Chapter 5: Delta Airlines

With over 2500 domestic flight legs daily, one major task faced by Delta Air Lines is deciding, in light of cost and other constraints, how to assign its fleet of over 500 aircraft along its daily routes so as to match the number of available seats to the expected number of passengers on any given flight. With the help of mathematical programming algorithms, experts at Delta have developed a solution to the fleet assignment problem that is expected to save Delta millions of dollars over the next few years.

Although the overall process of cost minimization is very complex to analyze, an important element is the attempt to minimize costs due to "spill" -- i.e., the number of passengers that cannot be accommodated because of insufficient aircraft capacity resulting from incorrect fleet assignments. While some of these spilled passengers might be "recaptured" on other Delta flights, those passengers who are lost to competing airlines represent lost revenue. Of course, assigning too large an aircraft for a particular flight could create the opposite problem -- losses due to empty seats. An important objective is therefore to estimate the expected size of the spill for any given size of aircraft.

Making the assumption that passenger demand is normally distributed (a common airline-industry assumption) with a given mean and standard deviation, spill is represented diagrammatically as the truncation of the passenger demand function at the point of an aircraft's capacity.

The following graph of airline passenger demand assumes a mean of 120 passengers and a standard deviation of 25 passengers.



The probability that there will be spill for a Boeing 727(which has a capacity of 148 passengers) is given by the shaded area under the normally distributed demand function to the right of 148 passengers. The shaded area is given by the following integral.



Thus, with the given passenger demand distribution, there is a 13 percent chance that there will be more passengers than can be accommodated by a Boeing 727. This probability can then be used to estimate the expected number of spilled passengers for a Boeing 727. Multiplying the expected number of spilled passengers by the percent that is expected to be lost to competitors (i.e., not "recaptured" on other Delta flights) gives the estimated total number of lost spilled passengers. Finally, when the number of lost spilled passengers is multiplied by the average revenue per spilled passenger, the result is an estimate of spill cost to Delta.

Source: The Operations Research Society of America and the Institute of Management Sciences, 290 Westminster Street, Providence, RI 02903. www.informs.org

What Would You Do?

1. Use a graphing utility to verify that the definite integral in the Case Study is approximately equal to 0.1314. Explain how you handled the infinite limit of integration.
   
   
   
2. Use the formula given in Section 4.2 for the Normal Probability Density Function to derive the definite integral above. Explain why the following definite integral is equal to 1.
   
3. With a given passenger demand function for a particular scheduled flight, explain what happens to the expected spill as larger-sized aircraft are assigned to that flight.
   
4. Repeat the analysis for a Boeing 727 with capacity of 162 passengers if the mean is 130 passengers with a standard deviation of 30.

[mean]: The mean or average of the n numbers x₁, x₂, x₃, ...,x_n is given by



[standard deviation]: The standard deviation of the n numbers x₁, x₂, x₃, ...,x_n is given by



The standard deviation measures how much a set of data varies from the mean.