Goal programming models and DSS for manpower planning of airport baggage service. The original publication is available at

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1 Title Goal programmig models ad DSS for mapower plaig of airport baggage service Author(s) Chu, SCK; Zhu, M; Zhu, L Citatio Lecture Notes I Ecoomics Ad Mathematical Systems, 2010, v. 634, p Issued Date 2010 URL Rights The origial publicatio is available at

2 Goal Programmig Models ad DSS for Mapower Plaig of Airport Baggage Service Sydey C.K. Chu 1, Miyue Zhu 2 ad Liag Zhu 3 1 Departmet of Mathematics, Uiversity of Hog Kog, Pokfulam Road, Hog Kog, Chia. schu@hku.hk 2 Departmet of Mathematics, Uiversity of Hog Kog, Pokfulam Road, Hog Kog, Chia. zhumiyue@gmail.com 3 Departmet of Mathematics, Fuda Uiversity, Shaghai, Chia. godloveme_zhu@hotmail.com Summary. Goal Programmig (GP) models ad Decisio Support System (DSS) are two powerful tools dealig with mapower plaig problems, ot oly o research level, but also as practical tools for idustrial implemetatio. Goal programmig is ofte useful as a optimizatio modelig techique for geeratig shift-duties of worker schedules. I our project for the baggage service agecy at the Hog Kog Iteratioal Airport, we proposed three model formulatios based o the basic fixed-legth shift duties geeratio model to approach various combiatios of goals of mapower plaig. Such a optimizatio modelig is built upo the essetial foudatio of a detailed data modelig ad its aalysis for all the drivig parameters ad demad/supply iput ecessary for umerical computatios. The data model ad GP model thus form the two itegral compoets of the overall automatio system the DSS, which is a automatic computer based ad user-friedly system to support maagemet o plaig decisios. Key words: Mapower plaig, gaol programmig, decisio support system. 1 Itroductio Goal Programmig (GP) is ofte useful i duties geeratio problems (DGP) dealig with crew schedules (Chu, 2001) (Chu ad Zhu, 2007). May models (Azmat ad Widmer, 2004)(Brusco ad Jacobs, 2000)(Caprara et al., 2003) have bee advocated i this area ad work well whe applied to practical projects (Chu, 2007). See for example, the reviews o methods, models ad applicatios i (Burke ad Petrovic, 2004)(Erst et al., 2004). The decisio support system (DSS), which realizes automatio, brigs mathematical optimizatio methods to a wide busiess use. This

3 2 S.C.K.Chu, M.Zhu ad L.Zhu paper provides a optimizatio approach with its DSS, ecompassig data modelig ad GP modelig for a fixed-legth shift-duties crew plaig problem for Hog Kog Airport Services (HAS) Limited (Chu ad Yue, 2003). HAS is the primary hadler of all groud services ad support fuctios, icludig aircrafts, passegers ad baggage at the Hog Kog Iteratioal Airport. Our project is to set the crew schedule for the baggage service agets BSA6 workers (beig bus drivers for baggage) of HAS. Whe a airplae is due to arrive at the Airport, the cotrol office of HAS Baggage Services will sed BSA6 workers to the apro to pull back the baggage of the passegers to either the baggage hadlig basemet level or hot trasfer area i a umber of trips. Similarly, whe a airplae is ready to depart, all baggage must be set o board. For operatioal purposes, the apro is cosidered as beig divided ito two zoes by their distaces to the basemet. Airplaes are classified ito two kids: wide-body ad arrow-body jets. There exist service stadards of baggage services dealig with each kid of airplae for each zoe, for example, the maximum time to fiish offloadig work, the least stadby time for the last baggage for a departure flight ad so o. Every BSA6 worker has a fixed-legth regular duty of 9.5 hours of daily work iclusive of a oe hour break preferable i the middle ad possibly a limited legth over-time duty immediately before or after the regular duty. This paper cosists of four further sectios. I Sectio 2, we will provide detailed data models which geerate demad data of work ecessary as iput for the proposed goal programmig model. I Sectio 3, we state three versios of multi-objectives goal programmig models as exteded from the basic DGP-GP model (Chu, 2001) to this problem. The, we provide i Sectio 4 some umerical results whe we apply the models to the HAS problem istace, together with some aalysis remarks. I Sectio 5, we itroduce the decisio support system developed for the HAS project as the fial product. Some cocludig remarks are give i the last sectio. 2 Data Modelig Sice the accurate estimatio o data of demad is essetial to the whole GP modelig approach, we put great efforts ito the data modelig step to geerate the data of demad as close as possible to the curret situatio while maitaiig flexibility for future chages i flight schedules or trasportatio facilities. We preset the overall demad patter by a histogram (or bar-chart) of the umber of required workers (or drivers i our problem here) over the operatioal time horizo of a day. This is doe i two steps. First, we set up a system to estimate the idividual demad for each flight o the flight schedule i the day separately; ad secodly, we sum them up over half-hourly time itervals ad make ay ecessary fie adjustmets to esure that the daily overall demad profile thus computed is practically realistic. 2.1 Idividual Demads Give the flight schedule, the first ad most importat thig to do is to estimate the umber of trips eeded for every flight, which is the foudatio for all later calculatios. Havig the umber the trips, we arrage the dispatchig times for drivers which meet the HAS service stadards. After readig a large umber of past records

