Copyright � 2001 INFORMS0092-2102/01/3101/0030/$05.001526–551X electronic ISSN

TRANSPORTATION—SCHEDULING-PERSONNELPROGRAMMING—INTEGER-APPLICATIONS

INTERFACES 31: 1 January–February 2001 (pp. 30–56)

Optimized Crew Scheduling at AirNew Zealand

E. Rod Butchers Network Logistics, Air New Zealand Ltd.Private Bag 92007, Auckland, New Zealand

Paul R. Day Optimal Decision Technologies Ltd.P.O. Box 128-233, Remuera, Auckland, New Zealand

Andrew P. Goldie Optimal Decision Technologies Ltd.

Stephen Miller Network Logistics, Air New Zealand Ltd.

Jeff A. Meyer Optimal Decision Technologies Ltd.

David M. Ryan Department of Engineering Science,University of Auckland, Private Bag 92019, Auckland,New Zealand

Amanda C. Scott Network Logistics, Air New Zealand Ltd.

Chris A. Wallace Network Logistics, Air New Zealand Ltd.

The aircrew-scheduling problem consists of two importantsubproblems: the tours-of-duty planning problem to generateminimum-cost tours of duty (sequences of duty periods andrest periods) to cover all scheduled flights, and the rosteringproblem to assign tours of duty to individual crew members.Between 1986 and 1999, Air New Zealand staff and consultantsin collaboration with the University of Auckland have devel-oped eight application-specific optimization-based computersystems to solve all aspects of the tours-of-duty planning androstering processes for Air New Zealand’s national and inter-national operations. These systems have saved NZ$15,655,000per year while providing crew rosters that better respect crewmembers’ preferences.

Commercial airlines must solve manyresource-scheduling problems to en-

sure that aircraft and aircrews are avail-able for all scheduled flights. Since aircraftand aircrews are among the most expen-sive of airline resources, their efficient util-ization is important. Because of the largesavings possible from using aircrews more

efficiently, many airlines have tried to de-velop optimization methods to solve theircrew-scheduling problems. Most early op-timization attempts failed because of in-adequate solution methods and lack ofcomputer power. Even now many airlinesstill use heuristic or manual methods tosolve crew-scheduling problems. Since the

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Figure 1: The aircrew-scheduling process consists of two distinct subproblems, the tours-of-duty problem and the rostering problem. The tours-of-duty (ToD) planning problem involvesconstructing the minimum-cost sequences of flights to crew the flight schedule. The ToDs fromthe ToD planning problem are passed to rostering four to six weeks before the start of the ros-ter period. The rostering process involves allocating the planned ToDs to individual crewmembers to form a line of work over a roster period. Aircrew rosters are typically up to onemonth long and are normally published to crews one to two weeks before the start of the rosterperiod.

mid-80s, improvements in optimizationtechniques and increasing computerpower have resulted in the developmentof optimization-based solution methodsfor aircrew-scheduling problems.

The aircrew-scheduling problem is usu-ally partitioned into two distinct subprob-lems (Figure 1).

The first, the tours-of-duty (ToD) plan-ning problem, calls for constructing se-quences of flights to crew the flight sched-ule, which are variously described as toursof duty, trips, pairings, or rotations. Insome situations, these ToDs can be one-day periods of work, but they can alsoconsist of sequences of flights and rest pe-riods spanning many days. The ToD plan-ning problem has prompted a great dealof research (Anbil, Forrest, and Pulleybank[1998] provide a useful summary), andover the past 20 years, airlines have regu-larly discussed the problem and varioussolution methods at meetings of the Air-line Group of the International Federation

of Operations Research Societies (AGI-FORS). Aspects of the ToD planning prob-lem for American Airlines were discussedby a previous Franz Edelman finalist[Anbil et al. 1991].

The second subproblem associated withaircrew scheduling is rostering. Rosteringinvolves allocating the planned ToDs fromthe first subproblem to individual crewmembers to form a line of work (LoW)over the rostering period. In contrast tothe extensive work on the ToD planningproblem, little work has been reported onsolution methods for rostering.

The two aircrew-scheduling subprob-lems can be considered from both man-agement and crew points of view. Man-agement seeks minimum-cost ormaximum-productivity solutions that arelegal in the sense that they do not breakany rules or agreements with the crew andare feasible in the sense that all the workis performed. Management also seeks toquantify the costs associated with satisfy-

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ing the rules and agreements. A furtherimportant feature of crew-scheduling solu-tions from the management point of viewis their operational robustness or sensitiv-ity to disruption of the planned flightschedule. For crews, the key issue is solu-tion quality. Among all legal and feasiblesolutions, crews will prefer solutions thatmaximize some quality measure definedby the particular crew group. Typicalquality measures are the fair distributionof work or satisfying the work preferencesof individual crew members in seniorityorder or the avoidance of unduly arduouswork patterns. While managers appreciatethe importance of solution quality in pro-moting crew satisfaction, the quality objec-tive remains the primary concern of crewmembers.The Tours-of-Duty Planning Problem

A tour of duty (ToD) is an alternating se-quence of duty periods and rest periods(or layovers). Each duty period comprisesone or more flights and may also includepassengering flights. A passengering flightis one on which a crew member travels asa passenger to get to a particular airportfor a subsequent operating flight. The first

duty period of a ToD must start at a crewbase, and the last duty period must end atthe same crew base. An airline might haveseveral bases (cities) at which crew mem-bers are domiciled. Each ToD will have anassociated crew complement made up of anumber of crew members of various ranksrequired to staff the ToD. For example, acrew complement might consist of oneB767 captain and one first officer, or oneinflight service director and three flight at-tendants (Figure 2).

In ToD planning, one constructs aminimum-cost set of generic ToDs thatcover all flights in an airline schedule withthe crew required (for example, captain,first officer, inflight service director, andflight attendants). Constructing ToDs doesnot include considering which individualcrew members will perform the ToDs. Air-lines that have stable and regular flightschedules can solve a standard daily orweekly representation of the flight sched-ule and then use that solution for the du-ration of the flight schedule. However, air-lines with variable flight schedules mustsolve a fully dated problem, and the ToDsolutions can differ from day to day or

Figure 2: In these example tours of duty, FRA004 is a 14-day international tour of duty (ToD)for pilots and CHC259 is a single-day national ToD for flight attendants. The top row of theToD indicates the airports visited (Auckland—AKL, Los Angeles—LAX, Frankfurt—FRA, Pa-peete—PPT, Christchurch—CHC, Rotorua—ROT), and the bottom row shows the flight num-bers. Px indicates a passengering flight and a dash indicates a day off overseas.

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week to week.The creation of legal duty periods and

ToDs is governed by a complex set ofrules. Some of these rules are specified byaviation regulatory authorities, such as theNew Zealand Civil Aviation Authority orthe Federal Aviation Administration.Other rules are specified by employmentcontracts and operational-robustness con-siderations. Typical duty period rules con-cern the length of a duty period, the latesttime at which a flight may start within aduty period, the maximum allowableflight time within a duty period, and thenumber of crew members required to op-erate a flight. Typical ToD rules concern

Many airlines still useheuristic or manual methods.

the minimum rest between duty periods,cumulative duty-time and flight-time lim-its over a rolling time period (for example,seven days), and in international opera-tions, the effects of time-zone changes oracclimatization at foreign ports. Opera-tional robustness rules are sensible rules ofthumb that minimize the impacts of dis-ruptions on the day of operation. For ex-ample, providing rest periods slightlylonger than the minimum legally requiredso that crews will still have their legal restperiods if their flights arrive late. WithinAir New Zealand and generally in mostairlines, the rules are different for pilotsand for flight attendants, and for nationaland for international operations.

