Journal of Marine Science and Technology, Vol. 16, No. 4,. 301-307 (2008) 301 THE MANPOWER SUPPLY PLANNING AND SHIFT SETTING FOR MASS RAPID TRANSIT CARRIAGE MAINTENANCE IN SHORT-TERM OPERATIONS Shangyao Yan*, Chia-Hung Chen**, and Miawjane Chen*** Key word: carriage maintenance, manower uly, mixed integer rogram, flexible trategy. ABSTRACT An effectie carriage maintenance lan cannot only reduce the ma raid tranit ytem oerating cot, but i alo directly related to it afety and chedule unctuality. Traditionally, the arrangement of the carriage maintenance manower uly ha been baed on taff exerience, and ha been erformed manually following imle rule. Thi i not only time-conuming but often ineffectie. Therefore, in thi reearch, we incororate flexible management trategie, different combination of rincile, and the related oerating contraint, to deelo a baic carriage maintenance manower uly lan model. Furthermore, we contruct an integrated model by modifying the baic model, which can hel an MRT lant more effectiely manage it maintenance manower. To ealuate the model in ractice, a cae tudy uing real oeration data from a Taiwan MRT maintenance facility i erformed, with the aitance of C comuter rogram and a mathematical rogramming oler. The reliminary reult are good, howing that the model could be ueful for lanning hort-term carriage maintenance manower uly. I. INTRODUCTION An effectie carriage maintenance lan cannot only reduce the ma raid tranit ytem (MRT) oerating cot, but i alo directly related to it afety and chedule unctuality. A good carriage maintenance manower uly lan not only meet all the afety requirement but will alo hel better manage maintenance reource and imroe erice quality. A a reult, it i a ery imortant iue to an MRT in determining a good carriage maintenance manower uly lan. Generally, the carriage maintenance manower uly lan Paer ubmitted 11/19/07; acceted 12/18/07. Author for correondence: Chia-Hung Chen (e-mail: chiahung@mail.tu.edu.tw). *Deartment of Ciil Engineering, National Central Unierity, Chung-Li 32001, Taiwan, R.O.C. **Deartment of Logitic Management, Shu-Te Unierity, Yen Chau 82445, Taiwan, R.O.C. ***Deartment of Finance, National United Unierity, Miao-Li, 36003, Taiwan, R.O.C. are claified into hort- and long-term one. The hort-term one, which inole the current MRT oeration, i uually erformed before the beginning of the next eriod (a eriod contain a coule of month). On the other hand, the long-term one i uually erformed at the beginning of the year (or eeral year in adance), according to the redicted carriage maintenance/manower demand baed on hitorical information. The contraint in thee two tye of lan are different. In articular, the carriage maintenance manower aailable for a hort-term lan i contrained by current carriage maintenance manower reource. How to et a good carriage maintenance manower uly lan for hort-term oeration i more difficult than for long-term one. Therefore, in thi reearch we focu on a hort-term carriage maintenance manower uly lan. Uing a Taiwan MRT maintenance facility (called X MRT hereinafter) a an examle, we introduce the rocedure for determining hort-term carriage maintenance manower uly lan in ractice. In the X MRT, two deartment are reonible for carriage maintenance, which are the reentie maintenance deartment () and the breakdown maintenance deartment (). The ha to deal with the reentie carriage maintenance, uch a daily and monthly check; while the i reonible for maintenance work after carriage breakdown occur. The demand for i a fixed demand, but that for may fluctuate from time to time, deending on unexected eent. In current ractice, an aerage demand i etimated and i ued for lanning. Furthermore, thee two deartment normally ue 3 hift and 2 hift (although ometime from 2 to 6 hift are occationally ued). In addition, a flexible working hour trategy allow X MRT to hire both full-time and half-time emloyee, although they uually hae full-time emloyee, mot often diided into 6 eole to a team/crew (occationally from 3 to 6 eole team are ued). In theory, more hift would increae the degree of freedom in cheduling, thu conering manower, but would of coure reult in a more comlicated manower uly lan. Therefore, for eae of cheduling, X MRT imlifie it maintenance manower uly lan to hae 3 hift and 2 hift. In current ractice, X MRT deend on taff exerience in etablihing the carriage manower uly lan, which i outlined a follow. The lanning rocedure for carriage man-
