Model formulations for the machine scheduling problem with limited waiting time constraints

Transcription

1 Model forulatios for the achie schedulig proble with liited waitig tie costraits Je-Shiag Che Departet of Idustrial Egieerig ad Maageet Far East College 49 Jughua Road, Shishr Shiag Taia 744 Taiwa R.O.C. Ji-Sha Yag Departet of Maageet ad Iforatio Techology Souther Taiwa Uiversity of Techology Taiwa R.O.C. Abstract This study cosiders the achie schedulig proble with liited waitig tie costraits. We exaie the achie eviroet of the ope-shop, job-shop, flow-shop, ad perutatio flow-shop, ad uses akespa as a easure perforace. Eight ixed biary iteger prograig odels are developed to optially solve these probles. Keywords : Schedulig, waitig tie, iteger prograig, ope-shop, job-shop, flow-shop. 1. Itroductio Most studies assue ifiite waitig tie betwee ay two cosecutive operatios of each job [9]. There are ay idustries where the liited tie costrait applies. For exaple, i a wafer fabricatio process, E-ails: jschec@s25.hiet.et, jsche@cc.fec.edu.tw Joural of Iforatio & Optiizatio Scieces Vol. 27 (2006), No. 1, pp c Taru Publicatios /06 $

2 226 J. S. CHEN AND J. S. YANG the waitig tie after the operatios i furace tubes is liited i order to prevet the absorptio of the particulates i air. Except wafer fabricatio, the proposed proble also exists i soe practical applicatios such as food productio [4], cheical productio, ad steel productio [12]. This study relaxes the o-wait costrait [2, 6, 11, 13, 15] that job ust be processed cotiuously without waitig tie betwee cosecutive achies. We cosider a schedulig odel which is siilar to, but soewhat ore flexible tha the o-wait odel. A review of the literature reveals that oly two papers have addressed the liited waitig tie costrait proble. Yag ad Cher [16] cosidered a two-achie flowshop sequecig proble with liited waitig tie costraits. The objective was to iiize the akespa. This proble was showed to be NP-hard ad a brach ad boud algorith was preseted to solve the proble. Su [12] cosidered a hybrid two-stage flow-shop with a batch processor i stage 1 ad a sigle processor i stage 2. The objective was to iiize the akespa. A heuristic algorith ad a ixed iteger progra were proposed. I this study, we cosider the ope-shop, job-shop, flow-shop, ad perutatio flow-shop schedulig probles with liited waitig tie costraits. The followig ters are used to defie the correspodig schedulig probles: Ope-shop (OS). There are achies. Each job has to be processed agai o each oe of the achies. However, soe of these processig ties ay be zero. There are o restrictios with regard to the routig of each job through the achie eviroet. The scheduler is allowed to deterie a route for each job, ad differet jobs ay have differet routes. Job-shop ( JS). I a job shop with achies, each job has its ow predeteried route to follow. Flow-shop (FS). There are achies i series. Each job has to be processed o each oe of the achies. All jobs have to follow the sae route (i.e., they have to be processed first o achie 1, the achie 2, etc). Perutatio flow-shop (PFS). I flow-shop, after copletio o oe achie, a job jois the queue at the ext achie. Usually, all queues are assued to operate uder the First I First Out (FIFO) disciplie-

3 MACHINE SCHEDULING PROBLEM 227 that is, a job caot pass aother while waitig i a queue. If the FIFO disciplie is i effect, the flow-shop is referred to as a perutatio flow-shop. This study assues that each job has a liited waitig tie costrait betwee ay two cosecutive operatios. Each achie ca process oly oe operatio at a tie ad differet operatios of the sae job caot be processed siultaeously. Each job visits each achie at ost oce. Jobs are ot preeptive. Buffer storage betwee ay two achies is uliited. All jobs are available siultaeously at tie zero. The purpose of this study is to cocetrate o ixed biary iteger prograig (BIP) forulatios of schedulig operatios i OS, JS, FS, ad PFS with liited waitig tie costraits. This study presets two ixed BIP forulatios for solvig the OS with liited waitig tie costraits probles, two ixed BIP forulatios for solvig the JS with liited waitig tie costraits probles, two ixed BIP forulatios for solvig the FS with liited waitig tie costraits probles, ad two ixed BIP forulatios for solvig the PFS with liited waitig tie costraits probles. The reaider of the paper is orgaized as follows. Sectio 2 provides preliiary iforatio. Sectio 3 addresses OS with liited waitig tie costraits. Sectio 4 addresses JS with liited waitig tie costraits. Sectio 5 addresses FS with liited waitig tie costraits. Sectio 6 addresses PFS with liited waitig tie costraits. Sectio 7 briefly draws coclusios. 2. Preliiaries Matheatical prograig-based schedulig research has received ore ad ore attetio fro researchers as coputer capacity icreases ad ore efficiet iteger prograig (IP) software becoes available [8]. Matheatical prograig forulatio is a atural way to attack achie schedulig probles [10]. Why are IP odels studied at all? Morto ad Petico [7] addressed it with the followig two reasos. First, there will always be certai special structure cases that are solvable. If we uderstad the geeral approaches, we ay recogize the. Secod, there are ofte various partial relaxatios of the equatios that ca be solved ad ay be useful. Blazewicz et al. [1] surveyed the developet