Airlines measure the quality of ToD so-lutions in terms of their dollar costs andmeasures of operational robustness andgood practice. For example, good practice

dictates that airlines use the layover loca-tions crew members prefer if this does notincrease costs. The costs incurred in oper-ating a flight schedule include the explicitcosts of crew members being at foreignports (hotel accommodation, meals,ground transport, and various other ex-penses and allowance entitlements), andthe implicit crew costs of salaries, over-time, and other pay-related airline-specificmetrics, such as flight time, trip credits, orincentive pay. Robustness quality mea-sures might favor crew members continu-ing to operate on the same aircraft (tominimize delays during disruptions) orcertain flight combinations over others.Airlines prefer solutions that are close tothe minimum dollar-cost solution but havebetter robustness or good practicefeatures.

A desirable feature of ToD solutions isthat all crew members on a flight performthe same sequence of flights for as muchof their duty period as possible and, evenbetter, that they also stay with the sameaircraft. This greatly reduces the propaga-tion of disruptions from one flight to otherflights on the day of operation. This iscalled unit crewing. ToD solutions with ahigh degree of unit crewing generally costmore than the minimum-cost ToD solutionbut are much more robust. Airlines mustfind some balance between cheaper solu-tions with less unit crewing and more ex-pensive solutions with more unit crewing.The unit-crewing idea is not normally ap-plied across crew types because pilotsand flight attendants work under dif-ferent rules and regulations that precludetheir operating exactly the same dutyperiods.

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Figure 3: In these example lines of work, PILOT01 is a 28-day international pilot line of work(LoW) and FA01 is a 14-day national flight-attendant LoW. The LoW activities include a simula-tor training session (SIM), a Frankfurt ToD (FRA004), days off (O), a Sydney ToD (SYD008),on-call days (CALL), and various national ToDs (three-digit codes).

The Rostering ProblemThe rostering problem consists of con-

structing a line of work (LoW) for eachcrew member. A LoW is a sequence of ros-ter activities and rest periods over a speci-fied roster period, such as 14 days or 28days duration. Roster activities includeToDs, training, leave, days off, and callduties (Figure 3).

The rostering process consists of assign-ing all roster activities to crew membersover the roster period, with the followingrequirements:—The roster assignment must be legal;that is, it must satisfy all rostering rules,including legislative rules, employment-contract conditions, operational or in-house rostering agreements and bestpractice.—The roster must be feasible; that is, allroster activities to be assigned must be as-signed to crew members of the correctrank. In particular, no flying ToDs shouldbe left unassigned in a feasible solution.—The roster quality should be maximized;that is, the best possible roster for the crewshould be obtained for any agreed mea-sure of roster quality.

The creation of legal LoWs is governedby a complex set of rules derived fromlegislation, crew-employment contracts,

and operational-robustness considerations.Typical rules concern the number of daysoff per roster period, cumulative dutytime and flight time limits over a rollingtime period (for example, 24 hours, sevendays, or 28 days), the rest period betweenToDs, and the qualifications required tooperate a particular ToD. Pilots generallyhave more complex operating experiencerequirements than flight attendants.

The earliest rostering approaches theairline industry used were based on bid-lines in which airlines constructed legalLoWs to produce feasible rosters withouttaking any account of the individual mem-bers of each crew rank. Crew membersthen bid for the LoWs they preferred. Theairline then processed crew members inseniority order and assigned each personhis or her most preferred LoW of those re-maining unassigned. Many rostering inef-ficiencies occurred because the airlinesconstructed LoWs without taking into ac-count crew availability, determined by an-nual leave, training, or carry-in activitiesfrom the previous period. This resulted inToDs remaining unassigned at the end ofthe rostering process. It is now recognizedthat assignment-based rostering systems,in which roster activities are assigned tospecific crew members, respecting their

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carry-in activities and other existing as-signments, are likely to produce muchmore efficient rosters. However, many air-lines still use bidline systems because oflong-standing industrial agreements.

Two major variants of assignment-basedrostering systems have evolved. Preferen-tial bidding by seniority (PBS) is based onsatisfying crew bids in strict seniority or-der. A crew bid is an expression of prefer-ence for work content or days off in aLoW. The seniority order results in long-employed crew members usually achiev-ing all or most of their bids, while more-recently-hired crew members are oftenunable to influence the content of theirLoWs. Equitability assignment, on theother hand, is designed to fairly andevenly spread satisfaction across all mem-bers of a crew rank, where the satisfactionmeasures the achievement of crew mem-ber’s individual bids and various agreedcollective quality measures.Air New Zealand

Air New Zealand is the largest nationaland international airline based in NewZealand. It employs over 2,000 crew mem-bers and operates flights to Australia,Asia, North America, and Europe, and be-tween the major centers within NewZealand.

Air New Zealand has two closely inte-grated business units, National and Inter-national. Air New Zealand National oper-ates eight B737-200 and nine B737-300aircraft on national flights and longerflights to Australia and the Pacific Islands.Air New Zealand International operates 13B767 and eight B747 aircraft on predomi-nantly international operations. The AirNew Zealand flight schedule is based on

an interconnected network, rather than ahub-and-spoke network like those oper-ated by many American airlines.

Air New Zealand has four major crewtypes: international pilots, internationalflight attendants, national pilots, and na-tional flight attendants. These differentcrew types have different employmentcontracts, different scheduling rules, anddifferent pay schemes. Each crew type canalso be partitioned into a number of ranks.For example, international flight atten-dants have the ranks of inflight service di-rector, inflight service coordinator, flightattendant premium service, and flight at-tendant Pacific class. An unusual indus-trial situation within Air New Zealand isthat pilots within the same fleet and rank

An optimization model canreliably detect infeasibility.

can belong to different contract groupsand therefore have different schedulingrules. This causes additional complexity informulating and solving crew-schedulingproblems.

The first contact with Air New Zealandoccurred in 1983, when David Ryan metwith senior Air New Zealand managers todiscuss their crew-scheduling problems.Despite a somewhat sceptical initial re-sponse from management, the first studentproject in 1984, investigating a small partof the national-tours-of-duty-planningproblem, produced impressive results, andso began the collaboration on which thispaper is based, and which continues to-day. Professor Ryan and postgraduate stu-dents at the University of Auckland havedeveloped the prototypes of the optimiza-

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tion systems implemented at Air NewZealand. In most cases, students were sub-sequently employed by Air New Zealandto extend their research work. By 1987, AirNew Zealand had hired two OR gradu-ates, and today it employs 11 graduates orpostgraduates, with a further five havingmade significant contributions during the1990s. In 1997, five of the early collabora-tors formed a company called Optimal De-cision Technologies, which now providesmuch of the support and development ofthe systems. We have developed threeToD optimizers:(1) for national pilots and flightattendants,(2) for international pilots, and(3) for international flight attendants.

These ToD optimizers each have specialcapabilities reflecting the route networkand business rules for that problem. Wehave also developed four rosteringoptimizers:(1) for national pilots,(2) for national flight attendants,(3) for international pilots, and(4) for international flight attendants.

These rostering optimizers implementdifferent rostering methods and use differ-ent roster quality measures, which we de-veloped and agreed upon with each crewgroup. The Air New Zealand rosteringproblems include both PBS for interna-tional pilots and equitability assignmentfor all other crew types. We developedseparate solvers because of fundamentaldifferences in the business problems or inthe problem behaviour.Models and Solution Methods forAircrew Scheduling

The set-partitioning model provides an

underlying mathematical model for boththe ToD planning and the rostering sub-problems of the aircrew-scheduling prob-lem. The set-partitioning problem (SPP) isa specially structured zero-one integer lin-ear program with the form

TSPP: minimize z � c xsubject to Ax � e

nand x � {0,1}

where e � (1,1,1,. . ., 1)T and A is a matrixof zeros and ones. Because of the compu-tational difficulties in solving very largeand practical instances of the set-partitioningproblem, many early attempts to use opti-mization solution methods to solveaircrew-scheduling problems, and in par-ticular the ToD planning problem, wereunsuccessful, and researchers resorted to avariety of heuristic solution methods.

