Birbil, Ş. İlker

Bülbül, Kerem

Muter, İbrahim

Taş, Duygu

Topçu, Y. İlker

Tüzün, Dilek

Yenigün, Hüsnü

Çoban, Elvin

Şahin, Güvenç

English

Sabanci University Research Database

Column Generation Approaches to a RobustAirline Crew Pairing Model For ManagingExtra Flights?Elvin C¸oban1, I˙brahim Muter1, Duygu Tas¸1, S¸. I˙lker Birbil1, KeremBu¨lbu¨l1, Gu¨venc¸ S¸ahin1, Y. I˙lker Topc¸u2, Dilek Tu¨zu¨n3, and Hu¨snu¨Yenigu¨n11 Sabancı University, Orhanlı-Tuzla, 34956 Istanbul, Turkey(elvinc, imuter, duygutas)@su.sabanciuniv.edu,(sibirbil, bulbul, guvencs, yenigun)@sabanciuniv.edu2 Istanbul Technical University, Mac¸ka-Bes¸iktas¸, 34367 Istanbul, Turkeyilker.topcu@itu.edu.tr3 Yeditepe University, Kayıs¸dag˘ı-Kadıko¨y, 34755 Istanbul, Turkeydtuzun@yeditepe.edu.tr1 IntroductionThe airline crew pairing problem (CPP) is one of the classical problemsin airline operations research due to its crucial impact on the coststructure of an airline. Moreover, the complex crew regulations andthe large scale of the resulting mathematical programming models haverendered it an academically interesting problem over decades. The CPPis a tactical problem, typically solved over a monthly planning horizon,with the objective of creating a set of crew pairings so that every flightin the schedule is covered, where a crew pairing refers to a sequence offlights operated by a single crew starting and ending at the same crewbase.This paper discusses how an airline may hedge against a certain typeof operational disruption by incorporating robustness into the pairingsgenerated at the planning level. In particular, we address how a setof extra flights may be added into the flight schedule at the time ofoperation by modifying the pairings at hand and without delaying orcanceling the existing flights in the schedule. We assume that the set ofpotential extra flights and their associated departure time windows are? This research has been supported by The Scientific and Technological ResearchCouncil of Turkey under grant 106M472.2 C¸oban et al.known at the planning stage. We note that this study was partially mo-tivated during our interactions with the smaller local airlines in Turkeywhich sometimes have to add extra flights to their schedule at shortnotice, e.g., charter flights. These airlines can typically estimate thepotential time windows of the extra flights based on their past experi-ences, but prefer to ignore this information during planning since theseflights may not need to be actually operated. Typically, these extraflights are then handled by recovery procedures at the time of opera-tion which may lead to substantial deviations from the planned crewpairings and costs. The reader is referred to [3] for an in-depth discus-sion of the conceptual framework of this problem which we refer to asthe Robust Crew Pairing for Managing Extra Flights (RCPEF). In [3],the authors introduce how an extra flight may be accommodated bymodifying the existing pairings and introduce a set of integer program-ming models that provide natural recovery options without disruptingthe existing flights. These recovery options are available at the plan-ning stage and render operational recovery procedures that pertain tocrew pairing unnecessary.The main contribution of this work is introducing a column gen-eration algorithm that can handle the robust model proposed in thenext section. This model poses an interesting theoretical challenge andis not amenable to a traditional column generation algorithm designedfor the conventional CPP. We point out that in [3] the authors explic-itly generate all possible crew pairings and solve the proposed integerprograms by a commercial solver. This approach is clearly not compu-tationally feasible for large crew pairing instances, and in the currentwork we present our preliminary algorithms and results for large in-stances of RCPEF. We demonstrate the proposed solution approacheson a set of actual data acquired from a local airline [2].2 Robust Airline Crew Pairing ProblemIn this section, we first introduce the proposed robust model and thendiscuss the difficulties that arise while solving this model by conven-tional methods. This leads us to the two solution approaches presentedin this paper.In [3], the authors examine several recovery options for managing theextra flights at the planning level. They classify the possible solutionsinto two