Best-Order Crossover in an Evolutionary Approach to Multi-Mode Resource-Constrained Project Scheduling

Transcription

1 Internatonal Journal of Computer Informaton Systems and Industral Management Applcatons. ISSN Volume 6 (2014) pp MIR Labs, Best-Order Crossover n an Evolutonary Approach to Mult-Mode Resource-Constraned Proect Schedulng Anca Andreca 1 and Camela Chra 2 1 Department of Computer Scence, Babes-Bolya Unversty Kogalnceanu 1, Clu-Napoca, Romana anca@cs.ubbclu.ro 2 Department of Computer Scence and Centre for the Study of Complexty Babes-Bolya Unversty Kogalnceanu 1, Clu-Napoca, Romana cchra@cs.ubbclu.ro Abstract: Mult-Mode Resource-Constraned Proect Schedulng s an NP-hard optmzaton problem ntensvely studed due to ts large area of applcatons. Ths paper concentrates on the evolutonary approaches to Mult-Mode Resource-Constraned Proect Schedulng based on permutaton encoded ndvduals. A new recombnaton operator s presented and a comparatve analyss of several recombnaton operators s gven based on computatonal experments for several proect nstances. Numercal results emphasze a good performance of the proposed crossover scheme whch takes nto account nformaton from the global best ndvdual besdes the genetc materal from parents. Keywords: mult-mode resource-constraned proect schedulng, evolutonary algorthm, crossover, permutaton based encodng, best ndvdual. I. Introducton Resource-Constraned Proect Schedulng Problem (RCPSP) requres the allocaton of lmted resources to dependent actvtes over tme, such that the makespan of the proect s mnmzed. RCPSP has been ntensvely studed due to ts large area of applcatons n ndustral engneerng, constructon engneerng, cvl engneerng, producton and servce ndustry. Mult-mode Resource-Constraned Proect Schedulng Problem (MRPSP) represents a generalzaton of RCPSP, and therefore even more dffcult to solve, n whch actvtes can be n varous modes affectng the resources needed. It has been proven that RCPSP belongs to the class of NP-hard optmzaton problems [3]. Snce exact solutons cannot be found n polynomal tme by any algorthm, there s a hgh nterest n developng good approxmaton methods for solvng RCPSP able to determne a near-optmal (or optmal) soluton usng reasonable resources. Evolutonary computaton provdes good approxmate methods for solvng problems belongng to ths class. There have been several bo-nspred approaches to ths problem among whch a genetc algorthm wth a new permutaton of prorty-based encodng scheme n [13], a permutaton-based eltst genetc algorthm n [7], a genetc algorthm wth two populaton and a mode optmzaton procedure n [10], an algorthm based on a prorty value encodng scheme [12], a new hybrd genetc algorthm [L, 2011]. An extensve descrpton of all bo-nspred methods for solvng RCPSP and MRCPSP s beyond the scope of ths paper. The role of the recombnaton operator durng the evolutonary search of the MRCPSP soluton s emphaszed n ths paper. Moreover, a new recombnaton scheme that uses genetc materal from the best ndvdual obtaned by the search process besdes genetc nformaton from the parents s presented n ths paper, together wth a proof that the precedence feasblty of solutons s preserved n offsprng f both parents and global best represent feasble solutons. Computatonal experments are performed for several RCPSP and MRCPSP nstances and results support a good performance of the proposed crossover scheme n comparson wth well-known recombnaton operators n an evolutonary search. The paper s organzed as follows: Secton 2 descrbes the Resource-Constraned Proect Schedulng Problem, the evolutonary approach wth an emphass on chromosome representaton s presented n Secton 3, several recombnaton operators for MRCPSP found n the lterature are descrbed n Secton 4, the new crossover n Secton 5, expermental results n Secton 6 and conclusons and drectons for further research n Secton 7. Dynamc Publshers, Inc., USA

