CD14-21
e)
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2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit 1 -1
Used Available
Resource A -1 3 0 <= 30
Resource B -3 1 0 <= 30
Activity 1 Activity 2 Total Profit
Number of Units 0 0 0
Resource Used Per Unit
The Solver message was that the Set Cell values do not converge. There is no optimal
solution, because a better solution can always be found.
14.14 a)
b) No. The objective function value is maximized by sliding the objective function line
up. This can be done forever, so there is no optimal solution.
CD14-22
d) No, solutions exist that will make Z arbitrarily large. This usually occurs when a
constraint is left out of the model.
e)
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2
3
4
5
6
7
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A B C D E F
Activity 1 Activity 2
Unit Profit -1 1
Used Available
Resource A 2 -1 0<= 20
Resource B 1 -2 0<= 20
Activity 1 Activity 2 Total Profit
Number of Units 0 0 0
Resource Used Per Unit
The Solver message was that the Set Cell values do not converge. There is no optimal
solution, because a better solution can always be found.
14.15 a)
CD14-24
b)
Corner Point
Corresponding Constraint
Boundary Equations
Corner Point Satisfies the
Equations
(0, 3)
x1 = 0
x1 + 2x2 = 6
0 = 0
0 + 2(3) = 6
(2, 2)
x1 + 2x2 = 6
2x1 + x2 = 6
2 + 2(2) = 6
2(2) + 2 = 6
(3, 0)
2x1 + x2 = 6
x2 = 0
2(3) + 0 = 6
0 = 0
(0, 0)
x2 = 0
x1 = 0
0 = 0
0 = 0
c)
Corner Point (x1, x2)
Its Adjacent Corner Points
(0,3)
(0,0) and (2,2)
(2,2)
(0,3) and (3,0)
(3,0)
(2,2) and (0,0)
(0,0)
(3,0) and (0,3)
CD14-25
d)
Corner Point (x1, x2)
Profit = 3x1 +2x2
(0,3)
6
(2,2)
10
(3,0)
9
(0,0)
0
e)
Corner Point
Profit = 3x1 +2x2
Next Step
(0, 0)
0
Check points (3, 0) and (0, 3).
(0, 3)
(3, 0)
6
9
Move to (3, 0).
Check point (2, 2).
(2, 2)
10
Stop, (2,2) is optimal. *
* the next corner point is (0, 3), which has already been checked
CD14-27
b)
Corner Point
Corresponding
Constraint Boundary
Equations
Corner Point Satisfies the
Equations
(0, 5)
x1 = 0
x1 + 3x2 = 15
0 = 0
0 + 3(5) = 15
(3, 4)
x1 + 3x2 = 15
2x1 + x2 = 10
3 + 3(4) = 17
2(3) + 4 = 10
(4, 2)
2x1 + x2 = 10
x1 = 4
2(4) + 2 = 10
4 = 4
(4, 0)
x1 = 4
x2 = 0
4 = 4
0 = 0
(0, 0)
x2 =0
x1 =0
0 = 0
0 = 0
c)
Corner Point (x1, x2)
Its Adjacent Corner Points
(0,5)
(0,0) and (3,4)
(3,4)
(0,5) and (4,2)
(4,2)
(3,4) and (4,0)
(4,0)
(4,2) and (0,0)
(0,0)
(4,0) and (0,5)
CD14-28
d)
Corner Point (x1, x2)
Profit = 3x1 +2x2
(0,5)
10
(3,4)
17
(4,2)
16
(4,0)
12
(0,0)
0
e)
Corner Point
Profit = 3x1 +2x2
Next Step
(0,0)
0
Check points (0, 5) and (4, 0).
(0,5)
(4,0)
10
12
Move to (4, 0).
Check point (4, 2).
(4,2)
16
Move to (4, 2).
Check point (3, 4).
(3,4)
17
Move to (3, 4).
Stop, (3,4) is optimal. *
* the next corner point is (0, 5), which has already been checked
CD14-29
14.20
CD14-31
Corner Point
Cost = 15x1 + 20x2
Next Step
(2, 4)
110
Check (0, 6) and (6, 2).
(0, 6)
(6, 2)
120
130
Stop, (2,4) is optimal.
14.23
Corner Point
Cost = 5x1 + 7x2
Next Step
(12, 6)
102
Check (21, 0) and (0, 18).
(21, 0)
(0, 18)
105
126
Stop, (12, 6) is optimal.
14.24 a) True (see solution concept number 6).
14.25 a) If the feasible region is unbounded then there may be no optimal solution.
14.26 a) The problem may not have an optimal solution.
CD14-32
c) The simplex method only chooses the origin as the initial corner point when it is a
feasible point.
14.27 a & c)
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10
A B C D E F
Activity 1 Activity 2
Unit Profit 2 1 ($million)
Used Available
Resource A 3 1 15 <= 15
Resource B 1 2 10 <= 10
Total Profit
Activity 1 Activity 2 ($million)
Number of Units 4 3 11
Resource Used Per Unit
b)
Corner Point (x1, x2)
Profit = 2x1 + x2
(0, 0)
0
(5, 0)
10
(4, 3)
11
(0, 5)
5
CD14-34
14.28 a)
b)
Corner Point
Profit = 3x1 + x2
Next Step
(0, 0)
0
Check (4, 0) and (0, 4).
(4, 0)
(0, 4)
12
4
Move to (4, 0).
Stop, (4, 0) is optimal. *
* the next corner point is (0, 4) which has already been checked
c)
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10
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12
13
14
A B C D E
Interior Point Algorithm for Problem 17.28c
Iteration
x1x2Z
0 1.00000 1.00000 4.0000
1 1.87500 1.12500 6.7500
2 2.69810 0.80190 8.8962
3 3.34396 0.40095 10.4328
4 3.66710 0.20047 11.2018
5 3.83305 0.10024 11.5994
6 3.91648 0.05012 11.7996
7 3.95824 0.02506 11.8998
8 3.97912 0.01253 11.9499
9 3.98956 0.00626 11.9749
10 3.99478 0.00313 11.9875
CD14-35
c) Examine (4.5, 0) first.
Corner Point
Profit = x1 + x2
Next Step
(0, 0)
0
Check (4.5, 0) and (0, 4.5).
(4.5, 0)
(0, 4.5)
4.5
4.5
Move to (4.5, 0).
Check (3, 3)
(3, 3)
6
Move to (3, 3)
Stop, (3, 3) is optimal. *
* the next corner point is (0,4.5) which has already been checked
Examine (0,4.5) first.
Corner Point
Profit = x1 + x2
Next Step
(0, 0)
0
Check (4.5, 0) and (0, 4.5).
(4.5, 0)
(0, 4.5)
4.5
4.5
Move to (0, 4.5).
Check (3, 3)
(3, 3)
6
Move to (3, 3)
Stop, (3, 3) is optimal. *
* the next corner point is (4.5, 0) which has already been checked
