# Sensitivity Analysis ?

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**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?
- Company Growth, Greater influence in the community, Maximizing Profit

- 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

**Variables:**

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.

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**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:
- Annual profit is insensitive to aircraft price, Interest rate, and proportion financed.

- 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.

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**Changed Influence diagram:**

Chance Nodes

constants

**Decision tree:**

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

**Uncertainties:**

- 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.