2 365 II. Mult-Mode Resource-Constraned Proect Schedulng MRCPSP consders a proect wth a set of actvtes and a set of avalable resources. The mult-mode feature of the problem, whch actually nduces a generalzaton to the Resource-Constraned Proect Schedulng Problem, refers to the fact that each consdered actvty can be n one of a set of modes and requres dfferent resources and has dfferent duratons when fndng tself n dfferent modes. The goal s to fnd a schedule of the actvtes wth mnmum makespan, but consderng both precedence and resources constrants. The problem s detaled and formalzed n what follows. Let J be the number of actvtes (also called obs) and a,..., a } the set of actvtes. The set of possble modes for { 1 J the actvty a s denoted by m, where M 1,..., M }. M { Jobs have to follow certan precedence constrants whch means that a ob can not start before all ts predecessors are fnshed. Let us denote by P the set of all predecessors of actvty a and by S the set of all ts successors. The precedence constrants are usually represented as an acyclc actvty-on-node network. Two addtonal actvtes are also normally consdered: an ntal actvty a 0, whch must precede all other actvtes of the proect and a fnal actvty a whch must be preceded by all actvtes of the proect. J 1 Fgure 1 depcts an acyclc actvty-on-node network for a proect nstance that we consder n order to better llustrate the Mult-Mode Resource-Constraned Schedulng Problem. Fgure 1. Acyclc actvty-on-node network If we denote by d m the duraton of actvty a n mode m, then the precedence relatons can be wrtten as: s d s, a S, (1) 1 1m where s represents the start tme of actvty a. In addton to the set of actvtes, there s also a set of avalable resources. There are three types of resources: renewable resources, whch are characterzed by a constant per-perod-avalablty non-renewable resources, whose avalablty covers the whole proect duraton doubly-constraned resources, whch are lmted both for each tmestep and for the whole proect In order to be executed, each actvty requres a certan amount of some of the resources. If we denote by R the set of renewable resources and by R the avalablty of resource { 1,..., R }, by N the set of non-renewable resources and by N the avalablty of resource l { 1,..., N }, by DC the set of l Andreca and Chra doubly-constraned resources and by DCk the avalablty of resource k { 1,..., DC }, then the lmted avalabltes of these types of resources could be wrtten as: r m a A( t) where actvty R, R, m M, (2) rm represent the quantty of resource R that a s usng n mode actvtes takng place at tmestep t; m and A(t) represent the set of nlm Nl, l N, {1,..., J}, m M, (3) a where actvty nlm represent the quantty of resource N l that a s usng n mode m. As stated above, both types of constrants are appled to doubly-constraned resources DC. It should be noted that both dummy actvtes (ntal and fnal) requre no tme and no resources. The goal s therefore to mnmze sj 1. A feasble schedule s one that comples wth both precedence and resource constrants. In order to llustrate a feasble schedule, let us consder the proect of 10 actvtes descrbed n Table 1. Each actvty of the proect can be n one of two modes, except the dummy actvtes whch only have one mode characterzed by 0 duraton and no resources needed. a m d m m r 1 r 2 m n 1 m n 2 m Table 1. Proect descrpton

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4 Best-Order Crossover n an Evolutonary Approach to Mult-Mode Resource-Constraned Proect Schedulng 366 Fgure 2. Proect descrpton The precedence relatons depcted n Fgure 1 apply to the actvtes belongng to ths proect. Also consderng the dummy actvtes results n the followng lst of actvtes: { 0,1,..., 10,11}. The tme duraton and amount of needed resources for each actvty a n each of the possble modes are presented n Table 1.There are 2 renewable resources wth the followng per-perod-avalablty: R 1 8, R 2 11 and 2 non-renewable resources wth the followng avalabltes: N 1 12, N A feasble schedule for ths proect s depcted n Fgure 2. III. Evolutonary Approach to Resource- Constraned Proect Schedulng Inspred by the process of natural evoluton, evolutonary algorthms (EAs) are search heurstcs wdely used to generate useful solutons to optmzaton problems. A populaton of ndvduals (also called chromosomes) s used to represent the search space of the problem. Each ndvdual encodes a canddate soluton to the problem and an evaluaton (ftness) functon s used to assess ts qualty. EAs evolve a populaton of ndvduals towards better solutons based on mechansms of selecton, recombnaton and mutaton [4]. A standard EA (SEA) s used to address the Mult-Mode Resource-Constraned Proect Schedulng problem. Some of the key aspects of the algorthm wll be descrbed n ths secton. A. Codfcaton The frst thng to consder n an evolutonary approach s how a soluton of the problem s to be encoded n a chromosome. There have been several dfferent codfcatons proposed n the lterature for the Resource-Constraned Proect Schedulng problem, among whch: actvty lst representaton, random key representaton, prorty rule representaton, shft vector representaton and schedule scheme representaton. Our work focuses on the actvty lst representaton whch means that a soluton of the problem s encoded as a lst of the actvtes whch represent ther executon order. Moreover, f one actvty a 2 appears after another actvty a1 n the actvty lst, t means that start tme of actvty a 2 s hgher or equal to start tme of actvty a 1 : s s. (4) a1 a 2 The lst of actvtes must be precedence feasble, meanng that each actvty must have a hgher ndex than each of ts predecessors. Ths representaton s sutable for RCPSP but can be easly extended for MRCPSP [1]. The extenson s done by also consderng a lst of modes, where each actvty wll have a correspondng mode (see Fgure 3). Fgure 3. Chromosome codfcaton (lst of actvtes and correspondng modes) When translatng an actvty lst nto a schedule, each actvty s assgned the smallest possble start tme, thus obtanng an actve schedule [9]. Ths means that no actvty can be left shfted wthout volatng the constrants. In order to llustrate the actvty lst representaton, let us consder the proect descrbed n Table 2. The lst of actvtes and correspondng modes presented n Fgure 4 s precedence feasble and can be translated nto the actve schedule depcted n Fgure 2. Fgure 4. Lst of actvtes and correspondng modes for proect descrbed n Table 2 B. Intal populaton Intal populaton s randomly generated n such a way that only precedence feasble ndvduals are obtaned. Therefore, the frst chromosome gene n the actvty lst s randomly selected from the entre actvtes set, and any other gene s randomly generated from the remanng set of actvtes only f all the predecessors of the chosen gene already exst n the chromosome confguraton obtaned at that pont. A mode s randomly generated for each actvty n the obtaned lst from all possble modes of the correspondng actvty. Ths mechansm does not ensure a resource feasblty of the chromosome. We are referrng of course to non-renewable resources, because the renewable resources constrants can be always ensured by consderng only non-overlappng actvtes. Each non-feasble chromosome wth respect to

5 367 non-renewable resources s subect to a repar procedure presented n the next subsecton. C. Repar operator Let us denote by ERR (N) the number of requested non-renewable resources that exceeded the capacty N l, l {1,..., N }, the so called excess of resource request [10]. ERR (N) s defned as n equaton (5). N l 1 J 1 ERR ( N) (mn(0, N l n lm )). (5) 0 A chromosome whch contans a lst of actvtes and correspondng modes wth ERR ( N) 0 s not feasble wth respect to non-renewable resources. Reparng such a chromosome means takng each actvty from the lst and searchng for a new mode that would mprove ERR(N). If such a mode s found, then t wll replace the old mode. Ths mechansm ensures a possble mprovement of the resource excess, but t does not ensure that only feasble solutons wll be obtaned. However, from our experments, more than 50% of the chromosomes become feasble solutons after beng subect to ths reparaton operator. At the frst glance, t mght not be a good dea to keep unfeasble solutons n the populaton. However, these chromosomes wll be penalzed when computng ther ftness, and therefore they wll not be consdered as relevant genetc materal for the search process. Nevertheless, they wll contrbute to the ncrease of the populaton dversty. D. Ftness functon There are two categores of potental solutons for the problem: feasble solutons wth respect to non-renewable resources non-feasble solutons wth respect to non-renewable solutons. As stated before, the problem of feasblty wth respect to renewable resources does not exst. Also, we only keep n the populaton solutons that are feasble wth respect to precedence constrants. Therefore, from ths pont onwards, the term non-feasble chromosomes wll refer to non-feasble solutons from the non-renewable resources pont of vew. For the feasble solutons the ftness s gven by the start tme of the last dummy actvty, whch actually represents the makespan of the correspondng schedule. It s computed as the total duraton of all actvtes, each actvty actng n the mode gven by the chromosome. If we consder the chromosome c depcted n Fgure 3, the ftness of such a feasble chromosome s gven by: J 1 Ftness ( feasble _ c) d a m a. (6) 0 For the non-feasble chromosomes the ftness s computed as the maxmum duraton of all actvtes (by choosng the modes that ensure the maxmum duraton of each actvty) plus the excess of resource request: J 1 0 Andreca and Chra Ftness ( non_ feasble _ c) max( d ) ERR( N) _ c, (7) where ERR(N)_c represents the excess of resource request for chromosome c and max( d ) represents the maxmum duraton of actvty, consderng all possble modes m. E. Selecton Roulette selecton s used for choosng whch ndvduals should enter the matng pool. The best ndvdual obtaned n one generaton wll always replace one randomly generated ndvdual from the next generaton. Ths mechansm ensures that the best ndvdual obtaned n the last generaton s actually the best ndvdual obtaned n all generatons of the algorthm. F. Recombnaton Several dfferent recombnaton operators wll be consdered for performng comparsons wth the one proposed n ths paper. These operators wll be descrbed n detal n secton IV. However, one feature that s common to all consdered crossover operators s the fact that they do not change the modes of the actvtes when performng recombnaton between parents chromosomes. They do change the order n whch actvtes appear n the actvty lst, but the mode of each actvty s kept the same. Another common feature of all consdered recombnaton operators s that they all create precedence feasble solutons of the problem f the parents represent feasble solutons of the problem. Also, each obtaned offsprng wll be subect to the reparng operator f the schedule t s encodng exceeds the avalable non-renewable resources. G. Mutaton A smple mutaton scheme s appled wth a certan mutaton probablty. Ths operator swaps two genes of a chromosome only f the obtaned permutaton stll represents a precedence feasble soluton of the problem. Mutaton obtans new ndvduals that recombnaton could not obtan. Mutaton s also responsble for changng the modes of the actvtes n the chromosome structure. Therefore, wth the same probablty of mutaton, the mode of each actvty can be changed to another mode from all possble modes of the correspondng actvty. The chromosome obtaned after mutaton wll be subect to the reparng operator f t exceeds the avalable non-renewable resources. IV. Crossover Operators In order to test the effcency of our proposed recombnaton operator, we consder some other crossover schemes proposed n the lterature for permutaton based encodng that preserves precedence relatons between actvtes, for Resource-Constraned Proect Schedulng. Three of the most popular crossover operators for permutaton based encodng are descrbed n what follows. They are smlar to the general crossover operators descrbed by [11] for permutaton based

6 Best-Order Crossover n an Evolutonary Approach to Mult-Mode Resource-Constraned Proect Schedulng 368 encodng, but they have been slghtly adapted so that they take precedence relatons nto account. In the paper where these operators are proposed [6] the authors also present a proof that they all create feasble offsprng f the parents represent feasble solutons of the problem. A. One-Pont Crossover The frst crossover operator s called one-pont crossover. Let us denote by P1 and P2 the parents consdered for recombnaton: P1 1 P1 J P1 {,..., }, P2 1 P2 J P2 {,..., }. An nteger q s randomly chosen so that 1 q J. The frst offsprng s obtaned by takng from the frst parent the entre P1 P1 sequence {,..., } and the rest of the genes are taken 1 q from the second parent so that the obs that have already been taken from the frst parent may not be consdered agan. Ths way the relatve postons of the actvtes n the parents confguraton s kept. Let us consder the followng example n order to llustrate the one-pont crossover: P 1 {1,2,6,4,3,5,7} P 2 {1,5,2,3,6,4,7} For q=3 we obtan the followng frst offsprng: O 1 {1,2,6,5,3,4,7} The second offsprng s obtaned s a smlar way, by changng the role of the parents. B. Two-Pont Crossover The two-pont crossover represents an extenson of the prevously descrbed recombnaton. Two nteger numbers q1, q2 are randomly chosen so that 1 g1 q2 J. The frst offsprng s obtaned by takng from the frst P1 P1 parent the sequence {,..., }, takng the next q2-q1 1 q1 postons from the second parent so that the obs that have already been taken from the frst parent may not be consdered agan and the rest of the postons {,..., } agan from the frst parent, wthout consderng the actvtes already taken at the frst two steps. For the two parents consdered before: P 1 {1,2,6,4,3,5,7} P 2 {1,5,2,3,6,4,7} For q1=3 and q2=5, the frst offsprng