Developing a quadratic programming model for time-cost trading off in construction projects under probabilistic constraint

Transcription

1 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, 2017 Developng a quadratc programmng model for tme-cost tradng off n constructon projects under probablstc constrant Abbas Mahmoudabad Department Of Industral Engneerng MehrAstan Unversty, Gulan, Iran mahmoudabad@mehrastan.ac.r Fateme Pakzad Industral Engneerng Department Mehrastan Unversty Gulan, Iran faeze32393@gmal.com Abstract Tme and cost are the man mportant attrbutes n constructon projects n whch they have tradng off relatons wth together. Therefore, decson makers are dealng wth the problem to organze the project completon tme based on ts overall cost. In the present research work, a non-lnear programmng model has been developed for consderng the relatonshp between tme and cost over the constructon projects whle the overall cost should be probably restrcted on the specfed base lne. Trapezodal fuzzy numbers have been used to defne the relatonshp between tme and cost n each actvty, so a quadratc programmng has been developed for consderng tme and cost tradng off as well as probablstc constrant on project overall cost. The proposed model has been appled n the case study of a constructon project named Wegh n Moton system (WIM) and results revealed that probablstc constrants and non-lnear relatons between tme and cost can be formulated usng a quadratc programmng approach. Keywords: Quadratc Programmng, Probablstc constrants, Project Management, tme-cost trade off, wegh n Moton System 1. Introducton Infrastructure projects and nfrastructure development projects are known as necessary for economc growth and development n the country and sgnfcant nvestments are accounted for them. Most project managers are tryng to plan the approved budget and n accordance wth the specfed tme to be fnshed. In 1961, Kelly Prce-tme balancng problem were dscussed for the frst tme by takng a lnear relatonshp between tme and expense actvtes (Shankar, 2011). Researches n ths area have led to the use of dfferent methods to solve the problem of balancng cost and tme. Methods and optmzaton algorthms are dvded nto two categores of accurate approxmaton algorthms and algorthms. Approxmaton algorthms nclude nnovatve methods and accurate algorthms of mathematcal methods. The success of the nnovatve ways n order to answer questons on projects depends on the type of problem and acheves the optmal soluton wll not be guaranteed. In general, the rules requred to develop and analyze these methods have been valdated wth expermental results. Innovatve methods can be provded by Fundal, Prager, Mosleh methods (Taha, 2008) wth ncreasng the sze and complexty of the ssues, whle the meta-heurstc method s very common to do the above concern. One of these methods s known as genetc algorthm [Km et. al & Azaron et al., 2005) and the others are optmzaton of brd populatons, Leap Frog and optmzaton of ACS noted. If mathematcal methods are able to solve the problem, the answer s to determne the absolute optmum. Mathematcal methods nclude the method used for lnear programmng, nonlnear programmng, nteger programmng and so on (Feng et al., 1997) whle the drect or ndrect algorthms can be dvded nto two categores. One of the examples of drect method s known as gradent algorthm whch s used to drect searchng the maxmum (mnmum) amount of the ssue by followng the most rapd rate of ncrease (decrease) n the objectve functon. In the ndrect method, usng optmzaton algorthms for solvng optmzaton problems wth the publc s lmted. Examples of these condtons nclude quadratc programmng, separable programmng and stochastc programmng (Taha, 2008), whle stochastc programmng s assumed that the data (parameters) unknown random varables [Ke & Lu, 2005) that have a certan probablty dstrbuton. Whle, fuzzy numbers n varous research projects and can be used to show uncertanty (Prad, 1979), ths nformaton s used to convert the plan nto a defnte equvalent (Wollmer, 1985) as well as one of the basc tasks n the estmated-duraton and cost actvtes s the use of fuzzy theory. Hers and Hejaz (2014) developed a model to balance cost and qualty and tme n ther artcle of a random scenaro-based modelng mult-objectve. Scenaro plannng, strategc plannng s a method used n some organzatons, to long-term plans s flexble. 2199

