Optmal Schedulng Algorthms to Mnmze Total Flowtme on a Two-Machne Permutaton Flowshop wth Lmted Watng Tmes and Ready Tmes of Jobs Seong-Woo Cho Dept. Of Busness Admnstraton, Kyongg Unversty, Suwon-s, 443-760, South Korea., Orcd: 0000-000-7946-990 Abstract In ths research, we consder a two-machne permutaton flowshop wth ready tmes and lmted watng tmes of obs n a semconductor fab. In the flowshop, each ob has ts correspondng ready tme (arrval tme) at machne (for the frst operaton) and lmted watng tme between the completon tme of the frst operaton on machne and the startng tme of the second operaton on machne. That s, the frst operaton of each ob cannot be started earler than ts ready tme on the machne, and the second operaton of each ob has to be started wthn ts correspondng lmted watng tme after ts frst operaton s completed on machne. In ths schedulng problem, the obectve s mnmzng the total flowtme of obs to be processed on the two-machne permutaton flowshop, and we suggested branch and bound algorthms to gve optmal solutons. To boost effcency of the suggested branch and bound algorthms, we developed several domnance propertes and three lower boundng schemes, and we used these n the branch and bound algorthms. In addton, we used the solutons of exstng heurstc algorthm as the ntal upper bounds of the branch and bound algorthms. To evaluate the performances of the branch and bounds, we executed computatonal smulaton test and showed the results. Keywords: lmted watng tme, ready tme, total flowtme, branch and bound algorthm INTRODUCTION In ths research, we consder a two-machne permutaton flowshop schedulng problem wth ready tmes and lmted watng tmes of obs n a semconductor fab, and the obectve functon s mnmzng the total flowtme of obs to be processed. In ths schedulng problem, the frst operaton of each ob cannot be started earler than ts ready tme on the machne and each ob has to start ts second operaton on machne wthn ts correspondng lmted watng tme after ts frst operaton on machne s completed. Ths schedulng problem can be denoted as F/r, maxwat/ (C -r ), whch s based on the three-feld notaton of Graham et al. [] n whch r and max-wat mean that obs cannot be processed on the frst machne earler than ther arrval tmes and have to be processed on the second machne wthn a certan perod of tme after those obs are completed on the frst machne, respectvely, and (C -r ) s the total flowtme of obs, and note that the typcal two-machne flowshop s a specal case of the schedulng problem of ths research [, 3, 4]. Such a two-machne permutaton flowshop can be found n a semconductor manufacturng lne (fab). For example, at the workstaton wth clean and dffuson operatons n a semconductor fabrcaton lne, after a chemcal treatment (clean) operaton for a wafer lot s completed on the clean machne (machne ), the next operaton (dffuson) for the wafer lot must be started on the dffuson machne (machne ) wthn ts pre-determned tme perod called lmted watng tme (the length of lmted watng tmes may be dfferent accordng to the recpes), and f the next operaton for the wafer lots s delayed, t must be abandoned or re-processed because the chemcal treatment s no longer effectve after the tme perod [, 4, 5]. On the other hand, snce the workstaton above may not be the frst operaton of semconductor fabs n general, obs arrve at ths workstaton dynamcally, that s, obs have dfferent postve arrval tmes at the machne (for the clean operaton) of ths operaton secton (clean-dffuson) [3, 6]. Many researchers have studed the typcal flowshop schedulng problems [7, 8 and 9]. In addton, as the followng researches, there are some studes for the flowshop schedulng problem wth lmted watng tmes and ready tmes of obs, respectvely. Tade et al. [0] and Potts [] suggested branch and bound algorthm and nvestgated the worst-case performance of fve approxmaton algorthms for mnmzng makespan on the two-machne flowshop wth release date, respectvely, and n the researches of Hall [] and Chu [3], a polynomal approxmaton scheme and a branch and bound algorthm were suggested for mnmzng makespan and total tardness on the two-machne flow-shop wth release date, respectvely [, 4]. Cho [, 3] suggested branch and bound 50
algorthms and heurstc algorthms for the schedulng problem wth ready tmes and lmted watng tmes of obs, and the obectve was mnmzng makespan. In addton, Cho [4] suggested varous heurstc algorthms for the same schedulng problem of ths research. However, there s no research whch suggested branch and bound algorthms, domnance propertes and lower boundng schemes for the schedulng problem on the two-machne flowshop wth ready tme, lmted watng tme and obectve functon of mnmzng total flowtme. PROBLEM DESCRIPTION In ths paper, the notatons below are used, and these were used n the research of Cho [, 3, 4]. p k ndces of obs, =,,..., n k ndces of machnes, k =, w r...m [] c k processng tme of ob on machne k lmted watng tme of ob ready tme of ob partal sequence partal schedule obtaned wth followed by obs,,...,m n ths order ob that s completed -th n a (partal) schedule completon tme of ob that s completed -th n a (partal) schedule on machne k C k() C -r (C -r ) completon tme of the last postoned ob n a partal schedule on machne k flowtme of ob total flowtme of all obs (obectve value of ths schedulng problem) In ths paper, only permutaton schedules (sequences) are consdered, n whch operaton sequences of obs are the same on the two machnes (machne and machne ), that s, the operaton orders of obs are not dfferenct on machne and. Although permutaton schedules are domnant n the ordnary two-machne flowshop schedulng problem wth the obectve of mnmzng makespan [4, 4, 5], t s not the case wth the lmted watng tme, the ready tmes and the obectve functon of mnmzng total flowtme, that s, there may be cases that a non-permutaton schedule (sequence) can be better than the best permutaton schedule (sequence). The example below s such a case. Example. (the case that permutaton schedules are not domnant) There are four obs to be scheduled, and the processng, lmted watng and ready tmes of obs are gven n Tables. In the fgure, we can confrm that the best permutaton schedule (b) can be obtaned by the operaton sequence (,, 3, 4) of obs, whle there s a better non-permutaton schedule, n whch the operaton sequence of obs s (, 3,, 4) on machne and the operaton sequence of obs s (,, 3, 4) on machne. Table : Processng, lmted watng and ready tmes of obs n the example = = = 3 = 4 p 4 p 4 w 0 0 0 0 r 0 3 (a) Non-permutaton schedule (b) Best permutaton schedule Fgure : An example that best permutaton schedule are domnated by non-permutaton schedule However, n the schedulng problem of ths research, only permutaton schedules are consdered snce permutaton schedules are preferred to non-permutaton schedules n most real systems because of the the ease of materal flow 50
management mplementaton, and non-permutaton schedules are not feasble n many cases because of techncal constrants of materal handlng systems (4, 4, 5). In ths two-machne permutaton flowshop schedulng problem, there are n obs to be processed n the machne and then machne, and we use the assumptons of Cho [3] as follows. ) In the two-machne flowshop schedulng problem, we have a gven set of obs wth dfferent arrval tmes, that s, each ob can be operated on the machne at ts arrval tme. ) The operaton tmes and lmted queue tmes of the obs are known and dfferent from each other. 3) Each ob should start ts operaton on machne wthn ts correspondng lmted queue tme after the ob s operated on the machne. Wth the notatons above, Cho [, 3] expressed the completon tme of each ob n a complete or partal schedules as follows, and we use theseequatons n ths research [4]. c [] =r [] +p [] c [] = c []+ p []=r [] +p []+ p [] (E.) (E.) c [] = max{max(r [], c [-]) + p [], c [-]-w []} for =,, n (E.3) c [] = max{c [], c [-]} + p [] for =,, n (E.4) Domnance Propertes In ths secton, seven domnance propertes are suggested, and they can be used to elmnate some partal schedules (sequences) that are domnated by other partal schedules. By usng the suggested domnance propertes, the number of sub-problems to be consdered can be reduced n the suggested branch and bound algorthms n ths research. The seven domnance propertes can be proved by usng the Johnson s algorthm [5] (n whch the Johnson s rule s used for sequencng obs, for two-machne permutaton flowshop schedulng problems, and accordng to the Johnson s rule, ob precedes ob n an optmal sequence f mn{p, p } mn{p, p }.), domnance propertes developed by Tade et al. [6] (for a two-machne flowshop schedulng problem wth ready tmes) and domnance propertes (for mnmzaton of makespan on a two-machne flowshop wth lmted watng tme and ready tme) developed by Cho [3]. Proposton. There s a gven partal schedule, f we have two obs () and () satsfyng the followng four condtons: (C.) C () r r, (C.), mn{p, p } mn{p, p }, (C.3), p p + w and (C.4) p + p p + p, then partal schedule s domnated by. Proof. If we can show that (A.) C k() C k() and (A.) F() F() for k =,, the proof of ths proposton s completed. Here, for k =,, (A.) C k() C k() was already proved under the condtons (C)-(C3) n the research of Cho [3]. Therefore, we need to show only (A.) F() F() for k =,. We have the followng equaltes related to the completon tmes of obs and n schedules and from the equaltes (E.)-(E.4) at the end of the prevous secton [3]. C () = max{max(r, C ()) + p, C () w } C () = max{c (), C ()} + p C () = max{max(r, C ()) + p, C () w } C () = max{c (), C ()} + p C () = max{max(r, C ()) + p, C () w } C () = max{ C (), C ()} + p C () = max{max(r, C ()) + p, C () w } C () = max{c (), C ()} + p (e.) (e.) (e.3) (e.4) (e.5) (e.6) (e.7) (e.8) To show that (A.) F() F() for k =,, that s, F() = C () r + C () r F() = C () r + C () r, we need to show only C () C () snce (A.) F() F() s reduced to C () + C () C () + C () and C () C () was already satsfed by (A.) [3]. Here, from the condton (C.) C () r r of ths proposton, (e.) and (e.6) are reduced to C () = r + p + p and C () = r + p + p, recpectvely. In addton, from the condton (C.4) p + p p +p, C () C () s satsfed. In concluson, we have (A.) C k() C k() and (A.) F() F() for k =,. Ths completes the proof. Proposton. There s a gven partal schedule, f we have two obs () and () satsfyng the followng four condtons: (C.5) r C (), (C.) mn{p, p } mn{p, p }, (C.6) C () C () p, (C.3) p p + w and (C.4) p + p p +p, then partal schedule s domnated by. Proof. The proof procedure of ths proposton s smlar to that of proposton. That s, If we can show that (A.) C k() C k() and (A.) F() F() for k =,, the proof of ths proposton s completed. Here, for k =,, (A.) C k() C k() was already proved 503
under the condtons (C), (C3), (C5) and (C6) n the research of Cho [3]. Therefore, we need to show only (A.) F() F() for k =,. To show that (A.) F() F() for k =,, we need to show C () C () (refer to the proof procedure of proposton ). From the condtons (C.5) r C () and (C.6) C () C () p, (e.) can be reduced to C () = C () + p + p. In addton, from (e.5) and (e.6), we have C () = max{c (), C ()} + p = max{max{max(r, C ()) + p, C () w }, C ()} + p = max{() r + p + p, () C () + p + p, (3) C () w + p, (4) C () + p } [3]. Here, from the condton (C.4) p + p p + p, we can confrm that (e.) = C () = C () + p + p () C () + p + p = C () of (e.6), that s, C () C () s satsfed. In concluson, we have (A.) C k() C k() and (A.) F() F() for k =,. Ths completes the proof. Proofs for the propostons (3-5) below are skpped n ths paper, snce t can be proven easly n the smlar way wth those of propostons -. Proposton 3. There s a gven partal schedule, f we have two obs () and () satsfyng the followng four condtons: (C.5) r C (), (C.) mn{p, p } mn{ p, p }, (C.7) C () C () p + w and (C.8) C () + p C () p p + w, (C.4) p + p p + p and (C.9) p p, then partal schedule s domnated by. Proposton 4. There s a gven partal schedule, f we have two obs () and () satsfyng the followng four condtons: (C.0) C () r C (), (C.) r r, (C.) mn{p, p } mn{p, p }, (C.) C () r + p, (C.3) p p + w and (C.4) p + p p + p, then partal schedule s domnated by. Proposton 5. There s a gven partal schedule, f we have two obs () and () satsfyng the followng four condtons: (C.0) C () r C (), (C.) r r, (C.) mn{p, p } mn{p, p }, (C.3) r + p C () r + p + w and (C.4) C () + p r p p + w and (C.9) p p, then partal schedule s domnated by. Proof. Let s assume that there s an optmal schedule n whch * s not placed n the frst poston. Then, the optmal schedule can be denoted as *, where and are partal sequences of the optmal schedule, and s a nonempty partal sequence. To prove the proposton, we show that F(* ) F( * ) for any par of and. From the condton r * + p * + p * mn *{r } of ths proposton, all obs (except for *) cannot start ther frst operatons earler than r * + p * + p *. Therefore, even f * s scheduled n the frst poston n a complete schedule (* ), the startng tmes and the completon tmes of the obs (n of * ) cannot be earler than those of the obs (n of * ). That s, f we assume that ob ( *) s scheduled n the frst poston of and we can show C k() = C k(*), k=,, the proof of ths proposton s completed. Here, we have C () = r + p and C () = r + p + p. On the other hand, we have C (*) = max{c (*), r } + p, C (*) = max{c (*), C (*)} + p, C (*) = r * + p * and C (*) = r * + p * + p *. In addton, from the condton r * + p * + p * mn *{r } of ths proposton, C (*) and C (*) can be reduced to C (*) = max{c (*), r } + p = max{r * + p *, r } + p = r + p and C (*) = max{c (*), C (*)} + p = max{r + p, r *+p *+p *} + p = r + p + p. In concluson, we have C k() = C k(*) for k =,. Ths completes the proof. From proposton 6, the followng corollary holds: Corollary. There s a gven partal schedule, f we have two obs () and () satsfyng the followng condton: C (*) mn *{r }, then schedules that start wth * domnate the others. Lower Boundng schemes To calculate a lower bound on the total flowtme of a complete schedule constructed from a gven partal schedule, we add lower bounds on flowtmes of the unscheduled obs (that are not ncluded to the partal schedule ) to the total flowtme of the scheduled obs n the partal schedule. Frst, we show a modfed verson (proposton 7) based on the proposton of Km [7], and t wll be used to obtan a lower bound on the total flowtme. Here, Let C, =,,..., n be nonnegatve real numbers. Also, let r ()(S) denote the ready tme of the ob to be completed -th n schedule S. Ths proposton can be proved by the proposton of Km [7]. Proposton 6. If there s ob *, where r * + p * + p * mn *{r }, there exsts an optmal schedule n whch ob * s placed n the frst poston. Proposton 7. If C C. C n, then for any schedule S' n a typcal two-machne permutaton flowshop, 504
max{0, C r() (S )} max{0, C r() (SERT )}, where S ERT s the earlest ready tme sequence, n whch obs wth earler ready tmes are placed earler. In ths proposton, f we let C be the completon tme of the ob that s placed th among obs n an arbtrary sequence S for the two-machne permutaton flowshop, C C. C n s satsfed (for any sequence). In addton, f we let r () (S) denote the ready tme of the ob that s placed th among obs n S, from the proposton 7, the total flowtme of sequence S s no less than max{0, C r() (SERT )}. Here, we cannot know the exact completon tmes of the (unscheduled) obs that are not ncluded n a partal schedule. Therefore, we calculate the lower bounds on the completon tmes of the unscheduled obs n a partal schedule by usng the followng propostons. In addton, the notatons below are used n the followng propostons. U partal schedule correspondng to a node beng consdered n the branch and bound algorthm set of obs not scheduled yet,.e., those that are not ncluded n w* the largest lmted watng tme of obs n U r* the shortest processng tme of obs n U p {}k P k m the -th shortest processng tme on machne k among obs n U sum of processng tmes of operatons n the order of the shortest processng tmes (on machne k) among obs n U,.e., P k p {q}k q lower bounds (m =,, 3, there are three versons of lower bounds) on the completon tme on machne of the ob that s completed - th among obs n U (n any complete schedule that starts wth a partal schedule ) In the proposton 8, we can obtan three lower bounds (, and 3 ) on the completon tmes of the unscheduled obs, assocated wth a partal schedule through the relaxaton of ready tmes and lmted watng tmes of obs n U. Proposton 8. In the two-machne permutaton flowshop, the completon tme of the ob that s completed -th among obs n U (n any complete schedule that starts wth ) s no less than = max[max(c (), r*) + p {}, C ()] + P or max[max(c (), r*), C () w* p {}] + P + = max [, ]. 