302 Journal of Marine Science and Technology, Vol. 16, No. 4 (2008) ower uly tyically conit of three tage: (1) the initial carriage maintenance demand etimation, (2) MRT maintenance manower uly lanning, and (3) crew aignment. Of the three tage, the firt tage i tyically imle, a tatitical work. The econd tage i comlicated and it reult ere a the bai of the third tage. In other word, crew aignment i handled after the manower uly roblem i oled. Not only doe a good MRT maintenance manower uly lan make crew aignment eaier, but alo reduce the MRT maintenance manower crew cot. Howeer, X MRT currently utilize a trial-and-error exerience-baed method with a rojected demand for carriage manower uly lan, without otimization from a ytem erectie. Such a trial-and-error method i neither efficient nor effectie. Therefore, thi reearch focue on lanning hort-term carriage maintenance manower uly lanning, which correond to the econd tage of the oerall lanning rocedure mentioned aboe. To hel the reader undertand the background, we now dicu ome of the at reearch on eronnel cheduling and flexible management trategie that are related to our. Much reearch ha already been deoted to eronnel cheduling roblem. For examle of recent reearch, ee Bruco and John [5], Lau [10], Naraimhan [11], Langerman and Ehler [9], Carara et al. [7], Beaumont [3], Bruco [6], Alfare [1], Higgin [8], Teodoroi and Lu i [12], Aykin [2], Yan and Tu [13], Yan et al. [16], Yan et al. [14], and Yan et al. [15]. Howeer, their roblem characteritic are not the ame a thoe needed for a carriage maintenance manower uly lan, meaning that the aboe model or olution are not alicable or can not be directly alied to our roblem. In addition, the flexible management trategie hae been widely recognized in the indutry and the reult indicate that flexible management trategie can increae the different degree of freedom in the chedule, thu conering manower (Blyton and Morri, [4]). The flexible management trategy, howeer, ha not yet been alied to MRT carriage maintenance manower uly lan. A a reult, we did not find any that addreed MRT carriage maintenance manower uly lanning roblem, with flexible management trategie in articular. An effectie combination of hift tarting time, the number of hift, and the number of crew i the key to a ucceful carriage maintenance manower uly lan. Therefore, in thi reearch, a baic carriage maintenance manower uly lan model (for conenience, hereinafter called the BM), incororating flexible management trategie, different combination of rincile, and the related oerating contraint, i deeloed. Moreoer, an integrated model (for conenience, hereinafter called the IM) i further deeloed by modifying the BM according to the imilar characteritic of the and the, which can hel X MRT lant more effectiely manage it maintenance manower. The model are formulated a mixed integer linear rogram and are oled uing a mathematical rogramming oler. Finally, to ealuate how the model erform in ractice, a cae tudy uing real oeration data from X MRT maintenance facility i erformed, with the aitance of C comuter rogram and a mathematical rogramming oler. The coe of thi reearch i confined to the ubject of MRT carriage maintenance manower uly and hift etting, gien the rojected manower demand, the uer/lower bound of the hift tarting time, the working hour for each hift, the team member for a work tye, and other related cot data. In addition, ince the X MRT timetable i rotated weekly, the lanning cycle i one week (een day). For the lanning week, hift tarting time hae to be the ame eery day of the week. It hould be mentioned that, in ractice, the alary tructure arie according to eronnel exerience, length of emloyment, education and other factor. It would be too comlicated to model minimum alary exene a an objectie function in the lanning tage, o for eae of modeling, thi tudy focue on the minimization of the maintenance manower uly in man-hour. The remainder of the aer i organized a follow. We firt decribe the roblem, and then formulate the model. Thereafter, we erform numerical tet to ealuate the erformance of the model. Finally we gie ome concluion. II. THE MODELS In thi ection, the modeling concet i firt dicued according to the X MRT current ractice. Then a baic model (BM) i formulated incororating three flexible management trategie. According to the imilar characteritic of the and the and the ue of exce maintenance manower between the two deartment, the BM i modified to further deelo an integrated model (IM). 1. Modeling Concet The tudy focue on modeling a comlicated hort-term carriage maintenance manower uly lan. According to the X MRT oeration, demand i gien and fixed in adance. The demand i gien by an exectedly aerage demand. Thee demand tyically ary in a week. Sometime in off-eak hour there may een be no demand which make flexible trategie a good idea for imroing efficiency of X MRT maintenance manower lanning. Taking into conideration X MRT ractice, three flexible trategie can be rooed baed on numerical and temoral concet: flexible hift, flexible team member, and flexible working hour. The flexible hift trategy allow an emloyer to chooe the referred number of hift tarting time er day and the hift tarting time. Thee need not be limited to the conentional three fixed hift (i.e., 0:00-8:00, 8:00-16:00, and 16:00-0:00) for the nor the two fixed hift for the (i.e., 8:00-16:00, and 16:00-0:00) currently utilized by X MRT. In the flexible team member trategy the number of team member can be adjuted in reone to change (from 3 to 6 in thi reearch) in demand in different time lot. The flexible working hour trategy allow X MRT to hire both full-time and half-time emloyee. Two tye of flexibility are introduced in our model: full-time (eight hour) hift and half-time (four hour) hift. Traditionally, the lanner only conider matching it main-
