Progress Risk Assessment for Spliced Network of Engineering Project Based on Improved PERT

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1 Available online a Sysems Engineering Procedia (0) nernaional Conference on Risk and Engineering Managemen (REM) Progress Risk Assessmen for Spliced Nework of Engineering Projec Based on mproved PERT Li Wenying a Liu Xiaojun b School of Managemen, Xiogy, Xi School of Managemen, Xi * Absrac Aimed a he deficiency of he radiional PERT mehod in he risk assessmen for engineering projec progress, his paper has improved PERT mehod, so as o make i more suiable for engineering projec risk assessmen of spliced nework. irsly, in calculaing projec aciviy duraion, i adops response managemen heory and inroduces double-buffering zone, so ha he calculaed projec aciviy duraion is more approximae o he realisic one. Meanwhile, he paper akes advanage of equivalen-weigh probabiliy mehod o revise he main pah, so ha he calculaed projec duraion is more suiable for he realisic siuaion. inally, he paper has furher demonsraed he feasibiliy of he mehod by an engineering projec case sudy. Key words: engineering projec; spliced nework; improved PERT; progress risk assessmen. nroducion n he lae 950s, nework plan appeared in he U.S.A., and in he mid-60s, our counry began o apply i ino engineering projec []. The spliced nework in he projec is characerized wih universaliy, while oher nework plans can be regarded as is special cases []. Due o uncerain facors in he engineering projec iself and is environmen, he aciviy duraion during projec implemenaion is uncerain. Malcolm, a scholar of he U.S.A., and ohers firsly proposed PERT (Program/Projec Evaluaion and Review Technique) [3]. Wih he wide applicaion of PERT mehod ino risk assessmen for engineering projec progress, people gradually realized is deficiency: () he formula of projec aciviy duraion and variance is oo fixed and uniary o change wih he acual siuaion; () due o lack of consideraion for he ineracion of he pahs in he nework plan, he expeced duraion calculaed is less han he acual and he variance is more han acual one; (3) PERT mehod can no be applied direcly ino he assessmen for spliced nework. Aimed a he aforemenioned problems, his paper will mainly analyze ha how o improve PERT mehod so as o do engineering progress risk assessmen for he spliced nework.. Transformaion of spliced nework *Liu Xiaojun.,fax: ;Li Wenying,el: address: xjun_liu@63.com;liwenyinglwy@6.com. Suppored by he key discipline of Shaanxi Province Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selecion and/or peer-review under responsibiliy of he Organising Commiee of The nernaional Conference of Risk and Engineering Managemen. doi:0.06/j.sepro

2 7 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) 7 78 PERT mehod is no suiable for progress risk assessmen for aciviy-on-arrow nework, and can no be applied direcly ino he spliced nework, herefore, i should be ransformed. There are four kinds of basic splice relaions for he spliced nework [4] : from sar o sar (STS); from sar o finish (ST); from finish o sar (TS); from finish o finish (T). Besides, here are wo kinds of mingled splice relaion: STS and T; ST and TS. or he spliced nework wih splice relaion of STS, we can use Gan char o represen he consrucion procedures of he neighbouring wo splices, hus using aciviy-on-arrow nework diagram o represen is splice relaion. However, for oher splice relaion, his mehod is no suiable. So we can use he following mehod o deal wih spliced nework [5]. Supposing here exiss STS splice relaion in aciviy and, he inerval STS is, such as ig. (a), hen we can esablish he corresponding nework relaion, such as ig. (b), in which node n represens end poin of aciviy-on-arrow nework, node and is respecively he sar nodes of aciviy and, he duraion of (, ) is, represening he inerval of aciviy and, dummy arrow represens virual work, showing he logical relaion during he aciviies, is duraion is 0. n ig.. (a) STS splice; (b) STS splice relaion afer reamen The oher similar splice relaion can be handled according o he aforemenioned mehods, he resuls are as follows: Table. igure of various splice relaion. Name of splice relaion Diagram of splice relaion Aciviy-on-arrow nework diagram wih broad sense STS n ST n TS T

