1 Airline Crew Pairing Optimization: Its challenges
Airline scheduling poses some of the most challenging problems in the entire Operations Research (OR) domain. In that, crew scheduling (CS) constitutes one of the most important and challenging planning activities. Notably, the crew operating cost is the secondlargest component of an airline’s total operating cost (after the fuel cost). Hence, its optimization promises enormous benefits, and even marginal improvements may translate to annual savings worth millions of dollars for an airline. However, CS is a combination of complex combinatorial optimization problems (with NPhard computational complexity), namely crew pairing and crew assignment, which are solved sequentially. Here, crew pairing optimization aims at generating a set of flight sequences (each called a crew pairing) to cover a finite set of flight legs from an airline’s timetable at minimum cost, while satisfying several legality constraints linked to the federations’ safety rules, airlinespecific regulations, labor laws, etc. Subsequently, crew assignment aims at assigning crew members to these optimal crew pairings. This research focuses on the critically fundamental step of CS, in that, a crew pairing optimization problem (CPOP) is formulated, and an optimization framework to solve the posed CPOP has been proposed. The distinctive contribution of this research relates to its tackling of a largescale complex flight network for an airline (leading to more than a billion legal pairings/variables), and validation of the results by the research consortium’s Industrial partner, GE Aviation.
2 Related Work
This research assumes significance owing to the unprecedented scale and complexity of the underlying problem. Notably, airline CPOP has received significant attention from the OR society, resulting which numerous optimization frameworks have been developed, and either published or commercialized as blackbox software. The available literature could be categorized in terms of smallscale and large scale CPOPs. In the former case, where enumeration of all legal pairings is computationally viable, the most widely adopted optimization class of techniques are Genetic Algorithms
(GA) which which are populationbased randomized search heuristics [
1, 2, 3, 4]. However, these GAbased approaches become obsolete when tested for largescale CPOPs [2, 4]. In the latter case, where enumeration of all legal pairings is computationally challenging, Column Generation (CG) is the most widely adopted technique, since it allows for guided exploration of search space based on the corresponding gain in the objective function(s). In that, only the pairings promising objective improvement are generated. However, as the scale of the problem grows, the exact CG implementation becomes intractable. This justifies the heuristic implementations of CG for medium and largescale CPOPs [5, 6]. Despite this progress, the much prevalent complex flight networks, incorporating multiple hubandspoke subnetworks and/or multiple crew bases, largely remain unaddressed. In that, the number of potential crew pairings (optimization variables) grow exponentially with the number of flights on account of not only its departure from a pointtopoint network; multiplicity of hubs as opposed to a single hub; but also the multiplicity of crew bases. Hence, the practical utility of the existing solutions is limited, and all the more questionable considering that air traffic is expected to grow double in 20 years with a 3.5% compound annual growth rate [7].3 Proposed Optimization Framework
The merit of this work lies in its attempt to overcome the existent research/utility gap by way of its addressal of a complex flight network through a customizable optimization framework, named as , tested and validated on realworld data, endorsing its higher technology readiness levels (TRL 89: System Test, Launch & Operations [8]). The research consortium’s Industrial sponsor, GE Aviation, has provided this realworld data set (from the US airlines’ flight network) for the testing and validation of the
. It is structured around the integration of deterministic optimization methodologies and airlinedomainknowledgebased heuristics. It is usable for optimizing airlines’ crew pairing costs, even for their large and complex flight networks (characterized by the presence of multiple hubandspoke subnetworks and/or multiple crew bases). It leverages the recent advancements in mathematical programming techniques (particularly, Column Generation, CG, and MixedInteger Linear Programming, MILP) along with enhanced datahandling capabilities and computational speeds. Furthermore, it is open to modifications, a freedom not generally available with commercial software. In doing so, it offers an opportunity for airlines to adapt and improvise the existing modules based on their domain knowledge or demandbase. Its higherlevel block diagram is presented in figure 1.
accepts a finite set of an airline’s flight schedule, its pairingcosting rules and pairinglegality constraints as input; and outputs a set of legal pairings, covering the given set of flights at minimal cost. As shown in figure 1, the proposed framework is based on multiple modules which are elaborated here. The novelty of AirCROP lies not just in the design of its building modules, namely Legal Crew Pairing Generation, Initial Feasible Solution Generation, and Optimization Engine; but critically in how these modules interact. In that:

Legal Crew Pairing Generation [9]:

Enumeration of all possible legal crew pairings from a given set of flights while satisfying a given set of pairinglegality constraints

Efficient constraintsatisfaction methodology by prioritizing the order in which a given set of pairinglegality constraints have to be satisfied

Timeefficient execution of this module using parallelization on multiple cores of a single processor, exploiting the multiple crew base characteristic of the input flight data set


Initial Feasible Solution Generation [10]:

Developed a computationally and timeefficient initialization heuristic to generate an Initial Feasible Solution (IFS; a manageable set of legal pairings covering all given flights) for largescale CPOPs

Utilizes a divideandcover strategy to decompose the given flight schedule into smaller flight subsets, and an Integer Programming (IP) technique to select pairings, constituting a minimalcost solution for each decomposed flight subset

It is ensured that the generated IFS has a balance between its characteristics, namely cost and degrees of searchfreedom, such that it favors an exploratory search in the subsequent module, Optimization Engine.

Generated IFS serves as input to the subsequent module, Optimization Engine.


Optimization Engine:
It involves recombination of input pairing set with new pairings promising lowercost solution with fullflight coverage. It is implemented using the following submodules:
Submodule I Optimization in continuous domain:

Developed an iterative optimization approach to search for a fullcoverage, lowercost, linearlyrelaxed solution (fractions of pairings are used to make up the fullcoverage solution) which utilizes existing/novel OR techniques in a computationally and timeefficient manner

In each of its iterations, needbase new legal pairings are generated using a domainspecific novel CG heuristic which attempts to maintain a balance between exploration (new pairings being generated from randomlyselected flights) and exploitation (new pairings being generated using domainknowledge and flightconnection information of the previous iteration’s solution).

In addition to this, another CG strategy has been implemented which has improved the performance of drastically. In that:

An archive dictionary is being maintained in which legal pairings being generated in each iteration of Submodule I are updated (items of this dictionary are unique flightpairs)

In each iteration of Submodule I, a set of critical flightpairs are identified (using the iteration’s dual solution) and for each of these selected flightpairs, a finite number of pairings are selected randomly from the archived dictionary. These pairings are added back to the LPsolution of the iteration.



Submodule II Optimization in integer domain:
A fullcoverage, lowercost integer solution (full pairings are used to make up the fullcoverage solution) is ensured through repetitive interactions between MILP technique (using available solvers) and Submodule I. Each of these interactions are referred to as a Reoptimization Loop.

Acknowledgments
This research is an outcome of an IndoDutch joint research project, supported by the Ministry of Electronics & Information Technology (MEITY), India [grant 13(4)/2015CC&BT]; Netherlands Scientific Research Organization (NWO), Netherlands; and General Electric (GE) Aviation. The authors would like to thank GE Aviation team members: Saaju Paulose (Senior Manager), Arioli Arumugam (Senior Director Data & Analytics), and Alla Rajesh (Senior Staff Data & Analytics Scientist) for their invaluable support in successfully completing this research work.
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