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I am sitting at the airport and I was wondering if there is an optimal number of passengers to try to board a plane through an interesting fare.

Flying an empty plane is obviously the worst situation: you have fixed costs and fuel costs that depend on the weight of the plane. For an empty plane your gain is 0% (all loss, 100% would mean breaking even and anything above that is pure gain).

As you start to add passengers, their weight is added (say, 110 kg per passenger and their luggage), which means more fuel to buy. But you also get some income so you are raising up from 0%.

Is the relationship gain(number_of_passengers_boarded) such that the weight passengers add is insignificant, semi-significant, or a key factor? Is the curve always raising (which means the more passengers - the better, until all space is exhausted)? This will certainly depend on the plane, so with two reasonable extremes (a typical regional jet vs a long-haul jet) are there key differences?

In other words: is it always interesting to lower the fare as the start date approaches? (purely from a gain perspective, I am not taking into account ways to optimize this for a passenger by, say, waiting until the last moment)

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If the cost of the ticket is less than the cost of the fuel necessary to carry the added weight of the passenger, his baggage and the extra fuel required to carry them, then no, it is not beneficial to fill another seat at any cost.

Additionally, cargo capacity not taken by passing passengers (self-loading-cargo) is often used to carry non-self-loading-cargo, so the capacity doesn't always go to waste. Cargo can be more profitable than passengers, so there can be limited pressure to fill the seats by lowering the price.

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Fill the plane up with payload, that's what they're designed for! The industry has high fixed costs, plus the variable costs per flight to carry the plane without payload to its destination. The revenue is from payload per flight only - so what is needed is maximising flights/year, and maximising payload/flight, to make revenue.

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The slide above is from this presentation, and lists the various cost categories - costs per year, per flight, per kilometre,per route, per seat. Revenue is per seat-kilometre. Fuel necessary for the trip is determined shortly before departure, while selling seats takes much longer before departure. All these variables create a multi-dimensional matrix for determining profitability, making the consideration of extra fuel burn per passenger kilo almost meaningless. Yes a 1% saving in weight results in a 1% fuel saving, as per this answer, but sell the seats! and control all costs!

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