A Column Generation based Heuristic for the Tail Assignment Problem
RResearch Paper:
A Column Generation based Heuristic for the Tail Assignment Problem
SAMBREKAR AKASH , ER RAQABI El Mehdi
1. Department of Mathematical and Industrial Engineering,Polytechnique Montreal, H3T 1J4, Canada
September 29, 2020 a r X i v : . [ c s . A I] S e p bstract This article proposes an efficient heuristic in accelerating the column genera-tion by parallel resolution of pricing problems for aircrafts in the tail assignmentproblem (TAP). The approach is able to achieve considerable improvement inresolution time for real life test instances from two major Indian air carriers.The different restrictions on individual aircraft for maintenance routing as peraviation regulatory bodies are considered in this paper. We also present avariable fixing heuristic to improve the integrality of the solution. The hy-bridization of constraint programming and column generation was substantialin accelerating the resolution process.
Keywords : Tail Assignment; Column generation; Constraint programming;Heuristic INTRODUCTION
The airline business . Since the availability of air transport, humans have beentraveling intensively using planes. Such a mean of transport is allowing trips withinthe same continent and between continents. In parallel to this increase in demand,the airline industry have been evolving on many aspects. First, many companiesemerged gradually in the industry. Some are targeting the international flights, i.e.between countries, while other are targeting the local flights, i.e. within the samecountry. Even on the local level, air transport seems to be competitive with othermeans of transport such as railway, cars, etc. With this growth in coverage, processesare continuously improved to reduce significantly costs, allow better flexibility forcustomers, and make the ticketing fully online. This ease in process is attractingmore customers than before. Consequently, the increase in demand has led to, andwill continue to lead, to large investments on many levels including the infrastructure,the fleet of planes, and the human resources. Furthermore, the increasing frequencyof flights has made the planning and scheduling of planes usage more complex.
The operational system . Airlines companies operate on a point-to-point system,a hub-and-spoke system or a hybrid system. In the first one, illustrated in Fig. 1,flights go directly from the origin to the destination. In the second one, illustratedin Fig. 2, a big airport is selected to be the center of the hub and the flights stopoverit before reaching the destination. The hub system may be composed of one hubor many hubs. In the hybrid system, there are both point-to-point flights and hubflights. The choice is based on many factors such as the operating costs as well asthe demand frequency. Each plane must respect international regulations in orderto fly over different places. In addition, it cannot stay more than 32 hours on thesky and must go through frequent maintenance processes once completed the cycle.After finishing a flight, it should at least remain 20 min on the ground before takinganother flight. Based on the historical data, companies plan the utilization of the3eet on a monthly basis.Figure 1 – Point system Figure 2 – Hub system
The planning complexity . In terms of size and complexity, planning in the airlineindustry is considered one of the most difficult problems known. For a given timehorizon, it has four main parts, see Fig. 3. The first part, i.e. flight scheduling, selecta set of flights with fixed departure and arrival periods while seeking expected profitmaximization. Then, fleet assignment is usually performed. It identifies the aircrafttype to be assign to each scheduled flight based on available aircraft and capacityrestrictions. Next, aircraft routing is tackled. Each aircraft is assigned to a set offlights while ensuring maintenance requirements. Finally, crew scheduling composedof crew pairing and crew rostering is solved. It assigns required crew personnel toeach of the planned flights in the schedule and at the same time satisfying the rulesas per the airline regulatory bodies such as FAR, DGCA, etc. Recovery scenariosare also elaborated to anticipate unexpected events such as delays, accidents, sickmembers, and airport closures. In such a case, the aircraft routing problem mustbe resolved to handle these perturbations. Survey articles by Gopalan and Talluri[1] and Barnhart et al. [2] contain good overview of airline planning problems, andthe use of operations research models to solve them. This research area has alsoa specific lexicon. A routing that starts and ends in the same airport is called arotation. Similarly, a pairing may start and end in the same crew base, i.e. wherethey actually live, spans from one to five days and is sometimes called itinerary. The4ain two resources for such problems are the fleet size and the available crew. Themain costs are fuel consumption and crew salaries. The goal is minimizing coststhrough an efficient usage of both airplanes and crew members while transportingas much customers as possible in each flight. The main constraints are internationalregulations and labour unions that must be respected in order to ensure long-termoperations. Figure 3 – Airline planning process
This research paper . In this paper, we focus on solving the aircraft routingproblem known in the literature [3] as the
Tail Assignment Problem (TAP) for oneof the largest airline companies in India. It incorporates all aspects of aircraft routingand maintenance requirements. When tackling this problem, the aim is to deal withthe operational perturbations, maintain feasibility, and determine an optimal fleetschedule that minimizes costs and provides the sequences of flights assigned to eachaircraft. This schedule must also satisfy maintenance constraints specific to eachindividual aircraft.