types:• Type A. Two pairings are selected and (partially) swapped to coveran extra flight.Robust Crew Pairing for Managing Extra Flights 3• Type B. One pairing with sufficient connection time between twoconsecutive legs is modified to cover an extra flight.In this work, we incorporate one Type A and two Type B solutionsas illustrated in Figure 1 where the estimated time window of the ex-tra flight k is depicted by the blue (shaded) rectangles. In Figure 1(a),the original pairings p and q, covering the flight legs i1, i2 and j1, j2,respectively, are partially swapped so that the extra flight k is insertedinto the flight schedule (Type A). The resulting pairings after swappingare illustrated in the figure where the term deadhead refers to a repo-sitioning of crew members to another airport as passengers on a flight,train, etc. In Figures 1(b) and 1(c), the original pairing p is modifiedto accommodate the extra flight k (Type B). The feasibility rules thatdefine both Type A and B solutions are explained in detail in [3].deadheadj2i2kj1City 1City 2i1(a) Pairings p,q are swapped.ki2deadheadCity 1City 2i1(b) Pairing p is modified.deadheadCity 1City 2i1 i2k(c) Pairing p is modified.Fig. 1. Recovery options for covering the extra flight k.The proposed robust mathematical model is given below:min∑p∈Pcpyp +∑k∈Kdkzk+∑k∈Kdk∑p∈P(1− yp)a¯kp +∑p,q∈P(1− xk(p,q))a¯pqk (1)s.t∑p∈Paipyp ≥ 1, ∀i ∈ F , (2)∑p∈Pa¯kpyp +∑p,q∈Pa¯pqkxk(p,q) ≥ 1− zk, ∀k ∈ K, (3)2a¯pqkxk(p,q) ≤ yp + yq, ∀p, q ∈ P, ∀k ∈ K (4)yp ∈ {0, 1}, p ∈ P, (5)zk ∈ {0, 1}, k ∈ K, (6)xk(p,q) ∈ {0, 1}, p, q ∈ P, k ∈ K, (7)4 C¸oban et al.where F is the set of all flights, K is the set of all extra flights, and P isthe set of all feasible pairings. Here, cp is the cost of pairing p, and dkis the opportunity cost of failing to cover extra flight k. Furthermore,we define the parameters aip = 1 if flight i is included in pairing p, and0 otherwise; a¯kp = 1 if extra flight k can be inserted into pairing p as aType B solution, and 0 otherwise; and a¯pqk = 1 if pairings p and q canform a Type A solution to cover extra flight k, and 0 otherwise. Thedecision variable yp is set to 1 if pairing p is selected, and 0 otherwise.Also, let xk(p,q) be an auxiliary binary variable that takes the value 1 iftwo pairings p and q forming a Type A solution for extra flight k areboth included in the solution, and 0 otherwise. Finally, we define thebinary variable zk equal to 1 if no Type A or B solution is present inthe solution for extra flight k, and 0 otherwise.The objective (1) minimizes the sum of the pairing costs and theopportunity costs of not accommodating the extra flights. Constraints(2) and (3) are the coverage constraints for the regular and extra flights,respectively. Observe that the model may opt for not covering an extraflight k if this is too expensive, setting zk to 1. Constraints (4) prescribethat a Type A solution for extra flight k formed by pairings p and q isonly possible if both of these pairings are selected.The formulation (1)-(7) has both exponentially many variables, onefor each pairing, and exponentially many constraints of type (4) whichmakes it both practically and theoretically challenging. Typical crewpairing models incorporate exponentially many variables, but have afixed number of constraints and are solved by traditional column gener-ation approaches where the pricing subproblem is a multi-label shortestpath problem solved over an appropriate flight/duty network. (See [1]for a review of these concepts.) In our proposed robust model, the num-ber of constraints (4) is not known a priori and depends on the pairingspresent in the model. Thus, ideally this formulation requires simulta-neous row and column generation. In the next section, we present ourpreliminary algorithms developed for the problem RCPEF.3 Solution ApproachesIn both approaches presented here, the primary goal is to fix the numberconstraints in the model before applying column generation.The Static ApproachIn the “static” approach, all pairings that construct Type A solutionsare generated a priori before column generation is applied to the linearRobust Crew Pairing for Managing Extra Flights 5programming (LP) relaxation of (1)-(7). To this end, we identify allpossible flights and connections that may appear in a pairing beforecovering an extra flight or its associated deadhead by a breadth-first-search and then construct