Planning of work schedules for toll booth collectors

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1 Lecture Notes n Management Scence (0) Vol 4: th Internatonal Conference on Appled Operatonal Research, Proceedngs Tadbr Operatonal Research Group Ltd All rghts reserved wwwtadbrca ISSN (Prnt), ISSN (Onlne) Plannng of work schedules for toll booth collectors Juthathp Vttawasakul and Juta Pchtlamken Department of Industral Engneerng, Kasetsart Unversty, Bangkok, Thaland jutap@kuacth Abstract We consder the workforce schedulng problem of toll booth collectors Integer programmng models are formulated to determne the number of toll booths requred to satsfy vehcle demands and then the manpower shfts Approprate work schedules may reduce the number of employees and save labour costs but stll respond to a mnmum number of employees requred n each perod The proposed schedule decreases the number of shfts and reduces the work hours of staff and substtutes by 04 (6%) and (455%) hours, respectvely However, ths new schedule wll ncrease the work hours of stand-by by 463 hours (64%) Total work hours of the new schedule s,5603 hours whch s reduced from that of the current schedule by 3 hours (0%) Keywords: data analyss; workforce plannng; workforce schedulng; nteger programmng; decson support system Introducton Traffc congeston s mportant because t wastes both tme and fuels The Expressway s a great alternatve to avod traffc jams, and expressway has served lots of drvers The average traffc volume s 4 mllon vehcles per day whch ncreases by mllon vehcles per day from the year 00 (Annual report 009 of Expressway Authorty of Thaland, EXAT) EXAT s a state enterprse under the msson to construct expressways for the Bangkok Metropoltan Area The expressway helps streamlne the traffc EXAT operates the expressway wth a total dstance of 09 klometers, ncludng the hghway route number and -way connectons Although the expressway reduces the number of cars on the road, but toll users may stll suffer from long wat tmes n rush hours Therefore, t should plan the number of toll booth collectors that match wth traffc volume durng peak and

2 J Vttawasakul and J Pchtlamken off-peak tmes and that also save labor costs Currently, EXAT has more than 3000 toll booth collectors The toll booths are open 4-hours a day, days a week The workforce s scheduled nto 3 shfts: mornng (6:0-4:0), afternoon (4:0- :0), and nght (:0-6:0) They do not work consecutve shfts However, the number of employees n each shft depends on the number of open toll booths whch consder the actual traffc data n each tme perod Toll booth collectors have 3 types of functons; staff (to be assgned n the toll booth), stand-by (to work on the rest perod of other staff), and substtutes (to work when the assgned staff are absent) We create work schedules by calculatng the number of toll booths requred to be open under the maxmum traffc per hours, calculatng the number of employees for each perod, and determnng the work days for employees wth 3 consecutve work days and non-consecutvely pared days off In ths paper, we develop nteger programmng models for desgnng a work schedule by consderng the largest toll staton at Dndang wth the data traffc from August to November 00 Lterature revew Workforce schedulng appears n many handbooks and survey papers Sngle-shft manpower schedulng problems are usually solved by specfyng the constrants of a partcular class of problems, generatng a lower bound on the workforce sze necessary for a schedule that satsfes the constrants, and then provdng an algorthm that constructs a schedule that provde necessary workforce For example, Burns and Carter (95) consder the seven-day work week, one shft per day model Burns and Koop (9) extend ths work and look at the seven days per week, multple shfts per day model Emmons (95), Emmons and Burns (99) and Emmons and Hung (993) consder related models The general nteger programmng formulaton on workforce schedulng appear n the lterature Ten and Kamyama (9) provde a systematc revew of the currenly avalable manpower schedulng algorthms under a common framework Jarrah et al (994) uses an nteger lnear program (ILP), a set of varables nvolved, and a heurstc method wth the workng hours requrement less than 4 hours on weekends Bllonnet (999) presents a soluton to the problem of schedulng employees wth an ILP, usng a smple one-pass method that frequently gves the least cost labor mx or unequal employee sklls Ln et al (000) present a development of workforce management system for 4 hours hotlne servce by applyng the regresson model or smulaton model for call centers wth a mxed nteger lnear program Further, ILP can provde optmal solutons to problems of schedulng for consecutve days off Tbrewala et al (9) show that a smple algorthm provdes optmal solutons to problems of schedulng men to meet cyclc requrements over perods where each man or machne must be dle for consecutve perods per cycle An example llustrates the applcaton of schedulng to meet seven dstnct daly requrements per week usng employees for 5 consecutve work days Rothsten (9) assumes a plannng perod of week by maxmzng the number of

