Integrated Flight Re-timing and Crew Pairing
Internship KLM
M.C. Krom
S2135469
Master’s Thesis Operations Research
Supervisor University of Groningen: Prof. M.H. van der Vlerk Co-assessor University of Groningen: Prof. K.J. Roodbergen
Supervisors KLM: M.M.H.J. Bolt
Integrated Flight Re-timing and Crew Pairing
M.C. Krom
S2135469
Abstract
Contents
1 Introduction 5 1.1 Literature Review . . . 6 2 Problem Formulation 7 2.1 Definitions . . . 7 2.2 Labour Agreement . . . 8 2.3 The Problem . . . 9 2.4 Modelling . . . 10 3 Solution Approach 12 3.1 Crew Pairing . . . 13 3.2 Initial Solution . . . 15 3.3 Re-timing . . . 16 3.4 Parameter Tuning . . . 18 4 Results 22 4.1 Crew Pairing . . . 23 4.2 Re-timing . . . 245 Sensitivity and What-If Analysis 28 5.1 Cost Proportions . . . 28
5.2 Briefing and Debriefing . . . 30
1
Introduction
The current airline planning, up to approximately 4 weeks before execution, is based on the phases as seen in Figure 1, and in the bullet list below Figure 1. The problems during the different phases are explained briefly. The traditional approach for solving the problems is to consider the phases sequentially and independently. The reason for this is that solving the separate parts can already be difficult. Mathematical optimization of decisions can help during the various stages (Barnhart et al., 2003). However, even when optimizing mathematically, working through the different phases sequentially means restricting the options for the next phase at each transition, and this leads to sub-optimal solutions. In practice, restricting part of the options arises naturally at deadline moments, both with external parties and internally, as decisions have to be made which are fixed from that moment on. Few changes can still be made close to the day-of-operation. Restricting options, or focusing solely on the sub-problem, simplifies the problem. Though it is then easier to solve, potentially good solutions may no longer be obtained.
Flight Planning Crew Planning
Flight Scheduling Fleet Assignment Aircraft Maintenance & Routing
Crew Pairing Crew Rostering
Figure 1: Sequence of planning • Flight Scheduling
Based on demand forecasts and expected profits decisions are made on the destinations to visit and the frequency of the visits. The departure and arrival times are also decided upon.
• Fleet Assignment Model (FAM)
The flights determined during Flight Scheduling are now linked to aircraft types. A good assignment takes into account the seating capacity and the expected revenues of the different types of aircraft. • Aircraft Maintenance & Routing Problem (AMRP)
The flights are linked to specific aircraft. An aircraft can only cover a flight if it can be at the designated departing station at the correct time. The obligatory maintenance of each aircraft is also taken into account at this stage.
• Crew Pairing Problem (CPP)
During each flight a crew, which is not yet a name based crew in this phase, must be present on board. The Crew Pairings are subject to part of the labour agreement.
• Crew Rostering
At this stage, individual crew members are assigned to the pairings. This may be via bidding or preferences. The full labour agreement must be adhered to.
The first three aspects, Flight Scheduling, Fleet Assignment, and Aircraft Maintenance & Routing, can be grouped under Flight Planning. The latter two, Crew Pairing, and Crew Rostering are part of the Crew Planning. Flight Planning solutions are created well in advance, but Crew Planning solutions are created relatively shortly before the day-of-operation. The most recent information, on for example the schedule and the available fleet is then used to create the Crew Planning.
the schedule. That freedom can be exploited by taking into account Crew Planning to make Flight Plan-ning decisions. The decisions made in the Flight PlanPlan-ning phase then anticipate the later planPlan-ning problems. The actual Crew Planning will take place a few weeks to a month before the actual execution. A good approximation of the Crew Planning can help improve the schedule created in the Flight Planning. Crew Rostering can be achieved as long as the pairings are relatively balanced in terms of size and composition of the duties. A feasible Crew Pairing is thus at least an approximation to the final total crew costs. Crew Pairing is considered to be difficult as the pairing possibilities increase rapidly in the number of flights flown. Especially for short- and medium-haul flights, which are of a moderate duration, the Crew Pairing Problem is difficult, as a duty can consist of several flights. Short- and medium-haul flights are usually planned separately from long-haul flights. This thesis will only study short- and medium-haul flights.
The largest Crew Pairing costs are those of pilots, as they are only allowed to fly a limited number of aircraft (based on type), their labour agreements are strict, and their salaries are relatively high. Hence, the focus in this thesis will be on the crew costs of pilots. Solving a Crew Pairing Problem whilst allowing for limited time changes in the schedule could reduce costs. This bridges the gap between Flight Planning and Crew Planning. Results may serve as a tool to fine-tune a final schedule based on the effects on pilot Crew Pairing costs. In this thesis, short- and medium-haul flights are re-timed by taking into account pilot Crew Pairing costs. A mathematical model will be presented and a problem for a weekly schedule will be solved using a heuristic solution approach.
As the number of possible feasible pairings is both large and hard to enumerate, an exact approach is hard and time consuming. Therefore, an Ant Colony Optimization (ACO) heuristic will be used for the re-timing of flights. The pilot Crew Pairing costs are approximated by a Large Neighbourhood Search (LNS) heuristic combined with a Set Partitioning model due to the large number of possible pairings.
1.1
Literature Review
An overview of the models used in Flight Planning and Crew Planning is given by Gopalan and Talluri (1998) and Gopalakrishnan and Johnson (2005) respectively. The five phases of Figure 1 have been the subject of many studies, proposing several modelling and solution approaches.
The Crew Pairing Problem is often modelled as a Set Covering problem (Marchiori and Steenbeek, 2000, and Muter et al., 2013) or Set Partitioning problem (Mingozzi et al., 1999, and Erdoˇgan et al., 2015) as the number of possible pairings can become extremely large.
Many papers are available in which Crew Pairing and some phase as in Figure 1 have been studied si-multaneously. Most natural is to examine the Crew Planning as a whole and thus to combine Crew Pairing and Crew Rostering (Souai and Teghem, 2009). However, also Aircraft Routing and Crew Pairing have been analysed jointly (Cordeau et al., 2001). There are even studies encapsulating Fleet Assignment, Aircraft Routing, and Crew Pairing simultaneously (Cacchiani and Salazar-Gonz´alez, 2013).
in the sense that they only consider a daily problem. Moreover, similar to Weide et al. (2010) they also take into account the full Aircraft Maintenance & Routing problem, which is beyond the scope of this thesis. This thesis aims to solve a flight re-timing and Crew Pairing problem relatively fast via a two-stage heuristic approach in which the method by Erdoˇgan et al. (2015) will be used as a basis to obtain the Crew Pairing solution. They use an LNS heuristic and are able to solve over 25 000 legs in on average 3 hours. This is still computationally large, but adjustments will be made to reduce computation time. These adjustments are required since the Crew Pairing solution will be called upon several times in the proposed re-timing heuristic. Moreover, the number of legs in the weekly short-and medium-haul schedule to be solved will at most around 2 000.
In Section 2, detailed assumptions and a mathematical description of the problem will be provided. In Section 3, the solution approach will be discussed. Next, in Section 4, the results are shown for several differently sized instances. Section 5 will be dedicated to a sensitivity analysis in which the impact of the labour agreement on the results is studied. In Section 6 a case study is presented. Finally, Section 7 concludes and provides ideas for further research.
2
Problem Formulation
First, a number of important concepts for understanding the problem will be defined. Next, a detailed problem description and corresponding mathematical model will be presented.
