Generating, scheduling and rostering of shift crew-duties: Applications at the Hong Kong International Airport

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Abstract

In the context of manpower planning, goal programming (GP) is extremely useful for generating shift duties of fixed length. A fixed-length duty consists of a fixed number of contiguous hours of work in a day, with a meal/rest break somewhere preferably around the middle of these working hours. It is such properties that enable the straightforward, yet flexible GP modeling. We propose GP models for an integrated problem of crew duties assignment, for baggage services section staff at the Hong Kong International Airport. The problem is solved via decomposition into its duties generating phase—a GP planner, followed by its GP scheduling and rostering phase. The results can be adopted as a good crew schedule in the sense that it is both feasible, satisfying various work conditions, and “optimal” in minimizing idle shifts.

Introduction

This paper advocates a general modeling framework for a complete crew assignment system. It arises naturally as a mathematical description for the staff deployment problem of their baggage handling agents at BSS-HAS, the Baggage Services Section of the Hong Kong Airport Services, Ltd. HAS of the Hong Kong International Airport at Chak Lap Kok of Lantau Island is the primary handler of all ground services and support functions, including aircrafts and passengers alike.

Our project of optimization modeling for staffing is motivated by the need to produce daily work plan of the baggage service agents at the passenger terminal. The complete BSS crew system consists of its three component GP models: the Duties Generation Problem (DGP), the Crew Scheduling Problem (CSP) and the Crew Rostering Problem (CRP). While such modeling may well be regarded as one among the vast literature of the commonly known area of workforce planning/scheduling (a very comprehensive review is given by Bodin et al. (1983)) and of timetabling and rostering (an excellent feature EJOR issue vol. 153(1) is edited recently by Burke and Petrovic (2004)), our decomposition approach has, for the actual case study, exhibited its significant impact albeit its modeling simplicity. The resulting preemptive (or non-preemptive) GP formulations have very satisfactorily addressed the planning/scheduling/rostering issues to handle frequent changes of flight schedules by flexibility in work patterns of agent duties (Yuen, 2000). Besides an illustration with this project’s sample results, randomly generated numerical problem instances are reported to give evidence of the DGP model’s robustness. To address further the integer computational issues of DGP, results on a comparative analysis between the GP and a competing heuristics are also included.

We give below a review of the literature on a complete crew assignment system, which consists of (often cyclical) scheduling and rostering stages as well, following the core shift duties generation (or the planning stage) of the overall system.

In the general area of routing and scheduling of vehicle and crew (Bodin et al., 1983), it is common to separate the overall problem into two steps consisting of the determination of the time tables—vehicle routing, followed by the staff assignment—crew scheduling.

Various useful models for Crew Scheduling Problem (CSP) aiming at differing merits and purposes have been proposed, such as (matching based) heuristics models of Ball et al. (1983); network models of Carraresi and Gallo (1984); and set partitioning models of Falkner and Ryan (1987). Among the mathematical programming approaches, there are work of Lessard et al. (1981); column generation approach of Desrochers and Soumis, 1989, Desrochers et al., 1992; integer programming approach of Ryan and Foster, 1981, Ryan and Falkner, 1987; decomposition approaches of Patrikalakis and Xerocostas, 1992, Vance et al., 1997, Caprara et al., 2003; and complementary approaches of Wren et al. (1985).

These quoted above constitute only a fraction of the vast literature, not to mention techniques of implementation for practical applications, notably computerized scheduling such as the various reported systems of “HASTUS” of Lessard et al. (1981); “CREW-OPT” by Desrochers et al. (1992); “EXPRESS” by Falkner and Ryan (1992); and that of Chu and Chan (1998) for various rail networks.

Successful real applications are extremely significant for the airlines. Besides the “household name” of SABRE, we mention two more recent “milestone” works of Vance et al., 1997, Mason et al., 1998.

The outcome of the crew scheduling phase is typically a set of daily staff assignments required to cover the (actual or forecast) demand. “In the (next) crew rostering phase, a set of working rosters is constructed that determine the sequence of duties that each single crew has to perform …, to cover everyday all the duties selected in the first phase” (quoted from Caprara et al., 1998). This has been referred to as the Crew Rostering Problem (CRP) by Caprara et al. in their FARO prize winning work for the Italian Railway Ferrovie dello Stato SpA, jointly sponsored by the Italian Operational Research Society during 1994–1995.

Similar to the case of crew scheduling, past work on CRP has seen numerous approaches and applications. There are optimization approaches such as that of Gamache and Soumis (1998); network model of Balakrishnan and Wong (1990); and column generation approach of Gamache et al. (1994). Novel heuristics approaches integrating set-covering and/or assignment/transportation problem are reported by Hagberg, 1985, Carpaneto and Toth, 1987, Caprara et al., 1999, Caprara et al., 2003. Recently, Valouxis and Housos (2002) proposed a quick heuristics for combined bus and driver scheduling, with the techniques of minimum cost matching, set partitioning and shortest path; whereas Chu and Yuen (2003) discussed a decomposed approach.

