Feasibility of Quantum Genetic Algorithm in Optimizing Construction Scheduling

Transcription

1 Feasibiliy of Quanum Geneic Algorihm in Opimizing Consrucion Scheduling Maser Thesis Baihui Song JUNE 2013 Commiee members: Prof.dr.ir. M.J.C.M. Herogh Dr. M. Blaauboer Dr. ir. H.K.M. van de Ruienbeek Design and Consrucion Processes Building Engineering Faculy of Civil Engineering and Geoscience Delf Universiy of Technology

2 Absrac The increasing scale of projecs in civil engineering indusries leads o he increase of he amoun and complexiy of daa ha need o be reaed and calculaed. As a new non-radiional way of compuing, quanum compuing shows exremely powerful capabiliies in huge daa analysis and processing. In his research we focus on opimizing civil engineering consrucion scheduling using quanum geneic algorihms which have he ousanding performance in opimizaion and daa processing, o creae he possibiliy of dealing wih civil engineering problem wih quanum compuing. This research consiss of an exensive lieraure sudy ha includes geneic algorihms, quanum geneic algorihms and consrucion scheduling. Exising applicaions of opimized design wih geneic algorihms and quanum geneic algorihms are he heoreical basis for he hypohesis. Based on he heoreical research, an improved quanum geneic algorihm which is used for consrucion scheduling opimizaion design wih he consrains of ime and human resourcing is esablished and named as AQGA (A quanum geneic algorihm). A null hypohesis is aken: AQGA canno help o improve he opimizaion design in civil engineering consrucion scheduling. Then a case sudy is done wih AQGA: hrough he analogy wih flow-shop scheduling, uni consrucion scheduling is chosen o be processed wih AQGA. A he end of he case sudy, he flowchar of uni consrucion opimized design wih improved quanum geneic algorihm AQGA is given, uni consrucion opimized design is realized hrough he improved quanum geneic algorihm There are some consrains and limiaions in he sudy and research. For example here are simplified and idealized assumpions since a realisic simulaion canno be realized because here is no available quanum compuer o run he simulaion. Bu he uni consrucion scheduling opimize problem is expressed in quanum compuing language. I is expeced o be a saring poin for fuure applicaion of quanum geneic algorihms wihin consrucion scheduling in civil engineering projecs and even more broad, quanum compuing wihin civil engineering. 1

3 Abbreviaions NP GA QGA AQGA MSP FSSP PID Non-deerminisic Polynomial Geneic Algorihm Quanum Geneic Algorihm A Quanum Geneic Algorihm Microsof Projec Flow-Shop Scheduling Problem Proporion Inegraion Differeniaion 2

4 Lis of Figures Figure 1 Explored field... 9 Figure 2 Geneic algorihm Figure 3 Concepual srucure Figure 4 Quanum roaion gae diagram Figure 5 Flow-shop scheduling Figure 6 QGA solving flowchar Figure 7 A qubi superposiion sae diagram: Bloch sphere Lis of Tables Table 1 Roaion angle selecion sraegy

5 Table of Conens Absrac... 1 Abbreviaions... 2 Lis of Figures... 3 Lis of Tables... 3 Table of Conens Inroducion Background Problem saemen Research focus Hypohesis Research mehod Research objecive and research quesion Limiaions of he research Srucure of he hesis Lieraure sudy Geneic algorihm Quanum geneic algorihm Consrucion scheduling Applicaions of opimized design wih geneic algorihms and quanum geneic algorihms Concepual framework Hypohesis Theoreical basis and foundaion AQGA Working mechanics Qubi encoding Decoding Quanum roaion gae Operaor design Workflow of AQGA Case sudy Background Modeling Opimizaion algorihm seps Resul Conclusion and recommendaion Answer o research quesion Achievemens Limiaions Fuure research

6 Nomenclaure References: Appendix 1 Quanum and Quanum Mechanics Appendix 2 Quanum Compuing A 2-1 Qubi A 2-2 Quanum gaes A 2-3 Developmen Saus of quanum compuing

7 1. Inroducion The coninuous developmen of he consrucion and civil engineering indusries and he increasing scale of such as PPP projecs has become an ineviable rend. The increasing scale of projecs will lead o an increased amoun of daa ha needs o be reaed and calculaed. I shows more and more ha using radiional mehods o deal wih hese issues is inadequae, which is likely o become less and less efficien. As a new non-radiional compuing mehod, quanum compuing has powerful capabiliies in large daa analysis and processing. Can his be a linking key poin of civil engineering and quanum compuing? The focus of his research is o invesigae he possibiliies of dealing wih civil engineering consrucion scheduling using quanum geneic algorihms because we are moivaed by he ousanding performance in opimizaion and daa processing of quanum compuing. The possibiliies of combining boh will be explored, o achieve opimized design of he consrucion scheduling wih quanum geneic algorihm Background For civil engineering projecs, planning and programming of projec consrucion is necessary. Consrucion scheduling is a key elemen of he planning and programming of projec consrucion. Consrucion scheduling is o make echnical schedule for provisions of he saring ime of consrucion preparaion work and he main projec, duraion of compleion and operaion, consrucion procedures, and srengh of consrucion. I aims o ensure he consrucion works can be compleed according o he conrac deadline.(jiao, 2012) The consrucion work needs o be carried ou in accordance wih he arrangemens of he consrucion scheduling. The principles of consrucion scheduling are coninuiy, rhyhm and balance.(jiao, 2012) The esablishmen of consrucion scheduling is carried ou based on assembly-line principles. There are many planning insrumens, and Gan char mehod and nework planning mehod are wo widely used mehods. The Gan char mehod is condensed and easy o undersand bu for paricularly complex projecs, he Gan char mehod is difficul o mee he needs of program managemen. Nework planning mehods which make up for he defecs of he Gan char are being used more in building indusry now. And during implemenaion i can also predic he impac of changes on duraion and subsequen work, which is helpful o ake imely counermeasures.(commiee, 2009) Currenly, here is an increasing amoun of large-scale projecs such as skyscrapers and infrasrucure projecs. In hese projecs, more arrangemens are needed for ime and human resource planning. Tha requires more opimized and efficien design for 6

