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Chapter 2 – Introduction to Optimization & Linear Programming : S-1
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Chapter 2
Introduction to Optimization & Linear Programming
1.
If an LP model has more than one optimal solution it has an infinite number of alternate optimal solutions.
In Figure 2.8, the two extreme points at (122, 78) and (174, 0) are alternate optimal solutions, but there are
an infinite number of alternate optimal solutions along the edge connecting these extreme points. This is
true of all LP models with alternate optimal solutions.
2.
There is no guarantee that the optimal solution to an LP problem will occur at an integer-valued extreme
point of the feasible region. (An exception to this general rule is discussed in Chapter 5 on networks).
3.
We can graph an inequality as if they were an equality because the condition imposed by the equality
corresponds to the boundary line (or most extreme case) of the inequality.
4.
The objectives are equivalent. For any values of X1 and X2, the absolute value of the objectives are the
same. Thus, maximizing the value of the first objective is equivalent to minimizing the value of the second
objective.
5.
a.
b.
c.
d.
e.
6.
linear
nonlinear
linear, can be re-written as: 4 X1 – .3333 X2 = 75
linear, can be re-written as: 2.1 X1 + 1.1 X2 – 3.9 X3๏ฃ 0
nonlinear
Chapter 2 – Introduction to Optimization & Linear Programming : S-2
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7.
8.
X2
20
(0, 15) obj = 300
15
(0, 12) obj = 240
10
(6.67, 5.33) obj =140
(11.67, 3.33) obj = 125
(optimal solution)
5
0
5
10
15
20
25
X1
Chapter 2 – Introduction to Optimization & Linear Programming : S-3
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9.
10.
Chapter 2 – Introduction to Optimization & Linear Programming : S-4
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
11.
12.
Chapter 2 – Introduction to Optimization & Linear Programming : S-5
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13.
X1 = number of softballs to produce, X2 = number of baseballs to produce
MAX
ST
14.
6 X1 + 4.5 X2
5X1 + 4 X2๏ฃ 6000
6 X1 + 3 X2๏ฃ 5400
4 X1 + 2 X2๏ฃ 4000
2.5 X1 + 2 X2๏ฃ 3500
1 X1 + 1 X2๏ฃ 1500
X1, X2๏ณ 0
X1 = number of His chairs to produce, X2 = number of Hers chairs to produce
MAX
ST
10 X1 + 12 X2
4 X1 + 8 X2๏ฃ 1200
8 X1 + 4 X2๏ฃ 1056
2 X1 + 2 X2๏ฃ 400
4 X1 + 4 X2๏ฃ 900
1 X1- 0.5 X2 โฅ 0
X1 , X2 โฅ 0
Chapter 2 – Introduction to Optimization & Linear Programming : S-6
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15.
X1 = number of propane grills to produce, X2 = number of electric grills to produce
MAX
ST
100 X1 + 80 X2
2 X1 + 1 X2๏ฃ 2400
4 X1 + 5 X2๏ฃ 6000
2 X1 + 3 X2๏ฃ 3300
1 X1 + 1 X2๏ฃ 1500
X1 , X2 โฅ 0
Chapter 2 – Introduction to Optimization & Linear Programming : S-7
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
16.
X1 = number of generators, X2 = number of alternators
MAX
ST
17.
250 X1 + 150 X2
2 X1 + 3 X2๏ฃ 260
1 X1 + 2 X2๏ฃ 140
X1, X2๏ณ 0
X1 = number of generators, X2 = number of alternators
MAX
ST
250 X1 + 150 X2
2 X1 + 3 X2๏ฃ 260
1 X1 + 2 X2๏ฃ 140
X1๏ณ 20
X2๏ณ 20
d. No, the feasible region would not increase so the solution would not change — you’d just have extra
(unused) wiring capacity.
Chapter 2 – Introduction to Optimization & Linear Programming : S-8
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18.
