ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 12, NO. 4 (2011) PAGES

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1 ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 12, NO. 4 (2011) PAGES DISCOUNTED CASH FLOW TIME-COST TRADE-OFF PROBLEM OPTIMIZATION; ACO APPROACH K. Aladn, A. Afshar and E. Kalhor School of Cvl Engneerng, Iran Unversty of Scence and Technology, Tehran, Iran Receved: 6 November 2010, Accepted: 1 February 2011 ABSTRACT Tradtonal tme-cost trade-off (TCTO) analyss n constructon management problems dsregards tme value of money. In fact, the value of money decreases wth tme and, therefore, dscounted cash flows should be consdered when solvng TCTO optmzaton problem. As a combnatoral optmzaton problem one may apply heurstcs and/or optmzaton technques to solve tme cost trade-off problems. The combnatoral nature of the dscrete TCTP, n whch the soluton space of the problem ncreases exponentally wth the ncrement n the number of actvtes and/or the number of potental mplementaton modes, demands specal soluton algorthm. A mult-objectve ant colony optmzaton (ACO) based model for project TCTO problem s developed, whch mnmzes project drect cost takng nto account dscounted cash flows. The model locates the near optmum Pareto front wth a set of non-domnated solutons n whch precse dscrete actvty tme-cost functon may be used. No smplfyng assumptons are needed to mplement the dscount factor nto the modelng structure and soluton procedure, as requred wth mathematcal optmzaton technques. Detals of model formulaton are llustrated by an example project. The results show that ncluson of dscounted cash flow results n dfferent modes of constructon as well as actvtes duratons and costs and consequently optmal project duraton. The proposed approach can help the practtoners n consderng net present value n tme-cost decsons leadng to dentfcaton of the best opton. Keywords: Optmzaton; dscounted cash flow; tme-cost tradeoff; multobjectve ant colony 1. INTRODUCTION In any constructon project, an actvty may be mplemented not only wth dfferent quanttes of the same resource but also wth varous types of resources. Selecton of approprate resources, ncludng crew szes, equpment, methods, and technologes to perform a project are challengng decsons to be made by constructon planners. These E-mal address of the correspondng author: a_afshar@ust.ac.r (A. Afshar)

2 512 K. Aladn, A. Afshar and E. Kalhor decsons ultmately affect the duraton and cost of the project. Major constructon projects often nvolve numerous actvtes, therefore evaluaton of all possble combnatons wthn a short perod of tme and a reasonable cost may not be feasble. As an example, the total number of alternatve combnatons of tme and cost for a project wth only 18 actvtes and 5 optons for each actvty has been estmated as cases. The mplementaton modes of actvtes and ther tme and cost parameters n a constructon project have been dentfed as major factors of the decson makng process. Snce cost can be expressed as a functon of tme, t s possble to determne the project tme-cost trade-off (TCTO) curve whch provdes the mnmum possble cost of completng project n ts feasble tme range. The total cost of a project conssts of drect and ndrect costs. Drect costs are ncurred because of the performance of project actvtes, whle ndrect costs nclude those tems that are not drectly related to ndvdual project actvtes. In general, ndrect cost ncreases as the project duraton ncreases and usually assumed as a percentage of project drect cost. In tradtonal TCTO analyss, the value of money s assumed to reman constant whatever the project tme span s. In other words, costs of actvtes are summed up to form project drect cost although actvtes are executed at dfferent tmes as ther start and fnsh tmes are scheduled. However, money has a tme value and therefore, t s mportant to consder the tme at whch costs are ncurred. For projects wth longer duraton consderaton of tme value of money becomes more mportant. Among many economcal measures the net present value s most commonly used one. Most of exstng technques for solvng TCTO problem do not consder tme value of money. Dscounted cash flow (DCF) analyss of TCTO problem s of great mportance for both owners and contactors. Generally speakng, for a fxed value of total bd, t s preferable for owners to select the bd wth the mnmum net present value. On the other hand, contactors prefer decdng on the tender wth the mnmum value to bd whle maxmzng the net present value as n unbalanced bddng strategy. In bref, a TCTP s a combnatoral problem whch nvolves fndng mplementaton modes for actvtes such that the optmal balances between project tme and cost are found [1], Eshtehardan et al.). Early attempts to solve TCTP were mathematcal and heurstcbased approaches (Feng et al., [2]). Despte ther ablty to produce optmal and near optmal results for TCTP wth lnear tme-cost relatonshp, such solutons have revealed a man weakness n solvng dscrete TCTP. It has been clamed that any exact soluton algorthm for dscrete tme-cost trade-off problem would very lkely exhbt an exponental worst case complexty (De et al., [3]). Ths s manly due to the combnatoral nature of the dscrete TCTP n whch the soluton space of the problem ncreases exponentally wth the ncrement n the number of actvtes and/or the number of potental mplementaton modes. The varety of assumed tme-cost functons for actvtes has led the problem to dfferent soluton approaches (Yang [4]). These nclude lnear programmng, nteger programmng, a hybrd of lnear and nteger programmng, and dynamc programmng. Recently, artfcal ntellgence (AI) technques such as genetc algorthms (GAs), ant colony optmzaton (ACO), and partcle swarm optmzaton (PSO) are ntroduced to overcome the problems assocated wth (1) large number of varables and constrants; (2) nonlnearty of tme-cost functons; and (3) mult-objectve optmzaton.

