$3.($Aggregate Planning$)$ FCC Company needs an aggregate plan for the coming year for its production. The company has developed the following data:

Inventory at the beginning of period $\backslash (1=0\backslash )$ units

Inventory at the end of period $\backslash (4=10\backslash )$ units

Number of workers employed in period $\backslash (4=39\backslash )$

Maximum working hours in a period $\backslash (=480\backslash )$ hours per worker

Worker hour needed to produce one unit $\backslash (=40\backslash )$

Cost to hire one worker $\backslash (=\backslash$$ $400\backslash )$

Cost to lay off one worker $=\backslash$$$600$

Regular time payment per worker $\backslash (=\backslash$$ $8\backslash )$ per hour

Overtime payment per worker $\backslash (=\backslash$$ $12\backslash )$ per hour

Cost to hold one unit in inventory per period $\backslash (=\backslash$$ $2\backslash )$

Overtime cannot exceed $\backslash (25\backslash \%\backslash )$ of the regular time work

Demand of a period should be satisfied on its period.

$($a$)$ Formulate a linear $($or an integer, if needed$)$ programming model to obtain an aggregate production plan. Clearly, define the indices, parameters, and decision variables, and write the model with its objective function and constraints. Your notation used for defining the decision variables should be the standard notation used in lecture notes. All constraints and the objective function should be written in closed form.

$($b$)$ Using GAMS, develop an optimal production plan for the model developed in part $($a$).$ Using the optimal solution, for each month tabulate the number of units of product produced, the number of units of product kept in inventory, the number of workers employed, the number of workers hired, and the number of workers fired. Do not forget to upload the GAMS file to the web site.

$($c$)$ Assume that the inventory level at the end of period $4$ is zero. Using appropriate tableaus, apply the following strategies and determine the best strategy to be implemented by the company.

Plan $1$: A constant workforce strategy that uses regular time work. Using Microsoft Excel, on the same graph, plot the cumulative production and cumulative demand curves over the time.

Plan $2$: A constant workforce strategy that produces $516$ units in regular time per period and uses overtime if needed.

$($a$)$ Compare Plans $1$ and $2$ and comment.