S. Yan et al.: The Manower Suly Planning and Shift Setting for Ma Raid Tranit Carriage Maintenance in Short-Term Oeration 303 tenance manower uly to the demand, neglecting the ue of exce maintenance manower. Howeer, in actual ractice, urlu maintenance manower may be hifted to other deartment, to further enhance efficiency. In addition, if the and the exected demand for continuou time lot fluctuate dratically (e.g., at the boundarie between eak and off-eak eriod), then ulying too much maintenance manower to meet the highet demand would reult in urlu manower and a wate of human reource, though they can be of ue in ome way. One way to deal with dratic fluctuation in maintenance manower demand in continuou time lot i to temorarily ulement manower from another maintenance deartment or off-duty eronnel, een if the unit man-hour cot i normally higher than the regular unit man-hour cot. Howeer, to determine what trade-off between urlu and inufficient maintenance manower i mot aroriate for each lot for each work tye i too comlicated to be oled without a ytematic model. We rooe two ytematic model that incororate three flexible trategie and two combination of manower uly rincile in order to roide effectie MRT maintenance manower uly lan. The lan are not only reonie to wide ariation in maintenance manower demand but can alo hel the X MRT maintenance facility manage it manower reource more effectiely, and conequently reduce it oerating cot. 2. BM Incororating flexible management trategie, different combination of rincile, and the related oerating contraint, the baic model (BM) i formulated. The notation for arameter and ariable ued in the model formulation are lited below. Parameter: D : the et of day in a week. Since our lanning eriod i one week (een day), D {0: Sunday, 1: Monday, 2: Tueday, 3: Wedneday, 4: Thurday, 5: Friday, 6: Saturday}. d : the th d day in a week; d D. S : the et of hift tarting time in one day. The length of a time lot i et to be one hour. Since there are 24 hour in a day; S {0, 1, 2, 3, 4, 5,, 23}. t : th the t time lot in a day; t S. : th the hift tarting time; S. A hift tarting time indicate when the hift (of different tye) tart. According to real ractice, the hift tarting time i the ame each day of the week. If an MRT garage doe not refer to ecify the tarting time (for examle, to reent the yielding of hift that tart too cloe together), then thee hift tarting time can be remoed from S. On the contrary, if the maximum number of hift tarting time i contrained due to ractical concern, then a contraint hould be imoed. P : the et of different team ize (i.e., different number of crew member). : tye maintenance team; P. m : the number of eron in a tye maintenance team. W : the et of different tye of work, to differentiate between full-time and art-time emloyee. w : tye w work; w W. h w : the working hour for tye w work. rc rc, : term oeration, ince the crew alary (the main art) ha already been aid, rc and rc are tyically not large. the regular manower cot er man-hour for the and for the, reectiely. In hort ec ec, : the, reectiely, which could be ued by the other deartment, and i thu in the form of the alue er urlu man-hour for manower uly exceeding demand for the and for a negatie cot. In ractice, ec i maller than rc, becaue ec i a deriatie alue. Similarly, ec i maller than rc, becaue ec i a deriatie alue. Note that if the exce manower i not well lanned in adance in hort-term oeration, then and ec are mall. ec ic, ic : the cot er temorary man-hour uly for inufficient manower for the and for the, reectiely. Inufficient manower may be ulemented temorarily from the other maintenance deartment or be trainee without qualification. In ractice, ic and ic are greater than rc and rc, reectiely, in hort-term oeration. u, u : the uer bound of the total number of hift in one day for both the and the. d : the maintenance manower demand in time lot t on It i meaured a the number of eron. bd : the maintenance manower demand in time lot t on It i meaured a the number of eron. : the uer bound of the manower which can be ulied in time lot t on b : the uer bound of the manower which can be ulied in time lot t on M : a ery large alue for eae of modeling.
304 Journal of Marine Science and Technology, Vol. 16, No. 4 (2008) H w : the et of all hift ulying tye w work in time lot t on The uroe of H i to rereent the relationhi between w the manower uly of hift for different work tye and the working time lot. Uing H, we can identify all the hift of w tye w work that will be on duty in time lot t on Variable: dw : the number of tye team for tye w work for tarting at hift on dw : the number of tye team for tye w work for tarting at hift on x : x = 1 if there are hift for tarting at, and 0 otherwie. x : x = 1 if there are hift for tarting at, and 0 otherwie. e, i : the exce manower and inufficient manower (in eron) for in time lot t on e, i : the exce manower and inufficient manower (in eron) for in time lot t on The BM i formulated a a mixed integer linear rogram a follow: Min z = rc h m H w H w w dw d D S + rc h m w dw d D S ST (1) m e + i = d, d D, t S (2) dw m e + i = bd, d D, t S (3) dw d D d D H w S S x u (4) x u (5) Mx, S (6) dw Mx, S (7) dw m, d D, t S (8) dw dw dw H w m b, d D, t S (9) dw x = 0 or 1, S (10) x = 0 or 1, S (11) 0 & I, d D, S, P, (12) dw 0 & I, d D, S, P, (13) dw e 0 & i 0, d D, t S (14) e 0 & i 0, d D, t S (15) Equation (1) i the objectie function that minimize the total ytem cot that include the regular manower cot for the two deartment, the negatie urlu alue of exce manower, and the temorary manower uly cot for inufficient manower. It hould be mentioned that the cot reulted from different number of hift tarting time i not ignificant in the current X MRT oeration and therefore can be neglected. If thi cot i ignificant in other alication, then the objectie function can be modified and the cot ( a x + b x ) S added to the objectie function, to reflect thi concern, where a and b denote the cot for the and the, reectiely, for etting the hift tarting time. Equation (2) and (3) tate that for the and for the, reectiely, the aigned crew member, ubtracting exce manower and adding temorary ulied manower, mut be able to meet the manower demand in eery time lot during their hift. Equation (4) and (5) contrain the maximum number of hift tarting time in each day for the and the, reectiely. Note that if the leat number of hift tarting time need to be conidered in other alication, then Equation (4) and (5) can be modified to become l S x u S l x u and, reectiely, where l and l de- note the leat number of hift tarting time for the and the, reectiely. Equation (6) and (7) enure that and maintenance team are aigned to a hift only when that hift exit. Equation (8) and (9) enure that the amount of and manower in eery time lot during the hift doe not exceed the aailable amount of manower. Equation (10), (11), (12) and (13) are integer contraint for the ariable. Equation (14) and (15) are ariable nonegatiity contraint. Note that e, i, e and e are not contrained to be integer. Howeer, due to contraint (2) and (3) and the integrality of d and bd, they are naturally integer. Gien the required demand and the maximum uly of manower for each time lot of each day, the work tye, the feaible range of hift and the related cot, the model determine the bet et of hift, each with a number of work tye, and the exce or temorary ulied manower in each time
S. Yan et al.: The Manower Suly Planning and Shift Setting for Ma Raid Tranit Carriage Maintenance in Short-Term Oeration 305 lot, in each day of the week. The model i oled uing the mathematical rogramming oler, CPLEX. 