3 DISCOUNTED CASH FLOW TIME-COST TRADE-OFF PROBLEM 513 Meta-heurstc and evolutonary algorthms have shown relatvely hgher effcency n handlng these problems. Although they do not necessarly guarantee the global optmal solutons, ther ablty to search the solutons space ntellgently, rather than completely, makes them capable of producng relatvely good solutons to large-szed problems. Among them algorthms, the genetc algorthms (GAs) and ant colony algorthm (ACO) have receved more attenton. In recent works, Feng et al. [2], L et al., [5] and Hegazy [6] adopted GAs for Tme- cost optmzaton problem. Employng modfed adaptve weght approach (MAWA) ntroduced by Zheng et al. [7], Xong and Kuang [8] presented an ACO model for solvng TCTP. They showed that the proposed ACO results n better solutons compared to GAs. In a smlar work, Ng and Zhang [9] adopted ACO to solve TCTP usng a MAWA approach n order to ntegrate optmzaton of tme and cost of the project nto a weghted objectve functon. Ther model showed hgher effcency over ts GA counterpart. Recently, Afshar et al. [10, 11] proposed a mult-objectve verson of an ACO algorthm to solve TCTP. Ther model showed satsfactory performance n locatng the Pareto front wth non-domnated solutons. Ammar [12] ponts out that the heurstc methods may not guarantee optmal solutons, hence proposes a nonlnear mathematcal optmzaton model for project TCTO problem whch mnmzes project drect cost and takes nto account dscounted cash flows. Dscussng the complcatons n handlng Dscount factor n the exponental form n a mathematcal optmzaton model, he uses a smplfed form of DCF whch ntroduces an approxmaton nto the model. Ths paper extends the tradtonal TCTO problem to account for tme value of money. Realzng the drawback of the commonly used mxed nteger nonlnear programmng algorthms to solve the general form of dscounted cash flow TCTO problem, a general modelng structure and soluton methodology s ntroduced. Prevously tested Nondomnated Archvng Ant Colony Optmzaton (NA-ACO) algorthm s employed to solve the TCTO problem wth DCF. The so called NA-ACO has the mert of consderng tme and cost of the project n a mult-objectve sense rather than ntegratng them nto a weghted objectve functon. The subject should be of nterest both for researchers and ndustry practtoners n academc and/or real feld problems. Dscounted cash flow (DCF) analyss of TCTO problem s of nterest for both owners and contactors. Developng hghly effcent and robust algorthms to solve hghly complex tme cost-tradeoff problems s stll a challengng subject for researchers. Practtoners are also wllng to have a relable trade-off functon between the tme and cost wth dscounted cash flow opton avalable for performng the project. 2. PROBLEM STATEMENT AND FORMULATION Consder a project wth n actvtes, where utlty data for project actvtes are represented by dscrete functons. Each dscrete pont represents a gven mode to mplement the actvty. The normal and crash dscrete ponts represent the two extremes of the actvty tme-cost functon. Obvously, for an actvty wth only one dscrete pont, the normal and crash ponts concde. The prmary nformaton obtaned from tradtonal schedulng are bascally actvtes early and late start and fnsh tmes. The best mode to carry out the actvtes, as well as ther

