Fuzzy LPT Algorithms for Flexible Flow Shop Problems with Unrelated Parallel Machines for a Continuous Fuzzy Domain

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2 The IE Nework Conference 4-6 Ocober 007 Fuzzy LPT Algorihm for Flexible Flow Shop Problem wih Unrelaed Parallel Machine for a Coninuou Fuzzy Domain Jii Jungwaanaki * Manop Reodecha Paveena Chaovaliwonge 3 Frank Werner 4,,3 Deparmen of Indurial Engineering, Faculy of Engineering, Chulalongkorn Univeriy, Bangkok Faculy of Mahemaic, Oo-von-Guericke-Univeriy, P.O. Box 40, D-3906 Magdeburg, Germany ii.@uden.chula.ac.h * Abrac A flexible flow hop problem can be conidered a a generalizaion of a pure flow hop problem in which he ob have o go hrough he age in he ame order. We conider a flexible flow hop problem wih unrelaed machine and eup ime, where he proceing ime depend on he choen machine and eup ime are equence-dependen. While for claical problem he proceing ime for each ob are aumed o be known exacly, in many real-world iuaion proceing ime vary dynamically due o human facor or operaing faul. In hi paper, fuzzy concep are ued in an LPT algorihm for managing uncerain cheduling. Given a e of ob ogeher wih a memberhip funcion for he andard proceing ime, he fuzzy LPT algorihm conruc a oluion by mean of a memberhip funcion for he final makepan. The propoed algorihm provide a more flexible mehod of cheduling ob han convenional cheduling mehod. The reul how ha he fuzzy LPT algorihm give a deviaion from he opimal makepan value of abou five percen for he mall-ize e problem. In addiion, he fuzzy LPT algorihm by uing he average value of he oal operaing ime a he la age (denoed by FLPT k ) give a good oluion for boh mall- and large-e problem. Keyword: Flexible Flow Shop, Unrelaed Parallel Machine, Fuzzy Se, LPT Algorihm. Inroducion Thi paper i primarily concerned wih a cheduling problem occurring in he producion indurie. They are eablihed a muli-age producion flow hop faciliie where a producion age may be made up of parallel machine. Thi i known a flexible flow hop or hybrid flow hop environmen, i.e. i i a generalizaion of he claical flow hop model. There are k age and ome age may have only one machine, bu a lea one age mu have muliple machine, and all ob have o pa hrough a number of age in he ame order. Moreover, in uch indurie, i i common o find newer or more modern machine running ide by ide wih older and le efficien machine. The older machine may perform he ame operaion a he newer one, bu would generally require a longer operaing ime for he ame operaion. Such a problem i called a flexible flow hop problem wih unrelaed parallel machine (ee Jungwaanaki, Reodecha, Chaovaliwonge, and Werner [-3]). In [-3], i ha been found ha he LPT algorihm i a good dipaching rule for he makepan problem. The laer work deal wih he iuaion ha he proceing ime for each ob are exacly given a deerminiic value. However, in many real-world applicaion, proceing ime may vary dynamically due o human facor or operaing faul. The eimaed proceing ime are no preciely known. Conequenly, everal concep uch a fuzzy e heory, probabiliy heory, DEMPSTER/SHAFER heory, a eniiviy analyi, and oher, have been ued o ake ino accoun he uncerainie. In hi paper, we will rea uncerainy by uing fuzzy e heory becaue of i impliciy and imilariy o human reaoning [4]. Such a heory ha been applied o many area uch a invenory conrol [5] and cheduling [6]. We apply fuzzy LPT algorihm o he problem under conideraion. Given a e of ob, each of which ha i memberhip funcion for he andard proceing ime, a cheduling reul wih a memberhip funcion for he final compleion ime i generaed. The remainder of hi paper i a follow. In Secion, he problem under conideraion i decribed. Secion 3 preen fuzzy LPT algorihm. A numerical example i dicued in Secion 4. Compuaional reul are dicued in Secion 5 and concluion are given in Secion 6.. Problem Decripion Flexible flow hop problem can be decribed a follow. There i a e J = {,,,, n} of n independen ob which need o be proceed, and he proceing yem i defined by a e O = {,,,, k} of k proceing age. A each age, O, here i a e M = {,, i,, m } of m unrelaed machine. Each ob, J, ha i releae dae r 0 and a due 08

3 dae d 0. Due o he unrelaed machine, he proceing ime p of ob on machine i a age i i equal o p / v, where i p i he andard proceing ime of ob a age, and v i i he relaive peed of ob which i proceed by he machine i a age. However, ince he andard proceing ime i uncerain, i i repreened by a fuzzy number. Conequenly, each ob ha a fuzzy andard proceing ime p for every age, O. There are proceing rericion of he ob a follow: () ob are proceed wihou preempion on any machine; () every machine can proce only one operaion a a ime; (3) operaion of a ob have o be realized equenially, wihou overlapping beween he age; (4) ob pliing i no permied. Seup ime conidered in hi problem are claified ino wo ype, namely a machinedependen eup ime and a equence-dependen eup ime. A eup ime of a ob i machine-dependen if i depend on he machine o which he ob i aigned. I i aumed o occur only when he ob i he fir ob aigned o he machine. ch i denoe he machine-dependen eup ime (or changeover ime) of ob if ob i he fir ob aigned o machine i a age. A equence-dependen eup ime i conidered beween ucceive ob. A eup ime of a ob on a machine i equence-dependen if i depend on he ob u compleed on ha machine. l denoe he ime needed o changeover from ob l o ob a age, where ob l i proceed direcly before ob on he ame machine. All eup ime are known and conan. Moreover, here i given a non-negaive machine availabiliy ime for any machine of a paricular age. The obecive i o minimize he fuzzy makepan Cmax which i equivalen o he fuzzy compleion ime of he la ob leaving he yem. 3. Fuzzy LPT Scheduling Algorihm In hi ecion, fuzzy e heory i ued in an LPT cheduling algorihm o chedule he ob wih uncerain andard proceing ime. Given a e of ob whoe proceing ime have heir memberhip funcion, he fuzzy LPT algorihm conruc a chedule by mean of a final compleion ime memberhip funcion. Fir, he relaed fuzzy e operaion are briefly reviewed. Then, he fuzzy LPT algorihm are propoed. 3. Relaed fuzzy e operaion Define wo fuzzy e A and B on he univere X. A given elemen x of he univere i mapped o a memberhip value uing a funcion-heoreic form. Such a funcion map elemen of a fuzzy e o a real-numbered value from he inerval [0,]. When he univere of he fuzzy e A i coninuou and infinie, he fuzzy e A i denoed by (ee [7]) μa( ) A = x () x One ype of he funcion-heoreic form ued in hi paper i a riangular memberhip funcion. I can be decribed by A = (a, b, c), where a b c (ee Figure ). μ A ( x ) X a b c Figure. A riangular fuzzy memberhip funcion for he fuzzy e A. For a riangular fuzzy memberhip funcion, he fuzzy e A and B can be repreened a follow: A = (a A, b A, c A ) and B = (a B, b B, c B ) () The um of he fuzzy e A and B i obained a follow: A + B = (a A +a B, b A +b B, c A +c B ) (3) Many fuzzy ranking mehod have been propoed for olving deciion and opimizaion problem uch ha a good oluion can be obained (ee e.g. [8] for a urvey). A ranking uing he averaging mehod i one of he mo widely ued mehod [9] and i adoped in hi udy. The ranking funcion RA ( ) i defined a follow: xμa( x) dx RA ( ) = μ ( xdx ) = A b a b a b a b a c b c b c b c b x a c x ( ) xdx + ( ) xdx x a c x ( ) dx + ( ) dx = ( a+ b+ c ) 3 Therefore, we ay ha A > B if RA ( ) > RB ( ). In hi paper, he operaion preened above are ued o chedule he ob wih uncerain andard proceing ime. A riangular memberhip funcion i applied o repreen he fuzzy andard proceing ime of each ob. I can be denoed by p = ( a p, b p, c p ), where a p b p c p. The average value of he fuzzy andard proceing ime i ave repreened by p. 3. Heuriic conrucion The fuzzy algorihm for he flexible flow hop problem wih unrelaed parallel machine i baed on he LPT rule, and i ue fuzzy concep o manage uncerainy. The andard proceing ime for each ob are defined by a fuzzy e. The propoed algorihm i a follow: (4) 09

4 Par : Finding he repreenaive: Sep : Selec he repreenaive of he peed / l and eup ime v / i for every ob and every age by uing he combinaion of he minimum, maximum, and average daa value. Par : Finding he ob equence: Sep : For each ob, find he repreenaive of he fuzzy operaing ime ( / ) and he oal fuzzy / operaing ime ( T ) baed on he riangular fuzzy addiion operaion by uing he following equaion: and = / / p + / v l i / T =, (5) (6) / O / where T = = ( a /, b /, c /) and a T T T T / b T / c T /. Sep 3: For each ob, find he average value of he repreenaive of he fuzzy operaing ime ( )of Par 5: Finding he be oluion each age and he oal fuzzy operaing ime ( T ) Sep 4: Reurn he be fuzzy oluion wih by uing he following equaion: ave ( a = Cmax, b Cmax, c Cmax ) and he average value Cmax. ( a / + b / + c / ), (7) 3 and T = Table Fuzzy andard proceing ime. ( a / + b / + c /) (8) 3 T T T p p Sep 4: Ue he fuzzy LPT algorihm o find he firage equence. Cae : Sor he ob in decending order of he average value of he repreenaive of he oal fuzzy operaing ime T ; if any wo ob have he ame T value, or hem in an arbirary order (hi algorihm i denoed by FLPT T ). Cae : Sor he ob in decending order of he average value of he repreenaive of he fuzzy / operaing ime of each age; if any wo ob have he ame / value, or hem in an arbirary order (hee algorihm are denoed by FLPT, FLPT,, FLPT k ). Par 3: Aigning he ob o he machine a he fir age Sep 5: Aign he fir ob [] in he ordered ob equence o he machine which ha he minimum average fuzzy compleion ime among all machine of hi age. Sep 6: Updae he availabiliy ( a ) of he eleced machine by uing he value of he fuzzy compleion ime of he ob aigned o hi machine. Sep 7: Remove he ob from he ordered ob equence. Sep 8: Repea Sep 5 o 7 unil he ob equence i empy. Par 4: Aigning he ob o he machine a he oher age Sep 8: Find he ob equence of he nex age. Cae : Se he ob equence for he age o be i equal o he ordered ob equence obained in Sep 4 (permuaion rule). Cae : Deermine he ob equence for he curren age by ordering he ob according o heir fuzzy compleion ime a he previou age (FIFO rule). Sep 9: Aign he fir ob [] in he ob equence in Sep 8 o he machine which ha he minimum average fuzzy compleion ime among all machine of he age. Sep 0: Updae he availabiliy ( a ) of he eleced machine by uing he value of he average fuzzy compleion ime of he ob aigned o he machine. Sep : Remove he ob from he ordered ob equence. Sep : Repea Sep 8 o unil he ob equence i empy. Sep 3: Conider he nex age and Repea Sep 8 o unil age k ha been conidered. i C max = Job (76, 85, 95) (8, 88, 94) Job (59, 67, 7) (49, 59, 6) Job 3 (88, 95, 99) (3, 33, 4) Job 4 (69, 78, 84) (9, 95, 0) Job 5 (6, 6, 68) (75, 76, 76) Table Relaive peed of he machine. Job Job Job 3 Job 4 Job 5 v v v Table 3 Sequence-dependen eup ime. Job Job Job 3 Job 4 Job 5 l X l 5 X l 36 X 7 4l X 6 5l X l X l 3 X l 34 5 X 0 4l X 6 5l X Table 4 Machine dependen-eup ime. Job Job Job 3 Job 4 Job 5 ch ch ch

5 4. A Numerical Example A numerical example i provided in hi ecion o illurae he algorihm propoed. Le n = 5, k =, and m = and m =. The releae dae of he ob are 9, 9, 0, 7 and 0, repecively. The machine availabiliie are 36 and 4 for he machine a he fir age and 04 for he machine a he econd age. Moreover, aume he fuzzy andard proceing ime given in Table. The relaive peed of machine are hown in Table. The equence- and machine-dependen eup ime are given in Table 3 and Table 4, repecively. The algorihm work a follow: Par : Finding he repreenaive: Sep : Selec he minimum value of he peed and he eup ime for every ob and every age. Par : Finding he ob equence: Sep : For each ob, find he repreenaive of he fuzzy operaing ime and he oal fuzzy operaing ime baed on he riangular fuzzy addiion operaion. The reul are hown in Table 5 and Table 6, repecively. Sep 3: Aume ha we ue he FLPT T algorihm. Find he average value of he repreenaive of he oal fuzzy operaing ime, he reul are hown in he la column of Table 6. Sep 4: Sor he ob in decending order of he average value of repreenaive of he oal fuzzy operaing ime. Thu, he ordered ob equence i {3,, 5, 4, }. Table 5 Repreenaive of he fuzzy operaing ime (by uing eup min and peed min ). / Job (95.69, 06.43, 8.365) (75.78, 8.39, 86.60) Job (8.566, 9.54, 96.59) (45.95, 54.54, 57.08) Job 3 (33.646, 43.56, 49.7) (63.770, , ) Job 4 (7.633, 80.54, 85.84) (88.53, 9.90, 97.03) Job 5 (9.93, 9.93, 00.5) (86.8, , ) Table 6 Repreenaive of he oal fuzzy operaing ime. / T T Job (70.870, 87.76, ) Job (7.58, , 53.6) Job 3 (97.46, , 3.567) 0.43 Job 4 (6.46, 7.46, 8.845) 7.5 Job 5 (78.3, 79.70, ) Par 3: Aigning he ob o he machine a he fir age Sep 5: Aign he fir ob [] (ob 3) in he ordered ob equence o he machine which ha he minimum average fuzzy compleion ime. The reul are a follow: ave ave ave p3 -on machine : max{ a, r3 } + ch3 + = 87.45; v3 ave ave ave p3 -on machine : max{ a, r3 } + ch3 + = ; v3 Thu, ob 3 i aigned o machine, ince he average fuzzy compleion ime of ob 3 aigned o / hi machine i lower han he average fuzzy compleion ime on he oher machine. I fuzzy compleion ime i (34, 4, 45). Sep 6: Se he availabiliy of machine o be (34, 4, 45). In addiion, we e he releae dae of ob 3 for he nex age o be (34, 4, 45) a well. Sep 7: Remove ob 3 from he ordered ob equence. Sep 8: Repea Sep 5 o 7 unil he ob equence i empy. For he nex ob (i.e. ob ), he average fuzzy compleion ime i calculaed a follow. ave ave ave p -on machine : max{ a, r } + ch+ = ; v ave ave ave p -on machine : max{ a, r } + 3+ = 6.830; v Again, ob i aigned o machine, ince he fuzzy compleion ime i horer in hi cae. All reul are hown in Table 7. Table 7 Fuzzy compleion ime a age. Job # Machine # Fuzzy compleion ime Average fuzzy compleion ime 3 (34.000, 4.000, ) (6.38, , 4.9) (7.6, 4.6, ) (3.770, , 6.736) (89.770, 3.409, ) Par 4: Aigning he ob o he machine a he oher age Sep 8: Find he ob equence of he nex age. Cae : For he permuaion rule, e he ob equence o be equal o {3,, 5, 4, } Cae : For he FIFO rule, e he ob equence o be equal o {, 3, 5, 4, } Sep 9: Again a in Sep 5, aign he fir ob [] (ob 3 in cae and ob in cae ) o he eleced machine which ha he minimum average fuzzy compleion ime. However, in hi example, here i only one machine a he econd age, o in cae ob 3 i aigned o he machine fir, wherea ob i aigned o he machine fir oherwie. Sep 0: Updae he availabiliy of he eleced machine by uing he value of he average fuzzy compleion ime of he ob aigned o he machine. Sep : Remove ob 3 in cae (or ob in cae ) from he ordered ob equence Sep : Repea Sep 8 o unil he ob equence i empy. Sep 3: Conider he nex age and Repea Sep 8 o unil age k ha been conidered (bu for hi example, we have k =). The reul are hown in Table 8. For he choen repreenaive of he fuzzy operaing ime, he FIFO rule generae a oluion which i beer han ha generaed by he permuaion rule, o we elec he oluion generaed by he FIFO rule and he fuzzy compleion ime i (59.83, 6.073, 65.35). Par 5: Finding he be oluion Sep 4: Repea Sep o Sep 4 again for he oher repreenaive, and reurn he be fuzzy oluion. The reul for hi example are hown in Table 9.

6 Table 8 Fuzzy compleion ime a age. Job # Machine # Fuzzy compleion ime Average fuzzy compleion ime Cae : Permuaion rule 3 (09.770, 8.884, 3.340) (34.947, 330., ) (40.8, 47.55, ) (5.74, 53.47, ) ( , , ) Cae : FIFO rule (0.35, 34.47, 48.53) (89.085, , ) ( , , 48.0) ( , , ) (59.83, 6.073, 65.35) Table 9 The be fuzzy oluion. Job # Machine # Fuzzy compleion ime Average fuzzy compleion ime Sage: 4 (03.57, 0.559, 5.7) (78.646, 88.56, 94.7) (3.64, 30.99, ) (76.55, 94.5, 36.89) (64.646, 8.34, ) Sage: 4 (96.53, , 6.48) (6.8, , 9.588) 75.4 ( , , ) (45.74, , ) ( , , ) Table 0 An opimal oluion. Cae of a p b p c p p ave p Compleion ime For an opimal oluion, we have ued he andard proceing ime by uing he value, a p b p, c p, and ave p a he andard proceing ime in he mahemaical model (ee Jungwaanaki, Reodecha, Chaovaliwonge, and Werner []). The reul of he compleion ime are hown in Table Compuaional Reul In our e, we ued problem wih 5, 0, 0, and 00 ob, and 5 machine per age, and and0 age. For each problem ize, en differen inance have been run. The andard proceing ime are fuzzy uch ha he value of b p are generaed uniformly from he inerval [0,00], a p = b p - 0 U[0,] and c p = b p + 0 U[0,], where U[0,] denoe a uniformly diribued random number from he inerval [0,]. The relaive peed are diribued uniformly in he inerval [0.7,.3]. The eup ime, boh equence- and machine-dependen eup ime, are uniformly generaed from he inerval [0, 50], wherea he releae dae are uniformly generaed from he inerval beween zero and half of heir oal andard proceing ime mean. Table Average performance of he fuzzy LPT algorihm for mall-ize e problem. Algorihm Cmax %deviaion FLPT T a b c ave FLPT a b c ave FLPT a b c ave Table Average value of he average fuzzy compleion ime ( C problem. Problem (n/m/)* ave max ) and average CPU ime for he e FLPT T FLPT FLPT FLPT 3 FLPT 4 FLPT 5 FLPT 6 FLPT 7 FLPT 8 FLPT 9 FLPT 0 CPU Time Average (m) (5//) (5//0) (0//) (0//0) (0/5/) (0/5/0) (0//) (0//0) (0/5/) (0/5/0) (00//) (00//0) (00/5/) (00/5/0) *(n/m/) = (number of ob/ number of machine per age/ number of age)

7 Fir, we preen he reul of he fuzzy LPT algorihm for mall-ize wo-age problem wih five ob and wo machine per age. We give he average deviaion from he opimal makepan value obained by uing a commercial mahemaical programming ofware, CPLEX and AMPL, wih an Inel Penium 4.00GHz CPU wih 56 MB of RAM. For he mall-ize problem, we obained he average opimal value in an average CPU ime of econd and he average makepan value of he fuzzy LPT algorihm a hown in Table. The average opimal makepan value uing he fuzzy andard proceing ime value a p, b p, c p, and ave p are , 83.77, , and , repecively. I can be oberved ha an FLPT algorihm ignificanly ouperform he oher, wherea an FLPT algorihm give poor oluion. The reul how ha he average percenage deviaion from he average opimal fuzzy makepan value of an FLPT algorihm are abou 5 percen. Nex, for he large-ize problem, we canno find an opimal oluion wihin a reaonable ime ince hee problem are NP hard [0]. Thu, we ued fuzzy LPT algorihm o find a be oluion inead of an opimal oluion. The reul in Table how he average value of he average fuzzy compleion ime ave ( Cmax ) of en differen inance for each problem ize and he average CPU ime (in milliecond). The reul how ha a fuzzy LPT algorihm by oring he ob in decending order