
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
(revenuescosts). 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
X_{1} be the number of soldiers produced each week and X_{2}
be the number of trains produced each week. 





Objective
Function: 


Max Z = 3X_{1} + 2X_{2} 


ST : 2X_{1} + X_{2}
<= 100 (Finishing Constraint) 


X_{1}
+ X_{2} <= 80 (Carpentry Constraint) 


X_{1}
<= 40 (Constraint on demand for soldiers) 


X_{1},
X_{2} >= 0 (sign restriction) 
