Sgle mahe stohast aotmet sequeg ad shedulg We develo algorthms for a sgle mahe stohast aotmet sequeg ad shedulg roblem th atg tme, dle tme, ad overtme osts. Ths s a bas stohast shedulg roblem that has bee studed varous forms by several revous authors. Alatos for ths roblem ted revously lude shedulg of surgeres a oeratg room (Deto, 007, Deto ad Guta,00), shedulg of aotmets a l (Robso ad Che, 00, Vade Bosh, 000), (Wag, 997) metos the alato of ths roblem to shedulg of shs a ort. Bege ad Queyrae (009) dsuss ths roblem the otext of shedulg exams a examato falty (MRI, Sas). I ths aer the roblem s formulated as a stohast teger rogram usg samle average aroxmato. A heurst soluto aroah based o Beders' deomosto s develoed ad omared to exat methods ad to revously roosed aroahes. Extesve omutatoal testg shos that the roosed methods rodue good results omared to revous aroahes. I addto e rove that the fte searo samle average aroxmato roblem s NP-omlete.. Itroduto The roblem e address assumes a fte set of jobs th stohast roessg tmes. It s assumed that the roessg tme durato of the jobs are radom varables th ko jot dstrbuto. The margal dstrbutos of job duratos are ot assumed detal. The roblem requres us to fd the sequee hh to erform the jobs, ad to assg a startg tme to eah job. A job may ot beg before ts sheduled startg tme or may t beg utl the revous job s omlete. If the th job the sequee fshes before the st job s sheduled to start, the there ll be dle tme o the mahe. Coversely, f the th job the sequee fshes after the st job s sheduled to start, the job ll ur atg tme. Further, f the last job fshes after a redefed deadle, there ll be overtme. The objetve s to determe the sequee ad sheduled startg tmes of jobs o the mahe that mmze a eghted lear ombato of job atg tme, mahe dle tme, ad overtme. We assume a searate ost (er ut tme) for eah job for both atg ad dle tmes, ad a sgle overtme ost. Se e exltly osder the radomess of the job roessg tmes, the objetve s to mmze total exeted ost here exetato s take th reset to the jot dstrbuto of surgery tmes. Ths roblem has bee alled the aotmet shedulg roblem beause t s easy to evso by aalogy to shedulg aotmets a hysa s offe. Jobs rereset atet aotmets, hle the dotor reresets the mahe. Watg tme s the tme atets must at beyod ther sheduled aotmet tme hle dle tme reresets tme the dotor s ot busy hle atg for the ext atet to arrve. The roblem a be deomosed to to arts. The frst s to determe the sequee hh the jobs ll be erformed. Gve a sequee oe must ext determe the amout of tme to alloate to eah job, or equvaletly assg eah job a sheduled startg tme. Ths seod roblem (hh e all the shedulg roblem) has bee studed revously uder the ame stohast aotmet shedulg. Prevous aroahes to ths roblem lude usg ovolutos to omute startg tmes (Wess, 990), samle average aroxmato ad the L-Shae Algorthm (Deto, 00), ad heursts (Robso ad Che, 00). We ll aroah ths roblem usg samle average aroxmato ad lear rogrammg a smlar fasho to (Deto, 007).
Gve reasoably effet methods to solve the shedulg roblem, e ext develo a algorthm for the sequeg roblem. Aordg to (Guta, 007) the sequeg roblem s stll a oe questo. The ma dea of the roosed method s based o a Beders' deomosto sheme. The master roblem s used to fd sequees ad the sub-roblems are the shedulg roblems (stohast lear rograms) as dsussed above. The Beders' master roblem beomes extremely hard to solve as uts are added, thus e tur to heursts to aroxmate ts soluto ad geerate romsg sequees. The remader of the artle s orgazed as follos. I the ext seto a bref reve of the lterature related to the sequeg ad shedulg roblems ad stohast teger rogrammg s rovded. I Seto, the model s desrbed ad formulated. I Seto 4 e omlexty results for the sequeg roblem are reseted. I Seto 5 algorthms are roosed to solve the sequeg ad shedulg roblems. I Seto 6 a method s roosed to hoose the umber of searos. I Seto 7 omutatoal results are reseted. I Seto 8 olusos ad future researh dretos are dsussed.. Lterature Reve Relevat revous ork a be dvded to ategores: stohast aotmet shedulg ad stohast teger rogrammg. We beg th revous ork o the stohast aotmet shedulg roblem. (Wess, 990) foud the otmal sequee he there are oly jobs (.e. ovex order) ad the shoed that ths rtero does ot guaratee otmalty he the umber of jobs s greater tha. (Wag, 997) assumed job duratos ere..d. radom varables follog a Coxa (hase tye) dstrbuto. Se duratos ere..d., sequeg as rrelevat. He assumed osts for atg tme ad total omleto tme. He develoed a effet umeral roedure to alulate mea job flo tmes the solved for the otmal sheduled startg tmes usg o-lear rogrammg. For examles th u to 0 jobs, he shoed that eve though job duratos are..d., the otmal startg tmes are ot equally saed. (Deto, 007) formulated the sequeg ad shedulg roblem as a stohast teger rogram ad the roosed smle heursts to determe the sequee. Oe a sequee s gve, the shedule of startg tmes as foud usg a samle average aroxmato (.e. searo based) aroah. The resultg shedulg roblem as sho to be a lear stohast rogram hh they solved by a L- shaed algorthm desrbed (Deto, 00). To determe a sequee they roosed three methods: sort by varae of durato, sort by mea of durato, ad sort by oeffet of varato of durato. These smle heursts ere also omared to a smle terhage heurst. The alato studed (Deto, 00) as shedulg surgeres a sgle oeratg room. They reorted results th real surgery tme data ad u to surgeres (jobs). They also assumed equal ealty osts aross surgeres for atg ad dle tme. They foud that sort by varae of durato gave the best results. (Kaador ad Koole, 007) assumed that job duratos ere Exoetally dstrbuted th dfferet meas ad that atet arrvals a oly be sheduled at fte tmes (every te mutes). Ther objetve futo luded atg tme, dle tme, ad overtme osts. Gve these assumtos, a queug theory aroah as used to alulate the objetve futo for a gve shedule of startg tmes. They further roved that the objetve futo as mult-modular th reset to a eghborhood that a move the start tme of jobs oe terval earler or later. Ths result guarateed that a loal searh algorthm ths eghborhood ll fd the otmal soluto. I (Kog, 00) the authors develoed a robust otmzato aroah to the aotmet shedulg roblem. They assumed that the dstrbutos of the serves