The research purpose . The purpose of this paper is twofold: (1) Present thetail assignment problem and a case study from an large airline company, and (2)Describe the mathematical formulation and the resolution approach that combinescolumn generation (CG), constraint programming (CP), and heuristics. The researchin this paper complements the available literature for this important problem byintroducing a new approach to handle maintenance called maintenance assignment,a parallel resolution of the sub-problems obtained from Dantzig-Wolfe decomposition.The practical approach can act as a basis for other solution approaches that seektackling large-scale instances of the problem.
The research organization . The rest of the paper is organized as follows: A5rief overview of the recent literature for the tail assignment problem is presentedin Sect. 2. Section 3 is devoted to a detailed description of the problem, while themathematical programming formulation is given in Sect. 4. The resolution approachas well as the heuristics implemented are described in Sect. 5. Section 6 presents thereal-world cases tested and the associated computational results. Finally, concludingremarks follow in Sect. 7.
In this section, we will study the research conducted in the area of tail assignmentand aircraft routing problems. We will also discuss the different optimization tech-niques implemented by researchers to solve these problems. Predominantly, the fleetassignment and aircraft routing problems have been studied extensively. But in re-cent years, TAP has garnered researchers’ attention in both industry and academiagiven its ability to handle disruptions in the airline planning stages. The AircraftRouting Problem (ARP) plans aircraft routes for specific aircraft types since aircraftsof same type follow the same routine of periodic maintenance checks. But ARP doesnot consider the individual maintenance regulations which vary with each aircraft.In [4], Desaulniers solves a daily aircraft routing problem for determining schedulesfor the aircraft’s with provision of varying departure times of flights. They proposeda set partitioning model and time constrained multi-commodity network flow model.The measure of improving the robustness of the routes to disruptions is consideredby several researchers. In [5], Liang et al propose a weekly line of flight (LOF) net-work model with an objective to improve the resilience to disruptions by providingadequate buffer times between connecting flights. Borndorfer [6] constructs routes byconsidering disruptions from preceding days that could affect the following plannedmaintenance activity. In [7], Basdere and Bilge propose an integer linear program-ming model based on connection network to plan feasible routes for individual tails,which maximizes the utilization of the total remaining flying time of aircraft fleet.6arac [8] addresses the different maintenance checks mandated by the FAA in theaircraft routing problem. They considered the legal flying hours limits and restric-tions at the maintenance bases. They resolve the problem using a Branch and Priceframework with follow on rule for fixing the variables. Feo and Bard [9] propose amodel to determine maintenance base locations and develop flight timetables thatmeet the demand for maintenance checks.Airlines prefer to follow the routes planned during ARP for better management butdisruptions make it difficult to follow the plan. It is certain that LOF’s have tobe restructured to recover from disruptions and to satisfy operational constraints.These routes planned by ARP can be used as an input for TAP. Several researcherslike Lapp and Maher have used the LOFs which are a priori generated by the ARP asan input for TAP. Lapp [10] proposed a integer programming model to build LOFssuch that maintenance reachability is maximized. Maher [11] proposed an iterativealgorithm that quickly provides a feasible solution by reducing the maintenance mis-alignments for the given input of LOFs compared to the column generation approachbut significant gap was evident from the optimal solution. Ruther [12] proposed anintegrated approach for aircraft routing, crew pairing and tail assignment problemby resolving close to day of operations using column generation.In Literature, some researchers have integrated the ARP and TAP by constructingthe routes for each individual aircraft. One way to solve such problems would beto enumerate all possible routes for each tail satisfying the individual aircraft con-straints and later modeling it as a set-partitioning problem. Well, its not possibleto generate all possible routes and solve such a large-scale problem with millions ofvariables. In such context, column generation can be employed for a dynamic genera-tion of routes by resolving a resource constrained shortest path problem for each tail.Extensive work in the area