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00:00:05,270 --> 00:00:08,109
Often, in linear
optimization problems,

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00:00:08,109 --> 00:00:11,260
we've estimated the data
we're using in the problem,

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00:00:11,260 --> 00:00:13,640
but it's subject to change.

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00:00:13,640 --> 00:00:17,290
Understanding how the solution
changes when the data changes

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00:00:17,290 --> 00:00:20,290
is called sensitivity analysis.

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00:00:20,290 --> 00:00:22,540
One way that the
data could change

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00:00:22,540 --> 00:00:24,740
is through marketing decisions.

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00:00:24,740 --> 00:00:27,300
Suppose that American
Airlines' management

9
00:00:27,300 --> 00:00:29,360
is trying to figure
out whether or not

10
00:00:29,360 --> 00:00:33,850
it would be beneficial to
invest in marketing its fares.

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00:00:33,850 --> 00:00:36,290
They forecast that
the marketing effort

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00:00:36,290 --> 00:00:40,560
is likely to attract one more
unit of demand, of each type,

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00:00:40,560 --> 00:00:42,940
for every $200 spent.

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00:00:42,940 --> 00:00:47,370
So for the discount fare,
the marketing cost per unit

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00:00:47,370 --> 00:00:50,670
is $200, and for
the regular fare,

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00:00:50,670 --> 00:00:54,690
the marketing cost
per unit is also $200.

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00:00:54,690 --> 00:00:57,250
We want to know how
much this will increase

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00:00:57,250 --> 00:01:02,590
our marginal revenue
for each type of fare.

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00:01:02,590 --> 00:01:05,950
This graph shows our current
feasible space and optimal

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00:01:05,950 --> 00:01:07,240
solution.

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00:01:07,240 --> 00:01:10,260
What would happen if we
increased the marketing

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00:01:10,260 --> 00:01:12,210
for discount fares?

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00:01:12,210 --> 00:01:15,060
The demand for discount
fares would increase.

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00:01:15,060 --> 00:01:18,120
But since we're not even meeting
the current demand for discount

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00:01:18,120 --> 00:01:20,670
fares with the
optimal solution, this

26
00:01:20,670 --> 00:01:23,850
doesn't give us
any extra revenue.

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00:01:23,850 --> 00:01:28,070
So we shouldn't add any
marketing for discount fares.

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00:01:28,070 --> 00:01:32,240
Actually, American Airlines
could decrease their budget

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00:01:32,240 --> 00:01:35,810
to market discount fares, and
even if the demand decreases,

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00:01:35,810 --> 00:01:38,020
it wouldn't change our revenue.

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00:01:38,020 --> 00:01:40,880
The demand could go
all the way down to 66

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00:01:40,880 --> 00:01:43,780
without affecting our decisions.

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00:01:43,780 --> 00:01:46,280
In sensitivity
analysis like this,

34
00:01:46,280 --> 00:01:50,710
we're often concerned with the
shadow price of a constraint.

35
00:01:50,710 --> 00:01:52,950
For a discount
demand constraint,

36
00:01:52,950 --> 00:01:55,620
this is the marginal
revenue gained

37
00:01:55,620 --> 00:01:58,630
by increasing the
demand by one unit.

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00:01:58,630 --> 00:02:01,940
In this case, the shadow
price is 0 for demand

39
00:02:01,940 --> 00:02:05,670
greater than or equal to 66.

40
00:02:05,670 --> 00:02:10,229
Now, let's look at what happens
when we market regular fares.

41
00:02:10,229 --> 00:02:13,120
If we increase the
demand for regular fares,

42
00:02:13,120 --> 00:02:16,020
our revenue increases.

43
00:02:16,020 --> 00:02:19,050
If we increase by
25 units of demand,

44
00:02:19,050 --> 00:02:24,380
our revenue
increases to $86,883.

45
00:02:24,380 --> 00:02:27,930
If we increase by another
25 units of demand,

46
00:02:27,930 --> 00:02:33,440
our revenue
increases to $96,358.

47
00:02:33,440 --> 00:02:36,650
So what's the shadow
price in this case?

48
00:02:36,650 --> 00:02:38,700
Remember that the
shadow price is

49
00:02:38,700 --> 00:02:42,550
the marginal revenue for
a unit increase in demand,

50
00:02:42,550 --> 00:02:45,440
in this case, of regular seats.

51
00:02:45,440 --> 00:02:49,790
From 100 to 125, the
revenue increased

52
00:02:49,790 --> 00:03:03,480
by $86,883 minus $77,408,
which is equal to $9,475.

53
00:03:03,480 --> 00:03:07,250
Since this was an increase
of 25 units of demand,

54
00:03:07,250 --> 00:03:12,040
the shadow price
is 9,475 divided

55
00:03:12,040 --> 00:03:16,540
by 25, which equals 379.

