Linear programming (LP) is an optimization method that finds the highest (or lowest) possible value of a set of linear equations given a number of linear constraints.
There are a number of objectives in Air Traffic Management that may be described by linear equations. Runway throughput for example: twice as many aircraft can be seen as "twice as good". Or delay: one minute of delay of a passenger aircraft with 400 passengers is as bad as 2 minutes of delay for a passenger aircraft with 200 passengers (both have 400 passenger-minutes-of-delay). Mathematical models based on these linear relationship can help to make decisions for air traffic controllers/planners.
There are obvious limitations to linear programming. Not everything can be modeled as a linear relationship. Often the relations are more complex and linear models will be a poor fit because the relationship is not linear at all. In ATFM modelling, usually binary integer variables are used that can only take values of zero or one, thus a Binary Integer Linear Program is solved (BIP). And is 40 minutes of delay of a 10 passenger aircraft as bad as 1 minute of delay of a 400 passenger aircraft? These kind of relationships tend to prevent the small aircraft from ever taking off: it always pays off to let a bigger aircraft jump the queue. Perhaps a quadratic component in the 'cost-function' would bring a solution here.
Therefore more advanced models are needed. Methods like Integer Linear Programming (ILP), Mixed Integer Linear Programming, Mixed Integer Quadratic Programming and more advance optimization algorithms are used as well.
Another problem is that while these optimization methods may be able to find a best possible strategy, they are not necessarily easy to understand. Air traffic controllers don't like to be just told what to do, they need to understand why a strategy is proposed so they can think ahead and create what-if scenarios. It is impossible to program every what-if scenario in the optimization model, so at times the controller will need to deviate from the proposed solution. Without understanding what the algorithm had planned, it is difficult for a controller to predict the knock-on effects of his deviation from the proposed plan.
Early arrival management software worked reasonably well when the weather was nicely playing along, but gave hopelessly wrong suggestions when a couple of thunderstorms would render the normal approach routes unusable. Flexibility and two way communication between the system and the controller is essential to make a good product, there is much more to it than choosing the right optimization algorithm.