A heuristic search algorithm for flow-shop scheduling

Transcription

1 Uversty of Wollogog Reserch Ole Sydey Busess School - Ppers Fculty of Busess 008 A heurstc serch lgorth for flow-shop schedulg Joshu P. F Uversty of Wollogog joshu@uow.edu.u Grh K. Wley Assupto Uversty Thld gkwley@sctech.u.edu Publcto Detls F J. P. & Wley G. K. (008). A heurstc serch lgorth for flow-shop schedulg. Ifortc: jourl of coputg d fortcs 3 (4) Reserch Ole s the ope ccess sttutol repostory for the Uversty of Wollogog. For further forto cotct the UOW Lbrry: reserch-pubs@uow.edu.u

2 A heurstc serch lgorth for flow-shop schedulg Abstrct Ths rtcle descrbes the developet of ew tellget heurstc serch lgorth (IHSA*) whch gurtees optl soluto for flow-shop probles wth rbtrry uber of jobs d chesprovded the job sequece s costred to be the se o ech che. The developet s descrbed ters of 3 odfctos de to the tl verso of IHSA*. The frst odfcto cocers thechoce of dssble heurstc fucto. The secod cocers the clculto of heurstc esttes s the serch for optl soluto progresses d the thrd deteres ultple optl solutos whethey exst. The frst odfctos prove perforce chrcterstcs of the lgorth d experetl evdece of these proveets s preseted s well s structve exples whch llustrte the use of tl d fl versos of IHSA*. Keywords heurstc serch lgorth for flow shop schedulg Dscples Busess Publcto Detls F J. P. & Wley G. K. (008). A heurstc serch lgorth for flow-shop schedulg. Ifortc: jourl of coputg d fortcs 3 (4) Ths jourl rtcle s vlble t Reserch Ole:

3 Ifortc 3 (008) A Heurstc Serch Algorth for Flow-Shop Schedulg Joshu Poh-O F Grdute School of Busess Uversty of Wollogog NSW Austrl E-l: joshu@uow.edu.u Grh K. Wley Fculty of Scece d Techology Assupto Uversty Bgkok Thld E-l: gkwley@sctech.u.edu Keywords: Adssble heurstc fucto doce flow-shop schedulg optl heurstc serch lgorth Receved: October Ths rtcle descrbes the developet of ew tellget heurstc serch lgorth (IHSA * ) whch gurtees optl soluto for flow-shop probles wth rbtrry uber of jobs d ches provded the job sequece s costred to be the se o ech che. The developet s descrbed ters of 3 odfctos de to the tl verso of IHSA *. The frst odfcto cocers the choce of dssble heurstc fucto. The secod cocers the clculto of heurstc esttes s the serch for optl soluto progresses d the thrd deteres ultple optl solutos whe they exst. The frst odfctos prove perforce chrcterstcs of the lgorth d experetl evdece of these proveets s preseted s well s structve exples whch llustrte the use of tl d fl versos of IHSA *. Povzetek: Ops je ov hevrstč skl lgorte IHSA*. Itroducto The optl soluto to the flow-shop schedulg proble volvg jobs d ches deteres the sequece of jobs o ech che order to coplete ll the jobs o ll the ches the u totl te (.e. wth u kesp) where ech job s processed o ches 3 tht order. The uber of possble schedules s (!) d the geerl proble s NP-hrd. For ths geerl proble t s kow tht there s optl soluto where the sequece of jobs s the se o the frst two ches d the sequece of jobs s the se o the lst two ches [4]. Cosequetly for the geerl proble wth or 3 ches there s optl soluto where the jobs re processed the se sequece o ech che d the optl sequece s og oly! job sequeces. However the optl soluto for the geerl proble wth ore th 3 ches the jobs re ot ecessrly processed the se sequece o ech che. Ths rtcle s cocered wth the developet of lgorth (IHSA * ) whch s gurteed to fd optl soluto for flow-shop schedulg probles volvg rbtrry uber of jobs d ches where the proble s costred so tht the se job sequece s used o ech che. Erly reserch o flow-shop probles s bsed ly o Johso s theore whch gves procedure for fdg optl soluto wth ches or 3 ches wth cert chrcterstcs [0] []. Other pproches for the geerl proble clude teger ler progrg d cobtorl progrg whch use tesve coputto to obt optl solutos d re geerlly ot fesble fro coputtol stdpot becuse the uber of vrbles creses expoetlly s the uber of ches creses [35]. Brch-d-boud ethods use upper or lower bouds to gude the drecto of the serch. Depedg o the effectveess of the heurstc d the serch strtegy ths ethod y retur oly er optl solutos but wth log coputto te [9] [9] [30] [36]. Heurstc ethods hve receved sgfct tteto [9] [8] [6] [7] [37] [40] [4] [4]. However eve the ost powerful heurstc ethod to-dte the NEH heurstc developed by Nwz et l. [3] fls to rech solutos wth resoble boud of the optl soluto soe dffcult proble cses [47]. A revew of pproches by Zobols et l. [47] dctes tht there hs bee strog terest rtfcl tellgece optzto ethods referred to s etheurstcs cludg: Sulted Aelg [3] [34]; Tbu Serch []; Geetc Algorths whch y gve optl soluto but due to the evolutory ture of ths pproch the coputto te s upredctble [] [3] [5] [38]; Fuzzy Logc [3] [4] [5] [6] [7]; At Coloy d Prtcle Swr Optzto [8] [43]; Iterted Locl Serch [39]; d Dfferetl Evoluto [33]. The strog terest etheurstcs geerted the developet of

4 454 Ifortc 3 (008) J. F et l. hybrd pproches whch cobe dfferet copoets of ore th oe etheurstc []. A tl verso of ew tellget heurstc serch lgorth (IHSA * ) for flow-shop probles wth rbtrry uber of jobs d ches d subject to the costrt tht the se job sequece s used o ech che hs bee proposed [6] [7] [8]. It s bsed o the Serch d Lerg A * lgorth preseted [44] [45] [46] whch s odfed verso of the Lerg Rel Te A * lgorth [4] [5] whch s tur odfed verso of the orgl A * lgorth [0] []. At the strt of the serch usg IHSA * dfferet ethods re cosdered for coputg esttes for the totl te eeded to coplete ll of the jobs o ll of the ches ssug tur tht ech of the jobs s plced frst the job sequece. It s show tht f there re ches the there re dfferet ethods tht should be cosdered. Aog the esttes ssocted wth ech ethod the sllest estte s referred to s the vlue of the heurstc fucto tht s ssocted wth tht ethod d t detfes the job tht would be plced frst the job sequece t the strt of the serch f tht ethod s used. If the vlue of the heurstc fucto does ot exceed the u kesp for the proble the the heurstc fucto s sd to be dssble d such cses IHSA * s gurteed to fd optl soluto provded the job sequece s the se o ech che. The proof of ths result for IHSA * s gve [8] d s slr to tht gve [] d [3] relto to the A * lgorth fro whch IHSA * s derved. The ter heurstc s used the ttle of IHSA * becuse the optlty of the lgorth d ts perforce depeds o the selecto of pproprte dssble heurstc fucto t the strt of the serch d ths fucto cotues to gude the serch to optl soluto. The purpose of ths rtcle s to descrbe the developet of IHSA * whch hs occurred sce the tl verso ws frst preseted [6]. For splcty of presetto the developet s descrbed ters of probles volvg rbtrry uber of jobs wth 3 ches. However the ottos deftos proofs d cocepts preseted y be exteded to probles volvg ore th 3 ches f the job sequece s the se o ech che d these extesos re oted t the pproprte plces throughout the presetto. Three sgfct odfctos hve bee de to the tl verso of IHSA *. The frst cocers the choce of dssble heurstc fucto t the strt of the serch. The secod cocers the clculto of heurstc esttes s the serch progresses d the thrd deteres ultple optl solutos whe they exst. Followg troducto to the tl verso of IHSA * ech of the 3 odfctos s preseted. Experetl evdece of proveets perforce chrcterstcs of the lgorth whch result fro the frst odfctos s provded the Appedx d dscussed secto 5. Istructve exples re gve to llustrte the tl d fl versos of IHSA * d these hve bee lted to 3 ches order to llow terested reders to flrze theselves wth the lgorth by reworkg the exples by hd. The proofs of results relted to the odfctos re preseted the Appedx. The tl verso of IHSA * Before presetg the tl verso of the lgorth ottos d deftos re troduced d the stte trsto process ssocted wth IHSA * s descrbed together wth the fetures of serch pth dgrs whch re used to llustrte the developet of optl job sequece.. Nottos d deftos The followg ottos d deftos re troduced for flow-shop proble volvg jobs J J J d 3 ches M M M 3. O j s the operto perfored o job J by che M j d there re 3 opertos. For job J the processg tes b d c deote the tes requred to perfor the opertos O O d O 3 respectvely d these processg tes re ssued to be o egtve tegers. If O j hs coeced but hs ot bee copleted the p j represets the ddtol te requred to coplete O j d t the te whe O j strts p j s oe of the vlues og b or c. The sequece φ st = {J s J t } represets sequece of the jobs wth J s scheduled frst d J t scheduled lst. T(φ st ) s the kesp for the job sequece φ st d S(φ st ) s the te t whch ll of the jobs φ st re copleted o che M. Usg these ottos d deftos Fgure llustrtes the er whch the opertos ssocted wth the job sequece φ st re perfored o the 3 ches. = b = c = Fgure : Processg the Job Sequece φ st S( φ st ) T( φ st ) A ethod for clcultg estte of the totl te to coplete ll of the jobs o ll of the 3 ches whe job J s the frst job the sequece s gve by + b + c. The the heurstc fucto ssocted wth ths ethod s H 3 where H 3 = [ + b + b s + b s + b ] c +. () If [ + b + b s + b s + b ] = s + b s the H 3 = s + b s + c d job J s would be

5 A HEURISTIC SEARCH ALGORITHM FOR... Ifortc 3 (008) scheduled frst the job sequece. It s see fro Fgure d proved the Appedx tht H 3 s dssble heurstc fucto d H 3 s the heurstc fucto tht ws used the tl verso of the IHSA *.. The stte trsto process The procedure for developg the optl job sequece usg IHSA * proceeds by selectg operto whch y be perfored ext o vlble che. At the te tht selecto s de ech of the 3 opertos s oly oe of 3 sttes: the ot scheduled stte; the progress stte; or the fshed stte d the operto whch s selected s og those the ot scheduled stte. Opertos ot the fshed stte re referred to s coplete. A stte trsto occurs whe oe or ore of the opertos ove fro the -progress stte to the fshed stte d t y te the stte level s the uber of opertos the fshed stte. IHSA * descrbes the procedure whch tkes the stte trsto process fro oe stte level to the ext d the developet of the optl job sequece s llustrted grphclly usg serch pth dgrs..3 Serch pth dgrs A serch pth dgr cossts of odes drw t ech stte level wth oe of the odes coected to odes t the ext stte level. Ech ode cots 3 cells whch re used to dsply forto bout opertos o the ches M M d M 3 respectvely. Whe stte trsto occurs oe of the odes t the curret stte level s expded d t s coected to the odes t the ext stte level where ech ode represets oe of the dfferet wys of strtg opertos tht re the ot scheduled stte. At ech of these odes cell s lbeled wth J to dcte tht the operto O j s ether progress or s oe of the opertos tht y strt o M j d p j whch s the te eeded to coplete J o M j. A blk cell dctes tht o operto c be perfored o tht che t ths te. Ner ech ode the heurstc estte (h) ssocted wth the ode s recorded. The heurstc estte s clculted usg the heurstc fucto chose Step of the lgorth cojucto wth the procedure used Step. It s estte of the te requred to coplete the operto t the ode detfed by the procedure Step s well s ll of the other opertos tht re ot the fshed stte. At ech stte level the ode selected for expso s the oe whch hs ssocted wth t the u heurstc estte og ll of the esttes for the odes t tht stte level. Ner ths selected ode the vlue of f = h + k s recorded where the edge cost (k) s the te tht hs elpsed sce the precedg stte trsto occurred. A coprso s de betwee f d h where h s the u heurstc estte t the precedg stte level. Bsed o tht coprso the serch pth ether bcktrcks to the ode expded t the precedg stte level or oves forwrd to the ext stte level. If bcktrckg occurs the the vlue of h s chged to the curret vlue of f d the serch oves bck to tht ode. If the pth oves forwrd the the vlue of the edge cost (k) s recorded below the expded ode. For coveece of presetto ew serch pth dgr s drw whe bcktrckg the prevous dgr s copleted. At stte level 0 there re root odes correspodg to the uber of jobs. The fl serch pth dgr represets the optl soluto d trces pth fro oe of the root odes where the u kesp s the vlue of h or f to terl ode where h = f = 0. The optl job sequece c be red by recordg the copleted opertos log the pth fro the root ode to the terl ode..4 IHSA * (tl verso) Step : At stte level 0 expd the ode detfed by clcultg the vlue of H 3 fro () d ove to the odes t stte level. If ore th oe ode s detfed the brek tes rdoly. For exple f H 3 = s + b s + c the the ode wth J s d s recorded the frst cell s the ode to be expded. Step : At the curret stte level f the heurstc estte of oe of the odes hs bee updted by bcktrckg use the updted vlue s the heurstc estte for tht ode d proceed to Step 3. Otherwse clculte heurstc estte for ech ode t the curret stte level usg Procedure d proceed to Step 3. Procedure s descrbed below. Step 3: At the curret stte level select the ode wth the sllest heurstc estte. If t s ecessry the brek tes rdoly. The sllest heurstc estte s dssble d uderesttes the u te requred to coplete ll of the coplete jobs o ll of the ches. Step 4: Clculte f = h + k where h s the sllest heurstc estte foud Step 3 d k (edge cost) s the te tht hs elpsed sce the precedg stte trsto occurred. Step 5: If f > h where h s the u heurstc estte clculted t the precedg stte level the bcktrck to tht precedg stte level d crese the vlue of h t tht precedg ode to the curret vlue of f d repet Step 4 t tht ode. Step 6: If f h the proceed to the ext stte level d repet fro Step. Step 7: If f = 0 d h = 0 the Stop. Procedure s used Step to clculte heurstc estte for ech ode t the curret stte level: () If cell s lbelled wth J the the heurstc estte h for the ode s bsed o the operto cell d s gve by h = + b + C ; for O the ot scheduled stte () p + b + c + C ; for O the -progress stte where C s the su of the vlues of c k for ll vlues of k such tht O k s the ot scheduled stte.

6 456 Ifortc 3 (008) J. F et l. (b) If cell s blk d cell s lbelled wth J the the heurstc estte h for the ode s bsed o the operto cell d s gve by h = b + C ; for O the ot scheduled stte (3) p + c + C ; for O the -progress stte where C s the su of the vlues of c k for ll vlues of k such tht O k s the ot scheduled stte. (c) If cell d cell re blk d cell 3 s lbelled wth J the the heurstc estte h for the ode s bsed o the operto cell 3 d s gve by h = C 3 ; for O 3 the ot scheduled stte (4) p 3 + C 3 ; for O 3 the -progress stte where C 3 s the su of the vlues of c k for ll vlues of k such tht O k3 s the ot scheduled stte. I Procedure the clculto of heurstc estte for ode s bsed o operto oly oe of the cells t the ode d opertos the other cells re ot tke to ccout. For exple f cell s ot blk the the estte s bsed oly o the operto cell. If cell s blk the the estte s bsed oly o the operto cell. A operto cell 3 s oly cosdered f the other cells re blk. The followg exple llustrtes the use of the tl verso of IHSA * to solve the flow-shop proble gve Tble. For structol purposes the proble s delbertely sple wth oly 3 ches d 3 jobs becuse t s teded to provde reders wth opportuty to becoe flr wth the lgorth usg exple tht c be reworked esly by hd. Tble : Exple of Flow-shop Proble Jobs/Mches M M M 3 J 0 J J For llustrtve purposes oly the frst serch pth dgr s preseted Fgure. At the strt of the serch H 3 = 6. I totl 5 serch pth dgrs re requred to fd the optl sequece J J J 3 wth u kesp of 6. Twety odes re expded 6 bcktrckg steps re requred d 43 steps of the lgorth re executed. 3 Modfctos to the tl verso of IHSA * The frst odfcto deteres the best heurstc fucto to use for gve proble d ffects Step of the lgorth. The secod odfcto ffects Procedure used Step d the thrd odfcto ffects Step 7 d ebles ultple optl solutos to be foud whe they exst. 3. A odfcto to step I the Appedx set of 6 heurstc fuctos re derved for the cse of 3 ches d proofs of the dssblty of these fuctos re preseted. It s show tht og ths set of 6 heurstc fuctos the oe whch Fgure : The Frst Serch Pth Dgr Usg the Itl Verso of IHSA* s dssble d hs vlue whch s closest to the u kesp wll be the oe og H H d H 3 whch hs the lrgest vlue where H = [b + c b + c b + c ] + H = [ + u + u + u ] + H 3 = [ + b + b + b ] + b c (5) where: u = [c c 3 c ]; u k = [c c c k- c k+ c ] for k ; d u = [c c c 3 c - ]. Choosg the dssble heurstc fucto og H H d H 3 wth the lrgest vlue the frst step of the lgorth esures tht the serch begs wth estte of the u kesp tht s ot greter th t but s the closest to t. Ths choce s expected to reduce the eed for bcktrckg t subsequet stge of

7 A HEURISTIC SEARCH ALGORITHM FOR... Ifortc 3 (008) the serch sce bcktrckg tkes the serch bck to prevous ode d creses the heurstc estte t tht ode to vlue whch s dssble but closer to the u te requred to coplete ll of the coplete jobs o ll of the ches. Cosequetly Step the tl verso of IHSA * s odfed d becoes: Step: At stte level 0 fro (5) choose the dssble heurstc fucto og H H d H 3 whch hs the lrgest vlue d f ecessry brek tes rdoly. I the cse where che M j dotes the other ches select H j. Expd the ode detfed by the chose dssble heurstc fucto d ove to the odes t stte level. If ore th oe ode s detfed the brek tes rdoly. The exple secto 4 below llustrtes the use of the odfcto to Step sple proble whch ebles the reder to rework the exple by hd. I the Appedx t s show tht the prtculr cse where che M j dotes the other ches the the best dssble heurstc fucto og H H d H 3 s H j d t hs vlue whch s greter th the vlue of ether of the other two fuctos by t lest ( )( ) where s the uber of jobs. Thus the cse of dot che the frst step of the lgorth there s o eed to clculte ech of the vlues of H H d H 3 order to choose the oe wth the lrgest vlue. Isted t s kow tht t wll be H j f che M j s the dot che. I the cse where there re ches d > 3 t s show the Appedx tht the best dssble heurstc fucto to use the frst step of the lgorth s the oe og F F F whch hs the lrgest vlue d f = 3 the F = H F = H d F 3 = H 3. Experetl evdece of proveets perforce chrcterstcs of the lgorth whch result fro usg the odfcto to Step s preseted the Appedx Tble A d s dscussed below secto A odfcto to step The secod odfcto to the tl verso of IHSA * cocers the clculto of heurstc esttes t odes o the serch pth whe the serch hs coeced. It s bsed o the prcple tht whe heurstc esttes for the odes t the se stte level re beg clculted Step t s desrble to obt the lrgest possble estte t ech of these odes before selectg the ode wth the sllest estte s the ode to be expded. The lrger the vlue of ths sllest estte the the less lkely t s tht the serch wll eed to bcktrck d ths s expected to prove the perforce chrcterstcs of the lgorth. As oted bove the use of Procedure Step the tl verso of IHSA * gves heurstc estte for ode bsed o operto oly oe of the cells whle opertos the other cells re ot tke to ccout. The secod odfcto ffects Procedure d volves clcultg heurstc esttes h h d h 3 t ode for cells d 3 respectvely. The x[h h h 3 ] s used s the heurstc estte (h) for the ode. Ths s doe for ech ode t the curret stte level d the s before Step 3 the u estte og these esttes detfes the ode to be expded. Cosequetly Procedure s replced by the followg Procedure : I Step of the lgorth for ode t the curret stte level () For cell : If the cell s blk the h = 0. Otherwse h s gve by (). (b) For cell : If the cell s blk the h = 0. (6) Otherwse h s gve by (3). (c) For cell 3: If the cell s blk the h 3 = 0. Otherwse h 3 s gve by (4). Procedure refers to clcultos specfed Procedure whch corporte curretly the heurstc fucto H 3. However () (3) d (4) re esly chged to corporte H or H for probles where oe of these fuctos hs bee selected usg the odfcto to Step of the lgorth. For exple f H s used the (3) d (4) re ot chged but () becoes h = + w + B ; for O the ot scheduled stte p + w + b + B ; for O the -progress stte where B s the su of the vlues of b k for ll vlues of k such tht O k s the ot scheduled stte d w s the sllest vlue of c k for ll vlues of k such tht O k s the ot scheduled stte. Procedure produces the se heurstc estte s Procedure f d oly f oe of the followg 3 codtos s stsfed: h = x[h h h 3 ]; h = x[h h h 3 ] d cell s blk; or h 3 = x[h h h 3 ] d cells d re blk. Uder y other codtos Procedure wll produce heurstc estte t ode whch s lrger th the estte gve by Procedure. Cosequetly usg Procedure wll ever produce estte tht s less th the estte produced by Procedure d prctce the estte usg Procedure s usully lrger d leds to reducto bcktrckg. Usg Procedure the tl verso of IHSA * odfes Step d t becoes: Step : At the curret stte level f the heurstc estte of oe of the odes hs bee updted by bcktrckg use the updted vlue s the heurstc estte for tht ode d proceed to Step 3. Otherwse t ech ode use Procedure to clculte h h h 3 d use x[h h h 3 ] s the heurstc estte for the ode. If there re ches d > 3 the there re cells t ech ode d estte s clculted for ech cell by extedg (6) d () (3) (4) to ccoodte the heurstc fucto F j (see Appedx) used Step of the lgorth. The exple secto 4 below llustrtes the use of the odfcto to Step sple proble whch ebles the reder to rework the exple by hd. Experetl evdece of ddtol proveets perforce chrcterstcs of the lgorth fro usg the odfctos to Steps d together s preseted the Appedx Tble A d s dscussed below secto 5.

8 458 Ifortc 3 (008) J. F et l. 3.3 A odfcto to step 7 For soe probles there re ultple optl solutos d t s ofte portt prctcl stutos to be ble to fd ll of the optl solutos sce t y be requred to fd optl soluto tht lso stsfes other crter. For exple optl soluto y be sought whch lso hs the lest wtg te for jobs tht re queug to be processed. Whe IHSA * s pleeted there re stutos whch dcte the possble exstece of ultple optl solutos. The frst stuto occurs whe there s ore th ode t stte level wth the sllest heurstc estte. I ths cse the tes re broke rdoly d oe of the odes s selected for expso d the serch cotues d produces optl soluto. At the copleto of the serch returg to tht stte level d selectg for expso oe of the other odes whch were ot selected whe the tes were broke y led to dfferet optl soluto. The secod stuto occurs whe t the copleto of the serch for optl soluto oe or ore of the odes t stte level 0 hs heurstc estte tht s less th or equl to the u kesp. I ths cse returg to those odes d begg the serch g y produce dfferet optl solutos. The odfcto to the tl verso of IHSA * tht ebles ultple optl solutos to be detered ffects Step 7. Ths odfcto s dfferet fro the prevous odfctos tht t does ot prove the perforce chrcterstcs of the lgorth but sted t s teded to fd ultple optl solutos f they exst. Cosequetly Step 7 becoes: Step 7: If f = 0 d h = 0 the optl soluto hs bee foud. If log the pth represetg the optl soluto there s ode whch ws selected for expso by brekg tes rdoly og odes t the se stte level wth the se u heurstc estte the retur to tht stte level d repet fro Step gorg y ode tht ws selected prevously for expso s result of brekg tes. If y of the vlues of h t root odes (stte level 0) s less th or equl to the u kesp the retur to stte level 0 d repet fro Step gorg root odes tht led to prevous optl soluto. Otherwse Stop. 4 The fl verso of IHSA * The fl verso of IHSA * corportes ech of the 3 odfctos: Step: At stte level 0 fro (5) choose the dssble heurstc fucto og H H d H 3 whch hs the lrgest vlue d f ecessry brek tes rdoly. I the cse where che M j dotes the other ches select H j. Expd the ode detfed by the chose dssble heurstc fucto d ove to the odes t stte level. If ore th oe ode s detfed the brek tes rdoly. Step : At the curret stte level f the heurstc estte of oe of the odes hs bee updted by bcktrckg use the updted vlue s the heurstc estte for tht ode d proceed to Step 3. Otherwse t ech ode use Procedure to clculte h h h 3 d use x[h h h 3 ] s the heurstc estte for the ode. Step 3: At the curret stte level select the ode wth the sllest heurstc estte. If t s ecessry the brek tes rdoly. Step 4: Clculte f = h + k where h s the sllest heurstc estte foud Step 3 d k (edge cost) s the te tht hs elpsed sce the precedg stte trsto occurred. Step 5: If f > h where h s the u heurstc estte clculted t the precedg stte level the bcktrck to tht precedg stte level d crese the vlue of h t tht precedg ode to the curret vlue of f d repet Step 4 t tht ode. Step 6: If f h the proceed to the ext stte level d repet fro Step. Step 7: If f = 0 d h = 0 the optl soluto hs bee foud. If log the pth represetg the optl soluto there s ode whch ws selected for expso by brekg tes rdoly og odes t the se stte level wth the se u heurstc estte the retur to tht stte level d repet fro Step gorg y ode tht ws selected prevously for expso s result of brekg tes. If y of the vlues of h t root odes (stte level 0) s less th or equl to the u kesp the retur to stte level 0 d repet fro Step gorg root odes tht led to prevous optl soluto. Otherwse Stop. If there re ches d > 3 the Steps d Step eed to be odfed ccordce wth the dscusso of ths cse preseted sectos 3. d 3. bove. The sple structve exple whch ws used to llustrte the tl verso of IHSA * (see Tble ) s used g to llustrte the fl verso of IHSA *. For ths proble fro (5) H 3 = 6 > H = 9 > H = 6 d usg the odfcto to Step of the lgorth H 3 s used Step of the lgorth. Sce o bcktrckg s ecessry optl soluto for the proble requres oly serch pth dgr whch s show Fgure 3 where the optl soluto hs u kesp of 6 d job sequece J J J 3. At ech ode the serch pth dgr shows the esttes h h h 3 d the heurstc estte for the ode h = x[h h h 3 ] whch result fro the use of the odfcto to Step of the lgorth. It s oted tht the u heurstc estte t stte level s 4 t both of the odes t tht stte level. I Step 3 of the lgorth the te ws broke rdoly d the ode t whch job J s scheduled o che M ws selected for expso. I Step 7 lthough for splcty secod serch pth dgr hs ot bee drw the serch returs to stte level d sted the

9 A HEURISTIC SEARCH ALGORITHM FOR... Ifortc 3 (008) chrcterstcs ssocted wth ech of these solutos ebled ssesset of y further proveets perforce chrcterstcs resultg fro the cluso of the odfcto to Step. Fro Tble A t s see tht for ech proble regrdless of the uber of jobs d ches the odfcto to Step whch volves usg the heurstc fucto wth the lrgest vlue Step leds to proveets ll of the perforce chrcterstcs. Furtherore ech proble usg the odfcto to Step together wth the odfcto to Step whch ffects the clculto of heurstc esttes s the serch progresses leds to further proveets the perforce chrcterstcs. Fgure 3: Serch Pth Dgr Usg the Fl Verso of IHSA * ode t whch job J 3 s scheduled o che M s expded. Ths gves secod optl soluto where the job sequece s J J 3 J. 5 Experetl evdece of proveets perforce Experetl evdece of proveets perforce chrcterstcs of IHSA * usg the odfctos to Steps d s preseted Appedx Tble A. The chrcterstcs cosdered re: the uber of odes expded; the uber of bcktrckg steps requred; d the uber of steps of the lgorth executed. I totl 4 probles re cosdered volvg: 3 5 d 0 ches; d d 40 jobs. Ech proble volvg 3 ches ws solved usg the heurstc fuctos H H d H 3 (5) whch re the se s F F d F 3 respectvely whe = 3. Probles volvg 5 d 0 ches were solved usg ther correspodg heurstc fuctos F F F 3 F 4 F 5 d F F F 3 F 0 respectvely. The solutos ebled proveets the perforce chrcterstcs resultg fro the use of oly the odfcto to Step to be ssessed. I ddto for ech proble the soluto ws obted usg the odfcto to Step together wth the odfcto to Step. The perforce 6 Cocluso Three odfctos to the tl verso of ew tellget heurstc serch lgorth (IHSA * ) hve bee descrbed. The lgorth gurtees optl soluto for flow-shop probles volvg rbtrry uber of jobs d ches provded the job sequece s the se o ll of the ches. The frst odfcto ffects Step of the lgorth d cocers the choce of dssble heurstc fucto whch s s close s possble to the u kesp for the proble. For probles wth rbtrry uber of jobs d 3 ches (M M M 3 ) set of 6 possble fuctos s derved (H H H 6 ) d ther dssblty s proved. It s show tht the fucto whch hs vlue tht s closest to the u kesp d s the best fucto to use Step of the lgorth s the fucto og H H d H 3 whch hs the lrgest vlue. I the prtculr cse where oe of the ches (M j ) dotes the other ches the best fucto s H j d there s o eed to clculte the vlues of the other fuctos. Furtherore ts vlue s greter th the vlue of ether of the other fuctos by t lest O( ) where s the uber of jobs. More geerlly for probles wth ore th 3 ches (M M M ) the best dssble heurstc fucto to use s the oe og F F F wth lrgest vlue d f che M j dotes the other ches the F j s the best heurstc fucto. The proofs of these ore geerl results y be obted followg the ethods used the proofs preseted the Appedx of the correspodg results for H H d H 3. The secod odfcto chges the procedure used Step of the tl verso of the lgorth to detere heurstc esttes t odes o the serch pth. The tl verso deteres heurstc estte t ode by cosderg operto oly oe of the cells t the ode whle opertos the other cells re ot tke to ccout. The odfed procedure deteres heurstc estte t ode by selectg the lrgest of the seprte esttes clculted for ech cell t the ode. The odfed procedure ever produces estte for ode tht s sller th the estte produced by the procedure used the tl verso of the lgorth d y cses t wll be lrger.

10 460 Ifortc 3 (008) J. F et l. The frst d secod odfctos esure tht t the strt of the serch d s the serch progresses heurstc esttes re dssble d re s close s possble to the u te eeded to coplete ll of the coplete opertos o ll of the ches. Ths reduces the chce tht the serch wll bcktrck d proves the perforce chrcterstcs of the lgorth. Experetl evdece fro probles volvg vrous ubers of ches d jobs dctes tht lthough the frst odfcto produces proveets perforce chrcterstcs of the lgorth these proveets re ehced whe the secod odfcto s cluded. The thrd odfcto reltes to Step 7 of the lgorth d cocers probles where there re ultple optl solutos. It ebles ll of the optl solutos to be foud d ths s coveet for stutos where ddtol crter y eed to be stsfed by optl soluto. Ths rtcle hs focussed o descrbg the developet of the fl verso of IHSA *. However there re severl res for future vestgto cludg coprso of the perforce of the lgorth wth other ethods such s brch- d-boud ethods d ethods for prug the serch tree order to prove eory geet durg pleetto. Refereces [] Blu C. Rol A. Metheurstcs cobtorl optzto: overvew d coceptul coprso ACM Coput. Surv [] Che C.L. Neppll R.V. Aljber N. Geetc lgorths ppled to the cotuous flow shop proble Coputers d Idustrl Egeerg 30: (4) [3] Cleveld G.A. Sth S.F. Usg geetc lgorths to schedule flow shop Proceedgs of 3rd Coferece o Geetc Algorths Schffer D.(ed.) S Mteo: Morg Kuf Publshg [4] Cowy R.W. Mxwell W.L. Mller L.W. Theory of schedulg Addso-Wesley Redg Msschusetts 967. [5] Etler O. Toklu B. Atk M. WlsoJ. A geetc lgorth for flowshop schedulg probles J. Oper. Res. Soc [6] F J.P.-O. The developet of heurstc serch strtegy for solvg the flow-shop schedulg proble Proceedgs of the IASTED Itertol Coferece o Appled Ifortcs Isbruck Austr [7] F J.P.-O. A tellget serch strtegy for solvg the flow-shop schedulg proble Proceedgs of the IASTED Itertol Coferece o Softwre Egeerg Scottsdle Arzo USA [8] F J.P.-O. A tellget heurstc serch ethod for flow-shop probles doctorl dssertto Uversty of Wollogog Austrl 00. [9] Fr J.M Ruz-Uso R. Leste R. Sequecg CONWIP flow-shops: lyss d heurstc It. J. Prod. Res [0] Gheoweth S.V. Dvs H.W. Hgh perforce A* serch usg rpdly growg heurstcs Proceedgs of the Itertol Jot Coferece o Artfcl Itellgece Sydey Austrl [] Grbowsk J. Wodeck M. A very fst tbu serch lgorth for the perutto flowshop proble wth kesp crtero Coput. Oper. Res [] Hrt P.E. Nlsso N.J. Rphel B. A forl bss for the heurstc deterto of u cost pths IEEE Trsctos o Systes Scece d Cyberetcs Vol. SSC-4: () [3] Hog T.P. Chug T.N. Fuzzy schedulg o two-che flow shop Jourl of Itellget & Fuzzy Systes 6: (4) [4] Hog T.P. Chug T.N. Fuzzy CDS schedulg for flow shops wth ore th two ches Jourl of Itellget & Fuzzy Systes 6: (4) [5] Hog T.P. Chug T.N. Fuzzy Pler schedulg for flow shops wth ore th two ches Jourl of Iforto Scece d Egeerg Vol [6] Hog T.P. Wg T.T. A heurstc Pler-bsed fuzzy flexble flow-shop schedulg lgorth Proceedgs of the IEEE Itertol Coferece o Fuzzy Systes Vol [7] Hog T.P. Hug C.M. Yu K.M. LPT schedulg for fuzzy tsks Fuzzy Sets d Systes Vol [8] Hog T.P. Wg C.L. Wg S.L. A heurstc Gupt-bsed flexble flow-shop schedulg lgorth Proceedgs of the IEEE Itertol Coferece o Systes M d Cyberetcs Vol [9] Igll E. Schrge L.E. Applcto of the brch d boud techque to soe flow shops schedulg probles Opertos Reserch Vol. 3: (3) [0] Johso S.M. Optl two- d three-stge producto schedules wth setup tes cluded Nvl Reserch Logstcs Qurterly : () [] Kburowsk J. The ture of splcty of Johso s lgorth Oeg-Itertol Jourl of Mgeet Scece 5: (5) [] Korf R.E. Depth-frst tertve-deepeg: optl dssble tree serch Artfcl Itellgece Vol [3] Korf R.E. Itertve-deepeg A * : optl dssble tree serch Proceedg of the 9th Itertol Jot Coferece o Artfcl Itellgece Los Ageles Clfor [4] Korf R.E. Rel-te heurstc serch Artfcl Itellgece Vol

11 A HEURISTIC SEARCH ALGORITHM FOR... Ifortc 3 (008) [5] Korf R.E. Ler-spce best-frst serch Artfcl Itellgece 6: () [6] L T.C. A ote o heurstcs of flow-shop schedulg Opertos Reserch 44: (6) [7] Lee G.C. K Y.D. Cho S. W. Bottleeckfocused schedulg for hybrd flow-shop It. J. Prod. Res [8] Lu B. Wg L. J Y-H. A effectve PSObsed eetc lgorth for flow shop schedulg IEEE T. Syst. M. CY. B [9] Lock Z. A brch d boud lgorth for the exct soluto of three che schedulg proble Opertol Reserch Qurterly 6: () [30] McMho C.B. Burto P.G. Flow-shop schedulg wth the brch d boud ethod Opertos Reserch 5: (3) [3] Nwz M. Escore Jr. E. H I. A heurstc lgorth for the -che -job flow-shop sequecg proble Oeg-It. J. Mge. S [3] Ogbu F.A. Sth D.K. The pplcto of the sulted elg lgorth to the soluto of the //Cx flowshop proble Coput. Oper. Res [33] Owubolu G.C. Dvedr D. Schedulg flowshops usg dfferetl evoluto lgorth Eur. J. Oper. Res [34] Os I. Potts C. Sulted elg for perutto flow shop schedulg OMEGA [35] P C.H. A study of teger progrg forultos for schedulg probles Itertol Jourl of Syste Scece 8: () [36] P C.H. Che J.S. Schedulg ltertve opertos two-che flow-shops Jourl of the Opertol Reserch Socety 48: (5) [37] Rvedr C. Heurstc for schedulg flowshop wth ultple objectves Eur. J. Oper. Res [38] Ruz R. Mroto C. Alcrz J. Two ew robust geetc lgorths for the flowshop schedulg proble Oeg-It. J. Mge. S [39] Stutzle T. Applyg terted locl serch to the perutto flowshop proble AIDA TU Drstdt FG Itellektk 998. [40] Tllrd E. Soe effcet heurstc ethods for the flow shop sequecg proble Eur. J. Oper. Res [4] Tllrd E. Bechrks for bsc schedulg probles Eur. J. Oper. Res [4] Wg C.G. Chu C.B. Proth J.M. Effcet heurstc d optl pproches for N//F/SIGMA- C-I schedulg probles Itertol Jourl of Producto Ecoocs 44: (3) [43] Yg K.C. Lo C.J. A t coloy syste for perutto flow-shop sequecg Coput. Oper. Res [44] Z M.R. Shue L.Y. Developg optl lerg serch ethod for etworks Scet Irc : (3) [45] Z R. Shue L.Y. Solvg project schedulg probles wth heurstc lerg lgorth Jourl of the Opertol Reserch Socety 49: (7) [46] Z M.R. A hgh perforce exct ethod for the resource-costred project schedulg proble Coputers d Opertos Reserch [47] Zobols G.I. Trtls C.D. Ioou G. Mzg kesp Perutto Flow Shop schedulg probles usg hybrd etheurstc lgorth Coputers d Opertos Reserch 008 do:0.06/j.cor Appedx Dervto of heurstc fuctos The purpose s to develop heurstc fuctos sutble for use IHSA *. I ech cse the objectve s to develop fucto whch uderesttes the u kesp (.e. dssble). Sx fuctos re developed d the proof of ther dssblty s preseted the ext secto. Fro Fgure S(φ st ) x [b t + s + T(φ st ) x [S(φ st ) + c t s + b s + es tht: T(φ st ) s + b s + c or T(φ st ) S(φ st ) + c t b t + c t + T(φ st ) s + c t + b or b c ] d ] whch (A) (A). (A3) Fro (A) two heurstc fuctos H 3 d H 6 re proposed: H 3 = [ + b + b + b ] + H 6 = [ ] + [b b b ] + c c d The rtole for the developet of H 3 s: select the job tht wll be fshed o M t the erlest possble te f t s plced frst the job sequece. Whe ths job s fshed o M [ + b + b + b ] uts of te hve elpsed d the ddtol te eeded to coplete ll of the jobs o ll of the ches wll be t.

12 46 Ifortc 3 (008) J. F et l. lest c uts of te. Sce [ + b + b + b ] [ ] + [b b b ] t follows tht H 3 H 6 whch s therefore lso plusble heurstc fucto. H d H 5 re derved fro (A): H = [b + c b + c b + c ] + H 5 = [b b b ] + [c c c ] + d The rtole for the developet of H s: select the job whch requres the lest totl out of te o ches M d M 3 (.e. [b + c b + c b + c ] uts of te) d suppose tht t s plced lst the job sequece whch es tht the erlest te tht t c strt o M s fter uts of te. Sce [b + c b + c b + c ] [b b b ] + [c c c ] t follows tht H H 5 whch s therefore lso plusble heurstc fucto. H d H 4 re derved fro (A3): H = [ + u + u + u ] + H 4 = [ ] + [c c c ] + b. d b where: u = [c c 3 c ]; u k = [c c c k- c k+ c ] for k ; d u = [c c c 3 c - ]. The rtole for the developet of H s: cosder ech job tur d suppose tht t s plced frst the job sequece d the fro og ll of the other jobs select the oe whch requres the lest out of te o M 3. Now for ech pr of jobs selected ths er detere the pr tht gves the lest totl te o M d M 3. Ths totl te plus the u totl te requred to fsh ll of the jobs o M s the vlue of H. Also [ + u + u + u ] [ ] + [u u u ] = [ ] + [c c c ] d t follows tht H H 4 whch s therefore lso plusble heurstc fucto. Adssblty Results d selected proofs relted to the dssblty of the heurstc fuctos H H H 3 H 4 H 5 d H 6 re preseted: R. H 3 H 6 d both re dssble. R. H H 4 d both re dssble. R 3. H H 5 d both re dssble. Oly proof for R s gve sce the reg proofs y be costructed the se er. Fro (A3) T(φ st ) s + c t + b for s t = wth s t d so prtculr T(φ t ) + c t + b b b T(φ t ) + c t + T(φ t ) + c t +. Hece f T * (φ st ) deotes the erlest te t whch y job sequece whch strts wth job J s s copleted o M 3 the T * (φ t ) [ + c + c 3 + c ] + T * (φ t ) [ + c + c 3 + c ] + b b T * (φ t ) [ + c + c + c - + c ] + b d the u kesp T * = [T * (φ t ) T * (φ t ) T * (φ t )] [ +u + u + u ] + b b = H [ ] + [c c c ] + = H 4. Cosequetly H H 4 d both re dssble. Fro the results R R d R3 t s see tht the heurstc fuctos H H H 3 H 4 H 5 d H 6 re ll dssble. However order to select the heurstc fucto og these tht s the closest vlue to the u kesp (.e. the best to use Step of IHSA * ) the choce should be de fro og oly H H d H 3 becuse the fucto og these 3 whch hs the lrgest vlue s dssble d hs vlue whch s lrger th y of the other 5 dssble fuctos. Cosequetly Step of IHSA * the vlues of H H d H 3 re clculted d the fucto wth the lrgest vlue s selected for use. Doce Mche M dotes the other ches f [ ] x[b b b ] d [ ] x[c c c ] d slr deftos pply f che M or che M 3 s dot. I the cse of dot che results R5 R6 d R7 detfy edtely whch heurstc fucto og H H d H 3 hs the lrgest vlue d s the best to use IHSA *. Also fro R8 t s see tht the best heurstc fucto hs vlue whch s greter th the vlue of ether of the other fuctos by O( ) where s the uber of jobs. R5. If che M dotes the H s the heurstc fucto wth the lrgest vlue R6. If che M dotes the H s the heurstc fucto wth the lrgest vlue R7. If che M 3 dotes the H 3 s the heurstc fucto wth the lrgest vlue. R8. If che s dot the the best heurstc fucto hs vlue whch s greter th the vlue of ether of the other fuctos by t lest ( )( ) where s the uber of jobs d 3.

13 A HEURISTIC SEARCH ALGORITHM FOR... Ifortc 3 (008) The proofs for R5 d R8 re gve otg tht proofs for the other results y be costructed the se er. Throughout these proofs ( ) = [ ] (b ) = [b b b ] (c ) = [c c c ] x( ) = x[ ] x(b ) = x[b b b ] d x(c ) = x[c c c ]. Suppose che M dotes d for = 3 : [r r + w ]; b [s s + l ]; d c [t t + d ] re dstct o egtve tegers fro tervls of wdths w l d d respectvely ech greter th or equl to (the uber of jobs). It follows tht the u vlues of c b re r + 0.5( ) s + 0.5( ) d t + 0.5( ) respectvely d whe these u vlues re tted ( ) = r (b ) = s (c ) = t x( ) = r + x(b ) = s + d x(c ) = t + for = 3. Also the xu vlues of c b re (r + w) 0.5( + ) (s + l) 0.5( + ) d (t + d) 0.5( + ) respectvely d whe these xu vlues re tted ( ) = r + w (b ) = s + l (c ) = t + d x( ) = r + w x(b ) = s + l x(c ) = t + d. Now H = [b + c b + c b + c ] + d slrly (b ) + (c ) + ( H x( ) + x(c ) + x( b d H 3 x( ) + x(b ) + x( ) (A4) ) (A5) c ). (A6) If (A4) (A5) (A6) re ll true the H s + l + t + d + r + 0.5( ) (A7) H r + + t + d + (s + l) 0.5( + ) (A8) H 3 r + + s + l + (t + d) 0.5( + ). (A9) Fro (A7) d (A8) s + l + t + d + r + 0.5( ) r + t d + (s + ) + 0.5( + ) = s s + l l + r r The best dssble heurstc fucto for rbtrry uber of ches For the cse where there re ore th 3 ches there s eed to chge the otto used prevously to represet the te tht ech operto O j requres o ech che so tht t j s the uber of uts of te requred by job J o che M j. If there re ches the the best dssble heurstc fucto wll be the oe wth the lrgest vlue og the set of fuctos F F F 3 F where F j = [ [ = j = + t t + u = = j t j t where for j u jk = [ [ [ t = t = j = = t + u j t for j t 3 = t = t = t = ] + = j = t = t + t u j for j= ] ] for k = t k t k + = ] for k t = t = = ] for k = d u k = 0 for k = 3. For proble wth ches where > 3 d the job sequece s the se o ech che the fucto og F F F 3 F wth the lrgest vlue s selected Step of IHSA *. If = 3 the usg t = t = b d t 3 = c for = 3 d represetg u j u jk d u j sply by u u k d u respectvely the 3 dssble heurstc fuctos H H d H 3 (5) whch hve bee used throughout the descrpto of the developet of IHSA * re gve by F F d F 3 respectvely = ( )[r (s + l) + ] ( )( ) 0 for d so H s greter th H by vlue whch s t lest ( )( ) for 3. I slr er t follows fro (A7) d (A9) tht H s greter th H 3 by vlue whch s t lest ( )( ) for 3 d ths copletes the proof of R5 d R8.

14 464 Ifortc 3 (008) J. F et l. Experetl evdece of proveets perforce chrcterstcs Tble A: Perforce of IHSA * : Modfcto to Step copred to Modfctos to Steps d. Note: For ech proble: () the hghlghted frst row ssocted wth the use of the odfcto to Step dctes the perforce chrcterstcs usg the best heurstc fucto; (b) the hghlghted lst row ssocted wth the use of odfctos to Steps & dctes the perforce chrcterstcs whe the best heurstc fucto s used together wth the odfcto to Step. No. of Mches 3 Nuber of Jobs Proble Modfcto Used & & & & & & & & & 40 0 Heurstc Fucto H H Vlue of Heurstc Fucto Nodes Expded Perforce Chrcterstcs 3 Bcktrcks 0 Algorth Steps H H H H H H H H H H H H 3 54 H H H H H H H H H H H H Mu Mkesp No. of Mches Nuber of Jobs Proble 3 4 Modfcto Used & & & & Heurstc Fucto Vlue of Heurstc Fucto Nodes Expded Perforce Chrcterstcs Bcktrcks Algorth Steps H H H H H H F F F3 F4 F Mu Mkesp F F3 F F4 F F6 F5 F9 F7 F8 F & F Tble A: Perforce of IHSA * 39 5

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