While most heuristic methods are fairlyeasy to implement and may have reason-ably inexpensive computer resource re-quirements, they suffer from two majordisadvantages: (1) they can provide nobound on the quality of any feasible solu-tion that they produce, and (2) they areunable to guarantee a feasible solution willbe found even if one exists. The heuristicmethod may fail to find a feasible solutioneither because the heuristic method is in-adequate or because the problem is trulyinfeasible. It is important to distinguishbetween these two possibilities, particu-larly in the rostering problem, which cansometimes be infeasible because of insuffi-cient crew. In contrast, an optimizationmodel can reliably detect infeasibility.During the past two decades, the develop-ment of optimization methods and tech-niques for the solution of set-partitioning

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problems and the increase in computerpower has meant that realistic-sized mod-els that arise in aircrew-scheduling prob-lems can be solved.The Tours-of-Duty-Planning Model

In the basic ToD-planning model, eachcolumn or variable in the SPP correspondsto one possible ToD that could be flownby some crew member. Each constraint inthe SPP corresponds to a particular flightand ensures that the flight is included inexactly one ToD. The elements of the Amatrix can then be defined as

a � 1 if the jth ToD (variable)ij

includes the ith flight (constraint),� 0 otherwise.

The value of cj, the cost of variable j, re-flects the dollar cost of operating the jthToD. The calculation of cj values is speci-fied by the particular problem being con-sidered but usually includes the cost ofpaid hours (both productive and unpro-ductive), ground transport, meals, and ac-commodation, and the cost of passenger-ing crew within the ToD. Many authors[Rubin 1973; Wedelin 1995] model theToD-planning problem using the set-covering formulation in which the equalityconstraints are replaced by greater-thanor-equals constraints. The overcover of aflight permitted by the set-covering con-straint can be interpreted as passengeringof the excess crew cover. This formulationresults in fewer variables, but it has themajor disadvantage of making it difficultto model accurately the rules and costs as-sociated with passengering crew. For ex-ample, duty-time limits for passengeringduties are generally longer than those foroperating duties. This cannot be correctly

modeled using set-covering constraints. Inthe Air New Zealand applications, weused set-partitioning constraints, which al-low us to accurately model passengering.This results in additional columns that ex-plicitly include passengering flights withaij � 0. Also each column or ToD mustcorrespond to a feasible and legal se-quence of flights that satisfies the rulesspecified in civil aviation regulations oremployment contracts or agreements.These rules or constraints can be thoughtof as being implicitly rather than explicitlysatisfied in the ToD-planning model. Forexample, rules imposing limits on totalwork time and rest requirements are em-bedded in the variable generation process.

The ToD-planning model is usually aug-mented with additional constraints thatpermit restrictions to be imposed on thenumber of ToDs included from each crewbase. Because these constraints typicallyhave nonunit right-hand-side values, wedescribe the ToD-planning model as ageneralized set-partitioning model.

Air New Zealand’s international-pilotand international-flight-attendant ToD-planning problems have additional com-plexities that cannot be met using the ba-sic ToD-planning model. We haveextended the basic ToD-planning model tohandle the additional complexities forthese different crew types, resulting in su-perior solutions to those from the basicToD-planning model. Air New Zealandforms and solves the ToD-planning modelindependently for each crew type and theflights they can operate.The Rostering Model

The rostering problem is to construct aLoW for each crew member in a rank so

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that each ToD is covered by the correctnumber of crew members from that rank.For each crew member, we can generate aset of many LoWs from which exactly onemust be chosen.

The rostering problem can also be mod-eled mathematically using a generalizedversion of the set-partitioning model. As-suming there are p crew members and tToDs, the model is naturally partitionedinto a set of p crew constraints, one foreach crew member in the rank, and a setof t ToD constraints corresponding to eachToD that must be covered. The variablesof the problem can also be partitioned tocorrespond to the feasible LoWs for eachindividual crew member. The A matrix ofthe rostering set partitioning model is a 0–1 matrix partitioned as

C C C . . . . . C1 2 3 pA � � �L L L . . . . . L1 2 3 p

and Ci � eieT is a (p � ni) matrix with ei

the ith unit vector and eT � (1,1,. . ., 1).The ni LoWs for crew member i form thecolumns of the (t � ni) matrix Li with ele-ments ljk defined as ljk � 1 if the kth LoWfor crew member i covers the jth ToD andljk � 0 otherwise. The A matrix has totaldimensions of m � ni where m � p �p�i�1t. The right-hand-side vector b is given bybi � 1, i � 1,. . ., p and bp�i � ri, i � 1,. . ., twhere ri is the number of crew membersrequired to cover the ith ToD. We refer tothe first p constraints as the crew con-straints, and the next t constraints as theToD constraints.

The cost vector c reflects the cost of eachLoW relative to all others. Since most air-lines do not use optimization systems forrostering, there is no obvious or tradi-

tional measure that can be used to dis-criminate among feasible solutions in anoptimization. Typically the rostering objec-tive reflects either the preferential biddingby seniority (PBS) or the equitable roster-ing philosophy. We have defined differentrostering objectives for each rostering sys-tem we have developed through consulta-tion with representatives of the crewgroups.

The rostering model has a special struc-ture that deviates from pure set partition-ing in that the right-hand-side-vector isnot unit-valued and some constraints neednot be equalities. The crew constraints ofthe A matrix also exhibit a generalizedupper-bounded structure which is notcommonly found in set partitioning.

Each column or LoW for a crew mem-ber must correspond to a feasible and le-gal sequence of ToDs that satisfies therules specified in civil aviation regulations,employment contracts, and agreements.These rules or constraints can be thoughtof as being implicitly satisfied in thevariable-generation process rather than ex-plicitly satisfied in the rostering model.

We have further enhanced the basic ros-tering model by adding additional con-straints to handle complicated qualifica-tion requirements within crew ranks. Forexample, at least a certain number ofhighly qualified people may be requiredon certain ToDs, or no more than twonewly trained crew members are allowedon the same ToD. Air New Zealand formsand solves rostering models indepen-dently for each crew base and rank.Solution of Generalized Set-PartitioningProblems

The generalized set-partitioning models

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arising in both ToD planning and roster-ing are solved using linear programming(LP) relaxation and branch and bound. Wesolved the Air New Zealand ToD planningand rostering models using a special pur-pose SPP solver, ZIP, developed by Ryan[1980] to take advantage of the characteris-tics of the SPP model. Over the course ofthe project, we have enhanced the revisedsimplex method (RSM) implementationwithin ZIP to include both partial-pricingand steepest-edge-pricing techniques[Forrest and Goldfarb 1992; Goldfarb andReid 1977]. The pricing step also includesa dynamic column-generator capabilitybased on resource-constrained shortest-path methods [Minoux 1984; Lavoie,Minoux, and Odier 1988; Desrosiers et al1991; Gamache et al 1999]. The nature ofthe underlying shortest-path network useddepends on the particular application.

We have also implemented constraintbranching [Ryan and Foster 1981] withinthe branch and bound. In contrast to theconventional variable branching withinbranch and bound, which forces a variableto value zero or one, constraint branchingpartitions sets of variables covering a pairof constraints (that is, the set of columnscontaining a one-element in either of therows corresponding to the constraint pair)into a set covering both constraints andthe complementary set covering just oneof the pair of constraints. The pair of con-straints will either be covered together bya single variable from the first set or becovered separately with two variablesfrom the second set. In a fractional solu-tion, it is always possible to find a pair ofconstraints that are covered fractionally byvariables from each set. In each crew-

scheduling application, there is a naturalchoice of constraint pairs to define thebranch.Integer Properties of Set-PartitioningProblems

Some set-partitioning problems are verydifficult to solve, whereas others are easierto solve. The LP relaxation of the ToD-planning and rostering models possess in-teresting structure and properties suggest-ing that they should be easier rather thanharder to solve. It is known that three par-ticular classes of zero-one matrices (uni-modular, balanced, and perfect) define set-partitioning polytopes that have nofractional extreme points. Of these threeclasses, balanced [Berge 1972] and perfect[Padberg 1974] are particularly importantfor understanding why the ToD-planningand rostering set-partitioning problemsare easier rather than harder to solve.