2.1
Definitions
The terms below are used throughout this thesis. They are expressions frequently used in the airline industry as well as in literature on airline problems.
2.1.1 Definitions regarding Flight Planning Definition Flight leg
A single non-stop flight with a departure and arrival time, and a departure and arrival station. Definition Slot
A time period in which a flight is allowed to arrive or depart. Definition Ground services
The department in charge of handling the aircraft whilst it is parked at an airport. This includes for instance refueling, cleaning, and catering.
2.1.2 Definitions regarding Crew Planning Definition Base station
The base station is the station where the crew is based. Definition Briefing (/Debriefing)
The instructions given to crew at the start (/end) of a working day. Definition Deadhead
Definition Duty
A sequence of flight legs covered by a crew on one day. This includes a briefing and debriefing respectively before the first and after the last flight.
Definition Layover
A complete day at an outstation without scheduled flights, which counts as a working day. Definition Night stop
An overnight stay at an outstation during which crew can rest. Definition Outstation
An outstation is any station which is not the base station. Definition Pairing
A sequence of duties separated by layovers and/or rest at base/outstation.
A generic example of a pairing is given in Figure 22 in the Appendix. The crew planned during a Crew Pairing problem, hence not on name basis, will hereinafter be called a functional crew or simply crew. A crew starts the day with a briefing, is able to fly several flights, and ends the day with a debriefing. Such a day is called a duty. The crew can work several duties in a row, which is called a pairing. A pairing is only feasible if the crew gets enough rest in between the duties. Rest can take place at the base station or at an outstation. In the latter case a hotel is arranged for the crew. A layover, which counts as a working day, occurs if the crew does not fly the day after the rest at an outstation.
A pairing is a collection of duties, where the first duty starts and the last duty ends at the base sta-tion. The solution to a Crew Pairing Problem consists of a set of pairings. The pairings in this set should cover all flights determined in the Flight Planning while adhering to the labour agreements. The aim is to minimize the total costs of these pairings.
2.2
Labour Agreement
Crew Pairings are only allowed when they are feasible with respect to the labour agreement. The following rules are commonly seen in labour agreements.
Rule 2.1 Maximal number of duties in a pairing.
Rule 2.2 Maximal number of hours spent flying per day. Rule 2.3 Maximal number of night flights per pairing.
Rule 2.4 Maximal number of early starts of a duty per pairing.
Rule 2.5 Maximal number of hours spent working based on briefing time and landings.
If the crew works during early hours or has to perform many landings, the number of hours they are allowed to work that day is lower.
2.3
The Problem
Flight Planning is an important phase in the airline business. The schedule created impacts the remainder of the planning as well as the final operation. However, the exact impact is hard to oversee. Hence, many decisions are currently made without sufficient knowledge of their effects. Though experience in the field may help in determining good decisions, the size of many airlines makes the process intractable.
Foremost, decisions on the departure and arrival times can potentially improve the results of subsequent problems. Deadlines regarding the schedule with external parties, such as airports and partner airlines, as well as the schedule reveal for the general public are often well before all further planning problems are solved. Before these deadlines, the departure and arrival times are relatively flexible. In the planning prob-lems which are solved after the deadlines this is no longer the case. For example, the planning of ground services and crew are planned after these deadlines. Yet, both are directly affected by schedule decisions. This thesis suggests allowing limited time changes in the schedule based on the effect they have on pilot Crew Pairing costs. This integrates the Flight Planning phase with the Crew Planning phase by studying the effect of current flight schedule decisions on future crew costs.
In the Crew Pairing problem, a set of pairings is required such that all flights in the schedule are covered. The pairings are subject to constraints by a labour agreement. The objective is to find a set with minimal costs. Crew Pairing can approached as a daily, weekly, or monthly problem. In the daily problem crew is scheduled for a single day, which is a simplification of the weekly planning. In the weekly planning the week-days often contain the same flights each day, which differs from the flights flown in the weekend. A monthly schedule can be generated by repeating the weekly schedule. A continuous weekly schedule is focused on in this thesis. In a continuous week the same flights are assumed to be flown each week. Pairings can then be formed across weeks such that a duty on Monday can follow a duty on Sunday. The week is fully covered if all flights of the week appear in a pairing.
The flights will be re-timed based on their flight code. A flight to or from an outstation which has the same departure and arrival time on several days in a single week is given the same flight code. A re-timed flight is then re-timed on all days the specific flight is flown.
Where the crew is restricted following from the labour agreement, the schedule is also subject to constraints. Major time changes in the schedule can be difficult to accomplish in the late planning stages. Acquiring the required time slots for new departures/arrivals and the capacity which ground services can handle are two examples of bottlenecks for major changes. Even some minor time changes may prove difficult or impossible to perform. Some of the schedule constraints are hard to quantify. The approach in this thesis is to find not only the best solution, but also a set of specific re-timings which have the desired effect on pilot crew costs. For each flight the re-time value, which can be not to re-time as well, proving to be most beneficial for costs is required. Experts on the different bottlenecks can then determine the re-timings which may be worthwhile to incorporate in the schedule.
2.3.1 Assumptions
A single base station is assumed for the airline, which serves as the starting point and the end point of a pairing. Adhering to practice, it is assumed that all flights have the base station either as the origin or the destination. Hence, no flights are flown from an outstation to another outstation. Furthermore, it is as-sumed that the flight schedules are created in discrete time with epochs of 5 minutes. This is an assumption without loss of generality as closer epochs or even a continuous schedule are unrealistic for planning purposes. Finally, to guarantee the Crew Pairing solution is feasible with respect to the Aircraft Maintenance & Routing, the minimal sit at the base station is chosen such that crew has enough time to change aircraft. Short aircraft - crew connections, in which the same aircraft is used for consecutive flights a single crew covers, are then not allowed. However, the resulting Crew Pairing solution will be feasible. At the outstation, the minimal sit is set shorter, as the only option will be to leave with the same aircraft.
2.4
Modelling
In the Crew Pairing problem, flight legs are assigned to a functional crew in the form of pairings. The method usually employed is to generate a large number of feasible pairings, as the complete set is too hard to enumerate due to the size of the set. The Crew Pairing Problem is then often modelled as either a Set Partitioning Problem or a Set Covering Problem. Table 1 and Table 2 provide an overview of the most important variables and parameters used in these models.
Table 1: Overview of variables Variable
xp decision whether or not pairing p is in the solution
yl decision on the number of deadheads for leg l
yω
l decision on the number of deadheads for leg l with schedule ω
zω
l decision whether or not leg l with schedule ω is in the solution
Table 2: Overview of parameters Parameter
cp costs of pairing p
d costs of a deadhead dl maximal deadheads for l
l / L flight leg index / set ω / Ω re-timings index / set p / P feasible pairings index / set
2.4.1 Set Partitioning Problem
In the Set Partitioning Problem, the set of pairings is partitioned such that the pairings in one subset cover all flight legs exactly once as is reflected in (2). This subset is a feasible solution to the Crew Pairing problem. The aim is to find the subset of pairings with the lowest total costs, as seen in (1). In (3) the pairings are such that they are either used or not in the solution. A pairing can include deadheads. alp is then 0, but
costs are incurred for the deadhead through cp.
Minimize X p∈P cpxp (1) subject to X p∈P alpxp= 1 ∀l ∈ L (2) xp∈ {0, 1} ∀p ∈ P (3) Here alp= (
1 if leg l is actively flown in pairing p, 0 otherwise.