3. IM According to the imilar characteritic of the and the in the current X MRT ractice, we further contruct an integrated model, IM. With the IM not only can we etablih thee two deartment manower uly lan earately, but we can alo allow them to uort each other, thereby increaing the flexibility of eronnel arrangement. Howeer, there are ome difference between the and the, in term of the leel of exertie and the ability. Therefore, to conform to real ractice, a manower reduction coefficient hould be incororated into the IM. In addition, the amount of manower aailable to uort the other deartment hould not exceed the maximum aailable manower. More effectie ue of manower reource lead to reduction in the IM cot. Additional arameter/ariable for the IM are defined below. a : the maximum number of manower unit aailable for uorting the in time lot t on a : the maximum number of manower unit aailable for uorting the in time lot t on k : the reduction coefficient for uing the manower to uort the demand. The alue i 1 between 0 and 1. k : the reduction coefficient for uing the manower to uort the demand. Similar to k, 2 1 the alue i between 0 and 1. The IM i formulated a a mixed integer linear rogram a follow: Min z = rc h m H w w dw d D S + rc h m w dw d D S ST m a + k1 a e + i = d, dw d D, t S (16) H w m a + k a e + i = bd, 1 dw d D, t S (17) H w H w (1) m a, d D, t S (18) dw m a, d D, t S (19) dw a a 0 & a I, d D, t S (20) 0 & a I, d D, t S (21) and equation (4) to (15). To ae ace, we only introduce below thoe IM equation which are different from the BM one. Equation (16) tate that the aigned crew member, minu the manower for uorting, lu manower ulied from, that i ubtracting exce manower and adding temorary ulied manower, mut be enough to meet the manower demand in eery time lot during thi hift. Similar to Equation (16), equation (17) tate that the aigned crew member, minu the manower for uorting, lu manower ulied from, that i ubtracting exce manower and adding temorary ulied manower, mut be enough to meet the manower demand in eery time lot during the hift. Equation (18) and (19) indicate that the total number of and manower unit in eery time lot ha to be enough to atify the larget amount required. Equation (20) and (21) are contrained to be integer ariable. Gien the required demand and the maximum uly of manower for each time lot of each day, the work tye, the feaible range of hift and the related cot, the model determine the bet et of hift, each with a number of work tye, the exce or temorary ulied manower in each time lot and the uorting manower in each time lot, in each day of the week. The model i oled uing the mathematical rogramming oler, CPLEX. III. NUMERICAL TESTS To tet how well the model may be alied in the real world, numerical tet mainly uing oerating data from X MRT maintenance facility are erformed. The aitance of C comuter language and the mathematical rogramming oler, CPLEX 8.1, are ued to build the model and to ole the roblem. The tet are erformed on a Pentium 4 2G with 1Gb of RAM in the enironment of Microoft Window XP. 1. Data Analyi The maintenance demand of the X MRT two maintenance deartment during Setember of 2006 are ued a our tet data. The inut, uch a the uer bound of maintenance manower uly in each time lot, the working hour of different work tye, and the uer/lower bound of the hift tarting time, are rimarily adoted from actual oerating data. According to the real ractice, two different work tye are et to make for a more flexible trategy (full-time and half-time). In addition, the normal number of hift tarting time ued in X MRT i three, although a many a ix hift tarting time may occationally be ued. Therefore, the feaible range i et from three to ix hift tarting time. Furthermore, the team ize ued in X MRT i ix, although ometime from three to ix eole team are occationally ued. Therefore, the team ize range i et from three to ix eole team. The three cot arameter alue, which are relatie to each other, for the BM and the IM are a follow:
306 Journal of Marine Science and Technology, Vol. 16, No. 4 (2008) Table 1. Tet reult. TAEM BM IM OBJ (NT$) 22080 21040 21030 Comutation time (econd) NA 1.31 72.38 Required manower (man-hour) 866 1016.81 866 1016.81 866 1016.81 Suly demand (man-hour) 1056 1152 996 1108 1004 1089 Exce manower (man-hour) 190* 131* 130 87 135.2 80.6 Inufficient manower (man-hour) 0** 0** 0 0 0.4 0 Suorting manower (man-hour) 0** 0** 0 0 22 12 Number of hift 3 2 5 3 5 3 Shift tarting time 09,10,23 07,15 02,03,09,10,23 07,11,15 *The exce manower for the and the for TAEM i externally calculated to be 190 and 131, reectiely. **The inufficient and the uorting manower for the and the for TAEM are both 0 by definition. 00,02,09,10,2 3 07,11,1 5 The regular maintenance manower cot er man-hour rc, rc ) i et to be 10. The alue er urlu man-hour ( for maintenance manower uly exceeding demand ec, ec ) i et to be zero, becaue X MRT doe not con- ( ider the ue of exce maintenance manower in their hort-term manower uly lan. Since it i not eay to imort temorary maintenance manower from the other maintenance deartment or from outide the comany in hort-term oeration, the cot er temorary man-hour uly for inufficient man-ower ( ic, ic ) i et to be 250, a the cloet reflec- tion of real ractice. Note that the three cot alue are adjutable to meet the requirement of other MRT. Excet for the aboe etting, in the IM model mutual uort i taken into account to increae the flexibility of eronnel arrangement. Howeer, there are ome difference between the and the, uch a the leel of exertie and the training of maintenance eronnel. Therefore, to conform to real ractice, a manower reduction coefficient i conidered in the IM. The leel of exertie and the training of the maintenance eronnel are better than thoe of the eronnel. Therefore, we et the manower reduction coefficient for the ( k 1 ) to be 0.9 and that of the ( k ) to be 0.8. 2 The two reduction coefficient are adjutable to meet other MRT oerating requirement. 2. Tet Reult Table 1 how the tet reult. Note that to reliminarily ealuate the rooed model, the reult are comared to thoe obtained uing the current X MRT trial-and-error exerience-baed method (denoted a TAEM). A hown in Table 1, OBJ rereent the ytem cot for the otimal olution obtained by CPLEX. The IM yield the bet olution, with an objectie alue of 21030. The BM erform lightly wore, with an objectie alue of 21040. The TAEM erformed the wort, with an objectie alue of 22080. The reult how that the two model outerformed the current trial-and-error method by a reduced cot of at leat 4.94% (=(22080-21040)/21040). Moreoer, for the BM, the exce and inufficient manower for the are 130 and 0 man-hour, reectiely, while the exce and inufficient manower for the are 87 and 0 man-hour, reectiely. For the IM, the exce and inufficient manower for the are 135.2 and 0.4 man-hour, reectiely, while the exce and inufficient manower for the are 80.6 and 0 man-hour, reectiely. From the aboe reult it can be een that the IM allowed for mutual uort between thee two deartment to comenate for inufficient manower. More ecifically, by uitably trading off the exce and inufficient manower for each lot of eery work tye and allowing the two deartment to uort each other mutually, the IM could hel manage manower more effectiely than the BM. For the number of hift, we ary the number of hift from 3 to 6, with tarting time that could be any time lot in a day for the two model. In theory, more hift would increae the degree of freedom in cheduling. Howeer, a hown in Table 1, all the model do not ue the uer bound of the hift tarting time. The reaon i that there i no demand in ome time lot in a day for the and the. For the comutation time, they are ery efficient, 1.31 econd for the BM and 72.38 econd for the IM. From the tet reult, both the BM and the IM could imroe oer the current manual trial-and-error method. The IM wa better than the BM in term of flexibility. In ummary, both the BM and the IM are efficient and effectie for X MRT carriage maintenance manower uly lanning and hift etting in hort-term oeration. IV. CONCLUSION Thi tudy incororate flexible management trategie, different combination of rincile, and the related oerating contraint to deelo a baic carriage maintenance manower uly lan model (the BM). By uitably modifying the BM, an integrated model (the IM) i further deeloed. The model are formulated a mixed integer linear rogram that are oled uing a mathematical rogramming oler. A cae tudy util-
