Handling Fuzzy Constraints in Flow Shop Problem

Transcription

1 Handlng Fuzzy Consans n Flow Shop Poblem Xueyan Song and Sanja Peovc School of Compue Scence & IT, Unvesy of Nongham, UK E-mal: {s sp}@cs.no.ac.uk Absac In hs pape, we pesen an appoach o deal wh fuzzy consans n a flow shop poblem. The fleble consans on job elease dae and due dae, as well as job pocessng me ae descbed by fuzzy ses. We popose a mehod o calculae he sasfacon degee of he schedule wh espec o fuzzy elease daes and fuzzy due daes of jobs. The objecve s o oban a schedule whch has mamum sasfacon degee. The poposed mehod can be used whn local seach heuscs whch seach fo he bes schedule. Keywods: Schedulng, Flow Shop Poblem, Fuzzy Ses, Fuzzy Consans. 1. Inoducon Flow shop schedulng concens pocessng jobs hough a sees of machnes n eacly he same ode. The eseach on flow shop schedulng poblem has dawn a gea aenon n he las decades wh he am o ncease he effecveness of ndusal poducon. In a eal flow shop pocess, jobs ae usually subjec o consans on elease dae,.e., he dae when a job can sa s pocessng, and due dae,.e., he dae when a job has o complee s pocessng. Fo eample, a job could only be pocessed when he equed aw maeal s avalable, o he compleon me of a job should espec he pefeed due dae of delvey o cusomes. A job sequence s consdeed o be feasble f s able o sasfy he consans of jobs on elease dae and due dae. Convenonal schedulng algohms whch opeae on csp daa may mss a feasble soluon due o nsgnfcan volaons of consans on elease dae o due dae. In eal wold poblems, he elease dae and due dae of a job could be elaed. If we allow some volaons o consans, we could oban a job sequence whch sasfes he elaed elease dae and due dae o a mamum degee.

2 A numbe of appoaches o schedulng poblems wh fuzzy consans have been developed and epoed n leaue. Ishbuch & Muaa [6] consde he fuzzy due dae n a flow shop poblem. The auhos defne a fuzzy due dae fo each job whch denoes he sasfacon gade of he decson make fo he compleon me of he job. The pocessng mes of jobs ae assumed o be deemnsc. They am o oban a job sequence whose compleon me sasfes he due dae o he mamum degee. Paul e al. [8] nvesgae a flow shop schedulng poblem wh fuzzy pocessng mes and fuzzy due daes. The objecve s o sa he job pocessng as lae as possble whle keepng he sasfacon gade fo he due dae of he job. Dubous e al. [1] deal wh fuzzy consans of a job shop poblem. The objecve s o oban a schedule ha sasfes he fuzzy elease dae and due dae o he mamum degee. They fomulae he mahemacal model of job shop schedulng poblem n he pesence of unceany based on he consaned popagaon echnque whch s used o seach fo a feasble schedule wh elease dae and due dae of jobs whn cean me wndows. In hs pape, we consde fuzzy consans n a flow shop poblem, wh he am o oban a schedule whch sasfes he consans o mamum. The consans consdeed n hs wok ae fuzzy elease dae and fuzzy due dae. In addon, we assume ha he pocessng mes of jobs ae uncean due o ncomplee knowledge o uncean envonmen n eal poducon schedulng envonmen. We use fuzzy ses o descbe he unceany of pocessng mes. The mahemacal model fo handlng consans n flow shop poblem ha s gven s nsped by Dubous e al. [1] ponee wok on fuzzy consaned job shop schedulng. Howeve, whls ha mehod s developed o be used whn consan popagaon echnque, ou mehod s developed o be used whn local seach echnques. The pape s oganzed as follows. Secon 2 pesens he fomulaon of he poblem. The mehod o calculae he sasfacon degee of a job sequence s gven n Secon 3 followed by an llusave eample n Secon 4. The concluson and fuue wok ae gven n Secon Poblem Saemen

3 Suppose we have a flow shop poblem wh n jobs, 1,,, and m machnes. A soluon s a sequence of jobs (he same on all machnes),.e., a pemuaon of n n elemens epesened by an n dmensonal veco,,..., ). Thee ( 1 2 n paamees ae assgned o each job: pocessng me, elease dae and due dae. Tangula membeshp funcons ae used o epess he fuzzy pocessng mes of jobs. Fgue 1 pesens a fuzzy pocessng me P, denoed by a ple ( a, b, c ). The membeshp value has s modal a b, whle a and c denoe he lowe bound and uppe bound of he pocessng me especvely. In pacce, a, b and c ae usually used o epess he pessmsc, modeae and opmsc values of he pocessng me on a machne. 1 P a b c Fgue 1: Tangula fuzzy pocessng me. The elease dae and due dae of job ae also descbed by fuzzy ses whch enable modelng of he pefeences of he decson make. As shown n Fgue 2, fuzzy elease dae R, denoed by a double ( nf, ), s descbed by a non-deceasng lnea membeshp funcon, and µ ) =0, µ ) =1. The membeshp ( nf R ( R funcon s nepeed n he followng way: s pefeed o elease he job afe, bu s no accepable o elease befoe nf. The sasfacon degee of he elease dae lnealy nceases n he me wndow [ nf, ]. Fuzzy due dae D, denoed by a double (, d ), s descbed by a deceasng lnea membeshp

4 funcon, and µ d ) =1, µ d ) =0. I means ha s pefeed o complee ( nf D ( D he job befoe, whls s unaccepable o complee afe d. The sasfacon degee of he due dae lnealy deceases n he me wndow [ nf d, d ]. Thus, he job should sa and end n he neval [ nf, d ] o fuzzy neval [ R, D ]. 1 R D nf d Fgue 2: Fuzzy elease dae and fuzzy due dae. Le Le R be he fuzzy elease dae of job ; P, j be he fuzzy pocessng me of job on machne j, j =1,2,, m ; Le C, j be he fuzzy compleon me of job on machne j, j =1,2,, m ;. The followng fomulae hold fo he calculaon of C, j : C 1,1 = 1 C,1 = R + P 1, 1 ; (1) ma ( R, C 1, 1 ) + P, 1, fo = 2,, n ; (2) C 1, j = C1, j 1 + P 1, j C, j = ma ( C 1, j,, j 1, fo j = 2,, m ; (3) C ) + P, j, fo = 2,, n ; fo j = 2,, m ; (4) whee + epesens he fuzzy addon, and ma epesens fuzzy ma.

5 Gven wo angula fuzzy numbes: P = ( a 1, b 1, c 1 ) and 1 P = ( a 2, b 2, c 2 ), hen 2 P P = ( a 1 + a 2, b 1 + b 2, c 1 + c 2 ); (5) Mamum of wo fuzzy numbes P 1 and P 2, ma ( P 1, 2 P ), has no necessaly a angula membeshp funcon. Fo smplcy, we apply he followng appomaon o peseve he angula shape: ma ( P 1, 2 P )= (ma( a 1, a 2 ), ma( b 1, b 2 ), ma( c 1, c 2 )); (6) The ask s o fnd such a sequence of jobs whch has mamum sasfacon degee wh espec o jobs elease daes and due daes. 3. Sasfacon Degee of a Job Sequence When elease daes and due daes ae fuzzy, he consans on he sa and end daes of jobs ae no necessaly sasfed o volaed, bu can be sasfed wh a cean degee. Le us denoe by R and D elease dae and due dae of job especvely. Gven a job sequence X =,,..., ), he sasfacon degee of X of he ( 1 2 n elease dae and due dae s calculaed as: Sa ( X ) = mn( Sa( )) ; (7) X Fo each job, le s and e be s sang me and endng me, especvely. Accodng o he possbly heoy poposed by Dubous & Pade [3], he sasfacon degee of he elease dae and due dae s calculaed as: Sa ( ) = mn ( µ s ), µ ( e ) ); (8) ( [ R, + ) (, D ] Le us consde pocessng me of he job. Le T be he oal fuzzy pocessng me

6 equed fo job o go hough he machnes. Le be he csp pocessng me of job, hen Sa ( ) = mn( µ ( e ), µ ( ), µ ( e )) (9) R T D Accodng o he well known eenson pncple, gven wo fuzzy numbes A and B, µ ( z) = mn( µ ( ), µ ( z )). Then, A+ B A B Sa ( ) = mn ( µ e ), µ ( e ) ) (10) ( [ R + T, + ) (, D ] Le C be he compleon me of job. Snce C = R + T, hen Sa ( ) = mn ( µ e ), µ ( e ) ). (11) ( [ C, + ) (, D ] The goal s o acheve he mamum sasfacon degee of he job. Gven job, pose R = ( nf, ) and D = (, d ). If we allow job o sa afe, we can noduce 0 such ha he elease dae of job s descbed by a apezodal membeshp funcon, R = ( nf,, +, + ), 0, as shown n Fgue 3(a); Paamee of couse affecs he shape of C (s gh pa). Accodng o fomula (11), he sasfacon degee of job wll acheve s mamum a he nesecon pon of C and D. The nesecon pon,.e., he mamum sasfacon degee of job, denoed by ρ, s no affeced by. Theefoe, we can consde only he lef pa of R, whch we denoe as eales

7 elease dae E ( R ) n Fgue 3(b) and consequenly he coespondng oal pocessng me E ( T ) and he eales compleon me denoed by E ( C ), E ( C ) = E ( R ) + E ( T ). Fgue 3(b) llusaes how o calculae ρ, mamum sasfacon degee when < + mod. The job mus sa a s = nf + ρ ( - nf ) and end a e o acheve he sasfacon degee. Fgue 4(a) and 4(b) show he case when + mod. The mamum sasfacon degee ρ s equal o 1. To acheve hs sasfacon degee, he job has o complee whn he neval [ + mod, ], and consequenly o sa whn he neval [, - mod ]. R T D C ρ nf s + nf mod mod nf nf e d + mod + mod Fgue 3(a). Compleon me and due dae when < + mod. E R ) E T ) E C ) ( ( D ( ρ nf s nf mod d nf + nf d nf + mod + e Fgue 3(b). Eales compleon me and due dae < + mod.

8 D C + + mod nf nf + mod + + d Fgue 4 (a). Compleon me and due dae when + mod. R E( C ) D nf d nf mod + + mod nf nf d + Fgue 4 (b). Eales compleon me and due dae + mod. 4. An Eample The descbed pocedue fo calculang he sasfacon degee fo job sequences wll be llusaed by an eample. Table 4.1 pesens a smple 2-machne 3-job flow shop poblem wh fuzzy elease daes, fuzzy due daes and fuzzy pocessng mes. Suppose he jobs ae fed no machnes n he followng ode In ode o oban he mamum sasfacon degee fo a job sequence, s necessay o oban he mamum sasfacon degee fo each job n he sequence, and hen he sasfacon degee fo he job sequence s obaned by akng no accoun all jobs n he sequence. To oban he mamum sasfacon degee fo each job, as menoned above, we need o calculae he eales compleon me of he job. The eales compleon me of a job s calculaed accodng o fomulae (1)-(4). Then, we employ he fomula (11) o ge he mamum sasfacon degee fo hs job. As shown n Table 4.2, he eales compleon me of job 1 s (9,14,16), and he

9 mamum sasfacon degee wh espec o s due dae s calculaed,.e., Sa ) = To acheve hs sasfacon degee, he job has o end a and o sa a Smlaly, he mamum sasfacon degee fo ohe jobs n he sequence ae also calculaed,.e., Sa ) =1 and Sa ) = The mamum sasfacon ( 2 degee fo hs job sequence s ( 3 ( 1 Table 4.1. A 2-machne 3-job flow shop poblem R P,1 P,2 D 1 (4,7) (1,2,3) (4,5,6) (12,20) (3,4) (5,7,9) (3,5,9) (21,26) 2 3 (5,8) (6,7,8) (5,8,9) (19,27) Table Sasfacon degee fo job sequence E ( R ) E ( C ) Sa ( ) e s 1 (4,7,7) (9,14,16) (3,4,4) (13,21,28) (5,8,8) (21,31,37) Concluson & Fuue Wok In hs pape, we consdeed fuzzy consaned flow shop poblem wh he am o acque a schedule whch can sasfy he consans o he mamum degee. The consans we consdeed ae fleble elease daes and due daes of jobs, whch ae descbed by fuzzy ses. The pocessng mes ae also uncean and descbed by fuzzy ses. Local seach echnques epoed n leaue, such as Smulaed Annealng, Tabu Seach, can oban hgh qualy soluons fo flow shop poblem wh vaed cea and consans. They seach he neghbohood of he cuen job sequence amng o fnd a bee sequence. Ou fuue eseach wok wll focus on he developmen of

10 Smulaed Annealng wh he sasfacon degee of he schedule as objecve funcon. Refeences [1] D. Dubos, H. Fage, and H. Pade (1995). Fuzzy Consans n Job-Shop Schedulng. Jounal of Inellgen Manufacung, volume 6, pages [2] D. Dubos, H. Fage, and P. Foemps (2003). Fuzzy Schedulng: Modellng Fleble Consans vs. Copng wh Incomplee Knowledge. Euopean Jounal of Opeaonal Reseach, volume 147, pages [3] D. Dubous, and H. Pade (1988). Possbly Theoy, New-Yok, Pleaum Pess. [4] D. Dubos and H. Pade (1978). Opeaons on Fuzzy Numbes. Inenaonal Jounal of Sysem Scences, volume 9(6), pages [5] P. Foemps (1997). Job-shop Schedulng wh Impecse Duaons: A Fuzzy Appoach. IEEE Tansacons on Fuzzy Sysems, volume 82, pages [6] H. Ishbuch and T. Muaa (2000). Flowshop Schedulng wh Fuzzy Duedae and Fuzzy Pocessng Tme. Schedulng Unde Fuzzness, Slownsk R., and Hapke M. (Eds.), Physca-Velag, Hedelbeg, pages [7] C. S. McCahon and E. S. Lee (1992). Fuzzy Job Sequencng fo a Fow Shop. Euopean Jounal of Opeaon Reseach, volume 62, pages [8] M. S. Paul, E. K. Russell and A. J. Jeff (1996). Schedulng Avals o a Poducon Sysem n a Fuzzy Envonmen. Euopean Jounal of Opeaon Reseach, volume 93, pages [9] W. Pedycz, and F. Gowde, (1998). An Inoducon o Fuzzy Ses Analyss and Desgn. The MIT Pess. [10] S. Peovc, and X. Song (2003). A New Appoach on Two-Machne flow shop poblem wh uncean pocessng me. In Poceedngs of he Inenaonal Symposum on Unceany, Modelng and Analyss, ISUMA 2003, pages , Unvesy of Mayland, College Pak, USA, Sepembe, [11] X. Song, S. Peovc (2004). Rankng of Makespans n Flow Shop Poblems wh Fuzzy Pocessng Tmes. In Poceedngs of he 5h Inenaonal Confeence on Recen Advances n Sof Compung (RASC2004), pages , Nongham, UK, Decembe, 2004.