4 514 K. Aladn, A. Afshar and E. Kalhor duratons and correspondng costs, are selected optmally by the TCTP model from ther utlty data to optmze the objectve functon and satsfy the mposed constrants. In ts general form, the objectve functon s usually set to mnmze the project cost. Assumng that ndrect cost ncreases lnearly wth project duraton, project drect cost only needs to be mnmzed. The project dscounted cash flow drect cost s the summaton of all actvtes present value costs expressed mathematcally by (Ammar [12]): n m SF Mnmze : dscounted PDC (1 r) c x 1j 1 (1) In whch m = avalable constructon modes for actvty (dscrete ponts n cost functon), x =zero-one varable belongs to the dscrete pont number j for actvty I, c = cost of actvty under mode j(dscrete pont number j), r=rate of nterest, SF = scheduled fnsh tme for actvty, and (1+r) SF s dscount factor expressed n terms of nterest rate (r). The zero-one varables are ntroduced to ensure that only one constructon method (dscrete pont) may be selected per actvty. The objectve functon s subject to the followng constrants: m x,,,...,n j (2) SF SF p m 1 d x 0 p 1, 2,...,NP SF k λ, k 1, 2,...,NE (4) The constrant number (2) s ncluded to force a sngle constructon method per actvty at a tme for an actvty. The number of zero-one varables needed s the sum of dscrete ponts for all actvtes (number of avalable constructon modes for all actvtes), whle requred number of zero-one constrants equals number of project actvtes, n. Network Logc Constrants are defned by Eq. (3), n whch the logcal relatonshp between any two consecutve actvtes and ts mmedate predecessor, p, s expressed mathematcally. Constrant number (4) defnes the Project completon s tme whch s controlled by the latest fnsh tme of endng actvtes. If the number of endng actvtes s denoted by NE, the project completon constrant(s) s gven by Eq. (4) n whch λ s the desred project duraton. To develop the set of non-domnated solutons one has to vary the project duraton and run the proposed mxed nteger NLP model accordngly. Generally speakng, TCTP s a specal case of mult-objectve optmzaton problem wth two objectves whch manly focuses on selectng optons wth correspondng tme and cost to complete an actvty so as to smultaneously mnmze the tme and total cost n largescale or complex networks. An effcent approach to a TCTP requres a mult-objectve optmzaton algorthm to allow for greater freedom n explorng possble solutons to reduce the lkelhood of beng trapped n local optma (Knowles et al., [13]). The proposed (3)