4 GP Models & DSS for Mapower Plaig of Airport Baggage Service 3 ad commuicatig with HAS supervisory staff, we pick out five major parameters which decide the umber of trips ad their trips workig times of the BSA6 drivers. The five parameters i the descedig order of importace are: a 1(0, 1)-departure or arrival flight; a 2 (0, 1)-aircraft type; a 3 (0, 1, 2, 3)-airlie carrier compay; a 4 (0, 1)- multiple or sigle destiatio/origi city; a 5(0, 1, 2, 3, 4, 5)-ature of the destiatio city for a departure flight / origi city for a arrival flight ( hot or ormal city with may or few flights). Each flight umber gives iput iformatio for the values of a 1, a 2, a 3, a 4, ad the destiatio/origial city immediately; ad these iformatio together geerate a 5. The we get the umber of trips of the flight (f(a)) accordig to these five parameters. The coefficiets of the fuctio are set based o the effect each parameter has, ad o past experiece ad records about the umber of trips. That is, the fuctioal form of f(a) is etirely derived from umerical fittig. Here we give this fuctio as follows: a a 5 = 0, a a 5 = 2, 3 f(a) = a 4 (1) 2.0 a 5 = a a 5 = 5 This method of obtaiig the umber of trips performs quite well whe validatig the computatioal values with the recorded umber of last year s operatios ad therefore we believe that it ca provide reliable umbers dealig with ew flights i the future. Havig the umber of trips of each flight, we set the operatio times for every trip, satisfyig the service stadards uder our pla. For example, the last trip to serve a flight will arrive at the basemet withi the permitted time stadard. 2.2 Overall Demads We divide oe day ito 48 time itervals, such that each iterval has half a hour. For every sigle flight, with the results we get i the idividual demad step, we obtai the umber of trips over each time iterval. Ad we sum up the umber over each iterval geerated by every such flight ad get the whole demad table for oe day s schedule. After that, we make some ad hoc fie adjustmets to the umbers to make the results closer to the real situatio. For example, two earby flight ca sometimes share a trip i peak hours. We have two ways of data processig dealig with the peak hour data which are also adopted i the DSS. Further data value aalysis eables us to omit eight time itervals (from 1:00 am to 4:59 am) with very little staff demad ad put the first two mid-ight itervals (from 00:00 am to 00:59 am) at the ed of the precedig day due to the usual patter of the airport workig hours. The we took the lower evelope ad upper evelope of the umbers over each iterval respectively to tur the computed fractioal values durig the whole process ito itegers. Hece, we have the fial demads we eed for the ext GP models. Sice the airport operates differet weekly flight schedules for witer ad summer, we work out totally 14 sets of daily data through the data modelig step. These are illustrated graphically i Figure 1, beig Moday (Witer) demad.

5 4 S.C.K.Chu, M.Zhu ad L.Zhu Fig. 1. Demad for Moday (Witer). 3 Goal Program Modelig As its ame implies, the Duty Geeratio Problem (DGP) model geerates duties (performed by workers) i a optimal way to meet the kow demad, over a cotiguous umber of time itervals. We describe below for completeess purpose its formulatio relevat for our BAS6 problem settig here. A detailed accout of origial DGP formulatios is give i a earlier paper (Chu, 2001) of oe of the authors. I this project, we have several differet goals: to cotrol the maximum deviatio of supply versus demad, to cotrol the total umber of regular workers required, ad to cotrol the total amout of over-time worker duties. We explore several model formulatios all exteded from the basic fixed-legth shift-duties model to approach various combiatios of goals. 3.1 Model 1 I model 1, we first add a large coefficiet W to the deviatio variable that cotrols the highest umber of over allocated workers i the workig time iterval over the day. The this goal is combied with the goal of miimum total staff cost. Both regular duties ad over-time duties are used ad all demad is covered uder this model. We use the followig otatios. Let J be the workig time horizo, ad j = 1... J idex the idividual hours. R j deotes the demad for iterval j ad d j represets the over allocatio (or over-achievemet deviatio variable i a goal programmig cotext) at iterval j. The legth of a duty is deoted by L. The primary decisio variable x ik is the umber of allocated staffs that start regular duty from iterval i ad breaks at the k th iterval after the start of duty, k = 1... L. Hece for a workig horizo of itervals 1... J, we have for the idex i = S... T. The earliest start iterval S is such that S 1; whereas the latest start iterval T is limited to T J L + 1 (to fiish work at iterval J). Normally, S = 1 as log as R 1 > 0

6 GP Models & DSS for Mapower Plaig of Airport Baggage Service 5 (there is demad for the very first iterval); ad T = J L + 1 wheever R J > 0 (there is demad for the very last iterval). I priciple the meal break could occur aywhere throughout the L time itervals. I practice, there are two agreed times: the earliest start of break (ESB) ad the latest start of break (LSB) times. This also requires aother restrictio o the break time idex from k = 1... L to beig k =ESB to LSB. For our case, the break time is required to be ot earlier tha 1 ad also ot later tha 6 hours after the start of duty. With half-hour time itervals, this traslates ito ESB=3 ad LSB=14. Aother primary decisio variable y m is the umber of allocated staffs who start their over-time duties at the m th iterval ad the duty lasts for itervals. Mathematically, the first model is give below. The weighted goal fuctio is: Mi W D + The costraits are: q LSB T LSB i=s k=esb x ik + j c k x ik + g J +1 m=l+1 y m (2) i=p k j i m=j +1 k j i + 1 y i+l, x i,k, i = 1... I (4) k y m d j = R j, j = 1... J (3) d j D, j = 1... J (5) Here p = max{j L + 1, S}, q = mi{j, T }, ad x ik, y m, d j are o-egative variables. We see that the LHS of the first costrait is the total work cotributio as a fuctio of staff duties cotaiig both regular ad over-time duties. The coefficiets c k represet the usual uit regular pay rates accordig to the break hour. g is the pay ratio of oe hour over-time pay over oe day regular pay. The secod costrait esures that the over-time duties ca oly be performed immediately before or after the regular duties. The sigle variable D of the last costrait records the maximum (over achievemet) deviatio over all time itervals, with weightig parameter W. 3.2 Model 2 I model 2, we set up realistic caps to the total umber of regular workers or overtime duties to get practically achievable plaig results uder differet mapower requiremet cotrols. All demad is covered uder model 2. The goal fuctio is the same as i model 1, but we add either or both of the followig two costraits: x ik c 0 (6) i k y m c 1 (7) m Here c 0 ad c 1 are two user-specified costats. We should otice that if we add a cap to the total over-time duties, we ca always get feasible solutios sice over-time

7 6 S.C.K.Chu, M.Zhu ad L.Zhu duties ca be replaced by regular duties ayway. But if we add a cap to the total regular duties ad the cap is too low, we sometimes caot get feasible solutios. This reveals that the over-time duties are ot as powerful as they seem to be. Usig the over-time duties are actually oly extedig the legth of duties of workers. I our project i HAS, c 0 ad c 1 are set slightly less tha the existig umbers of regular ad over-time duties, respectively. 3.3 Model 3 I the third model, we set realistic caps to the umber of total regular ad overtime duties ad we use a ew kid of staffs the part-time workers. The goals are approached i two steps. I the first step, we set caps to the umber of total regular ad over-time duties ad we allow shortage of mapower, which meas that ot all the demad is covered i step oe. Ad we cotrol the maximum shortage. The system is give as follows: The costraits are: q LSB i=p k j i k j i + 1 x ik + Mi Maxe (8) j m=j +1 y m + e j R j, j = 1... J (9) e j Maxe, j = 1... J (10) x ik c 0, y m c 1 (11) i k m Here, e j idicates the shortage of mapower i the iterval j, ad Maxe is the maximum shortage amog all the operatig itervals of the day. I the secod step, we fill up ay such shortage e j by part-time duties i aticipatio of the part-time workers to be subsequetly made available. The part-time workers are assumed to be of totally flexible work duratio. Thus, i this way, all the demad is also covered uder model 3. I this step, the goal fuctio is the same as i model 1: T LSB J +1 Mi W D + c k x ik + g y m (12) The costraits are: q LSB x ik + i=s k=esb j m=l+1 i=p k j i m=j +1 k j i + 1 y i+l, x i,k, i = 1... I (14) k y m d j + e j = R j, j = 1... J (13) d j D, e j c 2, j = 1... J (15) Here, c 2 is the computed result from step oe.