4 228 J. S. CHEN AND J. S. YANG of atheatical prograig forulatios i schedulig. Most IP probles i schedulig ivolve ixed BIP, i which soe variables are biary ad soe are cotiuous. This study eploys the ixed BIP techique to describe the foratios for the achie eviroet of the ope-shop, job-shop, flow-shop, ad perutatio flow-shop. To describe the proble, we itroduce the followig otatios: Sybol defiitio J i = job uber i; M k = achie uber k; O i j = operatio uber j of J i (oly used for the job-shop probles); Proble paraeters M = a very large positive uber; = uber of jobs for processig at tie zero; = uber of achies i the shop; L i = the upper liited waitig tie of ay two cosecutive operatios of J i ; p ik = the processig tie of J i o M k ; r ik = 1 if J i requires M k ; 0 otherwise (oly used for the ope-shop probles); r i jk = 1 if O i j requires M k ; 0 otherwise (oly used for the job-shop probles); N i E k = uber of operatios of J i, that is, N i = probles ad N i = j=1 r ik for the ope-shop r i jk for the job-shop probles (oly used for the ope-shop ad job-shop probles); = uber of operatios that ight ever be processed o M k, i.e., E k = N i r i jk (oly used for the job-shop probles); j=1 Decisio variables C ax = axiu copletio tie or akespa; C ax = ax C i; g ir = the startig tie of the rth operatio of J i (oly used for the ope-shop probles); h kq = the startig tie of the operatio i the sequece positio q o M k (oly used for the job-shop, flow-shop, ad perutatio flow-shop probles);

5 MACHINE SCHEDULING PROBLEM 229 w irk = 1 if the rth operatio of J i is scheduled o M k ; 0 otherwise (oly used for the ope-shop probles); x ikq = 1 if J i is scheduled i the qth positio for processig o M k ; 0 otherwise (oly used for the job-shop ad flow-shop probles); s ik = the earliest start tie of J i o M k (oly used for the job-shop, flow-shop, ad perutatio flow-shop probles); z ii k = 1 if J i precedes J i o M k (ot ecessarily iediately); 0 otherwise (oly used for the ope-shop, job-shop, ad flow-shop probles); z ii = 1 if J i precedes J i (ot ecessarily iediately); 0 otherwise (oly used for the perutatio flow-shop probles); y iq = 1 if J i is assiged to sequece positio q; 0 otherwise (oly used for the perutatio flow-shop probles). 3. Ope-shop with liited waitig tie costraits I ope-shops, there are o restrictios with regard to the routig of each job through the achie eviroet. We ust to deterie a route for each job, ad differet jobs ay have differet routes. This sectio presets two ixed BIP forulatios, aely OS-1 ad OS-2, for solvig ope-shop schedulig probles with liited waitig tie costraits. 3.1 The OS-1 odel The OS-1 odel bases o the precedece relatioship betwee operatios of the sae job o cosecutive achies i the processig requireets. The biary w irk that the odel uses is restricted, ad specifies the order i which the r th operatio of J i is processed o M k. The followig odel eploys the cocept of oe-operatio-oe-positio to describe the ope-shop schedulig probles with liited waitig tie costraits. Miiize C ax (1) Subject to w irk Mr ik i = 1, 2,..., ; r = 1, 2,..., N i ; k = 1, 2,..., (2) N i w irk = r ik i = 1, 2,... ; k = 1, 2,..., (3) r=1 g ir Mw irk i = 1, 2,..., ; r = 1, 2,..., N i ; g ir + k = 1, 2,..., (4) p ik w irk g i,r+1 i = 1, 2,..., ; r = 1, 2,..., N i 1 (5)

6 230 J. S. CHEN AND J. S. YANG ( ) g i,r+1 g ir + p ik w irk L i i = 1, 2,..., ; r = 1, 2,..., N i 1 (6) g i,ni + p ik w i,ni,k C ax i = 1, 2,..., (7) C ax 0; g ir 0 i = 1, 2,..., ; r = 1, 2,..., N i ; w irk is biary i = 1, 2,..., ; r = 1, 2,..., N i ; k = 1, 2,...,. (8) I the above odel, costrait sets (2) ad (3) describe the feasible value of w irk. Costrait set (4) eforces g ir = 0 whe w irk = 0. Costrait set (5) guaratees that the startig tie of O i,r+1 is later tha its fiish tie of O ir. Furtherore, costrait set (6) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (7) gives the defiitio of C ax which is to be iiized i the objective fuctio (1). Fially, costrait set (8) specifies the o-egativity of C ax ad g ir, ad sets up the biary restrictios for w irk. 3.2 The OS-2 odel This odel uses biary variable z ii k to express the either-or relatioship for the o-iterferece restrictios o M k. The followig odel applies the cocept of o-iterferece to solve ope-shop schedulig probles with liited waitig tie costraits. Miiize C ax (9) Subject to g ir + p ik g i,r+1 + M(1 r ik ) i = 1, 2,..., ; r = 1, 2,..., N i 1; k = 1, 2,..., (10) g ir + p ik g i r + M(2 r ik r i k) + M(1 z ii k) 1 i < i ; r = 1, 2,..., N i ; r = 1, 2,..., N i ; k = 1, 2,..., (11) g i r + p i k g ir + M(2 r ik r i k) + Mz ii k 1 i < i ; r = 1, 2,..., N i ; r = 1, 2,..., N i ; k = 1, 2,..., (12) g i,r+1 (g ir + p ik ) L i + M(1 r ik ) i = 1, 2,..., ; r = 1, 2,..., N i 1; k = 1, 2,..., (13)

7 MACHINE SCHEDULING PROBLEM 231 g i,ni + p ik C ax + M(1 r ik ) i = 1, 2,..., ; k = 1, 2,..., ; (14) C ax 0; g ir 0 i = 1, 2,..., ; r = 1, 2,..., N i ; z ii k is biary 1 i < i ; k = 1, 2,...,. (15) Costrait set (10) guaratees that the startig tie of O i,r+1 is later tha its fiish tie of O ir. Costrait sets (11) ad (12) together eforce the requireet that oly oe operatio ay be processed o a achie at ay tie. That is, either g ir + p ik g i r + M(2 r ik r i k) or g i r + p i k g ir + M(2 r ik r i k). If r ik = r i k = 1, the either g ir + p ik g i r or g i r + p i k g ir. Meawhile, costrait set (13) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (14) gives the defiitio of C ax which is to be iiized i the objective fuctio (9). Fially, costrait set (15) specifies the o-egativity of C ax ad g ir, ad sets up the biary restrictios for z ii k. 3.3 Copariso of OS-1 ad OS-2 Frech [3] stated that the speed with which IP probles ca be solved depeds upo the uber of variables ad costraits i the proble, ad that the doiat factor is the uber of biary variables. Wilso [14], ad Liao ad You [5] showed that, if two forulatios have the sae uber of biary variables, the the uber of costraits is the ext ost ifluetial eleet. Accordigly, Table 1 suarizes the sizes of these two odels. The OS-2 odel has the sae uber of cotiuous variables as that of the OS-1 odel, but has (N i (1/2) 2 + (1/2)) fewer biary variables ad ( 2 1 i =i+1 ) N i N i 2 2N i + ore costraits cotiuous variables. Thus, the OS-2 odel is theoretically better tha the OS-1 odel. Table 1 Copariso of OS-1 ad OS-2 Model No. of biary No. of costraits No. of cotiuous variables variables OS-1 N i 2N i + + 2N i + 1 N i + 1 OS-2 (1/2) 2 (1/2) 2 1 i =i+1 N i N i N i N i + 1