8 consrucion scheduling o achieve ime opimizaion and maximizaion of benefis. In he design and consrucion of he large-scale projecs, uni design and consrucion are more and more used. Unis mean some pars which are similar or even idenical o each oher, like uni façade, uni beam, uni column, uni floor, and so on. These unis always have he same srucures, same maerials, same consrucion order, ec. For example, uni façade is widely applied in some skyscrapers. Judging from he appearance i looks like assembled by unis. Every uni has same srucure like column, beam, shaf, box, ec. and same maerials like concree, seel, glass, ec. And hey also have he same consrucion order. For example firs verical columns, second horizonal beam, hen sequenially insallaion of differen pars, and so on. For he uni consrucion if a sysemaic consrucion scheduling can be done, i will save los of ime and make he work much more efficien. Uni design can be a simplified and opimized designed for consrucion scheduling and i also can be a saring poin o connec and combine civil engineering and quanum compuing, consrucion scheduling and quanum geneic algorihms. Now here are some exising opimizaion mehods for consrucion scheduling such as sofwares Primavera, Microsof Projec ha are widely used. These sofwares have go good managemen performance, and opimize he progress of he projec, which makes he projec work more efficienly. However, he opimizaion mehods for consrucion scheduling can be improved more. Calculaion and opimizaion capaciy of quanum compuing is ofen far higher han any of he radiional ools. In fac, he consrucion scheduling is an NP problem; also belong o muli-objecive opimizaion problem. The NP problem and muli-objecive opimizaion problem can be solved by quanum geneic algorihms. NP problem is he problem which can be solved wih non-deerminisic machine in polynomial ime.(yun, 2005) NP means non-deerminisic polynomial. If and only if an NP-complee problem L can be Turing-reducible o H in polynomial ime, he problem H is NP-hard.(Garey and Johnson, 1979) Typical NP problem includes shor-pah problem, scheduling problem, graph coloring problem, daabase search problem, and so on. Consrucion scheduling of complex projecs is also an NP problem. Quanum compuing is a mehod for performing calculaion based on he heory of quanum mechanics. Quanum compuing provides new possibiliies o solve he complex problems such as he NP problem. Quanum geneic algorihm is a combinaion of quanum compuing and geneic algorihm. Geneic algorihm is a search algorihm used o solve opimizaion in compuaional mahemaics. In recen years, he geneic algorihms have been successfully applied o reliabiliy opimizaion, flow shop scheduling, job shop scheduling, machine scheduling, equipmen layou, image processing, daa mining, and so on.(mai, T., 2010) Wih he combinaion of quanum compuing and geneic algorihm, quanum geneic algorihm has a good abiliy of solving NP problems like Travelling Salesmen Problem, scheduling problem, knapsack problem, and so on.(ma and Wang, 2005) 7

9 More informaion abou quanum and quanum compuing can be found in appendices 1 and Problem saemen There are more and more large-scale civil engineering projecs and he conen relaed o hese projecs become more and more, he consrucion schedule becomes increasingly complex.(liu, Wang, e al., 2007) In one projec, differen ime and human resourcing planning can be creaed, and hey will lead o differen consrucion resuls. Therefore, he progress of he projec design becomes criical. I is a key poin of a projec o make he opimal consrucion scheduling design. To opimize he consrucion scheduling design, an opimal consrucion sequence is a very imporan sep for he good compleion of a projec. Through radiional mehods, he opimal one needs o be esed one by one from he huge amoun of choices. This requires los of work and is less efficien, and he opimal consrucion sequence canno be found quickly. In mos cases, i is no possible o check all of he possibiliies. Therefore, consrucion scheduling design wih radiional mehods becomes more and more difficul o mee he requiremens. However, as he cenral elemen of he planning and programming of projec consrucion, i is necessary o find he opimal choice since a successful consrucion scheduling plan is a very imporan sep o realize a successful projec. Tha will promoe he projec a lo. To achieve ime opimizaion, maximizaion of benefis and saisfy cliens differen requiremens, opimal design of he consrucion scheduling is required o realize he opimal arrangemen of ime and human resource planning Research focus In he firs sep, he research focus is mainly on he quanum geneic algorihms and wha can be solved by quanum geneic algorihms. Then a null-hypohesis will be proposed of a new improved quanum geneic algorihm. A he same ime he research focus will shif o he new improved quanum geneic algorihm. The work mechanics and workflow of he improved quanum geneic algorihm will be explained and a case sudy will be presened o explore how o apply he improved quanum geneic algorihm on opimal consrucion scheduling design. The figure1 shows he fields explored and he research focus. 8

10 Consrucion Scheduling AQGA Geneic Algorihm QGA Quanum Compuing Figure 1 Explored field 1.4. Hypohesis There is an algorihm called A quanum geneic algorihm (AQGA). This algorihm is an improved quanum geneic algorihm. The null hypohesis of he research is: AQGA canno help o improve he opimizaion design in civil engineering consrucion scheduling Research mehod The exising quanum geneic algorihms will be used as he heoreical base of he research. The exising applicaions of quanum geneic algorihms will be used as he saring poin o discover he possibiliy of connecing and combining civil engineering and quanum compuing. Lieraure sudy will be he main mehod o collec he informaion and esablish he hypohesis and furher heoreical analysis. A case sudy will help o show how o inpu consrucion scheduling in quanum geneic algorihm and how o express consrucion scheduling in quanum geneic algorihm language. Finally we will have our conclusion and recommendaions based on he case sudy, as well as expecaions and possible fuure developmen. Due o some limiaions an acual quanum compuaion as a par of he case sudy canno be performed. This will be explained in Chaper

