$**$ SHOW ALL WORK AND FORMULAS IN EXCEL, INCLUDING SOLVER $**$ A furniture company produces tables and chairs. A table is made of $40$ board feet of

wood and a chair is made of $30$ board feet of wood. The company has ordered $42,000$

boards of wood at the cost of $$1$ per board foot $($and currently has them available in

the inventory$).$ It takes $2$ hours of labor to make an unfinished table or unfinished chair.

$3$ more hours of labor is required to turn unfinished table into a finished table and $2$

more hours of labor to turn an unfinished chair into a finished chair. Based on the

number of employees, there are $6000$ hours of labor available in a week that the

company pays at the rate of $$15$ per hour. The price for each product is given below:

a$)[10$pt$]$ Formulate a linear programming problem $m$ lematically to find the

production quantities that maximize the profit $($revenue $-$ wood cost $-$ labor

cost$).$ Clearly define the decision variables, objective and constraints.

Assume that all variables are continuous, i$.$e$.,$ you can produce a fractional

number of tables and chairs.

Decision Variables:

Primal Formulation

b$)[10$pt$]$ Transform the primal problem in part a$)$ into a dual formulation.

Do not solve the dual problem. This is just an exercise to learn dual

transformation.

c$)[5$pt$]$ Solve the primal linear programming problem defined in part a$)$ using

Excel's solver. Report the values of the decision variables, the objective function,

allowable increase and decrease in unit prices without changing the optimal

solution and the shadow prices corresponding to a board foot of wood and an

hour of labor at the optimal solution.

d$)[10$pt$]$ Is it profitable for the company to buy more wood at the rate of

$$\frac{1\mathrm{.}5}{?}$ board foot of wood for furniture production? Justify your answer with the

results from the sensitivity report.