Mathematical modeling and multi-objective evolutionary algorithms applied to dynamic flexible job shop scheduling problems Shen, Xiao-ning; Yao, Xin

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1 Mathematca modeng and mut-objectve evoutonary agorthms apped to dynamc fexbe job shop schedung probems Shen, Xao-nng; Yao, Xn DOI: /j.ns Lcense: Other (pease specfy wth Rghts Statement) Document Verson Peer revewed verson Ctaton for pubshed verson (Harvard): Shen, X & Yao, X 2015, 'Mathematca modeng and mut-objectve evoutonary agorthms apped to dynamc fexbe job shop schedung probems' Informaton Scences, vo. 298, pp DOI: /j.ns Lnk to pubcaton on Research at Brmngham porta Pubsher Rghts Statement: NOTICE: ths s the author s verson of a work that was accepted for pubcaton. Changes resutng from the pubshng process, such as peer revew, edtng, correctons, structura formattng, and other quaty contro mechansms may not be refected n ths document. Changes may have been made to ths work snce t was submtted for pubcaton. A defntve verson was subsequenty pubshed as X-N. Shen, X. Yao, Mathematca modeng and mut-objectve evoutonary agorthms apped to dynamc fexbe job shop schedung probems, Informaton Scences (2014), do: /j.ns Genera rghts Uness a cence s specfed above, a rghts (ncudng copyrght and mora rghts) n ths document are retaned by the authors and/or the copyrght hoders. The express permsson of the copyrght hoder must be obtaned for any use of ths matera other than for purposes permtted by aw. Users may freey dstrbute the URL that s used to dentfy ths pubcaton. Users may downoad and/or prnt one copy of the pubcaton from the Unversty of Brmngham research porta for the purpose of prvate study or non-commerca research. User may use extracts from the document n ne wth the concept of far deang under the Copyrght, Desgns and Patents Act 1988 (?) Users may not further dstrbute the matera nor use t for the purposes of commerca gan. Where a cence s dspayed above, pease note the terms and condtons of the cence govern your use of ths document. When ctng, pease reference the pubshed verson. Take down pocy Whe the Unversty of Brmngham exercses care and attenton n makng tems avaabe there are rare occasons when an tem has been upoaded n error or has been deemed to be commercay or otherwse senstve. If you beeve that ths s the case for ths document, pease contact UBIRA@sts.bham.ac.uk provdng detas and we w remove access to the work mmedatey and nvestgate. Downoad date: 16. Oct. 2018

2 Accepted Manuscrpt Mathematca modeng and mut-objectve evoutonary agorthms apped to dynamc fexbe job shop schedung probems Xao-Nng Shen, Xn Yao PII: S (14) DOI: Reference: INS To appear n: Informaton Scences Receved Date: 29 May 2014 Revsed Date: 12 October 2014 Accepted Date: 22 November 2014 Pease cte ths artce as: X-N. Shen, X. Yao, Mathematca modeng and mut-objectve evoutonary agorthms apped to dynamc fexbe job shop schedung probems, Informaton Scences (2014), do: /j.ns Ths s a PDF fe of an unedted manuscrpt that has been accepted for pubcaton. As a servce to our customers we are provdng ths eary verson of the manuscrpt. The manuscrpt w undergo copyedtng, typesettng, and revew of the resutng proof before t s pubshed n ts fna form. Pease note that durng the producton process errors may be dscovered whch coud affect the content, and a ega dscamers that appy to the journa pertan.

3 Mathematca modeng and mut-objectve evoutonary agorthms apped to dynamc fexbe job shop schedung probems Xao-Nng Shen a,*, Xn Yao b a Schoo of Informaton and Contro, Nanjng Unversty of Informaton Scence and Technoogy, Nanjng , Chna b CERCIA, Schoo of Computer Scence, Unversty of Brmngham, Edgbaston, Brmngham B15 2TT, Unted Kngdom Abstract Dynamc fexbe job shop schedung s of sgnfcant mportance to the mpementaton of rea-word manufacturng systems. In order to capture the dynamc and mut-objectve nature of fexbe job shop schedung, and provde dfferent trade-offs among objectves, ths paper deveops a mut-objectve evoutonary agorthm (MOEA)-based proactve-reactve method. The novety of our method s that t s abe to hande mutpe objectves ncudng effcency and stabty smutaneousy, adapt to the new envronment qucky by ncorporatng heurstc dynamc optmzaton strateges, and dea wth two schedung poces of machne assgnment and operaton sequencng together. Besdes, a new mathematca mode for the mut-objectve dynamc fexbe job shop schedung probem (MODFJSSP) s constructed. Wth the am of seectng one souton that fts nto the decson maker s preferences from the trade-off souton set found by MOEA, a dynamc decson makng procedure s desgned. Expermenta resuts n a smuated dynamc fexbe job shop show that our method can acheve much better performances than combnatons of exstng schedung rues. Three MOEA-based reschedung methods are compared. The modfed ɛ-moea has the best overa performance n dynamc envronments, and ts computatona tme s much ess than two others (.e., NSGA-II and SPEA2). Uttes of ntroducng the stabty objectve, heurstc ntazaton strateges and the decson makng approach are aso vadated. Keywords Metaheurstcs; Schedung; Evoutonary computatons; Mathematca modeng; Decson makng 1. Introducton Job shop schedung probem (JSSP) s we-known as a strongy NP-hard combnatona optmzaton probem [17], whch many deas wth fndng out the best sequences for processng jobs on each operabe machne to acheve the requred objectves subject to precedence and processng tme constrants. In JSSP, each operaton of a job shoud be processed on a predefned machne ony once n a fxed operaton sequence. However, the wde empoyment of mut-purpose machnes n the rea-word job shop makes t more genera that an operaton can be managed by severa machnes,.e., there are aternatve routngs, whch s the so-caed fexbe job shop schedung probem (FJSSP). FJSSP s a * Correspondng author. Address: No.219, Nngu Road, Nanjng, , Chna. Te.: Ema address: sxnystsyt@gma.com