2.4.2 Set Covering Problem
The Set Covering model below differs from the Set Partitioning model above mainly in the approach to deadheads. In the Set Partitioning model above, deadheads are not counted as covering a flight and costs for a deadhead are incorporated in cp. In the Set Covering model below, more pairings can cover a single
flight, which is shown in the difference between (2) and (5). Every flight which is covered more than once is thus used for a deadhead. A way of modelling this is to represent the number of deadheads on a single flight l by yl. Deadhead costs, d, are included in (4) and are now no longer part of the costs of the specific
pairing, cp, as it was in (1). Flights which are covered more than once are counted as deadheads in (5) via
yl. The number of deadheads is integer and at least 0 as can be seen from (7). Similar to (3), (6) ensures a
pairing is either used or not.
Minimize X p∈P cpxp+ d X l∈L yl (4) subject to X p∈P alpxp− yl= 1 ∀l ∈ L (5) xp∈ {0, 1} ∀p ∈ P (6) yl∈ N0 ∀l ∈ L (7) Here alp= (
1 if leg l is covered in pairing p, 0 otherwise.
2.4.3 Crew Pairing with re-timing
Based on the idea by Mercier and Soumis (2007), who model a re-timing, routing, and crew scheduling problem using a path formulation, ‘copies’ of flights with different departure and arrival times are added. The set of re-timed legs for flight l, including the original flight, is of size |Ωl|. Hence, |Ωl| different versions
is chosen for each flight by (10) and (11) guarantees only deadheads with this schedule are allowed. dmax is
a parameter which can be set to a sufficiently high number if no actual limit is present. Similar to (4), but summing over the different re-timed versions is (8). No fractional solutions can be made due to (12) - (14).
Minimize X p∈P cpxp+ d X l∈L X ω∈Ωl yωl (8) subject to X ω∈Ωl X p∈P aωlpxp− yωl = 1 ∀l ∈ L (9) zωi l ≤ (1 − z ωj l ) ∀l ∈ L, ∀ωi, ωj ∈ Ωl: ωi6= ωj (10) ylω≤ dmaxzlω ∀l ∈ L, ∀ω ∈ Ωl (11) xp∈ {0, 1} ∀p ∈ P (12) ylω∈ N ∪ {0} ∀l ∈ L, ∀ω ∈ Ωl (13) zlω∈ {0, 1} ∀l ∈ L, ∀ω ∈ Ωl (14) Here aωlp= (
1 if leg l with re-timing ω is covered by pairing p, 0 otherwise.
The decision variables xp represent the possible Crew Pairings which are used or not in the final solution.
In these yes/no decision problems, a model version with continuous variables is impractical, since fractional pairings are meaningless in Crew Planning. Fractional pairings cannot be carried out, and rounding the frac-tional pairings could result in double coverage of a flight or not covering a flight at all. The decision variables for the re-timing chosen, zω
l, is also binary, since a flight can only have one re-timing. The number of
dead-heads is integer and follows from the chosen pairings. The resulting model is a Mixed-Integer Problem (MIP). In the two-stage approach of this thesis, for fixed values of ω ∈ Ωl, the Crew Pairing problem will be solved.
The resulting costs are used as feedback in generating the next fixed setting of the schedule. The fixed setting results in solving a Set Covering as in Section 2.5.2 or, depending on the approach towards deadheads, a Set Partitioning as in Section 2.5.1. The heuristic approach in this thesis solves a Set Partitioning as in (1)-(3). A lower bound on an MIP is given by relaxing the problem resulting in a linear program without integer constraints. However, the set of pairings is still too large to enumerate all possibilities. Hence, no lower bound will be presented.
3
Solution Approach
As schedule constraints may be hard to quantify, not necessarily the best solution is important, but rather specific re-timings which have the desired effect on pilot crew costs. Certain re-timings may be infeasible with respect to schedule constraints such as slots, opening times of stations, and ground services. Furthermore, there may be re-timings which are feasible, but undesirable because of connecting intercontinental flights. The ACO heuristic is able to indicate re-timings which appear in solutions with low costs, hence it will be used for the re-timing of flights. Moreover, the ACO heuristic provides the possibility to include biases for re-timings which are preferred, hence allowing the schedule changes to be steered in a desired direction. In the following subsections, the proposed Crew Pairing heuristic is discussed, several initial solutions for the Crew Pairing heuristic are proposed, the re-timing heuristic is presented, and the parameters for the proposed heuristics are tuned.
3.1
Crew Pairing
A Crew Pairing solution is a set of feasible pairings covering all flights. The feasibility of a pairing is de-termined through the rules mentioned before. The objective in the Crew Pairing problem is to find a set of feasible pairings with minimal costs. The heuristic proposed aims to find a cost efficient Crew Pairing solution in a fast manner. A fast approximation to the Crew Pairing costs can help make decisions in the Flight Planning phases. A fast approximation can help determine the quality of a schedule in terms of the Crew Pairing costs. Moreover, if the method is fast, it can be included in further optimization.
The solution approach is based on Erdoˇgan et al. (2015), who base their solution quality on the costs of deadheads, and different hotel costs dependent on the outstation. They assume fixed salaries for the crew, which are independent of the number of flights flown. Deadhead costs will also play a role in this thesis, as will the costs for a hotel, which will be the same for all outstations. Moreover, though crew salaries may be fixed, minimizing the number of duties will minimize the amount of crew required. Especially in the planning phase this is a desired result. Hence, in this thesis, costs are incurred per duty.
Furthermore, Erdoˇgan et al. (2015) assume the following in terms of labour agreement 1. Pairings start and end at the base station.
2. Max landings in a duty. 3. Min duties in a pairing. 4. Max duties in a pairing.
5. Max duty time based on begin duty. 6. Min time between flights.
7. Min rest time.
8. Total duty time in a pairing.
Of the above, rules 1, 4, 5, 6, and 7 are applied directly in this thesis. 8 will not be incorporated, as balanced pairings are made during the actual Crew Planning and are not of importance in this phase.
Contrary to Erdoˇgan et al. (2015), who have selected flights for potential deadheads beforehand, in this thesis deadheads will only be introduced when there is no feasible pairing with the remaining flights. A single deadhead flight is then combined with the flight that must be covered to create a feasible pairing, and hence a feasible solution.
3.1.1 Method for Crew Pairing heuristic
Algorithm 1, the Crew Pairing heuristic, tries to find a cost efficient solution to the Crew Pairing problem. Below, the heuristic is briefly discussed, for further explanation the Appendix is referred to.
For a set of legs, the heuristic finds a solution, in the form of a set of pairings, which covers all flights. The initial solution (see Section 3.2) is an addition to the solution approach by Erdoˇgan et al. (2015). Using a fast initial solution greatly decreases the computation time.
A percentage of pairings are deleted from the solution randomly. For the legs which are then no longer covered, pairings which cover these flights are created to form a new solution which covers all legs. These pairings are created via a recursive function which explores many possible pairings. The possible duties for these pairings are generated in an enumerative manner whilst taking into account labour constraints. To save on computation time, only consecutive duties, in which at least 1 leg is present, are generated in this thesis, no layovers are thus created in the possible duties.
In the Combine pairings function, larger pairings are created with the already existing pairings. Once the set of legs remaining to be covered is relatively small, only relatively small pairings can be created. It may be pos-sible to append these pairings to an already existing pairing, thereby saving a duty. A visual example of this is given in Figure 23 in the Appendix. When a number of unique pairings have been found, a Set Partitioning model as discussed in Section 2.4.1 will be solved, this has also been implemented by Erdoˇgan et al. (2015), who find it significantly improves the solution. The pairings used are the pairings found in the best solutions. If the resulting solution is better than the best found solution so far, the best found solution so far is up-dated. If the solution is not the best so far, its neighbourhood can still be used to continue the search with a certain probability. Not necessarily choosing the best solution’s neighbourhood for exploration reduces the probability of ending up trapped in a local optimal. The method used to accomplish this is known as Simulated Annealing (SA). SA is another addition to the model by Erdoˇgan et al. (2015). Its theoretical result of converging to the global optimal solution with certainty (Granville et al., 1994) is attractive, though of course the computing time to ensure a global optimal with a significant probability can be extremely high. Further explanations can be found in the Appendix.
Algorithm 1 Crew Pairing heuristic 1: S,..π,..S∗ ...← initial solution 2: for it = 1 to numItLNS do
3: Remove θ% from S and randomly order removed legs 4: for all removed legs do
5: if leg not covered by S then 6: p,..p∗ ...← ∅
7: Function for pairings 8: S ...← S ∪ p∗
9: end if 10: end for
11: Combine pairings 12: π ...← π ∪ S
13: if S covers all legs and cost(S) < cost(S∗) then 14: S∗ ...← S
15: end if
16: if S 6= S∗ and randomP rob > e−(cost(S)−cost(S∗ ))temperature then
17: S ...← S∗
18: end if
19: if number of pairings in π ≥ numPair then 20: S∗,..S,..π ...← Set Partitioning 21: end if
22: end for
23: Set Partitioning with all solutions which have resulted from previous Set Partitionings
S represents the local solution, π a set of unique pairings, S∗ the best found solution, p the local pairing, and p∗ the best found pairing. For the Function for pairings the ‘RCMM algorithm’ as in Erdoˇgan et al. (2015) is referred to.
3.2
Initial Solution
If Algorithm 1 is run without an initial solution, the Pairing heuristic will fill a new solution with pairings until a feasible solution is created. This, however, is very time consuming. A feasible initial solution can be made, after which the heuristic is applied.
Base Outstation
Night stop Night stop
Base Outstation
Night stop Night stop
Figure 2: Example overnight at outstation
between their flights for the night stop connections.
A solution for this problem is to connect the mid-day flights to the overnight flights. The afternoon arriving crew leaves with the early morning flight, and the crew which arrives late at night will depart with the afternoon return flight. The bottom row of Figure 2 shows this solution graphically. The leftmost flight is now paired with the night stop flight retuning from the out station. The second flight is paired with an earlier flight not shown in the figure. The night stop flight arriving at the outstation is paired with a mid-day return the next day. As a continuous week is assumed, such pairs can solve the entire week.
Several initial solutions are proposed in this thesis. The first is to use the Crew Pairing heuristic until a feasible solution is created. This is expected to take a long time, especially for a large number of flights. The second is to partition the flights in 7 parts by flight code and use the Crew Pairing heuristic on the different parts, then merge for a final feasible solution. This is expected to take less time, since the possibilities are fewer. It is also expected to give a better result because the similar flights have similar flight codes. The third initial solution pairs up two connecting flights, in a 1- or 2-duty pairing. These pairs are found based on the flight code, the time of the flight, and the destination. It tries to make as many combinations as possible as in the bottom row of Figure 2. Since this method deterministically pairs flights it does not require comparisons in pairings, but combines efficient pairings nonetheless, it is expected to perform both well and fast. Table 24 in the Appendix shows the performance of the different initial solutions proposed.
0 20 40 60 80 100 2500 4000 Iterations A v er age cost (€)
First Second Third
Figure 3: Average cost after a number of iterations starting with different initial solution
After a number of iterations, for all initial methods, costs will decrease as can be seen in Figure 3. The final proposed initial solution, which is represented by the lowest dashed line, has a good solution quality and is relatively fast. Hence, it will be used as an initial solution for the Crew Pairing heuristic.
3.3
Re-timing
The objective is to find small time changes in the flight schedule, re-timings, which have a desired effect on the crew costs. Though it may be possible to re-time all flights individually, in this thesis flights to the same destination with the same arrival and departure times are re-timed simultaneously. The proposed heuristic, however, is easily adapted to individual flight re-timing.
Ant Colony Optimization (ACO) is the method used to solve the problem, as it can indicate the re-timings which appear in solutions with low costs. ACO is based on the natural behaviour of ants. Ants distribute themselves over an area looking for food. On their way back to the nest they leave a pheromone trail. When new ants go searching for food, they partially base their direction on the pheromone trails left by previous ants. The shortest paths leading to food are likelier to be chosen as the pheromone trails on these paths are stronger.
which certain re-timings are likelier due to higher pheromone levels. These schedules are assumed to be fixed and the Crew Pairing problem is solved using the heuristic in Section 3.1. The resulting costs are used as feedback on the schedule quality.
3.3.1 Method for Re-timing heuristic
For a number of iterations, represented by the number of ‘ants’, several schedules are created as each ant receives its own re-timing. For the resulting schedule of each ant, a Crew Pairing solution is found. This process is repeated for a number of iterations as well. The updated pheromone levels ensure that in the new set of ants, the ‘good’ re-timings of the previous sets of ants are more likely to be chosen, and the ‘bad’ re-timings are less likely to be chosen.
Several systems based on ants have been formulated in existing literature. Two of the better performing sys-tems (St¨utzle and Hoos, 2000) are known as the Max Min Ant System (MMAS) and the Ant Colony System (ACS). In a hybrid version of the two, called the Min Max Ant Colony System (MMACS), a combination of pheromone level bounds, local updates, and global updates is used.
The structure of the hybrid version can be found in Algorithm 2. The general idea is explained in the follow-ing text. Further explanation of the functions used in the Ant Colony heuristic is provided in the Appendix. A flight re-timing solution is either based on the ‘value function’, or on probabilities proportional to the function values of the ‘value function’. This ‘value function’ depends on the pheromone levels. Hence, re-timings are chosen based on feedback received on previous re-timing solutions through pheromone levels. If a re-timing has been solved before, the Crew Pairing will not be solved again, but the previously found Crew Pairing solution will be used. If a re-timing is introduced which causes the schedule to be unsolvable, the Crew Pairing heuristic returns sufficiently high costs.
The local pheromone update is applied after each re-timing and is independent of the solution found for the re-timing. Pheromone levels are brought closer to their original starting level such that more diverse schedules are created rather than only schedules near the current best solution.
Algorithm 2 General structure hybrid heuristic 1: for it = 1 to numItAnt do
2: for ant = 1 to numAnts do
3: L .← Create flight re-timing solution 4: CP P ...← CPP(L)
5: if Cost(L, CP P ) < Cost(Local best solution so far) then 6: Save this solution as the local best solution
7: end if
8: Local pheromone update for ant solution 9: end for
10: if Cost(Local best solution) < Cost(Global best solution found so far) then 11: Save this solution as the global best solution
12: end if
13: Global pheromone update for the best local solution 14: end for
3.4
Parameter Tuning
For 10 data sets, of various sizes and seasons, the parameters which are free to change are tuned to find the general best settings. The different data sets are tuned to avoid overfitting a single data set. Moreover, a generally good performing setting is used for the parameters for the same reason.
3.4.1 Crew Pairing
The parameters in Table 3 are free to change for the Crew Pairing.
Table 3: Parameter tuning Crew Pairing Parameter
θ percentage of pairings to remove
numItLN S total number of iterations for the Large Neighbourhood Search numP air number of pairings needed before a Set Partitioning problem is solved initT emp initial temperature Simulated Annealing
∆temp reduction in temperature Simulated Annealing
The costs as in Table 4 are used as a basic setting. The proportions are realistic, where the costs are normalized such that the costs of a hotel night are equal to 1. Furthermore, the briefing time, the debriefing time, and the minimal sit between flights are shown in Table 5. The min sit of 35 to 45 minutes at an outstation is dependent on the type of aircraft.
Table 4: Basic cost setting Type Value (e) Crew salary 20 Hotel night 1
Deadhead 5
Table 5: Basic time setting
Type Value (minutes)
Briefing time 60
Debriefing time 30 Min sit base station 60 Min sit outstation 35 / 45
decrease rapidly, costs hardly decrease further after approximately 10 - 20 iterations. For all other instances, a similar number is found for convergence, where all instance have converged after approximately 25 iterations. No significant relationship has been found between the number of flights and the number of iterations required for convergence. 0 20 40 60 80 100 3700 3850 4000 Iterations Cost (€)
Average Min / Max
Figure 4: Iterations before convergence
Ceteris paribus, an increase in the number of legs exponentially increases the time required until conver-gence, which is taken to be 25 iterations. Figure 5 shows the time until convergence for several instance sizes. The time required for the heuristic to complete increases approximately linearly in the number of iterations performed as can be seen in Figure 6.
200 400 600 800 1000 4 8 12 16 Number of legs Time (s)
Figure 5: Time until convergence
0 20 40 60 80 100 0 5 10 15 Iterations Time (s)
Figure 6: Time vs iterations
A parameter which does show a relation with the number of flights is the number of pairings before a Set Partitioning is solved. The number of pairings in a solution generally increases when the number of flights increases.
A lower value for numP air corresponds to a lower threshold before a Set Partitioning is solved. More Set Partition problems are thus solved if this value is lower. This will result in better solutions at the cost of higher computation times as can be seen in Figure 7, where the x-axis shows the multiple of the number of legs, nl, in the problem for numP air. However, the number of pairings should not be too low, otherwise
1 2 3 4 5 3700 3850 numPair Cost (€) (a) Cost 1 2 3 4 5 0 4 8 numPair Time (s) (b) Time
Figure 7: Effect of numP air, the number of pairings before solving a Set Partitioning
The number of legs has been used to determine the threshold, numP air. The exact number of legs in a data set are used as the threshold for the set partition.
During extensive parameter tuning each parameter can take on several different values, which can be found in the Appendix in Table 25. These settings have been tested on multiple instances from which can be concluded that the, on average, best setting for the Crew Pairing heuristic is as in Table 6. ∆temp is relatively high, hence the temperature decreases quickly. With a lower temperature, the probability is higher that the best solution is chosen to remove pairings from. Hence, the relatively high value of ∆temp causes a continued search in the neighbourhood of the best found solution more frequently.
Table 6: Crew Pairing setting
θ numItLN S numP air initT emp ∆temp
0.15 25 1 × nl 10 1.5
3.4.2 Re-timing
For the re-timing heuristic, the parameters in Table 7 are free to change.
Table 7: Parameter tuning re-timing Parameter
α weight of pheromone levels β weight of impact of υj
η(υj) impact of υj
li flight leg i, l ∈ L
numAnts number of ants per iteration
numItAnt total number of iterations for the ant colony φ decay coefficient
q0 probability threshold for arbitrary / random re-timing
ρ evaporation rate
τ0(li, υj) original pheromone levels
τmin lower bound pheromone level
τmax upper bound pheromone level
The total number of schedules created, and hence the total number of Crew Pairing problems solved, is equal to the number of iterations multiplied by the number of ants in each iteration. Different settings with a high, balanced, or low number of iterations and/or ants are possible. Figure 8 shows the costs for different settings of the heuristic. The x-axis shows the total number of times the Crew Pairing heuristic is solved per setting. From Figure 8 it can be seen that solving a total of 50 Crew Pairing problems, shows a convergence in the re-timing heuristic. The other instances show similar convergence. There is no clear relation between the number of legs and the total number of Crew Pairing problems before convergence.
0 20 40 60 80
3550
3700
Number of Crew Pairing solutions
Cost (€)
Figure 8: Total crew pairings solved
η(υj) is the function parameter by which a preference for a certain re-timing can be specified. A good method
to determine potential functions for this parameter is to perform a study under experts. Another possibility is to base it on phenomenons found through analysing historical data. This is out of the scope of this thesis. This thesis will not take into account a preference for certain re-timings. Hence, η(υj) = 1 ∀i, consequently
the value of β does not matter. Furthermore, τ0(li, υj) = τ0(lk, υm) ∀i, j, k, m for the same reason. The
starting value for the pheromone levels will thus be equal both across the re-timings and across the flights. In further research, or further development of the proposed solution approach, varying these values may be a worthwhile addition.
Next to the preferences for certain re-timings, the probabilities for re-timing (see Equation 18 in the Ap-pendix) are also based on the pheromone levels. α can be used to tune the importance of the differences in pheromone levels for the re-timings of a single flight code. A value of 1 corresponds to a linear dependence. A value lower than 1 decreases the importance of the difference in pheromone level whereas a value higher than 1 increases the importance of the difference.
Table 8 shows the settings required for a wide and a deep search for the re-timing heuristic.
Table 8: Re-timing setting parameter effects Parameter Wide search Deep search
numItAnts Low High
numAnts High Low
φ High Low
ρ Low High
τ0(li, υj), τmin, τmax Narrow Wide
for a deepening search. The decay rate and the evaporation rate, respectively φ and ρ, also correspond to deepening and widening the search. A high value for φ will keep the pheromone levels close to their original starting value. A low value for φ will allow for deep search before gradually returning to the original values. A high value for ρ will steer the search in the direction of the best found solution as the other pheromone levels are decreased drastically. A low level for ρ will keep the pheromones closer to their current level, making the relative difference with the best found pheromone level small.
Extensive parameter tuning shows the, on average, best setting for the re-timing heuristic to be as in Table 9. A relatively balanced setting between a deep search and a wide search gives the best results. Though some steering is recommended, good solutions are not necessarily in a close neighbourhood of other good solutions, hence the balanced setting works well.
Table 9: Re-timing setting
α β η(υj) numItAnts numAnts φ q0 ρ τ0(li, υj) τmin τmax
1 - 1 10 5 0.5 0.3 0.5 5 2.5 7.5
Rather than using previously found solutions for new re-timings, all Crew Pairing solutions start with the fast initial solution proposed. A previously found solution must be checked for feasibility. Moreover, since the set of possible solutions is large, a previously found solution may not be a good initial solution for the new re-timing. The initial solution proposed is similar for all re-timings, yet only creates feasible solutions.
4
Results
The calculations are run with an Intel(R) Core(TM) i5 M 520 @ 2.40 GHz processor and 4GB RAM. The heuristic is implemented using C++ and the coin-OR (Lougee-Heimer, 2003) cbc solver is used to solve the MIP. A time restriction of 20 seconds, based on testing, is put on the cbc solver as a compromise between the conflicting objectives of solution quality and computation time.
The data sets used in this section are different from the data sets used in the previous section to show results of the approach on unexplored data. The data sets in these sections have been created by splitting, combining, and adjusting actual historic planned flight schedules.
To compare solutions in terms of quality, key performance indicators (KPIs) must be defined. The dif-ferent kind of costs are shown below and have also been used in the previous sections.
• Crew salary • Hotel night costs • Deadhead costs These costs can be calculated when the following KPIs are available.
Table 10: Labour agreement
Rule Definition Value
2.1 Maximal number of duties in a pairing 5
2.2 Maximal number of hours spent flying per day 8
2.3 Maximal number of night flights per pairing 1
2.4 Maximal number of early briefings per pairing 3
2.5 Maximal number of hours spent working based on briefing time and landings Table 35 (Appendix) 2.6 Minimal number of hours hours rest after a duty within a pairing 10
2.7 Maximal hours sit at base station 3
For the setting as in Table 4, Table 5, and Table 10 the problem is solved for a number of flight schedules of various sizes, different from those in Section 3.4.
4.1
Crew Pairing
The costs and the computation times are shown in Table 11. These are the results of 10 runs of Algorithm 1. The standard deviation is on average 2% of the mean costs. This confirms that the heuristic performs relatively stable in terms of costs. The variation in the time spend on solving the problem is larger: the standard deviation is on average 18%. Such additional information is available in Table 28 in the Appendix.
Table 11: Average Crew Pairing costs and time results for winter (W) and summer (S) schedules # flights CPP cost(e) Time(s)
W S W S 200 2414 2990 2.82 3.04 400 4676 3970 4.69 4.95 600 5807 5290 7.98 7.49 800 7842 8183 11.01 11.90 1000 11442 10517 18.26 18.77
The average costs per flight are on average approximatelye10. Since duty costs were set at e20, on average, at least 2 flights are covered in a duty. Table 12 shows the split of costs.
Table 12: Average Crew Pairing costs split over cost components for winter (W) and summer (S) schedules
# flights CPP cost(e)
W S
Duty Hotel Deadhead Duty Hotel Deadhead
200 2302 42 70 2844 76 70
400 4540 126 10 3872 98 0
600 5566 141 100 5134 113 43
800 7512 238 92 7872 204 107
1000 10678 323 441 9948 216 353
W200 S200 W400 S400 W600 S600 W800 S800 W1000 S1000 Deadhead Hotel Duty Instance P ercent 80 90 100
Figure 9: Breakdown of the costs
Besides the performance indicators on the costs, it is interesting to see the distribution of the number of duties in a pairing, and the number of legs in a duty. Figure 10 shows the average of the former, and Figure 11 shows the average of the latter for a single instance. The other instances show similar distributions.
1 2 3 4 5 Number of duties Frequency 0 10 20 30 40 50 60
Figure 10: Histogram number of duties
1 2 3 4 5 Number of legs Frequency 0 20 40 60 80 100 120
Figure 11: Histogram number of legs in a duty Most of the duties consist of 3 flight legs as can be seen in Figure 11. This means that the crew either starts or ends at an outstation. Furthermore, as no flights are from an outstation to another outstation, the crew also either starts or ends at the base station. Since pairings are forcibly ended when they end at the base station, the three leg duties are either the start or the end duty of a pairing.
Although pairings of 5 duties are allowed, Figure 10 shows that few 5-duty pairings are created. This is because the pairings are forcibly ended when a duty ends at the base station. In a further Crew Rostering problem, smaller pairings can be combined to create larger pairings. However, for the Crew Pairing costs, the number of pairings is irrelevant, as costs are only incurred for the number of duties. Increasing the number of duties allowed in a pairing is unlikely to generate better results. Moreover, labour rules such as the number of early flights and the number of night flights which are allowed in a single pairing may force the pairings to be small. Relaxing these constraints is more likely to be beneficial than the rule on the maximal number of duties in a pairing.
4.2
Re-timing
The results of the re-timing heuristic are shown in Table 13. The Crew Pairing Problem (CPP) costs are the costs of the best re-timing found. These are the results of 10 runs of Algorithm 2.
Table 13: Re-timing heuristic results for winter (W) and summer (S) schedules # flights CPP cost(e) Time(s) Difference from original schedule
W S W S ...W S... 200 2060 2619 146 273 ...-14.6% -12.4%... 400 4513 3778 259 260 ...-3.5% -4.8%... 600 5374 4907 421 193 ... -6.7% -6.9%... 800 7618 7785 449 276 ... -2.5% -7.1%... 1000 11077 10195 935 950 ... -3.2% -3.1%...
Additional information, such as minima, maxima and standard deviations, concerning the results in Table 13 can be found in Table 30 in the Appendix.
Since the re-timing heuristic repeatedly calls for the Crew Pairing solution, the computation time is larger. It can also be larger than simply repeating the Crew Pairing heuristic a number of times due to the over-head of the re-timing heuristic. However, if a schedule is created which has been solved before or for which no feasible Crew Pairing solution exists, the heuristic may be faster. The deviation in the time needed to complete the re-timing heuristic is smaller than the time needed to complete the Crew Pairing heuristic. The variance of the mean time needed to solve the problem decreases as several Crew Pairing problems are solved in the re-timing heuristic.
It can be seen that the costs are lower for the re-timing heuristic than for the Crew Pairing heuristic of the original schedule. The percentual change per instance differs, as some schedules are already good in terms of crew costs whilst for others major cost savings can be achieved potentially.
Large differences are found especially for the instances with 200 legs. Many of the flights arriving at the base station differ less than an hour from following departures from the base station. Since the number of flights is low, there are hardly any alternative flights to continue the duty. Hence, many different duties, potentially with deadheads, will be required. If the flights are re-timed such that more connections are possible, the resulting Crew Pairing solution costs are significantly lower than those of the original schedule.
Table 14: Average Crew Pairing costs split over cost components for winter (W) and summer (S) schedules ACO CPP cost(e)
# flights W S
Duty Hotel Deadhead Duty Hotel Deadhead
200 2018 42 0 2536 73 10
400 4368 126 19 3680 98 0
600 5158 142 74 4748 114 45
800 7334 238 46 7498 200 87
1000 10436 324 317 9714 213 268
any of the instances, the largest instances still show a relatively high percentage of costs due to deadheads. Due to the size of the instance, more possibilities exist to form very efficient pairings. The remaining flights, however, may need to be solved using deadheads. Figure 12 shows the average percentual change between the average solution for the original schedule and the average solution for the re-timed schedule for the different cost components.
Duty Hotel Deadhead
P
ercentage change
−30
−15
0
Figure 12: Histogram percentage change of duties, hotel nights, and deadheads
The schedule is mainly re-timed such that deadheads can be avoided as well as such that fewer duties are required as can be seen from Figure 12. Since the largest fraction of all costs were incurred due to the duties, especially this decrease contributes to lower costs.
Analysis of the resulting schedules shows that flights which are to and from a certain outstation without a night stop in between are often re-timed such that they are closer to one another. Recall the example depicted in Figure 2, in which night stop flights and day flights are combined to form efficient pairings. The day flights are re-timed such that the resulting time between the flights is very close to the minimal sit at an outstation. If the day flights are not flown by the same crew, the flights are often re-timed so close together that a single aircraft could no longer fly both flights. Ground services will not be able to handle the arriving and departing flights in such a short time frame. The same crew does not fly both flights, for example, if the day flight is connected to a night stop flight as in the bottom row of Figure 2.