Most recently, the February 2004 feature issue vol. 153(1) of EJOR mentioned above provides a comprehensive review of the areas of “Timetabling and Rostering”. Significant scientific interests are evidenced by the success of the EURO Working group on Automated Timetabling (WATT) and the international series of conferences on the Practice and Theory of Automated Timetabling (PATAT). A dozen or so papers in this special issue report on a wide range of rostering applications, with an editorial by Burke and Petrovic (2004). Examples include review paper of staff scheduling and rostering by Ernst et al. (2004); nurse rostering problem by Bellanti et al. (2004); local search for shift design by Musliu et al. (2004); timetabling for sport leagues by Schönberger et al. (2004); university course timetabling system of Dimopoulou and Miliotis (2004); and a case study of single shift planning and scheduling by Azmat and Widmer (2004).

The distinctive feature of a fixed-length duty implies that the decision variables can be defined in terms of its starting time and the time interval for break. It is therefore natural to find the usage of set-covering models a popular approach, especially in applications whenever the number of different duties can be made relatively small. These effective efforts have been well illustrated by authors such as Caprara and others mentioned above. However, when advantages can be taken of special problem structures, more “compact” implicit integer-programming formulations (in the terminology of Brusco and Jacobs, 2000) often lead to substantial saving in computational times. The issue of contrasting this with a “usual” set-covering modeling for break and start-time flexibility is well addressed by Brusco and Jacobs (2000). This message is echoed by Caprara et al. (2003) in their recent decomposed approach into Phase 1: set-covering, and Phase 2: heuristic transportation problems.

We have independently been adopting such a decomposition approach exploiting the special problem structure (of fixed-length duties) such that the modeling formulation of DGP that we put forth here can be interpreted as the basic core—the planner—of a more sophisticated DGP/CSP/CRP integrated model in the following sense. DGP in its simplest form (computes and) allocates duties (of given fixed structure of work pattern, rather than crew or staff needing further varying requirements of scheduling) to cover known demands. Demands are given, for equally spaced (such as hourly or half-hourly) time intervals of (the working time of) a day. As such, DGP is the prerequisite to CSP and CRP in that it provides the planning inputs needed in subsequent scheduling and rostering of staff. The logical flow of their relationships can be summarized below, where the indices t = hour of day, j = day of week, p = weekly work pattern and a = agent,HourlyDemandsD(t)Planner:DGPModelDutiesX(i,k)AllocationsR(j,t)DailyRequirementsR(j)=tR(j,t)Scheduler:CSPModelDailyStaffingS(j)AllocationsR(j,t)Roster:CRPModelDutyRostersI(p,a,j,t)The flow chart indicates how the three stages are sequentially related. Hourly demands (D(t)) are fed as input to a planning model: the DGP. Shift duties of fixed length are generated in terms of the numbers of agents (X(i, k)) of start/break work pattern (i, k). These numbers aggregate into hourly allocations (R(j, t)) and then daily allocations (R(j)). The (7-day) outputs of this First Stage are the inputs to a scheduler model: the CSP. It takes the form of a simple cyclic weekly staffing model, producing daily staffing requirements (S(j)). Finally, these Second Stage outputs form (part of) the inputs to a roster model: the CRP. It is an integer linear programming formulation that determines the indicator variables (I(p, a, j, t)) which represent the detailed rostering decisions. Even though there is more emphasis of this paper on the First Stage—the more sophisticated and computationally challenging DGP, numerical results of the last two CSP/CRP stages are shown as an illustration for the whole of the staff assignment system. Since the combined problem is too complex, both in modeling and computation, such stage-wise decompositions for real applications are often the norm (Patrikalakis and Xerocostas, 1992, Vance et al., 1997, Caprara et al., 2003).

Section snippets

Goal programming models

As its name implies, DGP allocates duties (performed by crew) in an optimal way to meet known demand over a contiguous number of time intervals. We state its base and extended formulations below. A more detailed account of its first definition of DGP and its application can be found in an earlier paper of Chu (2001).

We use the following common notations for all the subsequent models. Let H be the working time horizon, and h = 1,  , H index the individual intervals (hours or half-hours). Rh denotes

Concluding remarks

The purpose of this paper is to illustrate by way of this DGP/CSP/CRP modeling and computational experience, the advantage of its readily producing significant improvement over existing manual staff assignment. Its usefulness is somehow, in our opinion and experience of actually applying it in real situations (Yuen, 2000), rather highly out of proportion with regard to its modeling simplicity. The core DGP model’s usefulness to the users is also strengthened by its computational robustness, in

Acknowledgements

This work was initiated by the Management of HAS (the Hong Kong Airport Services, Limited) among other operational projects of similar logistics nature we have been conducting at the Hong Kong International Airport. Deep appreciation is expressed for their provision of information and data, as well as numerous useful discussions. The work is partially supported by the HKU Small Project Funding Programme 2003–04: a/c 10205106/06772/25500/323/01. Virtual-BASIC programming and spreadsheets for the

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