A NEW HEURISTIC FOR THE PERMUTATION FLOWSHOP PROBLEM

Transcription

1 A NEW HEURISTIC FOR THE PERMUTATION FLOWSHOP PROBLEM Pawel J. Kalczynsi MS#03, College of Business Adinistation, Univesity of Toledo, Toledo, OH , Jezy Kabuowsi MS#03, College of Business Adinistation, Univesity of Toledo, Toledo, OH , ABSTRACT Fo ove twenty yeas the NEH heuistic of Nawaz, Enscoe, and Ha has been coonly egaded as the best heuistic fo iniizing the aespan in peutation flow shops. We pesent a new heuistic that outpefos NEH. The coputational coplexity of both heuistics is the sae. Keywods: Scheduling, flow shop, aespan, heuistic. INTRODUCTION The ancient flow shop poble is defined as follows. A set of n obs, J = {,,...,n}, available at tie zeo has to be pocessed in a shop with achines MBB, MBB,,MBB. Each ob is pocessed fist on MBB, next on MBB, and so on, and lastly on MBB. No achine can pocess oe than one ob at a tie, no peeption is allowed, all setup ties ae included into the ob pocessing ties, and thee is unliited stoage between the achines. The poble, coonly efeed to as F CBaxB, is to deteine a schedule that iniizes the copletion tie of the last ob, also nown as the aespan. The schedule with the sae ob odeing on evey achine is called a peutation schedule, and the peutation flow shop poble, F pu CBaxB, is to find a ob sequence that iniizes the aespan. Fo = and =3, the seach of the optial schedule can be esticted to peutation schedules, but the optial schedules ay have diffeent ob odeings on diffeent achines when > 3; see e.g. Potts et al. (99). Both F CBaxB and F pu CBaxB can be solved in O(nlogn) tie by the well nown algoith of Johnson (954), but F3 pu CBaxB is stongly NP-had; see Gaey et al. (976). Moeove, thee is no a polynoial tie appoxiation schee fo solving F pu CBaxB, unless P = NP; see Williason et al. (997). Since the pionee s wo of Johnson (954) dozens of heuistics have been poposed fo solving F pu CBaxB, but the O(nP P) NEH heuistic of Nawaz, Enscoe & Ha (983) is nown to easily outpefo othe heuistics, and be even faily copetitive with etaheuistics; see e.g. Watson et al. (00), Fainan et al. (004), Ruiz & Maoto (005), Reza Heazi & Saghafian (005), and Kalczynsi & Kabuowsi (007). The pupose of this pape is to pesent a new siple O(nP P) heuistic that outpefos NEH. 550

2 PnP fo BACKGROUND Let pbib be the pocessing tie of ob on achine MBiB i =,,, and =,,,n. The aespan, CBaxB(π), of a ob sequence π = (π(),π(),, π(n)) is coputed by: CBaxB(π) = n ax [ p + p + + p ].,π(),π() = = = This aespan can be also epesented by the length of a citical (longest) path in an acyclic netwo; see e.g. Pinedo (00). The netwo fo coputing CBaxB(π), whee fo siplicity π = (,,,n), is depicted in Figue ; the nodes ae nubeed by the coesponding ob pocessing ties. Thus, CBaxB(π) can be coputed in O(n) tie by the citical path ethod (CPM). The F pu CBaxB poble is to find a sequence π* with the iniu aespan. n,π() pbb pbb pb,n-b pbnb pbb pbb pb,n-b pbnb pb-,b pb-,b pb-,n- pb-,nb pbb pbb pb,n-b pbnb Figue. Netwo fo coputing the aespan of CBaxB(π), whee π = (,,,n) To copae the pefoance of heuistics, we eployed the following lowe bound fo CBaxB(π*), h LB = in ax{ p + L ((,λ {,s},s)) + p }, s h< i g hi hi gs i+ whee λbhib is the sequence obtained by Johnson s algoith applied on the ties abhib = p g and h i i bbhib = p fo J, and LBhiB(π) = g ax [ p h,π() + p g,π() + pi,π() ]. Ou LB can be n h+ = h+ = deteined in O(P P) tie, and we clai that it is the shapest polynoially coputed lowe bound nown so fa; copae with the bounds suaized by Ladhai & Haouai (005). HEURISTICS Suppose that a ob is to be inseted into the -th position of ρ = (ρ(),ρ(),...,ρ(k)). Then the esulting (K+)-eleent sequence is denoted by ρ(,). Clealy, ρ(,) = (,ρ), ρ(,) = (ρ(),...,ρ(-),,ρ(),...,ρ(k)) fo =,3,...,K, and ρ(,k+) = (ρ,). i 55

3 MBB 4 MBB 93 MB3B 67 Bwhee TNEH HeuisticT Step. Sequence the obs by thei deceasing total pocessing ties TBB = i= p to get the pioity ode γ. Set ρ = (γ()) and K = cad(ρ) =. Step. Let = γ(k+). Fo evey =,,...,K+, copute CBaxB(ρ(,)) to find the subsequence ρ(,*) with the iniu aespan. (In the case of ties, * is the fist index fo which the iniu is achieved.) Set ρ = ρ(,*) and K =: K+. Step 3. If K < n, etun to Step. Othewise, πbnehb = ρ is the final sequence. The pioity ode γ used to build up the subsequent ρ was shown to be supeio to 76 diffeent odes exained by Fainan et al. (003). The ustification of the insetion phase of NEH is athe staightfowad with the exception that the assued tie-beaing ethod sees to be poo. This is the eason why NEH is not optial fo F pu CBaxB; see Kalczynsi & Kabuowsi (007). The heuistic pesented below eploys a bette pioity ode as well as a bette tie beaing ethod that secues the optiality fo F pu CBaxB. KK Heuistic Step. Copute a = i and b = 3 + ipi fo J. Sequence the obs by p i i= thei non-inceasing cbb,b c = a (c = b ) if a b (a > b ), to get the pioity ode γ. Set ρ = (γ()) and K = cad(ρ) =. Step. Let = γ(k+). Fo evey =,,...,K+, copute CBaxB(ρ(,)), to find the subsequence ρ(,*) with the iniu aespan. (In the case of ties, * is the fist (last) index fo which the iniu is achieved if a b (a > b ).) Set ρ = ρ(,*) and K =: K+. Step 3. If K < n, etun to Step. Othewise, πbkkb = ρ is the final sequence. Steps of NEH and KK equie O(n+nlogn) tie. If CBaxB(ρ) is coputed by both fowad and bacwad passes of CPM, ρ(,*) can be found in O((K+)); see Taillad (990). Theefoe, each epetition of Step equies O((K+)), which secues the oveall coplexity of O(nP P). Conside the following instance of F3 pu CBaxB with n = 6 obs. i= i Job Job Job 3 Job 4 Job 5 Job The pioity ode found in Step of NEH is γ = (,,4,3,6,5). The subsequent epetitions of Step of NEH yield (,), (,,4), (3,,,4), (3,6,,,4), and πbnehb = (3,6,,5,,4) with CBaxB(πBNEHB) = 463. The pioity ode found in Step of KK is γ = (,4,,3,6,5). The subsequent epetitions of Step of KK yield (,4, (,,4), (,,4,3), (,6,,4,3), and πbkkb = (,6,,4,3,5) with CBaxB(πBKKB) = 44. The unique optial sequence π* = (6,,,4,5,3) has a aespan of

4 COMPUTATIONAL RESULTS The pefoance of KK and NEH was fist exained on popula bencha instances chosen by Taillad (993). The ob pocessing ties wee andoly geneated fo a unifo distibution ove the integes,,,99. Fo each of twelve pais (,n), Taillad selected N = 0 instances that seeed to be the hadest ones. These 0 data sets can be downloaded fo the OR-Libay at Hhttp://scga.s.ic.ac.u/info.htlH, whee one can also find the coesponding optial o the best nown sequences π*. Fo evey Taillad s instance, we coputed the pecentage incease ove CBaxB(π*) by: PI(π) = [(CBaxB(π) - CBaxB(π*))/CBaxB(π*)] 00%, whee π is the sequence found by the heuistic unde study. The aveage PIs (APIs) ae shown in Table. The oveall APIs fo πbkkb and πbnehb ae 3.% and 3.33%. Add hee that, accoding to Ruiz & Maoto (005), the second best so fa (afte NEH) heuistic has a athe poo oveall API of 6.%, and the best etaheuistic, whose initial sequence is πbnehb, enoys an oveall API of.3% unde the stopping citeion of 50,000 aespan evaluations. M N API(πBKKB) API(πBNEHB) Table. The API(πBKKB) and API(πBNEHB) fo Taillad (993) instances. Since the esults obtained on Taillad s data sets can be biased because of N = 0, we pefoed additional siulation expeients in which we andoly geneated N =,000 instances fo evey pai (,n), whee = 3,4,5,6,8,0,,5,0 and n = 0,30,...,00,50,00. The ob pocessing ties wee assued to be unifoly distibuted integes fo {,,...,99}. Fo evey instance, we coputed the lowe bound LB, to edefine PI as PI = [(CBaxB(π) - LB)/LB] 00%. The obtained APIs ae shown in Table. We also coputed the pecentages, PALB(πBKKB) and PALB(πBNEHB), of achieving the coesponding LB s; see Table 3. Note hee that PALB s ae lowe bounds of the optiality ates. The obtained esults validate the excellent pefoance of both KK and NEH when is sall. This pefoance ipoves with an incease of n. In fact, it is not a supise because, accoding to Taillad (993), and Potougal & Sith (00), any F pu CBaxB schedule is asyptotically optial. The quality of the used lowe bound LB stongly affects the esults of Tables and 3. Clealy, ou LB weaens apidly with an incease of. Add hee that the best banch-and-bound algoith of Ladhai & Haouai (005) ay handle the instances with up to 000 obs but only 7 achines. 553

5 n Table. Aveage pecentage of inceases ove the lowe bound LB, API(πBKKB)/API(πBNEHB) n Table 3. Pecentages of achieving the lowe bound LB, PALB(πBKKB)/PALB(πBNEHB). FINAL REMARKS The sequence πbnehb is assued to be the initial one in the best etaheuistics fo solving F pu CBaxB; see Fainan et al. (004) and Ruiz & Maoto (005). The eplaceent of πbnehb by πbkkb can only ipove thei pefoance. Siilaly, this eplaceent will be also useful in the banch-and-bound algoith of Ladhai & Haouai (005) in which NEH povides the initial uppe bound. Futheoe, the best etaheuistics fo solving the elated pobles unde the blocing and no-wait constaints, F bloc CBaxB and F no-wait CBaxB, also assue the initial sequence found by NEH odified to these pobles; see Nowici (999) and Gabowsi & Pepea (005). We stongly believe that it is possible to develop constuctive heuistics fo solving F bloc CBaxB and F no-wait CBaxB, which would outpefo the NEH esults. The wost case pefoance atio of NEH eains unnown, but it was shown to be bounded fo below and above by / and (+)/; see Nowici & Sutnici (993). We conectue 554

6 that this atio fo KK is /. Add hee that the appoxiation algoith of Sviideno (004) enoys the best nown atio of O( log), but its pactical ipotance is vey liited. REFERENCES Fainan, J. M., Gupta, J. N. D. & Leisten, R. (004). A eview and classification of heuistics fo peutation flow-shop scheduling with aespan obective. Jounal of the Opeational Reseach Society, 55, Fainan, J. M., Leisten, R. & Raendan, C. (003). Diffeent initial sequences fo the heuistic of Nawaz, Enscoe and Ha to iniize aespan, idle tie o flowtie in the static peutation flowshop poble. Intenational Jounal of Poduction Reseach, 4, Gaey, M. R. D., Johnson, D. S. & Sethi, R. (976). The coplexity of flow shop and ob shop scheduling. Matheatics of Opeations Reseach,, 7-9. Gabowsi, J. & Pepea, J. (005). Soe local seach algoiths fo no-wait flow-shop poble with aespan citeion. Coputes & Opeations Reseach, 3, 97-. Johnson, S. M. (954). Optial two- and thee-stage poduction schedules with setup ties included. Naval Reseach Logistics Quately,, Kalczynsi, P. J. & Kabuowsi, J. (007). On the NEH Heuistic fo Miniizing the Maespan in Peutation Flowshops. Oega, The Intenational Jounal of Manageent Science, 35(), Ladhai, T. & Haouai, M. (005). A coputational study of the peutation flow shop poble based on a tight lowe bound. Coputes & Opeations Reseach, 3, Nawaz, M., Enscoe J., E. & Ha, I. (983). A heuistic algoith fo the -achine, n-ob flow-shop sequencing poble. Oega, the Intenational Jounal of Manageent Science,, Nowici, E. (999). The peutation flow shop with buffes: A tabu seach appoach. Euopean Jounal of Opeational Reseach, 6, Nowici, E. & Sutnici, C. (993). New esults in the wost-case analysis fo flow-shop scheduling. Discete Applied Matheatics, 46, -4. Pinedo, M. (00). Scheduling: Theoy, Algoiths, and Systes (nd ed.). Uppe Saddle, N.J.: Pentice Hall. Potougal, V. & Sith, J. L. (00). The asyptotic convegence of soe flow-shop scheduling heuistics. Asia-Pacific Jounal of Opeational Reseach, 8,

7 Potts, C. N., Shoys, D. B. & Williason, D. P. (99). Peutation vs. non-peutation flow shop schedules. Opeation Reseach Lettes, 0, Reza Heazi, S. & Saghafian, S. (005). Flowshop-scheduling with aespan citeion: a eview. Intenational Jounal of Poduction Reseach, 43, Ruiz, R. & Maoto, C. (005). A copehensive eview and evaluation of peutation flowshop heuistics. Euopean Jounal of Opeational Reseach, 65, Sviideno, M. I. (004). A note on the peutation flow shop poble. Annals of Opeations Reseach, 9, Taillad, E. (990). Soe efficient heuistic ethods fo the flow shop sequencing poble. Euopean Jounal of Opeational Reseach, 47, Taillad, E. (993). Benchas fo basic scheduling pobles. Euopean Jounal of Opeational Reseach, 64, Watson, J.-P., Babulescu, L., Whitley, L. D. & Howe, A. E. (00). Contasting stuctued and ando peutation flow-shop scheduling pobles: seach-space topology and algoith pefoance. INFORMS Jounal on Coputing, 4, Williason, D. P., Hall, L. A., Hoogeveen, J. A., Huens, C. A. J., Lensta, J. K., Sevastianov, S. V., et al. (997). Shot shop schedules. Opeations Reseach, 45,