as uko, ad mmzed the orst ase exeted value over a famly of dstrbutos to determe the shedule. I (Vade Bosh, 000) ad (Vade Bosh, 00) the authors also assumed dsrete sheduled startg tmes (at 0 mute tervals over hours) ad luded ealtes for atg tme ad overtme. They assumed three lasses of atets a outatet aotmet shedulg settg here duratos ere..d. th lass but dfferet betee lasses. They used Phase Tye ad Logormal dstrbutos to model the three durato dstrbutos. Gve a sequee, they roosed a gradet based algorthm to fd the otmal shedule of startg tmes based o submodularty roertes of the objetve futo. They roosed a all-ars sa-based steeest deet loal searh heurst to fd a sequee. They stoed the searh after a fxed umber of teratos or he a loal mmum s foud. They reorted testg th smulated data for ases th 4 ad 6 jobs ad oluded that the heursts rodued good results terms of teratos ad otmalty ga he omared th exhaustve eumerato. The aroah to the roblem take ths aer s stohast teger rogrammg, thus revous aroahes to smlar roblems are relevat. There s a rh lterature o stohast teger rogrammg. I (Shulz, 00) a thorough reve of methods for solvg stohast rogrammg roblems th teger varables as gve. For the tye of roblem e address, there are several methods that mght seem to aly. Our roblem has both teger varables (for sequeg) ad otuous varables (for sheduled startg tmes) the frst stage, otuous varables the reourse futo (atg ad dle tmes), ad omlete reourse. For fte searo roblems, (Laorte, 99) roosed the Iteger L-Shaed Method, a algorthm that s sutable for solvg Stohast Programs here the frst stage varables are bary ad the reourse ost s easly omutable. The method uses Beders' Deomosto ombed th uts that dffer from tradtoal Beders' uts. Aother aroah based o searo deomosto as roosed (Caroe, 997). After deomosg the roblem by searos, they the solved a roblem th relaxed o-atatvty ostrats to get a loer boud th a brah ad boud sheme. Fally there s the dely used Beders' deomosto aroah. I Beders' aroah oe may deomose by fxg the teger varables or by fxg the set of all frst stage desos. The roblem e address s suh that f the teger (sequeg) varables are fxed, the otuous (shedulg) varables a be omuted th relatve ease by solvg a lear rogram usg a teror ot method. Ths makes Beders' deomosto artularly attratve for our roblem. We also exermeted brefly th the Iteger L-shaed method but foud that t offered o advatage over the Beders' aroah ths ase. Searo deomosto s ot arorate for our roblem se solutos to the sgle searo roblems rovde o useful formato about the overall soluto. Ths s beause ay searo subroblem, the startg tmes are smly set equal to the fsh tme of the revous job alays resultg zero ost. The urret lterature also dstgushes betee fte searo roblems ad fte searo roblems. For the fte searo ase, to methodologes are otetally useful. I (Home de Mello, 00) the authors roosed a algorthm for solvg the samle average aroxmato (fte searo roblem). They aled ths roedure may tmes utl stog rtera related to statstal bouds are fulflled. The other method amed at the fte searo ase s Stohast Brah ad Boud (Ruszzysk, 998). Ths method arttos the teger feasble sae ad omutes statstal uer ad loer bouds, the uses these bouds the same ay tradtoal brah ad boud uses uer ad loer bouds to fd the true otmal value th robablty oe. Solvg our samle average aroxmato roblem turs out to be very tme osumg, thus ether of these fte searo aroahes are ratally vable our ase.
. Problem Statemet We assume a fte set of jobs th duratos that are radom varables. We assume these job duratos have a ko jot dstrbuto, ad are deedet of the osto the sequee to hh the job s assged. Oly oe job may be erformed at a tme, ad overtme s urred he job roessg exteds ast a deadle reresetg the legth of the ork day. To sets of desos must be made, frst the job sequee must be determed, the a startg tme must be assged to eah job. I alato to surgery shedulg, the startg tme a be thought of as the tme the atet s sheduled to arrve thus a job (surgery) may ot beg before ts sheduled startg tme. The objetve futo ossts of three omoets, atg tme (the tme a atet must at betee hs/her sheduled startg tme ad atual startg tme), dle tme (the tme the O.R. s dle hle atg for the ext atet to arrve) ad overtme. Gve a sequee, startg tmes for eah job, ad the durato dstrbutos, the exeted atg tme ad dle tme before eah job ad the over tme a be estmated by averagg over a samle of searos. The objetve futo s a eghted lear ombato of these three exeted osts. Note that atg ad dle osts may be dfferet for eah job. Ths roblem has bee modeled as a to stage stohast rogram th bary ad otuous varables the frst stage desos (Deto, 007). They ororated the roessg tme uertaty to the model usg a samle average aroxmato (.e. searo based) aroah. The bary varables defe hh job (surgery) should be laed the th osto the sequee. The startg tmes ad the bary varables are all luded the frst stage desos. Ths roblem a be formulated as sho belo. Ths model s smlar to (Deto, 007).. Problem formulato Ides ad Sets J Jobs to be sheduled j,...,. I Postos the sequee,...,. K Searos to be osdered k,...,k.
