Resource-constrained Project Scheduling with Fuzziness

Transcription

1 esource-constraned Project Schedulng wth Fuzzness HONGQI PN, OBET J. WIIS, CHUNG-HSING YEH School of Busness Systems Monash Unversty Clayton, Vctora 368 USTI bstract: - esource-constraned project schedulng (CPS s often a challengng ssue n practce, due to ts combnatoral nature and uncertanty. Ths paper presents the framework of a heurstc approach to fuzzy CPS problems usng a fuzzy parallel schedulng method. Ths approach can handle both fuzzy and crsp numbers whch commonly coexst n many realstc CPS problems. Key-Words: - esource-constraned Project Schedulng; Prorty ules; Fuzzy Sets; Parallel Schedulng. Introducton In the real world, many CPS problems are often nherently uncertan due to the vagueness of actvty duraton tmes. Ths uncertanty should be consdered n many realstc CPS approaches. Tradtonally, the uncertanty was handled by stochastc approaches usng a probablstc-based PET method []. Ths knd of uncertanty s assocated wth randomness. However, n many stuatons, t s mpossble to get the dstrbuton of probabltes of actvty duraton tmes because such projects may not have been carred out prevously. The duraton tmes have to be estmated by a knowledgeable decson maker (DM. Such human expertse often ncludes ambguous nformaton, whch cannot be modelled usng the probablstc approaches. Fuzzy set theory has proven to be an effectve way of handlng such vague nformaton [2]. It can be employed to express the DM s both optmstc and pessmstc vews n the determnaton of actvty duraton tmes. In addton, CPS problems are one of the NP-hard class due to the complexty of the combnatoral nature [3, 4]. For even moderately-szed problems, obtanng an optmal soluton n reasonable computatonal tme can be very dffcult. However, heurstc-based approaches can produce a reasonable soluton to CPS problems [5, 6]. Schedulng concerned wth fuzzness s stll a new and challengng feld wth only a lmted number of publshed papers. Prade frst ntroduced the concept of fuzzy sets nto PET n 979 [7]. Followng that some papers appled ths concept further to project plannng [8, 9, ]. However, resource constrants were not taken nto account. Fu and Wang [] proposed a fuzzy resource allocaton model whch uses lnear programmng when resources are nsuffcent and Wlls et al [2] appled the fuzzy goal programmng approach n CPS, but such an approach s only sutable for small szed projects. Hapke et al. [3] adopted the smulated annealng technque n ther attempt to solve the multple objectve combnatoral optmsaton of fuzzy CPS. orterapong [4] and Hapke et al. [5] presented heurstc-based approaches applyng fuzzy sets to deal wth fuzzy actvty duraton tmes of a project. Ths approach has the advantage of beng robust, ntutve, and effcent n computaton wth respect to ts NP-hardness. Our approach s based on a smlar dea usng the fuzzy parallel schedulng method wth an effectve comparson of fuzzy numbers. In ths paper, a real case of an overhaul schedule s consdered as a fuzzy CPS problem. Most of these actvty duraton tmes have to be estmated by the DM, usng hs/her knowledge and experence. The company requres a flexble result n the overhaul schedulng whch ensures that t can be completed wthn a gven tme perod. In the remander of ths paper, we descrbe the fuzzy number arthmetc and fuzzy comparson, and then cover the fuzzy heurstc approach wth prorty rules. Fnally, we present a numercal example and come to a concluson. 2 Fuzzy arthmetc

