Multi-Criteria Methods for Unit Crewing in the Airline Tour-of-Duty Planning Problem

In crew scheduling, the aim of unit crewing is to keep crew members from different crew ranks together to operate a sequence of flights as a unit as much as possible. When the crew members of different ranks operate together for a sequence of flights, the sequence is unit crewed. A unit crewed solution is considered to be more robust in the sense that aircraft departures are less likely to be delayed due to waiting for a member of a crew. It is usual to solve the ToD planning problem for each crew rank as a separate problem. To maximize unit crewing in the Tour-of-Duty (ToD) solution, a set of ToDs for one rank in terms of minimal cost is constructed and a weighted sum method is used to penalize any tour for the other rank that does not contain the same flight sequence. We believe that a model that solves the two ToD planning problems simultaneously can improve the number of unit crewing sequences without increasing the total crew cost. We use the number of unit crewed flight as a second objective and a branch and bound strategy to ensure that as many unit crewing sequences can be selected as possible. 1 Airline Crew Scheduling Commercial airlines are required to solve many resource scheduling problems to ensure that aircraft and aircrew are available for each scheduled flight. The aircrew scheduling problem consists of two distinct sub-problems, the tour-of-duty (ToD) planning problem and the rostering problem. The ToD planning problem consists of finding sequences of flights to partition the flight schedule into ToDs that can be flown by one crew member. The rostering process involves the allocation of the planned ToDs from the first ToD planning problem to individual crew members to form a line-of-work (LoW) for each crew member over the rostering period. A ToD is an alternating sequence of duty periods and layovers, where duty periods comprise one or more flights, and may also include passengering flights. A passengering flight is one on which a crew member travels as a passenger in order to be positioned at a particular airport for a subsequent operating flight or to return to their home base. The first duty period of a ToD must start at a crew base, and the last duty period must end at the same crew base. An airline might have several bases (cities) at which crew are domiciled. Each ToD will have an associated crew complement made up of a number of crew of some ranks. A set partitioning model provides an underlying mathematical model for the ToD planning and the rostering sub-problems of the aircrew scheduling problem. The set partitioning problem (SPP) is a specially structured zero-one integer linear programme with the form: Minimise z = cx subject to Ax = e (SPP) x ∈ {0, 1} where e = (1, 1, ., 1) and A is a matrix of zeros and ones. In the basic ToD planning model, each column or variable in the SPP corresponds to one feasible ToD that can be flown by some crew member. Each constraint in the SPP corresponds to a particular flight and ensures that each flight is included in exactly one ToD. The elements of the A matrix can then be defined as aij =    1, if the j ToD includes the i flight as operating flight,