The ToD-planning SPP model generatedwith severely limited subsequence matrix-generation techniques [Ryan and Falkner1988] has the properties of a near-balancedproblem. Limited subsequence is a heuris-tic technique that restricts the choice ofsubsequent activities following any givenactivity. We have observed that modelsgenerated with limited subsequence havefewer variables than the full model but,when solved to optimality, exhibit stronginteger properties in that many of the ba-sic feasible solutions near the optimal so-lution have many variables at integervalue and few at fractional values. The ob-jective function of the ToD planningmodel penalizes the idle time betweensuccessive activities in a ToD and thus isconsistent with the concept of limited-subsequence generation techniques. The

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definition of limited subsequence may bechanged dynamically during the optimiza-tion convergence.

In the rostering SPP model, the crewconstraints are in the form of generalizedupper-bound constraints that have the ef-fect of creating perfect blocks of variables(LoWs) for each crew member. Padberg[1974] has shown that a perfect block ofvariables has no fractional extreme points.This implies that fractional solutions in therostering model can never occur withinthe crew member’s own variables but

A conservative estimate of thesavings is NZ$15,655,000 peryear.

must always occur through two or morecrew members competing for a ToD. Ourchoice of crew versus ToD constraintbranching is based on this fractional-solution structure in the rostering model.Degeneracy and Set-PartitioningProblems

The solution of the LP relaxation ofSPPs is often difficult because near-integerbasic feasible solutions are very degener-ate. Our use of limited-subsequence tech-niques in the generation of the constraintmatrix results in many near-integer basicfeasible solutions being visited during theLP convergence, and basic feasible solu-tions with as many as 80 percent of the ba-sic variables at zero value are common. Atsuch degenerate bases, the RSM tends tostall, sometimes for many thousands ofdegenerate pivots, even when conven-tional techniques, such as maximizing thepivot element, are used to determine theleaving variable. This degeneracy phe-

nomenon in the SPP has been discussedby a number of authors (for example,Albers [1980], Falkner [1988], and Marstenand Shepardson [1981]). To avoid the diffi-culties, Marsten [1974] solved the relaxedLP using a dual algorithm in his SETPARcode. For problems with very large num-bers of variables or when dynamic columngeneration is used, this is not an attractiveoption. Other authors have avoided theproblems of degeneracy by developing al-ternative bounding strategies based on La-grangian relaxation or dual-variableadjustments.

We overcame problems of degeneracyin our solution of the ToD planning androstering models by using steepest-edgepricing [Boyd 1995] and Wolfe’s method[Ryan and Osborne 1988; Wolfe 1963] andby using a carefully chosen right-hand-side perturbation scheme. Wolfe’s methodprovides a guaranteed termination of thestall, but we use it only when we detect asequence of degenerate pivots. The pertur-bation scheme of the rostering model con-sists of setting bi � 1 � e, i � 1,. . ., p,e � 0, thus creating a tension between thecrew constraints of the model and the ToDconstraints that still have integer right-hand sides. For a small value of e, such as10�7, the values of basic variables are per-turbed sufficiently to avoid degeneracy,and in practice, few truly degenerate piv-ots are observed during the convergenceof the RSM. Typically, as few as 100 to 200degenerate pivots occur in RSM conver-gences of more than 10,000 iterations, andWolfe’s method is seldom required.National ToD Planning

In the national ToD-planning system,we construct ToDs for both flight atten-

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dants and pilots. All flights are operatedby B737 aircraft with identical crew re-quirements. We build ToDs separately forflight attendants and pilots to cover be-tween 300 and 900 flights per week de-pending on crew type. ToDs are betweenone and four days in duration. The crewbases are in Auckland, Wellington, andChristchurch. We produce fully dated ToDsolutions because the flight schedule dif-fers from week to week. The major com-plications of the national ToD planningproblem include different pilot and flight-attendant rule sets, multiple home baseswith different numbers and types of ToDsrequired at each home base, crew-base-rank imbalances causing nonunit crewing,a dynamic schedule, and the expectedcombinatorial complexity of the SPPmodel. ToDs must satisfy many contractrules and soft rules, which can be complexin terms of definition or implementation.For example, “A maximum duty time of11 hours is allowed in any rolling 24 hourperiod” is a rule that is conceptually sim-ple but difficult to implement. An exampleof a rule that is both difficult to define anddifficult to implement concerns the ToDmeal-break requirements. In principle,crew members must be provided withmeal breaks every six hours. However, thedefinition of what constitutes a meal breakdiffers for pilots and flight attendants andalso depends on where the meal occurswithin the duty period. Meal breaks canoccur only at certain airports and times ofthe day. Pilots may also be provided withinflight meals on certain flights, and thereis some inconsistency in the exact posi-tioning of meal breaks within a ToD. Thecontract provides only superficial detail,

and so the rules in the system must reflectcomplex interpretations based on customand precedence.

The system generates optimized ToDsfor all national crew types, ranks, andbases. The main objective is to minimizethe total dollar cost, which includes salarycosts based on the number of crew days inthe solution and expenses from operatingthe ToDs. We use a flight-based shortest-path network in the dynamic column gen-erator. The constraint branch used withinbranch and bound is based on consecutiveflight pairs, particularly same-aircraftflight pairs, as this increases operationalrobustness. A typical problem consists ofbetween 600 and 2,000 constraints, withbetween 7,000 and 25,000 variables gener-ated by the end of the optimization.

We developed the original ToD plan-ning system in 1984 and 1985 and imple-mented it as a mainframe computer sys-tem in 1986, replacing a manual process.We believe the system was one of the firstfull optimization-based ToD solvers inproduction in an airline at that time. AirNew Zealand used the system extensivelyafter its introduction in response to the ar-rival of competition (Ansett New Zealand)in the domestic market. Using the ToDsystem, the airline could create new sched-ules and develop its associated crew plansin just two days in response to Ansett ini-tiatives. The system remained in produc-tion essentially in its original form until1997 when we redeveloped it to includeimproved optimization methods and im-plemented it on a Unix workstation. Sig-nificant benefits and improvements result-ing from the optimization system include—higher quality ToD solutions produced

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in shorter time,—manpower savings (reduced ToD crewdays and fewer ToD planning staff),—the ability to analyze flight schedules,—the ability to repair ToD solutions whenthe schedule changes or crew availabilitychanges,—the ability to produce reports for opera-tional and budgeting purposes, and—its use as a management tool in crew-basing studies, what-if scenarios, andaward negotiations.

Features of the system include the abil-ity to force or ban partial ToDs, hot start-ing from any previous solutions, and theautomatic construction of fully dated solu-tions. The national ToD-planning system isfully integrated with Air New Zealand’sschedule-data system and rostering sys-tem, and includes a graphical user inter-face that provides flexible user controls.

Air New Zealand uses the national ToDoptimizer every day to evaluate proposedschedules and every two weeks immedi-ately prior to the beginning of the nationalrostering phase to produce the ToDs cov-ering that 14-day roster period.National Rostering

The national rostering system constructsrosters for both flight attendants and pilotsover a 14-day roster period, resulting in aLoW for each crew member. There are tworanks for flight attendants (pursers andflight attendants) and two ranks for pilots(captains and first officers). The airlinebuilds separate fortnightly rosters for eachcrew type, rank, and base. The largestnational-pilot roster group is Auckland-based captains with 50 pilots, and the larg-est national-flight-attendant roster groupis Auckland-based flight attendants with

85 crew members. These problems coverfew crew members compared to the inter-national problems, but their combinatorialcomplexity makes them the most difficultto solve of all Air New Zealand’s rosteringproblems. Most ToDs last only one or twodays and are fundamentally similar inwork content. This means that each crewmember could be allocated any one of say25 alternative activities (ToDs, day off,training and so forth) on each of the 14days in the LoW. While many of the 2514

possible LoWs will be illegal because theyviolate some rostering rule, an extremelylarge number of alternative legal LoWswill remain valid for each crew member.