Parameters atg tme ealty for surgery j. dle tme ealty for surgery j. overtme ealty. d tme beyod hh overtme s urred., are suffetly large umbers. durato of job j searo k. Varables t sheduled startg tme for the surgery osto. atg tme searo k he surgery j s osto. dle tme searo k he surgery j s osto. overtme searo k. g k slak varable that measures the earless th reset to tme d. x j a bary varable deotg the assgmet of surgery j to osto. Costrats (),() defe the atg ad dle tme for every surgery ad searo. (4),(5) assure eah surgery s assged to oe osto the sequee. (6),(7) are logal ostrats that fore atg ad dle tmes to orresod to o-zero x j. Proertes of the formulato Oe the sequee s fxed the atg ad dle tmes ad tardess a be omuted for every searo as a futo of the sheduled startg tmes as sho belo (Deto, 007). I the stohast rogrammg frameork the omutato of these varables a be see as the reourse futo. Further, ay sequee ll yeld a fte objetve futo value therefore e have omlete reourse. Thus e ll oly eed otmalty uts to solve ths roblem usg Beders' deomosto. The atg tme ad sheduled startg tme of the frst surgery the sequee are both assumed to be zero.
. Stregtheg the MIP Formulato If e aly tradtoal brah ad boud to the roblem formulated seto, the eakess of the formulato aused by the bg M ostrats ll egatvely affet erformae. Se e ll omare the erformae of our algorthms to brah ad boud, t s mortat to stregthe the MIP formulato to the extet ossble. Through straghtforard but legthy aalyss e foud fte values for M ad M that reserve otmalty. Proof of the valdty of the results may be foud (Malla et al., 009). Here e smly state the results. We set M (hh aears the slak tme ostrats (6)) as follos: We set the values that aear the atg tme ostrats (7) as follos: here orresods to the jth largest value terms of -. Wth these "bg M" values, the formulato a be somehat tghteed. 4. Problem Comlexty Prevous authors (Deto, 007), (Guta, 007) have seulated that the samle average aroxmato sequeg ad shedulg roblem (SAA-SSP) s NP-Comlete, but to the best of our koledge the questo s stll oe. I ths seto e rove SAA-SSP th to searos ad th equal dle ost but dfferet atg osts for eah job s NP-Comlete. The roof uses oets smlar to those (Garey, 976). SAA-SSP Feasblty Verso Gve a olleto of jobs I dexed by, th duratos searo k gve by 0, ad a budget B of shedule ost, does there exst a sequee ad sheduled startg tme for the jobs I hose ost does ot exeed B? We ostrut a olyomal trasformato to the -Partto roblem to sho our roblem s NP-Comlete. The -Partto roblem as defed belo s ko to be NP-Comlete (Garey 976). Defto -Partto (Garey 976):
Gve ostve tegers, R, ad a set of tegers A {a, a,...,a } th a R R R ad a 4 < < for, does there exst a artto < A, A,.,...A > of A to -elemets sets suh that, for eah,, a R? a Theorem A The Samle Average Aroxmato Sequeg ad Shedulg Problem (SAA-SSP) th to searos s NP-Comlete he the atg osts are alloed to dffer betee jobs. Costruto We sho that -artto olyomally redues to a artular SAA-SSP th searos. We ostrut a stae of SAA-SSP th 4 jobs ad searos hh the frst jobs (job set G dexed by,,) have duratos by,, 4) have duratos ξ a adξ 0 for umber) The dle ost ealtes are hose as. The remag jobs (job set V dexed ξ H, ad ξ H R for 4 s. ( H s a teger 0 KR for all 4 jobs. The atg ost for jobs R V are 5 for,..., 4 ad for jobs set G: for,...,. The budget of the 5R shedule ost s B. 4 Itutvely, the ma dea of the roof a be see Fgure. The shedule ossts of bloks th eah blok otag oe of the arttos searo. The dle ost s set hgh eough to guaratee that there ll be o dle tme a otmal soluto to the SAA-SSP. The sheduled startg tme for every job s set equal to ts atual startg tme searo. We frst sho that f a -artto exsts, the shedule belo ll meet the budget B. We the sho that f the shedule s ot the "-Partto shedule" as sho Fgure, or f o artto exsts, the the budget to be exeeded. Blok Blok R R Searo H a, a, a, H a, a, a, Searo HR HR Sheduled startg tmes Sheduled startg tmes Zero roessg tme G jobs Zero roessg tme G jobs Fgure : Searo Case: (a, meas artto the frst elemet)
Proof of Theorem We start by shog that f a soluto to the -Partto roblem exsts, the t leads to a feasble soluto of SAA-SSP that meets the budget B. Assume there s a -artto {A, A,...,A } of A that fulflls the odto A {a,, a,, a, } ad R a, α α,. From ths artto e ostrut the sequee ad shedule as sho Fgure. Reall that searo the G jobs all have zero durato. The ost Z* for the shedule Fgure a be sho to be bouded above by the budget B 4 5R, as follos: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( 4 5 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0), os ( 0 4 5 ) 0 ( 0) (,,,,,,,,,, 4 4 4 4 4 4 * α s s s s a o boud uer R R R R artto from erfet shedule ostruted a a a R from ostruto searo G jobs of tmes at a a a a a a tmes V at ts at G all R searo tmes at all tmes dle all s s s s Z Lemma For ay sequee of jobs, the otmal sheduled startg tmes for the set of 4 jobs th dle ost ealtes KR s 0 (K s the umber of searos), at ost ealtes { },..., ad for R 4,..., 5 s gve by: { } k k K k t M t * * ξ. Ths lemma basally says that f the dle ost s set hgh eough, the otmal shedule ll be the oe that otas o dle tme. Ths s equvalet to settg the sheduled start tme of the th job the sequee equal to the earlest fsh tme ay searo of the - st job the sequee. Proof: We ll rove ths lemma by otradto.