8 232 J. S. CHEN AND J. S. YANG 4. Job-shop with liited waitig tie costraits I a job-shop, each job has its ow processig order ad this ay bear o relatio to the processig order of ay other jobs. This sectio presets two ixed BIP forulatios, aely JS-1 ad JS-2, for solvig job-shop schedulig probles with liited waitig tie costraits. 4.1 The JS-1 odel The JS-1 odel bases o the precedece relatioship betwee jobs o cosecutive achies i the processig requireets. The biary x ikq that the odel uses is restricted, ad specifies the order i which jobs are processed o the achie. The followig odel eploys the cocept of oe-job-oe-positio to describe the job-shop schedulig probles with liited waitig tie costraits. Miiize C ax (16) Subject to E k N i x ikq = r i jk i = 1, 2,..., ; k = 1, 2,..., (17) q=1 j=1 x ikq = 1 k = 1, 2,..., ; q = 1, 2,..., E k (18) h kq + r i jk h kq + p ik x ikq h k,q+1 k = 1, 2,..., ; q = 1, 2,..., E k 1 (19) r i jk p ik r i, j+1,k h kq + M ( 2 r i jk x ikq r i, j+1,k x ikq i = 1, 2,..., ; j = 1, 2,..., N i 1; q, q = 1, 2,..., E k (20) ) r i jk h kq + r i jk p ik r i, j+1,k h kq ( L i + M h k,ek + ( 2 r i jk x ikq r i, j+1,k x ikq i = 1, 2,..., ; j = 1, 2,..., N i 1; q, q = 1, 2,..., E k (21) p ik x i,k,ek C ax k = 1, 2,..., (22) ) )

9 MACHINE SCHEDULING PROBLEM 233 C ax 0 h kq 0, k = 1, 2,..., ; q = 1, 2,..., E k x ikq = 0 or 1 i = 1, 2,..., ; k = 1, 2,..., ; q = 1, 2,..., E k. (23) Costrait set (17) describes the feasible value of x ikq. Costrait set (18) esures that each of J i ust be placed i a uique positio of M k. Costrait set (19) eforces the techological requireets of each job. Furtherore, costrait set (20) restricts that the startig tie of O i, j+1 to be o earlier tha its fiish tie of O i j. Costrait set (21) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (22) gives the defiitio of C ax which is to be iiized i the objective fuctio (16). Fially, costrait set (23) specifies the o-egativity of C ax ad h kq, ad sets up the biary restrictios for x ikq. 4.2 The JS-2 odel The JS-2 odel utilizes iteger biary variables z ii k to express the either-or relatioship for the o-iterferece restrictios for idividual achie. The followig odel applies the cocept of o-iterferece to solve job-shop schedulig probles with liited waitig tie costraits. Miiize C ax (24) Subject to r i jk (s ik + p ik ) r i, j+1,k s ik i = 1, 2,..., ; j = 1, 2,..., N i 1 (25) s i k + p i k s ik + M(1 z ii k) 1 i < i ; k = 1, 2,..., (26) s ik + p ik s i k + Mz ii k 1 i < i ; r i, j+1,k s ik k = 1, 2,..., (27) r i jk (s ik + p ik ) L i i = 1, 2,..., ; j = 1, 2,..., N i 1 (28) r i,ni,k(s ik + p ik ) C ax i = 1, 2,..., (29) C ax 0, s ik 0 i = 1, 2,..., ; k = 1, 2,..., z ii k = 0 or 1 1 i < i ; k = 1, 2,...,. (30)

10 234 J. S. CHEN AND J. S. YANG Costrait set (25) restricts that the startig tie of O i, j+1 to be o earlier tha its fiish tie of O i j. Costrait sets (26) ad (27) eforce the requireet that oly oe job ay be processed o M k at ay tie. If both J i ad J i are processed o M k the either s i k + p i k s ik or s ik + p ik s i k. Costrait sets (26) ad (27) together guaratee that oe of the costraits ust hold whe the other is eliiated sice z ii k is a biary variable ad M is a large eough positive uber. Furtherore, costrait set (28) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (29) defies C ax to be iiized i the objective fuctio (24). Fially, costrait set (30) specifies the o-egativity of C ax ad s ik ad the biary restrictios of z ii k. 4.3 Copariso of JS-1 ad JS-2 The JS-1 odel has (E k (1/2) 2 + (1/2)) ore biary variables tha that of the JS-2 odel, has (2N i (E k ) 2 2(E k ) 2 + 2E k N i + ) ore costraits ad (E k ) ore cotiuous variables. Thus, the JS-2 odel is theoretically better tha the JS-1 odel. Table 2 Copariso of JS-1 ad JS-2 Model No. of biary No. of costraits No. of cotiuous variables variables JS-1 E k 2N i (E k ) 2 2(E k ) 2 E k E k JS-2 (1/2) 2 (1/2) 2 + 2N i Flow-shop with liited waitig tie costraits All jobs have to follow the sae route i a flow-shop. This sectio presets two ixed BIP forulatios, aely FS-1 ad FS-2, for solvig flow-shop schedulig probles with liited waitig tie costraits. 5.1 The FS-1 odel The FS-1 odel bases o the precedece relatioship betwee jobs o cosecutive achies i the processig requireets. The biary x ikq that the odel uses is restricted, ad specifies the order i which jobs are