4 generazaton of JSSP. It has more compexty than JSSP because the machne assgnment probem whch seects an aternatve machne for each operaton shoud aso be addressed, besdes the sequencng probem. Hence, FJSSP s aso consdered to be strongy NP-hard [15]. In rea-word manufacturng systems, t s often the case that the workng envronment changes dynamcay by unpredctabe rea-tme events, such as one machne fas to work suddeny, and new jobs arrve n a stochastc way, etc. A prevousy optma schedue may get poor system performance or even becomes nfeasbe n the new envronment. Moreover, some nformaton about the job shop s prevousy unknown. For exampe, the due date and processng tme of the new job are not gven unt the job arrves. Ths knd of probems s generay known as dynamc schedung [14]. As ndcated n [29], dynamc schedung s of great mportance to the successfu mpementaton of rea-word manufacturng systems. In the terature reported, there are many three categores of dynamc schedung technooges, whch are competey reactve, predctve-reactve, and pro-actve schedung [29]. Among them, predctve-reactve schedung s the most commony used. It has a schedung/reschedung process where prevous schedues are revsed to adapt to the new envronment caused by dynamc events. Most of the exstng research generated a new schedue by mnmzng the effect of dsrupton on shop effcency ke make-span [2,5]. However, t may produce a new schedue totay dfferent from the orgna one. For exampe, some remanng operatons n the prevous schedue whch have not begun processng at the tme of reschedung may have ther startng tme acceerated or deayed. It has a serous mpact on other producton actvtes panned based on the orgna schedue, and brngs nstabty and ack of contnuty n the shop system [33]. Thus, both the performances of effcency and stabty shoud be consdered n predctve-reactve schedung. Above a, FJSSP n the rea word has the dynamc and mut-objectve nature. A few terature have reschedued dynamc job shops based on mutpe objectves. Some of them ony consdered the performances of effcency [2,5,32], e.g. make-span and tardness. The others ncorporated both effcency and stabty [14,33,62]. A the above studes used a weghted sum approach to convert mutpe objectves to a snge functon. However, n most rea-word cases, t woud be dffcut to dentfy sutabe weghts for each objectve. On the other hand, mutpe objectves such as make-span, tardness and stabty are usuay confcted wth each other. It s better to hande mutpe objectves wth knowedge about ther Pareto front. The varous trade-offs among dfferent objectves provded by the Pareto front s very usefu n makng an nformed decson n dynamc

5 schedung. Evoutonary agorthms (EAs) have been recognzed to be we suted for mutobjectve optmzaton probems due to ther capabty to evove a set of soutons smutaneousy n one run. In the past 20 years, MOEA receved much attenton, and ots of success has been acheved [39]. So far, varous EAs have been apped to sove manufacturng optmzaton probems. To optmze cuttng parameters n the mut-pass turnng operatons, a comparatve study of ten popuaton-based optmzaton agorthms was performed n [47], and an artfca bee coony agorthm [48] and a hybrd Taguch-dfferenta evouton agorthm [49] were proposed, respectvey. To seect optma machnng parameters n mng operatons, a hybrd dfferenta evouton agorthm and a cuckoo search agorthm were presented n [50] and [51], respectvey. As to the structura and shape desgn optmzaton probem, dfferent EAs have been nvestgated, such as the hybrd of mmune agorthm and Taguch method [52], the hybrd dfferenta evouton agorthm [53], the harmony search agorthm [54], the hybrd partce swarm optmzaton agorthm [55-57], Cuckoo search agorthm [13], genetc agorthm [58], the mmune agorthm combned wth a h cmbng oca search [59,60], and the hybrd of mmune and smuated anneang agorthm [61]. To our best knowedge, n the terature reported, MOEA has not yet been adopted to regenerate new schedues n a predctve-reactve way when shop envronments change. The prmary am of ths paper s to sove MODFJSSP based on an MOEA n a modfed predctve-reactve schedung manner. Wth the am of coverng the shortage of exstng methods, three aspects are studed: () the mathematca mode for MODFJSSP s constructed. In the mode, four objectves ncudng both the performances of effcency and stabty are consdered smutaneousy. Besdes, constrants to the search space change dynamcay when rea-tme events occur, whch are aso addressed n the mode deveoped; () a new MOEAbased reschedung method s proposed, whch do not regenerate a new schedue from scratch, but ncorporate severa heurstc methods n creatng the nta popuaton, and use probem specfc genetc operators for varaton; and () n order to seect one approprate souton from the trade-off souton set found by an MOEA, a dynamc decson makng procedure s desgned. To evauate the effectveness of the proposed methods, a reastc dynamc fexbe job shop s smuated wth three purposes: (1) comparng the job shop performance produced by the MOEA-based reschedung method to that of the combnatons of exstng heurstc rues and that of the exstng statc agorthms; (2) anaysng dfferent trade-offs among the four objectves, and comparng the overa performances n dynamc envronments produced by three MOEAs (ε-moea [9], NSGA-II [10], SPEA2 [64]); and (3) nvestgatng the mpact

6 and utty of the stabty objectve, heurstc ntazaton strateges and the decson makng approach. The remander of ths paper s organzed as foows. Secton 2 presents a short overvew of the exstng reated work. Secton 3 descrbes the probem formuaton whch ntroduces the reschedung mode and constructs the mathematca mode of MODFJSSP. In Secton 4, the new MOEA-based reschedung method for MODFJSSP and the dynamc decson makng approach are descrbed n deta. Expermenta resuts are dscussed n Secton 5. Fnay, concusons are drawn n Secton Reated work Mathematca mode s very usefu for understandng the probem structure, thus a few terature have focused on mathematca formuatons for statc FJSSP. A mathematca mode was presented n [15] to acheve optma souton for sma sze probems. A mxed-nteger near programmng mode was deveoped for FJSSP n [30]. In [12], modes formuated for FJSSP n terature were revewed whch categorsed them as sequence-poston varabe based mode, precedence varabe based mode, and tme ndexed mode. As to the dynamc fexbe job shop schedung probem (DFJSSP), there have been few studes descrbng the mathematca mode. In [63], a dynamc reschedung mode based on Mut-Agent System was proposed. A mathematca mode for DFJSSP whch mnmzed a weghted sum of two objectves (make-span and stabty) was deveoped n [14]. It used bnary varabes to form constrants, whch woud ntroduce a ot of bnary parameters. Besdes, the defnton of make-span at a specfc reschedung pont s not gven. In ths paper, a dynamc mutobjectve optmzaton mode for DFJSSP whch can capture the dynamc characterstcs of both objectves (reated to effcency and stabty) and constrants are constructed. DFJSSP s formuated n our mode n a more comprehensbe way. Stabty measures the devaton between revsed and orgna schedues, and there s no unversa defnton for stabty. In [6], stabty was defned as the number of tmes reschedung occurred. In [1,44], stabty was defned as the startng tme devaton and operaton sequence devaton. In [14,33], stabty had two dmensons. One was the startng tme devaton, and the other refected how cose to the current tme changes were made. In our mode, a more sophstcated defnton for stabty s presented whch captures the devaton of operaton startng tme acceeratng, startng tme deay and competon tme deay between two successve schedues, respectvey. Generay, there are many four research drectons on mut-objectve dynamc job shop schedung n the exstng terature. The frst cass deveoped customzed rues before the runnng of a job shop. A co-evoutonary genetc programmng method was deveoped n [26]

7 for smutaneous desgn of dspatchng rues and due-date assgnment rues. Gene express programmng was adopted n [27] to evove machne assgnment rues and job dspatchng rues together n DFJSSP. Ths knd of methods s sutabe for off-ne optmzaton. The second cass beongs to competey reactve schedung. In [32], at each schedung pont, an exstng dspatchng rue that performed best was determned by ookng up the dotypc network mode constructed n advance. A heurstc was proposed n [28] to mpement the reactve schedung n a dynamc producton envronment where jobs arrve over tme. A mutpe attrbute decson makng technque whch used grey numbers to dea wth uncertantes was gven n [45] to determne whch ot was sutabe to be processed next when a machne was free. [38] focused on the mpementaton concept of a dscrete event smuaton based onne near-rea-tme dynamc schedung system usng conjunctve smuated schedung. Competey reactve schedung s quck to mpement, but t consders ony the oca nformaton, so the shop performance cannot be guaranteed. It s sutabe for onne dynamc schedung. The thrd cass s caed the predctve-reactve schedung, whch uses the goba nformaton and searches n a arger souton space n comparson wth competey reactve schedung [29]. In [2], an adaptve varabe neghbourhood search was trggered n respond to a random event. A conventona genetc agorthm was used n [5] to regenerate a new schedue whenever a dynamc event occurred. However, t s often neffcent to restart the optmzaton process wth a totay new popuaton [34]. In order to sove the nstabty probem nduced by unrestrcted reschedung, b-objectves of stabty and effcency were consdered smutaneousy n [14,33,62], respectvey. However, the weghted sum method was adopted n a the above studes to dea wth mutpe objectves. Predctve-reactve schedung s aso sutabe for onne dynamc schedung. In our paper, we sove MODFJSSP based on an MOEA n a modfed predctve-reactve schedung way. The fourth cass can be categorzed as the pro-actve schedung, whch buds predctve schedues n advance. [21] ntroduced four dfferent probabty dstrbutons to mode stochastc processng tmes, and proposed three uncertanty handng methods to estmate the ftness of a souton. A mut-objectve mmune agorthm s gven n [66] to produce robust schedung soutons of uncertan schedung probems descrbed by the workfow smuaton schedung mode. A smpfed mut-objectve genetc agorthm was proposed n [23] for the stochastc JSSP wth exponenta processng tmes. A robust and stabe predctve schedue for one machne schedung, JSSP, and FJSSP wth random machne breakdowns was generated by a genetc agorthm n [24,20,3], respectvey. A robust genetc agorthm

8 was proposed n [4] to mnmze the make-span of a parae machne schedung probem wth fuzzy processng tmes. In [46], parae machne schedung wth earnng effects and fuzzy processng tmes was soved by the smuated anneang agorthm and the genetc agorthm. In ths cass of methods, optmzaton s performed offne. Decson makng s an mportant process n evoutonary mut-objectve optmzaton, especay n the dynamc case. However, few attentons have been pad to ths aspect. In [8,11], ony probems wth b-objectves were consdered, and a precse weght vaue of each objectve shoud be provded by the decson maker (DM). However, the DM usuay does not have enough knowedge about objectve functons. When expressng preferences, they prefer to empoy the quatatve anguage ke Objectve A s more mportant than objectve B to descrbe the reatve mportance between two objectves [38]. For ths reason, n our paper, ngustc terms are used to represent the DM s vague thought nstead of requrng them to gve numerca vaues so as to reduce hs/her cogntve overoad. 3. Probem formuaton 3.1 Reschedung mode In order to nfuse more reaty n job shops, random new job arrvas and machne breakdowns (repars) are consdered. Among them, urgent job arrvas, machne breakdowns and repars are regarded as crtca events, and reguar job arrvas are uncrtca events. A modfed predctve-reactve dynamc schedung s adopted. A producton schedue for a the jobs at the nta tme s generated at frst. In order to reduce the reschedung frequency, a crtca-event-drven mode s empoyed. Once a crtca event occurs, the reschedung method s trggered. The tme at whch a new schedue s constructed s caed the reschedung pont, and the tme span between two successve reschedung ponts s named the reschedung nterva. Besdes, a speca case s consdered. Suppose by the tme nstant t * after a specfc reschedung pont t, a the schedued jobs have fnshed so that a the avaabe machnes are de, and the next crtca event has not occurred. If the number of reguar job arrvas between t and t * s arger than an upper bound, then a new schedue s constructed for these new uncrtca jobs to make fu use of the machne resources. We ca t the resource-de-drven mode. The upper bound s set to be 5 here. 3.2 Mathematca modeng of MODFJSSP Some notatons used for descrbng the mathematca mode are sted n Tabe 1. Gven the extremey hgh compexty of MODFJSSP, some common assumptons are made n ths paper.

9 (1) A job can be processed by ony one machne each tme and each machne can perform at most one operaton at a tme. (2) Once an operaton has begun on a machne, t must not be nterrupted, except for the machne breakdown. (3) The machne setup tme for two consecutve jobs s ncuded n the processng tme. (4) There s no trave tme between machnes. Jobs are avaabe for processng on a machne mmedatey after competng processng ts prevous operaton. (5) Jobs can wat to be processed n an unmted buffer of a machne. The mathematca mode for MODFJSSP at a specfc reschedung pont s formuated. At the reschedung pont t ( t > t0 ), consderng a the current nformaton gathered from the job shop foor, whch ncudes attrbutes of the avaabe machnes, a the unprocessed job operatons from the prevous schedue, and the new arrva reguar or urgent jobs snce the t 0 : the nta tme nt ( ) : Number of jobs that contan unprocessed and avaabe operatons at t Tabe 1 Notatons used for descrbng the mathematca mode t : The reschedung pont (=1,2, ) J ( t ): The th job at t, = 1,2, L, nt ( ) mt ( ) : Number of avaabe machnes at t M ( t ) : The k th avaabe machne at t k, k= 1,2, L, mt ( ) n : Tota number of operatons n job J ( t ). It s predefned and unchanged wth t C ( t ) : The competon tme of a the current avaabe operatons of job J ( t ) at t DD( t ): Due date for a the current avaabe operatons of job J ( t ) at t n ' ( t ) : Number of unprocessed and avaabe operatons eft n job J ( t ) at t, ' 1 n ( t ) n S ( t ): The startng tme of the frst unprocessed operator of job J ( t ) at t a : The nta arrva tme of job J ( t ) n the job shop. It s unchanged wth w : The mportance weght of job J ( t ). It s unchanged wth t I ( t ): Index of the frst unprocessed operaton n job J ( t ) at t, c ( ) : The competon tme of the ast operaton of job J ( t ) I( t)-1 that has begun processng before t TMA : The set of a the machnes that can process operaton j O ( t ). It s predefned and unchanged wth t j MA ( t ) : The set of avaabe machnes that can process operaton j O ( t ) at t j. MAj( t ) ÍTMAj A_ p ( t ): The actua processng tme of O ( t ) on ts assgned j machne at t c ( t ) : The competon tme of operaton O ( t ), j c ( t )= s ( t )+ A_ p ( t ) j j j k q ( t ): Number of operatons assgned to the machne M ( t ) at t, k= 1,2, L, mt ( ) O kr j j k t ' I ( t ) ³ 1 and I ( t ) + n( t )- 1 n O ( t ) : The j th operaton of job J ( t ) whch s avaabe to be j processed durng the reschedung nterva of t, j = I t I t I t n t - ( ), ' ( )+1, L, ( )+ ( ) 1 Average _ p : The average processng tme of operaton O ( t ) for the machnes n j TMA. It s unchanged wth j kk p ( t ) : The processng tme of O ( t ) on the machne j M ( t ) Î MA ( t ), kk = 1,2, L, MA ( t ) ( means cardnaty kk j of a set) s ( t ) : The startng tme of operaton O ( t ) j k _ ast c ( t -1) : The competon tme of the ast operaton processed on M ( t ) before t k kr O ( t ): The r th operaton schedued on the machne M ( t ) at t k, k r = 1,2, L, q ( t ) kr p ( t ) : The processng tme of operaton O ( t ) O kr kr c ( t ) : The competon tme of operaton O ( t ) R ( t ) : The nta reease tme of job J ( t ) reschedung nterva of t durng the A ( t ): The nta avaabe tme of machne M ( t ) durng the k reschedung nterva of t j j j t k j

10 prevous reschedung pont t -1, a new schedue whch represents the operatons assgned to each machne and the correspondng sequence s constructed by optmzng the foowng objectves wth respect to both shop effcency and stabty. mn F =[ f ( t ), f ( t ), f ( t ), f ( t )] (1) where f 1 ( t ) represents make-span whch means the eapsed tme requred for fnshng a the current jobs reschedued at t ; f ( t ) s the tardness measure whch gves penates to deays 2 from the due date; f ( t ) denotes the maxma machne workoad whch s to avod assgnng 3 too much work to a snge machne; f ( t ) 4 ndcates the stabty whch measures the devaton between the new and orgna schedues. The formua of each objectve s gven beow. f ( t )= max ( C ( t )) - mn ( S ( t )) 2 1 = 1,2, L, nt ( ) = 1,2, L, n( t ) å f ( t )= w max(0, C ( t )-DD( t )) = 1,2, L, n( t ) 3 k=1,2, L, mt ( ) qk ( t ) O kr f ( t )= max ( p ( t )) å r= 1 f ( t )= s ( t ) s ( t ) s ( t ) s ( t ) c ( t ) c ( t ) g j j -1 j j -1 j j -1 Oj Îrush Oj Îdeay Oj Îdeay startng startng devery å å å (5) where s ( t ) and c ( t ) are the startng tme and competon tme of operaton O n the j -1 j -1 j prevous schedue generated at the reschedung pont t -1, respectvey. The make-span measure f ( t ) n Eq. (2) s cacuated as the dfference between the 1 maxmum competon tme and the mnma startng tme of a the current jobs at t. Here, C ( t ) represents the competon tme of a the current avaabe operatons of job J ( t ). Ths s because by t, some operatons mght have become unavaabe due to the occurrences of dynamc events. For exampe, the operaton O cannot be processed because a of ts 47 aternatve machnes broke down before or at t, and have not gotten repared by t. The succeedng operatons O and O become unavaabe ether, because no pre-empton s aowed. Thus at t, C ( t ) s the competon tme of the ast avaabe operaton O The tardness measure f ( t ) n Eq. (3) s defned as the weghted sum of dfferences 2 between the competon tme and due date of each job n whch ts competon tme s arger than the due date. Smar to C ( t ), DD ( t ) s the due date for a the current avaabe operatons of job J ( t ). Accordng to ths characterstc, t s generated by a modfcaton to the commony used tota work content (TWK) rue [2]: (2) (3) (4)

11 DD ( t ) = a + K * å Average _ p j j= 1,2, L, ' I( t ) + n ( t )-1 (6) where DD ( t ) s equa to the sum of the job arrva tme and a mutpe of the tota average processng tme from the frst operaton to the ast avaabe operaton. The mutpe caed the tghtness factor and s reated to job characterstcs. In our research, norma dstrbuton wth the mean of 1.5 and varance of 0.5. K s K foows the The maxma machne workoad f 3 ( t ) n Eq. (4) s cacuated as the maxmum workng tme spent at any avaabe machne. The stabty measure f 4 ( t ) n Eq. (5) has three terms. The frst term s the startng tme devaton of operatons whch are reschedued to start processng at an earer tme mutped by a weght g, where g > 1. It gves more penates to brng forward the startng tme of an operaton, because rush order cost s ncurred f the devery of materas s requred earer than panned based on the orgna schedue. Here, we set g =1.5. The second term s the startng tme devaton of operatons whch are reschedued to begn processng at a ater tme. Ths case may ead to carryng costs because materas are devered earer than requred. The thrd term s the competon tme devaton of operatons whch have ther endng tme deayed, and t causes the deteroraton of devery performance. It shoud be mentoned that at the nta tme t 0, ony three objectves whch are makespan, tardness and the maxma machne workoad (wthout stabty) are to be optmzed. In the dynamc fexbe job shop, constrants to the search space change dynamcay wth occurrences of random events. They are sted as foows. 1) Machne set constrants { 1 2 ( ) } ' MA ( t ) Í M ( t ), M ( t ), L, M ( t ), for = 1,2, L, nt ( ) j m t, j = I ( t ), I ( t )+1, L, I ( t )+ n ( t )-1 (7) The machne avaabe set MA ( t ) contans a avaabe machnes that can process the j operaton O ( t ). It may change at dfferent reschedung ponts due to machne breakdowns or repars. j 2) Processng tme constrants kk For each machne M ( t ) Î MA ( t ) ( kk = 1,2, L, MA ( t ) ), a processng tme p ( t ) s kk j assocated wth operaton O ( t ). If O ( t ) s assgned to M ( t ) n the schedue at t, then ts j j kk actua processng tme _ ( )= kk ' A p t p ( = 1,2, L, nt ( ) j j, j = I ( t ), I ( t )+1, L, I ( t )+ n ( t )-1). j j

12 3) Precedence constrants Operatons of each job shoud be processed through the machnes n a partcuar order. At t, job J ( t ) conssts of a sequence of n ' ( t ) operatons, and each operaton O ( t ) can be j processed on any machne out of ts machne avaabe set MA ( t ) ( = 1,2, L, nt ( ) j, j = I t I t L I t n t - ). ' ( ), ( )+1,, ( )+ ( ) 1 4) Inta state constrants R ( t ) = max( t, c( )) for = 1,2, L, nt ( ) (8) I ( t )-1 A t = t c t for k= 1,2, L, mt ( ) (9) k _ ast k( ) max(, ( -1)) Eq. (8) gves the nta reease tme of each job. It guarantees that a the operatons that have begun processng before t not be consdered n the reschedung mode. If I ( t ) = 1, then c ( I ( t )-1) = 0. Eq. (9) gves the nta de tme of each machne. It ndcates that one machne s avaabe unt t has fnshed a the operatons that have begun before t. If there s no operaton processed on the machne M ( t ) before t k, then 5) No preempton constrants 5.1) No preempton n a snge job k _ ast c ( t -1)=0. An operaton of a job cannot be processed unt ts precedng operatons are competed. If O ( t ) s the frst unprocessed operaton of job J ( t ) at t j,.e. j = I ( t ), then t shoud start processng after the nta reease tme R ( t ): R ( t ) s ( t ), for = 1,2, L, nt ( ), j = I ( t ) (10) j Otherwse, ( ) ( ) ' c t s t, for = 1,2, L, nt ( ) ( -1) j, j = I ( t )+1, L, I ( t )+ n( t )-1 (11) j 5.2) No preempton n a snge machne An operaton can be processed on ts assgned machne unt the machne has fnshed ts prevous schedued operatons. Suppose at t, the operaton O ( t ) s assgned to the machne j M ( t ), and t s schedued as the r th operaton on M ( t ),.e., O ( t ) = O kr ( t ). k k j If r = 1, then O ( t ) shoud start processng after the nta machne avaabe tme A ( t ): If r ³ 2 k( r-1) O c t ( ), then j A ( t ) s ( t ), for 1 k j, suppose the competon tme of the O k( r-1) c t s t j r =, Î{ 1,2, L, ( )} (12) k mt k ( ) ( ), for r 2,, q ( t ) th ( r -1) operaton schedued on M ( t ) s = L, Î{ 1,2, L, ( )} (13) k mt k k

13 5.3) Startng tme of an operaton From Eq. (10) - Eq. (13), t can be concuded that the startng tme of operaton O ( t ) j (suppose t s schedued as the r th operaton on M ( t ) at t k ) s: ìmax( R( t ), A ( t )) for j = I ( t ), r = 1 k ï k( r-1) O k ïmax( R( t ), c ( t )) for j = I ( t ), r = 2, L, q ( t ) s ( t ) = j í ' (14) ï max( c ( t ), A ( t )) for j = I ( t )+1, L, I ( t )+ n( t ) - 1, r = 1 ( j-1) k ï ( 1) ' max( c ( t ), O k r- k î c ( t )) for j = I ( t )+1, L, I ( t )+ n ( t )- 1, r = 2, L, q ( t ) ( j-1) 6) Interrupton mode constrants At t, f one machne breaks down, and an operaton s beng processed on t, then the work has to stop and wat unt the machne has been repared. On the other hand, f a broken machne gets repared at t, then the prevousy nterrupted operaton (f any) w resume to be processed on t wth the nterrupt-resume mode, or be processed from scratch wth the nterrupt-repeat mode [1]. In ths paper, the nterrupt-resume mode s used. Compared to the exstng mode n [14], superortes of our mode can be summarzed as foows: (1) Mut-objectve handng method. A dynamc mut-objectve optmzaton mode for DFJSSP s constructed, where three effcency objectves and one stabty objectve are optmzed smutaneousy based on Pareto domnance, nstead of beng converted nto a snge one n [14]; (2) Objectve defntons. In our mode, consderng the dynamc feature of DFJSSP, objectve defntons are formuated specfcay for each reschedung pont Snce some unprocessed operatons mght become unavaabe temporary due to occurrences of random events (e.g. machne breakdowns), the four objectves (make-span, tardness, maxma machne workoad and stabty) at t. t are defned ony for current avaabe operatons. In contrast, ony two objectves of make-span and stabty were consdered n [14], and the defnton of make-span at a specfc reschedung pont was not gven; (3) Consderaton of dynamc constrants. In the dynamc fexbe job shop, n addton to objectves, constrants to the search space aso change dynamcay wth occurrences of random events. We cassfy these dynamc constrants nto sx categores, and gve straghtforward and comprehensbe defntons to them whch can capture dynamc features of constrants. In contrast, [14] used bnary varabes to form constrants, whch woud ntroduce a ot of extra bnary parameters. Besdes, the nta state constrants, the earest startng tme of an operaton, and nterrupton mode constrants were not consdered n [14]; and (4) Defnton of the stabty objectve. A more sophstcated defnton for stabty s presented whch captures the devaton of operaton startng tme acceeratng, startng tme

14 deay and competon tme deay between two successve schedues, respectvey, snce they have dfferent mpact on the producton pan. 4. A predctve-reactve schedung method to sove MODFJSSP 4.1 Framework of the predctve-reactve schedung method The fowchart of our predctve-reactve schedung n MODFJSSP s summarzed n Fg. 1. At the nta tme of the shop foor, a predctve schedue s generated by an MOEA consderng three objectves whch are make-span, tardness and the maxma machne workoad. Then durng the mpementaton of the schedue, at each reschedung pont, an MOEA-based reschedung method s trggered to construct a new schedue by consderng four objectves whch are make-span, tardness, the maxma machne workoad and stabty smutaneousy. The newy generated schedue s mpemented n the job shop unt the next reschedung pont comes, at whch tme the reschedung method s trggered agan. In short, MODFJSSP s a dynamc process formed by a sequence of mut-objectve FJSSPs wth dfferent sets of job operatons and machnes to be schedued. Ths process contnues unt a the jobs appearng n the shop foor have fnshed. At each schedung pont, a set of nondomnated soutons are obtaned by an MOEA. Thus one souton that fts nto the DM s preferences s seected by a decson makng procedure and mpemented n the shop foor. start Consder jobs & machnes that exst at the start pont n shop foor Generate a set of non-domnated predctve schedues by an MOEA consderng three objectves of makespan, tardness and maxma machne workoad Seect one predctve schedue to be mpemented by the decson makng procedure Move to the next reschedung pont Update the job shop state Have a the jobs fnshed? Yes Stop Obtan a set of non-domnated schedues by an MOEA-based reschedung method consderng four objectves of makespan, tardness, maxma machne workoad and stabty No Seect one schedue to be mpemented by the decson makng procedure Fg. 1. Fowchart of the proposed predctve-reactve schedung n the mut-objectve fexbe job shop. 4.2 The ε-moea-based reschedung method for MODFJSSP As ndcated n Secton 4.1, MODFJSSP can be seen as a dynamc process formed by a sequence of mut-objectve FJSSPs. However, we shoud not just treat MODFJSSP as a sequence of ndependent statc probems and use the exstng statc MOEAs to sove t. There

15 are many four reasons for that: (1) these probems are not ndependent and they are reated to each other. At each reschedung pont, a new FJSSP wth an updated set of job operatons and machnes s formed and to be schedued. Most operatons n the current probem are composed of unprocessed operatons eft from the prevous schedue, and most of the machnes are the same as those of the pror probem; (2) n a rea-word job shop system, stabty and contnuty whch means there shoud be a sma dfference between the new generated schedue and the orgna one are very mportant. So when reschedung, arrangements n the prevous schedue shoud be taken nto account; (3) MODFJSSP s a dynamc probem thus some dynamc optmzaton strateges shoud be ntroduced to make the agorthm adapt to the changng envronments qucky. Here, the features of dfferent dynamc events can be utzed to gude the searchng drecton; and (4) as ndcated n [34], t s often neffcent to restart the dynamc optmzaton process wth a totay new popuaton. Thus, we shoud nvent a new dynamc agorthm for sovng MODFJSSP, whch can capture the correatons between the sequence of probems, and avod producng a new schedue totay dfferent from the orgna one. ε-moea s an ε-domnaton based steady-state MOEA. It empoyed effcent parent and archve update strateges, and was vadated that t s a baanced agorthm whch can produce good convergence and dversty wth a very sma computatona effort, especay when deang wth many objectves (3 or more) [9]. MODFJSSP studed n ths paper s a dynamc probem wth four objectves. In order to sove t n an effcent way, an ε-moea-based reschedung method s proposed. Meanwhe, to keep the system stabty and contnuty n mnd, and to expot the nformaton eft from the orgna schedue and the characterstcs of dfferent dynamc events, our ε-moea-based reschedung method s featured wth three ponts: (1) some heurstc strateges are ncorporated n constructng the nta popuaton of ε-moea at each reschedung pont; (2) new ndvdua representatons and two knds of probem specfc varaton operators are desgned so that the proposed method can hande operaton sequencng and machne assgnment smutaneousy; and (3) the stabty objectve s consdered together wth the shop effcency (make-span, tardness, the maxma machne workoad) for mut-objectve optmzaton n our approach. The procedure of ɛ-moeabased reschedung method at the reschedung pont t ( t > t0 ) s presented beow. Step 1: Intazaton: Construct the nta popuaton Pt ( ) by some heurstc strateges accordng to the updated job shop state at t. Then mut-objectve evauatons are performed,

16 and a the non-domnated soutons are determned to form the nta archve popuaton At ( ). Set the counter of objectve evauaton numbers ct = popuaton _ sze. Step 2: Popuaton seecton: One ndvdua sp s chosen from the popuaton Pt ( ). Here, the tournament seecton method s used. Two ndvduas are pcked up unformy at random from the popuaton, and check the domnaton of each other. If one domnates the other, the former w be chosen. Otherwse, one of them s seected at random. At ( ). Step 3: Archve seecton: One souton e s chosen unformy at random from the archve Step 4: Varaton: Two offsprng sc and sc are generated from sp and e by two knds of 1 2 probem specfc varaton operators. Step 5: Decodng and objectve evauaton: Evauate the offsprng ndvduas sc and sc. 1 2 Step 6: Update of the popuaton: Offsprng ndvduas sc and sc are ncuded n Pt ( ) 1 2 usng a pop_acceptance procedure. Step 7: Update of the archve: Indvduas sc and sc are ncuded n At ( ) usng an 1 2 archve_acceptance procedure. Step 8: Termnaton: If the termnaton crteron s not satsfed, set ct = ct + 2 and go to Step 2, ese output At ( ), and seect one souton from At ( ) as the mpementaton schedue based on a decson makng procedure. In the above Steps 6 and 7, the pop_acceptance and archve_acceptance procedures are the same as those n [9]. The termnaton crteron s the counter ct acheves a predefned maxmum number of objectve evauatons. It shoud be mentoned that at the nta tme t 0 of the job shop, the ε-moea used to generate a set of predctve schedues s aso based on the procedure ntroduced above. The dfferences are that the random popuaton ntazaton s used n Step 1 nstead of the heurstc popuaton ntazaton, and when evauatng an ndvdua, ony three objectves (wthout stabty) are consdered. Detas of our mpementaton for the ε-moea-based reschedung method and the decson makng procedure w be dscussed beow Representatons In MODFJSSP, both the operaton sequence vector and the machne assgnment vector are used to represent a compete schedung ndvdua. Fg. 2 gves an exampe of such a representaton.

17 O76 fo81 fo69 fo82 fo91 fo77 fo92 fo83 fo fo84 93 Fg. 2. An exampe of the representaton of a chromosome. For the operaton sequence vector, a job-based representaton [18] s used. A the operatons from the same job are denoted by the job number. Take Fg. 2 as an exampe. Suppose at a specfc reschedung pont, the operatons O 69, O 76 and O 77 from jobs 6 and 7 are eft unprocessed from the prevous schedue, and there are two new jobs 8 and 9. Thus, the operaton sequence vector contans the job numbers 6, 7, 8, and 9. Each operaton s nterpreted accordng to ts order of appearance n the sequence vector. For exampe, the frst appearance of number 8 represents O 81, the second appearance of 8 means O 82, and so on. So the operaton sequence vector n Fg. 2 can be nterpreted as: where a O fo fo fo fo fo fo fo fo fo f b means operaton a jons the watng queue of ts assgned machne frst, then operaton b s schedued. The machne assgnment vector represents the assgned machne of each operaton. The order s from the frst remanng operaton of the odest job (wth the mnmum job ndex) to the ast remanng operaton of the newest job (wth the maxmum job ndex). For exampe, n Fg. 2, suppose the current avaabe machnes are 2, 3, 5, 6, 9, and 10. In the machne assgnment vector, the frst eement of 3 means the frst remanng operaton O 69 of the odest job 6 w be assgned to machne 3, the second eement of 6 represents the frst remanng operaton O 76 of the second odest job 7 w be assgned to machne 6, and so on. Thus, the machne assgnment vector can be nterpreted as: O machne 3, O machne 6, O machne 5, O machne 3, O machne 2, O machne 2, O machne 6, O machne 9, O machne 10, O machne where means the operaton s assgned to the correspondng machne Decodng For the convenence of objectve evauaton, a schedung souton shoud be decoded nto the form of Gantt chart. Fg. 3 gves the Gantt chart of the chromosome represented n Fg. 2. Assume the reschedung pont n Fg. 2 s t =10, and at t =10, two operatons O 68 and O 75 n the prevous schedue are beng processed on machne 6 and 5, respectvey. The predefned processng tme of each operaton on the assgned machne s sted n Tabe 2. From Fg. 2,

18 we can see that the sequence of operatons s: O (6) fo (3) fo (3) fo (2) fo (9) fo (5) f O (10) fo (2) fo (5) f O (6), where the number n the parentheses s the machne assgnment of each operaton. The process of constructng the Gantt chart n Fg. 3 s descrbed as foows. Frst, take the frst operaton O 76 n the operaton sequence nto account. Snce O 76 s aso assgned to machne 6, t can be processed unt both O 75 (ts prevous operaton n the same job 7)and O 68 (ts prevous operaton n the same machne 6) have fnshed due to the no preempton constrants. Then, snce the second operaton O 81 s the frst operaton of job 8 and ts assgned machne 3 s de, t s schedued at t =10 for a processng tme of 0.5 tme unts. Next, the thrd operaton O 69 s to be processed on machne 3 at the maxmum vaue of the competon tme of O 68 and O 81. The foowng operatons n the operaton sequence are schedued to the assgned machnes foowng the same method. Fg. 3. Decoded schedue of the chromosome n Fg. 2. Tabe 2 Processng tme of each operaton on the assgned machne n Fg. 2 operaton O 76 O 81 O 69 O 82 O 91 O 77 O 92 O 83 O 93 O 84 assgned machne processng tme It shoud be noted that the de tme nserton method whch nserts an operaton nto the frst avaabe de tme nterva of ts assgned machne s used to make fu use of the machne resources. For exampe, O 93 goes after O 77 on machne 5 accordng to the operaton sequence n Fg. 2. But n Fg. 3, there s an nterva of de tme on machne 5 between the competon tme of O 92 (11.2) and the startng tme of O 77 (11.8). Meanwhe, the processng tme of O 93 (0.4) s smaer than the ength of the tme nterva ( =0.6). Thus O 93 s nserted nto ths nterva and begns at the competon tme of O 92 (11.2). Let Lt ( ) be the tota number of operatons to be processed at t, and W(, j) be the ocus of O ( t ) n the j machne assgnment vector. The pseudo code of the decodng procedure s shown n Fg. 4.

19 Procedure: Decodng Procedure Input: JJ 1 nt ( : the array of avaabe jobs at t ), chromosome v s (u), v m (w) // exampes of v s (u), v m (w) s gven n Fg. 2, nt ( ) s the tota number of avaabe jobs at t Output: a schedue for = 1 nt ( ) j 0; // j means the number of operatons aready assgned for job JJ() JJ() JJ() end for for u = 1 Lt ( ) // Lt ( ) s the tota number of operatons to be processed at t v s (u), M v m (W(, I ( t ) + j ) ); // I ( t ) s the ndex of the frst unprocessed operaton n job J ( t ) search an avaabe de tme nterva on machne M from eft to rght for operaton O ; ( I ( t ) + j ) f such a tme nterva s found, then the operaton s nserted there; ese the operaton s schedued at the end of machne M; end f j j +1 end for return the schedue; Fg. 4. Pseudo code of the decodng procedure Update of the job shop state Once the reschedung procedure s trggered, the shop state shoud be updated at frst. () At t, nformaton eft from the prevous schedue shoud be coected, whch ncudes the remanng unprocessed operatons, and the operatons that are beng processed on each machne at t. Meanwhe, nformaton about new arrva jobs snce the prevous reschedung pont t -1 and the current avaabe machnes must aso be gathered. () Update the machne avaabe set MA ( t ) for a the current operatons. By t j, f some machnes have broken down and thus become unavaabe, they shoud be removed from the machne set of each operaton. For a specfc operaton, f there s no machne avaabe to process t, then t w not enter the reschedung mode at t. On the other hand, f a broken machne has been repared by t, t must rejon the machne set of correspondng operatons. Meanwhe, a the operatons that cannot be processed temporary due to the prevous breakdown of the repared machne, must be added to the reschedung mode at t. () Update the machne avaabe tme A ( t ) and job reease tme R ( t ) accordng to the nta state constrants (Eq. (8) and (9)) for a the current machnes and jobs Constructon of the nta popuaton n reschedung k At each reschedung pont, a new mut-objectve FJSSP wth an updated set of job operatons and machnes s formed and to be schedued. In order to gude the search of the ɛ- MOEA-based reschedung method and acceerate the convergence speed so that the method can adapt to the new envronment qucky, some heurstc methods are ncorporated n

20 creatng the nta popuaton of ε-moea, whch makes the proposed reschedung methods dfferent from those competey reschedung approaches that regenerate a new schedue from scratch [14,33,62]. () Make use of the characterstcs of dfferent dynamc events. As ndcated n [29], schedue repar refers to oca adjustments of the orgna schedue, and t can preserve the system stabty we. Hence, three schedue repars are empoyed here to expot the dynamc event characterstcs. Frsty, for machne breakdowns, a modfcaton to the parta schedue repar [1] s desgned. A the unaffected operatons reman unchanged both for ther machnes and startng tmes. The drecty affected operatons (prevousy schedued on the broken machne and unprocessed) are moved to another aternatve machne f possbe, and the ndrecty affected operatons are assgned to the same machnes as before. Ony the sequences of the affected operatons are reschedued. Secondy, for machne repars, some operatons are shfted to the repared machne so as to baance the machne workoad. These operatons shoud satsfy that they can be processed by the repared machne, and the shft w not affect the startng tme of other operatons. Thrdy, for new job arrvas, a new job s schedued as soon as one of ts aternatve machnes becomes de. The resut of schedue repar s caed the schedue repar souton. () Make use of the hstory nformaton. At each reschedung pont, nformaton eft from the prevous schedue s regarded as the hstory nformaton whch can be expoted. The sequence and machne assgnment vector of a the remanng unprocessed operatons n the od schedue s caed the hstory souton. () Make use of the heurstc machne assgnment rues. Two machne assgnment rues foowng the approach of ocazaton [22] are adopted. The frst rue searches for the goba mnmum n the processng tme tabe [14]. Then t fxes that assgnment, and updates the machne workoad on every other operaton. The second rue randomy permutes jobs and machnes n the processng tme tabe at frst. Then for each operaton, t fnds the machne wth the mnmum processng tme, fxes that assgnment, and updates the machne workoad. The frst rue determnes the machne assgnment for each operaton unquey, whe the second rue fnds dfferent assgnments n dfferent runs of the rues. (v) Incorporaton of random ndvduas. In order to ntroduce dversty, some random ndvduas are created n the nta popuaton. Sequence vectors are generated by permutng a the current operatons at random. For haf of these random ndvduas, machne assgnments are determned accordng to the two rues descrbed above. Each operaton n another haf s assgned to a randomy chosen machne from ts machne avaabe set.

21 In ths paper, 20% of the nta popuaton are formed wth the hstory souton and ts varants by mutaton (as ntroduced n Secton 4.2.5), 30% wth the schedue repar souton and ts varants, and 50% wth the random ndvduas Probem specfc genetc operators I) Sequence based varaton operators In order to preserve the feasbty of the generated offsprng, a specazed crossover operator s desgned for the operaton sequence vectors n the ndvduas. It works as foows. Step : A the current avaabe jobs at the reschedung pont t are dvded unformy at random nto two groups: G 1 and G 2. Step : The operatons from the frst job group G 1 are pcked from Parent 1, and recorded n a new array R 1 as ther orgna postons n Parent 1. The operatons from G 2 are pcked from Parent 2, and recorded n a new array R 2 as ther orgna postons n Parent 2. Step : A of the recorded operatons n R 1 and R 2 are merged accordng to ther orgna sequences to generate an offsprng. Another offsprng s generated usng the same method descrbed above, except that the operatons from G 1 are pcked from Parent 2, and the operatons from G 2 are pcked from Parent 1. Ths procedure s ustrated n Fg. 5. When mergng, f two operatons have the same postons n the parents, such as jobs 6 and 7 n the thrd order of each parent n Fg. 5, ther sequence n the offsprng s generated unformy at random. Fg. 5. An exampe of the crossover for the operaton sequence vectors n the schedung ndvduas. The sequence based mutaton operator s the commony used swap and nsert operators. The swap operator seects two operatons n the operaton sequence vector at random, and exchanges the postons of them. The nsert operator nserts one randomy seected operaton before another one. When performng a mutaton on an ndvdua, ether the swap or the nsert s chosen wth the possbty of 0.5. It shoud be noted that when performng the sequence based varaton operators, the machne assgnment vector s kept unchanged. Snce our representaton of the machne assgnment vector gven n Secton s not reated to the operaton sequence, the varaton of the sequence vector w not affect t. Ths w not cause the potenta nfeasbty probem when performng the swap or nsert operator, whch woud otherwse be

22 faced f we had used the representaton where each machne corresponds to each operaton n the operaton sequence vector as n [14]. II) Machne based varaton operators In the representaton of the machne assgnment vector gven n Secton 4.2.1, the machnes n the same postons of two parents correspond to the same operaton. So the tradtona snge pont crossover can be used. The mutaton operator s performed as foows. An aee s chosen randomy, and the machne on whch the operaton s to be processed s repaced wth one of the aternatve machnes. Smary, when performng the machne based varaton operators, the operaton sequence vector s kept unchanged n the offsprng Parameters We aso appy NSGA-II and SPEA2 to expore the Pareto front of non-domnated schedues at each reschedung pont n order to understand the mpact of dfferent agorthms on the performance of MODFJSSP. The popuaton ntazaton and varaton operators ntroduced above are aso used n NSGA-II and SPEA2. Parameters used by the three MOEA-based reschedung methods are gven n Tabe 3. SPEA2 had a tournament sze of 2 for matng seecton and the archve sze of 100. Each agorthm stopped after objectve evauatons had been performed. 4.3 Decson makng n DFJSSP Tabe 3 Parameter settngs of MOEA-based reschedung methods Popuaton sze 100 Sequence based crossover possbty =0.45 Machne based crossover possbty =0.45 Sequence based mutaton possbty =0.1 Machne based mutaton possbty =0.1 maxmum number of objectve evauatons At each reschedung pont, once a set of trade-off soutons are found by the MOEA method, one souton that fts nto the DM s preferences shoud be seected, and mpemented n the shop foor. Here, a decson makng method nspred by the Anaytc Herarchy Process (AHP) [35,43] and the Mut-attrbute Utty Theory (MAUT) [16] s proposed, and the procedure s gven as foows. Step : Constructon of the parwse comparson matrx. Suppose there are N_O objectves to be optmzed. As n AHP, the parwse comparson questons of How mportant s the objectve f reatve to f? (, j = 1,2, L, N_ o, j > ) are answered by the DM a pror. So there are totay N_ o ( N_ o- 1)/2 comparsons. The answers use the foowng nne-pont scae whch descrbes the degree of the preference for one objectve versus another [16], j