11 1.6. Research objecive and research quesion A proper ime and human resource planning is an imporan sar for a projec. As menioned in he problem saemen, he research objecive is o realize he opimal design of he consrucion scheduling and o realize he opimal arrangemen of ime and human resource planning using quanum compuing. Research quesion: How o apply quanum geneic algorihm o consrucion scheduling design? 1.7. Limiaions of he research The bigges limiaion of he research is ha acual quanum operaion and simulaion are no available for his research due o here is no an available quanum compuer which can run he operaion and simulaion. In addiion, he algorihm used in he process is an assumed algorihm. Alhough he hypohesis has been done based on adequae heoreical basis from exising algorihms, i sill have no been developed ye and here is no deailed working mechanism and funcions for every sep in he selecion process. So if here is an available quanum compuer he simulaion also canno be done Srucure of he hesis In his research quanum compuing and civil engineering are expeced o be combined. Afer research on los of applicaions of quanum compuing, quanum geneic algorihm from quanum compuing and consrucion scheduling from civil engineering were chosen o be combined. These are briefly inroduced in background in chaper1. And in chaper 1, he problem, research focus, research mehod, research objecive and limiaions of he research are also inroduced. In chaper 2, firs geneic algorihms, quanum geneic algorihms and consrucion scheduling are inroduced. There is some informaion abou he definiions, working mechanics, curren siuaion, ec. More informaion is available in he appendix. Then some applicaions of opimized design wih geneic algorihms and quanum geneic algorihms are inroduced, including indusrialized use, parameer opimizaion, combinaorial opimizaion, muli-objecive opimizaion, ec. In chaper 3, an improved quanum geneic algorihm is esablished and named as AQGA. 10

12 I is used for consrucion scheduling opimizaion design wih he consrains of ime and human resourcing. A null hypohesis is aken for his research and he hypohesis is: AQGA canno help o improve he opimizaion design in civil engineering consrucion scheduling. And he hypohesis is described in more deail. The working mechanics and seps of AQGA are explained. In chaper 4, a case sudy is done wih AQGA. Through he analogy wih flow-shop scheduling, uni consrucion scheduling is chosen o be processed. A model is buil and a finess funcion is defined. Then he iniial populaions will be measured, decoded and evaluaed. Populaions are updaed wih quanum roaion gae. Caasrophic operaor, crossover operaor and muaion operaor are used in he process. Then circulaory sysem is esablished and sars, unil he erminaion condiion is me and he opimal individual is found. A his poin, he uni consrucion scheduling opimizaion design wih AQGA is compleed. In chaper 5, conclusion and recommendaions are given. The main research resuls are summarized and he oulook of furher research work is elaboraed. 11

13 2. Lieraure sudy This chaper presens heoreical undersandings of geneic algorihms, quanum geneic algorihms, applicaions of quanum geneic algorihms, consrucion scheduling, and he possible connecion and combinaion of quanum geneic algorihms and consrucion scheduling design. This chaper consiss of five secions. The firs par and he second par inroduce he geneic algorihms, quanum geneic algorihms, and he applicaions of hem respecively. Then consrucion scheduling and radiional mehods of consrucion scheduling design are inroduced. Followed he fourh par some successful applicaions of opimized design wih quanum geneic algorihms are saed, which show he possibiliy of he combinaion of quanum geneic algorihm and consrucion scheduling design. The las par presens he concepual framework Geneic algorihm Geneic algorihms are search algorihms used o solve opimizaion problem in compuaional mahemaics. The geneic algorihm is one of he evoluionary algorihms. I is developed based on some phenomena in evoluionary biology, including geneic, muaion, naural selecion and hybridizaion. The geneic algorihms are usually implemened by compuer simulaion. For an opimizaion problem, a number of absrac represenaions (called chromosomes) of candidae soluions (called individuals) evolve o a beer soluion. The evoluion sars from a populaion of compleely random individuals, and hen occurs generaion by generaion. By he finess evaluaion, a pluraliy of individuals are randomly seleced from he curren populaion (based on heir finess), and new populaions are generaed by naural selecion and muaion.(melanie, 1999) This process coninues unil he erminaion condiions are saisfied. Generally erminaion condiions are: - Resricions on he number of imes of evoluion; - Resource consrains on he cos of he calculaions(e.g. calculaion ime and memory occupied); - An individual has me he condiions of he opimal value, i.e. he opimal value has been found; - Finess has reached sauraion, coninued evoluion will no produce beer finess individuals; - Human inervenion; - A combinaion of wo or more of he above.(ji, 2004) The basic operaions of he geneic algorihm process are as follows: - Iniializaion: Se he evoluion algebra Couners = 0, se he maximum evoluion 12

14 algebra T, and M individuals randomly generae as he iniial populaion P (0). - Individual Evaluaion: Calculae he finess of individuals in groups P (). - Selecion: apply he selecion operaor o groups. The purpose of he selecion is o make he opimizaion individuals direcly o he nex generaion, or new individual paired cross hen inheried o he nex generaion. The selecion is based on he assessmen of he individual's finess. - Crossover: apply he crossover operaor o groups. The so-called crossover refers o he operaion of replacing and resrucuring he par of he srucure of he wo paren individuals and generaing new individuals. Crossover plays a cenral role in he geneic algorihm. - Muaion: apply he muaion operaor o groups. Tha changes he value of cerain gene locus of individual sring in groups. - Groups of P (), afer selecion, crossover and muaion operaion, generae he nex generaion of groups P ( 1). - Judgmens of Terminaion condiion: if T = T, oupu he obained individual wih he larges finess by he process of evoluion as he opimal soluion, and erminae he calculaion.(ji, 2004) Randomly generae iniial populaion Mee he convergence crieria? N Y Oupu search resuls Perform selecion operaor Random [0,1] < Pc N Y Perform crossover operaor N Random [0,1] < Pm Y Perform muaion operaor Figure 2 Geneic algorihm (Yi, M., 2012) Compared wih radiional opimizaion algorihms, geneic algorihm has a few differences. Geneic algorihm doesn' have a direc effec on he parameer se bu uses some 13

