One Machine Scheduling Problem With Release dates and Tow Criteria يسأنح جذ نح ان اك ان احذ ت ج د أ قاخ تحض ش

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1 Journal of Th-Qar Unversty number2 Vol.6 March/2011 One Machne Schedulng Problem Wth Release dates and Tow Crtera Al-Zuwan Mohammad Kadhm Math. De. College of Comuter Scence and Mathematcs, Th-qar Unversty Mohanned M. Kadum Com. De. College of Comuter Scence and Mathematcs, Th-qar Unversty يسأنح جذ نح ان اك ان احذ ت ج د أ قاخ تحض ش يح ذ كاظى انض ي ذ يح ذ كاظى قسى انش اص اخ قسى انحاسثاخ كه ح عه و انحاسثاخ انش اض اخ كه ح عه و انحاسثاخ انش اض اخ جايعح ري قاس جايعح ري قاس ان ستخهص: قذو زا انثحث خ اسصي ح انتفشع انتق ذ نتشت ة يج ع ي ان تاجاخ عهى ان اك ح ان احذ, ان ذف تصغ ش انكهفح انكه ح نضي ا س اب ان تاجاخ عذد ان تاجاخ ان تأخش ع ذيا ك نه تاجاخ اصي تحض ش غ ش يتسا ح. تض انثحث ق ذ اد ى حه ل كف ء نثعض انحاالخ انخاصح ق اعذ نهحذ ي تفشعاخ شجشج انثحث ف طش قح انتفشع انتق ذ. قذ جذ انحم األيثم نغا ح 04 تاج. 1

2 Journal of Th-Qar Unversty number2 Vol.6 March/2011 Abstract. Ths aer resents a branch and bound algorthm for sequencng a set of obs on a sngle machne schedulng wth the obectve of mnmzng total cost of flow tme and number of tardy obs, when the obs may have unequal ready tmes. Lower bound; effcent solutons, domnance rules for ths roblem and a comutatonal exerence wll also be ncluded. Comutatonal exerence wth nstances havng u to 40 obs shows that the lower bound s effectve n restrctng the search. Key words. Flow tme, tardy obs, schedulng, release date 1. Introducton: The roblem of sequencng n obs on one machne under dfferent assumtons and multle crtera are consdered extensvely. The obectve functon to be mnmzed conssts of two crtera wth unequal ready tmes: sum of flow tme denoted by F lus total number of late obs denote byu. We assume that the two crtera have the same mortance. Denote ths roblem by 1/ r / ( F u ). Problem F 1 / r 0/ F s well nown and can olynomally solved (Smth 1956). For 1 / r / roblem was shown NP-comlete by Lenstra et al. (1977)[8]. Mason and Anderson (1991)[10] examne the statc sequencng roblem of orderng the rocessng of obs on a sngle machne so as to mnmzed the average weghted flow tme. It was assumed that all ob had zero ready tmes, and that the obs are groued nto classes. Kellerer et al. [6] rovde an aroxmaton algorthm for the corresondng1/ r / F roblem, wth a rato guarantee of O ( n). Zghar (2000) [15] used un effcent branch and bound technque wth a sutable lower bound and roved some domnance rules for 1 / r / F roblem For u 1 / r / roblem has been shown NP-hard [2]. Several secal cases yeld olynomal algorthm. The 1// u roblem can be solved n O(n log n) stes by usng Moor ' s algorthm [11]. Peter and Mhal (1996) [12] studed the sngle machne schedulng roblem to mnmze the weghted number of late obs. Shao and Mlan (2003) [13] consdered a sngle 2

3 Journal of Th-Qar Unversty number2 Vol.6 March/2011 machne schedulng to mnmze the number of late obs under uncertanty. They roosed a rather general model based on an algebrac aroach. Greogoro et al. (2004) [5] used a schedulng technques to mnmze the number of late obs n worflow systems by organzatons to control and mrove busness rocesses. Ther wor resent some of the roblems of usng schedulng results n orderng cases n a worflow and tacles two of them: the uncertantes on the cases, rocessng tmes and routng. Maran and Han (2008) [9], resent a new model to deal wth the stochastc comleton tmes, whch s based on usng a chance constrant to defne whether a ob s on tme or late. They have studed mnmzng the number of late obs roblem for four classes of stochastc rocessng tmes. The u 1/ r / roblem can be solved f there are agreeable due dates (.e. there s a renumberng of the ob so that r r 1 and d d 1 ( 1, 2,..., n -1) n O(n 2 ) stes by Kse et al. [7]. For the comoste crtera, Emmons (1975) [4] consdered a multle crtera, n whch the rmary crteron s to mnmze the number of tardy obs whle the secondary crteron s to mnmze the sum of comleton tmes. The comutatonal exerments he carred out ndcated that the addtonal comutatonal effort to contnue to otmalty to be remarably lttle. Van Wassenhove and Luda (1980) [14] consder the roblem of sngle machne schedule to mnmzed the flow obs and maxmum tardness. Abdul-Razaq and zghar [1] consdered the total cost of comleton tme and the number of tardy obs. They assumed that all obs avalable for rocessng at tme zero (.e. r =0 for each ob), and they solved the roblem wth u to 20 obs. In ths aer we extend the wor n [1] and [4]. The am n ths study s to mnmze the total cost of flow tme and number of tardy obs wth unequal release dates, that s we assume that the obs are avalable for rocessng wth dfference tmes(. e. r 0 ). Then our roblem s strongly NP-hard, snce the 1/ r / ( C u ) roblem NP-hard [1, 4]. 2. Problem Formulaton To state our schedulng roblem more recsely, we are gven a set N of obs, N= {1, 2,..., n} s to be rocessed one ob at the tme, on a sngle machne. For each ob, the rocessng tme, the due date d and the ready tme r are gven. For a gven rocessng order of obs the 3

4 Journal of Th-Qar Unversty number2 Vol.6 March/2011 comleton tme C, for ob, ( N ) be comuted. The obectve s to fnd a rocessng order of the obs that mnmzes the sum of the total cost of flow tme F and the number of tardy obsu, wth release dates. Ths s denoted by 1/ r / ( F u ). Let : the set of ermutaton schedules ( n!). : a ermutaton schedule, ( ) F : the flow tme of ob, ( ). C ( ) : the total comleton tmes of schedule, ( C ( ) c ) F ( ) ): the total flow tmes of schedule, ( F ( ) F ) U ( ) : No. of tardy obs of schedule, ( U ( ) u ). Where C 1 r1 1, C max{ C 1, r }, 2,..., n F C r 1 f C u 0 o. w. d Then the obectve s to fnd a schedule of the obs ( ) that mnmze total cost Z ( ), where: Z( ) F( ) U( ) The mathematcal form of our roblem can be formally stated as: Mn Z( ) Subect to : C, r and d 0...() For defne such a sequence more recsely. A sequence * s otmal n roblem () f there no sequence such that: * Z( ) Z( ) Smlarly, we say that a sequence 1 domnates a sequence 2 when, ( 1) Z( 2) 4 Z.

5 Journal of Th-Qar Unversty number2 Vol.6 March/ Domnance Theorems. If t can be shown that an otmal soluton can always be generated wthout branchng from a artcular node of the search tree, then that node s domnated and can be elmnated. Domnance rules usually secfy whether a node can be elmnated before ts lower bound s calculated. Clearly, domnance rules are artcularly useful when a node can be elmnated whch has a lower bound that s less than the otmum soluton [15]. Theorem (3-1) For roblem (). If r r, and d d then n otmal soluton. Let be a sequence of obs n whch the ob receds ob, r r and d d. Let, T be a comleton tme of ob ( 1) n. Let be a sequence has the same obs order of excet the ob recednd ob. There are three cases: Case 1: r r T Let a=r - r and b=t-r (F + F ), = a+2b+2 +, (F + F ), = a+2b+2 + (F + F ), - (F + F ), = - < 0.(1) f ob s late, then ob s late and, are late, then (1) s hold. If ob s early and s late, that s, u 1. If obs, are early, then, are early also. So (1) s satsfed. Case 2: r T r () f T r, then ( F ), F = a c, ( F F ), a b 2 Where a T r, b r - T and c T r F - F F ) = c ( b) <0 (2) ( F ), (, If ob s late then, ob s late and, are late also. In f s early and s late, then n ob s ether early or late and ob s late and (2) s hold. If early, then s early, s ether early or late and (2) s hold., are 5

6 Journal of Th-Qar Unversty number2 Vol.6 March/2011 () f r T, then Case 3 F - ( F F ), = ( b), where b r ( T ). ( F ), In f ob s late then ob s late and early and ob then T r r,, are late. If ob s s late, then obs, are late. If, are early, s early and ether early or late. () f b, b r r, then F - F F ) = ( b) < 0 ( F ), (, () f b <, then F - F F ) = ( 2b) < 0 ( F ), (, Snce, r and r ntegers and u {0, 1}, then the theorem s hold., Theorem (3.2) Let be a artal schedule, N, for, N and C C( ) 1 C, f C, then (, ) domnates, ). r ( Dessouy and Deogun [3] showed that f C r, then (, ) domnates, ) for ( 1 1 / r / F roblem. n Snce C r, then there exst the dle tme (. e. the machne wll be wated untl receve the ob, we denoted by I ) n the case (, ), I (because C r. So (, ) domnates (, ), for 1/ r / ( F u ) Theorem (3.3) Let be a artal sequence, N, for, and C C( ) 1 C, f C r, C r and <,then ob recedes ob (.e. ). 6

7 Journal of Th-Qar Unversty number2 Vol.6 March/2011 Zghar [15] showes that for the F 1 / r / roblem under the same condtons, snce < and u {0, 1}. So for 1/ r / ( F u ). Theorem (3.4) If for two adacent obs and, (, N) we have and r r, then we only conceder schedule n whch. A roof by the method of adacent ar wse nterchange s analogous to the roof of result. Let I, I denote to dle tmes of obs and resectvely then we can state the followng Theorem (3.5) If be a artal sequence, N, for,, such that I < I and r + < r +. Then n otmal schedule. Wth the same condtons [15], shows that for roblem1 / r / F. Exstence the dle tmes I and I means that r > C ( ) and r > C ( ). Snce I I, and r r, Hence n (, ) the ob s wated. Let a be denote to the wat, snce a 1 and the number of late obs s the same n (, ) and (, ), then n 1/ r / ( F u ) 4. Otmal Soluton: In ths secton we shall gves otmal solutons for our roblem () when the data of roblem satsfy some condtons. Theorem (4.1) If For 1/ r / ( F u ) roblem: r r and d d, for every N. Then SPT rule gve an otmal soluton. 7

8 Journal of Th-Qar Unversty number2 Vol.6 March/2011 Let (,,, ) be a schedule where, are tow artal schedules and, are tow obs wth. Let (,,, ) be another schedule has the same ob ' s order as n excet the obs and, and let T be a comleton tme of obs n subschedule. Let G F U Frst: f r T, then let a T r (I) f d T P we have G 2 2, a 2 2, G a G G 0,, (II) f T d T, then G G 0,, 1 2 ( and are late) 2 ( and are late) (III) f T d T, then G G 0,, (IV) f T d, then the obs and are early n Second: f and, the theorem s hold. r T, then n the same way above we show that the theorem s true (ntegral) by uttng a 0 and t r. So SPT rule gves an otmal soluton for roblem 1/ r r, d d / ( F u ) Theorem (4.2) Moor ' s algorthm (MA) gves an otmal soluton for roblem 1/ r r, / ( F u ). Snce all obs have constant release date r and constant rocessng tme. Then any sequence gve mnmum sum of flow tme, and n n F C 1 1. nr, where C, 1, 2,..., n 1 8