5 DISCOUNTED CASH FLOW TIME-COST TRADE-OFF PROBLEM 515 approach n ths study utlzes a mult-objectve optmzaton approach whch not only could provde the satsfactory soluton, but also determne the non-domnated set that s benefcal for the further decson-makng process. In ts mult-objectve form a TCTP may be presented as: ( k ) ( k ) Mnmze T Mn Max t x (5) L r L L r where (k ) Mnmze PDC ( 1 SF (k) (k) r ) dc x (6) A (k) ST: 1 (7) x k 1 t represents the duraton of actvty when employng the kth mode of constructon; and x s the zero-one varable for actvty when employng the kth mode of (k ) constructon (=1, f actvty s constructed wth opton k ; zero otherwse). The actvty sequence on the r th th r path s demonstrated by L r ={lr, 2r,..., nr} where r represents the sequence number of actvty on the rth path. The set of all paths of a network s presented by L={L r r=1,2,..., m}, n whch m symbolzes the total number paths n the network; dc = drect cost of actvty under the k th opton; c = ndrect cost (daly cost); and (k ) A= set of actvtes n the network. As defned by Afshar et al. [10], equaton 5 may be presented n the form of Eqs. (8 and 9): Mnmze T(project Duraton) max p path [T 1, T2,...,Tp,..., TP ] (8) In whch, all paths of a network are presented by p ({p p 1,2,..., p}) ; P symbolzes the number of all paths of the network. Each path connects the start actvty to the fnal actvty passng a seres of drectly connected actvtes. T p s the total duraton of path p and can be shown as: n p T P t 1 Where n p represents the number of actvtes on path p and t denotes the duraton of actvty when mplemented n mode j. (9) 3. NON-DOMINATED ARCHIVING ANT COLONY OPTIMIZATION ALGORITHM Frst proposed by Dorgo [14], Ant Colony Optmzaton (ACO) apples several generatons of artfcal ants to search for good solutons of an optmzaton problem. Good solutons are

6 516 K. Aladn, A. Afshar and E. Kalhor the product of the ants cooperatve communcaton. Ants make a soluton by addng benefcally defned soluton components to partal soluton under constructon. Consequently, ACO s a sutable procedure for problems of combnatoral type. In order to optmze a combnatoral problem va ACO, a graph should be defned as to present potental alternatves at each decson space (.e. actvtes) n the way that the entre soluton space s comprehensvely covered. The man body of ACO s comprsed of tour constructon, and update of pheromone trals. Frst proposed by Afshar et al. [10, 11], Non-domnated Archvng Ant Colony Optmzaton (NA-ACO) s a verson of mult-objectve ant colony based algorthm. Generally, when applyng ACO to an optmzaton problem, a graph should represent the problem so that the ants can travel between nodes avalable at each decson pont (.e. Decson varables). In ths paper, the decson varables are mode of constructon for each actvty whch must be selected from a gven lst of feasble modes. Ths paper ntends to mnmze the project duraton and total dscounted cost smultaneously. To do t, combnatons of modes of constructon and tmng for dfferent actvtes must be evaluated for two nomnated objectves. Assocate wth each objectve, a colony of ant searches for optmum solutons. In the proposed model, one colony attempts to mnmze the duraton tme and the other concerns mnmzaton of total dscounted cost of the project. Hereafter, these colones are called Tcolony and Ccolony, respectvely. Adapted from Afshar et al [10, 11 ], the structure of NA-ACO for tme-cost optmzaton can be brefly addressed n followng 5 steps; Step 1: Each agent of Tcolony constructs a path from frst actvty to the fnal actvty passng one and only one node at each decson stage (.e. actvty). Each agent may select any mode of constructon at each decson pont. The solutons found by ths colony are transferred to the Ccolony for evaluaton and pheromone updatng. Step 2: In the Ccolony, the dscounted project costs resulted from mported solutons are evaluated, and the best soluton s marked. The complete path for the best soluton receves pheromone, and pheromone on all edges s updated as defned by Eq.9: (1 ) k k 1 Where (0,1] s the evaporaton rate, and k s the pheromone concentraton on edge at K th teraton. To reduce the pressure of past generatons, the pheromone on all the edges s partally evaporated. The ncrement n pheromone conncentraton on edges assocated wth the best soluton found at K th k teraton s defned as ; whch s mathematcally shown as: k (9) Qk th k f path s travelled by k teraton's best ant k f ( B) (10) 0 otherwse Where Q k s a constant, and f k (B) s the best ftness value found n k th teraton.