2 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, 2017 Hu and He n hs artcle equlbrum model to provde qualty cost and tme nvolved n constructon projects wth resource allocaton approach. Total Qualty model projects of two researchers each actvty s calculated as a weghted average qualty. On the other hand the qualty of each actvty as weghted average qualty of raw materals, equpment, labor and management s calculated. New technques wll not be possble and the scale of the problem s ncreasngly ncreases. For ths purpose, the author of the three-storey buldng your model that only ncludes 13 actvtes to have solved wth genetc algorthm (Hu & He, 2014). Hua and Junje (2014) n hs artcle of Modelng project tme cost trade-off n fuzzy random envronment For complex envronment wth more than one type of uncertanty, ths paper presents three types of tme cost trade-off models, n whch the project envronment s descrbed va ntroducng the fuzzy random theory. The expected value and the chance measure of fuzzy random varable are ntroduced for modelng the problem under dfferent decson-makng crtera. for solvng the tme cost trade-off problem n mxed uncertan envronment wth randomness and fuzzness, the fuzzy random cost mnmzaton model, the fuzzy random expected cost mnmzaton model and the fuzzy random chance maxmzaton model were bult. To solve the models, the hybrd ntellgent algorthm ntegratng the fuzzy random smulatons and GA was desgned. The effectveness of the proposed algorthm was llustrated by three numercal experments (Hua & Junje, 2014) Hua Ke, Wemn Ma and Xaowe Chen n hs artcle of Modelng stochastc project tme cost trade-offs wth tme-dependent actvty duratons, n some projects, actvty duratons show ther complexty wth tme-dependence as well as randomness. a stochastc tme cost trade-off problem wth tme-dependent actvty duratons was formulated wth objectve of mnmzng the project cost wth completon tme lmts. For solvng the problem, three decson-makng crtera were ntroduced, based on whch the expected cost mnmzaton model, the a-cost mnmzaton model and the \ probablty maxmzaton model were establshed to satsfy dfferent practcal managng requrements. To solve the models, an ntellgent algorthm ntegratng the stochastc smulatons and GA was bult. The effectveness of the proposed GA-based ntellgent algorthm was llustrated by numercal experments (Ke et al., 2012). Karen et al., n ther artcle of Multstage quadratc stochastc programmng, Quadratc stochastc programmng (QSP) n whch each subproblem s a convex pecewse quadratc program wth stochastc data, s a natural extenson of stochastc lnear programmng. Ths allows the use of quadratc or pecewse quadratc objectve functons whch are essental for controllng rsk n fnancal and project plannng. Two-stage QSP s a specal case of extended lnear-quadratc programmng (ELQP).The recourse functons n QSP are pecewse quadratc convex and Lpschtz contnuous. Moreover, they have Lpschtz gradents f each QP sub-problem s strctly convex and dfferentable. Usng these propertes, a generalzed Newton algorthm exhbtng global and super-lnear convergence has been proposed recently for the two stage case. We extend the generalzed Newton algorthm to multstage QSP and show that t s globally and fntely convergent under sutable condtons. We present numercal results on randomly generated data and modfed publcly avalable stochastc lnear programmng test sets. Effcency schemes on dfferent scenaro tree structures are dscussed. The large-scale determnstc equvalent of the multstage QSP s also generated and ther accuracy compared (Karen et al., 2001). Zheng and Ng (2005) presented a new approach for tme cost optmzaton model by ntegratng fuzzy set theory and non-replaceable front wth genetc algorthms, where fuzzy set theory was ntroduced to model the managers predcton on actvty tmes and costs as well as the assocated rsk levels. In the current ssue of the balance between the cost and duraton of projects nonlnear programmng technque used to optmze the duraton of the project. The lmt for completon of the project cost s determned a probable range. The relatonshp between actvty and costs for quadratc equaton s consdered. Axle weghng system movng n Isfahan Nan mode s used for the study. 