pairings over this reduced network. We nextrun pairwise feasibility checks on these generated pairings to determineType A solutions. Thus, all constraints (4) are identified and added tothe model along with the associated auxiliary variables xk(p,q) before thecolumn generation procedure is invoked to identify Type B solutionsand new pairings that may lower the objective function value. Upontermination of the column generation procedure, a primal heuristic isapplied if the LP optimal solution is not integral.We point out that the static approach is an exact method for solvingthe LP relaxation of (1)-(7) because all constraints (4) are explicitlyincluded in the model. Clearly, the computational effort for this algo-rithm will be excessive for large instances of RCPEF.The Dynamic ApproachIn the static approach, all constraints (4) are incorporated into theformulation prior to column generation. In the “dynamic” approach, weopt for the complete opposite for speed. We exclude all variables xk(p,q)and constraints (4) from the formulation and dynamically generatepairings that reduce the objective and yield Type B solutions. After thecolumn generation terminates, we check whether the available pairingsyield any Type A solutions and add the associated constraints andvariables to the model. Next, we solve the LP relaxation of (1)-(7) withthe available constraints and variables and invoke a primal heuristic, ifnecessary, in order to obtain an integer feasible solution to RCPEF.The proposed dynamic approach does not necessarily provide anoptimal solution to the LP relaxation of (1)-(7) because pairings leadingto Type A solutions may be missed during the column generation. Inorder to reach a compromise between speed and solution quality, wepromote that at least N (partial) pairings that may potentially formType A solutions are kept on each node during the pricing subproblem.At the end of the pricing subproblem, such pairings are added to aspecial pool. This pool is examined for Type A solutions after thecolumn generation terminates.4 Computational ResultsIn this section, we present our preliminary results on the proposedstatic and dynamic approaches. Our primary goal is to illustrate the6 C¸oban et al.trade-off between robustness (as indicated by the number of Type Aand B solutions obtained) and computational effort. We conducted anumerical study on two sets of actual data. The results are presentedin Table 1 where in each cell the number of Type A solutions identifiedis followed by the solution time in parentheses. No more than 2 TypeB solutions per extra flight were obtained in all cases.Dynamic Static| F | | K | N = 0 N = 10 N = 50 N = 100 N = 5001 0(0.17) 10(0.31) 14(0.62) 18(0.71) 18(0.96) 18(0.50)42 2 0(0.17) 10(0.32) 86(0.67) 98(0.75) 118(1.03) 118(0.70)3 0(0.20) 10(0.42) 93(0.71) 106(0.81) 128(1.31) 128(1.34)1 10(0.71) 40(1.07) 64(1.39) 64(1.70) 64(2.57) 64(1.83)96 2 20(0.86) 80(1.15) 128(1.64) 128(1.98) 128(2.87) 128(5.86)3 38(0.86) 60(1.25) 136(1.76) 141(2.06) 141(3.42) 141(9.70)Table 1. Comparison of the number of Type A solutions and CPU times forthe dynamic and static approaches.Two trends are clear from Table 1. First, the performance of thedynamic approach depends critically on the value of N . There is athreshold value for N above which extra solution time is spent withno additional benefit. Second, the dynamic approach outperforms thestatic approach for large problem instances.5 Future ResearchThe results in Section 4 point to a clear need for simultaneous rowand column generation for solving the proposed model. We are goingto pursue this interesting direction in the future.References1. Klabjan D (2005) Large-scale models in the airline industry. In: De-saulniers G, Desrosiers J, Solomon MM (eds) Column Generation.Springer2. C¸oban E (2008) Column generation approaches to a robust airline crewpairing model for managing extra flights. MS Thesis, Sabancı U, Istanbul3. Tekiner H, Birbil S¸I˙, Bu¨lbu¨l K (2008) Robust crew pairing for managingextra flights. To appear in Computers and Operations Research, DOI:10.1016/j.cor.2008.07.005

Column Generation Approaches to a Robust Airline Crew Pairing Model For Managing Extra Flights

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Column Generation Approaches to a Robust Airline Crew Pairing Model For Managing Extra Flights

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