Sensitivity Analysis

1
Sensitivity Analysis

• Dr. Yan Liu
• Department of Biomedical, Industrial Human
  Factors Engineering
• Wright State University

2
Introduction

• Before Implementing a Decision, We Should
• Verify that numbers were entered correctly and
  calculations were performed properly
• Identify the specific assumptions behind the
  analysis
• What is Sensitivity Analysis?
• A systematic study of how the solution to a
  decision model changes as the assumptions are
  varied
• Varying one, two, or all the parameters
  simultaneously
• Also known as what-if analysis
• what would happen if
• Importance of Sensitivity Analysis
• Helps to obtain a fuller understanding of the
  dynamics of the decision problem
• Whether the best decision strategy is robust
• Helps to identify the important elements in the
  decision problem
• To construct a requisite decision model

3
Introduction (Cont.)

• Applications of Sensitivity Analysis
• Problem identification level
• Is this the right problem to solve? (the error of
  the 3rd kind solving a wrong problem)
• Problem structure level
• Is there any piece of the puzzle missing?
• How large is the impact of each variable?
• How important is the uncertainty of a variable?

4
Eagle Airlines Case

• Dick Carothers, the president of Eagle Airlines,
  wants to expand his operation. Eagle airlines
  owns 3 aircrafts and provides 50 charter flights
  and 50 scheduled commuter service (only 90 min.
  of flying time on average). He has a news that a
  small airline in the Midwest is selling an
  airplane (Piper Senecca).
• The owner of Senecca has offered 1) sell the
  airplane outright at price 95K (Carothers could
  probably buy it for 85K 90K) or 2) sell an
  option to purchase the airplane within a year at
  a specified price (the cost of the option is
  2.5k 4k).
• The properties of the airplane 1) new engines,
  FAA maintained 2) contains all the needed
  equipment 3) 5 seats 4) operating cost
  245/hour 5) fixed cost 20k yearly insurance
  finance charges
• Finance charges borrow 30-50 of the price at 2
  above the prime rate (currently 9.5, but subject
  to change)
• Revenue 1) charter flights at 300 - 350 per
  hour 2) scheduled flights at around 100 per
  person per hour (planes are 50 full on average)
  3) expect to fly the plane 800 1000 hours per
  year (50 charter flights)
• Carothers can always invest his cash in the money
  market at 8 yearly interest rate
• Variables in control 1) the price he is willing
  to pay 2) the amount financed
• Variables not in control 1) insurance cost 2)
  operation cost

5
Decision Elements

• Objectives

Maximize profits
Profits Revenue Cost
Revenue

• Revenue from charters (charter flight
  ratio)(hours flown)(charter price)
• Revenue from scheduled flights (1-charter
  flight ratio)(hours flown)(ticket
  price)(seats)(capacity of scheduled flights)

Cost

• Fixed cost insuranceFinanceinsurance(purchase
  price)( financed) (interest rate)
• Variable cost (hours flown)(operating
  cost/hour)

6

• Decisions to Make and Alternatives

• Purchase decision
• 1) Purchase the airplane outright
• 2) Purchase the airplane a year later with the
  option
• 3) Put money in the money market

• Proportion financed
• Any amount between 30 and 50

• Charter price
• Any amount between 300 and 350/hour

• Ticket price for scheduled flights
• Any amount between 95 and 108/hour

7

• Uncertain Events (Relevant to the Decisions)

• Charter flight ratio
• Hours flown
• Capacity of scheduled flights
• Insurance
• Purchase price
• Interest rate
• Operating cost/hour

8
Influence Diagram
Uncertainty
9
Sensitivity Analysis
10
One-Way Sensitivity Analysis

• Examine whether a variable really makes a
  difference in the decision by varying its value
  while keeping other variables at their base
  values (best guesses)

Question Under what condition is this procedure
adequate?
One-Way Sensitive Analysis of Hours Flown
11
Tornado Diagram

• A bar (or line) is used to represent the range of
  payoffs due to the variation of an input variable
• Allows us to compare one-way sensitive analyses
  for many input variables at once

Capacity of scheduled flight
Operating cost
Hours flown
Charter price
Interest rate
Purchase price
Profits
Money Market
Tornado Diagram of Eagle Airlines Decision
12
Two-Way Sensitivity Analysis

