$($Show excel formulas for solve$)$Jumbo manufactures bicycles for all ages. The demand fore$-$cast for the coming year is as shown in Table $9-5.$ Capacity at Jumbo is limited by the number of employees it hires. Employees are paid $$10$ per hour for reg$-$ular time and $$15$ per hour for overtime. Each bicycle requires $2$ hours of work from one employee. The plant works $20$ days a month and $8$ hours a day of regular time. Overtime is restricted to a maximum of $20$ hours per employee per month. Jumbo currently has $250$ employees and prefers not to change that number. Each bicycle uses $$35$ of material. Carrying a bicycle in inventory from one month to the next costs $$4.$ Jumbo starts with $4,000$ bicycles in inventory and wants to end the year with $4,000$ bicycles in inventory. Bicycles are currently sold to retailers for $$80$ each. The market is shared between Jumbo and its competitor, Shrimpy. Jumbo is in the process of making its production plan$-$ning and promotion decisions. Jumbo wants to consider only plans without any stockouts. One option is to drop the sale price by $$3($from $$80$ to $$77)$ for one month in the year. The outcome of this action by Jumbo is influenced by the action taken by Shrimpy. If neither firm promotes, the forecast demand for Jumbo is as shown in Table $9-5.$ If Jumbo pro$-$motes in a given month but Shrimpy does not, Jumbo sees consumption $($this does not include forward buying$)$ in that month increase by $40$ percent and forward buying of $10$ per$-$cent from each of the two following months. If Shrimpy pro$-$motes in a given month but Jumbo does not, Jumbo sees consumption in the month drop by $40$ percent, with no change in other months. If both promote in a given month, neither sees an increase in consumption but both see forward buying of $15$ percent from each of the two following months. The debate within Jumbo is whether to promote, and if so$,$ whether to do it in April or June. For the following questions, assume that Shrimpy and Jumbo have similar demand.

a$.$ What is the optimal production plan for Jumbo, assuming no promotion by either Jumbo or Shrimpy? What are the annual profits for Jumbo?

b$.$ What are the profits for Jumbo if Shrimpy promotes in April but Jumbo does not promote throughout the year $($because it follows everyday low pricing$)?$ What are the profits for Jumbo if it promotes in April but Shrimpy does not promote throughout the year? Comment on the bene$-$fit from promoting versus the loss from not promoting if the competitor does.

c$.$ What are the optimal production plans and profits if both promote in April? Both promote in June? Jumbo pro$-$motes in April but Shrimpy in June? Jumbo promotes in June but Shrimpy in April?

d$.$ What is the best decision for Jumbo if it can coordinate its decision with Shrimpy?

e$.$ What is the best decision for Jumbo if it wants to maxi$-$mize its minimum profits no matter what Shrimpy does?

Table $9-5$

Month $-$Demand

Jan $12,000$

Feb $11,000$

Mar $14,000$

Apr $20,000$

May $25,000$

Jun $27,000$

Jul $24,000$

Aug $20,000$

Sep $15,000$

Oct $10,000$

Nov $11,000$

Dec $10,000$