Genetic algorithm for project time-cost optimization in fuzzy environment

Transcription

1 Journal of Industral Engneerng and Management JIEM, (2): Onlne ISSN: Prnt ISSN: Genetc algorthm for project tme-cost optmzaton n fuzzy envronment Khan Md. Arful Haque, Md. Ahsan Akhtar Hasn Rajshah Unversty of Engneerng & Technology (Bangladesh) arf99pe@yahoo.com, aahasn@pe.buet.ac.bd Receved: October 2011 Accepted: August 2012 Abstract: Purpose: The am of ths research s to develop a more realstc approach to solve project tmecost optmzaton problem under uncertan condtons, wth fuzzy tme perods. Desgn/methodology/approach: Determnstc models for tme-cost optmzaton are never effcent consderng varous uncertanty factors. To make such problems realstc, trangular fuzzy numbers and the concept of -cut method n fuzzy logc theory are employed to model the problem. Because of NP-hard nature of the project schedulng problem, Genetc Algorthm (GA) has been used as a searchng tool. Fnally, Dev-C has been used to code ths solver. Fndngs: The soluton has been performed under dfferent combnatons of GA parameters and after result analyss best values of those parameters have been found for the near optmal or sustanable soluton. Research lmtatons/mplcatons: For demonstraton of the applcaton of the developed algorthm, a project on new product (Pre-pad electrc meter, a project under government fnance) launchng has been chosen as a real case. The formulaton of the model of the problem s developed under some assumptons that have been mentoned n secton 5. Practcal mplcatons: The proposed model leads decson makers to choose the desred soluton under dfferent rsk levels

2 Orgnalty/value: Reports reveal that project optmzaton problems have never been solved under multple uncertanty condtons. Here, the functon has been optmzed usng Genetc Algorthm search technque, wth vared level of rsks and fuzzy tme perods. Keywords: tme-cost optmzaton, fuzzy tme functons, rsk levels, genetc algorthm 1. Introducton A project s a combnaton of nterrelated actvtes whch must be executed n a certan order before the entre task s completed. The actvtes are nterrelated n a logcal sequence whch s known as precedence relatonshp. For the analytcal purpose, the actvtes of a project are represented n a network dagram mantanng precedence relatonshp to get solutons for schedulng and controllng. The longest contnuous path of a project network s called crtcal path whch determnes the project duraton. The most commonly used project management technques are Gantt chart, Mlestone, Crtcal Path Method (CPM) and Project Evaluaton and Revew Technque (PERT). The major objectve of project schedulng s to complete the entre project wthn budget and tme constrants. Tradtonal project schedulng problems manly focus on actvtes assumng determnstc or probablstc tme duratons. Project schedulng s the converson of a project acton plan n an operatng tmetable. It serves as the bass for montorng and controllng project actvty. Taken together wth the plan and budget, t s probably the major tool for the management of projects. The basc approach for all schedulng technques s to form a network of actvty and event relatonshps. Crtcal path of ths network s an mportant ssue for project schedulers, because t refers the duraton of whole project. Project managers are hghly concerned wth ths crtcal path for on tme completon of a project, especally when an extenson wll ncur a penalty (ether n lqudated damages, opportunty costs and goodwll losses). When some delay have been occurred, there may be necessary to compress the crtcal actvty by ncrementng the actvty resources above the normal level. It s often the case that the performance of some or all actvtes can be accelerated, or the duraton crashed, by allocatng more resources at the expense of hgher actvty drect cost. Ths crashng of actvtes can be acheved by multple shft work, extended work days, usng larger and more productve equpment and ncreasng the sze of labour crews. So, project schedule planners manly focus on fndng the most cost effectve way to complete a project wthn a specfed completon tme. Ths class of problem s usually called tme-cost trade off. In real constructon projects, tme and cost of actvtes may face sgnfcant changes due to exstng uncertantes such as nflaton, economcal and socal stresses, labour performance,