of TAP was carried out by Gronviskt [3] in his doctoralthesis. Gronviskt resolved TAP by combining constraint programming and columngeneration, which significantly reduced the resolution time due to reduced network7ize. He employed different techniques like Dual Re-evaluation scheme and suggesteda randomized order of resolving the pricing problems to generate dissimilar columnsin column generation iterations. Rather than implementing a complex branch andprice heuristic, the paper suggests integer fixing heuristics to obtain an integer solu-tion since the objective was to reach a good feasible solution as quickly as possible.Several researchers like Rousseau [13], Gabteni [14] and Gualandi [15] worked onhybridization of column generation and constraint programming to accelerate theconvergence of resolution process.Since TAP is to be deployed at the operational level that handles immediate dis-ruptions, the user expects resolution in few minutes. The paper presented hereprovides quicker resolution time to TAP by resolving the pricing problems in a par-allel way while ensuring the selection of dissimilar columns. It also incorporates theLOFs planned during the ARP and restructures them if necessary to handle anydisruptions. Based on our review of existing studies, we realized that only few stud-ies related to maintenance regulations like maintenance checks, assigning plannedmaintenance and creating maintenance opportunities along the route of the tail havebeen considered.
The TAP includes determining the routes for a set of individual aircraft that areidentified by tail numbers that should cover a set of flights from the planned schedule.The routes have to satisfy various actual operational constraints like flight connectionconstraints, maintenance constraints and activity restrictions. They are discussed indetail below. In our problem, we plan both flights and maintenance operations.Hence, we use the term activity to refer to both of them.8 .1 Flight connection constraints
If the arrival base of a first leg and departure base of a second leg is same andsufficient ground time is available, then it is possible to connect these two legs by aflight connection. Minimum ground time (MGT), also as called turn-around-time,is the minimum time required by an aircraft to be ready for its next take-off afterit has just landed on a particular airport. The MGT varies considerably from 25to 45 minutes depending upon the level of aircraft activity at a particular airport.Similarly, the airlines also have Maximum connection time (MCT) to restrict anaircraft from being idle for a long duration.
Maintenance constraints are mainly of two types: pre-assigned maintenance activityand periodic maintenance checks. The pre-assigned activities usually represent amaintenance activity specific to an individual tail to be carried out a particularmaintenance base between specified times. This pre-assigned has to be necessarilyassigned in the given time-frame with a flexibility in its start time. This activity isplanned based on the recommendation of aircraft manufacturers, usually Airbus andBoeing, but sometimes, the maintenance and engineering team (M&E) of airlinesmight be more restrictive in its maintenance planning. These maintenance activitiesmay usually span for multiple days. On the other hand, periodic maintenance checksare not restricted to be done on a specific time-frame or on a particular base. Inour problem, we consider two maintenance checks. The flying hours (FH) track thenumber of consecutive flight hours flown by an individual tail from the previousmaintenance check. Similarly, flying Cycle (FC) tracks the consecutive number oflandings made from the previous check. These maintenance checks vary dependingon the type of aircraft. The Table below shows an example for the FH and FC checksfor a specific tail. The tail number 485 needs a maintenance check before the tail hasbeen used for consecutive 150 flying hours and 500 cycles. It cannot exceed these9estrictions. Thus, these periodic maintenance checks have to be planned accordinglydepending upon the maintenance opportunities available throughout the route ofa specific tail. The periodic checks could be done at different maintenance basesdepending upon the hangar facilities of the bases.Tail number FH(hours) FC(cycles)Tail 485 150 500Tail 979 350 250Table 3.1 – FH and FC checks example
These restrictions forbid aircrafts from flying in certain sectors depending upon theirqualification. For extended operation (ETOP) sectors, usually a twin engine is re-quired. It may be that the specific aircraft does not have enough in-flight entertain-ment systems to serve certain sectors. Usually, some tails have runway restrictions onparticular airport. There are also situations when certain routes/flight connectionshave to be assigned to a specific tail to adhere crew connections.
