TECHNICAL REPORT Compuaonal resuls on new saff shedulng enhmark nsanes Tm Curos Rong Qu ASAP Researh Group Shool of Compuer Sene Unersy of Nongham NG8 1BB Nongham UK Frs pulshed onlne: 19-Sep-2014 las updaed: 06-O-2014. Ths repor lss resuls of applyng he algorhms presened n [2] o he saff shedulng prolem enhmark nsanes 1..24 [3]. The algorhms are an ejeon han meaheurs and a ranh and pre mehod. The ranh and pre mehod was shown o e ery effee on smaller and medum szed nsanes ofen fndng he opmal soluon. Is weakness s on he larger nsanes on whh may run ou of memory ryng o sole a su-prolem. The meaheurs s a more rous and praal mehod. Alhough s ouperformed on he smaller nsanes wll sll fnd good soluons on he larger nsanes f gen suffen me. For adonal omparsons we hae also nluded he resuls of applyng Guro 5.6.3 [1] o an neger programmng formulaon. Insanes Many of he orgnal enhmark nsanes aalale a [3] an now e easly soled [2]. Mos of he orgnal nsanes are also que dfful o use due o her real world naure. They onan many dfferen ypes of onsrans and ojees whh are omplaed o model and mplemen whaeer ype of solng approah s eng used (neger programmng meaheurs e). For hese reasons he olleon of nsanes has een reenly supplemened wh a new se of nsanes. The new nsanes are desgned o refle real world requremens and shedulng senaros ye sll e easy o use. They are also desgned o represen a range of dffuly: from ery easy o ery hallengng. To make hem easer o use and es he numer of onsran and ojee ypes has een redued o a ore of onsrans found ommonly n saff roserng prolems. The new nsanes are also gen n a plan e forma whh s a lo smpler o parse and use. Ths allows researhers o spend less me wrng ode for parsng he nsanes and more me on deelopng he algorhms and produng resuls. Tale 1 lss he nsanes and her dmensons. They range from ery small (8 saff 2 weeks 1 shf ype) o ery large (150 saff 52 weeks 32 shf ypes). Insane Plannng horzon (weeks) Saff Shf ypes Insane1 2 8 1 Insane2 2 14 2 Insane3 2 20 3 Insane4 4 10 2 Insane5 4 16 2 Insane6 4 18 3 Insane7 4 20 3 Insane8 4 30 4 Insane9 4 36 4 Insane10 4 40 5 Insane11 4 50 6 Insane12 4 60 10
Ineger Programmng Formulaon Insane13 4 120 18 Insane14 6 32 4 Insane15 6 45 6 Insane16 8 20 3 Insane17 8 32 4 Insane18 12 22 3 Insane19 12 40 5 Insane20 26 50 6 Insane21 26 100 8 Insane22 52 50 10 Insane23 52 100 16 Insane24 52 150 32 Tale 1 Benhmark nsanes An neger programmng model for he prolem s gen elow. All nsanes sar on a Monday and he plannng horzon h s always a whole numer of weeks (h mod 7 = 0). Parameers: I h D W T se of employees. numer of days n he plannng horzon. se of days n he plannng horzon = {1 h}. se of weekends n he plannng horzon = {1...h/7}. se of shf ypes. R se of shf ypes ha anno e assgned mmedaely afer shf ype. N l se of days ha employee anno e assgned a shf on. lengh of shf ype n mnues. ma m mamum numer of shfs of ype ha an e assgned o employee. mn mnmum numer of mnues ha employee mus e assgned. ma mamum numer of mnues ha employee an e assgned. mn mnmum numer of onseue shfs ha employee mus work. mn mamum numer of onseue shfs ha employee an work. mn o mnmum numer of onseue days off ha employee an e assgned. ma a mamum numer of weekends ha employee an work. q penaly f shf ype s no assgned o employee on day d. p penaly f shf ype s assgned o employee on day d. u preferred oal numer of employees assgned shf ype on day d. mn wegh f elow he preferred oer for shf ype on day d.
ma wegh f eeedng he preferred oer for shf ype on day d. Deson arales: k w 1 f employee s assgned shf ype on day d 0 oherwse 1 f employee works on weekend w 0 oherwse y oal elow he preferred oer for shf ype on day d. z oal aoe he preferred oer for shf ype on day d. Consrans: 1. An employee anno e assgned more han one shf on a sngle day. 1 d D 2. Shf roaon. A mnmum amoun of res s requred afer eah shf. Therefore eran shfs anno follow ohers. For eample an early shf anno follow a lae shf. 1 h 1} T u R ( d 1) u 3. The mamum numers of shfs of eah ype ha an e assgned o employees. For eample some employees wll hae onras whh do no allow hem o work ngh shfs or only a mamum numer of ngh shfs. m ma T 4. Mnmum and mamum work me. The oal mnues worked y eah employee mus e eween a mnmum and mamum. These lms an ary dependng on wheher he employee s full-me or par-me. mn l ma I 5. Mamum onseue shfs. The mamum numer of shfs an employee an work whou a day off. For eample par-me employees somemes do no work as many onseue shfs as full-me saff. d ma j= d j ma h ma } 6. Mnmum onseue shfs. Ths an e modelled y preenng eery sequene of onseue shfs elow he mnmum. For eample f he mnmum numer of onseue shfs s four hen we mus no allow any of he sequenes: {off-on-off off-on-on-off off-on-on-on-off} where off s a day whou a shf and on s a day wh a shf assgned. d s s j j= d 1 ( d s 1) > 0 s {1... mn 1} h ( s 1)}
7. Mnmum onseue days off. Ths an e modelled n a smlar way o he mnmum onseue shfs onsran. For eample f he mnmum numer of onseue days off s hree hen we mus no allow any of he sequenes: {on-off-on on-off-off-on}. 1 d s j= d 1 j 1 ( d s 1) > 0 s {1... o mn 1} h ( s 1)} 8. Mamum numer of weekends. A weekend s onsdered as eng worked f he employee has a shf on he Saurday or he Sunday. k w ( 7w 1) (7w) 2kw w W w W k w a ma I 9. Days off. These are days ha employees anno work eause for eample hey are on aaon. 10. Coer requremens. Ojee funon: Mnmse I I q = 0 d N T z ( 1 y ) = u I p d D T y mn z ma The ojee funon models he requremen o mamse he alloaon of employee shf requess and mnmse under and oer saffng. The parameers q and p are he weghs for shf on and shf off requess respeely. For eample an employee may reques o work a eran shf ype on a parular day. The hgher he wegh he more mporan he reques s o he employee. If here s no reques hen he parameer has he alue zero. The arales y and z are he oal numers of saff elow and aoe he preferred oer leel for eah shf ype on eah day d. The parameers of mnmsng under and oer oerage. Resuls mn and ma are weghs o represen he mporane To prode oher researhers wh resuls o ompare agans we hae used he wo esng algorhms presened n [2] and Guro 5.6.3 [1] and appled hem o he new nsanes. All he epermens were performed on Inel Core 2 Duo 3.16GHz 8GB ram. The Guro soler was lmed o a sngle hread and a mamum me of 1 hour. Tale 2. lss he resuls. Known opmal soluons are n old.
Ejeon han Branh and Pre Guro 5.6.3 Insane Weeks Saff Shfs 10 mn 60 mn LB Sol. Tme (s) LB Sol. Tme (s) Insane1 2 8 1 607 607 558 607 0.27 607 607 1.62 Insane2 2 14 2 923 837 828 828 0.13 828 828 5.22 Insane3 2 20 3 1003 1003 1001 1001 0.45 1001 1001 13.54 Insane4 4 10 2 1719 1718 1716 1716 1.50 1716 1716 158.99 Insane5 4 16 2 1439 1358 1141 1160 25.61 1143 1143 1520.24 Insane6 4 18 3 2344 2258 1949 1952 10.46 1950 1950 440.93 Insane7 4 20 3 1284 1269 1055 1058 93.73 1056 1056 2152.48 Insane8 4 30 4 2529 2260 1297 1308 11831.06 1281 1323 3599.83 Insane9 4 36 4 474 463 406 439 76.99 247 439 3599.85 Insane10 4 40 5 4999 4797 4631 4631 113.44 4631 4631 244.20 Insane11 4 50 6 3967 3661 3443 3443 19.11 3443 3443 109.92 Insane12 4 60 10 5611 5211 4040 4046 1336.4 4040 4040 2303.84 Insane13 4 120 18 8707 3037 Ou of memory - - 1346 3109 3600.55 Insane14 6 32 4 2542 1847 Ou of memory - - 1277 1280 3600.13 Insane15 6 45 6 6049 5935 Ou of memory - - 3806 4964 3600.00 Insane16 8 20 3 4343 4048 3224 3323 265.02 3211 3233 3599.99 Insane17 8 32 4 7835 7835 Ou of memory - - 5726 5851 3600.00 Insane18 12 22 3 6404 6404 Ou of memory - - 4351 4760 3599.99 Insane19 12 40 5 6522 5531 Ou of memory - - 2945 5420 3605.90 Insane20 26 50 6 23531 9750 Ou of memory - - 4743-3600.05 Insane21 26 100 8 38294 36688 Ou of memory - - 20868-3600.21 Insane22 52 50 10-516686 Ou of memory - - - - 3600.19 Insane23 52 100 16-54384 Ou of memory - - - - 3600.43 Insane24 52 150 32-156858 Ou of memory - - Ou of memory - - Tale 2. Resuls Referenes 1. Guro Opmzaon In. Guro Opmzer Referene Manual. 2014; Aalale from: hp://www.guro.om. 2. E.K. Burke and T. Curos New Approahes o Nurse Roserng Benhmark Insanes European Journal of Operaonal Researh 237(1) (2014) 71 81. 3. T. Curos. Employee Shf Shedulng Benhmark Daa Ses. 2014; Aalale from: www.s.no.a.uk/~e/nrp.