2 et and B be fuzzy numbers, and denote any basc fuzzy arthmetc operatons such as fuzzy addton, subtracton and multplcaton. ny operaton B can be defned a fuzzy set on and expressed n the followng form [6]: η ( z = max{ mn[( µ ( x, µ ( y] } ( B B x y= z In the schedulng, two lnear approxmatons for trapezodal and trangular fuzzy numbers are used. trapezodal number s a flat fuzzy number whch can be represented by 4-tuples ( a, a2, a3, a4 as shown n Fgure, where a and a 4 are the lower and upper bounds of the support of fuzzy number, whle a 2 and a 3 are the lower and upper modal values. However, a trangular fuzzy number can be consdered a specal case of the trapezodal fuzzy number where the lower and upper modal values are same ( a2 = a3 as shown n Fgure. ll the followng fuzzy operatons are based on trapezodal numbers. Degree of Membershp. µ a a 2 (a 2 =a 3 Fgure Trapezodal and trangular fuzzy numbers If and B are defned as trapezodal fuzzy numbers, the followng arthmetc operatons wll be used n the schedulng: + B = ( a + b, a + b, a + b, a + b (2 B = ( a b, a b, a b, a b ( B = ( a b, a b, a b, a b a ( (5 max(, B = ( ( a, b, ( a, b, ( a, b, ( a, b mn(, B = ( ( a, b, ( a2, b2, ( a3, b3, ( a4, b4 (6 where +, and represent fuzzy addton, subtracton and multplcaton respectvely, and, symbolse maxmum, and mnmum operatons for fuzzy numbers respectvely. Detals regardng fuzzy arthmetc can be found n Klr and Yuan [7]. The rankng of fuzzy numbers s another mportant ssue n the schedulng. Whle many rankng methods have been proposed to date, there s no sngle approach that can produce a satsfactory result n every stuaton: some may generate counterntutve results and others are not dscrmnatve enough [8]. To overcome such problems, Cheng a 4 [9] developed a new dstance approach for fuzzy number comparsons based on the calculaton of the centrod pont ( x, y to obtan the dstance ndex, where x and y are centrod values both n the horzontal and vertcal axes respectvely. et a trapezodal fuzzy number be represented as the followng membershp functon: x a µ ( x =, a x a2 a2 a a2 x a3 µ = (7 a4 x µ ( x = a x a 3 4 a4 a3 otherwse where µ :[ a, a ] [, ] 2 s the strctly contnuous left spread, and ts correspondng nverse functon s denoted by g ( x. µ ( x:[ a, a ] [, ] 3 4 s the strctly contnuous rght spread, and ts correspondng nverse functon s symbolsed by g ( x. ll the functons can be ntegrable due to ther contnuty. Therefore, the centrod pont ( x, y of a fuzzy number can be defned n the followng form: a2 3 4 [ µ ] + + [ µ ] x ( x dx xdx x ( x dx x ( a a 2 a3 = a2 a3 a4 µ ( x dx + dx + µ ( x dx a y ( = a a2 [ $ ( ] + ( ] yg y dy yg y dy g ( y dy + g ( y dy [ a a3 The rankng ndex can be expressed as: ( = ( x + ( y 2 2 (8 (9 For trapezodal fuzzy numbers, formula (8 of calculatng the centrod pont can be smplfed as: a4 + a3 a a2 + a3 a4 a a2 x = 3 ( a + a a a a + 2 a2 + 2 a3 + a4 y = 3 ( a + a2 + a3 + a4 ( However, for trangular fuzzy numbers, the value of a 2 s the same as that of a 3 due to the same modal value when applyng formula (. Where, j are any fuzzy numbers n set, the comparson of fuzzy numbers has the followng propertes when obtanng rankng ndces by formula (9: 2

3 ( f ( > ( j, then > j, (2 f ( = ( j, then = j, (3 f ( < (, then <. j 3 Fuzzy heurstcs Bascally, heurstc approaches for solvng CPS problems have fve methodologes: sngle and multple pass prorty rule-based schedulng [2, 2]; truncated branch and bound procedures [5]; dsjunctve arc concepts [22]; local search technques [23]; and nteger programmng based heurstcs [24]. lthough, a prorty rule-based approach s one of the oldest heurstcs, ths approach s stll an effcent technque for solvng PCSP problems. It s easy to use and fast n computatonal effort [6]. In the prorty rule-based approach, there are two basc schedulng methods: seral and parallel. In the seral method, only one actvty s selected and scheduled n each stage. In the parallel method, however, actvtes whose predecessors have been completed, can be consdered to be scheduled n a stage dependng on the avalablty of resources. In ths paper, the parallel method s used because t seems to work better [5]. There are two versons of the parallel method: Kelley s algorthm [25] and Brooks algorthm [26]. Our schedulng method s based on Brooks algorthm (BG. In the fuzzy parallel method, we assume that there are, at most J stages, n each of whch a set of actvtes s scheduled. Four sets are defned n the fuzzy schedulng scheme. complete set C( t n s a set of actvtes that were scheduled and are completed up to the schedule tme t n. ctvtes whch have already been scheduled, but have not been completed at that scheduled tme t n, are defned n the actve set ( t n. ctvtes whch have not been scheduled and whose mmedate predecessors have been completed by t n, are n the decson set D( t n. However, other remanng actvtes not belongng to the above three sets are n the remanng set ( t n. The partal schedule of each stage s composed of actvtes that are n the complete and actve set. The order of actvtes n the decson set s determned by a specfc prorty rule. The schedule tme of a stage s equal to the earlest completon tme of actvtes n the actve set belongng to the prevous stage. j In order to descrbe the parallel schedulng method, the followng notaton s ntroduced: Z s a group of actvtes n a gven project; d represents the fuzzy duraton of an actvty Z ; π r and k r denote the left-over capacty and an actvty Z s resource requrement, respectvely, of the renewable resource r. P and S are, respectvely, a set of actvtes mmedately precedng and succeedng actvty Z. ST and FT express the fuzzy start and fuzzy fnsh tmes of actvty Z. The general scheme of the fuzzy parallel schedulng can be presented n the followng form: INITIISTION: n: =, : =, D( : = {}, π r : = r r ( = C( : = φ, ( : = { Z} GOTO Step (2 DO WHIE n <J DO STGE n BEGIN ( : = mn{ FT Z ( } ( : ( \ { Z Z ( =, FT } C( t : C( = t { Z FT t } n n n πk = K k, r r r r ( t n D( t = { Z Z { C( t ( t, P C( t n n n n ( t = Z \ { C( t ( t D( t } n n n n (2 Orderng the prorty lst n D( t n DO WHIE kr π Kr ST : = FT : = ST + d ( : = ( Z Update πk r, D( t, and ( t n: = n + END r n n The tradtonal crtcal path method (CPM s an effectve tool frequently used to dentfy a crtcal path of a project whle solvng CPS problems. In the fuzzy CPM, fuzzy operatons are employed to calculate the fuzzy early start ( EST and the fuzzy late fnsh ( FT tmes of an actvty Z through the forward and backward pass n the project network: 3

4 EST = max[ EST + d ] Z P j j j ( FT = mn[ ST ] (2 Z S j j EFT = mn[ FT d ] Z S j j j (3 where EFT and ST represent the early fnsh and late start tmes of actvty Z. In the backward pass calculatons, t s mportant to note that some lower bounds of ST and FT may be negatve when fuzzy subtractons are taken. For practcal purposes, the negatve numbers have no physcal meanng, and these negatve numbers should be changed to zero. Heurstcs based on prorty rules have been tested by many researchers [5, 6, 2, 27]. However, there s no sngle prorty heurstc that always gves a better result n every schedulng stuaton. It s useful to apply a set of prorty rules smultaneously, to select the best result among them. In the scheme, popular prorty rules are employed, takng nto account tme, resources and the successors of current scheduled actvty.. Fuzzy early start tme ( EST : mn EST D ( t 2. Fuzzy early fnsh tme ( EFT : mn EFT D ( t 3. Fuzzy late start tme ( ST : mn ST D ( t 4. Fuzzy late fnsh tme ( FT : mn FT D ( t 5. Mnmum fuzzy slack ( MFSK : mn ST EST D ( t 6. Greatest fuzzy resource demand ( GFD : max d k D ( t r r 7. Shortest fuzzy processng tme ( SPT : mn d D ( t 8. ongest fuzzy processng tme ( PT : max d D( t 9. Most mmedate successors ( MIS : max S D( t. east mmedate successors ( IS : mn S D ( t In order to employ these rules, the fuzzy early and late start tmes, and the fuzzy early and late fnsh tmes for each actvty can be calculated by Equaton Numercal Example The project presented here s a real overhaul schedulng problem and s composed of 4 actvtes. The precedence relatons among actvtes s represented by a precedence network, as shown n Fgure 2, where SP and EP are the dummy actvtes representng the begnnng and end of the project SP Fgure 2 The Precedence dagram EP In the schedulng, three dfferent knds of renewable resources are requred throughout the project. However, the amount of these resources s lmted. The avalabltes of resources, 2 and 3 are 2, 8 and 9 respectvely n each perod. Due to the nature of uncertanty n many actvtes, these duraton tmes are estmated by trangular or trapezodal fuzzy numbers, and only a few duraton tmes of some actvtes can be determned as crsp numbers. The data of the project are lsted n Table where n represents the n th renewable resource. Table Duraton tmes and resource requrements ctvty No. Duraton 2 3 (5,8,2, (,2,5, (8,, (2,25,3, (,3,6, (6,9,5 2 7 (5,8,, (8,22,25, (,5,2, (5,8,2 2 2 (8,,2,

5 3 (4,6, The algorthm has been wrtten usng objectorented programmng language VB6. The mddle resultng data requred by the prorty rules are lsted n Tables 2 and 3. Table 2 The st part of data requred by rules ctvty No EST EFT ST (5,8,2,25 2 (,2,5,2 (,9,28,53 3 (8,,,5 (,27,4,66 4 (5,8,2,25 (35,43,5,6 (5,8,2,25 5 (,2,5,2 (2,25,3,4 (3,24,4,63 6 (,2,5,2 (6,2,24,35 (8,3,44,67 7 (8,,,5 (3,8,2,3 (4,37,5,74 8 (35,43,5,6 (53,65,75,9 (35,43,5,6 9 (2,25,3,4 (3,4,5,65 (28,45,64,8 (3,8,2,3 (8,26,28,42 (36,52,62,8 (3,8,2,3 (2,28,32,45 (29,47,59,79 2 (8,26,28,42 (23,3,33,47 (48,6,7,86 3 (2,28,32,45 (25,34,38,54 (44,59,69,87 4 (53,65,75,9 (63,75,85, (53,65,75,9 Table 3 The 2 nd part of data requred by rules ctvty No FT ST FT d (5,8,2,25 (2,44,6,2 2 (3,24,4,63 (,9,28,53 (6,72,9,2 3 (4,37,5,74 (,27,4,66 (32,4,4,6 4 (35,43,5,6 (6,2,24,288 5 (23,4,53,73 (,9,28,53 (6,78,96,2 6 (23,4,53,73 (,6,32,57 (8,27,27,45 7 (29,47,59,79 (,27,4,66 (25,4,5,75 8 (53,65,75,9 (44,76,2,24 9 (53,65,75,9 (,4,39,6 (6,9,2,5 (48,6,7,86 (6,32,44,68 (25,4,4,6 (44,59,69,87 (,27,4,66 (4,5,6,75 2 (53,65,75,9 (6,32,44, (53,65,75,9 (,27,4,66 (6,24,24,36 