Single Machine Scheduling with Stochastic Processing Times or Stochastic Due-Dates to Minimize the Number of Early and Tardy Jobs

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1 Iteratoal Joural of Operatos Research Iteratoal Joural of Operatos Research Vol. 3, No. 2, 9 8 (26 Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs H. M. Soush * Departmet of Statstcs ad Operatos Research, College of Scece, Kuwat Uversty, P.O. Box 5969, Safat 36, Kuwat Receved December 25; Revsed February 26; Accepted March 26 Abstract We study a sgle mache schedulg problem whch processg tmes due-dates are o-egatve depedet radom varables ad radom weghts ( pealtes are mposed o both early ad tardy obs. he obectve s to fd a optmal sequece that mmzes the expected total weghted umber of early ad tardy obs. We exple three scearos of the problem cludg a scearo wth determstc processg tmes ad stochastc due-dates, a scearo wth stochastc processg tmes ad determstc due-dates, ad a scearo wth stochastc processg tmes ad stochastc due-dates. hese problem scearos are NP-hard to solve; however, whe there are specal structures o the stochastcty of processg tmes due-dates, we establsh certa codtos uder whch the varous resultg cases are solvable exactly. We also approxmate the solutos f the geeral versos of these cases. he proposed exact ad approxmate soluto methods as well as our llustratve examples demostrate that varatos processg tmes, due-dates, ad earless/tardess pealtes affect schedulg decsos. Furtherme, we show that the problem studed here s geeral the sese that ts specal cases such as the stochastc problem of mmzg the expected weghted umber of tardy obs ad the stochastc problem of mmzg the expected weghted umber of early obs are both solvable by the proposed exact approxmate methods. Keywds Schedulg, Sgle mache, Stochastc, Number of early ad tardy obs. INRODUCION he sgle mache schedulg has bee extesvely studed me tha four decades f varous perfmace measures (e.g., Baer, 974, 995; Coway et al., 967; Frech, 982; Mto ad Petco, 993; Pedo, 22. he problem s cocered wth fdg a sequece amog obs as they proceed through a sgle mache der to optmze some perfmace obectves. he sgfcat of the problem s due to ts mptace developg schedulg they me complex ob shops, ad ts practcal aspects cosderg tegrated processes as sgle mache systems. May researchers have studed the sgle mache schedulg problem wth the obectve of fdg a sequece that mmzes the weghted umber of tardy obs. hs problem, whch we refer to as the problem, s ow to be NP-hard (e.g., Lestra et al., 977. Most of the avalable lterature o the problem deals wth the determstc case where ob attrbutes (e.g., setup tmes, processg tmes, due dates are ow wth certaty (e.g., Baptste, 999; Dauzere-Peres ad Sevaux, 24; Jola, 25; Moe, 968. I cotrast to the determstc problem, the amout of lterature o the stochastc problem where some of ob attrbutes are radom varables s lmted. hese studes cosder specal cases of the problem; f example, Balut (973 presets a chace-costraed fmulato of a case where processg tmes (whch may clude setup tmes are depedet mal radom varables. Boxma ad Fst (986 study a case where processg tmes ad due dates have depedet ad detcal dstrbutos. De et al. (99 exame a case wth radom processg tmes ad a expoetally dstrbuted commo due date. Ca ad Zhou (25 cosder a case wth expoetal processg tmes ad radom due dates. Assumg obs have a commo determstc due-date ad a commo tardess pealty, Pedo (983 aalyzes a case wth expoetal processg tmes, whle Jag (22 ad Seo et al. (25 exame a case where processg tmes have mal dstrbutos. Wth the excepto of La ad Mosheov (996 ad Soush (26, to the best of our owledge, o atteto s gve to the sgle mache schedulg problem where the obectve s to mmze the weghted umber of both early ad tardy obs, whch we refer to as the - problem. La ad Mosheov (996 study the determstc - problem by cosderg dfferet early-tardy (- pealty structures f obs cludg ob-depedet (.e., the - pealtes f all obs are Crespodg auth s emal: hsoush@uc.uv.edu.w 83-73X copyrght 26 ORSW

2 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 9 equal to oe, ob-depedet ad symmetrc (.e., the - pealtes f each ob are detcal, ad ob-depedet ad asymmetrc (.e., the - pealtes of each ob are dfferet. hey show that the frst two problem-classes are solvable polyomal tme, whereas the last class s NP-hard eve whe all obs have a commo due date. Soush (26 exames a stochastc - problem whch processg tmes are radom varables but due dates ad - pealtes are ow fxed quattes ad the obectve s to mmze the expected total weghted umber of early ad tardy obs. He proposes certa codtos uder whch ths problem s solvable exactly ad also presets a very effectve ad effcet heurstc f the geeral case of the problem. he stochastc - problem studed ths paper s a broad exteso of that of Soush (26 ad s defed as follows. here s a set of obs that are smultaeously avalable to be processed sequetally o a cotuously avalable sgle mache. Assume that o dle tme serto s allowed ad oce the processg begs o obs ca be pre-empted ad the sequece remas uchaged utl all obs are fshed. I ths stochastc problem, processg tmes ad/ due dates are stochastc, ad each ob s pealzed by a radom earless weght (f the ob s early ad a radom tardess weght (f the ob s tardy. he radom weghts are depedet of the amouts of tme that obs are early tardy, that s, obs mssg ther due dates by sht log perods are pealzed by the same amouts. he obectve s to fd a optmal sequece that mmzes the expected total weghted umber of early ad tardy obs o a sgle mache. he mptace of the proposed problem stems from the fact that may real-wld stochastc schedulg systems, each early/tardy ob s pealzed by the same pealty o matter how early/tardy the ob s. F example, varous dustres, raw materals parts are ofte eeded at specfc tmes. Smlarly, ar space flght schedulg, tass eed to be perfmed o exact tme pots durg partcular tme wdows der to esure the success of a flght. Also, the pealty fuctos the producto of pershable tems such as food, drugs, etc., have smlar structures. I addto, pc-up ad delvery systems, tems should be pced up delvered at certa tmes. herefe, whe obs (e.g., raw materal, tass, tems are early late, pealtes are curred o matter how early late the obs are (e.g., La ad Mosheov, 996. We fmulate the stochastc - problem Secto 2. hree scearos of the problem are expled Secto 3 cludg a scearo wth determstc processg tmes ad stochastc due-dates, a scearo wth stochastc processg tmes ad determstc due-dates, ad a scearo wth stochastc processg tmes ad stochastc due dates. Uder some structures o the stochastcty of processg tmes due-dates, we preset exact soluto methods f the varous resultg cases of the three scearos. he solutos f the geeral versos of these cases are also approxmated. Fally, a summary ad a few cocludg remars are gve Secto PROBLM NOAION AND FORMULAION he stochastc - problem studed ths paper s as follows. A set N = {,, } of obs s avalable at tme zero to be processed sequetally wthout preempto ad o dle tme sertos o a cotuously avalable sgle mache. Let r = [],, [],, [] be a sequece amog obs N where [], =,,, dcates the ob occupyg the -th posto r R ad where R the set of all! sequeces. he processg tmes p [], =,,, are o-egatve depedet radom varables wth probablty desty fuctos (pdf f [](. (.e., p []~f [](. ad cumulatve dstrbuto fuctos (cdf F [](.. he, the completo tme t [] f ob [], a radom varable, s defed as t [ ] = p[ l ]. l= he due dates ξ [], =,,, are also o-egatve depedet radom varables wth pdfs g [](. (.e., ξ []~g [](. ad cdfs G [](.. Let w [ ] ad w[ ] deote the radom - weghts (pealtes f obs [], =,,, where the expected values ω [ ] = w ( [ ] ad ω [ ] = ( w[ ] exst. Meover, p [], ξ [], w [ ] ad w [ ], =,,, are statstcally depedet of each other. F each ob [], =,,, let X [ ] be a earless dcat varable ad X[ ] be a tardess dcat varable where X ad X [ ] [ ], f ob[ s ] early wth probablty Pr( t < ξ, =, otherwse;, f ob[ s ] tardywth probablty Pr( t > ξ, =, otherwse. he expected total weghted umber of early ad tardy obs a sequece r R (deoted by W r, whch we refer to as the expected weghted umber of - obs r, s defed as W = [ [ w X + w X ] r] r = [ ω Pr( t < ξ + ω Pr( t > ξ ] r, r = ω + λ < ξ < λ [] < (2 W Pr( t r, where λ ω ω ( = ad t [] s defed by (. Note that the obectve fucto (2 of the stochastc - problem s