7 DISCOUNTED CASH FLOW TIME-COST TRADE-OFF PROBLEM 517 Step 3: In the Ccolony, agents select new edges usng pheromone nformaton. In fact, ants select each edge usng a probablty functon. Standng at th opton of S th actvty, the probablty of choosng an opton for (s+1) th actvty s represented by P where: P n 0 j f j s an allowable start tme otherwse Where s a heurstc value, and are parameters whch nduce the planners vew towards the mportance of selecton usng pheromone or heurstc value. j s consdered allowable start only f startng actvty (s+1) at j does not volate ether network logc or other constrants. The solutons produced n the Ccolony are sent to Tcolony for evaluaton and pheromone updatng. Ths recprocal exchange s called a cycle-teraton. Step 4: At the end of any cycle teraton, all the solutons found are sent to an offlne archve. In the archve, solutons are evaluated based on both objectves and non-domnated solutons are found. In a mnmzaton problem, a vector x (1) s partally less than another vector x (2), (x (1) < x (2) ) when no value of x (2) s less than x (1) and at least one value of x (2) s strctly greater than x (1). A soluton the vector of whch s partally less s a domnated soluton and a soluton whch cannot be domnated throughout an exstng soluton set s called a non-domnated soluton or a Pareto optma. A set of all non-domnated solutons found wthn a soluton set s called Pareto front. Step 5: Next, the pheromone values on all edges are re-ntated to o and pheromone trals are updated accordng to the non-domnated solutons n the archve. The total process contnues untl the stoppng crteron s met. Once agan the pheromone matrx s updated usng the non-domnated solutons n the offlne archve, and edges are explored by the Tcolony. 4. MODEL APPLICATION In order to llustrate the concept and performance of the proposed algorthm, an 18-actvty network confguraton s used as presented n Fgure 1. The number of avalable optons for each actvty along wth the cost and tme for each mode of constructon are gven n Table 1. For ths smple example, there are more than 3.6 bllon possble combnatons of constructon modes for delverng the entre project (El-Rayes and Kandl [15]). The prevously descrbed 18 actvty problem s solved and wthout ndrect cost. The number of ants, cycle teraton, and total teraton are set to 100, 30, and 200, respectvely. Dsregardng the ndrect cost Fgure 2 shows the Pareto front wth the sets of nondomnated solutons, both for total drect cost and dscounted cash flow cost wth dfferent nterest rates. As t was expected, DCF analyss has sgnfcant mpact on project drect cost and may change the optmum mode of constructon and ther sequencng. It s nterestng to (11)

8 518 K. Aladn, A. Afshar and E. Kalhor note that very many non- domnated solutons are dentfed when the total project duraton changes from 120 to 169 weeks for whch the total dscounted cost ranges from $51204 to $ Fgure 1. Network of test problem In other words, one may speed up (or slow down) the project mplementaton by 49 weeks wth a mnor ncrease (or decrease) n the total dscounted cost (.e. $51204 $49529= $1675). On the other hand, for duraton of 110 weeks, there s a dscountnuty n the Pareto front ndcatng that one week ncrease (or decrease) n project duraton wll sgnfcantly decrease (or ncrease) the total dscounted cost. As an example, for nterest rate of 9% the change n dscounted cost wll be n the range of $8000 ($68000 to $60000). In order to have some detaled nformaton about the generated Pareto solutons correspondng actvtes of generated solutons for nterest rate of 10%, 8% and 0% have been selected from the archve and shown n Table 2. When the dscounted cash flow s of nterest, the solutons are senstve to the nomnated optons n a few more actvtes compared to the case wth non dscounted total cost (Table 2). As llustrated, modes of constructon for actvtes number 1 and 17 have changed from 1 and 3 to 5 and 1, from total cost case and dscounted cost case, respectvely. In other words, when dscounted cash flow s consdered, the optons wth less costs and hgher duratons are more preferred at the early stages of the project mplementaton, whereas, the optons wth hgh costs and lower duratons are of more nterest n later stages. Ths s well llustrated for actvtes number 1 and 17 for whch constructon optons number 5 and 1 are selected, respectvely. When tme value of money s dsregarded (r=0%), the optmal project duraton (correspondng to mnmum total cost wth weekly ndrect cost of $1000) s 110 weeks. At a value of 10% nterest rate, the correspondng optmal project duraton s 117 weeks. These results are depcted graphcally n Fgure 3. It s clearly apparent from Fgure 3 that there s a dstnct value for optmal project duraton f tme value of money s gnored. From the results obtaned, t s obvous that gnorng tme value of money n the analyss of TCTO problem can produce msleadng decsons. The selected duraton for actvtes and consequently the optmum project duraton depend on nterest rate value chosen and ndrect cost rates. For the example project on hand, the optmum project duraton for DCF exceeds that of gnorng tme value of money. Ths may not be the case for projects wth dfferent characterstcs. One should therefore be careful n decdng the values of nterest rate and ndrect cost values.