15 encoding of parameer se. The geneic algorihm sars he search from a group of poins insead of a single poin. The geneic algorihm uses he finess value wihou derivaive or oher auxiliary informaion. The superioriy of he geneic algorihm are, firs, i hardly fall ino local opima in he search process. Even if he definiion of he finess funcion is no coninuous, irregular or noise, i can also find he overall opimal soluion wih grea probabiliy. Secondly, due o he inheren parallelism, geneic algorihm is very suiable for large-scale parallel compuer.(yi and Liu, 2001) Geneic Algorihms specializes in solving he global opimizaion problem. In recen years, he geneic algorihm has been successfully applied o reliabiliy opimizaion, flow shop scheduling, job shop scheduling, machine scheduling, equipmen layou, image processing, and daa mining.(mai, T., 2010) Geneic algorihm is a bionics algorihm. There is no absolue guaranee for is convergence. I inroduces "naural selecion" principles ino he opimizaion process. Because i has fewer resricions for problems such as problem objecive funcion and consrains are neiher required o be differeniable nor required coninuous, only required o be compuable, and i always search hroughou he soluion space and is able o find near-global opimal soluion. I has a wide range of applicaions in nework planning opimizaion. Compared wih oher search algorihms, geneic algorihm is compuaionally efficien; especially for complex planning opimizaion problem is excellen performance is more powerful. Wih radiional mehods he opimizaion process easily fall ino local minimum pligh bu GA overcomes ha. I is a very effecive algorihm wih superior performance. I is paricularly suiable for solving complex problems.(wang, Qin, e al., 2004) 2.2. Quanum geneic algorihm Quanum geneic algorihm is a produc of he combinaion of quanum compuaion and geneic algorihms. Quanum geneic algorihm is a newly developed opimizaion mehod based on he principle of quanum compuing. Based on some conceps and heories of quanum compuing, quanum geneic algorihm uses quanum bi encoding o represen chromosomes, and realizes he evoluionary search by quanum gaes and quanum gaes updaing. Quanum geneic algorihm has excellen convergence speed and global search abiliy. And he performance of he algorihm will no be affeced wih a small populaion size.(zhang, Li, e al., 2004) Based on he quanum sae vecor represenaion, quanum geneic algorihm applies he qubi probabiliy ampliude o he coding of he chromosome, so ha one chromosome can express muliple superposiions. The updae of he chromosome is realized by quanum roaion gae and quanum non-gaes. In his way quanum geneic algorihm can achieve opimizaion soluion. (Yang and Zhuang, 2003) 14

16 The same as he radiional GA, QGA qubi probabiliy ampliude obain he bis hrough comparaive Law, o calculae he finess value.(zhou and Qian, 2008) Yang uses he binary code of geneic algorihm o realize qubi encoding for problem of polymorphic. For example, one qubi is used o encode wo saes, wo qubis are used o encode four saes. The advanage of his mehod is good versailiy and simple.(yang and Zhuang, 2003) A qubi can be expressed as: φ =α 0 +β 1 Where α and β are complex numbers, and α 2 + β 2 =1 Where 0> and 1> denoe spin up and spin down saes respecively. So a qubi can simulaneously conain informaion of sae 0> and 1>. Using qubis encoded a chromosome can express muliple superposiion, ha makes quanum geneic algorihm has a beer diversiy characerisics han he classic geneic algorihm. Beer convergence can also be reached by using qubis coding. Wih he α 2 or β 2 ending o 0 or 1, quanum bis encoded chromosome will converge ino a single sae.(yang and Zhuang, 2003) In heory, ha anyhing ha can be solved by geneic algorihm is applicaion objec of quanum geneic algorihm.(yang and Zhuang, 2003) Compared wih he radiional GA which uses crossover and muaion operaion o mainain he diversiy of he populaion, QGA uses quanum gaes acing on qubis probabiliy ampliude o mainain he diversiy of he populaion, so he updaing of quanum gaes is he key of QGA.(Zhou and Qian, 2008) Because qubi chromosomes are able o characerize he superposiion sae, compared wih he radiional GA, QGA has beer diversiy of populaion and he abiliy of global opimizaion.(feng and Su, 2011) Narayanan and Moore proposed quanum-inspired geneic algorihm (Narayanan and Moore, 1996), Han K H and Kim J H proposed geneic quanum algorihm (Han and Kim, 2000) and Han K H, Park K H, Lee C H and Kim J H proposed parallel quanum-inspired geneic algorihm (Han, Park, e al.,, 2001). These algorihms are used o solve combinaorial opimizaion problems. The resuls show ha, he performance of quanum geneic algorihm is much beer han convenional geneic algorihm.(zhang, Li, e al., 2004) In he compuaional models of quanum compuing and quanum circuis, a quanum gae (or quanum logic gaes) is a quanum circui which has basic operaion on a small number of qubis. I is he basis of he quanum circui, jus like he relaionship beween he radiional logic gaes wih digial lines.(nielsen and Chuang, 2000) Commonly used quanum gaes include Hadamard gae, Pauli-X gae, Pauli-Y gae, Pauli-Z gae, Phase shif gaes, Swap gae, Conrolled gaes, Toffoli gae, ec. The quanum gaes which are mainly used in his sudy are he quanum NOT gae and 15