Setting the sit time at an outstation as low as possible is efficient for the aircraft and crew utilization. More flights can be assigned to a single aircraft and a crew can fit more flights in their duty, resulting in lower costs. Re-timing the flights to and from an outstation for the night stop can also contribute to reducing the crew costs. Since crew requires a minimal amount of rest at the outstations, a late incoming flight at the outstation could mean that crew is unable to fly a return flight the following day. In that case a layover may be required which means incurring a useless duty. Moreover, night stop flights without mid-day flights should be avoided as these are always part of layovers and even re-timing does not help in creating more efficient pairings covering these flights.
4.2.1 Distribution of duties and legs
1 2 3 4 5 Original Re−timed Number of duties Frequency 0 10 20 30 40 50 60
Figure 13: Histogram number of duties
1 2 3 4 5 Original Re−timed Number of legs Frequency 0 20 40 60 80 100 120
Figure 14: Histogram number of legs in a duty Figure 14 shows the difference in the average distribution of the number of legs in a duty between the original and the re-timed schedule. In the distribution of the number of legs, compared to the original schedule, the 1-, 2-, and 5-leg duties show a decrease, whereas the 3-, and 4-leg duties show an increase. The average number of legs in a duty has increased, which is the reason why fewer duties are required and thus costs for duties are lower. The flights are re-timed such that the longer duties, which are duties with more flights, are possible. This results in fewer 5-leg duties as well, but the savings from replacing 1-, and 2-leg duties by 3-, and 4-leg duties make up for the slight loss.
4.2.2 Greedy approach
To assess the solution quality of the proposed heuristic, a greedy approach is introduced. The proposed heuristic is compared to the greedy approach in terms of costs and computation time. The approach created for comparison consists of two steps: randomly re-timing a schedule a predetermined number of times, and solving the crew problem using the proposed Crew Pairing heuristic. The solution with the lowest costs is the final solution of this greedy approach.
In Figure 15 the proposed heuristic is depicted by the solid dots. The greedy approach results in the open dots. Note that these are 10 independent runs for each of the proposed solution approaches. The middle solid line represents the average costs found for the Crew Pairing problem with the original schedule. The dashed lines represent the highest and lowest costs for the original schedule.
5000
6000
Re−timing solutions
Cost (€)
Original Min / Max Proposed Greedy
Observe, that the random re-timing does not even find schedules with costs close to the original schedule. However, with more re-timings the probability that a good schedule is found increases. Though this will mean more solutions for the Crew Pairing problem and thus a higher computation time. Table 15 shows that the iterations, and hence the time, needed until the average solutions found by the greedy algorithm are approximately equal to the proposed heuristic is over 20 times larger than the proposed heuristic using Ant Colony Optimization.
Table 15: Cost proportions
Average costs (e) Time(s) Proposed heuristic 5374 420
Greedy 50 × 6018 495
Greedy 100 × 5855 924
Greedy 500 × 5482 5323
Greedy 1000 × 5436 9851
Re-timing the flights provides added value as the proposed heuristic finds schedules with costs below the costs of the original schedule consistently as is seen in Figure 15. Hence, the heuristic is able to re-time flights such that the quality of the schedule improves in terms of crew costs.
Because of the many possibilities for the Crew Pairing problem, the proposed heuristic does not always give the same re-timings. Many re-timings are also possible, of which several are likely to be favourable for the crew costs. Due to re-timing a flight forward, it can be connected to later flights to form a duty. Re-timing forward can also generate enough rest in between flights such that the flight can be connected to earlier flights to form a duty. For a backward re-timing, the same may hold with other sets of flights.
Since the labour rules imposed are still adhered to, yet cost savings can be made, taking into account costs of the subsequent Crew Pairing problem results in a cost efficient scheduling decision. The quality of the schedule improves in terms of crew costs when the flights are re-timed.
The Crew Pairing heuristic improves upon the initial solution used by approximately 30 - 40%. A feasible crew solution, in which costs are minimized, is found relatively quickly. A greedy approach, making use of the proposed Crew Pairing heuristic, is greatly outperformed by the re-timing heuristic. Both in terms of computation time and solution quality, the re-timing heuristic performs better than the greedy approach. In the next section, labour rules will be relaxed and tightened and their effects on the results of the Crew Pairing heuristic as well as the re-timing heuristic are studied. Moreover, aspects such as briefing and debriefing time, which have been assumed to be fixed, are now varied and the effects are studied.
5
Sensitivity and What-If Analysis
5.1
Cost Proportions
Table 16: Cost proportions
Duties Hotel nights Deadheads
Original 278 141 20
Expensive hotel 278 140 21
Expensive duty 278 141 20
Expensive deadhead 277 141 20
Costless deadhead 278 141 20
As expected, the number of duties, hotel nights, and deadheads are not very different from the original results when part of the costs are high. Average results of a single instance can found seen in Table 16. Similar results were shown over all instances. The distribution of the duties over the pairings, and the legs over the duties also do not show significant changes. The results are thus relatively insensitive to the proportions of the costs. Note that the results do depend on costs. If all costs are excluded, the problem is reduced to a feasibility problem, as there is no incentive to minimize the number of duties, hotel nights, or deadheads. The initial solution would already be an optimal solution in this case, as the initial solution is feasible and the relatively high costs for this solution no longer matter.
Deadheads are already relatively expensive in the original cost setting. Moreover, a deadhead does not cover a flight, but does count as work for a duty. A deadhead is thus inefficient on its own, regardless of the costs. Hence, making a deadhead costless hardly changes the solution either. Since deadheads do count as work, including them, even costless, indirectly generates costs as more legs are included in the duties and costs are incurred for the number of duties.
The number of hotel nights does not change either. Since a number of aircraft stay at an outstation overnight, a crew must also stay at the outstation overnight, as there is no return flight. Since the time between the evening flight to the outstation and the morning departure from the outstation is too short for the same crew to fly both flights, a second crew must have an overnight stay at an outstation for each aircraft with an overnight stay. These hotel nights cannot be avoided.
5.2
Briefing and Debriefing
An increase in the briefing time, compared to the original 60 minutes, limits the possibilities of feasible duties, which results in higher costs as well as a change in the number of legs in a duty. Apart from requiring more time to be briefed, hence leaving less time for flying, also the start of a duty is earlier. This is because even when flights are re-timed the briefing time is increased such that the crew must start earlier than with the original briefing time. Hence, the total amount of hours a crew is allowed to work, based on Table 35, may decrease as well, based on the start of the duty. For the mentioned reasons, the average number of legs flown in a duty decreases. Similarly, the average number of legs in a duty increases when the briefing time decreases. 1 2 3 4 5 30 min 60 min 90 min Number of legs Frequency 0 20 40 60 80 100 120
(a) briefing time
1 2 3 4 5 0 min 30 min 60 min Number of legs Frequency 0 20 40 60 80 100 120 (b) Debriefing time Figure 16: Histogram number of legs per duty
From Figure 16a it is clear to see that when the briefing time is long, an increase is seen, compared to original, in 1-, 2-, and 3-leg duties. The 5-leg, and 4-leg duties face a decrease. These type of duties are no longer feasible with the longer briefing time. When the briefing time is short, an increase is seen, compared to original, in 1-, 4-, and 5-leg duties. The decrease is now present in the 2-, and 3-leg duties. More flights can fit in a duty as the briefing time becomes shorter, creating duties with more flights and requiring fewer duties in total. Especially the 3-leg duties decrease, as most duties consisted of 3 legs as can be seen from Figure 11 on page 24. Many of these 3-leg duties can now include a 4th leg. Since many 4-leg duties can include a 5th leg, the increase in 4-leg duties is limited compared to the original. An increase in the efficient
4-, and 5-leg duties also induces an increase in the 1-leg duties as flights remain to be covered. For the debriefing time, originally 30 minutes, similar observations are made from Figure 16b. Only the increase is smaller for 4-, and 5-leg duties with a shorter debriefing time that it is for a shorter briefing time. The debriefing time only influences the end of a duty, whereas the briefing period influences the start of a duty, which in turn determines the maximal hours spent working.