GP Models & DSS for Mapower Plaig of Airport Baggage Service 7 4 Numerical Results ad Commets 4.1 Numerical Results Here we provide some umerical results whe applyig the model to our problem.

8 GP Models & DSS for Mapower Plaig of Airport Baggage Service 7 4 Numerical Results ad Commets 4.1 Numerical Results Here we provide some umerical results whe applyig the model to our problem. For our problem istace, the umber of workig hours of a day is 20, which meas J = 40. The legth of duty is 9.5 hours, which meas L = 19. The break is restricted to ESB=3 ad LSB=14, as metioed before. We prefer staff if possible takig break close to the middle of the duty, which meas c k is assiged a smaller value whe k is closer to the middle betwee 3 ad 14. g = = 2, 3, 4, 5, 6, which meas that the over-time duties ca last 1 to 3 hours. Havig all these, we code the model i LINGO (Schrage, 2005) to compute the solutio to the optimizatio problem ad we plot the results i chart form ad show the detailed values i table form. They are illustrated as follows: Fig. 2. Schedule for Suday (Summer, Model 1). I Figure 2, the dark bars represet the demad for the time itervals ad the latticed bars represet the over allocated duties. Together, a allocatio is plotted by the stacked dark ad latticed bars. The lie graph is the origial pla used by the airport. I Table 1, the first four rows are related to regular duties ad the last row refers to over-time duties. For example, (530,2.5) 6 meas that there are 6 allocated workers who start their regular duties at 5:30 ad take the break 2.5 hours after the start. We should otice that the over-time duty is immediately after the regular duty, which ca also be see i this table. For example, the 5 duties startig at 21:30 ad lastig for 1.5 hours are therefore actually after their regular duties startig at 12:00. Likewise, the 7 duties startig at 22:30 ad lastig for 1 hour are after the (1+6=) 7 duties startig at 13:00.

9 8 S.C.K.Chu, M.Zhu ad L.Zhu Table 1. Schedule for Suday (Summer, Model 1). (500,4) 4 (530,2.5) 6 (530,5.5) 3 (600,4) 2 (630,3.5) 4 (630,7) 5 (700,2.5) 2 (1200,4.5) 5 (1300,1.5) 1 (1300,5.5) 6 (1330,5) 3 (1330,6) 3 (1430,5.5) 1 (1430,6) 2 (1500,2) 6 (1530,3) 7 ) (2130,1.5) 5 (2230,1) Aalytical Commets ad Compariso of the Models By aalyzig the data ad the computed results, we fid that the demad patters are similar amog the seve days of a week. Each day, there are 3 or 4 peaks at aroud 7:30, 13:00, 15:00 ad 22:30, respectively. I fact, these are the actual peak hours at the airport. Uder the origial pla ow i use by the airport, most of the tasks ca be completed accordig to stadards except for some peak hours, especially i the morig ad i the late eveig. May workers, however, ca be foud idle i most of the other workig hours. This situatio is reflected i Figure 2. The mai objective achieved by our project is to geerate the daily crew schedulig pla which esures that all tasks ca be completed with miimum huma resources. It should ot be used as the detailed roll list by the duty maager sice flights may delay or duties ca be completed faster or slower tha estimated. It should be treated as support for decisio makig o the daily plaig, ad for total umber of staffs i the log ru. Compariso is made amog results from Model 1 ad Model 3, as show i Table 2. Here, X is the umber of regular workers, Y is the umber of over-time workers ad Z is the umber of part-time workers (with a part-time duty lastig for 4 hours). Table 2. Compariso of Model 1 ad Model 3. Model 1 Model 3 Stadard Day X Y X Y Z X Y Mo Tue Wed Thu Fri Say Su Total ) Whe model 3 is used, the X ad Y ca be cotrolled uder the operatig stadards. Here Stadard i the table refers to operatig stadard, or actually the

