Giapetto’s Woodcarving, Inc., manufactures two types of

Giapetto’s Woodcarving, Inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable labor and overhead costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable labor and overhead costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. Gepetto wants to maximize weekly profit (revenues – costs). Gepetto’s LP was

(x₁= soldiers and x₂= trains). After adding slack variabless₁, s₂, and s₃, the optimal tableau is as shown in Table.

Use this optimal tableau to answer the following questions:

a. Show that as long as soldiers (x₁) contribute between$2 and $4 to profit, the current basis remains optimal. If soldiers contribute $3.50 to profit, find the new optimal solution to the Giapetto problem.

b. Show that as long as trains (x₂) contribute between$1.50 and $3.00 to profit, the current basis remains optimal.

c. Show that if between 80 and 120 finishing hours are available, the current basis remains optimal. Find the new optimal solution to the Giapetto problem if 90 finishing hours are available.

d. Show that as long as the demand for soldiers is at least 20, the current basis remains optimal.

e. Giapetto is considering manufacturing toy boats. A toy boat uses 2 carpentry hours and 1 finishing hour. Demand for toy boats is unlimited. If a toy boat contributes $3.50to profit, should Giapetto manufacture any toy boats?