of he average value of he oal operaing ime / k (i.e. an FLPT k algorihm) give good oluion. Alhough an FLPT 5 algorihm ouperform he oher for he e problem wih 5 ob, machine per age, and 0 age, i value i only lighly beer han he value of an FLPT 0 algorihm. In general, he qualiy of an FLPT algorihm improve wih an increaing value of. 6. Concluion In hi paper, fuzzy LPT algorihm have been inveigaed for minimizing he makepan for he flexible flow hop problem wih unrelaed parallel machine and eup ime, which i ofen occurring in real world problem. Such algorihm are baed on he li cheduling principle by developing ob equence for he fir age and aigning and equencing he remaining age by boh he permuaion and FIFO approache. In addiion, proceing ime under uncerainy have been conidered. We have olved hi problem by uing fuzzy e heory. In paricular, we ued a riangular memberhip funcion for he andard proceing ime o ge a more real-world applicaion. Thu, fuzzy LPT algorihm are propoed o manage ob wih uncerain andard proceing ime. Thi approach generae a cheduling reul wih a memberhip funcion compleion ime. The reul how ha he recommended fuzzy LPT algorihm give a deviaion from he opimal makepan value of abou five percen for mall-ize e problem. In paricular, an FLPT k algorihm ha ue he average value of he oal operaing ime give good oluion for boh mall- and large-ize e problem. In he fuure, we will ue oher algorihm for hi problem and ry o apply oher characeriic of fuzzy e o he cheduling area. For inance, we can apply oher ype of memberhip funcion o hi or oher cheduling problem. Acknowledgmen Thi work wa uppored in par by INTAS (proec ) and he Deparmen of Indurial Engineering, Chulalongkorn Univeriy. Reference [] Jungwaanaki, J., Reodecha, M., Chaovaliwonge, P., and Werner, F Algorihm for flexible flow hop problem wih unrelaed parallel machine, eup ime, and dual crieria. The Inernaional Journal of Advanced Manufacuring Technology, (in pre) DOI 0.007/ [] Jungwaanaki, J., Reodecha, M., Chaovaliwonge, P., and Werner, F Conrucive and abu earch algorihm for hybrid flow hop problem wih unrelaed parallel machine and eup ime. Inernaional Journal of Compuaional Science, (): [3] Jungwaanaki, J., Reodecha, M., Chaovaliwonge, P., and Werner, F. and Werner, F Conrucive and imulaed annealing algorihm for hybrid flow hop problem wih unrelaed parallel machine. Thammaa Inernaional Journal of Science and Technology, (): 3-4. [4] Han, S., Ihii, H., and Fuii, S One machine cheduling problem wih fuzzy duedae. European Journal of Operaional Reearch, 79(): -. [5] Gen, M., Tuimura, Y., and Zheng, D An applicaion of fuzzy e heory o invenory conrol model. Compuer & Indurial Engineering, 33(3-4): [6] Kuroda, M., and Wang, Z Fuzzy ob hop cheduling. Inernaional Journal of Producion Economic, 44(-): [7] Ro, T.J Fuzzy logic wih engineering applicaion, McGraw-Hill, New York, U.S. [8] Chang, P.T., and Lee, E.S Ranking of fuzzy e baed on he concep of exience. Compuer & Mahemaic wih Applicaion, 7(9-0): -. [9] Sun, K.T Job cheduling uing ranking fuzzy number mehod. Proceeding of 998 nd Inernaional conference on Knowledge-Baed Inelligen Elecronic Syem, Adelaide, SA, Auralia, Apr. -3, 998: [0] Gupa, J.N.D Two-age, hybrid flowhop cheduling problem. Journal of he Operaional Reearch Sociey, 39(4):

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