Share a runway's cost among users needing different runway lengths. If user needs a facility costing , a nonempty coalition pays , and . This is a subadditive cost game because the maximum of two nonnegative costs does not exceed their sum. A cost-core allocation always exists: charge the whole cost to one longest-runway user and zero to all others. The Shapley value of an airport game gives a symmetric alternative that also lies in the cost core.
Suppose the distinct required runway lengths are , with users of type , constant cost per unit length, and suffix counts . Then every type- user has the displayed Shapley value. For layer , exactly users require it. The first of them in a uniformly random permutation contributes its entire cost . Each requiring user has probability of being first. Linearity of the Shapley value sums these expected marginal contributions across the layers that user needs.
The allocation lies in the cost core. Its payments sum to the full runway cost. For any coalition , its total payment toward layer is , where is the set of users requiring that layer. This is zero if needs no such layer, and at most the full layer cost otherwise. Summing gives . Suffix counts followed by prefix sums compute all distinct type payments in arithmetic operations.
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The "airport problem" generally refers to a variety of optimization problems related to the operation and management of airports, particularly those involving scheduling, capacity management, and resource allocation. Depending on the context, it can involve several specific issues: 1. **Flight Scheduling**: Determining optimal schedules for arrivals and departures to minimize delays while maximizing the utilization of runways, gates, and other resources.