10 GP Models & DSS for Mapower Plaig of Airport Baggage Service 9 curret practices, which do ot guaratee full demad coverage (at all hours). The umbers of part-time workers are ot sigificat except for Moday. This idicates that the officials of HAS could cosider addig a ew mode of part-time duty to the mapower staffig with a reasoable pay. 5 Decisio Support System Sice the GP model solves the core problem of mapower plaig give the demad ad optimizatio goals, it geerates the mapower supply schedule formig the essetial part of a itegratig decisio support system (DSS) towards actual implemetatio. I this project of baggage service agets (workers) of the Hog Kog Iteratioal Airport, our DSS tool is desiged as a computer ad kowledge based iformatio system to support user-friedly decisio makig activities. It is developed as a PC-based system to geerate crew schedules automatically givig the iput iformatio of flights schedule ad their supportive operatioal parameters. Whe the cliets iput or import the flights schedule, the data model system compoet builds up the daily demad profile (over idividual half-hourly time itervals) of workers automatically, which is also the iput data for the ext GP model. Sice the optimizatio GP system compoet is built upo the essetial foudatio of such a detailed data modelig characterized by its aalysis for all the drivig parameters ad demad/supply iput ecessary for umerical computatios, data model ad GP model thus form the two itegral compoets of the overall automatio system, ad fially the DSS. Here i Figure 3 is a illustrative scree capture of the mai iput page of the DSS. Fig. 3. Mai iput page of the DSS.

11 10 S.C.K.Chu, M.Zhu ad L.Zhu User ca iput or paste the flights schedule to the iput area ad simply click the Make Pla butto to get the detailed ad overall schedule i about 25 secods. The user-friedly spreadsheet based iterface also provides the ultimate flexibility for users to make chages to the origial characteristics of problem iformatio, ad to perform what-if aalysis, o top of its geeric calculatios of the mapower plaig fuctio. 6 Cocludig Remarks Goal Programmig (GP) models ad Decisio Supportig System (DSS) are used as practical tools for implemetatio i our project dealig with mapower plaig problems of HAS at the Hog Kog Iteratioal Airport. The overall automatio DSS has bee desiged with a view of beig wholely computer based ad userfriedly to support maagemet uses for loger-term plaig decisios, rather tha real-time dispatchig actios. The buildig ad usig of the system have sice the start of the project bee a joit effort with the decisio makers, the seior maagemet of the Baggage Services Sectio of HAS. Detailed flight ad workload data were provided ad discussed at workig meetigs at the HAS Airport office. Our cetral GP modelig idea of duties geeratio to cover required workload was also explaied ad uderstood. The reactio to our methodology ad the opiios o our model fidigs were rather favourable. This was partly due to the fact that our umerical results tallied well with their expectatio o possible operatioal improvemet they were lookig ad hopig for. The mai beefit as we could ascertai later was maily i terms of their improved future plaig o the staffig level ad the ease of the DSS uses for aswerig what-if type staffig questios i geeral. Ackowledgemet. This work is partially supported by the Hog Kog RGC Competitive Earmarked Research Grat (CERG) Award: HKU 7126/05E. Refereces Azmat, C. S. ad Widmer, M. (2004). A case study of sigle shift plaig ad schedulig uder aualized hours: A simple three-step approach. Europea Joural of Operatioal Research, 153, Brusco, M. J. ad Jacobs, L. W. (2000). Optimal models for meal-break ad starttime flexibility i cotiuous tour schedulig. Maagemet Sciece, 46, Burke, E. ad Petrovic, S. (2004). Timetablig ad rosterig. Europea Joural of Operatioal Research, 153, 1 2. Caprara, A., Moaci, M., ad Toth, P. (2003). Models ad algorithms for a staff schedulig problem. Mahtematical Programmig, 89, Chu, S. C. K. (2001). A goal programmig model for crew duties geeratio. Joural of Multi-criteria Decisio Aalysis, 10,

12 GP Models & DSS for Mapower Plaig of Airport Baggage Service 11 Chu, S. C. K. (2007). Geeratig, schedulig ad rosterig of shift crew-duties: Applicatios at the hog kog iteratioal airport. Europea Joural of Operatioal Research, 177, Chu, S. C. K. ad Yue, C. S. (2003). Plaig ad schedulig staff duties by goal programmig. I T. Taaka ad M. Iuiguchi, editors, Multi-Objective Programmig ad Goal-Programmig, pages Advaces i Soft Computig, Spriger-Verlag Berli Heidelberg. Chu, S. C. K. ad Zhu, M. (2007). Data ad optimaizatio modelig for mapower plaig of a airport baggage service. I Proceedigs of the Istitute of Idustrial Egieers Aual Coferece Idustrial Egieerig Research Coferece (IERC 2007), Nashvill, TN, USA, May 2007, pages Erst, A. T., Jiag, H., Krishamoorthy, M., ad Sier, D. (2004). Staff schedulig ad rosterig: A review of applicatios, methods ad models. Europea Joural of Operatioal Research, 153, Schrage, L. (2005). Optimizatio Modelig with LINGO. Lido Systems Ic., Chicago, 5th editio.

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