### Solution of Large Scale Crew Scheduling Problems

#### Problem description

A set of basic activities or tasks – typically, to fly an aircraft from A to B – has to be assigned to sets of crew members such that all basic activities are covered.

The basic activities are grouped into so-called pairings that are sequences of tasks usually starting and ending at the same home base. Additionally, complex rules and regulations coming from legislation and contractual agreements have to be met by the solutions. These often cannot be modelled as linear constraints.

Fig. 1: Two among large number of feasible solutions of a small-sized crew scheduling problem.

#### Mathematical model

Let be the number of flights and  be the total number of feasible pairings. The pure IP formulation of the problem is:

The ’s are 0-1 decision variables that indicate which pairings belong to the solution. The coefficient equals 1 if pairing contains flight , otherwise, is 0.

The number of feasible pairings can be several millions even for only medium-sized problems. Effcient methods have to be applied to attack such problems with a huge number of variables.

#### Contact

Tran Van Hoai

Im Neuenheimer Feld 368
69120 Heidelberg
Email:
hoai@iwr.uni-heidelberg.de

#### Methods

·        Parallel Branch and Cut on the pure IP formulation with:

o       Strong cutting planes, such as:

§        Clique inequalities:

§        Odd cycle inequalities:

o       Preprocessing, such as:

§        Eliminate duplicated columns,

§        Eliminate dominated rows,

§        Reduced cost fixing

o       Searching B&B tree in parallel with the master-slave model.

Fig. 2: performance of parallel branch and cut to solve small-sized crew scheduling problems.

·        Branch and Price with:

o       Column generation method on flight network by solving:

§        Constrained Shortest Path problems,

§        K-Shortest Path problems with black-box rule checking,

§        Heuristic Enumeration

o       Column generation by Constraint Logic Programming, utilizing the propagation of non-linear constraints on feasible region.

·        Local Search Heuristics to improve the solution, (see Fig. 3)

 Merge 2 pairings to a pairing Exchange segments of 2 pairings together Scroll pairing over schedule horizon

Fig. 3: three local search heuristics to reduce cost function.