17 quanum roaion gae.(wang, 2009) Quanum NOT gae is Pauli-X gae. I acs on a qubi. This is a quanum gae equivalen o he logical NOT gae. I can exchange he probabiliy ampliude of l0> and l1>, which means l0> is replaced by l1> and l1> is replaced by l0>, and αl0> +βl1> is replaced by αl1> +βl0>. This gae can be expressed in a marix: X=[ ]. Quanum roaion gae is expressed wih marix as: U=[ cos θ i sin θ i ]. The parameers sin θ i cos θ i of quanum roaion gae can be adjused so i is more versaile. Therefore, in he updaing of populaion of quanum geneic algorihm, he quanum roaion gae is mainly used. The process is as follows: [ α i β ]=U i [ α i i β ] =[ cos θ i i sin θ i sin θ i ] [ α i cos θ i β ] i In he formula, [ α i β ] is he qubi before he updae, [ α i i β ] is he updaed qubi, θ i is he i roaion angle, and is size and direcion deermine he performance of he algorihm.(yang, Lin, e al., 2003) More abou quanum, quanum compuing and quanum gaes please refer o Appendix1 and Appendix Consrucion scheduling Consrucion scheduling is a cenral elemen of he planning and programming of projec consrucion. Consrucion scheduling is making consrucion plans which give he provisions of commencemen ime, compleion ime and consrucion sequence of he various projecs. I needs o ensure he compleion ime of consrucion works according o he conrac deadline. Consrucion schedule is he basis of he execuion of all he consrucion work. The process of designing consrucion schedule is: (Commiee, 2009) ⑴ Divide he consrucion sages ⑵ Calculae workload ⑶ Deermine he amoun of labor and equipmen ⑷ Deermine he consrucion ime of differen sages (days or weeks) ⑸ Make iniial program of he consrucion schedule ⑹ Check and adjus he consrucion schedule iniial program 16

18 Gan char mehod and nework planning mehod are wo widely used consrucion scheduling design mehods in pracical civil engineering projecs. Gan char mehod is a ime-agged abular form plan. I is a radiional consrucion scheduling mehod and i has been use for a long ime in civil engineering projecs. I has he advanages like condensed and easy o undersand. Bu for he paricularly complex projecs, i shows he disadvanages ha i canno represen he relaionship beween he various asks and he key poin of compleion of he plan in he progress schedule. So nework planning mehod is more used nowadays. This is a mehod consiues arrows from lef o righ according o he sequence and flow direcion of he work process. Wih nework planning mehod he consrucion managemen saff can easily find he key poin and focus on i. During execuion i can predic he impac of changes on duraion and subsequen work in order o ake imely counermeasures. Currenly here are some exising opimizaion mehods for consrucion scheduling. Some sofwares like Primavera, Microsof Projec, ec. are widely used. Primavera is a projec plan managemen sofware. Primavera is a powerful, inegral and easy-o-use soluion which includes prioriizing, planning, managemen, and execuion of projec, program and projec porfolio.(jian, C., 2008) Primavera can help o make beer porfolio managemen decisions, evaluae projec risks and reurns, deermine he resources and skills required o complee he work. As a firs-class soluion, Primavera has he execuion and conrol capabiliies of finishing he delivery of he projec wihin specified ime, budge, qualiy and scope of design. Primavera is involved in he fields of projec managemen including: scope managemen, schedule managemen, cos managemen, resource managemen, risk managemen, and communicaion managemen.(li, H., 2011) Microsof Projec (or MSP) is a projec managemen sofware program developed and sold by Microsof. This sofware is designed o assis he projec manager o develop plan, assign resources o asks, rack progress, manage budges and analyze workload. The applicaion can generae he criical pah schedule, and he key chain can be made Gan char visualizaion.(ding, W., 2007) Microsof Projec combines usabiliy, powerful feaures and flexibiliy ogeher perfecly. Microsof Projec provides a reliable projec managemen ools, which helps consrucion managemen saff manage projecs more efficienly. Microsof Projec can be inegraed wih he familiar Microsof Office sysem programs, which can help consrucion managemen saff say informed, conrol projec work, schedules, and finances, mainain close cooperaion wih projec working group, and improve he efficiency.(lai, Z., 2008) 17

19 2.4. Applicaions of opimized design wih geneic algorihms and quanum geneic algorihms Geneic algorihms have been successfully applied o reliabiliy opimizaion, flow shop scheduling, job shop scheduling, machine scheduling, equipmen layou, image processing, and daa mining in recen years. The research of quanum geneic algorihms is sill in is infancy, and hus is relaed research is confined o limied aspecs.(yang and Zhuang, 2003) The heoreical applicaions are also more han he pracical applicaions. In a heoreical applicaion, Han K H used he geneic quanum algorihm solving he 0-1 knapsack problem and verified he efficiency of he algorihm. 100, 250 and 500 were used as he sandard daa of objecs in he experimen, and opimizaion soluions are obained by applying geneic quanum algorihm.(han, 2000) Ll B, Zhuang Z used he quanum geneic algorihm for funcion opimizaion. De Jong's unimodal funcion and Rosenbrock's mulimodal funcion opimizaion were seleced o opimize by quanum geneic algorihm. The resul was significanly beer han he resul of convenional geneic algorihm and geneic quanum algorihm.(li and Zhuang, 2002) Li B applied quanum geneic algorihm o financial informaion daa mining solving frequen paern discovery problem in ime series. The resuls were beer han convenional geneic algorihm, niche geneic algorihm and geneic quanum algorihm.(li, 2001) Yang J, Zhuang Z combined quanum geneic algorihm and muli-universe parallel quanum geneic algorihm o independen componen analysis algorihm, and proposed a new blind source separaion algorihm based on quanum geneic algorihm and muli-universe parallel quanum geneic algorihm. The simulaion resuls showed ha he compuing efficiency of quanum geneic algorihm and muli-universe parallel quanum geneic algorihm is 5 o 15 imes higher han ha of convenional geneic algorihms.(yang and Zhuang, 2003) Geneic algorihms and quanum geneic algorihms also go a number of applicaions in indusrial producion. In he field of indusrial process opimizaion simulaion he quanum geneic algorihm is applied. In he applicaion of supercriical waer oxidaion echnology o rea organic wasewaer, Zhou C and Qian F gave an improved quanum geneic algorihm which can be used o esimae he kineic parameers effecively in he opimizaion of he objecive funcion wih he presence of more han one kineic parameer. Removal efficiency is he mos imporan indicaor in he applicaion of supercriical waer oxidaion echnology o rea organic wasewaer. I is affeced by he reacion emperaure, pressure, residence 18