9 DISCOUNTED CASH FLOW TIME-COST TRADE-OFF PROBLEM 519 Table 1: Detals of the case example (adapted from Eshterardan et al. [1]) Actvty Opton Tme Cost Actvty Oopton Tme Cost

10 520 K. Aladn, A. Afshar and E. Kalhor Fgure 2. Non-domnated solutons for dfferent nterest rates for dscounted cash flow case Fgure 3. Total cost and dscounted cash flow cost wth $1000 weekly ndrect cost Table: 2 Modes of constructons for selected decson ponds from non-domnated solutons Actvty number T r % TC DC

11 DISCOUNTED CASH FLOW TIME-COST TRADE-OFF PROBLEM CONCLUSIONS Most of the tme-cost trade-off (TCTO) analyss n constructon management has dsregarded tme value of money. In ths study a mult-objectve ant colony optmzaton based model for TCTO problem was developed and tested whch mnmzes the dscounted cash flow of drect cost. The general form of dscount factor was embedded nto the modelng structure wthout any smplfyng assumptons. The model accepts the precse dscrete actvty tme-cost relatonshp, effectvely generates non-domnated set of solutons, and accounts for tme value of money. Although the model s formulated to account for DCFs n the determnstc envronment, effect of nflaton and stochastc nature of the problem can be ncorporated n further studes. Applcaton of the model showed that ncluson of dscounted cash flow would result n dfferent modes of constructon as well as actvtes duratons and costs and consequently optmal project duraton. From the results obtaned, t was obvous that gnorng tme value of money n the analyss of TCTO problem could produce msleadng decsons. The proposed approach can help the practtoners n consderng net present value n tme-cost decsons leadng to dentfcaton of the best opton. It can be concluded that DCF should be consdered n the analyss of TCTO problem, especally for projects span over tme perods more than 1 year. TCTO analyss wth DCF produces realstc results and consequently sound decsons REFERENCES 1. Eshtehardan E, Afshar A, and Abbasna R. Fuzzy-based MOGA approach to stochastc tme-cost trade-off problem, Automaton n Constructon, 18(2009) Feng CW, Lu L, and Burns SA. Usng Genetc algorthm to solve Constructon Tme- Cost Trade-Off problems, Journal of Computng n Cvl Engneerng, 11(1997) De P, Dunne EJ, and Wells CE. The dscrete tme-cost trade-off problem revsted, Europan, Journal of Operatonal Research, 81(1995) Yang, I. Usng eltst partcle swarm optmzaton to facltate B-crteron tme-cost trade-off analyss, Journal of Constructon Engneerng and Management No. 7, 133(2007) L H, Cao JN, and Love PE. Usng machne learnng and GA to solve tme-cost trade-off problems, Journal of Constructon Engneerng and Management, 125 (1999) Hegazy T. Optmzaton of tme-cost trade-off analyss: usng genetc algorthms, Canadan Journal of Cvl Engneerng, 26(1999) Zheng DXM, Ng ST, Kumaraswamy MM. Applyng a genetc algorthm-based multobjectve approach for tme cost optmzaton, Journal of Constructon Engneerng and Management. 130(2004) Xong Y, and Kuang Y. Applyng an ant colony optmzaton algorthm-based multobjectve approach for tme-cost trade-off, Journal of Constructon Engneerng and Management, 134(2008) Neg ST, and Zhang Y. Optmzng constructon tme and cost usng ant colony optmzaton approach, Journal of Constructon Engneerng and Management, No. 4,

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