X1 = proportion of beef in the mix, X2 = proportion of pork in the mix
MIN
ST
19.
.85 X1 + .65 X2
1X1 + 1 X2 = 1
0.2 X1 + 0.3 X2๏ฃ 0.25
X1, X2๏ณ 0
T= number of TV ads to run, M = number of magazine ads to run
MIN
ST
500 T + 750 P
3T + 1P ๏ณ 14
-1T + 4P ๏ณ 4
0T + 2P ๏ณ 3
T, P ๏ณ 0
Chapter 2 – Introduction to Optimization & Linear Programming : S-9
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
20.
X1 = # of TV spots, X2 = # of magazine ads
MAX
ST
15 X1 + 25 X2
5 X1 + 2 X2 8
0.2 X1 + 0.25 X2> 6
0.15 X1 + 0.1 X2> 5
X1, X2๏ณ 0
(cost)
(copper)
(zinc)
(magnesium)
Chapter 2 – Introduction to Optimization & Linear Programming : S-10
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
22.
R = number of Razors produced, Z = number of Zoomers produced
MAX
ST
23.
70 R + 40 Z
R + Z ๏ฃ 700
R โ Z ๏ฃ 300
2 R + 1 Z ๏ฃ 900
3 R + 4 Z ๏ฃ 2400
R, Z ๏ณ 0
P = number of Presidential desks produced, S = number of Senator desks produced
MAX 103.75 P + 97.85 S
ST
30 P + 24 S ๏ฃ 15,000
1 P + 1 S ๏ฃ 600
5 P + 3 S ๏ฃ 3000
P, S ๏ณ 0
Chapter 2 – Introduction to Optimization & Linear Programming : S-11
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
24.
X1 = acres planted in watermelons, X2 = acres planted in cantaloupes
MAX
ST
256 X1 + 284.5 X2
50 X1 + 75 X2๏ฃ 6000
X1 + X2๏ฃ 100
X1, X2๏ณ 0
X2
(0, 80) obj =
100
75
(60, 40) obj =26740
(optimal
50
25
(100, 0) obj =
0
0
25.
25
50
75
100
125
X1
D = number of doors produced, W = number of windows produced
MAX
ST
500 D + 400 W
1 D + 0.5 W ๏ฃ 40
0.5 D + 0.75 W ๏ฃ 40
0.5 D + 1 W ๏ฃ 60
D, W ๏ณ 0
Chapter 2 – Introduction to Optimization & Linear Programming : S-12
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
26.
X1 = number of desktop computers, X2 = number of laptop computers
MAX
ST
600 X1 + 900 X2
2 X1 + 3 X2๏ฃ 300
X1๏ฃ 80
X2๏ฃ 75
X1, X2๏ณ 0
Case 2-1: For The Lines They Are A-Changinโ
1.
200 pumps, 1566 labor hours, 2712 feet of tubing.
2.
Pumps are a binding constraint and should be increased to 207, if possible. This would increase profits
by $1,400 to $67,500.
3.
Labor is a binding constraint and should be increased to 1800, if possible. This would increase profits
by $3,900 to $70,000.
4.
Tubing is a non-binding constraint. Theyโve already got more than they can use and donโt need any
more.
5.
9 to 8: profit increases by $3,050
8 to 7: profit increases by $850
7 to 6: profit increases by $0
6.
6 to 5: profit increases by $975
5 to 4: profit increases by $585
4 to 3: profit increases by $390
7.
12 to 13: profit changes by $0
13 to 14: profit decreases by $760
14 to 15: profit decreases by $1,440
Chapter 2 – Introduction to Optimization & Linear Programming : S-13
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
8.
16 to 17: profit changes by $0
17 to 18: profit changes by $0
18 to 19: profit decreases by $400
9.
The profit on Aqua-Spas can vary between $300 and $450 without changing the optimal solution.
10. The profit on Hydro-Luxes can vary between $233.33 and $350 without changing the optimal solution.

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