3 = p {} or Proof. P k s a lower bound on the tme needed to operate obs among obs ncluded n U on machne k. In the case of, snce the obs ncluded n U cannot start ts second operaton (on machne ) earler than max(c (), r*) + p {} or C (), the completon tme of the ob that s completed -th among obs ncluded n U s no less than max[max(c (), r*) + p {}, C ()] + P. Also, n the case of, snce the obs ncluded n U cannot start ts frst operaton (on machne ) earler than than max(c (), r*) or C () w* p {}, the completon tme of the ob that s completed -th among obs ncluded n U s no less than max[max(c (), r*), C () w* p {}] + P + p {}. Fnally, snce the proofs of 3 and are completed, (max [, ]) s obvous. Ths completes the proof. Usng the two propostons above (proposton 7 and 8), we can obtan a lower bound on the total flowtme of any complete schedule that can be constructed from a partal schedule as σ max{0, c U m (σσ r} max{0, r[] (SERT )} for m=,, 3, where r[] (SERT ) denotes the ready tme of the ob completed -th n an earlest ready tme sequence of obs n U, and therefore, we can obtan the three versons of lower bounds (LB, LB and LB3) by usng Branch and Bound Algorthms, and 3. In ths research, we suggest varous versons of branch and bound algorthms for the consdered schedulng problem. The suggested branch and bound algorthms are based on the typcal branch and bound algorthm of Baker [8]. In the branch and bound algorthms, a branch and bound tree s constructed to generate all possble schedules (sequences), that s, a node n the branch and bound tree means ts correspondng partal sequence (schedule), and a node at the k-th depth n the branch and bound tree means ts correspondng partal schedule for the frst-postoned k obs n a complete schedule [3, 4]. 505
In the branch and bound algorthms, to obtan ntal upper bounds, AGLS (Appled Greedy Local Search) algorthm of Cho [4] s used. For the same schedulng problem of ths research, Cho [4] suggested four heurstc algorthms, H, H, AN (Appled NEH) and AGLS. Among the four heurstc algorthms above, AGLS showed the best performance. AGLS algorthm was developed based on the extended neghborhood search algorthm of Km [9]. In AGLS algorthm, the ntal soluton (sequence) obtaned from AN algorthm s then mproved by nterchangng a par of obs n the sequence, and all pars of obs are consdered for nterchanges at each attempt for mprovement, and then ths algorthm s termnated when the current soluton cannot be mproved any more [ and 4]. The AN algorthm apples MN (modfed NEH) algorthm of Cho and Km [4] to the schedulng problem of ths research. Also, n the suggested branch and bound algorthm, chld nodes are generated from a selected parent node, and we prune some chld nodes (domnated by others) by usng the developed domnance propertes n ths research. In addton, for the nodes that are not pruned by the domnance propertes, we calculate the developed three lower bounds (LB, LB and LB3), and the we elmnate the nodes wth larger lower bounds than the current upper bound. Fnally, when we construct the branch and bound tree, the depth and frst rule s adopted, n whch we select a node wth the bggest number of obs among the consdered nodes (partal schedules) for the next branchng [3]. Smulaton Tests To confrm the effectveness of the suggested lower boundng schemes, the developed domnance propertes and the exstng ntal upper bound, and to show the performnaces of the branch and bound algorthms, we executed some smulaton tests on randomly generated problems. The smulaton test problems and the branch and bound algorthms were coded n C++ language, and computatonal experments were done on a desktop computer wth a 5 processor of.3 GHz clock speed. We generated the processng tmes on the two machnes (the frst and second machnes), the ready tmes and the lmted watng tmes of obs n the same way wth those of Cho [4] as follows. That s, the processng tmes (on the machne and the machne ) were randomly generated from DU(, 50), where DU(a, b) means the dscrete unform dstrbuton wth a range of [a, b], and lmted watng tmes of obs were generated from DU(, 00), and the ready tmes of obs were randomly generated from DU(0, LB3), n whch LB3 s the lower bound suggested n ths research [4]. To evaluate the effectveness of the suggested lower boundng schemes, the developed domnance propertes and the exstng ntal upper bound, we randomly generated 60 test problems, 0 problems for each of sx levels (0,,, 3, 4 and 5) for the number of obs. By conductng the tests above, we suggested the best branch and bound algorthm, and to see the the performnace of the best branch and bound algorthm, we generated 90 addtonal test problems, 0 problems for each of nne levels (6, 8, 0,, 4, 6, 8, 30 and 3) for the number of obs. In tables -5, for a test nstance, the CPU tme for a branch and bound algorthm means the CPU tme (n seconds counter) to fnd optmal soluton, and f a branch and bound algorthm cannot gve the optmal soluton wthn 3600 seconds ( hour), ts CPU tme becomes 3600 seconds. For a branch and bound algorthm, as average CPU tmes (ACPUT) and medan CPU tmes (MCPUT) decrease, the performances of the branch and bound algorthms get better. In addton, n tables -4, for a test nstance, the rato of CPU tme for a branch and bound algorthm means ts relatve rato of CPU tme n comparson wth that of other branch and bound algorthm (BB3), and the average rato of CPU tme for a B&B algorthm s the mean of the rato values of CPU tmes (ARCPUT) for 0 nstances for each of sx levels for the number of obs. For example, n a test problem, the rato of CPU tme of BB/BB3 s the relatve rato of CPU tme of BB n comparson wth that of BB3 (CPU tme of BB/CPU tme of BB3). That s, as ARCPUT of BB/BB3 ncreases, the performance of BB gets worse n comparson wth BB. Table :Results of the tests on the effectveness of the lower bounds n 0 3 4 5 ACPUT (MCPUT ) ARCPUT* BB BB BB3 BB/BB3 BB/BB3 0.77 0.006 0.006 (0.30) (0.00) (0.00) 5.9 0.04 0.08 (0.65) (0.00) (0.00) 4.47 0.04 0.0 (.05) (0.006) (0.006) 0.866 0.09 0.08 (.700) (0.00) (0.006) 58.37 0.49 0.09 (30.95) (0.030) (0.030) 336.44 0.803 0.550 (47.45) (0.040) (0.040) 33.767.80 5408.65.58 307.830.80 007.546 3.300 3867.70.58 93783.888.07 average CPU tme or lower bound on the average CPU tme, whch was computed assumng that the CPU tme for an nstance that has not been solved to optmalty wthn 3600 seconds s 3600 seconds medan CPU tme * average rato of CPU tmes of a branch and bound algorthm n comparson wth those of BB3 506
Frstly, we performed three branch and bound algorthms (BB, BB and BB3) wth dfferent lower bounds (LB, LB anc LB3), respectvely. In BB, BB and BB3, all the suggested domnance propertes were used n the branch and bound algorthms, and the soluton of GLS s used as ntal upper bound. In the Table, BB3 shows the best performances n terms of the average CPU tme and the medan CPU tme. Also, n term of average rato of CPU tme, ARCPTs of BB/BB3 are larger than those of BB/BB3, whch means that BB outperforms BB (LB s more effcent than LB). Secondly, we evaluated the effectveness of the developed domnance propertes. For ths evaluaton, we compared BB4 (n whch none of the domnance propertes s used) wth BB3 (n whch all domnance propertes are used). The results of the tests are shown n Table 3. When the suggested domnance propertes are not used, the CPU tmes ncrease, that s, the suggested domnance propertes are effectve. n 0 3 4 5 Table 3:Results of the test on the effectveness of the domnance propertes ACPUT (MCPUT ) ARCPUT* BB3 BB4 BB4/BB3 0.006 0.006 (0.00) (0.00) 0.08 0.034 (0.00) (0.006) 0.0 0.09 (0.006) (0.00) 0.08 0.044 (0.006) (0.00) 0.09 0.6 (0.030) (0.050) 0.550 0.766 (0.040) (0.090).753.9.933.8.74.6 Table 4:Results of the test on the effectveness of the ntal upper bound n 0 3 4 5 ACPUT (MCPUT ) ARCPUT* BB3 BB5 BB5/BB3 0.006 0.07 (0.00) (0.030) 0.08 0.7 (0.00) (0.050) 0.0.70 (0.006) (0.40) 0.08 39.88 (0.006) (0.85) 0.09 944.656 (0.030) (0.400) 0.550 807.385 (0.040) (6.070),, * See the footnotes of Table. 86.00 63.76 46.590 3696.537 0399.859 6803.30 Thrdly, we compared BB5 (n whch ntal upper bounds were not used) wth BB3 (n whch the ntal upper bounds were used). The results of the tests are shown n Table 4. When the ntal upper bounds are not used, the CPU tmes ncrease sgnfcantly, that s, the brand and bound algorthm wth the ntal upper bounds becomes effectve. Fnally, to show the performances of the best branch and bound algorthm (BB3), we conducted the addtonal smulaton tests. Therefore, we randomly generated 90 addtonal test problems, 0 problems for each of nne levels (6, 8, 0,, 4, 6, 8, 30 and 3) for the number of obs, and n the Table 5, we show the ACPUT, MCPUT and NINS (the number of nstances that have not been solved to optmalty by BB3 wthn 3600s among 0 nstances). As you see n Table 5, BB3 could fnd optmal solutons for problems wth up to 8 obs n a reasonable amount of CPU tme.,, * See the footnotes of Table. 507
Table 5:Results of the test on the BB3 algorthm n ACPUT MCPUT NINS # 6 0.5 0.30 0 8.36 0.30 0 0 3.553.90 0 304.05 8.80 0 4 685.476 6.630 6 38.969 355.640 8 803.43 803.43 4 30 3.95 87.0 5 3 88.403 3600.00 6, See the footnotes of Table. # number of nstances among 0 nstances that have not been solved to optmalty by the branch and bound algorthm wthn 3600s CONCLUDING REMARKS In ths research, we suggested the branch and bound algorthms for the schedulng problem wth the obectve functon of mnmzng the total flowtme n a two-machne permutaton flowshop wth ready tmes and lmted watng tmes of obs n semconductor fabs. To boost the effcency of the suggested branch and bound algorthms, we developed several domnance propertes and lower boundng schemes, and the upper bounds from the exstng heurstc algorthms were used n those. From the results of the smulaton tests, we could confrm that the branch and bound algorthms became more effcent by usng the developed domnance propertes, lower boundng schemes and ntal upper bounds, and the best branch and b ound algorthm (BB3) could fnd optmal solutons for problems wth up to 8 obs n a reasonable amount of CPU tme. We can (need to) extend the results of ths research n varous drectons. For example, t may be necessary to develop domnance propertes and lower bounds for the m-machne permutaton flowshop schedulng flowshop problems or flowshops wth the other obectve functons (mnmzng makespan or total tardness). Also, more general cases can be consdered, n whch setup tmes are requred between the operatons of obs wth dfferent recpes. We can fnd such schedulng problems easly n a semconductor manufacturng lne. ACKNOWLEDGMENTS Ths work was supported by Kyongg Unversty Research Grant 05. REFERENCES [] Graham, R. L., Lawler, E. L., Lenstra, J. K., and Rnnooy Kan, A. H. G., 979, Optmzaton and approxmaton n determnstc sequencng and schedulng, Annals of Dscrete Mathematcs, 5, pp. 87-36. [] Cho, S. W., 06, Heurstcs for a two-machne permutaton flowshop wth lmted queue tme constrant and arrval tmes n a semconductor manufacturng lne, Internatonal Journal of Appled Engneerng Research, (8), pp. 585-5855. [3] Cho, S. W., 05, Optmzaton Algorthms for a Two-Machne Permutaton Flowshop wth Lmted Watng Tmes Constrant and Ready Tmes of Jobs, Journal of Informaton Technology Applcatons & Management,, pp. -7. [4] Cho, S. W., 07, Mnmzng total flowtme n a twomachne flowshop wth lmted watng tme constrant and ready tme n a semconductor fabrcaton plant, Submtted to Internatonal Journal of Operatons and Quanttatve Management. [5] Joo, B. J., and Km, Y. D., 009, A branch-and-bound algorthm for a two-machne flowshop schedulng problem wth lmted watng tme constrants, Journal of the Operatonal Research Socety, 60, pp. 57-58. [6] Cho, S. W., Lm, T. K., and Km, Y. D., 00, Heurstcs for Schedulng Wafer Lots at the Deposton Workstaton n a Semconductor Wafer Fab, Journal of the Korean Insttute of Industral Engneers, 36, pp. 5-37. [7] Chen, T. C. E., Gupta, J. N. D., and Wang, G., 000, A revew of flowshop schedulng research wth setup tmes, Producton and Operatons Management, 9, pp. 83-30. [8] Framnan, J. M., Gupta, J. N. D., and Lesten, R., 004, A revew and Classfcaton of heurstcs for permutaton flow-shop schedulng wth makespan obectve, Journal of Operatonal Research Socety, 55, pp. 43-55. [9] Gupta, J. N. D., and Stafford, J. E., 006, Flowshop schedulng research after fve decades, European Journal of Operatonal Research, 69, pp. 699-7. [0] Tade, R., Gupta, J. N. D., Della Croce, F., and Cortes, M., 998, Mnmzng makespan n the two-machne flow-shop wth release tmes, Journal of Operatonal Research Socety, 49, pp. 77-85. [] Potts, C. N., 985, Analyss of heurstcs for twomachne flow-shop sequencng subect to release dates, Mathematcs of Operatons research, 0, pp. 576-584. 508
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