Air New Zealand uses a “fair and equi-table” basis for national rostering. Once itachieves the management requirement ofcrewing all ToDs, the main objective is tomaximize crew satisfaction and LoW ro-bustness. During system development, weinvited the flight-attendant-crew union toidentify appropriate crew-satisfactionmeasures, such as the number of days be-tween onerous (or difficult) ToDs, themaximum number of onerous ToDs in aroster period, and the granting of requestsfor days off, early finishes, and late starts.We incorporate robustness within LoWsby inserting additional rest between adja-cent work activities. Some additional com-plications of the national-rostering prob-lems include the complex rule sets, theexistence of part-time and casual flight-attendant staff, and a variety of qualifica-tion requirements.

We build the national rosters in twostages, using a days-off optimization fol-lowed by a ToD assignment optimization.This approach is possible because, for the

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national rostering problems, the dailycrew-resource requirements can be accu-rately calculated prior to roster construc-tion. The initial days-off optimization isnecessary to reduce the combinatorial sizeof the rostering problem.

Initially, the days-off optimization con-siders only the rules that apply to days offto build a legal days-off LoW for eachcrew member. Each days-off LoW containsall of the crew member’s pre-assignmentsplus the required total of five days off. Wemodeled the days-off optimization prob-lem mathematically as a set-partitioningproblem. We perform an a priori genera-tion of all possible days-off LoWs for eachcrew member and then solve the resultingoptimization to find a days-off roster en-suring that the correct number of crewmembers are available to work each day.The objective of this problem takes into ac-count the days-off patterns crew memberspreferred, for example a days-off LoWcontaining two blocks of two days off andone single day off is preferred to a days-off LoW containing one block of two daysoff and three single days off.

We also modeled the ToD assignmentoptimization as a set-partitioning problemwith additional constraints to handle somecomplex qualification requirements. Forexample, at least one of the crew memberson a ToD must have a premium-servicequalification. The solution procedure startswith an “optimal” days-off LoW for eachcrew member but can use an alternatedays-off LoW if this is required to gener-ate a legal LoW. Because of the combinato-rial size of the problem, it is not possibleto build the full fortnightly roster in onestep, so subroster builds are used. A sub-

roster is a practical-sized subsection of theroster (usually five to seven days long)that can be formulated as one tractable set-partitioning problem. For each crew mem-ber, we generate a subset of partial LoWsover the subroster period using a limited-subsequence filtering technique. Limited-subsequence filtering restricts the choice ofsubsequent ToDs that will be consideredeach day. The crew-satisfaction andoperational-robustness measures that con-tribute to the cost for each partial LoWwere chosen in collaboration with crewmembers. The partial LoWs generatedmust be consistent with any previous sub-roster solutions, the days-off LoW, andpre-assignments, such as leave and train-ing. The constraint branch used withinbranch and bound is based on crew-ToDpairs. A typical five-day subroster prob-lem consists of 300 constraints and 100,000variables. If the subroster is feasible (thatis, all ToDs are covered and all crew areassigned a legal partial LoW), we confirmthe subroster solution and consider thenext subroster period. If the subroster isinfeasible, we generate a new set of partialLoWs based on the dual information fromthe LP solution. If the subroster still re-mains infeasible, we unconfirm the lastday of the previous subroster period andstep back one day, removing all ToD allo-cations on that single day. We then con-sider a new subroster period starting onthe unconfirmed day. The cause of the in-feasibility may be related to ToD assign-ments on the day just unconfirmed, so thenew subroster period may now be feasi-ble. This subroster confirm-or-step-backprocedure (Figure 4) continues until theroster is completed or the roster is infeasi-

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ble. We report detailed information aboutan infeasible roster to the roster builder sothat he or she can make an appropriatedecision to remove the infeasibility, for ex-ample, to deny a crew request. Since crewsatisfaction and LoW robustness aremainly subjective measures, the systemaims for high-quality solutions rather thanthe “optimal” solution. Features of thenational-rostering systems include flexibleuser control of the solution procedure anddetailed reports of infeasibilities and rostersolutions.

Two previous attempts [Clarke 1989;Mueller 1985] to solve the national roster-ing problem were unsuccessful, but Dayfinally solved the flight-attendant crew-rostering problem in 1992 using the sub-roster procedure during PhD researchsponsored by Air New Zealand [Day 1996;Day and Ryan 1997]. Air New Zealandimplemented a production system basedon this research in 1993. In 1998, Scottadapted the same solution methodology toproduce pilot rosters under quite differentoperating rules. The two national rostering

Figure 4: In the subroster solution procedure, the step-back process unconfirms the last day ordays of the previous subroster to overcome infeasibility. The remaining subrosters are thenbuilt to complete the whole roster. Cells with dark borders represent confirmed days, or sub-rosters that have been built.

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systems are now fully integrated into theAir New Zealand rostering system. Thenational rostering systems produce rostersof excellent quality for all crew ranks andbases in less than one hour of computa-tion. Two roster builders now constructrosters in one to two days immediatelyprior to roster publication, 10 days in ad-vance of the first day of the roster period.They manually resolve infeasibilitiescaused by pre-assignments and then usemultiple optimization runs to further im-prove roster quality. Using the previousmanual rostering methods, eight rosterbuilders took two weeks to build the na-tional rosters.International Flight Attendant ToDPlanning

The international-flight-attendant ToD-planning system constructs ToDs for aninternational schedule containing approxi-mately 300 flights per week. ToDs canlast up to 15 days and are mostly forAuckland-based crew, although Air NewZealand has recently established a newbase for flight attendants in London. Gen-erally the airline constructs a ToD solutionfor a standard week because of the stabil-ity of the international schedule. Theinternational-flight-attendant ToD-planning problem is particularly difficultin that flight attendants are qualified tooperate on different aircraft types. Thecomplexity arises because the different air-craft types require different numbers ofcrew members. For example, Air NewZealand requires a crew of 14 on the B747-400, eight on the B767-300, and seven onthe B767-200. The airline can make consid-erable savings by having some membersof a crew work a different set of flights

from the other crew members. This isknown as crew-complement splitting. Forexample, a full B747-400 crew may splitup so that some members of the crew fly aB767 flight in their next duty period. Theremaining members of the B747-400 crewcould combine with other crew membersto make up the crew for some other flightor they could fly as passengers to anotherairport to join other crew members. Crewsplitting is further complicated by the exis-tence of four ranks for international flightattendants and the ability to substitutecrew members of higher rank in place ofcrew members of a lower rank, called rankovercover. This crew-splitting complica-tion does not exist for pilots, who are usu-ally qualified to fly just one aircraft type.

We developed a unique optimizationsolver for international-flight-attendantToD planning that automatically incorpo-rates crew splitting within the optimiza-tion process [Wallace 2000]. We are un-aware of any other optimization-basedToD-planning system that incorporatesthis feature. The flight constraints forflights that might be covered by multipleToDs are expanded during the optimiza-tion into multiple constraints to modelcrew splitting. The extra constraints allowalternative complement splits and also en-sure that only one alternative is actuallychosen for each flight. We implementedthis capability using row generation, inwhich flight-complement splits are evalu-ated (and appropriate constraints, or rows,added) during the optimization procedure.We have applied a pricing calculationsimilar to that used in column-generationpricing but involving multiple columns.Ideally the flights that are candidates for

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crew splitting should be carefully chosento minimize the number of splits and toencourage unit crewing. In practice, split-ting is permitted everywhere and incurs apenalty cost.

