ANKEP SENSITIVITY ANALYSIS

Decision Analysis Industrial Engineering University of Brawijaya

SENSITIVITY ANALYSIS

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EAGLE AIRLINES

Dick Carothers, president of Eagle Airlines, had been considering expanding his operation, and now the opportunity was available.

An acquaintance had put him in contact with the president of a small airline in the Midwest that was selling an airplane.

Many aspects of the situation needed to be thought about, however, and Carothers was having a hard time sorting them out.

Eagle Airlines owned and operated three twin-engine aircraft. With this equipment, Eagle provided both charter flights and scheduled

commuter service among several communities in the eastern United States.

Scheduled flights constituted approximately 50% of Eagle's flights, averaging only 90 minutes of flying time and a distance of some 300 miles.

The remaining 50% of flights were chartered. The mixture of charter flights and short scheduled flights had proved

profitable, and Carothers felt that he had found a niche for his company. He was aching to increase the level of service, especially in the area of

charter flights, but this was impossible without more aircraft.

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EAGLE AIRLINES

A Piper Seneca was for sale at a price of $95,000, and Carothers figured that he could buy it for between $85,000 and $90,000.

This twin-engine airplane had been maintained according to FAA regulations.

In particular, the engines were almost new, with only 150 hours of operation since a major overhaul.

Furthermore, having been used by another small commercial charter service, the Seneca contained all of the navigation and communication equipment that Eagle required.

There were seats for five passengers and the pilot, plus room for baggage.

Typical airspeed was approximately 175 nautical miles per hour (knots), or 200 statute miles per hour (mph).

Operating cost was approximately $245 per hour, including fuel, maintenance, and pilot salary.

Annual fixed costs included insurance ($20,000) and finance charges.

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EAGLE AIRLINES

Carothers figured that he would have to borrow some 40% of the money required, and he knew that the interest rate would be two percentage points above the prime rate (currently 9.5% but subject to change).

Based on his experience at Eagle, Carothers knew that he could arrange charters for $300 to $350 per hour or charge a rate of approximately $100 per person per hour on scheduled flights.

He could expect on average that the scheduled flights would be half full.

He hoped to be able to fly the plane for up to 1000 hours per year, but realized that 800 might be more realistic.

In the past his business had been approximately 50% charter flights, but he wanted to increase that percentage if possible.

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EAGLE AIRLINES

The owner of the Seneca has told Carothers that he would either sell the airplane outright or sell Carothers an option to purchase it within a year at a specified price. (The current owner would continue to operate the plane during the year.)

Although the two had not agreed on a price for the option, the discussions had led Carothers to believe that the option would cost between $2500 and $4000.

Of course, he could always invest his cash in the money market and expect to earn about 8%.

As Carothers pondered this information, he realized that many of the numbers he was using were estimates.

Furthermore, some were within his control (for example, the amount financed and prices charged) while others, such as the cost of insurance or the operating cost, were not.

How much difference did these numbers make? What about the option? Was it worth considering? Last, but not least, did he really want to expand the fleet? Or was there something else that he should consider?

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Sensitivity Analysis: A Modeling Approach Sensitivity analysis answers the question, "What makes a

difference in this decision?

Requisite decision model is a model whose form and content are just sufficient to solve a particular problem.

The issues that are addressed in a requisite decision model are the ones that matter, and those issues left out are the ones that do not matter.

Determining what matters and what does not requires incorporating sensitivity analysis throughout the modeling process.

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Problem Identification and Structure

A decision-analysis process flowchart

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Problem Identification and Structure Solving the wrong problem sometimes is called an "error of

the third kind.

An error of the third kind, or Type III error, implies that the wrong question was asked; in terms of decision analysis, the implication is that an inappropriate decision context was used, and hence the wrong problem was solved.

How can one avoid a Type III error?

The best solution is simply to keep asking whether the problem on the surface is the real problem.

Is the decision context properly specified?

What exactly is the "unscratched itch" that the decision maker feels?

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Problem Identification and Structure In the case of Eagle Airlines, Carothers appears to be eager to

expand operations by acquiring more aircraft.

Could he "scratch his itch" by expanding in a different direction?

In particular, even though he, like many pilots, may be dedicated to the idea of flying for a living, it might be wise to consider the possibility of helping his customers to communicate more effectively at long distance.

To some extent, efficient communication channels such as those provided by computer links and facsimile service, coupled with an air cargo network, can greatly reduce the need for travel.

Pursuing ideas such as these might satisfy Carothers's urge to expand while providing a more diversified base of operations.

So the real question may be how to satisfy Carothers's desires for expansion rather than simply how to acquire more airplanes.

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Problem Identification and Structure A decision situation might be represented in a variety of

different ways.

Sensitivity analysis can aid the resolution of the problem of multiple representations by helping to identify the appropriate perspective on the problem as well as by identifying the specific issues that matter to the decision maker.

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Problem Identification and Structure Is problem structuring an issue in the Eagle Airlines case? In

this case, the alternatives are to purchase the airplane, the option, or neither.

Although Carothers might consider a variety of fundamental objectives, such as company growth or increased influence in the community, in the context of deciding whether to purchase the Seneca, it seems reasonable for him to focus on one objective: maximize profit.

Carothers could assess the probabilities associated with the various unknown quantities such as operating costs, amount of business, and so on.

Thus, it appears that a straightforward decision tree or influence diagram may do the trick.

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Influence diagram representing the Eagle Airlines decision

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Input variables and ranges of possible values for Eagle Airlines aircraft-purchase decision.

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The annual profit would be the total annual revenue minus the total annual cost

Problem Identification and Structure Thus, using the base values, Carothers's annual profit is

estimated to be $230,000 - $220,025 = $9,975.

This represents a return of approximately 19% on his investment of $52,500 (60% of the purchase price). 25

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Sensitivity to Probabilities

Modelling the uncertainty surrounding the critical variables:

Capacity of Scheduled Flights

Operating Cost

Hours Flown

Charter Price a decision variable set by Carothers.

Example:

Assume that, in an initial attempt to model the uncertainty, Carothers chooses two values for each variable, one representing an optimistic and one a pessimistic scenario.

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Sensitivity to Probabilities Influence diagram of Eagle Airlines decision. Note that only three variables are considered to be uncertain, and that Hours Flown and Capacity are considered to be probabilistically dependent.

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Decision tree for Eagle Airlines with uncertainty for three variables. Profit is calculated with all other variables held at their base values.

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A relevance arc leads from "Capacity of Scheduled Flights" to "Hours Flown" in the influence diagram, and in the decision tree the value for r = P(Low Hours | Low Capacity) may not be the same as the value for s = P(Low Hours | High Capacity).

In fact, our argument suggests that r will be greater than s.

On the other hand, Operating Cost is judged to be independent of the other variables.

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Assessing some values for probabilities p, q, r, and s.

Suppose that Carothers is comfortable with an assessment that p = 0.5, or that Operating Cost is just as likely to be high ($253) as low ($237).

Suppose that Carothers feels that a reasonable way to represent the dependence between Hours and Capacity is to let s be 80% of r.

If Capacity is high (55%), then the probability that Hours = 650 is only 80% of the probability that Hours = 650 when Capacity is low.

With these two specifications, we now have only two unspecified probabilities left to consider, q and r.

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Eagle Airlines' decision tree with probabilities substituted for p and s. This decision tree is now ready for a two-way sensitivity analysis on q and r.

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We now can create a two-way sensitivity graph for q and r. As with the two-way sensitivity analysis above, the graph will show regions for which the expected value of purchasing the Seneca is greater than investing in the money market.

To create the graph, we first write out the expected value of purchasing the airplane in terms of q and r, including the specifications that p = 0.5 and s = 0.8r.

This equation comes from solving the decision tree:

EMV(Purchase) = 0.5{q[-9725r - 4225(1 - r)] + (1 - q) [6525(0.8r) + 18,275(1 - 0.8r)]} + 0.5{q[675r + 10,175(1 - r)] + (1 - q)[16,925(0.8r) + 32,675(1 - 0.8r)]}

EMV(Purchase) = q(3500r - 22,500) - 11,000r + 25,475

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We would want to purchase the airplane if EMV(Purchase) > 4200.

q(3500r - 22,500) - 11,000r + 25,475 > 4200

25,475 - 4200 - 11,000r > q(22,500 - 3500r)


Two-way sensitivity graph for Eagle Airlines.

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The area below the line contains points where q < (21,275 - ll,000r)/(22,500 - 3500r); for these (q, r) points, EMV(Purchase)>$4200.

The graph makes sense because q and r are probabilities of the pessimistic scenarios low Capacity and low number of Hours Flown.

If Carothers thinks that the pessimistic scenarios are likely (q and r close to 1), then he would not want to buy the airplane.

Carothers may not have especially firm ideas of what the probabilities q and r should be.

Suppose, for example, that in the process of coming up with the probabilities he feels that q could be between 0.4 and 0.5 and that r could be between 0.5 and 0.65. These probabilities are represented by the points inside Rectangle A in Figure

5.8. All of these points fall within the "Purchase Seneca" region, and so the conclusion is that the Seneca should be purchased.

The decision is not sensitive to the assessment of the probabilities.

If, on the other hand, Carothers thinks that reasonable values of q and r fall in Rectangle B, then the optimal choice is not clear. In this situation, he could reflect on the chances associated with Capacity and

Hours Flown and try to refine his model of the uncertainty.

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The value of the two-way sensitivity graph is to provide guidance in determining how much effort is needed to model uncertainty in a decision problem.

Looking at it another way, the graph can reveal whether the decision is sensitive to the uncertainty in the problem and to the modeling of that uncertainty.

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Sensitivity Analysis in Action

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Sensitivity Analysis in Action Two-way sensitivity analysis for the heart disease treatment decision.

Source: Macartney et al. (1984).

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Warming Up

An orange grower in Florida faces a dilemma. The weather forecast is for cold weather, and there is a 50% chance that the temperature tonight will be cold enough to freeze and destroy his entire crop, which is worth some $50,000. He can take two possible actions to try to alleviate his loss if the temperature drops. First, he could set burners in the orchard; this would cost $5000, but he could still expect to incur damage of approximately $15,000 to $20,000. Second, he could set up sprinklers to spray the trees. If the temperature drops, the water would freeze on the fruit and provide some insulation. This method is cheaper ($2000), but less effective. With the sprinklers he could expect to incur as much as $25,000 to $30,000 of the loss with no protective action.

Compare the grower's expected values for the three alternatives he has, considering the various possible loss scenarios for the burners and the sprinklers. Which alternative would you suggest the grower take? Why?

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NEXT . . .

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Creativity and Decision Making

Theories of Creativity

Chains of Thought

Phases of the Creative Process

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ANKEP SENSITIVITY ANALYSIS

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