Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 o
ID: 459624 • Letter: T
Question
Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor and $30 on each lawn mower, and they sell all they can produce. The time requirements in the machine shop, fabrication, and tractor assembly are given in the table. How many tractors and saws should be produced to maximize profit, and how much profit will they make? Determine the sensitivity range for the profit for tractors. What is the shadow price for assembly? What is the shadow price for fabrication ? What is the maximum amount a manager would be willing to pay for one additional hour of machining time. A breakdown in fabrication causes the available hours to drop from 120 to 90 hours. his impact the optima! number of and mower, produced? What is the range for the shadow price for assembly.Explanation / Answer
a) From the graphical solution it can be seen the Max profit is achieved at y=30, x=15
hence maximum profit=30*15+30*30=450+900=1350
b) For sensitivity range for profit of tractors
Slope of objective function=-30/30 (: Y=mx+c; m=slope)
slope of machine shop=-2
slope of fabrication=-2/3
hence for range of optimality for tractors, let Ct be profit coefficient for tractors and Cl be profit coefficient for lawn mover
then the slope=-Ct/Cl
-2<= -Ct/Cl<=-2/3
When Cl=30 the range of optimality for Ct is
-2<=-Ct/30<=-2/3
-60<=Ct<=-20
hence 20<=Ct<=60 the range for profit of tractor is between 20 and 60
similary for lawn movers
-2<=30/Cl<=-2/3
-1/15<=1/Cl<=-1/45
15<=Cl<=45 hence range of optimality for profit coefficent for law mover is bewteen 15 and 45
c) for assembly, given x<=45 the shadow price will be zero since the constraint is non binding
x=15 which is less than 45 hence is non binding the dual price will be zero.
d)for fabrication given 2x+3y<=120, let increase the contraint hours by 10 ie. 2x+3y<=130
solving this equation with other two i.e 2x+y<=60 and x<=45
y=70/2=35
x=25/2=12.5
hence new objective function value=30*12.5+30*35=1425
increased profit=1425-1350=75
therefore profit increases by rate=75/10=7.5 per hour
the objective function increases by 7.5/hr increase in fabrication hours
the shadow price is 7.5
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