The TAP is an extension of aircraft routing problem but resolved at the operationallevel so as to handle any disruptions without changing the routes that were plannedat the Aircraft Routing Problem (ARP). After the aircraft routing problem has beenresolved and routes have been planned and assigned to an aircraft line, the TAPtries to assign the flights that were planned by the aircraft routing problem on thesame aircraft line. To achieve this, we provide bonuses to the generated routes thatfollow the same routes or sub-routes planned during the ARP.10
MODEL
Referring to the description given in the previous section, the problem consideredcan be formulated using a mixed integer programming (MIP) model. The problemis subject to two sets of constraints: flight coverage constraints and tail constraints.
The sets used in the mathematical formulation are as follows: F : set of activities to be covered, f ∈ F . G : set of routes generated, g ∈ G . T : set of tails/aircrafts, t ∈ T . V t : set routes valid for each tail, t ∈ T . The TAP has parameters that are available a priori. This data is incorporated intothe mathematical formulation as constants. They are written in bold style in themodel. C i : penalty cost for uncovered activity i ∈ F . C r : cost of each route, r ∈ G . a ir : equals 1 if route r ∈ G covers activity i ∈ F , 0 otherwise. The decision variables considered in the TAP are as follows: f i : binary variable equal to if activity i ∈ F is uncovered, 0 otherwise. x r : binary variable equal to if route r ∈ G is selected, 0 otherwise.11 .4 Constraints The model constraints are as follows:
Flight coverage : ensures that every activity has to be covered only once or can beleft uncovered. f i + (cid:88) r ∈G a ir .x r = 1 ∀ i ∈ F (4.1) Tail constraint : indicates that at most one route can be assigned to each tail. (cid:88) r ∈V t x r ≤ ∀ t ∈ T (4.2) Binary restrictions : on TAP variables f i and x r . In the TAP, we seek minimizing the following objective function:
Min (cid:88) i ∈F C i .f i + (cid:88) r ∈G C r .x r (4.3) We solve TAP by column generation technique. In this technique, we decompose theproblem into two components: a restricted master problem (RMP), which contains arestricted number of variables and a pricing problem (PP), which generates negativereduced cost routes for the tails. The resolution process is summarized in the Fig. 4.At first, we initialize the RMP and provide its dual solution to the pricing problems.The routes generated by pricing problems are added to the RMP and updated dual12olution is found after resolving the RMP. The process is continued until there areno negative reduced cost routes. After the RMP converges, we start fixing fewvariables and connections to ensure that paths, which are better in a standpointview of integrality, are generated. After fixing a significant part of the problem, werestore the integrality constraints on the model and solve it as a integer programmingmodel. Figure 4 – Resolution approach13 .1 Restricted Master Problem
The RMP is initialized with artificial slack variables covering each activity with a highpenalty cost because initial feasible primal and dual solution are necessary to initiatecolumn generation. The RMP formulation is the same as the one presented in thesection above. The only difference is the relaxation of the binary variables. The RMPis solved using Gurobi LP solver. The dual solution is known to be highly unstableover the initial iterations of the column generation leading to poor columns beinggenerated. To tackle such instability, different stabilization methods like BoxStepmethod [16] and Du Merle [17] method have been developed in the literature. Wesolve the RMP with barrier linear programming (LP) method without crossover.Its has been seen across literature that barrier method seems to be more efficientto stabilize the duals. The dual solution is centered since it points to an interiorsolution.
