EMGT 269 - Elements of Problem Solving and Decision Making
5. Sensitivity Analysis
Used through the entire modeling process
 Purpose of Sensitivity Analysis
1. To analyze what really matters in the decision problem
2. To construct a requisite decision model
 Examples of Sensitivity Analysis Techniques in DA
1. Determine if deterministic dominance or stochastic
dominance is present
2. Identifying the important variables through Tornado
Diagrams
3. Identify interaction effects between important variables
4. Identify the importance of probability assessments.
THE EAGLE AIRLINE CASE:
Dick Carothers wants to expand his operation
 Mid West has to offer:
1. An airplane @ price $95000.
(He can probably by the plane for $85K-$90K)
2. An option to buy the airplane a year later
(Cost of the option $2.5 – $4k)
 Currently:
1. Eagle Airlines (=Dick Carothers) owns 3 plains
2. 60% of flights are chartered flights and 40% are
scheduled
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 75
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Cost Data Mid West Plane:
1. New Engines, FAA Maintained
2. Contains all equipment that Eagle Airlines needs
3. 5 seats
4. Operation Cost: $245 per hour
5. Fixed Cost: $20k (=Yearly Insurance) + Finance Charges
Finance Charges: Borrow 40% of the price at 2% above the
prime rate (=9.5%, but subject to change).
 Revenue Data:
1. Chartered Flights: $300 - $400 per hour
2. Scheduled Flights: $100 per person per hours, plains are
on average 50% full
3. Expected number of hours flown with new plane 8001000.
 Variables in control:
1. The price you are willing to pay
2. The amount financed
 Variables not in control:
1. Insurance Cost
2. Operation Cost
 Carothers could always invest his cash @8% yearly
interest rate
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 76
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
1. Sensitivity Analysis: Problem Identification Level
 Are we solving the right problem?
Error of the 3rd kind: Solving the wrong problem
 How to avoid: continue to be skeptical if the problem on
the surface is the real problem
Eagle Airlines Case:
Carothers wants to expand his operation. The fact that he
owns an airline company does not mean he has to expand
by buying another plane. He could expand by investing in
computer industry.
2. Sensitivity Analysis: Problem Structure Level
 Are any of pieces of the puzzle missing?
 Is this a single or multiple objective problem?
2.1. Sensitivity Analysis: Dominance Considerations
 Ask whether one alternative could end up better than
another. If not, ignore that alternative.
Eagle Airline Case:
“Buying the option” is consider never better than “the buying
the plane alternative” as asking price a year from now will be
adjusted be similar.
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 77
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
2.2. Sensitivity Analysis: Importance of Variables
Eagle Airlines Case:
 Objective: Maximize Profit. Consider Annual Profit,
Ignore Taxes
1. Annual Profit = Annual Total Revenue – Annual Total Cost
2. Total Revenue = Revenue from Charters + Revenue from
scheduled flights
3. Total Cost = Variable Cost + Fixed Cost
Revenue from Charters:
(Charter Ratio)*(Hours flown per year)*Charter Price
Revenue from Schedules Flights:
(1-Charter Ratio)*(Hours flown per year)*(Ticket price per
hour)*(Number of Seats)*(Average Occupancy)
Fixed Cost:
Insurance + (Purchase Price)*(% Financed)*(Interest Rate)
Variable Cost:
(Hours flown per year)*(Operating Cost)
STEP 1:
1. Determine a range for every decision variable and a best
guess. (Low, Base, High)
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 78
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Eagle Airlines Case:
800
500
$325.00
$300.00
$100.00
$95.00
50.00%
40.00%
50.00%
45.00%
$245.00
$230.00
$18,000.00 $20,000.00
0.4
0.3
11.50%
10.50%
$85,000.00 $87,500.00
C h ar ter P r ice p er h o u r
T icket P r ice p er h o u r
O ccu p an cy R ate o n S ch ed u led F lig h ts
% o f C h ar ter F lig h ts
O p er atin g C o st p er H o u r
In su r an ce
P r o p o r tio n F in an ced
In ter est R ate
P u r ch ase P r ice
U sin g L o w V alu es
T o tal R even u e
T o tal C o st
T o tal P r o fit
R even u e F r o m C h ar ter s
R even u e F r o m S ch ed u led F lig h t
F ixed C o st
V ar iab le C o st
H ig h
B ase
Low
H o u r s F lo w n
U sin g B ase V alu es
1000
$350.00
$108.00
60.00%
70.00%
$260.00
$25,000.00
0.5
13.00%
$90,000.00
U sin g H ig h V alu es
$119,750.00
$135,677.50
-$15,927.50
$230,000.00
$220,025.00
$9,975.00
$342,200.00
$290,850.00
$51,350.00
$67,500.00
$52,250.00
$20,677.50
$115,000.00
$130,000.00
$100,000.00
$24,025.00
$196,000.00
$245,000.00
$97,200.00
$30,850.00
$260,000.00
 One Way Sensitivity Analysis
STEP 2:
1. Select a particular variable = free variable
2. Set all other variables to their best guesses (=base
values)
3. Set free variable to its lowers value and calculate payoff
4. Set free variable to its highest value calculate payoff
5. Set free variable to some intermediate values and
calculate payoff
6. Draw results in a one way sensitivity analysis graph
Eagle Airlines Case:
Fix all variables, except hours flown.
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 79
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
S e n s itiv ity G r a p h fo r T o ta l P r o fit
H o u rs F lo w n
500
6 0 0 664 7 0 0
800
900
1000
15000
10000
H o urs F lo w n
5000
0
Money Market
T o tal P ro fit ($)
20000
-5 0 0 0
STEP 3:
1. Perform a one way sensitivity analysis for all variables
2. Plot results in a Spider Diagram or Tornado Diagram
Eagle Airlines Case:
 Spider Diagrams
Hours Flown
Spider Graph for Total Profit/D5
Charter Price per hour
% Change in Input
40.00
%
60.00
%
$35,000.00
$30,000.00
$25,000.00
$20,000.00
$15,000.00
$10,000.00
$5,000.00
$0.00
-$5,000.00
-$10,000.00
-$15,000.00
Ticket Price per hour
Change in Profit
-60.00 -40.00 -20.00
20.00
%
%
%
0.00%
%
Occupancy Rate on Scheduled
Flights
% of Charter Flights
Operating Cost per Hour
Insurance
Proportion Financed
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 80
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Powerful Technique: Tornado Diagrams
STEP 4:
1. Calculated Payoff Range is a measure of uncertainty in
payoff due to uncertainty in the free variable
2. Plot the payoff ranges in a Tornado Diagram
3. Visually determine the important variables.
Tornado Graph for Total Profit
O c c u p a n c y Ra t e o n
S c h e d u le d Flig h t s
- 2 0 0 . 5 0 13 % to + 2 0 0 . 5 0 13 %
O p e ra t in g Co s t p e r
Ho u r
- 12 0 . 2 5 2 6 % to + 12 0 . 2 5 2 6 %
Ho u rs Flo wn
- 12 7 . 8 19 5 % to + 8 5 . 2 13 0 4 %
Ch a rt e r P ric e p e r
ho ur
- 10 0 . 2 2 0 6 % to + 10 0 . 2 2 0 6 %
% o f Ch a rt e r Flig h t s
- 3 0 . 0 7 5 19 % to + 12 0 . 3 0 0 8 %
Tic ke t P ric e p e r h o u r
- 5 0 . 12 5 3 1% to + 8 0 . 2 0 0 5 %
In s u ra n c e
- 5 0 . 12 5 3 1% to + 2 0 . 0 5 0 13 %
$2,500,000
$2,000,000
$1,500,000
$1,000,000
$0
-$500,000
- 1. 15 4 0 3 7 % to + 1. 15 4 0 3 7 %
-$1,000,000
P u rc h a s e P ric e
-$1,500,000
- 5 . 2 6 17 5 8 % to + 3 . 5 10 5 2 4 %
$500,000
- 10 . 0 8 7 7 2 % to + 10 . 0 8 7 7 2 %
In t e re s t Ra t e
-$2,000,000
-$2,500,000
P ro p o rt io n Fin a n c e d
Change in Profit
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 81
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Two Way Sensitivity Analysis
One way sensitivity analysis ignores the effect of
changing multiple variables at the same time
"One Way" under estimates sensitivity due to:
 Additive effects of varying more than one variable
 Multiplicative effects of varying more than one variable
Eagle Airlines Case:
 Tornado Diagram indicates that Occupancy Rate (OR)
and Operating Cost (OC) on scheduled flight are critical.
 Determine Annual Profit (AP) as a function of OR and OC:
AP = R*H*CP + (1-R)*H*TP*NPS*OR-H*OC-I-PP*F*IR
 Set all other parameters at their base values
AP = $130000 + $20000*OR-800*OC-$24025
 For what values of OR and OC is "buying the plane"
worse than "putting money in the savings account"
AP < $4200
$130000 + $200000*OR-800*OC-$24025 < $4200
$200000*OR < $800*OC -$101775
OR < 0.004*OC - 0.509
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 82
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Draw graph of values of OR and OC such that one is
indifferent between "buying the plane" and "the saving
account".
OR = 0.004*OC - 0.509
Base Case
245
0.5
Operating Cost
230
235
240
245
250
255
260
Occupancy Rate
0.41
0.43
0.45
0.47
0.49
0.51
0.53
BASE VALUES
0.54
Occupancy Rate
0.53
AP > 4200
0.52
0.51
0.50
0.49
0.48
0.47
0.46
0.45
0.44
0.43
0.42
0.41
0.40
230
AP < 4200
235
240
245
250
255
260
Operating Cost
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 83
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Sensitivity to Probabilities
Thus far, one way sensitivity and two sensitivity
acknowledge that parameters are uncertain by
indicating a range for the difference variables
Uncertainty Analysis:
 Parameters are uncertain by assessing a range and by
specifying how uncertain these parameters are by
assessing probability distributions for uncertainty of input
parameters
 Given the uncertainty distributions of the input
parameters and the model calculate the uncertainty
distribution of the output parameters
INPUT UNCERTAINTY
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.00
0.20
0.40
0.60
0.80
1.00
OC
MODEL
3 .5 0
3 .0 0
2 .5 0
BLACK
BOX
MODEL
2 .0 0
1.5 0
1.0 0
0 .5 0
0 .0 0
0 .0 0
0 .2 0
0 .4 0
0 .6 0
0 .8 0
1.0 0
OUTPUT
UNCERTAINTY
1.40
1.20
1.00
0.80
0.60
OR
0.40
0.20
0.00
0.00
0.20
0.40
0.60
0.80
1.00
AP
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.00
0.20
0.40
0.60
0.80
1.00
H
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 84
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Eagle Airlines Case: Simplified Approach
 Sensitivity Analysis on Probabilities
From Tornado Diagram it follows that critical variables are:
 Occupancy Rate
 Operating Cost
 Hours Flown
 Charter Price
Decision Variable
STEP 1: Develop an Uncertainty model for variables
over which one has no control
 Occupancy Rate
 Operating Cost
 Hours Flown
UNCERTAINTY MODEL
Operating
Cost
Occupancy
Rate
Hours
Flown
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 85
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Uncertainty Model Operating Cost:
$253 (p)
Operating Cost
$237 (1-p)
Uncertainty Model (Occupancy Rate, Hours Flown):
Occupancy Rate
- Flights Cancelled - Hours Flown
650 (r)
45% (q)
Hours Flown
900 (1 - r)
Occupancy Rate
650 (s)
55% (1-q)
Hours Flown
900 (1 - s)
 r = Pr(Low Hours | Low Occupancy Rate)
 s = Pr(Low Hours | High Occupancy Rate)
 Low hours when occupancy rate is low is more likely than
Low hours when occupancy rate is high.
Assumption: s = 0.8 r, p=0.5
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 86
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Incorporate Uncertainty Model in Eagle Airlines Decision:
Max Profit
650 (r)
- $ 9725
45% (q)
Hours Flown
900 (1 - r)
- $ 4225
$253 (0.5)
Occupancy Rate
650 (0.8 r)
$ 6525
55% (1-q)
Hours Flown
900 (1 - 0.8 r)
Purchase
Plane
Operating Cost
$ 18275
650 (r)
45% (q)
$ 675
Hours Flown
900 (1 - r)
$ 10175
$237 (0.5)
Purchase
Decision
Occupancy Rate
650 (0.8 r)
$ 16925
55% (1-q)
Hours Flown
900 (1 - 0.8 r)
$ 32675
Do not Purchase Plane, Earn 8% on $52500
$ 4200
STEP 2: Assess sensitivity of purchase decision to q and r.
 Express EMV(Purchase) in terms of q and r
EMV(Purcha se)  0.5q 9725r  4225(1  r ) 
(1 - q)6525(0.8r )  18725(1  0.8r )
 0.5 * q675r  10175(1  r ) 
(1 - q)16925(0.8r )  32675(1  0.8r )
or
EMV(Purcha se)  q(3500r - 22500) - 11000r  25475
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 87
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 For what values of q and r is "buying the plane" worse
than "putting money in the savings account"
EMV(Purcha se)  4200
or
q(3500r - 22500) - 11000r  25475  4200
or
q
11000r - 21275
3500r - 22500
 Draw graph of values of q and r such that one is
indifferent between is "buying the plane" and putting
money in the savings account"
q
r
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
q
0.95
0.93
0.91
0.89
0.88
0.86
0.84
0.82
0.80
0.78
0.76
0.74
0.72
0.70
0.68
0.66
0.63
0.61
0.59
0.56
0.54
11000r - 21275
3500r - 22500
1.00
0.90
Savings Account
B
0.80
0.70
0.60
q 0.50
A
0.40
0.30
0.20
Purchase Plane
0.10
0.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
r
EMV(Purchase) = 25475 for r=0, q=0
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 88
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
 Two Way Sensitivity Analysis for Three Alternatives
Read and study pages 169 - 172 and answer the following
question.
PROBLEM DESCRIPTION
In trying to make an investment a decision analyst draws the
following decision tree. The decision analyst is not sure of
his assessment of the probability p or q. When pressed, he
says he is sure that 0.3  p 0.7 and 0.1 q 0.4, but he is
not sure of the actual values. Can he make the decision?
PROFIT
Outcome 1
20,000
(p)
Option A
Option B
Outcome 2
(q)
2,000
Outcome 3
(0.2)
-8,000
Outcome 1
(p)
Outcome 2
(q)
Outcome 3
(0.2)
Option C
8,000
6,000
-2,000
5,000
Instructor: Dr. J. Rene van Dorp
Session 5 - Page 89
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen