# Example 2: Giapetto’s Woodcarving Problem

 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 labor 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.  Giapetto wants to maximize weekly profit (revenues-costs).    Formulate a mathematical model of Giapetto’s situation that can be used to maximize Giapetto’s weekly profit. (From Introduction to Mathematical Programming, by Wayne L. Winston. P.45) Formulate the Problem: Let X1 be the number of soldiers produced each week and X2 be the number of trains produced each week. Objective Function: Max Z = 3X1 + 2X2 ST :         2X1 + X2 <= 100  (Finishing Constraint) X1 + X2 <= 80 (Carpentry Constraint) X1 <= 40 (Constraint on demand for soldiers) X1, X2 >= 0 (sign restriction)