We have one pricing sub-problem for each tail.The structure of the pricing problemfor each tail is as shown in the Fig. 5. The network for the pricing problem for atail is represented by a connection flight network. The source node represents thecarry-in flight for the tail which specifies the present availability of the tail at aparticular base at a particular period of time. A sink node is added to the networkand several restrictions can be imposed through it. If a particular tail has to return toa particular airport in the night, it can be enforced through the sink node. The validflights that can be qualified to be served by the particular tail and the pre-assignedactivities are represented by the nodes in the figure. The flight nodes have inter-arcs between them which represent the flight connections that are possible whichare known a priori. In the Fig. 5, the source is connected to flights for which thedeparture base is same as the arrival base of the carry-in flight. All the flights areconnected to the sink. The connections are determined a priori based on several14usiness rules mainly MGT and max connection time. Some business heuristics arehelpful in reducing the density of the network without affecting the solution quality.The dual information π = { π , π ,..., π f }, β = { β , β ,..., β t } where π corresponds tothe duals related to the activity constraints and β corresponds to the tail constraints.They are obtained after solving the RMP and used to solve the pricing problem.Figure 5 – flight connection network Labeling Algorithm
To solve the problem, we employ the label-setting algorithmproposed by Desrochers and Soumis [18]. Each label at particular node representthe different paths of reaching the node from the source. A label at a particularnode label L = { ¯ c j , r , r ,..., r k } keeps tracks of the information like reduced costs andresource consumption accumulated along its path. In the label, ¯ c j is reduced costof the route and r k is the consumption of resource k. The labeling algorithm worksby pushing labels from the predecessor to all its successor and updating the reducedcosts and resources consumptions. The nodes to be treated are ordered accordingto the starting times of activities and for each node, we push the labels to all itssuccessors. At every node, we retain only few significant labels based on dominanceamong labels. A label p dominates label q if ¯ c p ≤ ¯ c q , and r kp ≤ r kq for every resource k.So we retain only non-dominated labels at a particular node. In reality, due to largenumber of resources we would end up with up large number of non-dominated labelson a particular node. In such cases, we adopt the lexicographical sorting strategyproposed by Gronviskt [3] to limit the number of labels stored. Lexicographical15rdering limits the number of labels stored at each node when dominance alone isnot enough. When the number of labels at a particular node are greater than 12and less than 20, we employ the lexicographical sorting technique. We do no retainmore than 20 labels at each node except the sink node. After all the nodes havebeen treated, the negative reduced cost paths from the sink node are added to theRMP as variables. These feasible routes are transformed as variables covering certainactivities with associated costs and are added to the RMP model. Handling Restrictions . Restrictions include pre-assigned activities, FH& FC re-strictions, and planning maintenance opportunities along the route. First, the pre-assignment activities, which are usually the long term planned maintenance, areensured by treating them as intermediate sink node and only labels from the inter-mediate node are extended ahead since they have to be necessarily assigned. Thepresence of pre-assigned activities further decomposes the shortest path problem.Secondly, the FH and FC restrictions that are specific to each tail are determinedby the M&E team well ahead of time. These restrictions are handled in the pricingproblem by using resource attributes in the label. We keep track of the resources oftype k consumed throughout. For each specific tail, the upper bounds on the resourceconsumption of resource k is different and only legal labels will be extended to thesuccessor that satisfies the upper bounds on the resource consumption. Third, sinceour problem also includes identifying maintenance opportunities when the aircraftis present at the maintenance hub and sufficient time is available for maintenancebefore the next flight. However, we can only perform the maintenance activity ifthere is sufficient capacity available at the hanger. For these reason, we extend twolabels from the node to its successor. Considering that the maintenance activity isperformed, the first label is extended and the relevant resources are to zero. Thesecond label is extended without maintenance and the accumulation of the resourceis retained as shown in Fig.6. We create a maintenance opportunity ’M’ between the16wo flights F2 and F3 to carryout the necessary maintenance activities. The connec-tions time of these two flights should be greater than the required maintenance timeand connection base must correspond to one of the maintenance bases.Figure 6 – Maintenance opportunity example
Parallel resolution of pricing problems . Since all the pricing problems utilize thesame dual information, all the pricing problems tend to generate columns coveringthe same activity which are good from a dual perspective. Eventually the activitywill be covered by one single tail in the optimal solution. To overcome this similarityof columns being generated, Gronviskt [3] suggests a dual re-evaluation scheme togenerate disjoint columns by penalizing the dual values of the activities coveredby the previous pricing problems. But this requires the pricing problems to besolved in a serialized order. Re-ordering the pricing problems in a random fashionor ranking them can assist being un-biased to each pricing problem. But insteadof serially resolving the pricing problems, we resort to solve the pricing problems inparallel. Since all the pricing problems can be solved independently, time is reducedsignificantly. Now to overcome the problem of similarity of columns being generated,we retain about 100 negative reduced costs paths for each tail at the sink. From thepool of paths generated from all the pricing problems, we select only few paths whichare disjoint in nature and add them to RMP as variables. Based on our study wefound that it is good to retain large number of non-dominated labels at sink nodefor each tail, and later select disjoint paths from the pool. We understand that itis possible that many poor columns are also generated but we add only paths thathave a high probability to be part of the solution of RMP. This helps us in solvingpricing problems in parallel and keeping the size of RMP less dense. The proposed17euristic for the disjoint path selection is as follows:
Algorithm 1:
Disjoint Path selection Heuristic Initialize a dual vector to retain penalizations: π pen = { , ..., } , C = { , ..., } for p ∈ P paths from solution pool S do Let F be the set of activities covered by path p if (cid:80) f ∈F π actualf ≥ (cid:80) f ∈F π penf then C ← C ∪ { p } ; for f ∈ F do π penf + = (cid:15).π actualf ; end else continue ; end end The value (cid:15) can vary between 0.8 to 1. The value 1 means that the selected pathsare completely disjoint in nature. Decreasing the value allows the overlapping ofactivities, thereby more columns are selected from the heuristic.
Integrality
Even though the column generation technique converges to a betterLP solution but still large integrality gap exists. Since routes which are better foran integral solution are yet to be generated. The branch-and-price framework isusually employed to provide better integral solutions for large scale problems. Theresolution time of the problem using branch-and-price would be high and also it isvery complex in its development. To resolve TAP in a reasonable time, we developa variable fixing heuristic as shown in Algorithm 2. After the column generationprocess converges, we initiate the variable fixing heuristic. We always tend to fixvariables whose LP solution values are above a certain threshold value. If we findany variables, we fix them to 1 and mark the pricing problems for these tails asinactive. The threshold value changes dynamically throughout the heuristic. When18e do not find any variables to fix, we reduce the threshold by 0.05 until we findsome variables to fix. Otherwise, even after decreasing the threshold value to anacceptable limit, we exit the fixing loop as no variables were found to fix. Then,we solve the RMP as an integer program with integrality constraints restored. Weconsider the set V representing the set of variables in RMP as well as an emptyset F . We also design three functions. First, Solve-RMP , which solves the RMPmodel and returns the dual solution verctor π . Second, Solve-PP( π ) , which resolvesall active pricing problems for each tail in parallel and returns a set of generatedcolumns. Third, DisjointPathSelection( π , P ) , which returns a subset of columns fromthe newly generated columns from PP as per the disjoint path selection heuristic.19e fix the initial threshold value to 0.95. Algorithm 2:
Variable fixing heuristic Initialize threshold = 0.95 while threshold ≥ do G = {} for v ∈ V \ F do if v ≥ threshold then G ← G ∪ { v } else continue ; end end if G = ∅ then threshold ← threshold − . else threshold = 0.95 F ← F ∪ { G } Fix the variables in G in the RMP; π =Solve-RMP() P =Solve-PP( π ) S = DisjointPathSelection( π , P ) Add S to RMP, V ∪ { S } Sovle RMP end endPreprocessing using constraint programming It was known from the paper[19] that simple pre-processing techniques based on Aircraft count balancing do notwork in case of flight schedules when there is possibility of unassigned flight activities.20ven to make the propagation filter work for CP model, we need to need add thesuccessor of every flight to be itself to allow unassigned activities. This would bevery weak in propagation, to handle this costGCC propagation is suggested. Butthe costGCC propagation depends completely on prior knowledge about the upperbound of the solution or number of flights that might be left unassigned. Thereforeit would be really difficult to apply it in practice. Therefore we suggest to use thepreprocessing only at the initial iterations of the column generation. This wouldhelp to accelerate the column generation in the initial stages and later use all theconnections for improvement stages. The commercial CP solvers are equipped withonly few traditional propagation capabilities like all-different , inverse , etc. In orderto implement certain tunneling constraints for resource consumptions along a pathand costGCC propagation technique, a specific CP solver must be implemented.Rather than implementing these propagation techniques, we make use of the readilyavailable capabilities of commercial solvers to propagate. The test instances used for the computational studies come from two major aircarriers in India. The information related to the instances is as shown in the Table6.1. The instances span from 2 to 15 days in window period. The instances aresolved at once without considering rolling time slices. The pre-connections refer toall the feasible flight connections. The Post-connections refer to the connectionspropagated through the CP model.For our studies, we consider post-connections only for initial 10 column generationiterations in the network of pricing problems to gain initial speedup. After 10 itera-tions, we consider all the pre-connections in the the network. The proposed disjointpath selection heuristic in Algorithm 1 significantly reduced the convergence of col-umn generation by about 40% for the test instances compared to serial resolution ofpricing problems. Even though the serial resolution took considerably few iterations21nstance