Linear Programming in Real$-$Life

Linear Programming also known as Linear Optimization is a mathematical solution that aims to find optimal solutions for issues where the relationship between variables is linear. Linear programming tries to find the best optimal solution within constraints.

Linear programming is a powerful tool which is applied in the fields of

Business planning: Optimizing production schedules, resource allocation, and management of inventory.

Finance: Managing loan risks or creating investment portfolios

Logistics: Can be used to design optimal transport routes or delivery schedules

Airlines use Linear programming extensively in their scheduling of flights and their crew$-$ pilots and aircrew schedules. Airlines typically want to maximize their earnings by optimizing their operations.

They achieve this by

Objective Function:

This defines what the airlines want to achieve. This could be minimizing costs, maximizing profits, and aircraft utilization.

Decision variables: These are the things the airlines can control like the type of aircraft to use depending on the number of passengers, the number of flights on a route whether to fly non$-$stop or the routes to fly, and the departure times.

Constraints: The constraints are the limitations the airline has to take into account like

$1.$ The number of planes and type of aircraft available.

$2.$ The availability of crew for each route and time. Pilot and co$-$pilot qualifications vary for different types of aircraft. Flight crews have restrictions on minimum rest periods, the maximum amount of flying time per day, and the length of mandatory rest periods between flights. Number of crew members required for a particular type or size of aircraft. $($Libretexts$,2021)$

$3.$ Passenger demand: Passenger demand varies from destination to destination and airlines need to find the optimal way of managing flights to high$-$traffic and low$-$traffic destinations

$4.$ Flight frequency: Airlines need to strike a balance to maintain flight frequency to high and low$-$demand destinations

$5.$ Airport capacity: Different airports have different takeoff and landing slots and capabilities. Not all airports can handle all types of aircraft.

$6.$ Boarding time: The Boarding of passengers is the longest time$-$consuming process in an airline$$s schedule. Airlines seek to minimize the boarding time by implementing an optimized boarding policy. The article below describes how Linear programming is used to reduce boarding times. $($Soolaki et al$.,2012)$

Specialized software can find optimal solutions for these decision variables. An optimized solution will find a solution to allocate resources taking into account all the constraints, which will help improve

Cost efficiency: will help airlines minimize operational costs like fuel, maintenance, and crew salaries

Improved scheduling: Optimize flight timings, frequency, and aircraft types to meet passenger demand efficiently.

Reduction in flight delays: An optimal solution will find ways to reduce flight delays due to airport congestion.

Even though Linear Programming helps airlines develop very effective schedules to maximize profit and customer satisfaction and minimize costs it is still not a perfect solution. Linear programming models cannot fully capture disruptions due to weather or any unplanned mechanical issues. But generally, Linear programming is a very effective tool for airlines to create cost$-$effective schedules.