11 MACHINE SCHEDULING PROBLEM 235 processed o the achie. The followig odel eploys the cocept of oe-job-oe-positio to describe the flow-shop schedulig probles with liited waitig tie costraits. Miiize C ax (31) Subject to x ikq = 1 i = 1, 2,..., ; k = 1, 2,..., (32) q=1 x ikq = 1 k = 1, 2,..., ; q = 1, 2,..., (33) h kq + p ik x ikq h k,q+1 i = 1, 2,..., ; k = 1, 2,..., ; q = 1, 2,..., 1 (34) h kq + p ik x ikq h k+1,q + M(2 x ikq x i,k+1,q ) i = 1, 2,..., ; k = 1, 2,..., 1; q, q = 1, 2,..., (35) h k+1,q (h kq + p ik x ikq ) L i + M(2 x ikq x i,k+1,q ) i = 1, 2,..., ; k = 1, 2,..., 1; q, q = 1, 2,..., (36) C ax = h + p i x i (37) C ax 0, h kq 0 k = 1, 2,..., ; q = 1, 2,..., ; x ikq = 0 or 1 i = 1, 2,..., ; k = 1, 2,..., ; q = 1, 2,...,. (38) Costrait set (32) esures that each operatio of J i is uiquely placed o M k. Costrait set (33) satisfies the requireet that each positio of M k has a uique operatio. Costrait sets (34) ad (35) eforce the techological requireets of each job. Meawhile, costrait set (36) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (37) gives the defiitio of C ax which is to be iiized i the objective fuctio (31). Fially, costrait set (38) specifies the o-egativity of C ax ad h kq, ad sets up the biary restrictios for x ikq. 5.2 The FS-2 odel The FS-2 odel utilizes iteger biary variables z ii k to express the either-or relatioship for the o-iterferece restrictios for idividual achie. The followig odel applies the cocept of o-iterferece to solve flow-shop schedulig probles with liited waitig tie costraits.

12 236 J. S. CHEN AND J. S. YANG Miiize C ax (39) Subject to s ik + p ik s i,k+1 i = 1, 2,..., ; k = 1, 2,..., 1 (40) s i k + p i k s ik + M(1 z ii k) 1 i < i ; k = 1, 2,..., (41) s ik + p ik s i k + Mz ii k 1 i < i ; k = 1, 2,..., (42) s i,k+1 (s ik + p ik ) L i i = 1, 2,..., ; k = 1, 2,..., 1 (43) s i + p i C ax i = 1, 2,..., (44) C ax 0, s ik 0 i = 1, 2,..., ; k = 1, 2,..., z ii k = 0 or 1 1 i < i ; k = 1, 2,...,. (45) Costrait set (40) eforces the techological requireets of each job. Costrait sets (41) ad (42) eforce the requireet that oly oe job ay be processed o M k at ay tie. If both J i ad J i are processed o M k the either s i k + p i k s ik or s ik + p ik s i k. Costrait sets (41) ad (42) together guaratee that oe of the costraits ust hold whe the other is eliiated sice z ii k is a biary variable ad M is a large eough positive uber. Furtherore, costrait set (43) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (44) defies C ax to be iiized i the objective fuctio (39). Fially, costrait set (45) specifies the oegativity of C ax ad s ik ad the biary restrictios of z ii k. 5.3 Copariso of FS-1 ad FS-2 The FS-1 odel has the sae uber of cotiuous variables as the FS-2 odel, but it has ((1/2) 2 + (1/2)) ore biary variables ad ( ) ore costraits. Thus, the FS-2 odel is theoretically better tha the FS-1 odel. Table 3 Copariso of FS-1 ad FS-2 Model No. of biary No. of costraits No. of cotiuous variables variables FS FS-2 (1/2) 2 (1/2)

13 MACHINE SCHEDULING PROBLEM Perutatio flow-shop with liited waitig tie costraits I a perutatio flow-shop, a job caot pass aother while waitig i a queue. This sectio presets two ixed BIP forulatios, aely PFS-1 ad PFS-2, for solvig perutatio flow-shop schedulig probles with liited waitig tie costraits. 6.1 The PFS-1 odel The FS-1 odel bases o the precedece relatioship betwee jobs o cosecutive achies i the processig requireets. The biary y iq that the odel uses is restricted, ad specifies the order i which jobs are processed. The followig odel eploys the cocept of oe-job-oepositio to describe the perutatio flow-shop schedulig probles with liited waitig tie costraits. Miiize C ax (46) Subject to y iq = 1 i = 1, 2,..., (47) q=1 y iq = 1 q = 1, 2,..., (48) h 11 = 0 (49) h 1q + h k1 + h kq + h kq + p i1 y iq = h 1,q+1 q = 1, 2,..., 1 (50) p ik y i1 = h k+1,1 k = 1, 2,..., 1 (51) p ik y iq h k+1,q k = 1, 2,..., 1; q = 2, 3,..., ; (52) p ik y iq h k,q+1 k = 2, 3,..., ; q = 1, 2,..., 1 (53) h k+1,q (h kq + p ik ) L i + M(1 y iq ) i = 1, 2,..., ; C ax = h + k = 1, 2,..., 1; q = 1, 2,..., (54) p i y i (55) C ax 0, h kq 0 k = 1, 2,..., ; q = 1, 2,..., ; y iq = 0 or 1 i = 1, 2,..., ; q = 1, 2,...,. (56)

14 238 J. S. CHEN AND J. S. YANG Costrait sets (47) ad (48) are the classical assiget proble, with (47) isurig that each job is assiged to just oe sequece positio, while (48) isures that each sequece positio is filled with oly oe job. Costrait sets (49), (50), ad (51) isure there is o idle tie o M 1, ad that J 1 processes o all achies without delay. Costrait set (52) isures that the start of each job o M k+1 is o earlier tha its fiish o M k. Furtherore, costrait set (53) isures that the job i positio q + 1 i the sequece does ot start o M k util the job i positio q i the sequece has copleted its processig o that achie. Costrait set (54) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (55) defies C ax to be the fiish tie of the last job processed o M. Fially, costrait set (56) specifies the o-egativity of C ax ad h kq, ad sets up the biary restrictios for y iq. 6.2 The PFS-2 odel The PFS-2 odel utilizes iteger biary variables z ii to express the either-or relatioship for the o-iterferece restrictios. The followig odel applies the cocept of o-iterferece to solve perutatio flowshop schedulig probles with liited waitig tie costraits. Miiize C ax (57) Subject to s ik + p ik s i,k+1 i = 1, 2,..., ; k = 1, 2,..., 1 (58) s i k+p i k s ik +M(1 z ii ) 1 i < i ; k = 1, 2,..., (59) s ik + p ik s i k + Mz ii 1 i < i ; k = 1, 2,..., (60) s i,k+1 (s ik +p ik ) L i i = 1, 2,..., ; k = 1, 2,..., 1 (61) s i + p i C ax i = 1, 2,..., (62) C ax 0, s ik 0 i = 1, 2,..., ; k = 1, 2,..., z ii = 0 or 1 1 i < i (63) Costrait set (58) isures that each job s start tie o M k 1 is o earlier tha that job s start tie o M k plus that job s processig tie o M k. Costrait sets (59) ad (60) are the paired disjuctive costraits which isure that J i either precede J i or follows J i i the sequece, but ot both. If both J i ad J i are processed o M k the either s i k + p i k s ik or s ik + p ik s i k. Costrait sets (59) ad (60) together guaratee that oe of the costraits ust hold whe the other is eliiated sice z ii is a biary variable ad M is a large eough positive uber. Meawhile,

