Show step by step working.

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:

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$).$