9 Journal of Th-Qar Unversty number2 Vol.6 March/2011 Then the mnmum of obectve functon deend only on u, but MA gves mnmum for u [11]. So MA gves an otmal soluton for roblem 1/ r r, / ( F u ). Let ERD be denoted to the earless release dates rule, accordng to ths rule, obs are sequenced from begnnng to end on the bass of an ascendng order of ther ready tme. Theorem (4.3) If for all ob * N, and d d *, Then ERD schedule gves an otmal soluton for roblem (). A roof by the method of adacent ar wse nterchange s analogous to the roof of 5. Branch and Bound Algorthm: We now gve the man feature of our branch and bound algorthm and ts mlementaton. Pror to ther alcaton, the ERD schedule as a heurstc s used to generate an uer bound on the total cost of an otmal schedule. Also, at the root node of the search tree an ntal lower bound on the total cost of an otmal schedule s obtaned by modfy the lower bound n [1]. Lower Bound: Let n 1 S 1 C whch s obtaned by SPT rule and S2 u 1 whch s obtaned by Moor ' s Algorthm. Abdul-Razaq and Zghar [1] rove that Z1 S1 S2 s a lower bound for 1/ r 0/ ( C u ) roblem. Snce roblem 1/ r 0/ ( C u ) s a satal case of our roblem P, then Z Z1 r s a n lower bound for roblem (P) because F C r, but ths lower bound s a wea. To modfy t, we shall use the relaxaton method as follow: Relaxed the release date by assumed that all obs have the same release date r *, * * where r mn{ }, and roblem P reduce to 1/ r r / ( F u ), and LB nr * r 9 Z s a lower bound of the orgnal roblem (P).

10 Journal of Th-Qar Unversty number2 Vol.6 March/2011 Our algorthm uses a forward sequencng branchng rule, for whch nodes at level L of the search tree corresond to ntal artal sequences n whch obs are sequenced n the frst L ostons. Before the branchng we see that: f the data of roblem satsfed the condtons of any theorem n secton (4), then ths roblem has an otmal soluton and not need to branch n search tree of the branch and bound method. If the lower bound LB equals the uer bound UB, then UB s otmal soluton and not need to branch. If not equals the followng attemt s made to elmnate nodes whch are based on () the domnance theorems whch are stated n secton (3) are aled from level one. () from the second level the domnance theorem of dynamc rogrammng s aled, an adacent obs nterchange to comare the total cost for the two obs most recently added to the fnal artal sequence wth corresondng total cost when these two obs are nterchanged n oston: f the former total cost s larger than the later, then the current node s elmnated, whle f both total cost are same, some conventon s used to decde whether the current node should be dscarded. For all nodes that reman after the domnance tests are aled, a lower bound s comuted. If the lower bound for any node grater than or equal to smallest of the revously generated uer bounds, then the node s dscarded. 6. Comutatonal exerence: In ths secton, we reort results of comutatonal tests to assess the effectveness of the branch and bound algorthm. Algorthm was coded n Fortran Power Statons(FORTRAN 90) and runs on Pentum IV HP. Comaq Comuter wth a 2.8 GHz rocessed and 256 Mb of RAM memores. Whenever a roblem remaned unsolved wth n tme lmt of 100 seconds, comutaton was abandoned for that roblem. Test roblems wth 5, 10, 15, 20, 25, 30, 35 and 40 obs were generated as fallows. For each ob N, an nteger rocessng tme was generated from the unform dstrbuton [1, 10]. An nteger den date d was generated for each ob n the same way as those of [1]. And an nteger ready tme r was generated for each from the unform dstrbuton [0, PR], where P n = P 1 and R{0.2, 0.5,0.8,1.1}. 11

11 Journal of Th-Qar Unversty number2 Vol.6 March/2011 Twenty roblems were generated for each value of n, results comarng the ntal lower bound(lb), and uer bound (UB) wth the otmal soluton on the total cost for n=35 are gven n table 1. Average comutaton tmes n second are gven n table 2. Table 1 gves the result of comutatons at the root node of the search tree for the 35 obs test roblem. One lower and uer bounds are gven for each for each roblem, together wth the otmal soluton value. We observe from table 1 that the lower and uer bounds were devatons from the otmum are very small for the other roblems. We observe from table 2 that roblems wth (5, 10, 15 and 20) obs are solved satsfactory, average comutaton tmes and numbers of nodes become large for n=25, 30, 35 and 40 wth one roblem not solve for 30 and 35 obs and two roblems not solve for n= 40. Also table 2 shows average comutaton tmes, number of unsolved roblems and the numbers of solved roblems that requre not more than 200 nodes, that requre over 200 nodes and not more than 500 nodes, that requre over 500 and not more 1000 nodes and that requre over 1000 nodes. Table (1) Comare the lower bound and uer bound wth otmal soluton for n= 35. No. LB UB Ot