56
00:03:16,540 --> 00:03:18,990
We can calculate that
this is the same shadow

57
00:03:18,990 --> 00:03:23,220
price from 125 to 150.

58
00:03:23,220 --> 00:03:25,990
So the marginal revenue
for every extra unit

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00:03:25,990 --> 00:03:33,670
of regular demand from
100 to 166 is $379.

60
00:03:33,670 --> 00:03:36,120
So given this
analysis, how can we

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00:03:36,120 --> 00:03:39,420
help the marketing department
make their decisions?

62
00:03:39,420 --> 00:03:45,020
The forecast was an extra unit
of demand for every $200 spent.

63
00:03:45,020 --> 00:03:48,040
For discount fares, this isn't
worth it, since the shadow

64
00:03:48,040 --> 00:03:51,510
price, or marginal
revenue, is 0.

65
00:03:51,510 --> 00:03:54,340
But for the regular
fares, this is worth it,

66
00:03:54,340 --> 00:03:58,180
since the shadow price is $379.

67
00:03:58,180 --> 00:04:00,020
So the marketing
department should

68
00:04:00,020 --> 00:04:02,590
invest in marketing
regular fares

69
00:04:02,590 --> 00:04:05,010
to increase the
demand by 66 units.

70
00:04:07,850 --> 00:04:11,410
Another sensitivity analysis
question in our problem

71
00:04:11,410 --> 00:04:14,760
is whether or not it would be
beneficial to allocate a bigger

72
00:04:14,760 --> 00:04:17,050
aircraft for this flight.

73
00:04:17,050 --> 00:04:19,290
This would change the
capacity constraint,

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00:04:19,290 --> 00:04:23,390
which currently limits
the capacity to 166.

75
00:04:23,390 --> 00:04:25,860
With our current
aircraft, the management

76
00:04:25,860 --> 00:04:30,950
knows that the cost
per hour is $12,067.

77
00:04:30,950 --> 00:04:37,680
So the total cost of the
six-hour flight is $72,402.

78
00:04:37,680 --> 00:04:41,450
With the 166 seats
filled, we get a revenue

79
00:04:41,450 --> 00:04:46,530
of $77,408 from our
optimal solution.

80
00:04:46,530 --> 00:04:51,490
If we increase the capacity
of the aircraft to 176 seats,

81
00:04:51,490 --> 00:04:57,030
the total cost would
increase to $76,590.

82
00:04:57,030 --> 00:05:00,310
But how much would this
increase our revenue?

83
00:05:00,310 --> 00:05:05,080
And if we increase the capacity
of the aircraft to $218,

84
00:05:05,080 --> 00:05:10,230
the total cost would
increase to $87,342.

85
00:05:10,230 --> 00:05:13,680
But how much would this
increase our revenue?

86
00:05:13,680 --> 00:05:17,080
For our analysis, let's assume
that the demand does not

87
00:05:17,080 --> 00:05:18,590
change.

88
00:05:18,590 --> 00:05:22,520
If we increase our
capacity to 176,

89
00:05:22,520 --> 00:05:25,170
the capacity constraint
will move right.

90
00:05:25,170 --> 00:05:27,900
And our optimal solution
will move right too.

91
00:05:27,900 --> 00:05:33,520
We now get a revenue of $79,788.

92
00:05:33,520 --> 00:05:37,480
If we then increase the
capacity to 218 seats,

93
00:05:37,480 --> 00:05:40,140
the capacity constraint
will move right again,

94
00:05:40,140 --> 00:05:45,890
and our revenue will
increase to $89,784.

95
00:05:45,890 --> 00:05:47,900
So let's look at
our extra profit

96
00:05:47,900 --> 00:05:51,630
from increasing the capacity
to see if it's worth it.

97
00:05:51,630 --> 00:05:57,940
With our current costs and
revenue, the profit is $5,006.

98
00:05:57,940 --> 00:06:01,700
If we increase the
capacity to 176 seats,

99
00:06:01,700 --> 00:06:07,350
our profit actually
decreases to $3,198.

100
00:06:07,350 --> 00:06:11,390
And if we increase the
capacity to 218 seats,

101
00:06:11,390 --> 00:06:16,580
our profit decreases
even more to $2,442.

102
00:06:16,580 --> 00:06:19,470
So even though our
revenue is increasing,

103
00:06:19,470 --> 00:06:21,210
the cost increases too.

104
00:06:21,210 --> 00:06:23,820
So it's not profitable
for us to increase

105
00:06:23,820 --> 00:06:26,610
the capacity of our aircraft.

106
00:06:26,610 --> 00:06:30,170
You can also see this by
using LibreOffice, which we'll

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00:06:30,170 --> 00:06:33,390
ask you to do in the
next quick question.