3 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) STS T, 3 4 ST TS, 3 4 Aiming a various splice relaion of spliced nework, we inroduce inerval process o adop he aforemenioned mehod in he above figure o deal wih he various splice relaion, hen obaining he progress nework wih broad sense including convenional process, inerval process and virual process. 3. The improved PERT mehod used o evaluae progress risk of spliced nework 3.. Duraion esimaion for he engineering projec aciviy by inroducing double-buffering zone The projec would be usually posponed for various reasons. Because he aferward conrol can no conrol he progress efficienly, he beforehand managemen mus be enhanced. The progress delay may come from he uncerainy during consrucion. Aimed a such progress risk, he applicaion of buffer zone brings good resul [6]. However, he projec planners seldom inroduce i ino he progress managemen. The non-buffering managemen heory proposed in he 960s holds ha buffer occupies exra space wihou creaing value, resuling in a wase. However, he recen research believes ha cerain buffer is quie imporan o reduce risk effecively. Double-buffering has wo advanages: () beforehand buffer prepares pre-esing process for each work. The people responsible for he projec, hrough pre-esing, are able o revise he hidden rouble before work so as o reduce he impac on he aferward work. () When he work is posponed, he delay can be eliminaed parially hrough buffer. Correspondingly, oher pars of delay can be eliminaed hrough beforehand buffer of he aferward work, which plays an imporan role in proecing he progress plan for engineering projec. or PERT mehod, he duraion of every aciviy in he projec is esimaed based on he exising informaion, he experience of he projec progress manager, and non-buffering managemen heory. Here hree esimaed values abou aciviy duraion are given: opimisic esimaed ime a, pessimisic esimaed ime b and he mos likely esimaed ime m. Therefore, when esimaing aciviy duraion by hree-poin mehod, we add double-buffering zone of each aciviy, so ha projec aciviy duraion will be more reasonable and equipped wih pracical significance. The lengh of he buffer zone is decided by aleraion probabiliy which is depended on former projec informaion and he experience of managers. The specific calculaion mehod can be learned from he bibliography [6]. B= SB= Cf ( CP)CP 0.4 Cf 0. CP 0.4 Cf CPCP 0.6 Cf 0.9 CP 0.6 n he above formula, B is he lengh of fron buffer zone; SB is he lengh of laer buffer zone; C f aleraion duraion of he work; CP is expeced aleraion probabiliy. is expeced () ()

4 74 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) The expeced value and variance calculaion of engineering projec aciviy duraion 3. Afer esimaing duraion of each aciviy of he projec, we need o calculae he expeced value and variance of he aciviy. supposed, in PERT mehod, ha projec aciviy y ha he expeced value and variance calculaed by his mehod has large deviaion, hus using i o assess he progress risk has cerain errors. Bibliography[8] summarized five mehods and presened some deviaion analysis, in which he expeced value formula Dij =0.63m +0.85(a+b) by Perry-Greig and variance formula ij =0.63(m-Dij) [(b-dij) + (a-dij) by Person-Turky has small deviaion in calculaing expeced value and variance of he aciviy duraion wih average deviaion of 0.0% and 0.5%. (4) Dij = 0.63m+0.85(a+b) (3) ij = 0.63(m-D ij ) +0.85[(b-D ij ) +(a-d ij ) ] n order o make he resuls of he PERT mehod more suiable for he acual siuaion, we can adop he aforemenioned formula o calculae he expeced value and variance of engineering projec aciviy duraion Calculaion of ime limi for engineering projec in improved PERT mehod by considering of he ineracion among pahs n he radiional PERT mehod, i is supposed here is a main pah in he nework. However, his mehod is lack of consideraion ha he oher non-dominan pah may have an effec on he main pah, so i is no reasonable. The calculaion mehod of he ime limi for engineering projec wih consideraion of ineracion in he nework pahs is as follows Compound analysis on PERT nework pahs () One pah inegraing ino he nodes The expeced value TEj and variance TEj of he earlies ime TE j of node is he expeced value and variance of compleion ime TE j for all predecessors. (5) (7) TEj=TE i + ij When he progress duraion ime yields o normal disribuion:, i<j TEj =TEi +D ij, i<j (6) TEj = TEi + ij i<j () us wo pahs inegraing ino he nodes There are jus wo pahs inegraing ino he nodes, he expeced value TEj and variance TEj of he earlies ime TE j of node j should be he compound value of wo pah duraion. Calculaing he expeced value TEj k and variance TEj k of he earlies ime from pah k(k=,) o node j and deermining he main pah. or pah k, is earlies compleion ime TE is: (8) (0) TEj k =TE i + ij, i<j, k=, When progress duraion yields o normal disribuion: j k TEj k = TEi +D ij, i<j (9) k TE j = TEi + ij i<j Calculaing difficuly coefficien

5 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) = Ts Te () n The smaller difficuly coefficien is he main pah. T s is he duraion of he longes pah. T e is duraion of he corresponding pah wih. n is sandard deviaion of he corresponding pah wih. Reversing he arrow direcion in PERT nework diagram, we search for he neares shun node of every pah inegraing ino he node. The aim is o eliminae he relaed pah s effec on he compound resuls in he process of synhesis. Calculaing he expeced duraion T k bj and corresponding variance ( bj k ) of pah k, relaive o he neares shun node b and he analyzed node j. j T = gh, b g<h j () bj k gb D ( bj k ) j = gh, b g<h j (3) gb Revising he expeced compleion ime of he main pah by equivalen probabiliy mehod. k Supposed ha he expeced value of relaive duraion of he main pah form shun node b o node j is T bj, he corresponding variance is ( bj k ), hen he compleion probabiliy by expeced value is 0.5, and he one of non-dominan is p. Due o he muual independence of he wo pah, he compleion probabiliy afer compound is 0.5p. is supposed o prolong i based on he expeced value T bj k of main pah duraion, is variance afer prolonged is he same, bu is duraion is prolonged by T. The revision mehod of his kind is called equivalen probabiliy revision mehod. Then T can be calculaed by he following formula. T = * bj * n he above formula, he value of can be obained by checking normal disribuion able wih value of 0.5p. bj is he sandard deviaion of he main pah. Upon he aforemenioned revision, we can obain he expeced value TE * j of he earlies ime of he node j. TE j * = TE j+ T (5) f we need o calculae he ime parameers of he nework wih he engineering projec aciviy as he objec, hen supposing here exising an added virual aciviy j aciviy is T, and he variance is 0. The aferward aciviy of he original node j is he one of he added virual aciviies. (3) More han wo pah inegraing ino he nodes is similar o he former kind of node; we need o firsly deermine he main pah, hen revising i by non-dominan pah. When he wo more pahs are muually independen, can be obained by checking normal disribuion able wih value of A=0.5pp pn, in which n is he number of he inegraed pah in he hird kind of nodes. When he wo more pah is dependen or parially independen, we can choose he sub-nework o revise so as o make i an independen pah and hen revise i Calculaion of ime limi for engineering projec Afer obaining he expeced value Dij of aciviy (ij) by equivalen probabiliy revision mehod, we can obain expeced ime parameers of various aciviies by CPM mehod of deermined nework. Supposed he pah duraion of nework planning diagram yield o he normal disribuion: E(T n )= ij (6) (4) n = ij (7) 3.4. Compleion probabiliy and risk of ime limi for engineering projec 3.4. Compleion probabiliy of engineering projec Supposing he required ime limi for engineering projec is T, and he pah duraion of nework is normally disribued, we can use he following formula o calculae projec compleion probabiliy.

6 76 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) 7 78 P( T s )= d d (8) n he formula, he inegral erm T e is he average value; n is probabiliy densiy funcion of sandard deviaion; T e is E (T n ); n is he sandard deviaion of he main pah, ha can be calculaed by formula (6) and (7). Obviously, if Ts=T e, he compleion probabiliy P=0.5. f T s >T e, P>0.5. f T s <T e, P<0.5. n order o make he calculaion simple, we can ransfer formula (8) ino sandard normal disribuion (Te=0 n =), hen using he normal disribuion able o calculae compleion probabiliy P. f =T, we can obain he following equaion by formula (8): P(T )= dt (9) According o he value of difficuly coefficien, we can obain he compleion probabiliy P under he required ime limi for projec T by checking from sandard normal disribuion able Progress risk for engineering projec The progress risk of projec can be described by risk probabiliy P r, and defined i as he uncompleed probabiliy wihin he required ime limi for engineering projec. Evidenly, if here is compleion probabiliy P, we can obain he calculaion formula of projec progress risk probabiliy P r. P=-P (0) r These and he Reference headings are in bold bu have no numbers. Tex below coninues as normal. 4. Case sudy A sub-nework progress plan of a pracical projec is as ig. 4, he planned ime limi for projec is T s =55 days. Supposing he duraion of projec process approximaely yield o he normal disribuion, we can do he progress risk assessmen as follows. B D Sar E S A C ig.. he spliced nework diagram of a pracical projec 4.. Transformaion of he spliced nework We ransform he spliced nework plan in ig. ino aciviy o arrow nework plan wih broad sense according o he mehod described in he paper as he following figure.3 shows. B 3 S BC BD D SS CE 5 S DE 8 E 9 A 6 C 4 AC 7