The main objective is to minimize totaldollar cost, which includes salary costsbased on crew days in the solution and ex-penses from operating the ToDs. The opti-mization system uses a duty-period-basedshortest-path network in the dynamic col-umn generator. The constraint branchused within branch and bound is based onconsecutive flight pairs, usually separatedby a rest period. A typical problem con-sists of between 300 and 800 constraintswith between 5,000 and 25,000 variablesgenerated by the end of the optimization.The ToD planners run the system in an it-erative manner, refining the solutions toachieve a balance between full- and split-complement patterns and cost savings. Weincorporate operational robustness withinToDs by including extra buffer timeswhen crew change aircraft during a ToD.We focus on reducing legal but undesir-able (arduous) flight sequences withinToDs and on distributing work equitablybetween ToDs. This is possible becausemany different ToD patterns exist at ap-proximately the same cost. The airlineuses the international-flight-attendant ToDoptimizer every day to evaluate proposedschedules and every four weeks immedi-ately prior to the beginning of theinternational-flight-attendant-rosteringphase to produce the ToDs covering that28-day roster period.International-Flight-Attendant Rostering

The international-flight-attendant-rostering problem, involving 1,500

Auckland-based flight attendants in fourcrew ranks, is the largest rostering prob-lem solved at Air New Zealand. The prob-lem has three processes.

The first international-flight-attendant-rostering process is to create and assigncall lines. A call line is a sequence of callduties and days off over a 28-day period.A call duty is a period of 12 hours duringwhich a crew member can be called out tofill a vacancy in a ToD. Call lines are allo-cated to crew on a strict rotational basis.Typically, 125 flight attendants will be oncall in each roster period. The sequence ofcall duties and days off within each callline must meet employment contract rules,and the set of all call lines must providethe required number of call duties speci-fied for each day. We have modeled thisproblem mathematically as a small gener-alized set-partitioning problem [Deaker1994], which we solve to generate calllines taking into account varying levels ofcall requirement for each day. We thensolve an assignment problem to allocatethe call lines to crew members due for call,maximizing achievement of crew requests.

The second international-flight-attendant-rostering process involves as-signing flight attendants who speak vari-ous languages (such as German, French,Mandarin, Cantonese, Thai, Japanese, andSamoan) to ToDs requiring those languageskills. We developed a languages-assignment optimization to perform thisprocess [Waite 1995]. The language re-quirements are specified in terms of thenumber of speakers of each language onselected flights. About 250 crew membershave language skills, and each roster pe-riod has approximately 700 language re-

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quirements. In addition, there are severalgrades of language skills (ranging fromgrade 1 for beginners to grade 5 for fluentspeakers), and some crew members mayspeak more than one of the languages. Thelanguage requirements can be complex,for example, “One person who speaksboth Mandarin and Cantonese at grade 5level, plus two people who can be eitherMandarin or Cantonese speakers so longas at least one of these is a grade 4 or bet-ter.” The basic problem is to build partialLoWs for language-skilled crew to coverthe language requirements. Language re-quirements are flight based and indepen-dent of crew rank so crews of differentranks and on different ToDs can be com-bined to cover the language requirementon a given flight. Since Air New Zealanddoes not have enough language-speakingflight attendants to cover all language re-quirements, the main objective is to mini-mize the undercoverage of language re-quirements. All language requirements aresolved as a single optimization problem.Crew members without language skillsmust also be considered because denyingtheir requested ToD pre-assignments mayimprove overall coverage of languagerequirements.

We modeled the problem mathemati-cally as a generalized set-partitioningproblem, containing crew constraints, ToDcomplement constraints, and language-requirement flight constraints. It incorpo-rates a penalty for undercoverage, and it isformulated to spread any undercoverageacross lower-priority flights. We includedother penalties that take into account crewrequest satisfaction and ToD variety in aLoW. The language-assignment optimiza-

tion system uses a ToD-based shortest-path network in the dynamic column gen-erator. The constraint branch used withinbranch and bound is based on crew-ToDpairs taking into account language re-quirements. A typical problem consists of4,000 constraints with 25,000 variablesgenerated by the end of the optimization.The solver can also be used to roster anyother flight-based skill requirements (forexample, service-level skills). Compared toprevious manual methods for language as-signment, the model can solve for morelanguages and meets more of the languagerequirements while automatically consid-ering crew requests. The previous manualprocess took two people several days,whereas one person can complete the pro-cess in several hours using the optimiza-tion system. This aspect of flight-attendantrostering provides Air New Zealand withimportant commercial benefits becausemany of its passengers, particularly thosefrom Asia and Europe, do not speak En-glish. By providing language-qualifiedcrew members on relevant flights, the ros-tering system supports Air New Zealand’sfocus on quality customer service. The sys-tem is fully integrated with the rosteringoptimization. Language ToD assignmentsare treated as pre-assignments in the mainrostering-optimization procedure.

The final international-flight-attendant-rostering process is the allocation of ToDsto form a LoW for each crew member. TheToDs can last one to 15 days, and typicallyeach crew member will be allocated threeor four ToDs over the 28-day roster pe-riod. The overall problem is therefore lesscombinatorial than the national-rosteringproblem. However, many more crew

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members are on the international roster.International flight attendants can requesta limited number of specific ToDs anddays off. The airline considers all of thecrew members in each rank to be equaland rosters them on a fair-and-equitablebasis.

This means that over time, all crewmembers should have approximately thesame levels of satisfaction of requestedToDs and days off and approximately thesame frequency of ToDs visiting desirableand undesirable ports. We use historicalstatistics for each crew member to deter-mine roster quality. A statistic is a recordof the number of times a crew member hasbeen assigned a particular activity and thedate of the last assignment. The statisticsrecorded for crew members include satis-faction of requested days off, ToD destina-tions, and ToDs that last more than 10days. The statistical rules are divided intotwo groups: hard statistical rules thatmust be enforced and soft statistical rulesthat can be violated with a penalty. Themain objective of the rostering system is tomaximize overall crew satisfaction basedon statistics. We also apply other soft rulesnot related to statistics, such as forcing anonerous through-the-night ToD to providea day off immediately before and after.The airline builds a 28-day roster for eachrank independently. There are two crewbases and four crew ranks, with the larg-est crew rank consisting of approximately550 crew members. Some additionalcomplications of the international-flight-attendant-rostering problem includecomplex rule sets and matching LoWs forspouses within the crew rank. Theinternational-flight-attendant-rostering op-

timization system uses a ToD-basedshortest-path network in the dynamic col-umn generator. The column generatordoes not generate LoWs that break hardstatistical rules and assigns a cost to eachLoW based on the violation of soft statisti-cal rules. The constraint-branch usedwithin branch-and-bound is based oncrew-ToD pairs. A typical problem con-sists of 1,100 constraints with 35,000 vari-ables generated by the end of the optimi-zation. Roster builders intervene to resolveroster infeasibility by altering pre-assignments or relaxing hard statisticalrules.

Air New Zealand implemented theoriginal international-flight-attendant-rostering system based on a priori genera-tion as a mainframe computer system in1989 [Ryan 1992]. We believe that this wasthe first optimization-based rostering sys-tem used by any airline. At the time of itsoriginal implementation, the optimized so-lution demonstrated that it was possible toconstruct rosters with a five percent reduc-tion in the number of flight attendants(approximately 30 fewer in a rank of 650)and, at the same time, significantly im-prove the quality of the rosters from acrew point of view. The reduction in crewnumbers occurred over time through attri-tion, and during this time, Air New Zea-land increased scheduled flying withouthiring additional crew members. We re-vised the system to incorporate column-generation methods in 1996, which re-sulted in a further two-percent savings(approximately 20 fewer crew members)and further improved achievement of re-quested ToDs and days off. The airlineachieved these overall savings by increas-

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ing crew productivity, reflected in the re-duction from an average 13.6 days off perroster in 1988 to an average of 10.4 daysoff in 2000. The legal minimum-days-offrequirement is 10 days off per roster. Be-cause extra days off are allocated to crewon call, we believe an average of 10.4 daysoff is very close to the achievable mini-mum. In developing and implementingthis system, we consulted representativesof the flight-attendant union, who definedthe roster quality measures that we in-corporated in the optimization. Theinternational-rostering optimizer is fullyintegrated into the Air New Zealand ros-tering system. It produces rosters in twoto six hours, depending on crew rank. Theairline runs the rostering optimization asclose to roster publication as possible, andit publishes rosters 10 days in advance ofthe first day of the 28-day roster period.International-Pilot ToD Planning

