The preemptive resource-constrained project scheduling problem subject to due dates and preemption penalties: An integer programming approach

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1 Journal of Indusral Engneerng 1 (008) The preempve resource-consraned projec schedulng problem subjec o due daes and preempon penales An neger programmng approach B. Afshar Nadjaf Deparmen of Indusral Engneerng, Islamc Azad Unversy of Qazvn, Qazvn, Iran, Correspondng Auhor, Afsharnb@Mehr.sharf.edu S. Shadrokh Deparmen of Indusral Engneerng, Sharf Unversy of Technology, Tehran, Iran, Shadrokh@Sharf.edu Absrac Exensve research has been devoed o resource consraned projec schedulng problem. However, lle aenon has been pad o problems where a ceran me penaly mus be ncurred f acvy preempon s allowed. In hs paper, we consder he projec schedulng problem of mnmzng he oal cos subjec o resource consrans, earlness-ardness penales and preempon penales, where each me an acvy s sared afer beng preemped; a consan seup penaly s ncurred. We propose a soluon mehod based on a pure neger formulaon for he problem. Fnally, some es problems are solved wh LINGO verson 8 and compuaonal resuls are repored. Keywords Projec schedulng; Resource consraned; Preempve schedulng; Earlness-ardness cos. 1. Inroducon Preempve projec schedulng problems are hose n whch he accomplshng of an acvy can be emporarly nerruped, and resared a a laer me. Consequenly n he leraure on preempve projec schedulng, preemped acves can smply be resumed from he pon a whch preempon occurred a no cos. However, hs suaon s no always rue n pracce. I s lkely ha n some cases, a ceran delay or seup cos mus be ncurred. The leraure on soluon mehods for he preempve resource consraned projec schedulng problem wh weghed earlness ardness and preempon penales (PRCPSPWETPP) s scan. Of course, several papers have been devoed o machne schedulng wh preempon penales. Pos and Van Wassenhove (Pos & Van Wassenhove, 199, ) suggesed consderng preempon penales under he lo-szng model. Then, Monma and Pos (Monma & Pos, 1993, ) and Chen (Chen, 1993, ) suded he preempve parallel machne schedulng problem wh bach seup mes. Zdrzalka (Zdrzalka, 1994, 60-71), Schuurman and Woegnger (Schuurman & Woegnger, 1999, ) and Lu and Cheng (Lu & Cheng, 00, ) suded preempve schedulng problems wh job dependen seup mes. Julen and e al. (Julen & e al, 1997, ) proposed more preempon models and appled hem o wo sngle machne schedulng problems. In projec schedulng feld, Vanhoucke (Vanhoucke, 001) and Vanhoucke and e al. (Vanhoucke & e al, 000a, ) have developed an exac recursve search algorhm for he basc form of weghed earlness-ardness projec schedulng problem (WETPSP) n he absence of resource consrans and preempon. The algorhm explos he basc dea ha he earlness-ardness coss of a projec can be mnmzed by frs schedulng acves a her due dae or a a laer me nsan f forced so by bndng precedence consrans, followed by a recursve search whch compues he opmal dsplacemen for hose acves for whch a shf owards me zero proves o be benefcal. Vanhoucke and e al. (Vanhoucke & e al, 000b) have exploed he logc of he recursve procedure for solvng he WETPSP n her branch and bound procedure for maxmzng he ne presen value of a projec n whch progress paymens occur. Kaplan (Kaplan, 1988) was he frs o sudy he preempve resource-consraned projec schedulng problem (PRCPSP). She formulaed he PRCPSP as a dynamc program and solved usng a reachng procedure. Demeulemeeser and Herroelen (Demeulemeeser & Herroelen, 1996, ) developed a branch and bound algorhm for he problem. In hs paper, we consder he projec schedulng problem of mnmzng he oal cos subjec o resource consrans, earlness-ardness penales and preempon penales, where each me an acvy s sared afer beng preemped; a consan seup penaly s ncurred. The paper s organzed as follows Secon descrbes he problem. An neger formulaon s gven n secon 3. A numercal example and compuaonal resuls are represened n secon 4 and 5, respecvely. Secon 6 conans he conclusons. 35/