3 executon errors of contractor, desgn errors, natural events such as clmate changes and etc. Therefore, total tme and cost of project may dffer sgnfcantly because of these uncertantes. Almost all of the projects, the requred nformaton for estmaton of project parameters are ether unavalable or ncomplete. In practce, the majorty of constructon companes do not systematcally record the duratons of project actvtes. In addton, t s commonly known that no two constructon projects are alke. Also n many cases the project s done for the frst tme, ths compels us to use expert opnon n forecastng the project parameters. Experts use ther own judgment, experence and project nformaton that s avalable to them. In practce, lngustc terms such as, approxmately, more or less, or about are commonly found n the statements used by these experts. These terms clearly exhbt some sort of mprecson that naturally leads to a range of possble values, rather than a defntve estmate usng a sngle value. So, uncertanty and project parameters are nextrcable. In thess stuaton, determnstc models of constructon tme-cost trade off are not realstc. Crsp decson makng n uncertan envronment causes loss of some parts of nformaton. Use of uncertan models, whsh s capable of formulatng vagueness of the dynamc condtons of real world gves more stablty to solve tme-cost trade off problem. Some probablstc technques are used n many cases to meet these uncertantes to some extent. But project parameters may not be n statstcal manner. That s why fuzzy set theory s approprate to consder affectng uncertantes n actvty duraton, drect and ndrect cost of a project. Snce dfferent alternatves of possble duratons and costs for the actvtes can be assocated wth a project, the problem s arsen to search the best soluton. As combnatoral optmzaton problems, fndng optmal decsons s dffcult and tme consumng consderng the number of possble permutatons nvolved. Therefore, any analytcal procedure may sut for small project, but t wll be neffcent for large project because of exhaustve enumeraton. To avod the problem of combnatoral exploson, heurstc models can be used. It does not ensure the optmalty but t gves better near optmal soluton wthout mathematcal rgor. Ths research amed at development of an effcent heurstc approach wth fuzzy actvty tme and cost for project tme-cost optmzaton ncorporatng the vagueness or fuzzness of the dynamc condtons of the real world. 2. Lterature revew Schedulng of project actvtes wth mnmum cost s one of the concerned felds of project management to avod the penaltes ncurred for delayng the project completon tme. Hence, t s becomng as one of the most fundamental and essental bases of research nterest of many researchers

4 Consderng the mportance of constructon tme-cost optmzaton, varous analytcal and heurstc methods have been proposed by many researchers n recent decades. Some examples of heurstc approaches are Fondahl s (1961) method, Prager s (1963) structural model, Semens s (1971) effectve cost slope model, and Moselh s (1993) structural stffness method. Semens has developed SAM (Semens Approxmaton Method) algorthm whch s deally suted for hand computaton and also s sutable for computer soluton. Lakshmnarayanan, Gaurav and Arun (2011) developed a meta-heurstc mult-colony ant algorthm for optmzaton of three objectves tme-cost-rsk as trade-off problem. Feng, Lu and Burns (1997) developed a GA model that s an mprovement of ther earler lnear programmng / nteger programmng model (Lu, Burns & Feng, 1995). The model by L and Love (1997), on the other hand, was formulated to produce the tmes, n real numbers, by whch each crtcal actvty should be reduced. The study dd not consder the formaton of other crtcal paths durng the crashng process and was lmted to contnuous, as opposed to dscrete, varables for crashng tmes. Hegazy (1999) developed a practcal GA model by mplementng the GA protocols wthn Mcrosoft Project. Ths model has demonstrated an mprovement over the prevous GA models. Zheng, Ng and Kumaraswamy (2004) has ntroduced a mult-objectve model by usng genetc algorthms ntegrates the adaptve weght derved from prevous generatons and nduces a search pressure toward an deal pont. Chua, Chan and Govnda (1997) proposed a tme-cost trade-off model usng genetc algorthm. L, Hussen and Le (2011) proposed a methodologcal framework ncludng optmzaton, senstvty analyss, and mproved GA for buld-operate-transfer projects. However, the above mentoned tme-cost trade-off models manly focus on determnstc envronment. Uncertantes (such as nflaton, economcal and socal stresses, executon errors of contractor, desgn errors, natural events as clmate changes, etc.) n the problem have receved less attenton due to ts complexty. These uncertantes cause changes n duratons and cost of actvtes and hence total cost of project may dffer sgnfcantly. Because of ts practcal relevance researchers have recently begun to pay specal attenton to uncertan schedulng. Han, Chung and Lang (2006) has used fuzzy crtcal path method to mprove fuzzy arport s ground operaton decson analyss assumng fuzzy actvty tmes as trapezodal fuzzy number. They also utlzed a fuzzy rankng method developed by Lng and Han (2004) n fuzzy CPM. Chanas and Zelnsk (2001) presented two of calculaton of the path degree of crtcalty wth applcaton of the extenson prncple of Zadeh. Lu (2003) formulated the crtcal path and the project crashng problems by lnear programmng wth fuzzy actvty tmes and then defuzzfy the fuzzy actvty tmes followng Yager s (1981) rankng method. Lorterapong and Moselh (1996) presented an approach of project-network analyss based on fuzzy sets theory. Hs approach overcomes the lmtatons of backward pass calculatons to compute project

5 completon tme wth uncertanty. To solve tme-cost trade-off problem, Ghazanfar (2008) developed an approach by possblty goal programmng wth fuzzy decson varables. It could not fully satsfy the real feld problem of tme-cost trade-off problem as crashng costs are crsp. Leu, Chen and Yang (2001) ncorporated fuzzy set theory wth genetc algorthms to model uncertanty n tme-cost trade-off problem. Project schedulng wth resource consderaton was also developed by Leu, Chen and Yang (1999). Abbasna, Afshar and Eshtehardan (2008) have nvestgated fuzzy logc based approach called Non-domnated Sortng Genetc Algorthm (NSGA) for tme-cost trade-off problem n uncertan envronment. Ths model cannot fully meet uncertanty of practcal problem. Other than fuzzy tme-cost trade-off problem, a probablstc model has been developed by Arazon, Perkgoz and Sakawa (2005) n PERT network usng a genetc algorthm. Another analytcal method that combnes fuzzy set theory wth the PERT technque for measurng the crtcalty n a project network was developed by Chen and Huang (2007). 3. Fuzzy set theory 3.1. Concept of fuzzy numbers There are dfferent forms of fuzzy numbers. Ths paper only consders trangular fuzzy A ~ a, b, c, the membershp functon s; numbers that can be defned by a trplet trangle 0, x a, b a (x;a,b,c) c x, c b 0, x a a x b b x c c x The parameter b s the centre pont, a and c are two base ponts as n Fgure 1. It shows that degree of membershp of a, b and c are 0, 1 and 0 respectvely

6 1 A (x) 0 a A b x Fgure 1. Trangular fuzzy membershp functon and -cut level 3.2. Concept of alpha ()-cut The -cut s a commonly used method to connect the prncples of fuzzy sets wth a collecton of crsp sets, whch can n turn be fed nto most of the exstng systems. Gven a fuzzy set A defned on X and any number [0,1 ], the -cut, A s the crsp set as defned by: A {x A(x) } That s, the -cut of a fuzzy set A s the crsp set A that contans all the elements of the unversal set X whose membershp grades n A are greater than or equal to the specfed value of whch s clearly explaned by Fgure 1. In ths paper, -cut s known to ncorporate the experts or decson makers confdence over hs preference or the judgments. =1 means the expert s hghly certan about hs knowledge regardng a phenomenon over whch he expresses hs preferences then the outcome wll be a sngle value havng the membershp 1 n the fuzzy performance set. When <1, t ndcates there exst uncertanty. =0 expresses the hghest level of uncertanty. Actually represents the degree of rsk that the managers s prepared to face. 4. Genetc algorthm (GA) Genetc algorthm s a heurstc search approach that provdes a means of optmzaton of NPhard problems. These procedures combne an artfcal survval-of-the-fttest strategy wth genetc operators abstracted from nature (Goldberg, 1989), to form a mechansm that s sutable for a varety of optmzaton problems. The theory behnd GA s that a populaton of certan speces wll, after many generatons of random evolutons, adapt to lve better n ts envronment. GA solves optmzaton problems n the same fashon. Ths procedure begns by