S. Yan et al.: The Manower Suly Planning and Shift Setting for Ma Raid Tranit Carriage Maintenance in Short-Term Oeration 307 izing real oerating data from a Taiwan MRT i conducted to reliminarily ealuate the model and the olution algorithm in the real world. The reult how that both the BM and the IM are an imroement oer the trial-and-error method ued in actual oeration. Moreoer, the IM i better than the BM in term of flexibility. In ummary, both the BM and the IM are efficient and effectie for the MRT carriage maintenance manower uly lanning and hift etting in hort-term oeration. Although the reliminary tet reult how that the model are otentially ueful for manower uly lanning and hift etting, eecially for the Taiwan MRT oeration, more tet or cae tudie hould be conducted, o that uer may gra their limitation, before utting them to ractical ue. The model and the cae tudy hould all be ueful reference for the MRT when determining the mot otimal hort-term maintenance manower chedule. Finally, in actual oeration, ome of the model arameter, uch a manower demand for, are tochatic. How to modify the determinitic model to become mixed determinitic and tochatic model for a cloer match of actual oeration could alo be a toic of future reearch. ACKNOWLEDGEMENTS Thi reearch wa uorted by a grant (NSC-95-2211-E-008-164) from the National Science Council of Taiwan. We would like to thank the Taiwan MRT for roiding the tet data and it aluable oinion on thi reearch. We alo thank Tung-Sheng Yeh for heling collect and analyze the data. Finally, we would like to thank the editor and the anonymou reiewer for their helful uggetion which greatly imroed the reentation of thi aer. REFERENCES 1. Alfare, H. K., An efficient two-hae algorithm for cyclic day-off Scheduling, Comuter and Oeration Reearch, Vol. 25,. 913-923 (1998). 2. Aykin, T., A comaratie ealuation of modeling aroache to the labor hift cheduling roblem, Euroean Journal of Oerational Reearch, Vol. 125, No. 2,. 381-397 (2000). 3. Beaumont, N., Scheduling taff uing mixed integer rogramming, Euroean Journal of Oerational Reearch, Vol. 98,. 473-484 (1997). 4. Blyton, P. and Morri, J., HRM and the limit of flexibility. in Blyton, P. and Turnbull, P. (ed.), Reaeing Human Reource Management,. 116-130 (1992). 5. Bruco, M. J. and John, T. R., A equential integer rogramming method for dicontinuou labor tour cheduling, Euroean Journal of Oerational Reearch, Vol. 95, No. 3,. 537-548 (1996). 6. Bruco, M. J., Soling eronnel tour cheduling roblem uing the dual all-integer cutting lane, IIE Tranaction, Vol. 30,. 835-844 (1998). 7. Carara, A., Fichetti, M., Toth, P., Vigo, D., and Guida, P. L., Algorithm for railway crew management, Mathematical Programming, Vol. 79,. 125-141 (1997). 8. Higgin, A., Scheduling of railway track maintenance actiitie and crew, Journal of the Oerational Reearch Society, Vol. 49,. 1026-1033 (1998). 9. Langerman, J. J. and Ehler, E. M., Agent-baed airline cheduling, Comuter and Indutrial Engineering, Vol. 33, No. 3-4,. 849-852 (1997). 10. Lau, H. C., On the comlexity of manower hift cheduling, Comuter and Indutrial Engineering, Vol. 23, No. 1,. 93-102 (1996). 11. Naraimhan, R., An algorithm for ingle hift cheduling of hierarchical workforce, Euroean Journal of Oerational Reearch, Vol. 96, No. 1,.113-121 (1996). 12. Teodoroi, D. and Lu i, P., A fuzzy et theory aroach to the aircrew rotering roblem, Fuzzy Set and Sytem, Vol. 95,. 261-271 (1998). 13. Yan, S. and Tu, Y. P., A network model for terminal cabin crew cheduling, Euroean Journal of Oerational Reearch, Vol. 140, No. 3,. 531-540 (2002). 14. Yan, S., Chen, C. H., and Chen, C. K., Long-term manower uly lanning for air cargo terminal, Journal of Air Tranort Management, Vol. 12, No. 4,. 175-181 (2006). 15. Yan, S., Chen, C. K., and Chen, C. H., Cargo terminal hift etting and manower ulying in hort-term oeration, Journal of Marine Science and Technology, Vol. 14, No. 2,. 109-118 (2006). 16. Yan, S., Yang, T. H., and Chen, H. H., Airline hort-term maintenance manower uly lanning, Tranortation Reearch Part A, Vol. 38,. 615-642 (2004).