2 B. Afshar Nadjaf e al. /The preempve resource-consraned projec schedulng problem subjec. Problem descrpon A non-regular performance measure, whch s ganng aenon n jus-n-me envronmens, s he mnmzaon ndvdual acvy due dae wh assocaed un earlness and un ardness penaly coss. In he classcal resource-consraned projec schedulng problem (RCPSP) here s no room for preempon of he acves n he projec. Preempve projec schedulng problems are hose n whch he accomplshng of an acvy can be emporarly nerruped, and resared a a laer me. Consequenly n he leraure on preempve projec schedulng, preemped acves can smply be resumed from he pon a whch preempon occurred a no cos. However, hs suaon s no always rue n pracce. I s lkely ha n some cases, a ceran delay or seup cos mus be ncurred. The deermnsc preempve resource-consraned projec schedulng problem wh weghed earlness ardness and preempon penales (PRCPSPWETPP) nvolves he schedulng of projec acves n order o mnmze he oal earlness-ardness and preempon penales of he projec n he presence of resource consrans. In sequen, assume a projec represened n AON forma by a dreced graph G = {N, A} where he se of nodes, N, represens acves and he se of arcs, A, represens fnsh-sar precedence consrans wh a melag of zero. The preempable acves are numbered from he dummy sar acvy 1 o he dummy end acvy n and are opologcally ordered,.e. each successor of an acvy has a larger acvy number han he acvy self. The fxed duraon of an acvy s denoed by d ( 1 n), whle h denoes s deermnsc due dae. The objecve of he PRCPSPWETPP s o schedule a number of acves, n order o mnmze he oal cos of he projec subjec o fnsh o sar precedence relaons wh a me-lag of zero, consraned resources and a fxed deadlne. 3. Problem formulaon We have he followng noaons for preempve resource-consraned projec schedulng problem wh weghed earlness ardness and preempon penales (PRCPSPWETPP) n A N d h EST LST EFT number of acves se of arcs of acyclc dgraph represenng he projec se of nodes of acyclc dgraph represenng he projec duraon of acvy due dae of acvy earles sar me of acvy laes sar me of acvy earles fnsh me of acvy of he weghed earlness-ardness penaly coss of he projec acves. In hs problem seng, acves have an LFT ak r T Z π ν τ E T laes fnsh me of acvy avalably of he kh resource ype resource requremen of acvy for resource deadlne of he projec objecve funcon each preempon penaly of acvy per un earlness cos of acvy per un ardness cos of acvy earlness of acvy (neger decson varable) ardness of acvy (neger decson varable) In our formulaon, 0-1 varables Xj are defned, whch specfy wheher jh un of duraon of an acvy fnshes a me or no. More specfcally, for every un j of duraon of acvy and for every feasble compleon me [ EST + j,lft ( d j)], Xj s defned as follows Xj = 1, f jh un of duraon of acvy fnshes a me Xj = 0, oherwse Also, 0-1 varables yj are defned, whch specfy wheher jh un of duraon of an acvy s preemped a me or no. More specfcally, for every un j of duraon of acvy and for every feasble compleon me [ EST + j,lft ( d j)], yj s defned as follows yj = 1, f jh un of duraon of acvy preemps a me yj = 0, oherwse The varables Xj and yj can only be defned over he me nerval of he acvy n queson. These lms are deermned usng he radonal forward and backward pass calculaons. The backward pass calculaon s sared from a fxed projec deadlne T. Inroducng he bnary decson varables Xj and yj, as well as he neger varables E and T denong he earlness and ardness of acvy,respecvely, and usng he above noaon, preempve resource-consraned projec schedulng problem wh weghed earlness ardness and preempon penales (PRCPSPWETPP) under he mnmum oal early-ardy and preempon penaly cos objecve can be mahemacally formulaed as follows n-1 ( = ) n 1 ( d LFT - ( d- j) E + + T y = j= 1 = EFT - ( j d- j) mn Z = ( 1 ) subjec o LFT = EFT E h X for =,..., n -1 d ( ) ( )) 36/

3 Journal of Indusral Engneerng 1 (008) LFT T X h for =,..., n -1 d ( ) 3 = EFT LFT X d = EFT LFT ( d j + 1) X ( j 1) = EFT ( d j + 1) LFT ( d ) j = EST + subjec o LSTj for all (, j) A LFT ( d j) + 1 X j = EFT ( d j),...,n -1 1 for j j = X j = = 1,...,d =,..., n -1 for j =,...,d ( 6 ) =,...,n -1 X j X ( j+ 1)( + 1) + yj forj = 1,...,d ( 7 ) = EFT ( d j),..., LFT ( d j) n 1 d 1 = j= 1 r X j a k k = 1 for = 1,..., T =,...,n -1 j = 1,...,d X j, yj {0,1} for ( 9) k = 1,...,m = EST + j,..., LFT ( d j) E, T n for =,...,n -1 ( 4) ( 5) ( 8) ( 10) The objecve n Eq. (1) s o mnmze he oal cos of he projec. Eq. () and (3) compue he earlness and ardness of each acvy. The consran se gven n Eq. (4) mposes he fnsh-sar precedence relaons among he acves. In Eq. (5) s specfed ha he fnsh me for every un of duraon of an acvy has o be a leas one me un larger han he fnsh me for he prevous un of duraon. Eq. (6) specfes ha only one compleon me s allowed for every un of duraon of an acvy. Eq. (7) guaranee ha f wo successve uns of duraon an acvy (.e. un j and j+1) are nerruped a me ; herefore correspondng decson varable y j mus se o 1. The resource consrans for every resource ype k are specfed n Eq. (8) by consderng for every me nsan and every resource ype k, all possble compleon mes for every uns of duraon of all acves such ha he acvy s n progress n perod. Ths consran se spulaes ha he resource consrans canno be volaed. Eq. (9) and (10) specfes ha he decson varables X j and y j are bnary, whle E and T are neger. Ths formulaon requres he defnon of a mos n T d = 1 bnary decson varables and of (n-) neger varables. Also, he number of consrans of he formulaon amouns o a mos n n n n T 1 ( ) + ( 1) / + (+ ) d+ mt. = ). The correspondng AON projec nework s shown n Fg.1. There are 7 acves (and wo dummy acves) and one resource ype wh an avalably of 5. The number above he node denoes he acvy duraon, whle he numbers below he node denoe he due dae, he un early-ardy cos (For ease of represenaon, we assume he un earlness coss o equal he un ardness coss) and he resource requremens, respecvely. Also, we assume he preempon penaly for all acves equal o 1. The opmal non-preempve schedule for hs example s presened n Fg.. Fgure 1. Problem nsance for he PRCPSPWETPP Fgure. Opmal non-preempve schedule o he problem example The proposed formulaon for hs problem example requres he defnon of 19 bnary decson varables and of 14 neger varables. The number of consrans and nonzero elemens of consrans marx equals o 108 and 580, respecvely. Usng he LINGO verson 8, based on branch and bound mehod, we obaned he opmal schedule of Fg.3 wh a cos of 3. Ths problem solved whn 1 second of CPU-me. Of course, hs schedule s presened a he level of he sub-acves, ha s, each acvy s dvded o d segmen wh duraon of 1 and resource requremen of r. 4. Numercal example In hs secon, we demonsrae he compuaon of he opmal PRCPSPWETPP soluon on a problem nsance ha s adaped from he Paerson se (Paerson, 1984, 37/

4 B. Afshar Nadjaf e al. /The preempve resource-consraned projec schedulng problem subjec Table 1 The average CPU-me and he sandard devaon needed o solve he PRCPSPWETPP Number of acves Number of problems Average CPU-me Sandard devaon Fgure 3. The opmal schedule a he level of he sub-acves problems solved o opmaly Translang hs opmal schedule n erms of he orgnal acves, he opmal preempve schedule of Fg.4 s obaned. I s should be obvous ha n hs opmal schedule h un of duraon of acvy s preemped a me. Fgure 4. The opmal preempve schedule a he level of he acves 5. Compuaonal resuls In order o valdae he neger programmng model for he preempve resource consraned projec schedulng problem wh preempon penales and weghed earlness ardness penales, a problem se conssng of 900 problem nsances was generaed. Ths problem se conssng of equally 300 nsances wh 10, 0 and 30 acves. The problem se was exended wh un earlness-ardness penaly coss and preempon penales for each acvy whch are randomly generaed beween 1 and 10. The due daes were generaed n he same way as descrbed by Vanhoucke and e al. (Vanhoucke & e al, 000a, ). Frs, a maxmum due dae was obaned for each projec by mulplyng he crcal pah lengh by 1.5. Subsequenly, we generae random numbers beween 1 and maxmum due dae. The numbers are sored and assgned o he acves n ncreasng order. Acvy duraons and acvy resource demand are randomly seleced beween 1 and 10. Maxmum number of predecessors and successors and number of resource ypes supposed 3. 0 second 15 second 30 second 60 second CPU me lm Fgure 5. Effec of he number of acves and he allowed CPU-me for he problem The problem se has solved usng he LINGO verson 8 under wndows XP on a personal compuer wh Penum 4, 1.7 GHz processor. Table 1 represens he average CPU-me and s sandard devaon n second for a dfferen number of acves wh a me lm of 60 second. 95% of problems wh 10 acves can be solved o opmaly whn second of CPU-me. For problems conssng 0 acves, 79% of he problems can be solved o opmaly when he allowed CPU-me s 15 second, whereas 91% of he problems can be opmally solved where he CPU-me lm s 30 second. For problems wh 30 acves, 46% of he problems can be solved whn 30 second of CPU-me whereas 80% of he problems can be solved o opmaly when he allowed CPU-me s 60 second. Fg.5 dsplays he number of problems solved o opmaly for a dfferen number of acves and allowed CPU-me. 6. Summary and conclusons Ths paper repors on an neger programmng based procedure for preempve resource consraned projec schedulng problem wh weghed earlness ardness and preempon penales (PRCPSPWETPP). The objecve s o schedule he acves n order o mnmze he oal cos of earlness-ardness and preempon penales subjec o he precedence consrans, resource consrans and a fxed deadlne on projec. Pure neger programmng model appled for solvng a numercal example. Fnally, some es problems are solved wh LINGO verson 8 and compuaonal resuls are repored. 7. References [1] C. N. Pos, L. N. Van Wassenhove, Inegrang schedulng wh bachng and lo-szng A revew of algorhms and complexy, Journal of he Operaonal Research Socey 43 (199) [] C. L. Monma,C. N. Pos, Analyss of heurscs for preempve parallel machne schedulng wh bach seup mes, Journal of Operaons Research 41 (1993) /

5 Journal of Indusral Engneerng 1 (008) [3] B. Chen, A beer heursc for preempve parallel machne schedulng wh bach seup mes, SIAM Journal of Compuaon (1993) [4] S. Zdrzalka, Preempve schedulng wh release daes, delvery mes and sequence ndependen seup mes, European Journal of Operaonal Research 76 (1994) [5] P. Schuurman, G. J. Woegnger, Preempve schedulng wh jobdependen seup mes, Proceedng of he 10 h ACM-SIAM Symposum on Dscree Algorhms (1999) [6] Z. Lu, T. C. E. Cheng, Schedulng wh job release daes, delvery mes and preempon penales, Informaon Processng Leer 8 (00) [7] F. M. Julen, M. J. Magazne, N. G. Hall, Generalzed preempon models for sngle-machne dynamc schedulng problems, IIE Transacons 9 (1997) [8] M. Vanhoucke, Exac algorhms for varous ypes of projec schedulng problems Nonregular objecves and me/cos rade-offs, Unpublshed Ph.D. Dsseraon, Kaholeke Unverse Leuven (001). [9] M. Vanhoucke, E. Demeulemeeser, W. Herroelen, An exac procedure for he resource consraned weghed earlness-ardness projec schedulng problem. Annals of Operaons Research 10 (000a) [10] M. Vanhoucke, E. Demeulemeeser, W. Herroelen, Maxmzng he ne presen value of a projec wh progress paymens, Research Repor 008 (000b) Deparmen of Appled Economcs, Kaholeke Unverse Leuven. [11] L. A. Kaplan, Resource consraned projec schedulng wh preempon of jobs, Unpublshed Ph.D Thess (1988) Unversy of Mchgan. [1] E. Demeulemeeser, W. Herroelen, An effcen opmal soluon procedure for he preempve resource consraned projec schedulng problem, European Journal of he Operaonal Research 90 (1996) [13] J. H. Paerson, A comparson of exac procedures for solvng he mulple consraned resource projec schedulng problem, managemen scence 30 (1984) /

6 B. Afshar Nadjaf e al. /The preempve resource-consraned projec schedulng problem subjec