• Study the joint impact of changes in two variables

In the Eagle Airlines example, suppose we want to
conduct a two-way sensitivity analysis of the two
most critical variables capacity of the
scheduled flights (CS) and operating cost (OC)
Set all the other input variables at their base
values, yielding Profit 130,000200,000CS-80
0OC-24,025
If Profit lt 4,200 (market value), then it is
less profitable to purchase the airplane If
Profit gt 4,200 (market value), then it is more
profitable to purchase the airplane
Solve equation Profit 4,200 ?
130,000200,000CS-800OC-24,025 4,200 ?
800OC 200,000CS 101,775 (indifference line)
13
(indifference line)
Two-way Sensitive Analysis of Eagle Airlines
Decision
14
Sensitivity to Probabilities

• Study the impact of uncertainties of events

License Technology
Patent Awarded
23M
Continue Development
Demands High
(p?)
43M
Develop Production and Marketing to Sell Product
(0.7)
Demands Med.
21M
(q?)
Development Result
Demands Low
3M
(1-p-q)
No Patent
Development Decision
-2M
(0.3)
Stop Development
0
Decision Tree of Research-and-Development
Decision
15
Decision Strategies
EMV(A)0
Strategy A Stop development
Strategy B Continue development, license
technology
EMV(B) 230.7 (-2)0.3 15.5M
Strategy C Continue development, develop
production marketing
EMV(U1) 0.7EMV(U2)(-2)0.3
U2
U1
EMV(U2) 43p21q3(1-p-q)
EMV(C)43p21q3(1-p-q) 0.7 (-2) 0.3
28p12.6q1.5
16
EMV(C) and EMV(B) are always greater than EMV(A),
so only EMV(B) and EMV(C) need to be compared to
find the best strategy
EMV(C) EMV(B) ? 28p 12.6q 14 (indifference
line)
Two-way Sensitive Analysis of Probabilities for
Research-and-Development Decision
17
Exercise
The town of Bedford is planning a celebration of
honor its gold-metal-winning Olympic
Cross-country skier. The celebration is set to be
held in the towns center for the day she
returns. Unfortunately, the weather report
predicts that the temperature might not be
conducive to an outdoor event on that day. As a
backup plan, they are thinking about reserving
indoor space. However, that reservation would
cost additional taxpayers money. Therefore,
Julie Bauer, the town manager, needs to decide
whether to reserve the indoor space.

• Draw the decision tree of this decision
• Suppose that the probability of a cold weather is
  estimated to be 40 and that holding the event is
  twice as important as saving taxpayers money,
  what would you suggest to Julie Bauer from the
  analyses of both the expected values and risk
  profiles?
• Julie Bauer is also wondering how her decision
  would be affected by the probability of the cold
  weather and the tradeoff between the two
  objectives, so what would you tell her by
  performing a two-way sensitive analysis for the
  two parameters? (Hint set the ratio of the
  weight of holding event to the weight of
  saving money, say k, as the parameter for
  tradeoff)

18
1.
Decision Tree
19
2.
Let wh and ws denote the weight of holding the
event and the weight of saving money,
respectively. Because wh 2ws and wh ws 1,
solving the equations, we can get wh 2/3, and
ws 1/3. Converting both attributes to 0 -100
scale, we can set holding event to 100,
canceling event to 0, saving money to 100,
and costing money to 0.
Not Cold
Not Cold
20
Reserve EV(Reserve) 0.4 200/3 0.6 200/3
200/3 Overall score 200/3(100) Dont
Reserve EV(Dont Reserve) 0.4 100/3 0.6
100 220/3 Overall score 100/3(40), 100(60)
No strategy dominates the other
Compared to the strategy of reserving the indoor
space, the strategy of not reserving the indoor
space has a slightly higher expected value of
the overall score but is riskier as well.
Cumulative Risk Profile
(Draw the risk profiles on your own)
21
3. Assume wh kws and wh ws 1, solving the
equations, we can get wh k/(k1), and ws
1/(k1).
Not Cold
EV(Reserve) 100k/(1k) EV(Dont Reserve)
p100/(k1) 100(1-p) EV(Reserve) EV(Dont
Reserve) ? 100k/(1k) 100p/(k1) 100(1-p) ?
kp1 (indifference curve) When kp gt1,
EV(Reserve) gt EV(Dont Reserve), and when When
kp lt1, EV(Reserve) lt EV(Dont Reserve)
22
EV(Reserve) gt EV(Dont Reserve)
EV(Reserve) EV(Dont Reserve)
C
EV(Reserve) lt EV(Dont Reserve)
(p0.4 k2)
Two-Way Sensitivity Analysis of the Probability
of Cold Weather and Weight Ratio of the Two
Attributes