* * k k Case : Assume that a job suh that M{ t } t k K that job has ostve atg tme every searo (.e. fat that ths s a otmal soluto se e a rease < ξ the otmal shedule. Ths meas k > 0 k K ). Ths otradts the assumed * t utl some k beomes zero thout affetg the atg tme of the other jobs. Ths mles that e ere a subotmal soluto. Case : t M t ξ t the otmal shedule ad Z* s > k K * k k * Assume that a job suh that { } δ the otmal objetve futo value. Ths meas that e have dle tme of legth δ for at least oe searo job (see Fgure ). Ths reases the objetve futo by s δ. The best ossble ase s that the atg tme of all future jobs all other searos s redued by δ (see Fgure ) thus redug the objetve futo by (K-) δ 4 j j. We have: 4 4 5R j δ ( K ) δ < ( K ) δ < ( 4 )( K ) δ Max ( 4 )( K ) j Thus s e set s ( 4 )( K ) 5R the otmal shedule ll ota o dle tme. -------------------------------------------------------- δ δ δ Fgure : best ossble ase f e rease * t δ Lemma For ay gve sequee of roblem SAA-SSP, the otmal shedule leads to a soluto that has a teger valued objetve futo. Proof of Lemma
By lemma e ko that the sheduled startg tmes follo a reursve formula that deeds o atg * * * k k tme ad duratos of jobs earler the sequee: t M{ t } t 0, k K ko that e a exress atg tme of the th job the sequee usg: k, k j k j ξ Further e 0 ξ t. Se the duratos are tegers, ad the tegers are losed uder addto ad subtrato, e a olude that all atg tmes are teger (ad all dle tmes are zero). Se atg osts are also teger, the objetve futo value of the otmal soluto to the shedulg roblem s a teger. Lemma The atual startg tmes of jobs set V must be the same eah searo otherse the budget ll be exeeded. Proof of Lemma Se o dle tme a exst (Lemma ), the shedule for searo must aear as belo. If the V jobs searo do ot start at the same tme as searo, there ll be o-zero atg tme for at least V job. The atg tme must be at least (from the teger lemma ), ad se atg ost for jobs V s 5R/, the budget s exeeded Thus the shedule for V jobs must look lke ths R R R Searo H H H Searo HR HR HR Fgure : Otmal sequeg atter. Lemma 4 There must be subsets of G jobs that ft erfetly to the frst - oe slots (of dth R) the shedule above f there exsts a erfet artto. These subsets osst of G jobs ad orresod to the erfet artto. Proof of Lemma 4 If do ot exst these subsets of G jobs there ll be dle tme the shedule, ad the budget ll be exeeded (by Lemma ). Therefore these subsets must osst of G jobs due to the R R bouds < a 4 <, otherse the subsets ll add to more or less tha R. The oly ossble soluto for these subsets s the erfet artto ad by defto of the artto roblem, the remag jobs must ft erfetly to the last oe slot. Wth Lemma 4 e a olude that the SAA-SSP th to searos s NP-omlete he the atg osts are alloed to dffer betee jobs.
5. Proosed Soluto Methodology The aroah e roose for the SAA-SSP uses a heurst method to fd good solutos a reasoable amout of tme. The Master roblem our Beders' Deomosto s a teger rogram, ad thus s dffult to solve ad eve more dffult as e add more uts every terato. Relaxg the sde ostrats (otmalty Beders uts) results a easy to solve assgmet roblem. We use ths roerty to ostrut a heurst to geerate good feasble solutos to the master roblem. The dea of solvg the master roblem heurstally has also bee roosed by several authors ludg (Cote, 984) ad (Aardal, 990). 5. Proosed Algorthm The bas outle of our algorthm s as follos; 0. start th a arbtrary sequee. Set the Uer Boud (UB) ad Loer Boud (LB)-.. Solve the LP shedulg subroblem for the urret sequee yeldg z*,x*,*.. Udate the UB f z*<ub.. Geerate a Beders' ut from the otmal extreme ot the dual (*). 4. Use our smlfed master roblem heurst to try to fd a feasble soluto to the master roblem. 5. If the e sequee s the same as the revous sequee, restart the heurst. 6. If (umber of teratos < max teratos) go to, else sto. Fgure 4: Flohart of the roosed algorthm.
