Suppose that an airline company has X planes, in general the companies keep a number of these aircraft (say 80% of X) in reserve in case of hazards. My question is how the airline companies compute the number of aircraft to operate and the ones to keep in reserve?

I cannot find any papers written about this problem on the literature.

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    $\begingroup$ Shouldn’t it be %20 in reserves? $\endgroup$ – Oguz Toragay Sep 9 '19 at 12:00
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    $\begingroup$ Are you asking how this is done in practice? Or for literature on this problem? Or for a solution to the specific problem you described? In either case, but especial the third, we might need a bit more info. $\endgroup$ – LarrySnyder610 Sep 9 '19 at 12:31
  • $\begingroup$ I would like to know both: how this is done in practice and what are the works that have been done on this subject in the literature. I don't have more information about this problem, but I know that the objective would be to find the right portion to avoid delays in the flights. $\endgroup$ – user109284 Sep 9 '19 at 13:22
  • $\begingroup$ While, in the short term, the number of planes owned would be fixed, on a longer time frame, it isn't. So the optimization wouldn't be on what percentage to hold in reserve, it would be on how many to buy for active service and how many to buy for reserve. $\endgroup$ – Acccumulation Sep 9 '19 at 19:10

In this document the airline fleet decision process has been categorized in the following bullets:

  • Forecast of expected traffic demand (RPK)
  • Planning average load factor (%)
  • ASK needed to be generated to meet the traffic demand
  • The productivity of the aircraft (ASK per day) results in the number of aircraft to be acquired and its financial impact (Costs)
  • Merging revenue estimates with costs results in different margin scenarios for aircraft types being considered

which in turn has the following main drivers:

  • Traffic forecasts
  • Yield forecasts
  • Operating costs estimation
  • Estimated aircraft productivity
  • Holistic drivers feed the decision process system

In addition to the mentioned document, you can also find the following papers that discuss the different aspects of the fleet planning problem:

  • Airline schedule planning: Integrated models and algorithms for schedule design and fleet assignment [p1].
  • Robust optimization for fleet planning under uncertainty [p2].

[p1]: Lohatepanont, Manoj, and Cynthia Barnhart. "Airline schedule planning: Integrated models and algorithms for schedule design and fleet assignment." Transportation Science 38.1 (2004): 19-32.

[p2]: List, George F., et al. "Robust optimization for fleet planning under uncertainty." Transportation Research Part E: Logistics and Transportation Review 39.3 (2003): 209-227.

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    $\begingroup$ thank you for your answer, but my question concerns the planes that are already owned by a company, how we can decide weather to use for example 80% of the planes and let 20% in reserve or to use 90% of the planes and let 10% in the reserve? $\endgroup$ – user109284 Sep 9 '19 at 17:08

The reason you're not finding anything about this in the literature is that airlines do everything they can to avoid having airplanes in reserve: an airplane on the ground is an airplane that's losing money.

An airline will determine the fleet size needed to serve the anticipated demand (see Oguz's answer for literature on how they do this), and will cover isolated problems (eg. an airplane with an engine that won't start) by deferring scheduled maintenance, by dispatching aircraft that have minor problems (eg. the backup compass isn't working), or by rebooking passengers on alternate flights.

See this question and answers on Aviation SE for more details.

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I think this is an interesting question from an operations research / simulation exercise. As such you could attack the question very simply to start with and then go on more and more detailed if it makes sense. Not very different really from many other OR questions.

A very first, very simple calculation can be done on the back of an envelope, say per week:

  • each week of flights creates $x$ dollars in profit per plane.
  • each plane set aside as spare costs $y$ dollars.
  • one problem takes $z$ weeks to fix.
  • the probability of a plane getting a problem needed to be fixed is $p$.

You can now calculate an optimum where (profit) - (costs for spare planes) is at maximum.

My guess though, is that the cost to keep a plane as a spare is high, and the profit per week is low, giving a profit maximum at $0$ spare planes.

Starting from this very simple back of the envelope calculation you can start adding details. Maybe it takes different amounts of time to fix the plane (with different probabilities). Maybe, even if you have a plane it is in the wrong airport so you will miss a few of the flights. Maybe you could rent a plane for a while. And on it goes. My guess is that rather soon you find that there are several stochastical variables and that the analytical answer will be difficult to find -- you then start doing simulations instead.

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