12 Journal of Th-Qar Unversty number2 Vol.6 March/ Table 2 No. of roblems solved wth out branchng; unsolved n lmt tmes and No. of nodes n search tree. No. of nods N OLB OUB NB UN _ 9 _ _ 15 1 _ _ _ 1 _ _ 1 _ _ 2 _ OLB: lower bound gves otmal soluton OUB: uer bound gves otmal soluton NB: roblems solved wth out branchng UN: roblems unsolved n the lmt tme -200 roblems requred to solved not more 200 nodes -500 roblems requred to solved over 200 not more 500 nodes roblems requred to solved over 500 not more 1000 nodes Problems requred to solved over 1000 nodes. 12

13 Journal of Th-Qar Unversty number2 Vol.6 March/ Concluson In ths aer, we have resented a branch and bound algorthm for sngle machne schedulng wth a comoste obectve. The sum of flow tme and the number of tardy obs wth release dates. 1/ r / ( F u ). Our branch and bound s able to solve roblems, wth u to 40 obs. Although we decomose the roblem n to subroblems wth smler structure and modfy the lower bound n [1] by usng relaxaton method. Our results ndcate that roblems wth a comoste obectve and unequal release date are stll much harder to solve. Ths bound s vald lower bound for 1/ r / ( F u ) roblem whch s a general case of our roblem. Prosect for the future are farly otmstc that more effectve and new lower bound technques and domnance rules are beng develoed for such NP- hard roblems. References [1] Abdul-Razaq T. S. and Zghar M. K.," One otmal machne schedulng to mnmze total coast of comlaton tme and number of tardy obs", J. Basrrh Researches. Vol. 25 Par (2000). [2] Chen B., Potts C. N. and Woegnger G. J., " A revew of machne schedulng: comlexty, Algorthms and Aroxmblty", Handboo of comnatoral otmzaton D. Z. Du and P. M. Pardalos (Eds) , 1998 Kluwer Academc Publshers. [3] Dessouy, M. L. and Deogun, J. S. "Sequencng obs wth unequal ready tmes to mnmze mean flow tme", SIAM, J. Comut. Vol. 10, No. 1 Feb. (1981), [4] Emmons, H. "One machne sequencng to mnmze mean flow tme wth mnmum number tardy", Naval Res. Logst. Quart 22 (1975), [5] Gregoro Baggo, Jacques Waner and Clarence Ells, "Alyng schedulng technques to mnmze the number of late obs n worflow systems." ACM symosum on aled comutng

14 Journal of Th-Qar Unversty number2 Vol.6 March/2011 [6] Kellerer H., Tautenhohn T. and Woegnger G. J. "Aroxmablty and nonaroxmablty results for mnmzng total flow tme on a sngle machne." Proceedngs of the 28 th Annual ACM symosum on theory of Comutng (1996), SIAM Journal on Comutng. [7] Kse H., Ibar T. and Mne H., "A Solvable Case of the one machne schedulng roblem wth ready and due tmes." Oer. Res. 26 (1978) [8] Lenstra J. K., "Sequencng by enumeratve methods" Mathematsch Centrum, Amsterdam (1977). [9] Maran Yan den Aer and Han Hoogeveen, "mnmzng the number of late obs n a stochastc settng usng a chance constrant." Journal of Schedulng V. 11 No. 1 Feb [10] Mason A. J. and Anderson E. J., "mnmzng flow tme on a sngle machne wth ob classes and setu tmes" Naval Research Logstcs, vol. 38, (1991). [11] Moore J. M. "An n ob, one machne sequencng algorthm for mnmzng the number of late obs.", Management Sc. 19(1973) [12] Peter Brucer and Mhal Y. ovalyov, "Sngle machne batch schedulng to mnmze the weght of late obs", Mathematcal methods of O. R. V.43, N.1 Feb.(1996) [13] Shao Chn Sung, Mlan vlach, "Sngle Machne schedulng to mnmze the number of late obs under uncertanty", Fuzzy set and systems 139 (2003) [14] Van Wassenhove L. N. and Gelders L. F., "Solvng a bcrteron schedulng roblem," Euroean Journal of Oer. Res. 4(1980) [15] Zghar M. K., "Sngle machne schedulng to mnmze total cost of flow tme wth unequal ready tmes", Journal of Basrah Researches V.25, ar. 3 (2000)