4 (63,75,85, 2 The tables show that some actvtes n the decson set D( t may have the same prorty values n a specfc prorty rule and generate a group of dfferent prorty lsts. For nstance, there are 24 dfferent combnatons of schedulng sequences, creatng 24 dfferent feasble schedules n the EST rule. However, the objectve of the k r schedulng s to mnmse the overhaul completon tme. By runnng the developed software usng the fuzzy parallel method, the shortest project completon tmes from each ndvdual rule are selected, as shown n Table 4, n whch we can see that n ths partcular overhaul schedulng, the EST, FT, SPT and IS rules produce better results than the others, and the fuzzy values of the project completon tme are qute close. But, when applyng the fuzzy number rankng method, the shortest project completon tme s ganed by the SPT rule, by whch each actvty s start and fnsh tme s dsplayed n Fgure 2. It provdes the DM wth detaled nformaton about every moment of the overhaul performance, and wth a reasonable schedulng result under the vague nformaton of actvty duraton tmes. Table 4 Project completon tmes under rules Heurstc ule Schedulng Sequence Project Completon Tme EST,2,3,7,5,6,4,,,2,9,3,8,4 EFT 3,2,,7,6,,5,, 2,3,9,4,8,4 ST,4,2,5,3,6,7,8,,9,,3,2,4 FT,2,3,4,6,5,7,,,8,9,3,2,4 MFSK,4,8,2,5,6,9,3, 7,,3,,2,4,4,8,2,5,6,9,3, GFD 7,,,3,2,4 3,2,,7,6,5,4,, SPT,9,8,2,3,4 PT,4,8,2,5,6,3,9,7,,,3,2,4,3,4,8,2,5,6,9,7, MIS,,3,2,4 2,3,,7,4,5,6,, IS,9,8,2,3,4 (68, 82, 96, 6 (76, 94, 7, 45 (76, 94, 5, 33 (6, 8, 93, 22 (, 25, 39, 78 (94, 8, 32, 7 (64, 78, 89, (9, 2, 29, 65 (87, 7, 2, 5 (69, 84, 94, 6 5 Concluson Practcal CPS s often a complex problem due to ts NP-hardness and uncertanty. Its combnatoral nature s dffcult to handle usng the exstng exact algorthms. Whle heurstc approaches appear to be attractve, they gve reasonable practcal solutons, tradtonal heurstc approaches suffer the major shortcomng of beng unable to handle such uncertanty. 5

6 Ths paper presents a fuzzy CPS heurstc approach that ncorporates fuzzy set theory to model the uncertan actvty duraton tmes estmated by the DM. To obtan a good soluton for the mnmsaton of the project completon tme, a group of prorty rules s employed n a fuzzy parallel schedulng method. Ths approach s smple and straghtforward n practcal applcatons. Therefore, ths study provdes the framework of a heurstc approach for solvng CPS problem nvolvng uncertan actvty duraton tmes modelled by fuzzy numbers. ctvty 4 ST 4 FT 4 ctvty 3 ST 3 FT 3 ctvty 2 ST 2 FT 2 ctvty ST FT ctvty ST FT ctvty 9 ST 9 FT 9 ctvty 8 ST 8 FT 8 ctvty 7 ST 7 FT 7 ctvty 6 ST 6 FT 6 ctvty 5 ST 5 FT 5 ctvty 4 ST 4 FT 4 ctvty 3 ST 3 FT 3 ctvty 2 ST 2 FT 2 ctvty ST FT Fgure 2 Fuzzy actvty start and fnsh tmes n SPT tme eference: [] D.G. Malcolm, J.H. oseboom, C.E. Clark and W. Fazar, pplcatons of a Technque for esearch and Develpment Program Evaluaton, Operatons esearch Vol.7, 959, pp [2] C. Carlsson, On the elevance of Fuzzy Sets n Management Scence Methodology, TIMES/ Studes n the Management Scences 2,984, pp.-28, Eds. H. -J. Zmmermann,.. Zadeh and B. G. Ganes, Elsever Netherlands. [3] M. Garey and D. Jonhon, Computers and Intractablty: Gude to the Theory of NP- Completeness. Freeman, SanFrancsco, C, 979. [4] J. Blazewcz, J.K. enstra and.g.h. nnooy Kan, Schedulng Subject to esource Constrants: 6

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