Math 164: Homework #4, due on Wednesday, April 29, 2009
•Review Sections 4.1-4.3 from the textbook and solve additional problems.
[1] Solve the following linear program graphically:
maximize z= 6x1−3x2,subject to
2x1+ 5x2≥10,
3x1+ 2x2≤40,
x1, x2≤15.
[2] Convert the following linear program to standard form:
minimize z=x1−5x2−7x3, subject to
5x1−2x2+ 6x3≥5
3x1+ 4x2−9x3= 3
7x1+ 3x2+ 5x3≤9
x1≥ −2, x2, x3free.
[3] Consider the system of linear constraints
2x1+x2≤100,
x1+x2≤80,
x1≤40,
x1, x2≥0.
(a) Write this system in standard form, and determine all the basic solu-
tions (feasible and infeasible).
(b) Determine the extreme points of the feasible region (corresponding to
both the standard form, as well as the original version).
[4] Consider a linear program with the constraints in standard form:
{Ax =b, x ≥0}.
Prove that if d6= 0 satisfies Ad = 0 and d≥0, then dis a direction of
unboundedness.
[5] Consider the linear program: minimize z=−5x1−7x2, subject to
−3x1+ 2x2≤30
−2x1+x2≤12
x1, x2≥0.
(a) Draw a graph of the feasible region
(b) Determine the extreme points of the feasible region
(c) Determine two linearly-independent directions of unboundedness.
(b) Convert the linear program to standard form and determine two
linearly-independent directions of unboundedness for this version of the prob-
lem. Verify that the directions of unboundedness satisfy Ad =~
0 and d≥~
0.
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