Read the following case and answer the question$($s$)$ that follow:

A company produces a certain product and has the capacity to produce $100$ units per month in

regular time, and an additional $15$ units per month in overtime. The production cost of the

product varies by month due to changes in the material costs over time. The quantities of the

product to be delivered over the next six months have already been set. The delivery

requirements and production costs are given in the following table:

It is possible that the units produced in a month exceeds the delivery requirement. In such a

case, the unsold units are then carried in stock to the following month. The cost of carrying

an unsold unit in stock is R$12$ per unit per month.

At the beginning of month $1$ there are $20$ units in stock and the company does not want to

have any unsold units at the end of month $6.$ The company must determine the number of

units of the product to produce in regular time and in overtime each month to meet the

delivery requirement at minimum cost.

$($a$)$ Define the decision variables that will be needed to formulate a linear programming $($LP$)$

model for this problem.

$(10$ marks$)$

$($b$)$ Formulate the Linear Programming $($LP$)$ model that represents this problem. All excel

tabs are to be shown

$(30$ marks$)$

$($c$)$ Solve the LP model using SOLVER. All excel tabs and solver screens are to be shown

$(20$ marks$)$

$($d$)$ Interpret your SOLVER solution within the context of the original business problem

$(20$ marks$)$