The international-pilot ToD-planningsystem constructs ToDs for Auckland-based B767 and B747 pilots. Each weekAir New Zealand operates about 50 B747flights and 250 B767 flights. ToDs can lastup to 14 days. Because of the reasonablystable nature of the international flightschedule, the airline can usually buildToDs for a standard week. That is, the air-line builds ToDs for one week and thenreplicates the one-week solution to createToDs for an entire roster (four weeks).When schedules vary, the airline mustbuild fully dated solutions. The standardcrew complement is two pilots (one cap-tain and one first officer). The major com-plication of the international-pilot ToD-planning problem is the potential use ofaugmented crew (or third pilot) ToDs,

which allow the airline to extend duty pe-riods by including extra pilot(s) on someflights. For example, the operation of twoflights separately may require two pilotsper flight, whereas the operation of bothflights within the same duty period mayrequire three pilots because of the longerduty time. The airline must then evaluatewhether it is better for the solution to con-tain two short duty periods, each requir-ing two pilots, or one long duty period re-quiring three pilots. Also, some pilot rulesconcern acclimatization in time zones,meaning that the number of pilots re-quired depends on the content of the en-tire ToD, not just the duty period. Becauseof the nature of the flight network, the so-lution to the pilot-ToD-planning problemfor Air New Zealand often requires three-and four-pilot crews. This problem is re-ferred to as the crew-augmentationproblem.

We have developed a ToD planningsolver that automatically constructsaugmented-crew ToDs within the optimi-zation. We believe this feature is unique;we understand that the ToD planning sys-tems used by other airlines constructaugmented-crew ToDs in a subsequentstep. We handled the augmented-crewToDs by incorporating additional con-straints in the optimization model. Theseconstraints ensure that when a ToD re-quiring additional pilots is in the solution,other ToDs will also be included in the so-lution that provide extra crew for the re-quired duty periods. The main objective isto minimize total dollar cost, which in-cludes salary costs based on crew days inthe solution and expenses from operatingthe ToDs. The optimization system uses a

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combination of a priori and dynamic col-umn generation. The dynamic column-generation subproblems construct legalToDs using a duty-period-based shortest-path network. The constraint branch usedwithin branch and bound is based on con-secutive flight pairs. A typical problemconsists of between 140 and 1,100 con-straints with between 60,000 and 275,000variables generated by the end of theoptimization. Additional features of thissystem include the capability to rank-overcover a crew requirement by allowingcaptains to operate as first officers, theability to force or ban duty periods andrest periods, the automatic creation offully dated solutions, and the ability tolock or confirm any partial ToD solutionsand reoptimize the remaining flights.

Air New Zealand implemented theinternational-pilot-ToD-planning system in1996 [Goldie 1996]. The system providessignificant benefits and improvementsover the previous manual procedure, in-cluding an estimated three-percent costsavings per year. The international-pilot-ToD-planning system is fully integratedwith Air New Zealand’s schedule datasystem and rostering system and includesa graphical user interface. The airline usesthe international-pilot-ToD optimizerevery day to evaluate proposed schedulesand every four weeks immediately priorto the beginning of the rostering phase toproduce the ToDs covering that 28-dayroster period.International-Pilot Rostering

Many airlines worldwide roster interna-tional pilot crews using preferential-bidding-by-seniority systems (PBS) thatare based on greedy-sequential-heuristic

roster-construction methods. In PBS sys-tems, crew members bid for work anddays off. The airline then constructs ros-ters by satisfying as many bids as possiblebut considering crew members strictly inseniority order within each crew rank. Theinternational-pilot-rostering system at AirNew Zealand constructs LoWs forAuckland-based B747 and B767 pilotgroups. Each crew group has three ranks:captains, first officers, and second officers.The system builds a 28-day roster for eachcrew rank and fleet. The airline has about80 B747 captains, 80 B747 first officers, 40B747 second officers, 110 B767 captains,115 B767 first officers, and 25 B767 secondofficers. The ToDs can last up to 14 days,so the overall problem is less combinato-rial than the national-rostering problem.However, its major complication is that itmust satisfy the maximum number of bidsin strict seniority order. That is, a solutionin which the most senior pilot achieves allof his or her bids and all other pilotsachieve no bids is preferred over one inwhich the most senior pilot achieves 90percent of his or her bids and all other pi-lots achieve all of their bids. Crew-member bids include requests for specificor generic ToDs or days off, bids to avoidcertain ToDs and bids for work-patterncharacteristics, such as a lot of work, ordays off grouped together. Crew memberscan make any number of bids and canspecify the relative importance of eachbid. The system satisfies these bids increw-member-seniority order, with theproviso that all of the ToDs must be as-signed and every crew member must havea legal LoW. Some additional complica-tions of the international-pilot-rostering

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problem include qualification and trainingrequirements.

Between 1992 and 1994, Thornley [1995]developed a new optimization model andsolution method for PBS. The solutionmethod incorporates a unique squeezeprocedure that violates the bids of morejunior crew members to satisfy the bids ofmore senior crew members. This guaran-tees that the airline can satisfy the maxi-mum number of bids in seniority orderand assign all ToDs. Other airlines useheuristic methods that cannot providesuch a guarantee. Crews negotiated thePBS objective as part of a crew contract.Unlike the national rostering system, theinternational rostering system cannot as-sign days off independently of the ToDs.The international-pilot-rostering optimiza-tion uses a ToD-based shortest-path net-work to generate legal LoWs for each crewmember. It gives each LoW a cost basedon bid satisfaction. The constraint branchused within branch and bound is based oncrew-ToD pairs. A typical problem con-sists of up to 500 constraints with 25,000variables generated by the end of the opti-mization. Users must intervene to resolveroster infeasibilities, that is, when no LoWexists for a crew member, or when no fea-sible roster exists with all ToDs assigned,ignoring all bids. The squeeze procedureguarantees satisfaction of bids in strict-seniority order by solving a series of LPproblems that eliminate all LoWs that donot provide the highest bid satisfaction forthe current crew member being consid-ered. If this causes the roster to become in-feasible, it makes the LoWs that providethe next-highest bid satisfaction for thecurrent crew member available until feasi-

bility is restored. When the current crewmember has a fractional LoW coverage atdifferent bid-satisfaction levels in a solu-tion, this implies integer infeasibility andthe system applies a back-up procedure. Itsequentially moves back to more-seniorcrew members and allows LoWs withlower bid satisfaction for them into the so-lution until the LoWs in the solution forthe current crew member all have identicalbid satisfaction. The squeeze procedurethen recommences, starting from themore-senior crew member for whom weadded the lower-bid-satisfaction LoWs.

The system implements a difficult PBScontract requirement and has providedhigh bid satisfaction for senior pilots. Re-cently we introduced soft rules to the sys-tem to improve the quality of days off al-located to all crew members prior to thebid-satisfaction squeeze procedure. Wehave also made a wider variety of bidchoices available to crew members.

Air New Zealand introduced the systemas part of a major cultural change it initi-ated in its employment contracts in 1991.Prior to the change, the airline builtinternational-pilot rosters manually usingfair-and-equitable assignment rosteringand with salary-based pay. The rosteringrules were very restrictive and were basedon historic practice. The new contractspecified that the airline would build ros-ters using an automated preferential-bidding-rostering system and would basepay almost exclusively on flight hours (in-centive pay). The more hours a pilotworked, the more he or she would bepaid. While PBS itself does not provideany direct savings, an acceptable PBS ros-tering system was a prerequisite for relax-

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ing the restrictive rostering rules. Air NewZealand’s successful implementation ofPBS between 1994 and 1997 enabled it tobuild rosters with 30 fewer internationalpilots. In addition, the more-restrictivehistoric-practice rostering rules requiredAir New Zealand to maintain a crew baseof about 30 pilots in Los Angeles to oper-ate flights to Europe. The airline no longerneeded this crew base once the rules wereremoved, leading to additional savings.The PBS system is fully integrated into theAir New Zealand rostering system. Ittakes about two days to produce rosters.The airline runs the rostering optimizationas close to roster publication as possible,and it publishes rosters 10 days in ad-vance of the first day of the 28-day rosterperiod.Implementation

Between 1986 and 1999, the aircrew-scheduling project developed eightoptimization-based systems to solve all as-pects of the planning and the rosteringprocesses for Air New Zealand’s nationaland international airlines (Table 1). Ini-tially the airline incorporated these sys-tems into its existing mainframe environ-ments. It has since integrated them intothe Sabre AirCrews database environment.