7 generatng an ntal populaton of random solutons that are codfed n the strngs of numbers called chromosomes. Each ndvdual chromosome represents one soluton that s better, or worse, than others n the populaton. The ftness of each soluton s determned by evaluatng ts performance wth respect to an objectve functon named as ftness functon. A selecton mechansm s appled to the current populaton to create an ntermedate populaton. Then crossover and mutaton operators are appled to the ntermedate populaton to form a novel populaton. Ths populaton s then used n the next teraton of the algorthm. Usually, ths process s contnued for a large number of offsprng generatons n whch the populaton keeps evolvng (better solutons replace unft solutons), untl a termnatng crteron s met. At the end of the process, the member of the populaton wth the best performance becomes the optmum soluton. The operatons of GA process are shown n Fgure 2. Start GA Create ntal populaton Select parents from current populaton Copy parents to offsprng Cros sover Yes Perform Crossover No No Replace current wth new populaton Put offsprng n new populaton Perform Mutaton Yes Muta ton No Stoppng crteron met? Yes Stop GA Fgure 2. The GA process 5. Problem formulaton The objectve of ths model s to fnd the optmal combnaton of actvty duratons wth mnmum total project cost at dfferent rsk levels (-cut levels) so that the project can be completed wthn a specfed project duraton selected from feasble project tme spectrum. The mathematcal model s expressed as follows: mn C C C C N C J (1) all

8 C C C N d 0 d N 0 C N f ( d f ( d low d low ) 0 d ) 0 Otherwse C J d N J Subject to T D (2) d d mn d (3) max d 0, = 1, 2, 3... Where, C = Total project cost at a specfc -cut level d = Duraton of actvty at a specfc -cut level C N = Normal cost of actvty at the duraton d C C = Crash cost of actvty at the duraton d C J = Indrect cost of actvty at the duraton d N = Normal cost rate of actvty at the duraton d N C = Crash cost rate of actvty at the duraton d N J = Indrect cost rate of actvty at the duraton d T = Project duraton at a specfc -cut level D = Permssble maxmum project duraton d mn = Mnmum duraton of actvty at a specfc -cut level fnshed n crash

9 d = Mnmum duraton of actvty at a specfc -cut level fnshed normally low d max = Maxmum duraton of actvty at a specfc -cut level fnshed normally Equaton (1) ndcates the total project cost. Equaton (2) restrcts the project must be fnshed wthn the targeted duraton. Equaton (3) ensures each actvty tme s restrcted between mnmum crash duraton and maxmum normal duraton. Ths model s formulated on the bass of some assumptons. These are all actvtes are crashable, normal and crash actvty duratons are characterzed by trangular fuzzy numbers, drect and ndrect cost rates are also characterzed by trangular fuzzy numbers, duratons of actvtes are consdered only nteger values wthn ts range, drect cost rates are decreased and ndrect cost rates are ncreased as actvty duraton ncreases n fuzzy nature, and resources are not lmted. 6. Fuzzy relatonshp between tme and cost Fgure 3 depcts the concept of actvty duratons and ther correspondng costs generaton procedure. The actvty duraton can be dvded nto three regons- crash tme (regon-1), normal tme (regon-2) and overlappng tme (regon-3) as n Fgure 3(a). Wthn the crash tme, the actvty needs to be performed n a crash mode,.e., t needs to spend great deal of effort to reduce the tme. Wthn the normal tme, the actvty can be fnshed n normal mode. The cost n a crash mode s generally hgher than that n a normal mode snce more resources are needed to nvest n the actvty so as to fnsh as early as possble. When the actvty tme falls wthn the overlappng regon, t can be performed n ether normal mode or crash mode. But t s reasonable to fnsh the actvty n normal mode as t wll be cost effectve. At a specfc -cut level, ponts h, g, f, e n Fgure 3 (b) represent correspondng drect cost rates of ponts a, c, b and d n Fgure 3 (a) respectvely. In case of ndrect cost, ponts and k n Fgure 3 (c) represent correspondng ndrect cost rates of ponts a and d n Fgure 3 (a) respectvely. Drect and ndrect cost rates of other duratons n the acceptable regon are then calculated. These cost rates are used to determne the total costs (drect cost ndrect cost) of each actvty whch are the nputs of the next steps. Wth the objectve functon, total project duraton s determned by the crtcal path and the total cost of the project can be calculated at a specfc -cut level. To test the feasblty of the soluton, project duraton s compared wth targeted completon tme. The fttest soluton would be the soluton whch has mnmum total cost