Figure 17 shows the percentual changes in costs for longer briefing and longer debriefing times. Nearly all 5-leg duties are no longer feasible when either the briefing of the debriefing time is longer. Approximately 40% fewer 4-leg duties remain with a longer briefing or debriefing time. However, the 4-leg duties face an increase due to the 5-leg duties no longer being feasible, hence more than 40% of the 4-leg duties are likely to become infeasible. The legs are now included in more 1-, 2-, and 3-leg duties. The percentual increase in these duties is small as most duties already consisted of 1, 2, or 3 legs.
and hence duties cover fewer legs. This effect is not possible for the increase in the debriefing time. For a decrease, the same effect is seen, but reversed. More longer duties are possible with a decrease in the briefing time than with a decrease in the debriefing time.
1 2 3 4 5 Brief Debrief Number of legs P ercentage change −100 −80 −60 −40 −20 0 20 40
Figure 17: Histogram percentage change from original schedule
5.3
Labour Agreement
In this section, the different values used in the labour agreement are varied and their effects are studied. Several rules with respect to the labour agreement will be relaxed and tightened to see the effects on the results. The results for the Crew Pairing heuristic are shown in Table 17. The results of the re-timing heuristic, in terms of costs and time, are shown in Table 33 in the Appendix. The observations for both are discussed below.
Table 17: Crew Pairing results for changes in the labour agreement
Cost(e) Time(s) Cost change(%)
Original 5807 7.98
-Max hours flying Relaxed 10 5799 8.08 - 0.1%
Original 8 Tightened 6 5815 6.90 + 0.1%
Max night flight Relaxed 3 5771 7.80 + 0.6%
Original 1 Tightened - - -
-Max early briefing Relaxed 5 5806 7.71 0%
Original 3 Tightened 1 6238 7.19 + 7.4%
Max hours work Relaxed + 1 5493 5.78 - 5.4%
Original Table 35 Tightened - 1 5930 6.94 + 2.1%
Min hours rest Relaxed 9 5879 7.79 + 1.2%
Original 11 Tightened 13 7149 8.03 + 23.1%
Max hours break base Relaxed 5 5278 7.61 - 9.1%
Original 3 Tightened 1 9240 12.16 + 59.1%
Maximal number of hours spent flying
Increasing the amount of hours per day that a crew can fly increases the possibilities, however, the maximal hours a crew can work is still restricted to the number of hours they are allowed to work based on their briefing time. An increase in the hours does not provide for much freedom if the hours they can work in total is low. Decreasing the amount of hours per day that a crew can fly could restrict the solutions and hence increase the total costs. As this increase is very small, it can be concluded that the constraint is not binding. The crew flies less than the original 8 hours, and mostly less than 6 hours per day as well.
Maximal number of night flights
An increase in the number of night flights allowed creates more possible solutions. However, only a limited number of flights qualify as a night flight. Hence, the effects of a relaxation are relatively small. Decreasing the number of night flights is not possible as it was originally 1. Decreasing it to 0 would create an infeasible problem as there are scheduled flights which must be flown at the times which make it a night flight. Maximal number of early briefings
Similar to the night flights, an early briefing only occurs with a limited number of flights, although this number may be larger than the number of night flights. Though more longer pairings are possible, no sig-nificant change is observed. The number of pairings required, unlike the number of duties required, does not bring about higher costs. In the Crew Rostering the size of pairings is more important, as days with-out work are required following a pairing. Relaxing the maximal number of early briefings may thus show larger improvements in the Crew Rostering problem. Restricting the maximal number of early briefings has a larger, negative effect. Multiple efficient pairings which are found under the original labour rules have become infeasible due to the tightening of the constraint.
Maximal number of hours spent working
Increasing the number of hours that a crew can work by 60 minutes increases the possibilities, especially for crew starting early, as the amount of work allowed with an early start is relatively low. Decreasing the working hours that a crew can fly by 60 minutes decreases the possibilities, and especially early flights are now harder to place in an efficient pairing. An increase in working hours can provide freedom up to the point where the maximal amount of hours that a pilot can fly per day, or any other binding constraint, is reached.
Minimal number of hours of rest between duties
Increasing the minimal rest, implies more time is required before a crew can start working again after ending a duty. Even with the possibility of re-timing flights, many combinations will not be allowed as too little rest will be between the flights. Layovers solve the problem, but are expensive and inefficient. Decreasing the minimal rest allows for more freedom, the Crew Pairing heuristic can create better pairings as there are more possibilities. However, the better created pairings leave flights to be covered which result in more inefficient pairings, leading to slightly larger costs for crew with the proposed heuristic.
Maximal number of hours between flights at the base station
Setting the minimal break at the base station to 20 minutes, such that all short aircraft - crew connections can be made leads to the results as in Table 18. Table 34 shows the results for the re-timing heuristic. The impact of relaxing the constraint can now be discarded. Even though the effects of tightening the constraint are approximately cut in half, they are still relevant.
Table 18: Crew Pairing results for changes in the labour agreement with minimal break at base 20 minutes. Cost(e) Time(s) Cost change(%)
Original 5076 6.08
-Max hours break Relaxed 5 5023 6.35 - 1.0%
Original 3 Tightened 1 6651 8.05 + 31.0%
Combination
A natural combination of constraints to consider are those of the maximal number of hours spent working and the maximal number of hours spent flying. Relaxing both does not further increase the savings made. As was seen from small effects of the maximal number of hours spent flying, this constraint is non-binding. Also in combination with the relaxed maximal number of hours spent working, further relaxation of the maximal number of hours spent flying has little effect. When re-timing, the effects of a combined relaxation are larger. Re-timing may allow a duty to start later, such that the bracket in which the duty starts is later. Then the maximal number of working hours increases, adding a relaxed maximal number of hours spent working doubles the effect. Long duties can then be made, which may use the relaxed maximal number of hours spent flying.
Similar to results found before, the re-timing heuristic shows a tendency to minimize the time between consecutive flights at an outstation. For some of these re-timings, the resulting schedule is infeasible with respect to schedule constraints such as ground services. Furthermore the negative effects of restrictions are reduced, as the re-timings can be steered in a favourable direction in the restricted case as well. Especially the maximal number of early briefings and night flights benefit from the possibility of re-timing. The tight-ened break at the base station and the tighttight-ened minimal rest, however, are both still very high. This is due to the small number of pairings which are feasible with these constraints.
The overall best results are seen in relaxing the maximal number of hours spent working based on the briefing time and relaxing the maximal break at the base station. The worst results are found by tightening the constraints above. The results from the maximal break at the base station are mainly important due to the long minimal time needed at the base station to ensure feasibility in the Maintenance & Routing problem. The short aircraft - crew connections cannot be made and thus few possibilities follow flights arriving at the base station. A relaxation allows for more feasible connecting flights at the base station, whereas a tightening makes most of the original possibilities infeasible as well.