15 MACHINE SCHEDULING PROBLEM 239 costrait set (61) esures that the waitig tie of ay two cosecutive operatios of J i caot greater tha its upper liited waitig tie L i. Costrait set (62) equate akespa to the axiu copletio tie of all jobs o the last achie. Fially, costrait set (63) specifies the oegativity of C ax ad s ik ad the biary restrictios of z ii. 6.3 Copariso of PFS-1 ad PFS-2 The PFS-1 odel has the sae uber of cotiuous variables as that of the PFS-1 odel, but has ((1/2) 2 + (1/2)) ore biary variables ad ( ) fewer costraits. Thus, the PFS-2 odel is theoretically better tha the PFS-1 odel. Table 4 Copariso of PFS-1 ad PFS-2 Model No. of biary No. of costraits No. of cotiuous variables variables PFS PFS-2 (1/2) 2 (1/2) Coclusios This study cosiders the ope-shop, job-shop, flow-shop, ad perutatio flow-shop schedulig probles with liited waitig tie costraits. Eight ixed BIP odels are proposed for solvig these schedulig probles. The odels of OS-1 ad OS-2 are for the ope-shop with liited waitig tie costraits. The odels of JS-1 ad JS-2 are for the job-shop with liited waitig tie costraits. The odels of FS-1 ad FS-2 are for the flow-shop with liited waitig tie costraits, while the odels PFS-1 ad PFS-2 are for the perutatio flow-shop with liited waitig tie costraits. This study eploys ILOG OPL ad CPLEX to verify the accuracy of the above eight proposed ixed BIP odels. Theoretically speakig, OS-2 is better tha OS-1, JS-2 is better tha JS-1, FS-2 is better tha FS-1, ad PFS-2 is better tha PFS-1. Refereces [1] J. Blazewicz, M. Dror ad J. Weglarz, Matheatical prograig forulatios for achie schedulig: a survey, Europea Joural of Operatioal Research, Vol. 51 (1991), pp

16 240 J. S. CHEN AND J. S. YANG [2] J. L. Bouquarcd, J. C. Billaut, M. A. Kubzi ad V. A. Strusevich, Two-achie flow shop schedulig probles with o-wait jobs, Operatios Research Letters, Vol. 33 (2005), pp [3] S. Frech, Sequecig ad Schedulig: A itroductio to the Matheatics of the Job-Shop, Ellis Horwood, Chichester, U.K., [4] A. Hodso, P. Muhlea ad D. H. R. Price, A icrocoputer based solutio to a practical schedulig proble, Joural of the Operatioal Research Society, Vol. 36 (1985), pp [5] C. L. Liao ad C. T. You, A iproved forulatio for the job-shop schedulig proble, Joural of the Operatioal Research Society, Vol. 43 (1992), pp [6] C. Y. Liaw, C. Y. Cheg ad M. Che, Schedulig two-achie owait ope shops to iiize akespa, Coputers ad Operatios Research, Vol. 32 (2005), pp [7] T. E. Morte ad D. W. Petico, Heuristic Schedulig Systes, Wiley, New York, [8] C. H. Pa, A study of iteger prograig forulatios for schedulig probles, Iteratioal Joural of Systes Sciece, Vol. 28 (1997), pp [9] M. Piedo, Schedulig: Theory, Algoriths ad Systes, Pretice-Hall, New Jersey, [10] A. H. G. Riooy Ka, Machie Schedulig Probles: Classificatio, Coplexity ad Coputatios, Martius Nijhoff, The Hague, Hollad, [11] C. J. Schuster ad J. M. Fraia, Approxiative procedures for owait job shop schedulig, Operatios Research Letters, Vol. 31 (2003), pp [12] L. H. Su, A hybrid two-stage flowshop with liited waitig tie costraits, Coputers & Idustrial Egieerig, Vol. 44 (2003), pp [13] J. B. Wag ad Z. Q. Xia, No-wait or o-idle perutatio flowshop schedulig with doiatig achies, Joural of Applied Matheatics ad Coputig, Vol. 17 (2005), pp [14] J. M. Wilso, Alterative forulatios of a flow-shop schedulig proble, Joural of the Operatioal Research Society, Vol. 40 (1989), pp [15] G. J. Woegiger, Iapproxiability results for o-wait job shop schedulig, Operatios Research Letters, Vol. 32 (2004), pp [16] D. L. Yag ad M. S. Cher, Two-achie flowshop sequecig proble with liited waitig tie costraits, Coputers & Idustrial Egieerig, Vol. 28 (1995), pp Received July, 2005