10 Crash duraton Normal duraton (x) a optmstc margn b c d pessmstc margn Duraton (a) Crash and normal actvty fuzzy tme Normal cost rate Crash cost rate (x) e f g h Cost rate (b) Crash and normal actvty fuzzy drect cost rate (x) j k Cost rate (c) Fuzzy ndrect cost rate Fgure 3. Fuzzy relatonshps between tme and cost of an actvty 7. GA based solver The operatonal archtecture of GA solver for fuzzy tme-cost optmzaton of a project s shown n Fgure 5. GA solver starts wth the creaton of an ntal populaton pool where each soluton s presented as a strng (chromosome) of actvty duratons. The ntal populaton number s taken as computer nput for the program. In Fgure 4, chromosome 1 (parent) and chromosome 2 (parent) show sample encoded chromosomes that have been used n ths paper. Numbers n cells represent actvty duratons of seven actvtes from A to G successvely. A value of -cut level and specfed project duraton (wthn feasble regon) are

11 also taken as nput. After evaluatng the ftness functon, probablty p and cumulatve probablty q for chromosomes (where = 1, 2, 3,.) have been calculated. Roulette wheel mechansm s used to select chromosomes whch wll undergo breedng to create next generaton. In order to explot and explore potental solutons, genetc algorthm generates possble chromosomes by usng crossover and mutaton operators. Sngle-pont crossover s used n ths soluton. A crossover rate pc s taken ndcates how many chromosomes wll undergo the crossover operaton. Fgure 4 shows sngle-pont crossover operaton. Here, offsprng 1 and offsprng 2 have been created from ther parent chromosomes by exchangng all bts after randomly chosen crossover pont (crossover pont s 3). Chromosome Chromosome Offsprng Offsprng Fgure 4. Sngle pont crossover operaton A mutaton rate pm s used to control the percentage of bts on whch mutaton s appled. The bts n chromosomes are selected at random whch wll undergo the mutaton operaton. The selected bts wll be swapped wth the randomly selected actvty duraton wthn ts acceptable regon at the specfed α-cut level. A generaton s completed and a new set of populaton (offsprng) has been created. Evaluate the ftness functon value for the new populaton and save the best value. Ths process of reproducton wll contnue untl predefned number of generatons (stoppng condton) s met. The overall GA problem solvng approach s represented n Fgure

12 Start Problem Inputs Project network Expected completon tme Fuzzy duratons Fuzzy cost rates α-cut levels SA nputs Encodng type Chromosome Selecton method Reproducton Termnaton condton GA Result Developng the objectve functon Acceptable actvty duraton generaton Fgure 5. GA problem solvng approach 8. A case study and result analyss The project on launchng of a new product (pre-pad electrc meter, a project under the government fnance) s presented wth a 7-actvty CPM network (AOA dagram) llustratng project schedulng wth tme-cost trade-off problem. The precedence relatonshps of the network are depcted n Fgure 6. The duratons, drect and ndrect cost rates for both normal and crash modes of each actvty are shown n Table 1. The optmstc and pessmstc project duraton margns wth dfferent values of -cut levels are defned n Table 2 and Fgure 7. Only nteger actvty tmes have been consdered to estmate project duraton. The regon bounded by the optmstc and pessmstc margns s the possble project duraton spectrum for constructon tme-cost trade-off. D 2 4 A G 1 C E 5 B 3 F Fgure 6. CPM network of the project on new product (Pre-pad electrc meter, a project under government fnance) launchng

13 Value of alpha-cut Journal of Industral Engneerng and Management Actvty No. A Name of the Actvty Forecast sales volume, usng market research Duraton (day) Drect cost rate (Tk./day) Crash Normal Crash Normal Indrect cost rate (Tk./day) 15,17,20 25,27,30 300,400, ,200,250 70,70,70 B Desgn pre-pad meter 10,12,15 13,16,20 200,250, ,120,130 20,20,20 C D Materals and component procurement Laboratory test of materals and component 15,16,20 16,20,26 200,250,300 80,120,170 60,60,60 2,4,5 5,7,8 150,220,280 50,90,120 80,80,80 E Proto-type Producton 2,3,5 4,6,7 290,370, ,160, ,100,100 F G Laboratory and commercal testng Commercal producton, sales and dstrbuton plannng 2,3,4 3,5,7 140,225,350 50,90,130 50,50,50 3,5,6 4,6,9 150,210,250 40,95,140 30,30,30 Table 1. Duratons and cost rates of actvtes of the project on new product Pre-pad electrc meter, under government fnance) launchng Crash duraton (day) Normal duraton (day) Mn Max Mn Max Table 2. Optmstc and pessmstc project duraton margns Project duraton (day) Fgure 7. Possble project duraton spectrums of the project The proposed GA based fuzzy tme-cost optmzaton solver was coded n Dev-C and run on a personal computer havng Intel(R) Pentum(R) Dual CPU 2.20GHz and 2GB

14 Optmum cost (Tk.) Journal of Industral Engneerng and Management of RAM. Two folders were created: one s nput folder and the other s output folder. All nputs are gven nto the nput folder. GA operatonal parameters are gven as nput through dalogs and the all output data are exported to output folders for data analyss. Table 3 shows the results produced wth the parameters set as number of ntal populaton = 100, generaton number = 10,000, project completon tme 55 days. Here, -cut value was vared. The best result was found at crossover rate = 0.8, and mutaton rate = 0.1. For -cut value=2, the result reveal that under tme constrant of 55 days, the project manager should try to complete ths project wthn 55 days wth cost of Tk consderng 0.2 (lower) rsk levels. Values of - cut level Duratons (day) of actvtes A B C D E F G Optmum project duraton (day) Optmum Cost (Tk.) Computaton tme (second) Table 3. Effect of -cut value on the optmum soluton An analyss found that as the alpha-cut value ncreases, the total cost of a project also ncreases (see Fgure 8). It s because, the choce become lmted for the selecton of smaller value from possble duraton of each actvty to reduce the total cost of the project. Alternatvely, t can be sad that total cost of a project ncreases wth the rsk level (-cut value) ncreases Alpha-cut level Fgure 8. Impact of alpha-cut level on project total cost 9. Convergence analyss Fgure 9 shows the convergence graph of the problem solvng wth the GA based solver. A convergence analyss s performed usng the best values n each generaton. Ths fgure s

15 Optmum cost (Tk.) Journal of Industral Engneerng and Management drawn from the data wth parameters of 0.2 alpha-cut level, project completon tme 55 days, 100 number of ntal populaton, P c = 0.8 and P m = 0.1. It s observed from the fgure that the optmum value s mprovng wth generaton number and t s convergng toward global optmum. After 4960 generatons the global optmum soluton has been found and ts value s Tk only Generaton number Fgure 9. Convergence graph of the project 10. Conclusons Ths research work customzed GA based project tme-cost optmzaton algorthm n a fuzzy envronment. It provdes an effcent computatonal technque for tme-cost optmzaton project schedulng problem ncorporatng uncertanty n network analyss. Because of dynamc stuaton of envronment, these proposed algorthms are effcent due to fuzzness of ts varables. Wth the concept of alpha-cut method of fuzzy theory, fuzzy nput varables were transformed nto crsp values. Due to NP-hard nature of the problem, a computer code of GA based solver was used to fnd the optmum soluton wthn project completon tme constrant at dfferent values of alpha-cut level. Decson makers (manly project managers) can use ths model to choose the desred (optmum) soluton for tme-cost trade off wthn a tme lmt under dfferent rsk levels that vares wth the values of ( =0 means lowest level of rsk and =1 means hghest level of rsk). The problem was resolved under dfferent combnatons of GA parameters. After analyss of results, optmum values of those parameters were found. The performance of the presented GA based algorthm can be further analyzed n terms of CPU tme by comparng t wth other best known algorthms for project tme-cost optmzaton