20 ime, he amoun of oxidan and caalys facors. Accurae esimaion of he kineics parameers can hus accuraely calculae he impac of various facors on he removal rae, and hus lay he foundaion for he design and opimizaion of he indusrial uni.(zhou and Qian, 2008) In chemical producion, due o he precision of measuring insrumens, measuremen mehods, and environmenal impac, he acual on-sie measuremen daa are unable o mee he heoreical maerial balance and energy balance consrains, so we need daa correcion o weed ou he error in he measuremen daa. In daa correcion, he daa classificaion and daa coordinaion require he equaion derivaion or marix conversion mehod, he calculaion process is complex and he radiional mehod is prone o error. Guo J applied an improved quanum geneic algorihm o daa correcion, and wih he simulaion sudy of he flue gas sysem daa correcion and he disillaion process daa correcion, ha he feasibiliy of he program was indicaed.(guo, 2011) In susainable use of waer resources evaluaion, Zhao X, Li Z, and Ding J used real-coded chaoic quanum geneic algorihm for parameer opimizaion in regional susainable use of he waer resources evaluaion model, and used he opimized model o make evaluaions for regional waer resources susainable use. Comparing he resuls of evaluaion of he measured daa of he 12 regions in China wih he resuls of oher evaluaion mehods, he model has pracicaliy and feasibiliy.(zhao, Li, e al., 2007) Improved quanum geneic algorihm-based parameer uning mehod for PID conroller design is also an applicaion of quanum geneic algorihm. Due o he simple principle, easy conceps o undersand, easy o implemen and good robusness, he PID conroller mees he general requiremens of he indusrial process and has been widely used in indusrial conrol sysems. The core issue of he PID conrol is PID parameers (proporional coefficien, inegral ime, derivaive ime). Zeng C, Zhao X inroduced quanum crossover, quanum variaion and groups caaclysm operaion in he basis of quanum geneic algorihm, proposed an improved quanum geneic algorihm. The PID parameer uning mehod based on improved quanum geneic algorihm ransform he PID conroller parameer uning ino a parameer opimizaion problem. Parameer uning was achieved hrough he evoluionary compuaion of improved quanum geneic algorihm. The comparison of he simulaion resuls wih oher parameers opimizaion mehods shows ha his mehod can ge beer qualiy conrol. The simulaion resuls also demonsrae he feasibiliy of his mehod.(zeng and Zhao, 2009) In he civil engineering indusry, he geneic algorihms are used in he diagnosis of srucural damage, and achieved some resuls, which we describe laer. The dynamical mehod of srucural damage diagnosis is based on he following inferences: here is a clear correspondence beween dynamic parameers of he srucure and siffness, qualiy and he maerial consiuive characerisics. Cach he dynamic parameers of he srucure by vibraion es mehod hen he work sae of he srucure 19

21 can be deduced.(yi and Liu, 2001) Using geneic algorihms in damage diagnosis, ake he measured srucural free vibraion frequencies and mode shapes as he basis, ake he reducion raio of a modulus of elasiciy of he respecive unis of he srucure as design variables, and change damage diagnosis from a srucure parameer idenificaion problem o a problem which only needs o conduc vibraion analysis and geneic algorihm.(li, W. and Chen, C., 2005) Geneic algorihm finds he opimal resuls wih is unique way of hinking. The damage diagnosis opimizaion mehod wih geneic algorihm, wih no much informaion from es, no only can quickly find he damaged pars and can accuraely simulae he exen of damage, even he opimizaion capabiliy of geneic algorihm will no be affeced when he mode may be los.(yi and Liu, 2001) YI W, Liu X proposed improved geneic algorihm which is proved can be effecively applied o he field of damage diagnosis. Compared wih he neural nework, i does no require raining samples, especially for large and complex srucure wihou a lo of raining samples. And he algorihm can idenify he damage locaion and exen of damage of fixed end beams, coninuous beams and frame srucure, raher han some mehods which mus be indicaed o he locaion of he injury.(yi and Liu, 2001) Li W, Chen C applied a geneic algorihm combined wih sensiiviy adjusmen operaion o srucural damage diagnosis. Compared wih simple geneic algorihm, geneic algorihm combined wih sensiiviy adjusmen can grealy reduce he geneic ieraion algebra, hereby reduce he compuaion ime and improve he compuaional efficiency, and has a higher accuracy.(li, W. and Chen, C., 2005) Cheng X, Wang Y, Wang X ook he elemen siffness reducion facor as he idenified parameers, ook he frequencies and mode shapes as objecive funcion, and calculaed he reducion facor wih geneic algorihms. And he effeciveness of recogniion on he pile damage had been verified by example compuaion.(cheng, Wang, e al., 2008) Zhang W, Liu J convered srucural damage diagnosis problem o he opimizaion problem based on offshore plaform measured naural frequencies and mode shapes, and used geneic algorihms o deal wih hem. Numerical simulaion and physical model experimenal resuls boh showed ha he mehod can accuraely diagnose srucural damage of offshore jacke plaform, improve he robusness of he evoluionary search, and improve he reliabiliy of offshore plaform srucural damage diagnosis.(zhang and Liu, 2011) Bu here are also limiaions o geneic algorihms in his area. Pracice has proved ha when he number of design variables is more han 20, he efficiency of simple geneic algorihm may be low due o a large design space need o deal wih.(yi and Liu, 2001) On he oher hand, he geneic algorihm works in he way of groups and generaions, he finess value of each individual in each generaion need o be calculaed firs in order o geneically manipulae o produce offsprings. In he field of civil engineering, he srucural analysis program of calculaing finess value is more complex, he ime used in he 20