3 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 92 me geeral tha that of ( the stochastc problem (.e., whe ω [ ] = ad λ [] = ω, [ ] =,, where oly the expected weghted umber of tardy obs s mmzed (e.g., Boxma ad Fst, 986; Ca ad Zhou, 997; Jag, 22, ad ( the stochastc earless ( problem (.e., ω [ ] = ad λ [] = ω [ ],,, where oly the expected weghted umber of early obs s mmzed. Meover, our stochastc - problem, ω [ ] f all obs [], =,,, are ether at most equal to at least equal to ther ω [ ] (.e., ether λ [] λ [] f all obs [], that s, < λ [] <, =,,. Hece, geeral, the problem ca be ether fmulated as a pure stochastc problem (.e., λ [], =,, as a pure stochastc problem (.e., λ [],,,. Utlzg the covetoal otato, the proposed stochastc - problem ca be represeted by // [ w X + w X ]. Defto. F // [ w X + w X ] a sequece r* R s optmal f r R r W r * = m{ W}, (3 where W r s gve by (2. Sce ω = ω s a costat ad s depedet of ob derg, usg (2, r* ca be equvaletly foud as λ ξ r R (4 r* = argm{ Pr( t < r}, < λ <+. Observe that // [ w X + w X ] s geeral the sese that ts lmtg specal cases reduce to some classcal sgle mache schedulg problems. F example, whe ω [ ] ω, [ ] =,,, we get the stochastc problem,.e., // [ w X ], where ω [ ] = λ [] ad p [] ad ξ [] are radom varables (e.g., Boxma ad Fst, 986; Ca ad Zhou, 25. I ths case, f ξ [] = d [], =,,, where d [] are ow fxed quattes, we have // w X where p [] are radom varables (e.g., Jag, 22; Seo et al., 25. Whe ω [ ] ω, [ ] =,,, we get the stochastc problem, that s, // where [ ] [ w X ], ω = λ [] ad p [] ad ξ [] are radom varables. Whe p [], ξ [],, w[ ] ad w [ ], =,...,, are all ow wth certaty, we have the determstc - problem, that s, // w X + w X (e.g., La ad Mosheov, 996 where X[ ] ad X[ ] are defed as, ft < d, X [ ] =, otherwse; ad, ft > d, X [ ] =, otherwse. I ths case, f w [ ] = wth certaty, we get the determstc problem,.e., // w X (e.g., Baptste, 999; Lestra et al., 977; Moe, 968. Whe Pr(t [] < ξ [] =, =,, (.e., all obs are early wth certaty, Pr(t [] > ξ [] =, =,, (.e., all obs are tardy wth certaty, ω = ω, =,,, usg (4, ay sequece r R s optmal f the proposed stochastc - problem. A ave approach to exactly solve // [ w X ] X + w s to ( eumerate all sequeces r R, ( derve the ot cdf of p [] f all obs [], =,, each r R, ( use ( to get the cdf of each t [], =,,, r R, (v compute Pr(t [] < ξ [], =,,, each r R, (v apply (2 to compute W r, r R, ad the (v use (3 (4 to fd r*. hs approach may be the oly oe f there are o specal structures o the stochastcty of processg tmes due dates. However, sce the geeral case of the determstc problem s NP-hard (e.g., La ad Mosheov, 996, the geeral case of the stochastc problem s eve harder to solve due to the addtoal dffculty of computg Pr(t [] < ξ [], =,,, whch requre complex tegratos of mult-varate dstrbutos. 3. PROBLM SCNARIOS AND SOLUIONS o aalyze // [ w X + w X ], cosder a sequece δ R where s a arbtrary sub-sequece of obs, excludg obs ad ad obs δ, appearg the frst q postos (.e., = [],..., [q ], obs ad are adacet whch respectvely occupy postos q ad q + (.e., [q ] = ad [q + ] =, ad δ s a arbtrary sub-sequece of obs, excludg obs ad ad obs, occupyg postos q + 2 to (.e., δ = [q + 2],,[]. he, the expected weghted umber of - obs δ, usg ( ad (2, s q δ = ω + λ [ l] < ξ l= W Pr( p + λ Pr( p + p < ξ + λ Pr( p + p + p < ξ + λ Pr( p + p + p + p < ξ, [ l] q+ 2 l= q+ 2

4 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 93 where p q = p. Iterchagg obs ad δ produces aother sequece δ R whose W δ ca be smlarly computed as W δ. he, δ s preferred to δ, deoted by δ δ, (.e., ob mmedately precedes ob f every obs N ad every choce of ad δ ff ΔW = W δ-w δ = λ [Pr(p + p < ξ Pr(p + p + p < ξ ] λ [Pr(p + p < ξ Pr(p + p + p < ξ ], (5 where Pr(p + p + p < ξ Pr(p + p < ξ ad Pr(p + p + p < ξ Pr(p + p < ξ due to o-egatve processg tmes ad o-egatve due dates. From (5, we observe that ΔW = ΔW does ot deped o obs δ at all; however, t depeds o obs but t s depedet of ther derg. Hece, some occasos, we replace ΔW by ΔW ( to remd the reader of ths depedece. Iequalty (5 s too geeral to allow the developmet of useful statemets to establsh the relato δ δ f every N ad every ad δ. However, whe there are specal structures o the stochastcty of processg tmes due-dates, we ca use ths equalty to vestgate ad solve exactly the varous resultg cases f three scearos of the problem cludg a scearo wth determstc processg tmes ad stochastc due-dates, a scearo wth stochastc processg tmes ad determstc due-dates, ad a scearo wth stochastc processg tmes ad stochastc due-dates. 3. Determstc processg tmes ad stochastc due dates Cosder the scearo /p = π, ξ ~g (./ [ w X ] X + w where p = π ad ξ ~g (., =,,, ad the o-egatve quattes π are ow wth certaty. Usg (5, δ δ f every N ad every ad δ (.e., ob mmedately precedes ob ff λ [G (π + π + π G (π + π ] λ [G (π + π + π G (π + π ], (6 where π q = π, G (π + π + π G (π + π, ad G (π [ ] + π + π G (π + π. he suffcet codtos to satsfy (6 are the as follow. ( λ λ ; (7 ( λ λ, G (π + π + π G (π + π G (π + π + π G (π + π ; (8 ( λ λ, G (π + π + π G (π + π G (π + π + π G (π + π. (9 Below, we use (7 (9 to exame some cases of /p = π, ξ ~g (./ [ w X + w X ]. 3.. Idetcally dstrbuted due-dates Suppose that ξ, =,,, are depedetly ad detcally dstrbuted (..d radom varables wth a geeral pdf g(. (.e., ξ ~g(.. heem. F /p = π, ξ ~g(./ [ w X ] X + w, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad π π ; ( λ λ ad π π. Proof. It mmedately follows from codtos (7 (9. Collary. F /p = π, ξ ~g(./ [ w X + w X ], a sequece [],, [l], [l + ],, [], l {,,, }, where obs [] ad [ + ] do ot exst, s optmal f ( λ [] λ [l] λ [l + ] λ [], ad ( π [] π [l] ad π [l + ] π []. Proof. Cosder a sequece r : πρ R, N, where = [],, [q ], = [q ], π = [q + ],, [q 2 ], = [q 2], ad ρ = [q 2 + ],, [], that s, r : πρ = [q ][q + ] [q 2 ][q 2]ρ. Suppose that obs r are arraged accdg to Collary. Iterchagg obs ad r, N, produces aother sequece r: πρ = [q 2][q + ]...[q 2 ][q ]ρ R. We show that swtchg obs N r creases the expected umber of - obs, that s, W r < W r. Usg (5, W r ca be wrtte as W r = = = = W [ q ]... + q q q 2 2 ρ + W [ q ][ q + ] W [ q ]... + q q q 2 2 ρ + W [ q ][ q + ] W [ q ]... + q q q 2 2 ρ + W [ q ][ q + ] + + W[ q ][ q ] + W 2 q q 2 W [ q ]... + q q q 2 2 ρ + W [ q ][ q + ] + + W[ q ][ q ] + W 2 q q + W[ q ][ q ] =W r W [ q ][ q ] 2 Wpq, ( ( pq, r r

5 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 where ( pq, r r W pq, the sum of ΔW pq f the adacet obs p q N terchaged r to get r, s gve as Wpq = W [ q ][ q + ] ( pq, r r + W[ q ][ q ] W [ q ][ q ] W + [ q ][ q ] 2 W[ q ] q 2 Sce obs r are arraged accdg to Collary (see also codtos ( ( of heem, based o (5, ΔW pq yeldg W where the strct equalty ( pq, r r pq holds f ad oly f obs p ad q are ot detcal. Usg (, the W r < W r f ad oly f the pars of adacet obs swtched r to get r are ot the same. I other wds, the terchage of obs p ad q r, p q N, creases the expected umber of - obs. herefe, sce the terchage of ay par of obs a sequece foud accdg to Collary creases the expected umber of - obs, the such a sequece tself must be optmal. I geeral, sce the optmalty codtos of Collary do ot hold amog obs, r* caot be foud. However, f these codtos are satsfed amog the obs a sequece problem f λ [] λ [] (.e., ω [] ω [ ] 94 ad π [] π []. Also, the stochastc problem f ω [ ] = ω, =,, (.e., obs have a commo mea tardess pealty, arragg obs o-decreasg der of π (.e., accdg to shtest processg tme (SP rule yelds r*. I the stochastc problem f ω [ ] = ω, =,,, (.e., obs have a commo mea earless pealty, arragg obs o-creasg der of π (.e., based o logest processg tme (LP rule gves r* Dstctly dstrbuted due-dates Whe ξ ~g (., =,,, t s me dffcult to develop useful statemets to establsh δ δ f /p = π, ξ ~g (./ [ w X + w [ ] X [ ] ] Nevertheless, such statemets ca be derved f the case. where ξ, =,,, are expoetally dstrbuted (.e., ξ ~exp(γ wth G (x = exp( γ x ad meas /γ. he use of expoetal dstrbuto shop schedulg s ustfed by, f example, Boxma ad Fst (986, Ca ad Zhou (997, 25, Jag (22, ad Pedo (983. [],, [l], [l + ],, [], l {,,, }, the λ []/π [] λ [l]/π [l] λ [l + ]/π [l + ] λ []/π [], λ []/π [] λ []/π []. he coverse of the latter may ot be true, that s, t s possble to have λ [p]/π [p] λ [q]/π [p], p < q =,,, such that the codtos of heem do ot hold f every ob [p] precedg ob [q]. Hece, we ca approxmate the soluto (.e., fd a caddate f r* f /p = π, ξ ~ g(./ [ w X + w [ ] X [ ] ] by arragg obs o- decreasg der of λ /π, =,,. I the case where λ /π = λ /π, N, the ob wth smaller λ, {, }, s placed befe the other ob. hs s due to the fact that λ /π = λ /π, N, f oly λ λ (.e., ω ω, =, ; thus, the ob wth smaller λ, {, }, s scheduled frst to avod large tardess pealty. Also, λ /π = λ /π, N, f oly λ λ (.e., ω ω, =, ; hece, the ob wth smaller λ, {, }, s scheduled frst to avod large earless pealty. Remar. Sce /p = π, ξ ~g(./ [ w X ](.e., the stochastc problem ad /p = π, ξ ~ g(./ [ w X ] (.e., the stochastc problem are specal cases of /p = π, ξ ~g(./ [ w X + ] w X (.e., the stochastc - problem (see Secto 2, based o Collary, a sequece [],, [] s optmal f the stochastc problem f λ [] λ [] (.e., ω [] ω [ ] ad π [] π [], ad s optmal f the stochastc heem 2. F /p = π, ξ ~exp(γ / [ w X ] X + w, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ, γ γ, ad π /γ π /γ ; ( λ λ, γ γ, ad π /γ π /γ. Proof. Usg G (x = exp( γ x (8, δ δ f every N ad every ad δ f λ λ ad [ exp( γ π ]exp[ γ (π + π ] [ exp( γ π ]exp[ γ (π + π ]. ( From (, f every N ad every, exp( γ π exp( γ π ff γ π γ π ad exp[ γ (π + π ] exp[ γ (π + π ] ff γ (π + π γ (π + π. he latter codto always holds f γ γ ad γ π γ π. Sce γ γ ad γ π γ π mply π π, the γ π γ π. Hece, usg (8 ad (, δ δ f every N ad every ad δ f λ λ, γ γ ad π /γ π /γ (due to γ π γ π. Smlarly, usg (9, we ca show that δ δ f every N ad every ad δ f λ λ, γ γ, ad π /γ π /γ. herefe, δ δ holds f every N ad every ad δ f ( λ λ ; λ λ, γ γ, ad π /γ π /γ ; ( λ λ, γ γ, ad π /γ π /γ.

6 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 95 Collary 2. F /p = π, ξ ~exp(γ / [ w X ] X + w, a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( λ [] λ [l] λ [l + ] λ [], ad ( γ [] γ [l], γ [l + ] γ [], π []/γ [] π [l]/γ [l], ad π [l + ]/γ [l + ] π []/γ []. Proof. Usg a approach smlar to that of the proof of Collary, we show that a sequece obtaed by arragg obs accdg to Collary 2 (see also codtos ( ( of heem 2 s optmal. Let us relax the optmalty codtos of Collary 2 by removg γ [] γ [l] ad γ [l + ] γ []. If the remag codtos of the collary hold amog the obs a sequece [],, [l], [l + ],, [], l {,,, }, the λ []γ []/π [] λ [l]γ [l]/π [l] λ [l + ]γ [l + ]/π [l + ] λ []γ []/π [] (.e., λ []γ []/π [] λ []γ []/π []. Hece, we ca approxmate the soluto (.e., fd a caddate f r* f /p = π, ξ ~exp(γ / [ w X + w X ] by arragg obs o-decreasg der of λ γ /π. Remar 2. Based o Collary 2, a sequece [],, [] whch λ [] λ [] s optmal f /p = π, [ ] [ ] ξ ~exp(γ / [ w X ] f γ [] γ [] ad π []/γ [] π [l]/γ [], ad s optmal f /p π, ξ ~exp(γ [ ] [ ] π []/γ []. / [ w X ] f γ [] γ [] ad π []/γ [] Remar 3. F /p = π, ξ ~g (./ [ w X ] X + w accdg to Collares 2, obs [],,, l, l +,,, l {,,, }, are arraged r* o-decreasg der of ω ω [ ] [ ](.e., λ[] where there are addtoal codtos mposed o π [] γ [] of obs [], =,, l (.e., obs wth ω [ ] ω [ ] λ[] as well as o those of obs [], = l +,, (.e., obs wthω [ ] ω [ ] λ []. Hece, /p = π, ξ ~g (./ [ w X + w ] X amog obs [],, [l], [l + ],, [] where < λ [] < + (.e., the stochastc problem s a mxture of /p = π, ξ ~g (./ [ w X ] amog obs [],, [l] where ω [ ] = λ[] (.e., the stochastc problem ad /p = π, ξ ~g (./ [ w X ] amog obs [l + ],, [] where ω [ ] = λ[] (.e., the stochastc problem. 3.2 Stochastc processg tmes ad determstc due dates Cosder the scearo /p ~f (., ξ = d / [ w X ] X + w where p ~f (. ad ξ = d, =,...,, wth d beg ow costats (see also Soush (26. he, usg (5, δ δ f every N ad every ad δ ff ΔW ( = λ [Pr(p + p < d Pr(p + p + p < d ] λ [Pr(p + p < d Pr(p + p + p < d ], (2 λ[ F * F( d F * F * F( d ] λ [ F * F ( d F * F * F ( d ], (3 where F ( x = F[] *...* F[ q ] ( x, F * F ( x, F * F ( x, ad F * F * F ( x are the covolutos of the cdfs of p [] f obs [], =,..., q,, f obs ad ob, f obs ad ob, ad f obs ad obs ad, respectvely. Meover, F * F * ( F d F * F( d ad F * F * F( d F * ( F d due to o-egatve processg tmes. Sce (2 (3 are too geeral to allow the developmet of practcal statemets to establsh δ δ, we aalyze the followg cases of /p ~f (., ξ = d / [ w X + w ]. X 3.2. Idetcally dstrbuted processg tmes Assume that p, =,,, are..d. wth a geeral pdf f(. (.e., p ~f(. ad ξ = d (ow costat. Usg (3, δ δ f every N ad every ad δ ff λ ( q* ( q+ * [ F ( d F ( d] λ (4 ( q* ( q+ * [ F ( d F ( d ] ( q where F * ( x s the covoluto of the cdfs F (. of ob ( q + ad the q obs, ad F * ( x s that of obs ad ad the obs. Iequalty (4 s stll dffcult to be expled f..d. processg tmes wth a geeral f(.. However, we ca exame a case where p ~exp(α wth F (x = exp( αx ad mea /α, =,,. heem 3. F /p ~exp(α, ξ = d / + [ w X w X ], δ δ f every N ad every ad δ f

7 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 96 ( λ λ ; ( d d ad ether δ δ ψ ψ where δ = λ d exp( αd, (5 ad ψ = λ d exp( αd ; (6 ( d d ad ether δ δ ψ ψ. Proof. F p ~exp(α, =,,, usg the relatoshp betwee Posso ad expoetal dstrbutos, (4 ca be equvaletly wrtte as λ [ [ Nd ( = ] [ Nd ( = ] = q = q+ [ λ [ Nd ( = ] [ Nd ( = ], = q = q+ ] ] (7 where N(t has a Posso dstrbuto wth mea rate α. Smplfyg (7, we have λ[ exp( α /!] λ [( exp( α /!], q q ( αd d q d α d q λd exp( αd λ d exp( αd, (8 q q where (8 depeds o obs N as well as the posto q of ob. F λ λ, N, (8 s always satsfed. F d d, N, (8 holds f q =,,, (ob occupes posto q + f ether ( λ d exp( αd λ d exp( αd ; ( λd exp( αd λ d exp( αd. F d d, N, (8 holds f ether ( λd exp( αd λ d exp( αd ; ( λ d exp( αd λ d exp( αd. Based o heem 3, the followg collary provdes the codtos uder whch a optmal sequece ca be foud. Collary 3. F /p ~exp(α, ξ = d / [ w X ], X + w a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ether ( δ [] δ [l] δ [l + ] δ [] where δ s gve by (5, d [] d [l] ad d [l + ] d []; ( ψ [] ψ [l] ψ [l + ] ψ [] where ψ s gve by (6, d [] d [l] ad d [l + ] d []. Proof. We use a approach smlar to that of the proof of Collary to show that a sequece foud by arragg obs accdg to Collary 3 (see also codtos ( ( of heem 3 s optmal. I geeral, the optmalty codtos of Collary 3 do ot hold amog obs ad thus r* caot be foud. However, f these codtos are satsfed amog the obs a sequece [],, [l], [l + ],, [], l {,,,}, the ether ( δ []d [] δ []d []; ( ψ []/d [] ψ [l]/d []. Hece, the soluto f /p ~exp(α, ξ = d / [ w X + w ] X a be approxmated (.e., ca fd a caddate f r* by arragg obs o-decreasg der of ether δ d ψ /d where δ ad ψ are defed by (5 ad (6. xample. Cosder the problem /p ~exp(α, ξ = d / [ w X w X ] of able where α =.5 (.e., + (p = 2 ad d are dfferet =,..., 5. Here, δ 4 < δ < < δ 3 < δ 5 < δ 2 where, usg (5, δ =.37,.95,.4,.439, ad.55, =,, 5, respectvely, ad d 4 > d ad d 3 < d 5 < d 2. Based o codto ( of Collary 3, r*: Sce t [] of each ob [], =,, 5, has a rlag pdf wth parameters ad α =.5, the Pr(t [] < d [] = α exp( αd [] - ( d. Hece, processg obs accdg! = to r* results obs [], =,, 5, beg early wth Pr(t [] < 8 =.982, Pr(t [2] < 7 =.864, Pr(t [3] < 5 =.456, Pr(t [4] < 5.5 =.297, ad Pr(t [5] < 6 =.85, respectvely. Note that the caddate foud by arragg obs o-decreasg der of δ d s also optmal. able. A stochastc - problem wth p ~exp(.5 ad ξ = d (ow costat Job d ω ω λ Remar 4. Based o Collary 3, a sequece [],,[] s optmal f /p ~exp(α, ξ = d / [ w X ] f ether δ [] δ [] ad d [] d [], ψ [] ψ [] ad d [] d []. Also, a sequece [],, [] s optmal f

8 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 97 /p ~exp(α, ξ =d / [ w X ] f ether δ [] δ [] ad d [] d [], ψ [] ψ [] ad d [] d []. Assumg obs have a commo ow fxed due date (.e., ξ = d = d, =,,, we have the followg collary. Collary 4. F /p ~exp(α, ξ = d/ [ w X ] X + w, a optmal sequece s foud by arragg obs a o-decreasg der of λ. Proof. It follows from Collary Dstctly dstrbuted processg tmes Suppose that p ~f (. ad ξ = d (ow costat, =,,. Usg (3, δ δ f every N ad every ad δ ff λ[ F * F( d F * F * F( d] λ [ F * F ( d F * F * F ( d ], where F * F * ( m{ F d F * F( d, F * F( d }. (9 heem 4. F /p ~f (., ξ = d/ [ w X ] X + w, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad F (y F (y, y d; ( λ λ ad F (y F (y, y d. Proof. Usg (9, δ δ f every N ad every ad δ f ( λ λ ; (2 ( λ λ ad F * F( d F * F( d; (2 ( λ λ ad F * F( d F * F( d. (22 hus, f (2, (2, (22 are satsfed, the (9 holds; however, the coverse may ot be true. he codto F * F ( d F * F ( d (2 ca be wrtte as d d F( d x f ( xdx F ( d x f ( xdx, (23 where f ( x = f[] *...* f[ q ] ( x s be the covoluto of the pdfs of p [] f obs [], =,..., q,. Lettg y = d x, y d, we ca wrte (23 as d d F( y f ( d ydy F ( y f ( d ydy, d [ F( y F ( y] f ( d ydy. (24 If F (y F (y, y d, the (24 holds ad thus F * F ( d F * F ( d. Smlarly, we ca wrte F * F ( d F * F ( d (22 as d [ F ( y F ( y ] f ( d ydy. (25 If F (y F (y, y d, the (25 holds ad F * F ( d F * F ( d. Accdgly, usg (2 (22, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad F (y F (y, y d; ( λ λ ad F (y F (y, y d. heem 4 ca be used to exame /p ~f (., ξ = d/ w ] X [ w X + f ay geeral processg tme pdfs. Below, we vestgate cases wth, f example, expoetal, webull, ufm dstrbutos. he use of these dstrbutos shop schedulg s ustfed by, e.g., Boxma ad Fst (986, Ca ad Zhou (997, 25, Jag (22, ad Pedo (983. (ve though the expoetal case s a specal stuato of the webull case, t s aalyzed due to the developmet of a geeral optmalty codto. xpoetal processg tmes Suppose that p ~exp(α wth cdf F (x = exp( α x ad meas /α, =,,. Collary 5. F /p ~exp(α, ξ = d/ [ w X ] X + w, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad α α ; ( λ λ ad α α. Proof. F p ~exp(α, =,...,, usg codto ( of heem 4, δ δ f every N ad every ad δ f λ λ ad exp( α y exp( α y, y d. However, ths equalty holds f y d f α α. Smlarly, usg codto ( of heem 4, δ δ f every N ad every ad δ f λ λ ad α α.