would be: O 1 {1,2,6,5,3,4,7}. The other offsprng s obtaned n a smlar way, by consderng the second parent for the frst and for the last sequence, and the frst parent for the second sequence n offsprng. C. Unform Crossover For obtanng the frst offsprng by unform crossover, a zero or a one s randomly generated for each poston n the chromosome. If the poston s 0, then the actvty wth the q2 J lowest ndex that has not been consdered yet s coped from the frst parent. Analogously, f the poston s 1, then the actvty wth the lowest ndex that has not been consdered yet s coped from the second parent. For the two parents consdered before: P 1 {1,2,6,4,3,5,7} P 2 {1,5,2,3,6,4,7} and array {0,1,1,0,1,0,0}, the frst offsprng s: O 1 {1,5,2,6,3,4,7}. The second offsprng s obtaned n a smlar way, by swappng the role of the two parents. V. Proposed Best Order Recombnaton Scheme An ntal varant of the best-order crossover has been proposed by the authors n [5]. The obtaned results proved the effcency of the proposed crossover for the well known Travelng Salesman Problem. The best-order operator has been adapted for Resource-Constraned Proect Schedulng, ensurng that obtaned offsprng represent feasble solutons of the problem. The proposed operator s descrbed n what follows, together wth a proof that feasble parents create feasble offsprng. A. Best-Order Crossover The proposed Best-Order Crossover (BOX) s usng nformaton from the best ndvdual obtaned by the search process, besdes usng genetc materal from the parents. We use ths nformaton n order to accelerate the search process by orentng t towards promsng regons of the search space. In order to be able to dfferentate between good and bad genetc materal, we have to decde what relevant genetc materal means when consderng permutaton based encodng. In TSP for example, t s not mportant that a certan cty has been vsted at a certan moment of tme, but rather the successon of vsted ctes, because the edges of a tour can be seen as the carrers of the genetc nformaton. In RCPSP, the order of the actvtes s also mportant because of the precedence constrants. The proposed BOX also explots the fact that the order of the actvtes s mportant, not ther postons. The man new feature of the proposed crossover operator s the use of genetc materal belongng to the best ndvdual together wth genetc nformaton from the two parents that are subect to recombnaton. Several cuttng ponts are randomly chosen. The number of cuttng ponts s randomly selected and can be even zero. Every two consecutve cuttng ponts (ncludng the begnnng and the end of the chromosome array) wll generate a sequence of alleles; when the number of cuttng ponts s 0, we wll only have one sequence contanng the whole chromosome. One of the followng values wll be assgned to each resultng sequence: -1, -2, -3. These values dentfy the source used for creatng the offsprng. A sequence dentfed by -1 means that the alleles wll be taken from the man parent. A sequence dentfed by -2 means that the alleles wll be taken from the other parent and when -3 s assgned to a sequence, the alleles wll be taken from GlobalBest.

7 369 For example, n order to create the frst offsprng, we consder the frst parent as the man parent. When we have a -1 sequence n offsprng, we take the correspondng postons n the same order from the man parent. For a -2 sequence, we take the correspondng postons from the man parent but the order s gven by the other parent. For a sequence dentfed by -3, we take the correspondng postons from the man parent but n the order mposed by GlobalBest. In Fgure 5, the sequence 4-5 s taken from Parent 1 s the same order, sequence from the frst parent s coped n Offsprng 1 but the order s gven by GlobalBest and the sequence s also taken from the frst parent but the order s gven by the second parent. Fgure 5. Best Order Crossover Operator B. Feasble Solutons THEOREM 1. The offsprng obtaned by BOX represent feasble solutons of the problem f the parents are also feasble solutons. Proof: Let P1 ad P2 be two feasble parents: P1 P1 P1 { 1,..., J }, P2 P2 P2 { 1,..., J } G G and let G { 1,..., J } be the global best obtaned by the search process up to that moment, also a feasble soluton. Let us assume that the frst obtaned offsprng s not a feasble soluton: { 1,..., J },, so that 1 k J and S. k The followng cases may appear: ) both and belong to the same parent or both of k them belong to the global best. Ths means that actvty k s before actvty k n that chromosome, whch s a contradcton of the parents/global best feasblty. ) and belong to dfferent parents or to one k parent and to the global best. Ths means that actvty s before actvty k n P1, because we take sequences of actvtes from P1 n an order gven by P1, P2 or G, whch s a contradcton of P1 feasblty. VI. Expermental Results Andreca and Chra The four recombnaton operators descrbed n Sectons 4 and 5 - One-Pont Crossover (1PX), Two-Pont Crossover (2PX), Unform Crossover (UX), Best Order Crossover (BOX) - are engaged n a standard evolutonary approach to MRCPSP n order to see how the results are affected by the applcaton of dfferent crossover schemes. The parameters used by the evolutonary algorthm are gven n Table 2. Parameter Value Populaton sze 100 Number of generatons 100 Mutaton probablty 0.05 Table 2. Evolutonary algorthm parameters For testng the performance of the proposed recombnaton scheme, several ProGen proect nstances have been consdered [8]. ProGen s a Proect Schedulng Problem Instance Generator of controlled dffculty, whch s focused on precedence constraned and resource constraned proect schedulng problems. Instances for both sngle-mode and mult-mode resource constraned schedulng problems are thus obtaned. Some results obtaned for the sngle-mode varant of the problem (RCPSP) for proects wth 30 actvtes have been presented n [2]. The experments performed n ths paper wll nclude both more complex sngle-mode schedulng problems (the complexty beng gven by the ncreased number of actvtes), and mult-mode schedulng problems wth a large number of actvtes. For the sngle-mode varant of the problem we have consdered 80 ProGen nstances havng between 30 and 120 actvtes. Table 3 presents the best and the average results over 10 runs of the standard evolutonary algorthm wth all four recombnaton operators for each of 20 randomly chosen nstances. For the mult-mode varant, 400 ProGen nstances wth a number of actvtes varyng between 10 and 30, have been consdered. The results obtaned for 20 randomly chosen nstances are presented n Table 4. These results refer to the best and average results over 10 runs of the evolutonary algorthm wth all four recombnaton operators for each consdered problem nstance. The bold values n both Tables 3 and 4 represent the best solutons obtaned for each nstance. When usng BOX, the algorthm s able to fnd the known soluton of the problem n the hghest percentage, compared to the other recombnaton operators. Also, f we take a look at the mean values obtaned after 10 runs, t can be notced that n most of the runs the result obtaned by the algorthm usng BOX s closer to the known soluton of the problem compared to the other ones. The convergence curve obtaned n one run of the algorthm for a sngle-mode nstance and one run for a mult-mode nstance are depcted n Fgures 6 and 7. It can be notced that not only the fnal results are better when usng BOX, but the soluton s detected n fewer generatons, whch reduces the tme and resources needed for detectng t. Ths behavor has

8 Best-Order Crossover n an Evolutonary Approach to Mult-Mode Resource-Constraned Proect Schedulng 370 been observed for most of the consdered nstances. Therefore, the test results ndcate the acceleraton of the search process when usng the proposed recombnaton operator. 1PX 2PX UX BOX Best Avg Best Avg Best Avg Best Avg 301_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Table 3. Best and average makespan obtaned after 10 runs for sngle-mode nstances 1PX 2PX UX BOX Best Avg Best Avg Best Avg Best Avg c1510_ c2111_ _ _ _ _ _ _ _ _ _ m411_ m510_ n310_ n310_ r110_ r311_ r510_ r511_

9 371 Andreca and Chra r511_ c1510_ Table 4. Best and average makespan obtaned after 10 runs for mult-mode nstances obtanng the offsprng from feasble parents. The man feature of the proposed operator s the use of genetc nformaton from the best ndvdual besdes the two parents consdered for recombnaton. Expermental results performed on ProGen proect nstances ndcate a superor performance of the proposed operator, thus emphaszng the role that recombnaton has n acceleratng the search n an evolutonary process. An extensve study of all recombnaton operators used for RCPSP and MRCPSP wll be performed and the experments wll be extended to more nstances wth more actvtes. Fgure 6. One run of the evolutonary algorthm for nstance 1201_1. The results for each recombnaton operator are depcted over generatons. Fgure 7. One run of the evolutonary algorthm for nstance 510_7. The results for each recombnaton operator are depcted over generatons. The proposed operator does not requre extra resources because an evolutonary algorthm usually keeps track of the best ndvdual obtaned by the search process at each tme step. VII. Conclusons A new recombnaton operator for permutaton based encodng has been proposed n ths paper. Ths operator s sutable to Resource-Constraned Proect Schedulng Problem and Mult-Mode Resource-Constraned Proect Schedulng Problem because t preserves the precedence constrants when Acknowledgments Ths research s supported by Grant PN II TE 320, Emergence, auto-organzaton and evoluton: New computatonal models n the study of complex systems, funded by UEFISCDI, Romana. References [1] Alcaraz, J., Maroto, C., A New Genetc Algorthm for the Mult-Mode Resource-Constraned Proect Schedulng Problem, [2] Andreca, A., Chra, C., The Role of Crossover n Evolutonary Approaches to Resource-Constraned Proect Schedulng, Proceedngs of the Internatonal Conference on Intellgent Systems Desgn and Applcatons (ISDA 2012), Koch, Inda, pp , [3] Blazewcz, J., Lenstra, J., Rnnooy Kan, A., Schedulng subect to resource constrants: Classfcaton and complexty. Dscrete Appled Mathematcs 5, pp , [4] Dumtrescu, D., Lazzern, B., Jan, L.C, Dumtrescu, A., Evolutonary Computaton, CRC Press, Boca Raton, FL., [5] Gog, A., Chra, C., Recombnaton operators n permutaton-based evolutonary algorthms for the travellng salesman problem, Chapter 10 n Logstcs Management and Optmzaton through Hybrd Artfcal Intellgence Systems, IGI Global, pp , [6] Hartmann, S., A Compettve Genetc Algorthm for Resource-Constraned Proect Schedulng, Naval Research Logstcs 45, pp , [7] Km, J.-L., Ells, R.D., Permutaton-Based Eltst Genetc Algorthm for Optmzaton of Large-Szed Resource-Constraned Proect Schedulng, J. Constr. Eng. Manage. 134, 904, [8] Kolsch, R., Schwndt, C., Sprecher, A., Benchmark nstances for proect schedulng problems; Kluwer; Weglarz, J. (Hrsg.): Handbook on recent advances n proect schedulng, pp , [9] Kolsch, R., Seral and parallel resource-constraned proect schedulng methods revsted: Theory and

10 Best-Order Crossover n an Evolutonary Approach to Mult-Mode Resource-Constraned Proect Schedulng 372 computaton. European Journal of Operatonal Research, 90, , [10] Peteghem, V., Vanhoucke, M., A genetc algorthm for the Mult-Mode Resource-Constraned Proect Scedulng Problem, workng paper, [11] Reeves, C.R., Genetc algorthms and combnatoral optmzaton. In V. J. Rayward-Smth, edtor, Applcatons of modern heurstc methods, pages Alfred Waller Ltd., Henley-on-Thames, [12] Ren, Y.H., Kong, D.C., Peng, W.L., A Genetc Algorthm Based Soluton wth Schedule Mode for RCPSP, Advanced Materals Research, vol , pp , [13] Zhang, H., A Genetc Algorthm for Solvng RCPSP, Computer Scence and Computatonal Technology, ISCSCT '08. Internatonal Symposum on, pp , Author Bographes Anca Andreca receved the BSc degree n 2001, MSc degree n 2002 and PhD degree n 2007, all degrees n computer scence from Babes-Bolya Unversty, Clu-Napoca, Romana. Anca s currently assstant lecturer wthn the Department of Computer Scence, Babes-Bolya Unversty, Romana. Her man research nterests nclude evolutonary computng, nature-nspred metaheurstcs and complex systems. Anca has publshed three books and over 50 papers. Camela Chra receved the BSc degree n computer scence from Babes-Bolya Unversty, Clu-Napoca, Romana n From 2000 to 2005 she was engaged n full-tme research at Galway-Mayo Insttute of Technology (GMIT), Galway, Ireland focusng n the area of agent-based systems and ontologes for dstrbuted collaboratve desgn envronments. She receved the MSc and PhD degrees from GMIT Ireland n 2002 and 2005, respectvely. Snce 2006, Camela s a researcher wthn the Department of Computer Scence and Centre for the Study of Complexty, Babes-Bolya Unversty, Romana, currently workng n the study and analyss of complex systems usng cellular automata, complex networks and supportng artfcal ntellgence models. Her man research nterests nclude computatonal ntellgence, swarm technques, complex systems and networks, machne learnng, mult-agent systems and ontologes wth applcatons rangng from dstrbuted cooperatve work, optmzaton, plannng, schedulng, routng and logstcs to socal network analyss, data mnng and bonformatcs. Camela Chra has publshed over 100 scentfc papers on these topcs n varous ournals and conference proceedngs, 5 book chapters and 2 books.