A major factor contributing to the suc-

cess of the crew-scheduling project at AirNew Zealand has been the close collabora-tion between the OR team and the groupsaffected by the introduction of the sys-tems: the scheduling staff, managers, andcrews.—Our collaboration with the schedulingstaff provided us with a comprehensiveunderstanding of all aspects of the ToD-planning and rostering problems. Also thescheduling staff gained confidence in theoptimization technology as we developedthe systems. This two-way communicationhelped us to develop customized systemsthat have a natural fit, both in supportingAir New Zealand’s business practices anddelivering high-quality solutions.—One Air New Zealand manager spon-sored the first student project in 1984 andhas continued to support development ofthe crewing optimizers. Since 1984, man-agers in many areas of the company haverecognized the power and benefits of theoptimization methods. They have also ac-cepted that considerable research and de-velopment are necessary before successfulimplementation of a production systemcan be considered. Also management sup-port has been ongoing because each sys-tem we have developed and implementedhas delivered more than was expected.

Pilots Flight Attendants

ToD Planning Rostering ToD Planning Rostering

National 1986Revised 1997

1998 1986Revised 1997

1993

International 1996 1994 1998 1989revised 1996

Table 1: Air New Zealand’s aircrew-scheduling project developed eight optimization systemsbetween 1986 and 1999 for tours-of-duty (ToD) planning and for rostering for both national andinternational pilots and flight attendants.

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—To gain acceptance of the systems, wehad to provide crew members, who are di-rectly affected by the solutions produced,with some understanding of optimizationand reasons to trust the solution technol-ogy. We maintained ongoing communica-tion with representatives of the crewgroups. “The accessibility of OR staff andtheir readiness to explain processes anddiscuss problems proved to be a key factorin the successful implementation of thesystem, and it is this relationship that willensure its continued success”, said Cap-tain David Allard, flight instructor on theB767 fleet and scheduling committeemember [Allard 2000].

A characteristic of optimization-basedmethods is that they produce solutionsthat exploit the rules. Air New Zealand’sintroduction of optimized ToD-planningand rostering systems has resulted in in-creased crew productivity, while still pro-viding very-high-quality solutions. Suchsystems can lead to solutions that are legalbut which contain undesirable features. Itis therefore important that solutions notonly be optimal from a financial point ofview but also crew friendly and safe. Weachieved this by collaborating with thecrew groups to identify additional softrules. This is of fundamental importancein maintaining the integrity of the crew-ing optimizers in a production environ-ment. Management and OR staff continueto work with the crew’s contract-management groups and unions to insti-tute appropriate soft rules to addresscrews’ quality issues. We can relax the softrules when we cannot find a feasible solu-tion, so they have no impact on crew pro-ductivity yet have a huge positive benefit

on solution quality from the crew’s per-spective. Air New Zealand has also collab-orated with NASA in pioneering work oncrew-fatigue measurement. We have incor-porated the results of these studies intothe ToD planning and rostering optimizersas additional constraints and rules.Impacts

The crew-scheduling optimizers providefinancial benefits to Air New Zealand bydirectly reducing the number of hotel-bednights, meals, and other expenses for crewaway overseas and by reducing the totalnumber of crew members required overall.Each optimization application has also re-duced the costs of constructing and main-taining the crewing solution for the flightschedule. Over the past 10 years, Air NewZealand’s aircraft fleet and route structurehave increased significantly in size yet thenumber of people needed to solve thecrew-scheduling problem dropped from27 in 1987 to 15 in 2000.

A conservative estimate of the savingsfrom the crew-scheduling optimizers isNZ$15,655,000 per year.

We have measured savings in relationto the original solutions produced prior toimplementation of the optimizations. Dur-ing implementation, we benchmarkedeach optimizer against the previous solu-tion methods (in most cases, manual) andidentified the cost benefits. We have ob-served that savings were made even be-fore the systems were fully implemented,because the users incorporated beneficialfeatures of the initial optimized solutionsthey were evaluating into their manual so-lutions. Air New Zealand has audited theprojects internally to validate the savingsclaimed for each system.

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While the estimated savings areNZ$15.6 million per year, the estimateddevelopment costs over 15 years were ap-proximately NZ$2 million. In 1999, thesavings represents 11 percent of the netoperating profit of NZ$133.2 million forthe Air New Zealand group (excludingone-off adjustments).

In addition to the direct dollar savings,the optimization systems provide manyintangible benefits:(1) Using the crewing optimizers, schedu-lers produce high quality solutions in min-utes; manual solutions took them two ormore days. For example, a B767 pilot ToDproblem can be optimally solved in ap-proximately 60 minutes, while the B747-400 pilot ToD problem can be solved inless than five minutes on a Unix worksta-tion. Roster builders now solve theinternational-flight-attendant-rosteringproblem for 550 crew members in onerank in less than six hours.(2) Crew schedulers now function as ana-lysts, preparing and validating data, andinterpreting and evaluating solutions.(3) The airline no longer depends on asmall number of highly skilled schedulerswith intimate knowledge of employmentcontracts and scheduling rules. These con-tracts and rules are now embedded in thecrewing optimizers.(4) Managers now use the crewing optim-izers to investigate strategic decisions forcrewing, for example, evaluating the costsof proposed rule changes or determiningideal crew numbers at crew bases.(5) With short build times, schedulers caneasily produce solutions close to the dayof operation, efficiently accommodatinglate schedule changes.

(6) The schedule planning group can ob-tain accurate, reliable feedback on crewcosts for proposed schedules, leading tothe development of more profitable andmore robust flight schedules. This infor-mation was formerly difficult to obtain be-cause manual ToD solutions were so timeconsuming.(7) Prior to 1986, Air New Zealand oper-ated fixed six-month winter and summerflight schedules. Now it operates flexibleflight schedules that can vary from weekto week, allowing quick response to mar-ket opportunities.(8) Air New Zealand can now repair solu-tions quickly to accommodate smallchanges in the schedule.(9) The rostering optimizers reflect crew-defined quality measures and soft rules.(10) The rostering optimizers satisfy over80 percent of all legal requests from inter-national flight attendants for ToDs anddays off and an even greater percentage ofrequests from national pilots and flightattendants.(11) The rostering optimizers can accu-rately identify and minimize roster infeasi-bility and can also identify days on whichthere are insufficient numbers of crew.(12) The international-flight-attendant-languages optimization has improved pas-senger service for which Air New Zealandis renowned. Air New Zealand has re-ceived the “Globe Award for the Best Air-line to the Pacific” from the top BritishTravel Weekly in 2000, 1999, 1997 and1996. The AB Road Airline Survey alsoranked it first for inflight service in Au-gust 1999.(13) The crewing optimizers guaranteeand demonstrate that the airline satisfies

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legislative and contractual rules. Manualsystems could not guarantee compliancewithout complicated and time-consumingchecking.

At the start of this project, Air New Zea-land employed no OR analysts. Now AirNew Zealand employs six permanent ORprofessionals in crew scheduling and con-tracts with up to five OR analysts for vari-ous projects and to provide support andmaintenance. The OR staff now focuses onexploiting the what-if potential of the ORtools, concentrating initially on crewing-related issues. However, management be-lieves that the OR team, in collaborationwith the University of Auckland and ORconsultants, will make its most valuablecontribution by further developing the un-derlying tools and techniques and apply-ing them in other areas of the company.

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