INEN 420 EXTRA REVIEW PROBLEMS
Chapter 4
Simplex Method
Page 213 Problem 7 (Also use the graphical method to verify
your optimal solutions)
One optimal solution is z = 2, x1 = 1/2, x2 = 0.
An alternative optimal solution z = 2, x1 = 1/3, x2 = 2/3.
This LP has an infinite number of optimal solutions (see your
graph!)
The Big-M and the 2-Phase Simplex Methods
Page 213 Problem 3
The optimal solution is z = 15, x1 = 3, x2 = 0.
Chapter 6
Sensitivity Analysis and Duality
Page 349 Problem 13
a. The optimal dual solution is w = 16, y1 = 0, y2 = 2.
b. The current basis remains optimal for c3 ≥ 1.
c. c1 ≤ 12.
Complementary Slackness
Page 328 Problem 3
The optimal primal solution z = 49/3, x1 = 5/3, x2 = 8/3, x3 =0.
Chapter 7
Transportation Problem
Page 408 problem 5:
In part (b) use the Northwest corner method.
Let xij = number of type j files stored on medium i. The cost of
40, for example, is derived from the fact that if one word
processing file is stored on a hard disk it will be accessed eight
times per month and require five minutes per access. Thus 8(5) =
40 minutes per month will be spent accessing a word processing
The optimal tableau is as follows:
WP
PP
16
8
DUMMY
0
16
4
2
0
80
32
12
0
40
HD
CM
TAPE
200
DATA
0
200
100
100
100
100
100
1
300
300
100
100
100
Page 408 Problem 10:
The optimal solution has z = 10(4) + 5(2) + 5(8) + 10(4) = 130.
Assignment Problem
Page 411 problem 28:
Multiplying by (-1) converts problem to a minimization yielding
P1
P1’
P2
P2’
P3
P3’
9 AM
-8
-8
-9
-9
-7
-7
10 AM
-7
-7
-9
-9
-6
-6
11 AM 1 PM
-6
-5
-6
-5
-8
-8
-8
-8
-9
-6
-9
-6
2 PM
-7
-7
-4
-4
-9
-9
3 PM
-6
-6
-4
-4
-9
-9
ROW MIN
-8
-8
-9
-9
-9
-9
Optimal Tableau and Assignment:
P1
P1’
P2
P2’
P3
P3’
9 AM
0
0
0
0
3
3
10 AM
1
1
0
0
4
4
11 AM
1
1
0
0
0
0
1 PM
2
2
0
0
3
3
2PM
0
0
4
4
0
0
3 PM
1
1
4
4
0
0
Six lines are now needed to cover the zeroes. The optimal solution
x66 = 1, x15 = 1, x21 = 1, x32 = 1, x44 = 1, x53 = 1 is now available.
Thus Professor 1 teaches 9 AM and 2 PM class, Professor 2 teaches
10 AM and 1 PM class, and Professor 3 teaches 11 AM and 3 PM
class.
Chapter 8
Shortest Path Problem
Page 418 Problem 1:
1-3-6 is the shortest path (of length 31) from node 1 to node
6.
Maximum Flow Problems
Page 430 Problem 4:
Maximum flow is 45. Min Cut Set = {1, 3, and si}. Capacity
of Cut Set = 20 + 15 + 10 = 45.
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Page 430 Problem 5:
Maximum flow = 9. Min Cut Set = {2, 4, si}. Capacity of Cut
= 3 + 1 + 3 + 2 = 9.
Minimum Spanning Tree
Page 473 Problem 5:
Begin at NY and include NY-Cleveland arc (length 400). Next
include NY-Nashville arc (length 800). Next include
Nashville-Dallas(length 600). Next include Dallas-St.
Louis(length 600). Next include Dallas-Phoenix(length 900).
Next include Phoenix-LA (length 400. Finally include Salt
Lake City-Los Angeles (length 600). Total length of minimum
spanning tree is 4300 miles.
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