2. Developng Mathematcal Model The mathematcal model s the frst step to better plan better sgns, parameters and decson varables and constrants and objectve functons mples ntroduced them. The duraton of each actvty trapezodal fuzzy numbers (a, b, c, and d) consdered and n any case cost also ncludes an arrangement wth C (a), C (b), C (c), C (d) expressed. By ncreasng the duraton of each actvty, cost ncreases (badsghted). on the other hand f the actvty s too low, operatng costs also ncreased (optmstc case). Contrbuton can be a curve of the second degree s. on the other hand any quadratc equaton contans coeffcents that ncludes power factor of the second varable power factor of varable and fxed value. Tme curve for actvty s formulated as equaton (1). Where n: C(T ) (T ) ( T) (1) Where: 2 : The second tme power factor for actvty n operatng cost. 1 : The frst tme power factor for actvty n operatng cost. : Constant coeffcent of cost for actvty. 0 Decson varable: The analyss model earlest start tme and duraton of each project actvty and the project completon tme as the decson varable n the model. The objectve functon s to mnmze the duraton of the entre project, whch s obtaned by mnmzng 2200

3 Cost Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, 2017 the tme requred to complete the project, so the project completon tme as a decson varable n the model. Startng tme s also consdered to be varable n the decson because the project completon tme, maxmum n s the earlest start tme. = The earlest start tme of n the actvty = Duraton of n the actvty = the maxmum amount of tme to start of n the actvty (completng tme of project) C(d) C(a) C(c) C(b) Legend a: Optmstc duraton tme d: Pessmstc duraton tme c: The lower lmt of the lkely tme range b: The upper lmt of the lkely tme range C(a): The cost of optmstc duraton tme C(d): The cost of pessmstc duraton tme C(c): The cost of lower lmt of the lkely tme range C(b): The cost of upper lmt of the lkely tme range a b c d Tme Fgure 1: The relatonshp between the cost and duraton of actvty Parameters: Coeffcents of quadratc curves assocated wth each actvty are consdered as one of the parameters. Tme optmstc, pessmstc tme, the range of possble actvtes, fuzzy mean and varance of operatng costs as well as the parameters of the model are: = Possble to carry out actvtes optmstc that the tme th and talked wth a trapezodal fuzzy numbers wll be dsplayed. = The lower lmt of the possble range of tme dong that n the trapezodal fuzzy numbers are shown wth the letter b. = The upper lmt of the possble range of tme dong that n the trapezodal fuzzy numbers are shown wth the letter c. = The longest tmes possble to complete the work th my cyncal tme n the trapezodal fuzzy numbers are defned wth the letter d. = Average cost of dong th, fuzzy numbers mean that optmstc, pessmstc and lkely range s obtaned. = Fuzzy varance cost, the varance of fuzzy numbers optmstc, pessmstc and lkely range s obtaned. = X varable coeffcents n the model are dscussed n the lmt J wth V_1. The coeffcents zero, (1) and (1) n the fall. Factor V1 to work am so that for the earlest onset of actvty am aganst (+1), for all the actvtes that prevously needed Actvty are, (1) and for the rest of the actvtes (the prerequste ) are zero. = The coeffcent model varable T_ the restrctons J wth V_2 shown. These factors nclude the zero and +1 s to work the coeffcent V_2 s the case for all actvtes that are a prerequste for actvty, the value of (+1) and for actvtes that are prerequstes Actvty do not, the value wll be zero. Frst objectve functon: The objectve functon s to mnmze the duraton of the project. Ths s acheved by mnmzng the maxmum amount X (the earlest start tme of ) s obtaned. Therefore, the maxmum value of the M X consdered that the objectve can be acheved by mnmzng the amount of M and by (2) s as follows: 2201

4 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, 2017 M Max ( X ) So the objectve functon for (3) s defned as: MnZ ( M ) (2) (3) Constrants: In order to acheve the goal of mnmzng the tme constrants that nclude: In the network structure of the project from legtmate actvtes as the actvtes project wll be used to maxmze ts earlest start tme the maxmum total duraton of the project. The lmt for (4) s shown: X ( ) M Any actvty on the network has prerequstes that are effectve n determnng the lmts. Ths lmtaton s requred by the actvty coeffcents and V 1, V 2 n secton (2-2) have stated, are defned as follows: As soon as should start as soon as the sum of (j) ( actvty s a prerequste) and actvty (j) s. fold (+1), a prerequste for actvty (j) to (1) s. The lmt as (5) s shown: V ( X X ) V ( T ) 1 2 j j V 2 Coeffcent equals to the tme of actvty +1. V 1 (4) Factor for the earlest start tme of - As a prerequste for startng any actvty, ths lmt accordng to (6) s assumed. X 0 (6) 1 The duraton of each actvty possble n the tme doman has a low (optmstc tme) and a top (the cyncal tme) s. That should not be too hgh than too low. The lmt s for 7 nto the model. (7) low ( T ) T upp ( T ) The mathematcal model of plannng wll become a quadratc programmng model. C(T ) (T ) ( ) 2 1 T Snce the tender stage s very mportant. Prce Check out the entre project should be n a range lkely to determne the amount does not exceed a certan, reassurng. So n ths study, the possblty that the amount of total C (T ) (total project cost) less than the total average cost of actvtes s fuzzy Z lkely to be checked. The project cost s lkely to be determned Z a plausble range. P MeanC C (T ) Snce the total average cost experences can fuzzy random varable E MeanC have: Where obtaned: P (T ) MeanC E MeanC C E MeanC Var MeanC Var MeanC C (T ) E MeanC Var MeanC Z and Var MeanC (5) (8) (9) s normally dstrbuted. So we X (10) standard normal s wth zero mean and varance one the relaton (11) s So random restrcton has converted nto defnte lmtatons and relatonshp s used n the model [5]. C ( T ) E MeanC Z Var MeanC So the objectve functon consderng the lmtatons mentoned, s optmal. General model: After dentfyng the decson varables, the parameters, lmtatons and objectve functon n the prevous secton, the general model s as follows: Mn Z M Subject to: X 0 1 X ( ) M 1, 2,..., n (11) (12) 2202

5 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, 2017 lowt T uppt 1, 2,..., n V ( X X ) V ( T ) 1, 2,..., n, j 1 j 2 j C ( T ) E MeanC Z Var MeanC 1, 2,..., n 3. Case Study In thus project we use Constructon of the weghng system n Isfahan-Naeen as case study for checkng recommended model that cost and tme would be seemed as fuzzy numbers. As we sad n before relaton between tme of actvty and ts cost would be related to one of them whch s shown n frst equaton. For example, supply actvtes sutable for road pavement to nstall the system wth the code D s shown, curve equaton (13) can be expressed as: 2 C(T ) (T ) ( T ) d d d By solvng the problem by software GAMS, the duraton of each project actvty and any of ts earlest start tme and the completon of the project (the decson) and the value of objectve functon (optmal duraton of the project) and lmt values of the ncludng completon of the project costs, potental costs of the project (95%), the total cost of the project actvtes and the total varance fuzzy average cost of the project actvtes are dentfed to be set n accordance wth the project plan. The objectve functon, the optmal duraton of the project can be acheved by takng lmts. In (Table 1) amounts earlest start tme of any actvty, duraton of each actvty and total project completon tme s shown: Table 1. Project plannng and mplementaton of actvtes code Actvty descrpton Start Duraton tme of project A Installng the necessary approval revew 0 1 B The whole system s nstalled on a network locaton 1 1 C Locaton system nstalled n the bow part 2 1 D Supply of road pavement perfect platform for system nstallaton 3 1 E The project s desgned to nstall n place 4 1 F Network communcatons and nformaton systems 5 1 G Operaton of constructon (foundaton and rg) 7 2 H Supply and nstallaton of detecton sensors weght, speed and vehcle class 8 1 I Power supply 6 1 J Communcatons nfrastructure 7 1 K Tender for the supply and nstallaton of hardware and software for the control canter 6 1 L Supply and nstallaton of hardware and software for the control canter 11 1 M Weghng supplyng software systems, lcense plate readng, by grade and speed 7 1 N Launchng systems, nstalled software and hardware 8 2 O calbraton 10 1 P Temporary delvery 11 1 Q end 12 0 Tme of endng project 12 The ssue of possble restrctons, the possble range of the cost of the project can be completed by takng 95% gans. In (Table 2) lmt values that nclude, the cost of completng the project, the potental cost of the project (95%), the total cost of the project actvtes and the total varance fuzzy average cost of project actvtes s shown: Table 2. Plannng the cost of mplementng actvtes Amount of Lmtatons lmtatons Total average phase project cost (mllons) 1327 The total project cost varance fuzzy (mllon) The total project cost s fuzzy standard devaton (mllon) Possble completon of the entre project cost (mllons) Out of the total project cost (mllons) Snce the completon of the total project cost amounts possble varables wth a normal dstrbuton. The cost of completng the project s wthn ± 3σ of the average fuzzy costs to determne the probablty of each of them to the area of the normal dstrbuton curve to the costs to be calculated. That conforms to the fgure 2. (13) 2203

6 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, Summary and Concluson Model usng non-lnear programmng problem was solved wth possble restrctons. And based on objectve functon, optmal project completon tme, and then takng nto account the normal probablty dstrbuton for the cost of the project, a probable range for the cost of the project was completed. GAMS software model to help the software that s powerful n solvng programmng problems were solved. The duraton of projects can take up to 12 months and the cost of dong t n the range of to mllon USD s lkely. The results ndcate that a possble approach to control the cost of the project wll be mplemented through non-lnear programmng stages. µ µ- σ µ-0.85 σ µ+ σ µ-2.99 σ µ-1.99 σ µ+1.64 σ µ+1.99 σ µ+2.99 σ Fgure 2. Values cost of completng the project n the normal dstrbuton curve References Azaron. A,, Perkgoz C,., Sakawa M,., A genetc algorthm approach for the tme cost trade-off n PERT networks, Appl. Math. Comput. 168 (2005) Feng, C., Lu, L. and Burns, S., Usng genetc algorthms to solve constructon tme cost trade off problems, 1997, Journal of Computer and cvl engneerng, 11(3): Hamdy A. Taha, Operatons Research, An Introducton, Prentce hall publcaton, Hers, G., Hejaz, S., A scenaro based stochastc mult-objectve modelng for tme-cost-qualty trade-off problem, 2014, Project Management Development, Practce and Perspectves, Thrd Internatonal Scentfc Conference on Project Management n the Baltc Countres, Rga, Unversty of Latva. Hu, W., He, X., An nnovatve tme-cost-qualty tradeoff modelng of buldng constructon project based on resource allocaton, the scentfc world journal. Hua, Ke., Junje Ma., Modelng project tme cost trade-off n fuzzy random envronment, Appled Soft Computng 19 (2014) Karen, K., Lau., Robert, S., Womersley., Multstage quadratc stochastc programmng, Ke H,., Lu B,., Project schedulng problem wth stochastc actvty duraton tmes,2005. Appl. Math. Comput. 168 (2005) Ke, H., Ma, W., Chen, X., Modelng stochastc project tme cost trade-offs wth tme-dependent actvty duratons, Appled Mathematcs and Computaton 218 (2012) Km K., Gen., Yamazak G., Hybrd genetc algorthm wth fuzzy logc for resource-constraned project schedulng, Appl. Soft. Comput. 2 (M, 2003) Mokhtar H,., Kazemzadeh R,B,., Salmasna A,., Tme cost tradeoff analyss n project management: An ant system approach,2011. IEEE Trans. Eng. Manage. 58 (2011) Prade H,., Usng fuzzy set theory n a schedulng problem: a case study, Fuzzy Set Syst. 2 (1979) Shankar R., Raju N., Srkanth M.M.K,, G., Bndu H., P., Tme, cost and qualty trade-off analyss constructon of project. Contemporary Engneerng Scences, 2011.Vol.4, no. 6, Wollmer R,D,., Crtcal path plannng under uncertanty, Math. Program. Stud. 25 (1985) Zheng D,X,M,., Ng S,T,., Stochastc tme cost optmzaton model ncorporat-ng fuzzy sets theory and nonreplaceable front,2005. J. Constr. Eng. M. 131 (2005) Bography Abbas Mahmoudabad, correspondng author (mahmoudabad@mehrastan.ac.r), s Ph.D. n Industral Engneerng and drector of Master Program n Industral Engneerng at MehrAstan Unversty, Gulan, Iran and deputy of Plannng and Coordnaton n Transport and Fuel Management Centre, at Road Mantenance and Transport Organzaton, Tehran, Iran. He acheved hs Ph.D. degree n January 2014 n the feld of optmzaton n Hazmat transportaton and receved Thess Dssertaton Award from IEOM socety n March 2015, Duba, UAE. He has publshed near 60 journal or nternatonal conference papers and one book chapter publshed n the feld of ndustral engneerng, transportaton, traffc and road safety. He teaches transport and ndustral engneerng courses at unverstes and has around 2204

7 Proceedngs of the Internatonal Conference on Industral Engneerng and Operatons Management Rabat, Morocco, Aprl 11-13, years of executve experences on traffc and road safety plannng n developng countres. He has also strong cooperaton wth natonal and nternatonal agences traffc safety and more wth nternatonal agences n the feld of ndustral engneerng. Some natonal transportaton projects have been mplemented under hs supervsory roles wth the results of fatalty reducton n ntercty transportaton. Fatemeh Pakzad has Bachelor and Master of Scence degrees n Industral Engneerng. She graduated from MehrAstan Unversty, Gulan, Iran n June Her thess dssertaton s on studyng the probablstc cost-tme tradng-off n cvl engneerng projects and publshed her papers n ths feld. 2205