Hw1 IE312 Solution
Section 3.1
1. Define: x1: number of acres of corn planted
x2: number of acres of wheat planted
max z = 30x1 + 100x2
s.t.
x1 + x2 7
(Land Constraint)
4x1 + 10x2 40 (Labor Constraint)
10x1 30
(Govt. Constraint)
x1 0, x2 0 (Sign Constraint)
2a. at (2, 3) check all the constraints until some constraint is violated.
x1 + x2 = 2 + 3 =5 7
OK
4x1 + 10x2 = 30 40 OK
10x1 = 10* 2 = 20 30 Violated
x1 0, x2 0
No, government constraint is violated.
2b. 4x1 + 10x2 = 46 40
No; Labor constraint is not satisfied.
2c. No, x2 0 is not satisfied.
2d. Yes, all constraints and sign restrictions are satisfied.
3. Define: x1: number of bushel of corn produced
x2: number of bushel of wheat produced
get the number of acres of corn planted: x1/10
get the number of acres of wheat planted: x1/25
max z = 3x1 + 4x2
s.t. x1/10 + x2/25 7
4x1/10 + 10x2/25 40
x1 30
x1 0, x2 0
(Land Constraint)
(Labor Const.)
(Govt. Const.)
(Sign Constraint)
4. Define :
x1 = Number of Type 1 Trucks produced daily
x2 = Number of Type 2 Trucks produced daily
max z = 300x1 + 500x2
s.t. x1/800 + x2/700 1
x1/1500 + x2/1200 1
(Paint Shop Const.)
(Assembly Shop Const.)
1
x1 0, x2 0
Painting shop finishes 800 truck 1 in one day. So it takes 1/800 day to paint a truck 1. Likewise, one
truck 2 need 1/700 day to finish it. One day, the total time spend in two truck types will be x 1/800 +
x2/700.
Section 3.2
1. EF is 4x1 + 10x2 = 40, CD is x1 = 3, and AB is x1 + x2 = 7. The feasible region is bounded by
ACGH.
Pick a feasible point, e.g. (4, 0), get Z = 120. We have 120 = 30x 1 + 100x2, pick x1 = 3, then x2 = 0.3,
draw isoprofit line 30x1 + 100x2 = 120 as shown in dotted line. Point G is optimal. At G the
constraints 10x1 30 and 4x1 + 10x2 40 are binding. Thus optimal solution has x1 = 3, x2 = 2.8 and z
= 30(3) + 100(2.8) = 370.
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2. AB is x1/800 + x2/700 = 1. CD is x1/1500 + x2/1200 = 1.
Feasible region is bounded by ABE. Dotted line is z = 300x 1 + 500x2 = 150,000. Moving isoprofit
line up and to right we find optimal solution to be where x1 0 and Paint Shop constraint are binding.
Thus x1 = 0, x2 = 700, z = 350,000 is optimal solution.
5.
Define:
xD = number of desks produced
xC = number of chairs produced
max z = 40xD + 25xC
s.t.
4xD + 3xC 20
2xD - xC 0
xD 0, xC 0
(Material Constraint)
(Marketing Constraint)
(Sign Constraint)
Graphical Solution:
AB is 4xD + 3xC = 20, CD is 2xD - xC = 0. The feasible region is bounded by AGC. Dotted line is z
= 40xD + 25xC = 120. Moving isoprofit line up and to right we find optimal solution at G (2, 4) where
Material Constraint and Marketing Constraint are binding.
The optimal solution is at xD = 2, xC = 4, z = 40*2+25*4 = $180.
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