16 Dfferent selecton mechansms can change the algorthm effcency. In ths problem Roulette wheel selecton method s suggested. A new selecton method can be created as an attempt to mprove the soluton qualty. Crossover and mutaton operators are mportant parameters of genetc algorthm. In ths paper, sngle pont crossover and random mutaton operatons were used n every generaton. Other new types of crossover and mutaton operatons can be developed whch may be respondent to the generaton qualty. The algorthm can also be extended by consderng resource lmted tme-cost optmzaton wth other types of fuzzy numbers. References Abbasna, R., Afshar, A., & Eshtehardan, E. (2008). Tme-cost trade-off problem n constructon project management, based on fuzzy logc. Journal of Appled Scences, 8(22), Arazon, A., Perkgoz, C., & Sakawa, M. (2005). A genetc algorthm approach for the tme-cost trade-off n PERT networks. Appled Mathematcs and Computaton, 168, Chanas, S., & Zelnsk, P. (2001). Crtcal path analyss n the network wth fuzzy actvty tmes. Fuzzy Sets and Systems, 122, Chen, C. T., & Huang, S. F. (2007). Applyng fuzzy method for measurng crtcalty n project network. Informaton Scences, 177, Chua, D. K. H., Chan, W. T., & Govnda, K. (1997). A tme-cost trade-off model wth resource consderaton usng genetc algorthm. Cvl. Eng. Syst., 14, Feng, C., Lu, L., & Burns, S. (1997). Usng genetc algorthms to solve constructon tme-cost trade-off problems. ASCE Journal of Computng n Cvl Engneerng, 11(3), Fondahl, J. W. (1961). A non-computer approach to the crtcal path method for the constructon ndustry. Tech. Rep. No. 9, The Constr. Inst., Dept. of Cv. Engrg., Stanford Unv., Stanford, Calf. Ghazanfar, M., Yousefl, A., Amel, M. S. J., & Amr, A. B. (2008). A new approach to solve tme-cost trade-off problem wth fuzzy decson varables. Int. J. Adv. Manuf. Technol, Do /s y

17 Han, T. C., Chung, C. C., & Lang, G. S. (2006). Applcaton of fuzzy crtcal path method to arport s cargo ground operaton system. Journal of Marne Scence and Technology, 14(3), Hegazy, T. (1999). Optmzaton of constructon tme-cost trade-off analyss usng genetc algorthms. Canadan Journal of Cvl Engneerng, 26(6), Klr, G. J., & Yuan, B. (2005). Fuzzy sets and Fuzzy logc: Theory and Applcatons. Prentce - Hall of Inda. Lakshmnarayanan, S., Gaurav, A., & Arun, C. (2011). Tme-cost-rsk trade off usng ant colony optmzaton. Journal of Constructon n Developng Countres, Prevew Manuscrpt. Leu, S. S., Chen, A. T., & Yang, C. H. (1999). Fuzzy optmal model for resource-constraned constructon schedulng. Journal of Computng n Cvl Engneerng, 13(3), Leu, S. S., Chen, A. T., & Yang, C. H. (2001). A GA-based fuzzy optmal model for constructon tme-cost trade-off. Internatonal Journal of Project Management, 19, L, H., Hussen, M. A., & Le, Z. (2011). Incentve genetc algorthm based tme cost trade-off analyss across a buld operate transfer project concesson perod. Can. J. Cv. Engrg., 38, L, H., & Love, P. (1997). Usng mproved genetc algorthms to facltate tme-cost optmzaton. ASCE Journal of Constructon Engneerng and Management, 123(3), Lang, G. S., & Han, T. C. (2004). Crtcal path analyss based on fuzzy concept. Internatonal Journal of Informaton and Management Scences, 15(4), Lu, L., Burns, S., & Feng, C. (1995). Constructon tme-cost trade-off analyss usng LP/IP. ASCE Journal of Constructon Engneerng and Management, 121(4), Lu, S. T. (2003). Fuzzy actvty tmes n crtcal path and project crashng problems. Cybernetcs and Systems, 34, Lorterapong, P., & Moselh, O. (1996). Project-network analyss usng fuzzy sets theory. Journal of Constructon Engneerng and Management, 122(4),

18 Moselh, O. (1993). Schedule compresson usng the drect stffness method. Can. J. Cv. Engrg., 20, Prager, W. (1963). A structural method of computng project cost polygons. Mgmt. Sc., 9(3), Semens, N. (1971). A smple CPM tme-cost tradeoff algorthm. Mgmt. Sc., 17(6), B Yager, R. R. (1981). A procedure for orderng fuzzy subsets of the unt nterval. Informaton Scence, 24, Zheng, D. X. M., Ng, S. T., & Kumaraswamy, M. M. (2004). Applyng a genetc algorthmbased multobjectve approach for tme-cost optmzaton. Journal of Constructon Engneerng and Management, 130(2), Journal of Industral Engneerng and Management, 2012 ( El artículo está con Reconocmento-NoComercal 3.0 de Creatve Commons. Puede coparlo, dstrburlo y comuncarlo públcamente sempre que cte a su autor y a Intangble Captal. No lo utlce para fnes comercales. La lcenca completa se puede consultar en