Computational results on new staff scheduling benchmark instances

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1 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 Ths repor lss resuls of applyng he algorhms presened n [2] o he saff shedulng prolem enhmark nsanes [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 [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 Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane

2 Ineger Programmng Formulaon Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane 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.

3 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)}

4 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 [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.

5 Ejeon han Branh and Pre Guro Insane Weeks Saff Shfs 10 mn 60 mn LB Sol. Tme (s) LB Sol. Tme (s) Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Insane Ou of memory Insane Ou of memory Insane Ou of memory Insane Insane Ou of memory Insane Ou of memory Insane Ou of memory Insane Ou of memory Insane Ou of memory Insane Ou of memory Insane Ou of memory Insane Ou of memory - - Ou of memory - - Tale 2. Resuls Referenes 1. Guro Opmzaon In. Guro Opmzer Referene Manual. 2014; Aalale from: hp:// 2. E.K. Burke and T. Curos New Approahes o Nurse Roserng Benhmark Insanes European Journal of Operaonal Researh 237(1) (2014) T. Curos. Employee Shf Shedulng Benhmark Daa Ses. 2014; Aalale from: