1
OR CHAPTER 3 SOLUTIONS
Section 3.2
1.
max z =
30x
1
+ 100x
2
s.t.
x
1
+ x
2
≤
7 (Land Constraint)
4x
1
+ 10x
2
≤
40(Labor Constraint)
10x
1
≥
30(Govt. Constraint)
x
1
≥
0, x
2
≥
0
EF is 4x
1
+ 10x
2
= 40, CD is x
1
= 3, and AB is x
1
+ x
2
= 7. The feasible region is bounded by ACGH. The dotted
line in graph is isoprofit line 30x
1
+ 100x
2
= 120. Point
G is optimal. At G the constraints 10x
1
≥
30 and 4x
1
+
10x
2
≤
40 are binding. Thus optimal solution has x
1
= 3, x
2
= 2.8 and z = 30(3) + 100(2.8) = 370.
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2
2.
x
1
=
Number of Type 1 Trucks produced daily
x
2
=
Number of Type 2 Trucks produced daily
Expressing profit in hundreds of dollars we
obtain the following formulation:
max z =
3x
1
+ 5x
2
s.t.
x
1
/800 + x
2
/700
≤
1 (Paint Shop Const.)
x
1
/1500 + x
2
/1200
≤
1 (Engine Shop Const.)
x
1
≥
0, x
2
≥
0
AB is x
1
/800 + x
2
/700 = 1. CD is x
1
/1500 + x
2
/1200 = 1.
Feasible region is bounded by ABE. Dotted line is z =
3x
1
+ 5x
2
= 1500. Moving isoprofit line up and to right
we find optimal solution to be where x
1
≥
0 and Paint Shop
constraint are binding.
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 Spring '07
 Mohaghegh,Abeledo
 Optimization, feasible region, dotted line

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