7 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) ig. 3. progress nework diagram wih broad sense of a pracical projec 4.. Esimaion aciviy duraion of he engineering projec The difference beween he inerval process and oher processes in characerisics is ha he inerval process doesn consume resources, and here is no essenial difference in oher aspecs. 4.. According o he aforemenioned mehods, we can use he formula () and () o calculae he lengh of double-buffering zone of every projec aciviy, hen esimaing he aciviy duraion wihin double-buffering zone by hree poin mehod. 4.. According o formula (3) and (4), we can calculae expeced value and variance of projec aciviy duraion. The calculaed resuls are as follows. Table. Characerisic value of he projec aciviy duraion. aciviy Expeced value of aciviy duraion (day) Sandard variance of aciviy duraion Variance of aciviy duraion A(,4) B(,) C(6,7) D(3,5) E(8,9) S BC (3,6) BD (,5) AC (4,7) S DE (5,8) SS CE (6,8) Calculaion projec compleion probabiliy and progress risk rae by equivalen probabiliy mehod n he nework as ig. 5 shows, node 5,6, and 7 is he inegraed node of he wo pahs, node 8 is he inegraed node of hree pahs. or node 5, he neares shun node is node. can be obained, hrough formula () and (3), ha expeced value of duraion of pah -5 and -3-5 is respecively 3 days and 5 days wih variance of Obviously, pah -3-5 is he main pah which needs o be revised. can be acquired ha =.86 by formula (), he corresponding probabiliy is by sandard normal disribuion able, T=0.005 days by formula (4), which shows ha pah -5 has lile effec on pah -3-5 and can be negleced. n he same way: or node 6, he main pah is can be acquired by calculaing ha T=0.67 days. or node 7, he main pah is 4-6-7, and he calculaed T is so small ha i can be negleced. or node 8, here are hree nodes o inegrae, which have he common neares shun node ha is node. Because pah and pah have common shun poin, we need o make a compound analysis on i and hen synhesize he resuls wih pah The average value of pah and pah are respecively 8 days and 0 days wih variance of 0.5 and Evidenly, pah is he main pah ha needs o be revised. can be acquired ha =.77 by

8 78 Li Wenying and Liu Xiaojun / Sysems Engineering Procedia (0) 7 78 formula () and T=0.003 days by checking sandard normal disribuion ale and formula (4) ha can be negleced. or pah , is average value is 8 days and is variance is or pah , is average value is 30 days and is variance is.. The laer is he main pah ha needs o be revised. is obained ha =.39 by calculaing and T=0.0 days by checking sandard normal disribuion able and formula (4). or his nework planning, we acquired he expeced ime limi for projec is days wih variance of.5 by improving PERT mehod. When he required ime limi for projec is 40 days, he compleion probabiliy is 89.% and he progress risk rae is P r =0.9%. 5. Conclusion This paper, by analyzing engineering projec spliced nework and applying he improved PERT mehod, has made a progress risk assessmen on i. The improved PERT mehod mainly has he following hree characerisics: () o calculae engineering projec aciviy ime and variance by formula nearer o pracical siuaion; () o revise he main pah o acquire more accurae compleion probabiliy and risk rae due o consideraion of he ineracion of he pahs in nework planning diagram; (3) o apply PERT mehod ino progress risk assessmen of spliced nework for engineering projec. inally, his paper analyzed he imporance of he improved PERT mehod in progress risk assessmen of spliced nework by a pracical example wih he hope ha i has some guidance for engineering projec progress risk assessmen in he fuure. References. Xie Xing-hao, Applicaion prospec of nework plan in consrucion, ournal of Xi &Technology,00(6), Chen Hu, Managemen of engineering projec, Beijing: China's consrucion indusry press, 00, Yuan Ting-wei, On applicaion of improved PERT in risk analysis of consrucion process, Shanxi archiecure,00(3),-. 4.Li Quan-yun, mproving Calculaing Mehod for he Time Parameer of Splice Nework Planning. Building science, 005(), Tong He-jing, Transforming Spliced Nework o AoA Nework, Technology economics,009 (0), Zhao Zhen-yu, Sudy on he Double-Buffering Mehod for Projec Schedule Conrol, ournal of Engineering Managemen, 00(5), Xu Zhi-sheng, Evaluaion of Consrucion Progress Risk Using PERT Mehod in Huge Bridge Projec, ournal of Disaser Prevenion and Miigaion Engineering,009(), KEEER.D.L,VERDN.W.A, Beer Esimaion of PERT Aciviy Time Parameers, Managemen science,993,39(9): Wang Zhuo-fu, Risk managemen and response of engineering projec, Beijing: The waer and elecriciy press,005:6-8.