Scheduling problem with uncertain parameters

Transcription

1 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters Schedung probem wth uncertan parameters by Wojcech Bożejo 1,3, Paweł Rajba 2, Meczysław Wodec 2,3 1 Wrocław Unversty of Technoogy, Poand, 2 Unversty of Wrocław, Poand, 3 Coege of Management "Eduacja" Wrocław, Poand Correspondng author: Meczysław Wodec Insttute of Computer Scence, Unversty of Wrocław u. Joot-Cure 15, Wrocław, Poand phone: (+48 71) e-ma: mwd@.un.wroc.p BSTRCT In the paper we consder a strong NP-hard snge-machne schedung probem wth deadnes and mnmzng the tota weght of ate jobs on a snge machne. Processng tmes are determnstc vaues or random varabes havng norma dstrbutons. For ths probem we study the toerance to random parameter changes for soutons constructed accordng to tabu search metaheurstcs. We aso present a measure (caed stabty) that aows an evauaton of the agorthm based on ts resstance to random parameter changes. Keywords: schedung, weght tardness, norma dstrbuton, tabu search, stabty. 1. INTRODUCTION In many appcatons serous dffcutes occur whe ndcatng parameters or when the data comes from naccurate measurement equpment. Due to short reazaton terms, short seres and producton eastcty there are no comparatve data and no possbty to conduct expermenta studes that woud enabe one to determne expct vaues of certan parameters. Furthermore, n many economy branches e toursm, agrcuture, commerce, budng ndustry, etc., the processes that occur have by ther nature random character (they depend on weather, maret condtons, accdents, etc.). Mang decsons n the condtons of uncertanty (ac of exact vaues of parameters) becomes quotdan. Pobems of tang decsons under uncertanty are soved by appcaton of probabstc method or through fuzzy sets theory. In the frst case (Dean [3], Vondrá [18]) nowedge of Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

2 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters dstrbuton of random varabes s of cruca mportance. Some pocesses are charactersed wth randomty by nature. They depend on weather condtons, traffc ntensty, number of accdents, geoogca condtons, devce s faure, etc. If they, nevertheess, posses certan hstory, t s possbe to defne ther dstrbuton on the bass of statstca data. In many ssues the uncertanty of data s not of random nature but t resuts from unqueness of a process, errorr n measurement, etc. In such a case a natura method of representng uncertanty are fuzzy numbers (Iscbuch et a. [6], Ish [7]). In ths case a huge probem s posed by a proper choce of membershp functon and deffuzfcaton method. They have cruca nfuence on the quaty of taen soutons. In ths paper we examne a schedung probem on a snge machne wth the atest possbe processng tmes and cost mnmzng for the beated jobs. The deays needed to accompsh the jobs are determnstc or random varabes wth norma dstrbuton. In ths case we study the resstance to random parameter changes on soutons constructed accordng to the tabu search metaheurstcs. We aso present a certan measure (caed stabty) that aows one to evauate the resstance of soutons to random data perturbatons. 2. PROBLEM DEFINITION In ths paper we consder the schedung probem on a snge machne. The machne can perform ony one job at a tme. For job ( 1 n ), et p, w, d be: the processng tme, a weght functon of costs and the deadne expected. If for a gven sequencng the deadne of job exceeds d, the deay U s 1, f not, U s 0. The probem of mnmzng the tota weght of ate jobs on a snge machne (TWLJ) conssts n fndng a job sequence that n mnmzes the sum of deay costs,.e. wu 1. The probem can be wrtten as 1 wu, and though ts formua s so smpe, t s NP-hard (Karp [8]). Such probems have been studed for qute ong together wth many varatons, especay wth poynoma computatona compexty. For the probem 1 p 1 wu (a the processng tmes are dentca) Monma [13] has presented an agorthm wth On ( ) compexty. Smary, for the probem 1 w c U, (where the cost functon factors are dentca) there s the Moore agorthm [14] wth O( nn n ) compexty. Lawer [11] has adapted the Moore agorthm to sove the probem Probems wth the earest startng tmes compose another 1 p p j w wj wu Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

3 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters group r. Kse et a. [9] have proven that even the probem of ate jobs mnmzaton (1 r U wthout the cost functon weght) s strongy NP-hard. They have aso presented a poynoma agorthm that has computatona compexty 2 On ( ) for a partcuar exampe, the 1 r r j d d j U probem. If a parta order reaton s gven on the set of jobs, the TWLJ probem s strongy NP-hard even when the job reazaton tmes are untes. Lenstra and Rnnoy Kan [12] have proven that f a parta order reaton s a unon of ndependent chans, the probem s aso strongy NP-hard. There have been ony a few exact agorthms sovng the TWLJ probem pubshed. They are based on the dynamc programmng method (Lawer and Moore [10] wth O( nmn{ p max{ d}}) compexty and Sahn [17] wth O( nmn{ p w max{ }} d )) compexty and on a mtaton and dvson method (Varrea and Bufn [18], Potts [15], Potts and Van Wassenhowe [14], Bożejo, Grabows and Wodec [1], Bożejo and Wodec [2] and Wodec [20]). The ast one s a parae agorthm. The schedung probem on a snge machne can be formuated as foows: The probem: There s a set J {1 2 n } of jobs that have to be processed wthout nterruptons on a machne that can wor on one job at a tme. The job can start at tme zero. For job J et p be the processng tme, d the expected deadne, and weght. We want to determne a job sequence that mnmzes the weght of ate jobs. For a gven sequence et the cost (penaty) of a ate job, where w costs functon C be the date of accompshng of job J. Then f ( C ) wu s 0, f C d, U () (1) 1, otherwse. Let be the set of permutatons of J. The cost of the permutaton s a defned as foows: n ( ) ( ) 1 W ( ) w U (2) where C( ) p ( ) s the processng tme of the job () J. The probem of j1 mnmzng the tota weght of ate jobs (TWLJ) bos down to fndng an optma permutaton whch satsfes Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

4 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters n ( ) ( ) 1 W ( ) mn w U Exact effcent agorthms to sove the TWLJ probem ony exst when the number of jobs does not exceed 50 (80 n a mutprocessor envronment Wodec [20]). That s why n practce we use approxmate agorthms (essentay the correcton type). 3. TBU SERCH METHOD In sovng NP-hard probems of dscrete optmzaton we amost aways use approxmate agorthms. The soutons gven by these agorthms are satsfactory appcatons (they often dffer from the best nown soutons by ess than 1a search methods. The tabu search method ( - proposed by Gover [4] and [5]) s a metaheurstc approach desgned to fnd a near-optma souton of combnatora optmzaton probems. The basc verson of starts from an nta souton x 0 The eementary step of the method performs, for a gven souton x a search through the neghborhood ( ) Nx of x The neghborhood ( Nx ) s defne by move (transtons) performed from x move transforms a souton nto another souton. The am of those eementary search s to fnd n Nx ( ) a souton 1 x wth the owest cost functons. Then the search repeats from the best found, as a new startng souton, and the process s contnued. In order to avod cycng, becomng trapped to a oca optmum, and more genera to conduct the search n good regons of the souton space, a memory of the search hstory s ntroduced. mong many casses of the memory ntroduced for tabu search (see. Gover [4]), the most frequenty used s the short term memory, caed the tabu st. Ths st recorded, for a chosen span of tme, soutons or seected attrbutes of these soutons (or moves).the search stops when a gven number of teratons or current neghborhood s empty In ths secton we present some propertes whch are the base of a new neghborhood s constructon and, further, very fast tabu search agorthm CLSSIC TBU SERCH LGORITHM In sovng NP-hard probems of dscrete optmzaton we amost aways use approxmate agorthms. The soutons gven by these agorthms are, n ther appance, fuy satsfyng (they often dffer from the best nown soutons by ess then 1%). Most of them beong to the oca search methods group. Ther actng conssts n vewng n sequence a subset of a set of Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

5 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters acceptabe soutons, and n pontng out the best one accordng to a determned crteron. One of ths method reazatons s the tabu search, whose basc crterons are: neghborhood - a subset of a set of acceptabe soutons, whose eements are rgorousy anayzed, move - a functon that converts one souton nto another one, tabu st - a st contanng the attrbutes of a certan number of soutons anayzed recenty, endng condton - most of the tme fxed by the number of agorthm teratons. Let be any (startng) permutaton, L a tabu st, W costs functon, and the best souton found at ths moment (the startng souton and gorthm Tabu Search () repeat Determnate the neghborhood N( ) of the permutaton ; Remove from N( ) the permutatons forbdden by the L st; Determnate the permutaton N( ), n whch W( ) = mn{ W( ): N( )}; f ( W( ) < W( ) ) then := ; Incude parameters on the L st; := unt (endng_condton). can be any permutaton). The computatona compexty of the agorthm depends mosty on the way the neghborhood s generated and vewed. Beow we present n detas the bascs eements of the agorthm THE MOVE ND THE NEIGHBORHOOD Let = ( (1),, ( n )) be any permutaton from the, and a set of ate jobs n. L( ) ={ ( ) : C > d }, ( ), m ( ), m Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

6 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters By ( =1,2,, 1, 1,, n ) we mar a permutaton receved from by changng n the eement ( ) and (). We can say at that pont that the permutaton was generated from by a swap move (s-move) s (t means that the permuta-ton = s ( ) ). Then, et M( ( )) be a set of a the s-moves of the ( ) eement. By M ( ) = M ( ( )), ( ) L( ) we mean an s-moves set of the ate eements n the pemutaton. The power of the set M ( ) s top-bounded by nn ( 1) / 2. The neghborhood s the permutaton set N( ) = s ( ) : s M ( ). Whe mpementng the agorthm, we remove from the neghborhood the permutatons whose attrbutes are on the forbdden attrbutes st L TBU LIST To prevent from arsng cyce too qucy (returnng to the same permutaton after some sma number of teratons of the agorthm), some attrbutes of each move are saved on socaed tabu st (st of the prohbted moves). Ths st s served as a FIFO queue, see r Bożejo, Grabows and Wodec [1]. Mang a move M ( ) (that means generatng permutaton r from ) j we save attrbutes of ths move, trpe ( ( ) ( r r j F )), on the tabu st. Let us assume that we consder a move M ( ) whch generates permutaton If there s a trpe ( r j ) such that ( ) r j on the tabu st, and F ( ) then such a move s emnated (removed) from the set M ( ) The dynamc ength of tabu st L s a cycc functon defned by the expresson: ow, f S( ) ter S( ) h, ( ter) ow, f S( ) h ter S( 1), where: ter s the number of teraton of the agorthm, 12 s the number of the cyce. Integer number 0, S( ) ( 1)( h H ), here S (0) 0. Low s the standard ength of the L st (by h teratons of the agorthm) and H s the wdth of the pc equa ow. j j Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

7 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters If decreases then a sutabe number of the odest eements of tabu st L s deeted and the search process s contnued. parameters of ength of the tabu st are emprca, based on premnary experments. Changng the tabu st s ength causes dversfcaton of the search process. 4. PROBBILISTICS JOBS TIMES Let = p, d, w be an exampe of determnstc data for TWLJ probem. We assume that tmes of executon of jobs J are ndependent random varabes wth a norma dstrbuton,.e. p ~ N( p, ). The expected vaue of tmes E( p ) p. Then data p, d, w, where p [ p ] 1,2,..., n s a matrx of random varabes, we ca a probabstc data, and the probem - probabstc (TWLJP n short). Let be some sequence of jobs executon at objects. In order to smpfy the cacuatons we assume that moments of competon of separate wors have aso a norma dstrbuton 2 C ~ N( m, ). The equvaent of deays (1) are random varabe U () j j 1 j1 1, f C > d, ( ) ( ) () = 0, f C( ) d( ). The mean of random varabe U, () 1,2,..., n, E( U ) x P( U x) ( ) ( ) x d m 0 P( C d ) 1* P( C d ) 1 ( ), ( ) j1 ( j) ( ) ( ) ( ) ( ) 2 j1 ( j) where s dstrbuant of random varabe wth a norma dstrbuton N(0,1). By sovng a TWLJ probem (wth a random tmes of jobs executon) for a cost functon (4) we assume j FP( ) E( F( )) E( w U ) n 1 ( ) ( ) n n d( ) m j1 ( j) w ( ) E U ( ) w ( ) j1 ( j) ( ) (1 ( ). (3) Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

8 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters Tabu search agorthm of sovng TWLJP probem (wth a goa functon (3)) we ca a probabstc one, P n short. 5. SUSTINBILITY OF LGORITHMS Sustanabty s some property whch enabes estmatng of the nfuence of data perturbaton on changes of goa functon vaues. We present a method of generatng a set of nstances as the fst prorty. Let = p, d, w, where: p = [ p ] 1,2,..., n, d = [ d ] 1,2,..., n and w = [ w ] 1,2,..., n are respectvey, the matrx: of wor executon tmes, competon tmes and penaty coeffcent, w be some nstances of (determnstc) data for TWLJ probem. By D ( ) we denote a set of data generated from through perturbaton of tme executon. Ths perturbaton conssts n changng of p = [ p ] 1,2,..., n, eements nto randomy determned vaues (.e. numbers generated n accordance wth certan dstrbuton, for nstance monotonous, etc.). ny eement of D ( ) set taes form of p', d, w where perturbed eements of matrx p p are determned randomy. Thus, D( ) set ncudes nstances of determnstc ' ' = [ ] 1,2,..., n, data for TWLJ probem, dfferent from one another ony by vaues of jobs executon tmes. Let {, P }, where and P agorthms are: determnstc, fuzzy and probabstc respectvey. By we denote a souton (permutaton) determned by agorthm for data. The vaue of expresson W ( ) s cost (4) for an nstance of determnstc data, when objects are executed n a sequence of (permutatons) (.e. n a sequence defned by agorthm for ) data. Then 1 W( ) W( ) ( D( )) D( ) ( ) D( ) W We ca a sustanabty of souton determned by agorthm on a set of D ( ) perturbed data. Determnng next, for a startng souton of agorthm was denoted and W( ) W ( ) 0. Thus, ( D ( )) 0. The vaue of expresson ( D( )) s an average reatve devaton of the best souton for the best set soutons, for every nstances of perturbed data D ( ). Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

9 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters Let be some set of determnstc nstances for TWLJ probem. Sustanabty coeffcent of agorthm on a set we defne as foows: 1 S( ) ( D ( )) (4) The smaer the coeffcent, the more sustanabe the soutons set by agorthm.e. sma changes n vaue of data cause sma changes of goa functon vaue. 6. NUMERICL EXPERIMEN The agorthms presented n ths paper have been tested on many exampes. Determnstc data for one machne probem wth deay cost mnmzaton 1 wt were generated n a randomzed way (see [20]), and are avaabe on the OR-Lbrary. For a gven number of n jobs ( n ,250,500) we have determned n trpes ( p w d ), 1 n, where the processng tme p and the cost w are the reazaton of a random varabe wth a unform dstrbuton, respectvey from the range [1, 100] and [1, 10]. Smary, the crtc nes are drawn from the range [ P(1 TF RDD 2) P(1 TF RDD 2)] dependng on the n parameters RDD TF , whe P p 1. For every coupe of parameters RDD, TF (there are 25 such coupes) 5 exampes have been generated. The whoe determnstc data set contans 525 exampes (125 for every n ). For every determnstc data exampe ( p w d ), 1 n, we have defned a probabstc data exampe ( p, wd ), 1 n, where p s a random varabe wth norma dstrbuton representng the processng tme (the exact descrpton n Secton 4). We denote the set of exampes by. The determnstc and probabstc P agorthms were started from dentty permutaton. Moreover, we have adopted the foowng parameters: 1. dynamc ength of tabu st ((ter): h n /4, H n /10, ow n, / 4 n, 2. the maxmum number of agorthm teratons: n 2 or n. The determnstc agorthm has been performed on, and the probabstc agorthm P on. In order to evauate the stabty coeffcent (4) of both agorthms, 100 exampes of perturbed data have been generated for every determnstc data exampe from (we have presented the way of generatng these exampes n Secton 4). Then, a these Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

10 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters exampes have been soved by the agorthm. Based on these cacuatons, we have determned the stabty coeffcent of both agorthms. The resuts are presented n Tabes 1. Tabe 1. Stabty coeffcent (reatve average error S (, ) ) for n /2 and n teratons. Number of jobs Determnstc agorthm Probabstc agorthm P n n 2 teratons n teratons n 2 teratons n teratons 40 0,094 0,111 0,033 0, ,118 0,139 0,042 0, ,289 0,303 0,046 0, ,403 0,362 0,053 0, ,415 0,487 0,076 0,094 avg. 0,261 0,280 0,050 0,062 The average stabty coeffcent ( n 2 teratons) for the determnstc agorthm s, S( ) and for the random agorthm S( P) Ths means that the perturbaton of the souton determned by the agorthm causes a target functon vaue deteroraton of about 26%. In the P agorthm the deteroraton s ony about 5%. So the medum error for the determnstc agorthm s more than 5 tmes that for the probabstc agorthm. Tabe 1 contans too the resuts of a two tmes bgger for n teratons. The fact that the stabty of both agorthms has sghty deterorated s a tte surprsng. The stabty dfference s more advantageous for the random agorthm n n 2 teratons, even f ths agorthm s st sgnfcanty more stabe than the determnstc one. In ths case the medum error of the agorthm s more than 10 tmes that for the P agorthm. Moreover, the data perturbaton causes (for the souton determned by the probabstc agorthm) a target functon vaue deteroraton of around 6,2%. We have aso made cacuatons for more teratons ( nog n, 2 n ). The stabty coeffcent for both agorthms have sghty deterorated, as we as the stabty dfference between the determnstc and the probabstc agorthm (even f the random agorthm mantans ts superorty). The number of n 2 teratons n the tabu search method s very sma (usuay we mae 2 n teratons). Based on the resuts obtaned, we can say that n ths Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

11 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters case t s not ony suffcent, but even optma. For ths reason the medum cacuaton tme for one exampe, on a persona computer wth a 2,6 GHz Pentum processor s very short and does not exceed one second. The experments conducted have shown wthout doubt that soutons determned by the probabstc agorthm are very stabe. The perturbaton (change) of the processng tme causes a medum deteroraton of a few percent (maxmum about a 11%). From the pont of vew of ts utty n practce, ths s competey satsfactory. 7. CONCLUDING REMRKS In ths paper we have presented a method of modeng uncertan data usng random varabes wth norma dstrbuton. We have presented an agorthm based on the tabu search method n order to sove a certan schedung probem on a snge machne. For ths probem, we have evauated the souton stabty, whch means ts resstance to random changes n job parameters. The experments have shown that the agorthm n whch the processng tmes are random varabes wth norma dstrbuton, s very stabe. The medum reatve error for perturbed data does not exceed 5% when the teraton number s sma, and the cacuaton tme s short. References [1] Bożejo W., Grabows J., Wodec M., Boc approach-tabu search agorthm for snge machne tota weghted tardness probem, Computers & Industra Engneerng, Esever Scence Ltd,. Voue 50. Issue1/2, 2006, [2] Bożejo W., Wodec M., parae metaheurstcs for the snge machne tota weghted tardness probem wth sequence-dependent setup tmes, Proceedngs of the 3rd Mutdscpnary Internatona Schedung Conference: Theory and ppcatons (MIST 2007, Pars), [3] Dean B.C., pproxmaton agorthms for stochastc schedung probems, PhD thess, MIT, [4] Gover F., Tabu search. Part I.ORS Journa on Computng, 1, 1989, [5] Gover F., Tabu search. Part II.ORS Journa on Computng, 2, 1990, [6] Ishbusch H., Murata T., Schedung wth Fuzzy Duedate and Fuzzy Processng Tme, n: Schedung Under Fuzzness, R.Słowńs and M.Hape (eds), Sprnger-Verag, 2000, [7] Ish H., Fuzzy combnatora optmzaton, Japanese Journa of Fuzzy Theory and Systems, Vo. 4, no. 1, 1992, [8] Karp R.M., Reducbty among Combnatora Probems, Compexty of Computatons, R.E. Merand and J.W. Thatcher (Eds.), Penum Press, New Yor, 1972, [9] Kse H., Ibara T., Mne H., sovabe case of the one-machne schedung probem wth ready tmes and due tmes, Operatons Research, 26, 1978, [10] Lawer E.L., Moore J.M., Functona equaton and ts appcatons to resource aocaton and sequencng probems, management scence, 16, 1969, [11] Lawer E.L., "pseudoponoma" agorthm for sequencng jobs to mnmze tota Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012

12 Bożejo W., Rajba P., Wodec M. Schedung probem wth uncertan parameters tardness, nnas of Dscrete Mathematcs, 1, 1977, [12] Lenstra J.K., Rnnoy Kan.H.G., Compexty resuts for schedung chans on a snge machne, European Journa of Operatona Research, 4, 1980, [13] Monma C.I., Lnear-tme agorthms for schedung on parae processor, Operatons Research, 30, 1982, [14] Moore J.M., n n-job, one machne sequencg agorthm for mnmzng the number of ate jobs, Menagement Scence, 15, 1968, [15] Potts C.N., Van Wassenhove L.N., Branch and Bound gorthm for the Tota Weghted Tardness Probem, Operatons Research, Vo. 33, 1985, [16] Potts C.N., Van Wassenhove L.N., gorthms for Schedung a Snge Machne to Mnmze the Weghted Number of Late Jobs, Management Scence, 34, 7, 1988, [17] Sahn S.K., gorthms for Schedung Independent Jobs, J.ssoc. Comput. Match., 23, 1976, [18] Varea F.J., Bufn R.L., Schedung a Snge Machne to Mnmze the Weghted Number of Tardy Jobs, IEE Trans., 15, 1983, [19] Vondrá J., Probabstc methods n combnatora and stochastc optmzaton, PhD, MIT, [20] Wodec M., Branch-and-Bound Parae gorthm for Snge-Machne Tota Weghted Tardness Probem, dvanced Manufacturng Technoogy, 37, 2007, Proceedngs of the Internatona Conference on ICT Management (ICTM 2012), Wrocaw, Poand September 17-18, 2012