9 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 98 Collary 6. F /p ~exp(α, ξ = d/ [ w X ] + w X, a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( λ [] λ [l] λ [l + ] λ [], ad ( α [] α [l] ad α [ l + ] α []. Proof. We use a approach smlar to that of the proof of Collary to show that a sequece obtaed by arragg obs accdg to Collary 6 (see also codtos ( ( of Collary 5 s optmal. F the expoetal processg tmes, as the followg theem shows, we ca provde a me geeral optmalty codto. heem 5. F /p ~exp(α, ξ =d/ [ w X ] + w X, a optmal sequece s foud by arragg obs a o-decreasg der of λ α. Proof. Sce p ~exp(α wth F (x = exp( α x, =,,, the F * F ( d d = F ( f ( d x xdx d = α [ exp( α ( d x] exp( α xdx α [exp( α d exp( αd] = exp( α d, ( α α α α. (26 Substtutg (26 to (9, we get ( λα λ α [exp( α d exp( α d]/( α α F ( d, α α, whch holds f ( λα λα [exp( α d exp( α d]/( α α, α α. (27 Sce the fracto the bracets of equalty (27 s o-egatve, the the equalty holds f λ α λ α, N. Usg a approach smlar to that of the proof of Collary, we ca show that a sequece [],, [] foud by arragg obs accdg to λ α λ α, N (.e., λ []α [] λ []α [] s optmal. hat s, r* ca be detfed by arragg obs a o-decreasg der of λ α. Remar 5. Usg heem 5, r* f /p ~exp(α, ξ = d / [ w X ] ca be foud by arragg obs o-creasg der of αω o-decreasg der of (p / ω (.e., accdg to the weghted shtest expected processg tme (WSP rule (e.g., Pedo, 983, ad f /p ~exp(α, ξ = d/ [ w X ] by arragg obs o-decreasg der of αω o-creasg der of (p /ω (.e., accdg to the weghted logest expected processg tme (WLP rule. Webull processg tmes Suppose that p, =,,, have Webull dstrbutos wth shape ad scale parameters α ad β (.e., p ~W(α, β ad cdfs F (x = exp[ ( α x β ]. Collary 7. F /p ~W(α, β, ξ = d/ [ w X ] X + w, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad F (d F (d where F (d = exp[ ( α d β ] ad β β ; ( λ λ ad F (d F (d where β β. Proof. F p ~W(α, β, =,,, usg codto ( of heem 4, δ δ f every N ad every β ad δ f λ λ ad exp[ ( α y ] hs equalty smplfes to y β β α β y exp[ ( α ]. β β / α whch holds f all y d f β β ad d β β α β / α β β (.e., ( α d β ( α d F(d F(d. Smlarly, usg codto ( of heem 4, δ δ f every N ad every ad δ f λ λ β ad exp[ ( α y ] to β β α β α β y β y exp[ ( α ], whch reduces /, y d. hs holds f β β β β ad ( α d ( α d F (d F (d. herefe, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad F (d F (d where β β ; ( λ λ ad F (d F (d where β β. Collary 8. F /p ~W(α,β, ξ = d/ [ w X ] X + w, a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( λ [] λ [l] λ [l + ] λ [], ad ( F [](d F [l](d ad F [l + ](d F [](d where F (d = exp[ ( α d β ], β [] β [l], ad β [l + ] β [].

10 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 99 Proof. We use a approach smlar to that of the proof of Collary to show that a sequece obtaed by arragg obs accdg to Collary 8 (see also codtos (-( of Collary 7 s optmal. I geeral, the optmalty codtos of Collary 8 do ot hold amog obs ad thus r* caot be detfed. However, f these codtos are satsfed amog the obs a sequece [],, [l], [l + ],,[], l {,,, }, the λ []F [](d λ []F [](d. herefe, the soluto f /p ~W(α, β, ξ = d/ [ w X + w X ] ca be approxmated (.e., ca fd a caddate f r* by arragg obs o-decreasg der of λ F (d. xample 2. Cosder the problem /p ~W(α,β, ξ = d/ w ] X [ w X,, 5. + of able 2 where d = 3, = able 2. A stochastc - problem wth p ~W(α, β ad ξ = d = 3 Job α β ω ω λ Here, F (3 =.855,.662,.87,.722, ad.652, =,, 5, respectvely. Sce λ 4 < λ 2 < λ 5 < < λ < λ 3, F 4(3 > F 2(3 > F 5(3, ad F (3 < F 3(3 where β 4 < β 2 < β 5 ad β > β 3, based o Collary 8, r*: he caddate obtaed by arragg obs o-decreasg der of λ F (d s also optmal. Remar 6. Accdg to Collary 8, a sequece [],, [] whch λ [] λ [] s optmal f /p ~W(α, β, ξ = [ ] [ ] d/ [ w X ] f F [](d F [](d where F (d = exp[ ( α d β ] ad β [] β [], ad s optmal f [ ] [ ] /p ~W(α, β, ξ = d/ [ w X ] f F [](d F [](d where β [] β []. Ufm processg tmes Suppose that p, =,...,, are ufm radom varables defed the tervals [a, b ], b > a (.e., p ~U[a, b ]. A ufmly dstrbuted processg tme provdes a tme wdow (.e., [a, b ] durg whch the ob s processed wth equal probablty. he cdf of p ~U[a, b ] s defed as, f x a, x a F( x =, f a x b, b a, f x b. (28 Collary 9. F /p ~U[a, b ], ξ = d/ [ w X ] + w X, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad F (d F (d where F (x s defed by (28; ( λ λ ad F (d F (d. Proof. F p ~U[a,b ], =,...,, =,...,, usg codto ( of heem 4, δ δ f every N ad every ad δ f λ λ ad (y-a /(b -a (y a /(b a y[(b a (b a ] a (b a a (b a, y d (29 If b a b a ad a (b a a (b a (.e., (b a /a (b a /a, the (29 holds f ay y d; hece, F (d F (d where F (x s defed by (28. If b a < b a, (29 ca be rewrtte as y [a (b a a (b a ]/[(b a (b a ], y d, whch holds as log as d [a (b a a (b a ]/[(b a (b a ] F (d F (d. hus, whe λ λ, (29 holds f ether ( b a b a ad (b a /a (b a /a, (2 b a < b a ad F (d F (d. Smlarly, usg codto ( of heem 4, δ δ f every N ad every ad δ f λ λ ad y[(b a (b a ] a (b a a (b a, y d. (3 If b a b a ad (b a /a (b a /a, the (3 holds f ay y d; thus, F (d F (d. If b a > b a, we ca wrte (3 y [a (b a a (b a ]/[(b a (b a ], y d, whch holds as log as d [a (b a a (b a ]/[(b a (b a ] F (d F (d. herefe, δ δ f every N ad every ad δ f ( λ λ ; ( λ λ ad F (d F (d; ( λ λ ad F (d F (d. Collary. F /p ~U[a,b ], ξ = d/ [ w X ], + w X a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( λ [] λ [l] λ [l + ] λ [], ad ( F [](d F [l](d ad F [l + ](d F [](d where F (x s defed by (28. Proof. We use a approach smlar to that of the proof of Collary to show that a sequece obtaed by arragg obs accdg to Collary (see also codtos ( ( of Collary 9 s optmal. Smlar to the case wth webull f (., we ca approxmate the soluto ( fd a caddate f r* f /p ~U[a, b ],

11 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 ξ = d/ [ w X + w [] X [ ]] by arragg obs o-decreasg der of λ F (d. F ( x* λ[ F( x F * F ( x] dg( x F ( x* λ[ F ( x F * F ( x] dg( x. (32 Remar 7. Based o Collary, a sequece [],, [] whch λ [] λ [] s optmal f /p ~U[a, b ], ξ = d/ [ w X ] f F [](d F [](d, ad s optmal f /p ~U[a, b ], ξ = d/ [ w X ] f F [](d F [](d. Remar 8. Based o Collares 6, 8, ad, f /p ~f (., ξ =d / [ w X + w X ], obs [], =,, l, l +,,, l {,,, }, are arraged r* o-decreasg der of ω ω [ ] [ ](.e., λ[] where the F [](d of obs wth ω ω [ ], =,, l, are o-creasg der whle the F [](d of obs wth ω [ ] ω [ ], = l +,,, are o-decreasg der. hus, /p ~f (., ξ = d / [ w X + w X ] amog obs [],, [l], [l + ],, [] where < λ [] < + (.e., the stochastc - problem s a mxture of / p ~f (., ξ = d / [ w X ] amog obs [],, [l] where ω = λ [] (.e., the stochastc problem ad of /p ~f (., ξ = d / [ w X ] amog obs [l + ],, [] whereω [ ] = λ [] (.e., the stochastc problem. 3.3 Stochastc processg tmes ad stochastc due dates Cosder the scearo /p ~f (., ξ ~g (. / [ w X + w X ]. he, δ δ f every N ad every ad δ, usg (5, ff λ [ F * F( x F * F * F ( x] dg ( x λ where [ F * F ( x F * F * F ( x] dg ( x. * * ( * * ( F F F x F F F x F * F ( x. (3 F * F( x ad Iequalty (3 s too geeral to allow the developmet of useful statemets to establsh δ δ. However, we ca utlze ths equalty to exple the followg cases Idetcally dstrbuted due-dates Suppose that ξ ~g(. ad p ~f (., =,...,. Usg (3, δ δ f every N ad every ad δ ff We use (32 to aalyze dfferet cases of / p ~f (., ξ ~g(. / [ w X + w X ]. xpoetal processg tmes Cosder the case where p ~exp(α wth F (x = exp( α x ad ξ ~g(., =,,. heem 6. F /p ~exp(α, ξ ~g(./ [ w X ] X + w, a optmal sequece ca be foud by arragg obs a o-decreasg der of λ α. Proof. Usg F (x = exp( α x ad (26 (32, δ δ f every N ad every ad δ ff exp( α x exp( αx F ( x( [ λα λα ( ] dg( x α α, α α. (33 Sce the fracto sde the paretheses of (33 s o-egatve, the (33 holds f λ α λ α, N. Usg a approach smlar to that of the proof of Collary, we ca show that a sequece [],, [] obtaed by arragg obs accdg to λ α λ α, N (.e., λ []α [] λ []α [] s optmal. hat s, r* ca be detfed by arragg obs o-decreasg der of λ α. Remar 9. Based o heem 6, r* f /p ~exp(α, ξ ~g(./ [ w X ] ca be foud by arragg obs o-creasg der ofα ω o-decreasg der of (p / ω (.e., WSP rule (e.g., Ca ad Zhou, 25, ad r* f /p ~exp(α, ξ ~g(./ [ w X ] ca be foud by arragg obs o-decreasg der of αω o-creasg der of (p / ω (.e., WLP rule. xpoetal due-dates Cosder the case where ξ ~exp(γ wth G (x = exp( γx ad p ~f (., =,...,. heem 7. F /p ~f (., ξ ~exp(γ/ [ w X ] X + w, a optmal sequece ca be foud by arragg obs a o-decreasg der of λ /[/L (γ ]

12 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 where L (γ deotes the Laplace-Steltes trasfm (LS of f (. evaluated at γ. Proof. Substtutg G(x = exp( γx to (32, δ δ f every N ad every ad δ ff λ exp( γxf ( x*[ F( x F * F ( x] dx λ exp( γxf ( x*[ F ( x F * F ( x] dx, λ L (γl (γ[ L (γ] λ L (γl (γ[ L (γ], (34 where L (γ = exp( γ x f( xdx = γ γ exp( xf ( xdx, q s the LS of f (. evaluated at γ, ad L (γ = L[ ]( γ. Iequalty (34 smplfes to λ L (γ[ L (γ] λ L (γ[ L (γ], λ /[/L (γ ] λ /[/L (γ ]. Usg a approach smlar to that of the proof of Collary, we ca show that a sequece [],, [] obtaed by arragg obs accdg to λ /[/L (γ ] λ /[/L (γ ] s optmal; thus, r* s obtaed by arragg obs a o-decreasg der of λ /[/L (γ ]. xample 3. Cosder the problem /p ~f (., ξ ~exp(γ/ w ] X [ w X + of able 3 where p has a mal pdf wth mea 5 ad varace 2 (.e., p ~N(5, 2, p 2 has a gamma pdf wth shape ad slope parameters 3 ad (.e., p 2~G(3,, p 3~exp(4, p 4~U[2, ], ad p 5 has a ch-square pdf wth degree of freedom 8 (.e., p 5~χ 2 (8. Also, let ξ ~exp(γ, =,..., 5, wth γ =. he use of mal, expoetal, ad ufm processg tmes shop schedulg s ustfed by, e.g., Balut (973, Bertrad (983, Ca ad Zhou (997, 25, Jag (22, Kse ad Ibara (983, Sar et al. (99, Soush (999, ad Soush ad Allahverd (25. Sce p has a mal dstrbuto, ts varace must be small eough relatve to the mea such that Pr(p. o accomplsh ths, a coeffcet of varato (CV of at most.32 s cosdered f p (.e., CV = σ /μ.32 to sure Pr(p >.999. able 3. A stochastc - problem wth p ~f (. ad ξ ~exp( Job f (. ω ω λ N(5, G(3, xp( U[2, ] χ 2 ( Usg the pdfs of p, =,, 5, we respectvely have L ( = exp( 4, L 2( =.25, L 3( =.8, L 4( = exp( 2[ exp( 8]/8, ad L 5( =.97. he, λ /[/L (γ ]=.56,.74, 4.,.2, ad.98, =,, 5, respectvely. Arragg obs o-decreasg der of λ /[/L (γ ] provdes r*: (see heem 7. Remar. Based o heem 7, r* f /p ~f (., ξ ~exp(γ/ [ w X ] ca be foud by arragg obs o-creasg der of ω /[/L(γ ] (e.g. Boxma ad Fst, 986, ad r* f /p ~f (., ξ ~exp(γ / w X [ ] ca be foud by arragg obs o-decreasg der of ω / [/L(γ ]. Idetcally dstrbuted processg tmes Cosder the case where p ~f(. ad ξ ~g(., =,,. Usg (32, δ δ f every N ad every ad δ ff + λ λ ( q * ( q ( [ ( * F x F ( x] dg( x. (35 heem 8. F /p ~f(., ξ ~g(./ [ w X + w X ], a optmal sequece ca be foud by arragg obs a o-decreasg der of λ. ( q + Proof. Sce * ( q * F ( x F ( x, x, usg (35, δ δ f every N ad every ad δ f λ λ. We ca use a approach smlar to that the proof of Collary to show that a sequece [],, [] foud based o λ [] λ [] s optmal, that s, r* ca be foud by arragg obs a o-decreasg der of λ. Commo expected dfferece betwee earless ad tardess pealtes Cosder the case where λ = w ( w = ω ω, < λ < (.e., ob expected earless pealtes dffer from ther expected tardess pealtes by a commo costat, p ~f (., ad ξ ~g(., =,...,. he, usg (32, δ δ f every N ad every ad δ ff λ F ( x*[ F( x F( x] dg( x. (36 Let the stochastc derg p st p (p st p deote F (x F (x (F (x F (x f all x. heem 9. F /p ~f (., ξ ~g(., λ = λ/ [ w X + w ], X a optmal sequece ca be foud by

13 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 2 arragg obs o-creasg stochastc derg of p (.e., p [] st st p [] f λ >, ad o-decreasg stochastc derg of p (.e., p [] st st p [] f λ <. Proof. Whe λ =, ay sequece r R s optmal (see Secto 2. Whe λ >, equalty (36 holds f every N ad every f F (x F (x f all x (.e., p st p. Smlarly, whe λ <, the equalty s satsfed f every N ad every f F (x F (x f all x (.e., p st p. Usg a approach smlar to that of the proof of Collary, we ca show that a sequece [],, [] obtaed accdg to p [] st st p [] whe λ >, accdg to p [] st st p [] whe λ < s optmal. Hece, r* ca be foud by arragg obs o-creasg stochastc derg of p f λ >, ad o-decreasg stochastc derg of p f λ <. xample 4. Cosder the problem /p ~N(μ,σ 2, ξ ~g(., λ = λ/ = ω ω [ w X w X ] of able 4 where λ + = 2, =,..., 5(.e., /p ~N(μ, σ 2, ξ ~g(., λ = 2/ [ w X ]. Sce p ~N(μ, σ 2, =,, 5, p are such that CV.32 to sure Pr(p >.999. Here, F (x = Pr(Z z where Z s a stadard mal radom varable ad z = (x μ /σ, =,, 5. If Pr[Z (x μ /σ ] Pr[Z (x μ /σ ] f all x, the p st p. hat s, p st p f (x μ /σ (x μ /σ x(σ σ μ σ μ σ f all x. Suffcet codtos to satsfy ths equalty (.e., p st p are σ σ ad μ σ μ σ (.e., CV CV. Sce σ σ 2 ad CV CV 2, the p st p 2. Smlarly, p st p 3, p st p 4, ad p st p 5 because σ σ 3 ad CV CV 3, σ σ 4 ad CV CV 4, ad σ σ 5 ad CV CV 5, respectvely. Aalogously, we ca show that p 4 st p 2, p 4 st p 3, ad p 4 st p 5; p 2 st p 5 ad p 2 st p 3; ad p 5 st p 3. Hece, p st p 4 st p 2 st p 5 st p 3 whch mples r*: (see heem 9. Note that f /p ~f (., ξ ~g(., λ = λ/ [ w X ] ad /p ~f (., ξ ~g(., λ = λ/ [ w X ], the optmal sequeces are respectvely foud by arragg obs o-decreasg ad o-creasg stochastc derg of p. able 4. A stochastc - problem wth p ~N(µ,σ 2, ξ ~g(., ad λ = ω ω = 2. Job µ σ 2 ω ω λ Idetcally dstrbuted processg tmes Suppose that p ~f(. ad ξ ~g (., =,...,. Usg (3, δ δ f every N ad every ad δ ff ( q * + ( q [ F ( x F * ( x][ λdg ( x λ dg ( x ]. (37 Sce t s dffcult to aalyze (37 f geeral g (., we cosder the case where ξ ~exp(γ wth G (x = exp( γ x. heem. F /p ~f(., ξ ~exp(γ / [ w X + ] w X, δ δ f every N ad every ad δ f ( λ λ ; ( γ γ ad ether η η φ φ where η = λ L(γ [ L(γ ], (38 ad φ = λ L - (γ [ L(γ ]; (39 ( γ γ ad ether η η φ φ. Proof. Usg (37, δ δ f every N ad every ad δ f ( q * ( q+ [ F ( x F * ( x][ λγ exp( γx λγ exp( γ x], that ca be equvaletly wrtte as λ L q (γ [ L(γ ] λ L q (γ [ L(γ ], λ L q (γ [ L(γ ] λ L q (γ [ L(γ ], (4 where L(γ = γ exp( γxfxdx ( s the LS of F(. evaluated at γ. Iequalty (4 shows that the relato δ δ depeds o every ob N as well as the posto q of ob. (Note that L(γ ( L(γ ff γ ( γ. F λ λ, N, (4 s always satsfed. F L(γ L(γ (.e., γ γ, N, (4 holds f q =,..., (ob occupes posto q + f ( λ L(γ [ L(γ ] λ L(γ [ L(γ ], ( λ L (γ [ L(γ ] λ L (γ [ L(γ ]. F L(γ L(γ (.e., γ γ, (4 holds f

14 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 3 ( λ L(γ [ L(γ ] λ L(γ [ L(γ ], ( λ L (γ [ L(γ ] λ L (γ [ L(γ ]. Collary. F /p ~f(., ξ ~exp(γ / [ w X ] X + w, a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( η [] η [l] η [l + ] η [] where η s gve by (38, γ [] γ [l] ad γ [l + ] γ []; ( φ [] φ [l] φ [l + ] φ [] where φ s gve by (39, γ [] γ [l] ad γ [l + ] γ []. Proof. We use a approach smlar to that of the proof of Collary to show that a sequece obtaed by arragg obs accdg to Collary (see also codtos ( ( of heem s optmal. I geeral, the optmalty codtos of Collary do ot hold amog obs ad hece r* caot be foud. However, f these codtos are satsfed amog the obs a sequece [],, [l], [l + ],, [], l {,,, }, the ether ( η []/γ [] η []/γ []; ( φ []γ [] φ [l]γ []. Hece, the soluto f /p ~f(., ξ ~exp(γ / [ w X + w X ] ca be approxmated (.e., ca fd a caddate f r* by arragg obs o-decreasg der of ether η /γ φ γ where η ad φ are defed by (38 ad (39. xample 5. Cosder the problem /p ~G(.5, 2, ξ ~exp(γ / [ w X + w X ] of able 5. F p ~G(.5,2, L(γ = [2/(2 + γ ].5, =,, 5. he, usg (38, η =.25,.49,.378,.378, ad.262, =,, 5, respectvely. Sce η 3 < η < < η 2 < η 5 < η 4, γ 3 < γ ad γ 2 > γ 5 > γ 4, based o codto ( of Collary, r*: Here, the caddate foud by arragg obs o-decreasg der of η /γ s also optmal. able 5. A stochastc - problem wth p ~G(.5, 2 ad ξ ~exp(γ Job γ ω ω λ λ Remar. A sequece [],, [] s optmal f /p ~f(., ξ ~exp(γ / [ w X ] f ether η [] η [] ad γ [] γ [], φ [] φ [] ad γ [] γ [l] (see Collary. Also, a sequece [],, [] s optmal f /p ~f(., ξ ~exp(γ / [ w X ] f ether η [] η [] ad γ [] γ [], φ [] φ [] ad γ [] γ [] Dstctly dstrbuted processg tmes ad due-dates Assumg p ~f (. ad ξ ~g (., =,...,, we aalyze the followg cases. xpoetal processg tmes ad ufm due-dates Cosder the case where p ~exp(α ad ξ ~U[a, b ], =,...,. Usg F (x = exp( α x, g (x = /(b a, ad (26 (3, δ δ f every N ad every ad δ ff λα ( α α ( b a b a [exp( α x exp( α x] F ( xdx λα b [exp( α x exp( αx] F ( xdx ( ( b a a α α (4 heem. F /p ~exp(α, ξ ~U[a, b ]/ w ] X [ w X ad every ad δ f +, δ δ f every N ( λ < < λ ; ( λ α /(b a λ α /(b a ad ether ( < λ < λ, a a, ad b b, (2 λ < λ <, a a, ad b b. Proof. We have [exp( α x exp( α x]/(α α f x. he, f λ < < λ, (4 always holds. F < λ < λ, usg (4, δ δ f every N ad every ad δ f λα ( b a λα ( b a b a b a [exp( α x exp( αx]/( α α F ( xdx (42. [exp( α x exp( α x]/( α α F ( xdx Iequalty (42 s satsfed f ts left had sde (LHS s at most equal to oe (.e., λ α (b a λ α (b a λ α /(b a λ α /(b a ad ts rght had sde (RHS s at least equal to oe (.e., a a ad b b. (By defto, (4 also holds f λ = λ = α = α. Hece, δ δ f every N ad every ad δ f < λ < λ, a a, b b, ad λ α /(b a λ α / (b a. F λ < λ <, δ δ f every N ad every ad δ f a equalty smlar to (42 but wth drecto holds. However, such a equalty s satsfed f ts LHS s at least equal to oe (.e., λ α /(b a λ α /(b a where λ < λ < ad

15 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 4 ts RHS s at most equal to oe (.e., a a ad b b. hus, δ δ f every N ad every ad δ f λ < λ <, a a, b b, ad λ α /(b a λ α /(b a. Collary 2. F /p ~exp(α, ξ ~U[a, b ]/ [ w X + w ] X, a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( λ [] λ [l] λ [l + ] λ [] ad λ []α []/(b [] a [] λ []α []/(b [] a [], ad ( a [] a [l], a [l + ] a [], b [] b [l], ad b [l + ] b []. Proof. Usg a approach smlar to that of the proof of Collary, we show that a sequece foud by arragg obs based o Collary 2 (see also codtos ( ( of heem s optmal. he optmalty codtos of Collary 2 ca be relaxed by removg λ [] λ [l] λ [l + ] λ [] ad codto ( leavg behd oly λ []α []/(b [] a [] λ []α []/(b [] a []. hs codto ca be used to approxmate the soluto (.e., fd a caddate f r* f /p ~exp(α, ξ ~U[a, b ]/ [ w X + w X ], that s, arragg obs o-decreasg der of λ α /(b a provdes a caddate f r*. Remar 2. Based o Collary 2, a sequece [],, [] where λ [] λ [] ad λ []α []/(b [] a [] λ []α []/(b [] a [] s optmal f /p ~exp(α, ξ ~U[a, b ] / [ w X ] f a [] a [] ad b [] b [], ad s optmal f /p ~exp(α, ξ ~U[a, b ]/ [ w X ] f a [] a [] ad b [] b []. Collary 3. F /p ~exp(α, ξ ~U[a, b]/ [ w X ], X + w a optmal sequece s foud by arragg obs o-decreasg derg of λ α, =,...,. Proof. It mmedately follows from Collary 2 (see also equalty (4. Note that Collary 3 ad heem 6 provde the same results. xpoetal processg tmes ad due-dates Cosder the case where p ~exp(α ad ξ ~exp(γ, =,...,. he, usg F (x = exp( α x, G (x = exp( γ x, ad (26 to (3, δ δ f every N ad every ad δ ff λαγ ( α α + γ α + γ α exp[ ( x] exp[ ( x] F ( xdx λαγ [ exp[ ( α + γ x ] ( α α ] exp[ ( α + γ x] F ( xdx. (43 heem 2. F /p ~exp(α, ξ ~exp(γ / [ w X ] X + w, δ δ f every N ad every ad δ f ( λ < < λ ; ( λ α γ λ α γ ad ether ( < λ < λ ad γ γ, (2 λ < λ < ad γ γ. Proof. We have [exp[ (α + γ x] exp[ (α + γ x]]/(α α f all x, {, }. Whe λ < < λ, (43 s satsfed. Whe < λ < λ, usg (43, δ δ f every N ad every ad δ f λαγ λαγ [exp( ( α + γ x exp( ( α + γ x] F ( xdx ( α α.(44 [exp( ( α + γ x exp( ( α + γ x] F ( xdx ( α α Iequalty (44 holds f ts LHS s at most equal to oe (.e., λ α γ λ α γ ad ts RHS s at least equal to oe (.e., γ γ. (By defto, (43 also holds whe λ = λ =, α = α, γ = γ ad λ α λ α. Hece, δ δ f every N ad every ad δ f < λ < λ, γ γ ad λ α γ λ α γ. F λ < λ <, δ δ f every N ad every ad δ f a equalty smlar to (44 but wth drecto s satsfed. But, such a equalty holds f ts LHS s at least equal to oe (.e., λ α γ λ α γ where λ < λ < ad ts RHS s at most equal to oe (.e., γ γ. Hece, δ δ f every N ad every ad δ f λ < λ <, γ γ ad λ α γ λ α γ. Collary 4. F /p ~exp(α, ξ ~exp(γ / [ w X ] X + w, a sequece [],, [l], [l + ],, [], l {,,, }, s optmal f ( λ [] λ [l] λ [l + ] λ [] ad λ []α []γ [] λ []α []γ [], ad ( γ [] γ [l] ad γ [l + ] γ []. Proof. We use a approach smlar to that of the proof of Collary to show that a sequece foud by arragg

16 Soush: Sgle Mache Schedulg wth Stochastc Processg mes Stochastc Due-Dates to Mmze the Number of arly ad ardy Jobs IJOR Vol. 3, No. 2, 9 8 (26 5 obs accdg to Collary 4 (see also codtos ( ( of heem 2 s optmal. We ca relax the optmalty codtos of Collary 4 by removg λ [] λ [l] λ [l + ] λ [] ad codto ( leavg behd oly λ []α []γ [] λ []α []γ []. hs codto ca be used to approxmate the soluto (.e., fd a caddate f r* f /p ~exp(α, ξ ~exp(γ / [ w X + w X ], that s, arragg obs o-decreasg der of λ α γ ca provde a caddate f r*. Remar 3. Accdg to Collary 4, a sequece [],, [] whch λ [] λ [] ad λ []α []γ [] λ []α []γ [] s optmal f / p ~exp(α, ξ ~exp(γ / [ w X ] f γ [] γ [], ad s optmal f /p ~exp(α, ξ ~exp(γ / [ w X ] f γ [] γ []. Collary 5. F /p ~exp(α, ξ ~exp(γ/ [ w X ] X + w, a optmal sequece s foud by arragg obs o-decreasg derg of λ α, =,,. Proof. It mmedately follows from Collary 4 (see also equalty (43. Observe that Collary 5 ad heem 6 provde the same results. Remar 4. F /p ~f (., ξ ~g (./ [ w X ] X + w, based o ths subsecto s dscusso, obs [], =,, l, l +,,, l {,,, }, are arraged r* o-decreasg der of ω ω [ ] [ ](.e., λ[] where there are addtoal codtos mposed o some other characterstcs of obs [], =,, l (.e., obs wth ω [ ] ω [ ] as well as o those of obs [], = l +,, (.e., obs wth ω [ ] ω [ ] / [ w X. Hece, /p ~f (., ξ ~g (. + w ] X amog obs [],, [l], [l + ],, [] where < λ [] < (.e., the stochastc - problem s a mxture of /p ~f (., ξ ~g (./ [ w X ] amog obs [],, [l] where ω [ ] = λ[] (.e., the stochastc problem ad of /p ~f (., ξ ~g (. / [ w X ] amog obs [l + ],, [] where ω λ [] (.e., the stochastc problem. = 4. SUMMARY AND SOM CONCLUDING RMARKS I ths paper, we have studed a stochastc sgle mache schedulg problem whch processg tmes due-dates are o-egatve depedet radom varables ad radom weghts (pealtes are mposed o both early ad tardy (- obs. hese radom weghts do ot deped o the amout of devatos of ob completo tmes from ther due dates, that s, the pealty f mssg a due date by a sht log perod s the same. he obectve s to fd a optmal sequece that mmzes the expected total weghted umber of early ad tardy obs. We have examed three scearos of the proposed stochastc - problem cludg a scearo wth determstc processg tmes ad stochastc due-dates, a scearo wth stochastc processg tmes ad determstc due-dates, ad a scearo wth stochastc processg tmes ad stochastc due-dates. hese problem scearos are NP hard to solve; however, based o some structures o the stochastcty of processg tmes due dates, we have solved exactly varous resultg cases of the three scearos (see able 6. We have also preseted methods to approxmate the solutos f the geeral versos of these cases. It s demostrated that the proposed stochastc - problem those obs whose mea earless pealtes are at most equal to ther mea tardess pealtes appear the optmal sequece befe those whose mea earless pealtes are greater tha ther mea tardess pealtes. Meover, the problem studed here s show to be geeral the sese that ts specal lmtg cases reduce to some classcal sgle mache schedulg problems cludg the stochastc problem of mmzg the expected weghted umber of tardy obs ad the stochastc problem of mmzg the expected weghted umber of early whch both are solvable by the proposed exact approxmate methods. hs research valdates oe of the prcples of sychroous maufacturg that statstcal fluctuatos ob characterstcs such as processg tmes, due dates, ad earless ad tardess pealtes affect schedulg decsos. A mmedate exteso of ths study s to exple the most geeral verso of the problem whe processg tmes ad due dates have dstct arbtrary dstrbutos. I addto, due to the mptace of research schedulg wth setup tmes (e.g., Allahverd et al., 999; Allahverd et al., 26; Allahverd ad Soush, 26, t s hghly recommeded to exame the proposed stochastc - problem by cpatg explctly ob setup tmes. ACKNOWLDGMNS he auth would le to express hs grattude to the three aoymous referees f ther valuable commets ad to the Offce of Vce Presdet f Research at Kuwat Uversty f fudg ths research uder grat SS3/. RFRNCS. Allahverd, A., Gupta, J.N.D., ad Aldowasa,. (999. A revew of schedulg research volvg setup cosderatos. OMGA: he Iteratoal Joural of Maagemet Sceces, 27: