}474\mathrm{.}974\mathrm{.}574\mathrm{.}774\mathrm{.}874\mathrm{.}9$

a$.$ If the weight specification limits for a block are $74\mathrm{.}5$ kg $($LSL$)$ and $75\mathrm{.}5$ kg $($USL$),$ what is the Cpk$?$ Is the production process capable? Why or why not? $[1\mathrm{.}5$ Marks$]$

b$.$ Using A$2,$ D$3,$ and D$4$ factors, determine the upper and lower control limits for the sample mean and range, and decide if the process is in control. State the reason for your decision. $[1\mathrm{.}5$ Marks$]$

$($Please use Excel for accurate results. Round final answers to $4$ decimal places.$)$

Task $3$: A fertilizer plant acquires phosphate from a single source. The plant$$s annual phosphate requirement has a normal distribution with a mean of $350,000$ metric tons and the standard deviation of $1,200$ metric tons. The lead time for phosphate supply is $1$ week. To place an order, an administrative employee spends around one hour reviewing the inventory level, preparing the order, tracking the delivery, and updating records. When the delivery is received, it takes roughly two worker hours to unload, inspect, and store the material in a warehouse. The administrative staff member is paid $$26$ per hour and the warehouse employees are paid $$16$ per worker hour. Delivery charges are $$600$ per shipment. The company purchases phosphate at $$500$ per metric ton. The annual cost of capital is $5\%.$ Other annual costs of holding inventory are estimated to be $5\%$ of the purchase cost. How much should the plant order at a time? What will be the Reorder Point if the desired service level during the lead time is $98\%?$ Z$0\mathrm{.}98=2\mathrm{.}05$

$($Consider $52$ weeks in a year for planning. Round answers to $4$ decimal places.$)[3$ Marks$]$

Problem $4$: A small Tennessee company makes fused magnesia and silica. To prepare input for the planning software, the company first uses a spreadsheet to determine the capacity requirements for the main equipment $($the fusion furnaces$)$ and the production schedule for main products. Product#$3050$ had the following forecasted sales and outstanding orders $($all in thousand pounds$)$:

January February

Week $2341234$

Forecast $55555566666666$

Outstanding order $40015$

The initial inventory is $80,000$ pounds and $60,000$ pounds are desired as the minimum safety stock at the end of each week.

Considering the holding and setup costs with EPQ, the company has found that the optimum production quantity is around $80,000$ units in each run. Use this production quantity to determine the production schedule $($MPS$)$ and Available to Promise inventory for the next $7$ weeks.

$[2\mathrm{.}5$ Marks$]$

January February

WK$2$ WK$3$ WK$4$ WK$1$ WK$2$ WK$3$ WK$4$

Forecast $55555566666666$

Committed orders $400150000$

Beg Inv. of each period $80$

Production

Projected on hand inventory

Available$-$to$-$promise

Task $5$: A medical facility does MRI scans for sports injuries. Occasionally a test yields inconclusive results and must be repeated. Using the following $8$ sample results for the number of retests in n $=200$ observations each, determine the control limits for the proportion of retests. Use three sigma $($standard dev$)$ limits$.$ Is the process in control? Why$/$why not?

$($Round answers to $4$ decimal places.$)$

$[1\mathrm{.}5$ Marks$]$

Sample: $12345678$

Number of retests: $23313230$