22 analysis process is relaively long, causing he larger he populaion size, he more cos of ime and he slower he rae of evoluion. When he populaion size reaches a cerain exen, he geneic algorihm runs oo slow and loses he original superioriy. So, in addiion o he groups downsizing and reducing evoluion algebra, improve he speed of convergence of he geneic algorihm is he key.(yi and Liu, 2001) Quanum geneic algorihm has corresponding advanages, such as he populaion size is small, fas convergence speed, shor compuing ime and beer opimizaion abiliy. Therefore, quanum geneic algorihm provides he possibiliy for furher solving he problem of srucural damage diagnosis. There are also some progresses in he applicaion of geneic algorihm in he consrucion aspecs, mainly in hydraulic engineering opimizaion. Waer conservancy projec is a complex sysem engineering, global managemen of waer conservancy projecs needs o follow cerain seps and procedures, and one of he main asks of he managemen is schedule conrol. In projec managemen, scienific and reasonable arrangemen of he schedule and conrol of he progress of he consrucion is he imporan facor of ensuring he projec schedule, qualiy and cos. In addiion o schedule conrol, here are some oher resricions need o mee such as resource. The resource-consrained resource arrangemens problem is an NP-Hard problem. Is purpose is o seek a planning program which can resul in he shores period and also mee he requiremens of resource consrains. Wih mahemaical mehods when he nework diagram of he process is increased, he imes of calculaion required will grow in accordance wih he series level. Geneic algorihm is less resricive for he problem iself. The objecive funcion and he consrains of he problem need o be compuable raher han differeniable or coninuous. Meanwhile, geneic algorihm can always search hroughou he enire soluion space and is able o find near-global opimal soluion, so i has a wide range of applicaions in he nework planning opimizaion. There is cerain superioriy o apply geneic algorihm in opimize waer conservancy projec consrucion schedule conrol analysis wih limied resources.(wang, 2008) Wang A applied geneic algorihm o solving he consrucion scheduling opimizaion problem of waer conservancy projecs wih limied resources. He also applied his designed "limied resources, he shores period opimizaion" mahemaical model o opimize Huaihe Linhuaigang flood conrol projec consrucion o seek he balance beween he duraion and cos. According o he principle of inpu parameers and he acual siuaion of seleced problems, 3 ses of inpu parameers and 3 ses of opimized programs are chosen. Through he comparison of he opimizaion resuls, he opimal soluion in he case of consan invesmen and oher resources will shoren he duraion from 232 monhs o 217.2monhs, which means compression duraions of 6.4% (14.8monhs). For he enire projec, he duraion a leas can be shorened by 3.6 monhs, accouning for 6% of he 60 monhs.(wang, 2008) 21

23 2.5. Concepual framework Quanum + Civil Engineering Quanum Geneic Algorihm AQGA Improved, opimized Consrucion Scheduling Analysis Possibiliies Applicaions of QGA Inernal connecions Exising Consrucion Scheduling Figure 3 Concepual srucure 22

24 3. Hypohesis 3.1. Theoreical basis and foundaion Due o he simulaion of survival of he fies in naural selecion, he geneic algorihm has srong self-organizing and adapive abiliy, inelligence, robusness, sysem opimizaion, ec. These enable ha geneic algorihms are widely used in various areas like machine learning, arificial inelligence, and economic forecass. In coninuous inegraion and developmen process, he geneic algorihm also has exensive and far-reaching applicaions in civil engineering like srucural damage diagnosis and process conrol. However, despie he geneic algorihms show grea advanages in solving problems, here are also limiaions like large number of ieraions and slow rae of convergence. In combinaorial opimizaion problems here are various consrains which he feasible soluion mus saisfy. As a resul, using radiional geneic operaors such as crossover, variaion may no longer be a feasible soluion. And in solving process due o various resricions, cross and oher operaors are likely o fall ino local opimizaion.(ma and Wang, 2005) When he number of design variables is larger, he design space which needs o deal wih will be correspondingly larger, and hen he simple geneic algorihm can be inefficien. On he oher hand, in he field of civil engineering, srucural analysis program which is used o calculae each individual's finess is more complicaed. When he populaion size is relaively large, he finess value of each individual in each generaion mus be calculaed and hen generae offsprings, his will make he cos of he ime is oo much and he speed of evoluion is oo slow. When populaion size reaches a cerain exen, he geneic algorihm will run oo slow and lose he original advanage. Ways o solve his problem are groups downsizing, reducing evoluion algebra, and he key poin is o improve he speed of convergence of he geneic algorihm.(yi and Liu, 2001) A his poin, quanum geneic algorihm shows he possibiliies o solve hese limiaions because i has corresponding advanages, such as he small populaion size, fas convergence speed, shor compuing ime and beer opimizaion abiliy. Quanum geneic algorihm is a produc of he combinaion of quanum compuaion and geneic algorihms. In heory, ha any problems ha can be solved by geneic algorihm can also be invesigaed by quanum geneic algorihm.(yang and Zhuang, 2003) So i can be expeced ha quanum geneic algorihm can also be applied in he fields in which geneic algorihm already applied, and because quanum geneic algorihm has beer populaion 23