5.. The LP Shedulg Subroblem Gve a sequee, fdg the otmal sheduled startg tmes s ko to be a lear rogram (Deto, 00, Deto, 007). Ths s the LP subroblem ste oe above. The sze of ths LP gros radly th the umber of searos. (Deto, 00) roosed solvg ths roblem usg the L-Shaed method. We develoed a etork flo lke aroah based o the fat that the dual s a etork flo roblem th sde ostrats. For roblems th 500 searos ad 0 surgeres the etork flo based method as sueror to both the Stadard L-Shaed Method ad Stadard Dual Smlex. Hoever, further testg revealed that teror ot methods ere at least oe order of magtude faster tha ay of the other methods. Thus, e use the Clex barrer algorthm to solve the subroblems th the algorthm. The LP subroblem hh, for a gve sequee, fds sheduled startg tmes that mmze total ost averaged over the searos s gve belo. Ths s the dual of the shedulg roblem, 5. The Master Problem Beders' Deomosto
We use a straghtforard alato of Beders' Deomosto o the mxed teger stohast rogram defed seto. The master roblem otas the teger sequeg varables (x j ). The master roblem formulato aears belo. The master roblem at terato T Beders' Deomosto. The ostrats () are the Beders' otmalty uts. The oeffets of these ostrats ome from the er rodut betee the dual varables (, ad q varables orresodg to the dual varables of the ostrats (6) ad (7)) ad the surgery duratos ad bg M s. These uts ota formato foud from solvg the LP subroblems ad allo us to mltly elmate sequees that ll ot mrove the objetve futo. To assure overgee of the teratve Beders' soluto aroah, all that s eeded s a feasble teger soluto to the master roblem at eah terato. Hoever, se ths roblem s omutatoally hard to solve, e ostrut a smlfed verso that s easy to solve, but does ot guaratee a feasble soluto to the master roblem. 5. Smlfed Master Problem
We remove the sde ostrats (Beders' uts) ad reate a e objetve futo based o the oeffets of these ostrats. The resultg smlfed master roblem (SMP) s a assgmet roblem ad thus a be easly solved. The dea s to ature the formato the ut ostrats suh a ay that e a fd a feasble soluto to the master roblem th reasoable relablty. We exermeted th several methods for ostrutg these objetve futo oeffets. Based o ths exermetato the best method as foud to be. here the ome from a modfed Beder s otmalty ut oeffets. Sefally, e ostrut based o Beders ut but th a slghtly modfed dual vetor. We set to zero the j ad q j dual varables. These dual varables orresod to ostrats (6) ad (7) the orgal formulato. Settg the j ad q j dual varables equal to zero s vald he the osts are equal, but does ot guaratee vald uts the uequal ost ase. Hoever, retag these dual varables adds ueessary ose to the formato otaed the dual the mortat ostrats () ad (). We foud that ths method rodues better results, ad se the roosed algorthm s heurst, e adot t. For a artular terato t the ba vetor s geerated as follos: The desrablty of usg the max oerator to aggregate the ostrats s suorted by the follog roosto. Proosto The otmal soluto to the SMP s a uer boud of the master roblem Beders' Deomosto. Furthermore, f the otmal objetve futo value of the SMP s less tha the urret uer boud, the otmal soluto s a feasble soluto to the master roblem.
Proof: I the frst terato (.e. th oe sde ostrat), t s lear that the smlfed roblem fds a feasble soluto to the master roblem. I subsequet teratos th more tha oe sde ostrat the master roblem a be defed as follos, Where S s the set of sequees ad F t s the left had sde ostrat t for sequee the Master Problem. Gve a sequee, exatly x j 's ll be equal to. Let (,.., ) be the oeffets ostrat t for the x j 's that are equal to uder sequee. Thus the Master Problem a be rtte as the summato of oeffets, ad ths form e a see that Thus the soluto of the SMP s a uer boud to the master roblem. If ths uer boud s less tha the urret boud, e have a feasble soluto to the Master Problem. Of ourse there s o guaratee that the uer boud rovded by the SMP mroves the overall uer boud. Whe to thgs a to hae, () e get a e sequee hh ase e otue teratg or, () e get the same sequee hh ase the algorthm ll rodue ths same sequee o subsequet teratos (.e. e are stuk). Whe the algorthm gets stuk e restart the algorthm from a e sequee usg oe of the restart rules desrbed the ext seto. 5.4 Restart Rules Oe the algorthm returs the same sequee for to oseutve teratos, t ll otue to do so ad ftum, therefore, e eed to restart from a e sequee. Gve the e restart sequee, e smly remove all revous Beders' uts ad start aga. We tred three restart rules as dsussed belo.
5.4. Worst Case Ths at-ylg rule s based o fdg a sequee "far ay" from sequees vsted se the last restart. To aomlsh ths e smle relae "mmze" th "maxmze" the SMP objetve futo. 5.4. Perturbato Ths at-ylg rule tres to slghtly erturb the sequee order to esae the yle. here. ad ba m ad U(0,) reresets seudo-radomly geerated Uform(0,) devates. The value of ba m as hose so that the sequee ll ot hage muh. Ufortuately, ths rule does ot guaratee a e restart sequee dfferee from all revous restart sequees. Thus e may reeat a revous terato. To avod ths e develoed the ext method. 5.4. Memory Radom Restart Ths at-ylg rule s based o fdg a sequee dfferet from all revous restart sequees. We store all restart sequees R,..,R q here q s the umber of tmes that e have restarted the algorthm. The dea of ths restart rule s to guaratee that e are ot gog to yle betee restartg ots ad therefore the algorthm ll vst at least oe e sequee eah terato. We formulated the follog feasblty teger roblem (M q ).
Where s radom umber geerated from ItUform(0,-). We use Clex to fd a feasble soluto. Clex fds a feasble soluto extremely qukly as t turs out, so that the mat o the algorthm's exeuto tme s mmal. The ostrats that ota the revous restart ots guaratee that e ll ot restart the ext terato from these revous sequees. Further, the radom umber serves as a kd of dstae from the set of restartg ots here he - e mght obta a feasble sequee for M q that dffers from restart sequees R r at most elemets. Whe 0 the e restart sequee ll dffer from every revous restart ot all the elemets. 6. Auray of the Fte Samle Average Aroxmato The ultmate goal s to solve the fte searo stohast rogrammg roblem. The method roosed ths researh attemts to solve a fte searo roblem. Further, the method does ot guaratee a otmal soluto to ths fte searo roblem. I ths seto e ll try to evaluate ho our algorthm ll erform o the fte searo roblem. (Lderoth, 00) develoed a ay to omute statstal uer ad loer bouds o the otmal soluto to the fte searo ase based o exteral samlg tehques. Ufortuately, ths method requres solvg a fte searo roblem may may tmes ad thus s omutatoal rohbtve our ase, eve for a small umber of searos. We therefore desged a smler exermet to quatfy the erformae of our algorthm o the fte searo ase. There are to ma ssues. The frst ssue s samlg error, that s: "Ho ell s the fte searo objetve futo aroxmated by the fte searo (samle average) objetve futo?" The seod ssue s: "Ho does the ru tme of the algorthm affet erformae th reset to the fte searo roblem?" There s a bas tradeoff e eed to evaluate. For a fxed omutato tme alloae, ho may searos should e use? If e use may searos, e ll oly have tme to geerate a fe addate sequees thus lmtg our ablty to "otmze". O the other had th fe searos, e a geerate may sequees, ad erhas eve solve the fte searo roblem to otmalty. Hoever, the otmal soluto to the fte searo roblem may be a oor soluto to the fte searo roblem he the umber of searos s small. To quatfy ths tradeoff e ostruted the follog exermet. We geerated 50 test roblems th 0 jobs eah. For eah of these 50 roblems e geerated fte searo staes th 50, 00, 50, ad 500 searos. For eah of these 00 staes e the ra the algorthm for 0,40,60,80,00,0,40 seods (the exermets ere oduted uder the same omutatoal odtos detaled seto 7). For eah ru of the algorthm e saved the best te sequees here "best" s th reset to the umber of searos used the urret ru. Ths resulted the geerato of 7x4x080 (ot eessarly all dfferet) total sequees for eah of the 50 test roblems. For eah of these sequees, e solved the LP shedulg sub-roblem th 0,000 searos ad reorted the best sequee S* 0000 ad objetve futo z* 0000 foud. Ths serves as our aroxmato to a overall best soluto to the fte searo roblem for eah of the 50 test ases. Our frst set of results s amed at uderstadg the samlg error. The bas questo e asked s: "Ho ofte does the sequee our algorthm thks s best tur out to be the best sequee the 0,000 searo ase?". For eah of the "0 best" sequees retured by the algorthm after 40 seods of rug, the lots belo sho the umulatve frequey for hh that soluto as best for 0,000 searos. For examle Fgure 5 shos that for 500 searos, the soluto judged to be the best of the to te by the algorthm as deed the best of the to te 40% of the tme. By examg the lot e see that 50 ad 00 searos ur sgfat samlg error se eah of the to te solutos has essetally the same robablty of beg best for the 0,000 searo ase. Wth 500 searos, e see
that solutos that are good for the fte searo roblem also ted to be good for the 0,000 searo roblem. We a olude that 00 searos does ot rovde a suffet aroxmato to the 0,000 searo ase. 500 searos seems to rovde a reasoable aroxmato hle 50 s borderle. Fgure 5: Cumulatve Frequey by Order Posto The seod set of results demostrates the tradeoff betee the umber of searos ad the umber of sequees geerated. For eah umber of searos ad ru tmes e take the best sequee (aordg to the algorthm), evaluate t th 0,000 searos usg the LP shedulg sub-roblem, the omute the eret error from z* 0000. The eret error s averaged over the 50 test staes ad sho the lot belo. Fgure 6 shos that 50 searos has the best erformae overall. The 50 ad 00 searo ases do ot erform ell for ay ru tme beause, as sho revously, they do ot aroxmate the 0,000 searo roblem ell. After 40 seods, the 500 searo ase has "aught u th" the 50 searo ase. If oe ere to ru the algorthm for more tha 40 seods, 500 searos ould lkely be the better hoe.
Fgure 6:Peret Error Results 7. Comutatoal Exeree for the Fte Searo Case To quatfy the erformae of our algorthms for solvg the fte searo roblem e erformed to sets of exermets. I the frst e omared solutos from the algorthms th the otmal soluto o smaller roblems. I the seod e omared algorthm erformae th the to smle behmark heursts over a broader set of test roblems. The exermets ere oduted o a Petum Xeo.0 GHz (x) th a 4000 (MB) server the Coral Lab at Lehgh Uversty. 7. Comarso th Otmal Solutos I ths seto, e omared the solutos foud by our memory restart algorthm to the otmal soluto foud by Clex usg brah ad boud. We geerated to ategores of test roblems, those th equal osts (.e. osts are equal aross surgeres) ad those th dfferet osts. It s orth otg that the ases th equal ost ere sgfatly easer for Clex to solve to otmalty tha ere the ases here the osts ere dfferet. We geerated 0 staes th 0 searos ad 0 staes th 00 searos eah ategory. I the ext seto e dsuss detal ho e geerated a large sute of test roblems by varyg a varety of fators. I ths seto e seleted staes so as to sa a de rage of these fators.
To fd otmal solutos e used Clex 0. to solve the stregtheed IP formulato dsussed subseto.. Wth Clex the "MIP emhass" arameter as set to "otmalty". We ra our memory restart algorthm for 000 ad 4000 teratos for eah test roblem. Table shos the results of ths exermet. The ru tme results ad gas are the average ru tme ad average ga er roblem stae. Note that some ases Clex took several days to fd the otmal soluto hle our roosed method took at most 4 mutes to fd solutos th average otmalty ga at most 5.8%. It s terestg to ote that the otmalty ga seems to be smaller for the roblems th more searos. Oe ossble exlaato for ths behavor s that as the umber of searos reases, the umber of teratos betee restarts of our algorthm also reases (see Table 6). Table : Otmalty Ga 7. Comarso th Behmark Heursts 7.. Behmark Heursts We mlemeted to smle heursts for omarso uroses. The frst s the ''sort by varae'' heurst roosed by (Deto, 007). Ths smly sequees jobs from smallest to largest varae. Ths heurst a be exeted to ork farly ell the equal osts ase, but there s o reaso to exet t to ork ell the uequal osts ase. The seod behmark heurst as a smle erturbato heurst based o sort by varae. A radomly geerated erturbato as added to eah job's varae, the jobs ere sequeed from smallest to largest erturbed varae. The erturbatos ere geerated from U, (,here s the total umber of jobs ad s the maxmum stadard devato of the jobs). The dea s to geerate a umber of sequees that are lose to the sort by varae sequee. We geerated the same umber of sequees as ere geerated by the Beders' based heursts ad reorted the best oe foud. Se omutato tme s domated by the tme requred to solve the LP subroblems, the ru tmes ere roughly equvalet. 7.. Test Problems We reated a set of test roblems based o fators that mght affet algorthm erformae. The fators hose ere: () umber of surgeres, () resee of overtme ost, () at ad dle ost struture, (4) umber of searos, ad (5) surgery tme dstrbuto (Table ). The umber of teratos as fxed at 000 (e study the affets of umber of teratos the seod exermet to follo).
Table : Exermet Desg Note that the des,,,4 the ossble values of the fator Surgery Dstrbuto are used as the labels of Data Fgure 7. The meas ad varaes of surgery durato dstrbutos ere based o real data from a loal hostal. We the geerated smulated surgery tmes from ormal dstrbutos (truated at zero) th arameters refleted the real data. I table uder surgery dstrbuto the symbol, meas all surgery duratos ere geerated usg the same mea ad stadard devato (86, 66 ). The symbol, meas that as set at 66 ad as set based o the oeffet of varato geerated from a uform(0.,.05)dstrbuto. The symbol, meas that as set at 86 the as set based o the oeffet of varato geerated from a uform(0.,.05) dstrbuto. The symbol, meas that as frst geerated from uform(90,00) dstrbuto the as set based o a oeffet of varato geerated from a uform(0.,.05) dstrbuto. I the equal ost ase e geerated a sgle atg ost ad dle ost eah from a uform (0,50) dstrbuto. These to osts ere the aled to every surgery. I the uequal ost ase dvdual dle ad atg osts ere geerated for eah surgery, aga from a uform (0,50) dstrbuto. Whe overtme s luded, the over tme ost s set to.5 tmes the average of the atg osts. The deadle as set equal to the sum (over surgeres) of the average (over searos) durato lus oe stadard devato (over searos) of ths sum. We reated a full fatoral exermetal desg ad erformed 5 relates for eah ombato of fator levels. Ths resulted 00 staes for eah of the fve algorthms tested: Aroxmate Beders' Deomosto (th the three dfferet restart rules), sort by varae, ad erturbed sort by varae th 000 teratos. We geerated 000 (ot eessarly uque) sequees for eah algorthm (exet sort be varae), solved the LP sub-roblem for eah to get the objetve futo, ad reort the best soluto foud. Thus for eah roblem stae e have fve solutos, oe for eah heurst. We take the best soluto of these fve, the omute the eret ga from ths best soluto for eah heurst for eah roblem stae. The overall average ga results aear table. Table : Average eretage ga over the best soluto foud v/s searos
Table shos the algorthm erformae as the umber of searos vares. It s terestg to ote that as the umber of searos reases, the dfferee erformae dereases. I artular for the equal osts ase, the smle sort by varae heurst erforms qute ell omared th the other methods. Se the ultmate goal s to solve the fte searo roblem, t ould seem that sort by varae s a effetve heurst the ase of equal osts. Whe osts are ot equal, sgfat mrovemet over sort by varae s ossble. The results of table sho that all three Beders' based algorthms sgfatly out erform the smler heursts. I order to vestgate ho the algorthm as affeted by the exermetal fators e erformed a aalyss of varae ad grahed terato lots. The terato lots, hh aear Fgure 7, sho that the erformae of eah algorthm as reasoably uform over the dfferet fators. Iterato Plot (data meas) for Ga 4 0 5 0 A Be Memory A Be Pert A Be Worst Pert Beh Sort Varae 0.0 Overtme 0 0.0 0.5 Overtme Data 0.5 0.00 Data 4 0.00 Cost 0.0 0.5 Cost 0 0.0 0.5 Surgeres 0.00 Surgeres 0 5 0 0.00 0.0 0.5 0.00 0 0 50 0 Searos 00 50 500 Algor 0.0 0.5 0.00 Searos Algor 0 50 00 50 500 A Be Memory A Be Pert A Be Worst Pert Beh Sort Varae Fgure 7: Iterato lot exermet
The revous exermet fxed the umber of teratos at 000. We oduted a seod exermet hh examed the erformae of the three algorthms over a varyg umber of teratos. I ths exermet the fators hose ere: umber of surgeres, umber of searos ad umber of teratos (000, 000, 000, 4000). We further assume o overtme ost, the dstrbutos of the job duratos are from the ase here ad µ have o restrtos, ad the ost oeffets are ot equal. A summary of the exermet s table 4. The results of the aalyss of varae sho that all ma effets (algorthm, teratos, searos, ad umber of surgeres) are statstally sgfat. I artular the "memory" restart heurst as (statstally) sgfatly better tha the other to. Iterestgly, o to fator or hgher teratos ere sgfat mlyg that algorthm erformae behaved a very uform maer aross the other fators. From ths e olude that the memory heurst aears to be the best a statstal sese all ases, although the atual dfferee erformae s farly small. The terato lots Fgure 8 also llustrate these results. We also reort the average ru tmes for the memory algorthm ad the average umber of restarts Tables 5 ad 6. I table 5 oe a observe a lear deedey betee the umber of searos ad umber of jobs ad the roessg tme. The majorty of the tme s set solvg the LP subroblem thus the roessg tme reases a lear fasho. We also observed that the amout of tme set the soluto of the smlfed master roblem does ot hage th the umber of teratos. Table 4: Desg for exermet Fgure 8 : Iterato lots for exermet
Table 5: Average roessg tme seods for memory algorthm over 000 teratos. 0 searos 50 searos 00 searos 50 searos 500 searos No OT OT No OT OT No OT Over No OT OT No OT OT 0 Surgeres 5.8 7. 8.4 8.9 4.7 9.0 97.4.6 00. 4.9 5 Surgeres 8.0 9.4 8.0 9.9 69.7 67. 47.0 04. 8. 45.0 0 Surgeres 0.5.9 40.9 56.9 99.5 04. 8.5 5.9 694.7 77.8 Table 6: Average restarts for memory algorthm over 000 teratos. 0 searos 50 searos 00 searos 50 searos 500 searos No OT OT No OT Over No OT Over No OT OT No OT OT 0 Surgeres 46.6 89.7.5 49.5. 50..6 49.4. 47.8 5 Surgeres 6. 64. 7..5 6.5.0 6.6. 6.7.0 0 Surgeres 8.7 68.0. 4.5 0. 4.6 0.0.7 0. 6.4 8. Colusos I ths aer e develoed e sequeg algorthms for the stohast aotmet shedulg roblem. Fe aers have addressed ths roblem, the loe exeto beg (Deto, 007). Deto tested some smle heursts ad shoed that "sort by varae" as the best of those tested. I ths aer e develoed e algorthms based o Beders' deomosto hh erform sgfatly better tha sort by varae ad a erturbed sort by varae heurst, eseally the uequal ost ase. To be far, the roosed algorthms utlze muh more omutg tme (a oule of mutes) tha sort by varae, but roughly the same tme as erturbed sort by varae. The algorthm ru tmes are more tha suffet for mlemetato o real roblems. Of the three Beders' based algorthms, the "memory restart" method rovdes the best results (statstally), but the dfferee betee the three methods s farly small from a ratal stadot. The results further shoed that the relatve erformae of the three algorthms s uform aross the several fators used to reate roblem staes th dfferet haratersts testg. It s orth otg that the geeral aroah used to reate our heursts may ork ell for other roblems here the master roblem thout Beders' uts s easy to solve, but the roblem th uts s hard. Ths aroah seems artularly ameable to stohast sequeg roblems here the master roblem (before Beders' uts) s a assgmet roblem. A fal otrbuto of ths aer s a formal roof that the sequeg roblem s NP-Comlete. Whle ths has bee alluded to by several authors, the questo, to the best of our koledge, as revously oe.
Referees Aardal, K., T. Larsso. 990. A beders deomosto based heurst for the herarhal roduto lag roblem. Euroea Joural of Oeratoal Researh 45() 4-4. Mehmet A. Bege ad Maure Queyrae. Aotmet shedulg th dsrete radom duratos. Uder reve (A verso of ths aer aeared the roeedgs of the Symosum o Dsrete Algorthms 009 (SODA09), Uversty of Wester Otaro, 00a. Caroe, C. C., R. Shultz. 999. Dual deomosto stohast teger rogrammg. Oeratos Researh Letters (4) 7-45. Cote, G., M. A. Laughto. 984. Large-sale mxed teger rogrammg: Beders-tye heursts. Euroea Joural of Oeratoal Researh 6() 7 -. Deto, B., D. Guta. 00. Sequetal boudg aroah for otmal aotmet shedulg. IIE Trasatos 5 () 00-06. Deto, B., J. Vaao, A. Vogl. 007. Otmzato of surgery sequeg ad shedulg desos uder uertaty. Health Care Maagemet See (0) Garey, M.R., D. S. Johso, R. Seth. 976. The omlexty of flosho ad jobsho shedulg. Math. Of Oeratos Researh () 7-9. Guta, D. 007. Surgal sutes oeratos maagemet. Produto ad Oeratos Maagemet 6 (6) 689-700. HFMA. 005. Ahevg oeratg room effey through roess tegrato. Health Care Faal Maagemet Assoato Reort. Kaador, G. C., G. Koole. 007. Otmal outatet aotmet shedulg. Health Care Maage See 0 7-9. Kleyegt, A., A. Sharo, T. Homem de Mello. 00. The samle average aroxmato method for stohast dsrete otmzato. SIAM Joural o Otmzato () 479-50. Laorte, G., F.V. Louveaux. 99. The teger l-shaed method for stohast teger rograms th omlete reourse. Oeratos researh letters () - 4. Lderoth, J. T., A. Sharo, S. J. Wrght. 006. The emral behavor of samlg methods for stohast rogrammg. Aals of Oeratos Researh 4 9{45. Malla, C., R. H. Storer. 009. Stohast sequeg ad shedulg of a oeratg room. Teh. re., Idustral ad System Egeerg, Lehgh Uversty. Nork, V., Y. Ermolev, A. Ruszzysk. 998. O otmal alloato of dvsbles uder uertaty. Oeratos Researh 46() 8. Kog Qgxa, Chug-Yee Lee, Chug-Pa Teo ad Zhhao Zheg. 00. Shedulg Arrvals to a Stohast Serve Delvery System usg Coostve Coes, orkg aer. Robso, L.W., R. R. Che. 00. Shedulg dotors aotmets: Otmal ad emrally-based heurst oles. IIE Trasatos 5 95-07. Shultz, R. 00. Stohast rogrammg th teger varables. Mathematal Programmg 97 85-09. Storer, R. H., D. S. Wu, R. Vaar. 99. Ne searh saes for sequeg roblems th alato to job sho shedulg. Maagemet See 8(0) 495-509. Vade Bosh, P. M., D. C. Detz. 000. Mmzg exeted atg a medal aotmet system.iie Trasatos 84-848. Vade Bosh, P. M., D. C. Detz. 00. Shedulg ad sequeg arrvals to a aotmet system. Joural of Serve Researh 4 () 55. Wag, P. Patrk. 997. Otmally shedulg ustomer arrval tmes for a sgle-server system. Comuters & Oeratos Researh 4(8) 70-76. Wess, E.N. 990. Models for determg the estmated start tmes ad ase ordergs. IIE Trasatos () 4-50.