Horizon(days) Tails Flights Pre-Connections Post-ConnectionsA-120 2 42 509 10057 1969A-129 2 42 539 12000 1333A-45 2 41 563 16404 3196A-3 3 52 955 45401 12903C-5 5 14 243 4270 2175C-7 7 14 335 7982 3349C-10 10 14 470 15369 5201C-15 15 14 715 35324 9364Table 6.1 – Instances informationto converge than parallel resolution but resolution times were higher. For the serialresolution, we tested the instances based on the ranking order rule as suggested in[3]. The comparison between parallel resolution and serial resolution is as shown inTable 6.2. Serial Resolution Parallel ResolutionInstance LP Obj Iter Time(s) Iter Time(s)A-120 3778764 18 129.7 25 92A-129 3729446 15 96 17 60A-45 1246737 30 990 35 660A-3 2101400 127 6502 133 4640C-5 604633 39 138.7 41 95C-7 704486 144 1014 149 780C-10 727383 168 5049 172 3270C-15 726777 556 38935 582 29950Table 6.2 – Comparison between parallel resolution and serial resolution22he integral solution to all the test instances with our solution approach is as shownin the Table 6.3. We conducted two set of experiments with and without CP propa-gation. The rest of the solution approach for both experiment remained same. Thesolution approach begins with resolving the pricing problems for each tail and con-verging relaxed LP version of TAP model as in section 4. Then, we initiate fixingthe variables by the heuristic presented in Algorithm 2. Later, when no candidatevariables are found to fix, we initiate connection fixing. The RMP, an LP program,is later converted to an Integer program by restoring the integrality constraints.Without CP propagation With CP propagationInstance Objective Time(s) Objective Time(s) RemarksA-120 3778764 1020 3778764 98.4 Two uncovered flightsA-129 3729546 840 3729313 22.8 Two uncovered flightsA-45 1252593 3480 1246937 709.2 Complete AssignmentA-3 2115250 4707 2115250 2105 One uncovered flightsC-5 637041 105 637041 55 Complete AssignmentC-7 704486 802 704486 104 Complete AssignmentC-10 727355 3760 727355 470 Complete AssignmentC-15 74627 29979 745627 3747.37 Complete AssignmentTable 6.3 – Instance ResultsThe results showed that, using CP propagation at the initial iterations of the columngeneration, we were able to converge quickly. Since the network problems with CPpropagation contain fewer connections, it helps in accelerating the column generationto a better solution. The CP propagation model is resolved considering that therewould be no unassigned flights in the final solution which is far from reality, as thereis a possibility of flights being left unassigned. So for these reasons, we consider thePost-connections only for 10 initial iterations in our approach.Its evident that withCP propogation we were able to converge to a better solution in less time for all the23nstances. 24
CONCLUDING REMARKS
This paper presents a combination of column generation and constraint programmingapproach supported with efficient heuristics used to tackle the TAP. This problem ismainly related to the operational perturbations that happen in the short term withinairline industry. In our paper, we innovatively contributed to the existing literaturethrough an innovative way to ensure the handling of restrictions. We also performeda parallel resolution of the the pricing problems, which proved to be more efficientthan the sequential resolution. To do so, we designed an heuristic that deals withthe integrality aspect. Finally, we highlighted the role that constraint programmingbrings to the formulation as well as the impact on the execution time. The testsresults proved the effectiveness of our approach in tackling rapid perturbations thathappen on the operational level. The paper also provides an exhaustive descriptionof the TAP as well as an updated review of literature related to the optimization inairline problems. The proposed approach has been deployed in production systemsat LAI in the Tail Assignment Optimizer (TAO).
Acknowledgements
The research work was conducted by the R&D team of Laminaar Aviation InfoTech.The company provides aviation-specific IT solutions to airlines across the globe. Wewould also like to extend our gratitude to the LAI team, Mr. Sujayendra Vaddagiri,Mr. Sunil Sindhu and Mr. Mayur Pustode for their continued co-operation andvaluable support. 25 eferences [1] Ram Gopalan and Kalyan T Talluri. Mathematical models in airline scheduleplanning: A survey.
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