25 diversiy and beer compuing parallelism, i has he possibiliy o make beer opimizaion resul. According o he lieraure sudy, quanum geneic algorihms have already been used in indusrialized use, for parameer opimizaion, combinaorial opimizaion, muli-objecive opimizaion, ec. In hese areas of applicaion, QGA shows srong abiliy and advanages o solve problems. Quanum geneic algorihm can deal wih problems wih small populaion size, have fas convergence speed, shor compuing ime and beer opimizaion abiliy. And because quanum geneic algorihms have beer populaion diversiy and beer compuing parallelism, hey offer he possibiliy o make beer opimizaion resul. So quanum geneic algorihm is chosen o apply in his research for he opimal design of consrucion scheduling. Consrucion scheduling is ofen affeced by muliple consrains. Therefore, opimizaions of muliple objecives need o be me in he consrucion scheduling opimizaion design. The main wo are he dual consrains of ime and human resources. According o he above, in he opimal design of he consrucion scheduling, quanum geneic algorihm can be used for opimized design, o realize consrucion scheduling opimizaion design under he condiion of he human resource consrain AQGA We assume he exisence of an algorihm called A quanum geneic algorihm (AQGA). This algorihm is an improved quanum geneic algorihm and is designed o solve he opimizaion problem under he condiions of muliple consrains. In his research, i is used for consrucion scheduling opimizaion design wih he consrains of ime and human resource. We ake a null hypohesis for his research: AQGA canno help o improve he opimizaion design in civil engineering consrucion scheduling Working mechanics Qubi encoding In quanum geneic algorihm, a chromosome is characerized by qubis. Wha a qubi expresses is no longer deermined, bu i conains all he informaion. A qubi can be in he 24

26 sae of "0", "1'' or heir superposiion sae. A chromosome of n qubis can expresses 2 n saes. If a chromosome consiss of n genes and a gene consiss of k qubis, he chromosome is expressed as: q j = [ α 11 β α12 11 β α 1k α21 12 β 21 β 1k α22 β 22 α 2k β 2k α n1 α n2 β n1 β n2 α nk ] β nk In he formula, represens he chromosome evoluion algebraic, q j represens he j-h individual chromosome of generaion. Qubi encoding allows a chromosome o realize simulaneous expression of muliple superposiions. Qubi encoded chromosome expresses he probabiliy disribuion of he soluion in soluion space Decoding Make a measuremen for he qubi chromosomes and conver i ino 0-1 binary srings Quanum roaion gae In general, he quanum chromosome complees qubi updae hrough quanum roaion gae, he updae process is as follows: [ α i β ]=U i [ α i i β ] =[ cos θ i i sin θ i sin θ i ] [ α i cos θ i β ] i In he formula, θ i is he roaion angle, and is size and direcion deermine he performance of he algorihm. The roaion angle θ i is deermined by: θ i = θ i s(α i, β i ) The adjusmen sraegy is shown in he following able: (Wang, Wu, e al., 2005) 25

27 x i b i f(x) < f(b) θ i α i, β i α i β i > 0 α i β i > 0 α i = 0 β i = False True False True 0.05π ± False 0.01π ± True 0.025π ±1 1 1 False 0.005π ±1 1 1 True 0.025π ±1 Table 1 Roaion angle selecion sraegy In he able, x i indicaes corresponding binary bi in he i-h quanum bis of curren chromosome x, b i indicaes corresponding binary bi in he i-h quanum bis of currenly bes individual chromosome b. f (x) indicaes he finess value of curren chromosome, f (b) represens he finess value of he curren bes individual. θ i and s(α i, β i ) respecively ensures ha he size and direcion of he roaion angle are correc. The figure below illusraes he srucure of he quanum roaion gae. Figure 4 Quanum roaion gae diagram (Neo, De Andrade Berner, e al., 2011) 26

28 Operaor design Quanum caasrophe Quanum geneic algorihm easily falls ino local opimum. Through he analysis of quanum geneic algorihm, i is found ha if he bes individual of he previous generaion is local exremum, i.e., when he bes individual doesn' change in successive generaions, caaclysm operaions need o be aken o add a larger disurbance o he populaions in he evoluionary process. Tha can help hem o be away from local exremum and sar a new search.(yang, Liu, e al., 2004) Se he caaclysm probabiliy as P z. When he bes individual does no change for successive generaions, keep he bes individual in he populaion as he firs chromosome of nex generaion. For oher individuals, selec and replace single qubi according o he caasrophe probabiliy P z or change he probabiliy ampliude of qubi gene hroughou he quanum. Of course, he oher individuals can also be rebuil.(wang, Wu, e al., 2005) Quanum crossover Crossover operaion can help achieve he srucural informaion exchange beween individuals. Se a crossover probabiliy, randomly selec cerain individuals from he populaion o reorder and make hem involved in crossover operaion o complee he exchange of chromosome srucure informaion Quanum muaion The main role of quanum muaion is o slighly disrup he evoluionary saus of some individuals in he curren populaion and improve he local search capabiliies. Choose he quanum muaion probabiliy P e, selec a quanum bi or a par qubi gene, and realize he muaion hrough quanum NOT gae.(xiong, Chen, e al., 2004) 27

29 3.4. Workflow of AQGA Sep1 Esablish iniial populaion If a chromosome consiss of n genes and a gene consiss of k qubis, he chromosome is expressed as: q j = [ α 11 β α12 11 β α 1k α21 12 β 21 β 1k α22 β 22 α 2k β 2k α n1 α n2 β n1 β n2 α nk ] β nk Each qubi (α i, β i ) on all chromosomes is iniialized o( 1 2, 1 2 )T, ha indicaes ha a chromosome expresses he equiprobable superposiion of is all possible saes. Sep2 Measuring Measure all he individuals of he iniial populaions and ge he binary sring R()={α 1, α 2, α n }. Sep3 Evaluaion Deermine a finess funcion and make evaluaion for each individual of he R (). Reain he bes individual in conemporary. Deermine wheher he erminaion condiion is me, if i is me hen sop algorihm, or else he following operaions coninue. Sep4 updae wih quanum roaion gae Updae he populaions wih quanum roaion gae U(), which means for every qubi of quanum chromosome run: [ α i β ]=U i [ α i i β ] =[ cos θ i i sin θ i sin θ i ] [ α i cos θ i β ] i Sep5 Quanum caasrophe If he bes individual of successive generaions does no change, make he caasrophic operaion. Sep6 Quanum crossover and quanum muaion Ac crossover operaor and muaion operaor on he quanum chromosome o updae populaions. 28