Sensitivity Analysis ?


  • Sensitivity analysis means varying the inputs to a model to see how the results change
  • Sensitivity analysis is a very important component of exploratory use of models
    • model is not regarded as “correct”
    • sensitivity analysis helps user explore implications of alternate assumptions
    • human computer interface for sensitivity analysis is difficult to design well
  • In many models we need to make assumptions we cannot test
  • Sensitivity analysis examines dependence of results on these assumptions


  • Sensitivity Analysis Answers the question:
    • What does make a difference in the decision?
  • Determining what does matters and what does not requires incorporating sensitivity analysis throughout the modeling process.
  • No optimal sensitivity analysis procedure exist for decision analysis: Model building is an Art!
  • The question that we ask performing SA is: Are we solving the right problem?
  • Type III Error: implies that the wrong question was asked or inappropriate decision context was used.



Example: Air Line Company:


  • Eagle Airlines has expansion plan. Currently 50% of flights are scheduled and 50% are chartered.
    • A new seneca airplane costs 85000-90000USD.
    • It has seats for 5 passenger. Operating cot is 245 per hour. Annual fixed cost is 20000 including insurances and finance charges.
    • The company needs to borrow 40% of the money with 9.5% interest rate.
    • The company may be able to charge 300-350$ per hour for charter or 100$ per person for scheduled flights. Scheduled flights on average is half full. Company hops that the airplane fly 1000 hour per year but 800 is more realistic.
  • Other options:
    • Invest in Bank with 8%
    • Rent airplane with 2500-4000$



Modeling the problem:


  • Alternatives:
    • Purchasing the airplane
    • Renting the airplane
    • Investing in a bank
  • Objectives?
  • If the probability of various unknown such as operating cost, amount of business ,etc is known then an decision tree or influence diagram can be used to structure the problem.

Initial influence diagram:


Intermediate calculations

Consequence node





The base Value: Initial Guess regarding the variables

Lower and Upper Bound: Absolute extremes ( variables can not fall beyond)



Annual Profit:


  • We can use input variables to calculate estimate of annual profit: 23000-220025= 9975$
  • This shows 19% on the investment (60% of plane)



One Way Sensitivity Analysis:


  • What variables really make a difference in terms of the decision in hand?
    • Do different interest rates really matter?
    • Does it matter that company can set the ticket price?
    • Hours Flown how much impacts on the profit?
  • For example in the case of Hours Flown company is quite unsure by setting bands between 500 and 1000. To show the effect of this variable we use a graph.



One way Sensitivity graph for hours flown:


The fact that the company believes that the hours flown could be above or below 664 suggests that this is a crucial variable.



Tornado Diagrams:


  • A Tornado Diagram allows us to compare one way SA for many variables at once.
  • Tornado Diagram tells us which variables we need to consider more closely and which ones we can leave at their base value.
  • We take input variables and wiggle them between high and low values to determine how much change is induced in profit.
  • Every thing is held at its base value except the variable under study.


Setting Capacity of scheduled flights at 40% instead of 50% implies a loss of 10025


  • The most sensitive variable ( one with the longest bar ) is set at top and the least sensitive at the bottom.
  • The vertical line at 4200 represents what could be earned by investment in Bank.
  • Interesting points:
  • Tornado Diagram tells us which variables we need to consider more closely and which ones we can leave at their base values.


Two way sensitivity Analysis:


  • Suppose we wanted to explore the impact of several variables at one time.
  • A graphical technique is available for studying the interaction of two variables.
  • For example suppose we want to consider the joint impact of changes in the 2 mot crucial variables( Operating cost and Capacity of scheduled flights)
  • Imagine a rectangular space taht represents all of the possible values that these two variables could take.
  • We have to find those values of 2 variables for which the annual profit would be less than 4200$.
  • The point labeled base value shows that when we plug in the base values for the capacity and operating cost, we get an estimated profit that is grater than 4200$ so the project looks promising.
  • However if we consider point C where operating cost is slightly more than base (248) and capacity is slightly less than base (48%) they lead to a situation which suggest not to buy the plane!



Sensitivity to probability:


  • The next step is to model the uncertainty surrounding the critical variables.
  • There are 4 critical variables in this example: Capacity of scheduled flights, Operating cost, Hours flown and Charter price, which we only need to think about 3 because charter price is decided by company.



Changed Influence diagram:


Chance Nodes




Decision tree:


  • This decision tree shows the pessimistic and optimistic values for the three uncertain variables.





  • Now that the problem is simplified, we can include consideration about interdependencies of the chance variables.
  • For example probability distribution of Hours flown depend on Capacity of scheduled flights. Thus r is greater than s in Decision tree.
  • The next step is to asses values to p,q,r, and s.



Sensitivity graph:


  • Now we can create a two way sensitivity graph for q and r.
  • We write the expected value of purchasing airplane in terms of q and r. We set p=0.5 and set s=0.8r.

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