$($PLEASE USE THE SOLVER ADD$-$IN ON EXCEL TO SOLVE THIS PROBLEM AND SHOW ALL OF THE INPUT$)$ Skycell, a major European cell phone manufacturer, is making production plans for the coming year. Skycell has worked with its customers $($the service providers$)$ to come up with forecasts of monthly requirements $($in thousands of phones$)$ as shown in Table $8-10.$ Manufacturing is primarily an assembly operation, and capacity is governed by the number of people on the production line. The plant operates for $20$ days a month, eight hours each day. One person can assemble a phone every $10$ minutes. Workers are paid $20$ euros per hour and a $50$ percent premium for overtime. The plant currently employs $1,250$ workers. Component costs for each cell phone total $20$ euros. Given the rapid decline in component and finished$-$product prices, carrying inventory from one month to the next incurs a cost of $3$ euros per phone per month. Skycell currently has a no$-$layoff policy in place. Overtime is limited to a maximum of $20$ hours per month per employee. Assume that Skycell has a starting inventory of $50,000$ units and wants to end the year with the same level of inventory

$($a$)$ Assuming no backlogs, no subcontracting, and no new hires, what is the optimum production schedule? What is the annual cost of this schedule?

$($b$)$ Is there any value for management to negotiate an increase of allowed overtime per employee per month from $20$ hours to $40?$