Question $4-$ Aggregate capacity management $(9$ marks$)$

A company is establishing a production capacity plan over four seasons. The normal operating hours is $8$ hours per day and $60$ days per season. The demand forecasts for Autumn, Winter, Spring and Summer are $100,000$ units, $75,000$ units, $80,000$ units, and $90,000$ units, respectively. At the beginning of Autumn, the company has $80$ employees and $12,760$ units of inventory. The productivity rate of each employee is $2$ units per hour, and all employees work at their full capacity within the normal operations hours. To have additional production capacity, the company is allowed to hire temporary $($seasonal$)$ employees who have similar productivity rate and salary rate compared with full time employees during Autumn and Summer. There is no cost for hiring or laying off the temporary employees. Using overtime is another option for adding the production capacity.

The operating costs that are applied in this company are as follows. Straight time labour cost is $$50$ per hour, and overtime labour cost is $$75$ per hour. Backorder cost is $$60$ per unit$-$season and inventory holding cost is $$30$ per unit$-$season.

a$)$ Construct a production plan and calculate the total costs over the four seasons under the following scenario:

Autumn

Winter

Spring

Summer

Temporary workers

Allowed

Not allowed

Not allowed

Allowed

Overtime

Not allowed

Allowed

Allowed

Not allowed

Backorder

Allowed

Not allowed

Not allowed

Not allowed

Ending inventory

Not allowed

Not allowed

Allowed

Not allowed

The total overtime is limited to $3,000$ hours per season. In your production capacity plan, show the calculations of the key variables, including the backorder units, the ending inventory units, the overtime hours, and the number of temporary employees that the company needs to hire $(4$ marks$).$ Explain the logic behind the production plan $(1$ mark$)$

b$)$ The company considers adopting level capacity strategy in its production plan where it has $85$ employees at the beginning of the Autumn season and plans to keep them over the four seasons without adding any temporary employees. Overtime $($limited to $3,000$ hours per season$),$ ending inventory, and backorders are allowed in all seasons except in Summer. Based on the scenario, construct a production plan over the four seasons that produces the lowest total costs $(3$ marks$).$ Explain the reason if you need to have backorder, inventory or overtime in any season $(1$ mark$).$