3 Lecture Notes n Management Scence Vol 4: ICAOR 0, Proceedngs consecutve days off Later, Rothsten (93) solves the ILP problem usng LP procedures and states that the soluton s ntegral because of the correspondence between the schedulng problem and the capactated network problem for whch there s always an optmal ntegral soluton Maer-Rothe and Wolfe (93) add the constrants that every worker s gven a weekend off at least every 3 weeks and that the length of a workng perod should not exceed 5 days Gopalakrshnan et al (993) develop and mplement a decson support system for plannng and schedulng part-tme workers at a local newspaper company Usng a heurstc to develop schedules, Ernst et al (004) present an overvew and the mportance of the work schedules Methodology Ths secton dscusses the staff problem schedulng of the toll booth collectors The Dndang toll staton has the capacty of channels All lanes are avalable for manual collecton wth the excepton of lanes 5 and because the electronc toll machnes are nstalled on these lanes The overall schedulng procedure s as follows: Survey the staffng patterns that EXAT currently uses Compute the mnmum workforce requred and the mnmum requrement of employees to cover the entre week Schedule the days off Determne the work days for employees Assgn shft schedules The detals of each step explaned below: Survey the patterns that EXAT currently uses as shown n Table,, and 3 Table Patterns of staff schedules Total Shft Pattern Number Hours of Work Number of Hours Mornng weekday 3 05:00-3:00 06:00-4:00 0:00-5:00 4 3:00-:00 Afternoon weekday 5 4:00-:00 6 4:00-:00 5:00-3:00 Nght weekday :00-06:00 Mornng weekend 9 06:00-4:00 0 0:00-5:00 Afternoon weekend 4:00-:00 5:00-3:00 Nght weekend 3 :00-06:00

4 J Vttawasakul and J Pchtlamken 9 Table Patterns of stand-by schedules Shft Pattern Number Hours of Work Mornng weekday 06:30-3:40 0 Afternoon weekday 3 4 5:00-3:00 5:00-:00 5:00-: Nght weekday 5 :0-06:0 Mornng weekend 6 0:00-3: Afternoon weekend 9 0 0:00-3:00 5:00-3:00 5:00-:40 5:00-:00 Nght weekday :0-06:0 Table 3 Patterns of substtute schedules Shft Pattern Number Hours of Work Mornng 06:0-4:0 Afternoon 4:0-:0 Nght 3 :0-06:0 Total Number of Hours Total Number of Hours Note: Under the current work schedule stand bys do not work on nght shfts, but under the proposed schedule, we assgn a work for stand bys nto the normal work hours (:0-06:0) n the nght shft Compute the mnmum workforce requred by solvng an nteger programmng problem, wth the objectve of mnmzng the number of employees (Ten and Kamyama 9) The decson varables of the schedulng problem are as follows: X m the number of employees who work on pattern m, m =,, s Other nput varables are defned as: bt the mnmum requrement of employees n hour t,t =,, 4 Cmt the number of employees who work on m n hour t The objectve functon s: s Mnmze Z X () Subject to: m m s Cmt X m bt () m s X m 0 and nteger m

5 0 Lecture Notes n Management Scence Vol 4: ICAOR 0, Proceedngs The objectve functon s to fnd the mnmum total number of employees pattern (Eq ) The constrant of ths ILP are: the employees who work n hour t must be at least the requred number of employee who work on that hour (Eq ) After we determne the mnmum number of employees The next step s to fnd the mnmum requrement of employees to cover the entre week (Burns and Carter 95) It s based on the seven days per week, one shft per day model Ths schedule starts on Sunday and ends on Saturday The followng constrants are satsfed a) The demand per day, n j, j =,,, (n s Sunday and n s Saturday) b) Each employee s gven A out of every B weekends off c) Each employee work exactly 5 out of days (from Sunday to Saturday) d) Each employee works no more than 6 consecutve days There are three smple lower bounds on the mnmum number of the workforce: the weekend constrant, the total demand constrant, and the maxmum daly demand constrant The workforce sze s determned by satsfyng all the three bounds Let W denote the mnmum number of employees and let max( n, n ) denote the maxmum weekend demand The weekend constrant s the average number of employees avalable each weekend, and the number of employees must be suffcent to meet the maxmum weekend demand: ( B A) W Bmax( n, n ) The total demand constrant s that the total number of employee each days of the week must be suffcent to meet the total weekly demand Snce each employee s gven two days off every week, and they work at most fve days per week: 5 W ( n n n n n n n ) The maxmum daly demand constrant s that the number of employees must be suffcent to meet the maxmum demand on any day: W max( n,, n ) Our algorthm yelds a schedule that requres a workforce of a sze equal to the largest of these three lower bounds 3 Schedule the days off for maxmzng the number of weekly assgnments wth consecutvely pared days off wth the mnmum workforce Rothsten (93) That we determne n () by solvng an nteger program We defne: X = the number of collectors who have Sunday and Monday off X = the number of collectors who have Monday and Tuesday off X = the number of collectors who have Saturday and Sunday off Other nput varables are defned as:

6 J Vttawasakul and J Pchtlamken b = the requrement days off on Monday b = the requrement days off on Tuesday b = the requrement days off on Sunday u = the number of collectors who have Monday non-consecutvely off u = the number of collectors who have Tuesday non-consecutvely off u = the number of collectors who have Sunday non-consecutvely off d = the number employees assgned to non-consecutvely pared days off The objectve functon s to: Maxmze Z X (3) Subject to: = X X u b ( =,,, where X X ) (4) X / b (5) u u j j ( =,, ) (6) d, u and X 0 for all The objectve functon s to fnd the mnmum number of employees wth two consecutvely pared days off (Eq 3) The constrant of ths ILP are: the total number of employees wth consecutvely and non-consecutvely pared days off but the same days off such as Monday equal to the total of employees who have days off that day (Eq 4); employee who have days off whether consecutvely pared or not must be equal to the number of employees at work where each employee must have two days off per week (Eq 5); and non-consecutvely pared days off won t be assgned to the same day (Eq 6) 4 Determne the work days for employees whch 5 consecutve work days and consecutve days off Sequence of assgnments for work days begn wth the day of the week wth the largest number of employee requrement and the next hghest and so on (Tbrewala 9) See Vttawasakul (0) for complete detals Assgnng the work days and the days off for employees, Tbrewala (9) and Rothsten (93) methods gve closely smlar results for assgnng the work days and the days off for employees The number of employees wth consecutvely

7 Lecture Notes n Management Scence Vol 4: ICAOR 0, Proceedngs pared days off by Rothsten (93) s 40 people; Saturday and Sunday, Sunday and Monday 6, Monday and Tuesday, Tuesday and Wednesday 9, Wednesday and Thursday, and Thursday and Frday 9 Tbrewala s algorthm yelds the dfferent results only on pared days off: Saturday and Sunday, and Saturday and Monday 5 Shft schedule assgnment s formulated as ILP Defne the decson varables: X employee number ( =,, 40) work of shft j ( j =,, 3) on day W jk jk k ( k =,, ; denotes Sunday and denotes Saturday) Requrement of employees work of shft j on day k The objectve functon s: 40 3 Mnmze Z X () j k jk Subject to: X jk ( =,, 40, k =,, ) () j Xjk Wjk ( j =,, 3, k =,, ) (9) X jk ( =,, 40, j =,, 3) (0) k X jk 0 and nteger, j, k The objectve functon to fnd employee no work of shft j on day k (Eq ) The constrant of ths ILP; employee no work of shft j only shft per day (Eq ), the number of employees work of shft j on day k must be equal to the requrement of employee work on that shft (Eq 9), and employee no work of shft j at least shft per day (Eq 0) Results Work schedules from the fve steps above are dfferent from the current schedule The number of employee s 4 people n current schedule and 40 people n the new schedule Table 4 Comparson the number of shft The number of shft Current schedule New schedule Percentage Mornng 000% Evenng 4% Nght %

8 J Vttawasakul and J Pchtlamken 3 Table 5 Comparson work hours of employee types (man hours) Work hours Current schedule New schedule Percentage Staff,3440, % Stand by % Understudy % From the Tables 4 and 5, the proposed schedule decreases the number of shfts and reduces the work hours of staff and substtutes by 04 (6%) and (455%) hours, respectvely However, ths new schedule wll ncrease the work hours of stand-by by 463 hours (64%) Total work hours of the new schedule s,5603 hours whch s reduced from that of the current schedule by 3 hours (0%) Concludng remarks Ths paper apples nteger programmng to plan for work schedules for toll booth collectors Usng a smple formula to solve the problem based on 5 consecutve work days and consecutve days off, our work schedules respond to the demand of toll users and the mnmum requrements of employees n each perod In the future, we wll develop a decson support tool based on ths methodology Acknowledgments We would lke to thank EXAT for provdng us wth data References Bllonnet A (999) Integer programmng to schedule a herarchcal workforce wth varable demands European Journal of Operatonal Research 4: 05-4 Burns RN and Carter MW (95) Work Force Sze and Sngle Shft Schedules wth Varable Demands Management Scence 3: Burns RN and Koop GJ (9) A Modular Approach to Optmal Multple Shft Manpower Schedulng Operatons Research 35: 00-0 Emmons H (95) Work-Force Schedulng wth Cyclc Requrements and Constrants on Days Off, Weekends Off, and Work Stretch IIE Transactons : -6 Emmons H and Burns RN (99) Off-Day Schedulng wth Herarchcal Worker Categores Operatons Research 39: Emmons H and Hung R (993) Multple-Shft Workforce Schedulng under the 3-4 Compressed Workweek wth a Herarchcal Workforce IIE Transactons 5: -9 Ernst AT, Jang H, Krshnamoorthy M and Ser D (004) Staff schedulng and rosterng: A revew of Applcatons, methods and models European Journal of Operaton Research 53: 3- Gopalakrshnan M, Gopalakrshnan S and Mller DM (993) A Decson Support System for Schedulng Personnel n a Newspaper Publshng Envronment Management Scences 3: 04-5

9 4 Lecture Notes n Management Scence Vol 4: ICAOR 0, Proceedngs Jarrah AIZ, Bard JF and deslva AH (994) Solvng Large-scale Tour Schedulng Problem Management Scence 40(9): 4-44 Ln CKY, La KF and Hung SL (000) Development of a workforce management system for a customer hotlne servce Computer & Operatons Research : Maer-Rothe C and Wolfe HB (93) Cyclcal schedulng and allocaton of nursng staff Soco-Economc Plannng Scences : 4-4 Rothsten M (9) Schedulng manpower by mathematcal programmng Industral Engneerng: 9-33 Rothsten M (93) Hosptal manpower shft schedulng by mathematcal programmng Health Servces Research (): Tbrewala R, Phlppe D and Browne J (9) Optmal Schedulng of Two Consecutve Idle Perods Management Scence : Ten JM and Kamyama A (9) On Manpower Schedulng Algorthms SIAM Revew 4: 5-