A Fuzzy Goal Programming Approach for a Single Machine Scheduling Problem
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1 Proceedngs of e 9 WSEAS Internatonal Conference on Appled Maematcs, Istanbul, Turkey, May 7-9, 006 (pp40-45 A Fuzzy Goal Programmng Approach for a Sngle Machne Schedulng Problem REZA TAVAKKOLI-MOGHADDAM, BABAK JAVADI AD IMA SAFAEI 3 Department of Industral Engneerng, Faculty of Engneerng Unversty of Tehran P.O. Box: 365/4563, Tehran Department of Industral Engneerng Mazandaran Unversty of Scence and Technology P.O. Box: 734, Babol 3 Department of Industral Engneerng Iran Unversty of Scence and Technology P.C.: 6844, armak, Tehran IRA Abstract: - Ths paper presents e fuzzy mxed-nteger goal programmng model for a sngle machne schedulng problem w b-obectves consstng of e mnmzaton of e total weghted flow tme and total weghted tardness. Ths model s solved by a fuzzy goal programmng approach to verfy and valdate e proposed approach for e above problem. Two test problems n small and large szes are generated at random and en e computatonal results demonstrate e effectveness of e proposed approach. Key-Words: - Sngle machne schedulng, Fuzzy goal programmng, Weghted flow tme, Weghted tardness Introducton Schedulng conssts of plannng and arrangng obs n an orderly sequence of operatons n order to meet e customer's requrements []. The schedule of obs and e control of er flows rough a producton process are e most sgnfcant role n any modern manufacturng systems. In a snglemachne schedulng, ere s only one machne to process all obs to optmze e obectve functon, say mnmzng e sum of e maxmum earlness and tardness []. It s well known at e optmal soluton of sngle obectve models can be qute dfferent to ose models consst of mult obectves. In fact, e decson maker often wants to mnmze e earlness/tardness penalty or total flow tme. Each of ese obectves s vald from a general pont of vew. Snce ese obectves conflct w each oer, a soluton may perform well for one obectve or t gves nferor results for oers. For s reason, schedulng problems have a multobectve nature. In decson makng stuatons, e hgh degree of fuzzness and uncertantes s ncluded n e data set. The fuzzy set eory provdes a framework for handlng e uncertantes of s type [3]. Bellman and Zadeh [4] presented some applcatons of fuzzy eores to e varous decson-makng processes n a fuzzy envronment. Zmmerman [5 and 6] presented a fuzzy optmzaton technque to a lnear programmng (LP problem w sngle and mult obectves. The fuzzy set eory has been appled to formulate and solve problems n varous areas such as artfcal ntellgence, mage processng, robotcs, pattern recognton, and e lke (Hannan, [7] and Yager, [8]. Dfferent approaches to mult-obectve sngle machne problems w fuzzy parameters have been presented n e lterature durng e last decade. Ish and Tada [9] consdered a sngle machne schedulng problem mnmzng e maxmum lateness of obs w fuzzy precedence relatons. A fuzzy precedence relaton relaxes e crsp precedence relaton and represents e satsfacton level w respect to precedence between two obs. Therefore, e problem to be solved consdered an addtonal obectve to maxmze e mnmum satsfacton level obtaned by e fuzzy precedence relatons. An algorm for determnng nondomnated solutons s proposed based on a graph representaton of e precedence relatons.
2 Proceedngs of e 9 WSEAS Internatonal Conference on Appled Maematcs, Istanbul, Turkey, May 7-9, 006 (pp40-45 Adamopoulos and Papps [0] presented a fuzzylngustc approach to mult-crtera sequencng problem. They consdered a sngle machne, n whch each ob s characterzed by fuzzy processng tmes. The obectve s to determne e processng tmes of obs and e common due date as well as to sequence e obs on e machne where penalty values are assocated w due date assgned, earlness, and tardness. Anoer approach to solve mult-crtera sngle machne schedulng was presented by Lee, et al. []. The proposed approach s used lngustc values to evaluate each crteron (e.g. very poor, poor, far, good, and very good and to represent ts relatve weght (e.g. very unmportant, unmportant, medum mportant, mportant, and very mportant. A tabu search meod s used as a stochastc tool to fnd e nearoptmal soluton w an aggregated fuzzy obectve functon. Ishbuch and Murata [] presented a flow shop schedulng problem w fuzzy parameters such as fuzzy due dates and fuzzy processng tmes, n whch e obectves are to mnmze e total flow tme, makespan, and e maxmum earlness and tardness of all obs. A mult-obectve genetc algorm s developed to handle ese fuzzy schedulng obectves. Thereafter, a number of researches have extended e fuzzy set eory to e feld of goal programmng proposed by arsmhan [3]. In fact, e fuzzy goal and mult-obectve programmng has a very extensve applcaton. For example, Snha, et al. [4] presented a fuzzy goal programmng n mult-crtera decson systems. Rao, et al. [5] proposed a fuzzy goal programmng approach for e structural optmzaton problem. Kumar, et al. [6] proposed a fuzzy goal programmng approach for a vendor selecton problem n supply chan. Mshra, et al. [7] presented a fuzzy goal programmng model of a machne-tool selecton and operaton allocaton problem n flexble manufacturng systems. Problem Formulaton The followng notatons and defntons are used to descrbe e sngle machne schedulng problem w mult obectves.. Indces and Parameters = number of obs. P = processng tme of ob (=,. R = release tme of ob (=,. D = due date of ob (=,. W = weght of ob (=,. M = a large postve nteger value.. Decson Varables f ob s scheduled after ob ; X = 0 oerwse., and C = completon tme of ob T = tardness of ob.3 Maematcal Model The mxed-nteger programmng (MIP formulaton of e sngle machne schedulng problem for bobectves and a set of constrants can be wrtten as follows: mn Z = WC ( = mn Z = WT ( = C R + P (3 X + X =, ; # (4 C C + M X P, ; # (5 T = max { 0, C D } (6 X 0,, ; # (7 { } The obectve functons ( and ( mnmze e total weghted flow tme and total weghted tardness respectvely. Constrant (3 ensures at e completon tme of e ob s greater an ts release tme plus processng tme. Constrant (4 specfes e order relaton when any two obs have already scheduled. Constrant (5 stpulates e completon tme relatvty of any two obs. M should be large enough for Constrant (6 so at t s always feasble. Constrant (7 specfes e tardness of each ob..4 Fuzzy Mxed-Integer Goal Programmng Model When vague nformaton related to e obectves s presented, en e problem can be formulated as a fuzzy goal-programmng problem. A typcal fuzzy mxed-nteger goal programmng problem (f-migp formulaton can be stated as follows:
3 Proceedngs of e 9 WSEAS Internatonal Conference on Appled Maematcs, Istanbul, Turkey, May 7-9, 006 (pp40-45 Fnd x =,,..,n ( ( ( to satsfy Z x Z l =,,...,L h x d =,,...,J S x = c k =,,...,K x k k 0 and nteger =,,...,n where, Z x s e l l ( ( ( h x s e S x s e k k l k l l goal constrant. nequalty constrant. equalty constrant. (8 Z s e target value of e l constrant. d c s e avalable resource of nequalty constrant. s e avalable resource of equalty constrant. In relatons gven n Eq. (8, e symbol ndcates e fuzzness of e goal. It represents e lngustc term about and t means at Z l (x should be n e vcnty of e aspraton l. The l fuzzy goal sgnfes at e decson maker wll be satsfed even for values Zl ( x Zl slghtly greater an (or lesser an up to a stated devatons sgnfed by e tolerance lmt. The system constrant h ( x d and e k system constrant S k ( x = ck are assumed to be crsp. The fuzzy set eory [3] s based on e extenson of e classcal defnton of e set. In e classcal set eory, each element of a unverse X eer belongs to a set A or not, whereas n e fuzzy set eory, an element belongs to a set A w a certan membershp degree. Defnton: A fuzzy set A n X s defned by: A = {( x, µ A ( x x X } where, µ A ( x : X [ 0,] s called e membershp functon of A and µ A( x s e degree of membershp to whch x belongs to A. By usng e approach proposed by Yang, et al. [8], e f-migp formulaton may be solved to determne e decson set and en to maxmze e set. Ths approach s based on a pecewse lnear approxmaton w e mn-operator for aggregatng e fuzzy goals. Once e membershp functons of e fuzzy obectves µ Z l ( x are known, e fuzzy optmzaton problem (f-migp formulaton s transformed nto an equvalent crsp Z formulaton (c-migp for e optmzaton problem. An equvalent crsp maematcal programmng (c- MIGP formulaton s gven as follows: max α α µ Z ( x, l l =,,...,L h ( x d, =,,...,J S k ( x = ck, k =,,...,K x 0 and nteger (9.5 Applcaton of f-migp Model An f-migp for a sngle machne schedulng problem formulaton s presented as follows: = = WC WT Z (0 Constrants (3 to (7. Z ( 3 Problem Soluton In s paper, e effectveness of e FGP technque for e sngle machne schedulng problem n a small sze s demonstrated rough a data set as shown n Table. For each ob, e processng tme, release tme, and weght of obs are chosen at random between 0 and 0. The correspondng due date s also computed by D =P (-M as gven n []. s e number of obs and M e unformly random number between 0 and. Table Input data for a small-szed problem Job P R D W The followng soluton procedure s employed to solve e above numercal example. Step. One obectve s taken at a tme and e rest of e formulaton s solved by usng e Lngo 8 software as shown n Table.
4 Proceedngs of e 9 WSEAS Internatonal Conference on Appled Maematcs, Istanbul, Turkey, May 7-9, 006 (pp40-45 Table Intermedate computatonal results Indvdual obectve functon for mnmzaton Z * Z * Total weghted flow tme 377 Z * Total weghted tardness 30 Step. Sutable membershp functons for all obectve functons are decded on e bass of ntermedate results n e solutons set of e ndvdual obectve functon. The membershp functons of e two fuzzy goals consstng of mnmzng e total weghted flow tme and e total weghted tardness of obs are constructed as gven n Eqs. ( to (3., f Z 377 ( Z 377 µ ( Z =, f 377 Z , f Z 47 (, f Z 30 ( Z 30 µ ( Z =, f 30 Z , f Z 40 (3 µ ( Z Z Fg. Membershp functon of Z max α 377 α α Constrants (3 to (7. The fnal computatonal results for e proposed model are optmal as shown n Table 3. Table 3 Fnal computatonal results Out put parameters values Degree achevement of fuzzy goal 0.76 Optmum of total weghted flow tme 389 Optmum of total weghted tardness 3 In addton, e assocated optmal sequence of obs s gven bellow: J J 5 J 4 J J 3 Table 4 shows e nput data for e large-szed problem. The sngle machne schedulng problem s solved to valdate e effectveness of e FGP approach. The ntermedate computatonal results for e ndvdual obectve functons are shown n Table 5. The membershp functons of e two fuzzy goals based on e ntermedate computatonal results of e ndvdual obectve functon are desgned as gven n Eqs. (4 and (5. µ ( Z Fg. Membershp functon of Z To solve e mult-obectve formulatons, e proposed model for e above-mentoned problem s wrtten as follows: Z Table 4 Input data for a large-szed problem Job P R D W Table 5 Intermedate computatonal results Indvdual obectve functon for mnmzaton Z * Z * Total weghted flow tme 07 Z * Total weghted tardness 04
5 Proceedngs of e 9 WSEAS Internatonal Conference on Appled Maematcs, Istanbul, Turkey, May 7-9, 006 (pp40-45, f Z 07 ( Z 07 µ ( Z =, f 07 Z 407 ( , f Z 407 Table 6 Fnal computatonal results Out put parameters values Degree achevement of fuzzy goal 0.83 Optmal of total weghted flow tme 59 Optmal of total weghted tardness 3 µ ( Z µ ( Z, f Z 04 ( Z 04 =, f 04 Z , f Z Z (5 4 Concluson Ths paper has proposed e new fuzzy mxednteger goal programmng model for a sngle machne schedulng problem w two obectves. In s paper, ese two obectves are to mnmze e total weghted flow tme and total weghted tardness smultaneously. Ths work has been done for e frst tme n solvng a b-crtera sngle machne schedulng problem. Due to e real-world stuaton and satsfacton of e decson maker for e above obectves, e proposed model s solved by a fuzzy goal programmng approach. The assocated computatonal results have been reported to show e effectveness of e proposed approach. µ ( Z Fg. 3 Membershp functon of Z The proposed formulaton of e large-szed problem s transferred and rewrtten nto e followng model: max α 07 α α Constrants (3 to (7. The fnal computatonal results of e large-szed problem are shown n Table 6. Z Fg. 4 Membershp functon of Z References: [] D.R. Sule, Industral Schedulng, PWC Publshng Company, 997. [] R. Tavakkol-Moghaddam, G. Mosleh, M. Vase and A. Azaron, Optmal schedulng for a sngle machne to mnmze e sum of maxmum earlness and tardness consderng dle nsert, Appled Maematcs and Computaton, Vol. 67, o., 005, pp [3] L.A Zadeh, Fuzzy sets, Informaton and Control, Vol. 8, 965, pp [4] R.E. Bellman and L.A. Zadeh, Decson makng n a fuzzy envronment, Management Scences, Vol. 7, 970, pp [5] H.J. Zmmermann, Descrpton and optmzaton of fuzzy systems, Internatonal Journal of General Systems, Vol., 976, pp [6] H.J. Zmmermann, Fuzzy programmng and lnear programmng w several obectve functons, Fuzzy Sets and Systems, Vol., 978, pp [7] E.L. Hannan, Lnear programmng w multple goals, Fuzzy Sets and Systems, Vol. 6, 98, pp [8] R.R. Yager, Multple obectve decson-makng usng fuzzy sets, Internatonal Journal of Man- Machne Studes, Vol. 9, 977, pp [9] H. Ish and M. Tada, Sngle machne schedulng problem w fuzzy precedence relaton, European Journal of Operatonal Research, Vol. 87, 995, pp
6 Proceedngs of e 9 WSEAS Internatonal Conference on Appled Maematcs, Istanbul, Turkey, May 7-9, 006 (pp40-45 [0] G.I. Adamopoulos and C.P. Papps, a fuzzylngustc approach to a mult-crtera sequencng problem, European Journal of Operaton Research, Vol. 9, 996, pp [] H.T. Lee, S.H. Chen and H.Y. Kang, Multcrtera schedulng usng fuzzy eory and tabu search, Internatonal Journal of Producton Research, Vol. 40, o. 5, 00, pp [] H. Ishbuch and T. Murata, Flow shop schedulng w fuzzy due date and fuzzy processng tme n R. Slowńsk and M. Hapke (Eds., Schedulng under Fuzzness, 000. [3] R. arsmhan, Goal programmng n a fuzzy envronment, Decson Scences, Vol., 980, pp [4] S.B. Snha, K.A. Rao and B.K. Mangara, Fuzzy goal programmng n mult-crtera decson systems: a case study n agrcultural plannng, Soco-Economc Plannng Scences, Vol., o., 988, pp [5] S.S. Rao, K. Sundarau, B.G. Prakash and C. Balakrshna, Fuzzy goal programmng approach for structural optmzaton, AIAA Journal, Vol. 30, o. 5, 99, pp [6] M. Kumara, P. Vratb and R. Shankar, A fuzzy goal programmng approach for vendor selecton problem n a supply chan, Computers and Industral Engneerng, Vol. 46, o. 5, 004, pp [7] S. Mshra, S. Prakash, M.K. Twar and R.S. Lashkar, A fuzzy goal-programmng model of machne-tool selecton and operaton allocaton problem n FMS, Internatonal Journal of Producton Research, Vol. 44, o., 006, pp [8] T. Yang, J.P. Ignzo and H.-J. Km, Fuzzy programmng w nonlnear membershp functons: pecewse lnear approxmaton, Fuzzy Sets and Systems, Vol. 4, pp
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