An Estimting Method for IT Project Exected Durtion Oriented to GERT Li Yu nd Meiyun Zuo School of Informtion, Renmin University of Chin, Beijing 100872, P.R. Chin buyuli@ruc.edu.cn zuomeiyun@263.net Abstrct. Reworking in develoing softwre exists widely in IT roject mngement. But the current estimting methods on roject durtion cnnot del well with this roblem. In this er, some key concets, such s djusted ctivity, djusted GERT network, exected roject durtion, exected criticl th, re firstly defined. Then uthors study estimting method on roject durtion for different rework modes, including seril, rtctic, nd combined structure by encsulting of ctivities. Finlly, generl estimting method on IT roject durtion oriented to GERT is resented, nd detiled stes re illustrted with n exmle roject on IT softwre develoment. Keywords: Project Mngement, Project Schedule Estimtion, Exected Durtion l. INTRODUCTION In roject mngement, roject schedule often fces with the roblem how to know roject durtion under the restriction of resource. PERTProgrm Evlution nd Review Technique nd CPM Criticl Pth Method re the most imortnt roject mngement techniques [1], which hve been creted out in the need of Western industril nd militry estblishments to ln, schedule nd control comlex rojects [2]. GERT Grhicl Evlution nd Review Technique is nother imortnt nlysis technique, being widely using in engineering mngement nd enterrise roducing for strong bility to describe reworking in roject mngement [1]. Reworking exists in IT roject mngement widely, nd GERT cn describe it well. But current method to estimte roject durtion only cn del with determinte roject network without reworking. So, the roblem how to estimte roject durtion with reworking becme key technique for the using of GERT. Currently, few such reserches re mde in the world. Very few reserch is mde on estimting roject durtion oriented to GERT by simulting method [3]. But it is difficult to comute ccurtely roject durtion, esecilly when times of simulting is few, the error is more obvious. The er tries to resolve the roblem by encsulting of ctivities. In next section, key concets, such s exected durtion nd exected criticl th etc, re resented. In section 3, uthor reserch on estimting method on IT roject durtion under ll kinds of reworking cses, including seril, rtctic, nd combined
1558 Li Yu nd Meiyun Zuo structure. Generl resolving method is given in section 4. In section 5, estimting method is used in roject exmle bout the develoment of informtion system roject. 2. EXPECTED CRITICAL PATH AND EXPECTED PROJECT DURATION Under the condition with reworking, criticl th cn not be determined becuse the number of reworking for certin ctivity is not determinte. So there my be lot of criticl ths which hve sme ctivities in roject network digrm with reworking. Since the number of ctivities consisting of criticl th is determinte nd ech ctivity hs determinte durtion, roject network digrm is determinte. Project durtion could be infinite when reworking of certin ctivity lwys hened. If no reworking hened, GERT network becme determinte network nd roject durtion is determinte. Project durtion cn be comuted by CPM nd PERT methods. But both of the methods re not dt to GERT network with reworking. Although criticl th nd durtion of the roject bsed on GERT re not determinte, exect durtion of ech ctivity could be comuted ccording to the robbility distribution of the reworking. Then, ccording to exected durtion of ech ctivity, criticl th of roject cn be comuted by using CPM/PERT methods. Becuse this kind of criticl th is bsed on exected durtion of ech ctivity, we cll it s exected criticl th of roject, nd corresonding durtion s exected roject durtion. They re very useful in roject schedule mngement when reworking could hen. 3. ESTIMATING OF EXPECTED PROJECT DURATION FOE DIFFERENT REWORKING MODES 3.1 Including One Reworking Activity Here, being suosing tht roject only includes one reworking ctivity A, nd its durtion is, the robbility of reworking is. As shown in Fig.1. Now, how to comute roject durtion? Noting the exected roject durtion s D, so D ( D) And D 1 2
An Estimting Method for IT Project Exected Durtion Oriented to GERT 1559 Figure 1. Project Including One Reworking Activity Figure 2. Seril Reworking In the following, two extremities be nlyzed, =0 In this cse, it mens tht no reworking exists, nd exected roject durtion cn be comuted s following, D 3 0 2=1 In this cse, it mens tht reworking lwys exists. As be showed in Eqution (4), exected roject durtion is +, D 1 4 Generlly, bigger shows higher reworking robbility, so exected roject durtion is longer nd D=D() incresed with incresing of reworking robbility. Sometimes, it is usul tht roject llows to times of reworking. Tht is to sy, fter to number of reworking, the ctivity must be finished. The mode exists widely in roject mngement rctice. In this mode, comuting exected roject durtion hs rcticl mening. Let s noting Dn s roject durtion while llowing n times of reworking for n ctivity, then the number list {Dn} (n=1,2 ) hs following equtions, D 0 = D 1 = + D 2 = ++ 2. D n = ++ + n So, n1 (1 ) Dn Esecilly, while n it mens there is no limit for times of reworking. So
1560 Li Yu nd Meiyun Zuo n1 (1 ) D lim Dn lim n n 1 This conclusion is consistent with bove the result. 5 3.2 Including Lots of Reworking Activities 3.2.1 Seril Reworking As shown in Fig.2, roject includes two seril reworking ctivities A nd B. Then D D A D B And DA nd DB cn be comuted s Equtionnd Eqution2, ( ) DA 2 DA D A 2 ( ) b DB b 4 DB D B 4 So D D A D B b 2 1 4 6 3.2.2 Prtctic Reworking As shown in Fig.3, roject includes two rtctic reworking ctivities A nd B. Then D mx{ D A, D } And DA nd DB cn be comuted s Equtionnd Eqution2so ( ) DA 2 DA D A 2 ( ) b DB b 4 DB D B 4 Then b D mx{ DA, DB} mx{, } B 2 4 7
An Estimting Method for IT Project Exected Durtion Oriented to GERT 1561 Figure 3. Prtctic Reworking Figure 4. Combined Reworking 3.2.3 Combined Reworking As shown in Fig.4, roject includes seril reworking nd rtctic reworking. Among of them, ctivities A nd B re rtctic, being seril with ctivity C. Then, exected roject durtion is D mx{ D, D } D According to Eqution (6) nd Eqution (7), D be comuted s following b c D mx{ DA, DB} DC mx{, } A B C 2 4 1 6 8 3.3 Reworking on Lots of Activities As shown in Fig.5, if ctivity B needs reworking, ctivity A lso must rework. Reworking reltes to two ctivities. According to Eqution (1), then D ( b) ( 2 D) So b D 9 2
1562 Li Yu nd Meiyun Zuo Figure 5. Reworking on Lots of Activities Figure 6. Multi-choices of Reworking 3.4 Multi-choices of Reworking In IT roject mngement, there re often lots of th choices for one ctivity. As shown in Fig.6, fter ctivity A is finished, reworking could be hened t once or ctivity B or C could be executed. Tht is to sy, fter ctivity A be finished, there re three th choices. In this cse how to comute roject durtion? In fct, this cse cn be trnsformed to two reworking choices mode. Activity B nd C cn be encsulted to one virtul ctivity D, s be shown in Fig.7. After encsulting ctivity B nd C, ccording to the robbility theory, the durtion of virtul ctivity D cn be comuted s following 2 * b 3 * c D D 10 So, roject durtion is DD * A D 1 2 2 3 3 2 * b3 * c 2 * b3 * c 2 3 1 Figure 7. After Encsulting Activity B nd C
An Estimting Method for IT Project Exected Durtion Oriented to GERT 1563 Figure 8. Encsulting Activity 4. ESTIMATING STEPS OF EXPECTED PROJECT DURATION ORIENTED GERT After discussing the comuting method on exected roject durtion under different modes, we rovide the generl estimting stes of exected roject durtion oriented to GERT in the following. 4.1 Encsulting Reworking Activities The key ste in estimting of exected roject durtion oriented to GERT is to trnsform GERT network with reworking ctivities to trditionl determinte network without reworking ctivities. So the first ste is to modify GERT network by encsulting reworking ctivities, s shown in Figure 8. Activity A with reworking is trnsformed to modified ctivity A* without reworking by encsulting. We cll ctivity A* s djusted ctivity of the ctivity A. 4.2 Comuting of Durtion of Adjusted Activity--- * D A According to Eqution (2), durtion of djusted ctivity A* is * D A 2 We cll it s djusted durtion of ctivity A. 12 4.3 Generting Adjusted GERT Network Doing s bove two stes for ll ctivities with reworking, we cn generte new GERT network without reworking. We cll the network s djusted network digrm. 4.4 Comuting of Exected Criticl Pth nd Exected Durtion After eliminting reworking ctivity by encsulting nd generting djusted GERT network, exected criticl th nd exected roject durtion cn be comuted by trditionl CPM/PERT methods.
1564 Li Yu nd Meiyun Zuo 5. EXAMPLE For better exlining the bove method, here we give IT develoment roject consisting of system design, buying hrdwre, softwre nlysis, buying softwre, hrdwre exmining, hrdwre setu, etc, s shown in Fig.9. Among these ctivities, two ctivities, hrdwre exmining nd system exmining, could exist reworking. If there exists questions in hrdwre exmining ctivity, hrdwre modifying ctivity hened. System exmining hens ccording to trouble degree. All ctivities durtion nd reworking robbility re shown in Fig.9. Now, we try to comute exected criticl th nd roject durtion. Figure 9. GERT Network of IT Project Figure 10. Modified Activity M fter Encsulting 5.1 Encsulting Reworking Activities In this exmle, there exist reworking in two ctivities. They cn be encsulted to two djusted ctivities M nd N, s shown in Fig.10 nd Fig.11. 5.2 Comuting of Exected Durtion of Modified Activity Exected Durtion of Adjusted Activity M According to Eqution (2), exected durtion of djusted ctivity M cn be comuted s following,
An Estimting Method for IT Project Exected Durtion Oriented to GERT 1565 DE DD 3 4 D M = 8. 75 P 0.8 1 2Exected Durtion of Modified Activity N Becuse ctivity K exist three th choices, ccording to Eqution (10), exected durtion of djusted ctivity N cn be comuted s following, P3*9 P5*18 4.5 D 16,21 = 7. 5 0.6 0.6 D 13,21 = D 13,14 + D 14,15 + D 15,16 +D 16,21 =4+0+6+11.25=21.25 Figure 11. Modified Activity N fter Encsulting Figure 12. Modified GERT Network Without Reworking 5.3 Generting Adjusted GERT Network After encsulting ll reworking ctivities nd comuting their djusted durtion, we cn get djusted GERT network without reworking, s shown in Fig.12. 5.4 Comuting of Exected Criticl Pth nd Exected Durtion Using CPM techniques, we cn get exected roject criticl th, A B C M F N Then, its exected durtion cn be comuted,
1566 Li Yu nd Meiyun Zuo D=30+4+3+8.75+6+21.25=73dys 6. CONCLUSIONS Reworking exists widely in IT roject mngement. The uthors studied estimting method on roject durtion in different modes bsed on GERT network digrm, nd got the generl stes of comuting roject durtion. The method not only dts to GERT network with reworking, lso dts to trditionl determinte roject network without reworking. Mturity of the method needs to be more verified in the future. ACKNOWLEDGEMENTS This reserch ws suorted by Oen Foundtion of Key Lbortory of Dt Engineering nd Knowledge Engineering, Ministry of Eduction P. R. Chin, Oen Foundtion of Key Lbortory of Informtion Mngement nd Economy, Ministry of Informtion Industry, P. R. Chin under grnts F0607-31, F0607-42, nd Reserch Foundtion of Renmin Univ. of Chin. REFERENCES 1. K. Kenzo nd N. Nishiuchi, Efficient Monte Crlo simultion method of GERT-tye network for roject mngement, Comuters nd industril engineering. Volume 42, Number 2-4,.521-531, (2002). 2. E.H. John, Alictions of simultion in roject mngement, in Proceedings of the 11th conference on Winter simultion (Sn Diego, Cliforni, United Sttes, 1979),.211-219. 3. K. Kim nd J.M.D.L. Grz,. Evlution of the resource-constrined criticl th method lgorithms, Journl of construction engineering nd mngement-asce. Volume 131, Number 5,.522-532, (2005). 4. A. Azron, H. Ktgiri, nd M. Skw, A multi-objective resource lloction roblem in PERT networks, Euroen Journl of Oertionl Reserch. Volume 172, Number 3,.838-854, (2006). 5. C.H. Cheng, Fuzzy reirble relibility-bsed on fuzzy GERT, Microelectronics nd Relibility. Volume 36,.1557-1563, (1996). 6. E.R. Clyton nd L.J. Moore, GERT vs. PERT, The Journl of System Mngement. Volume 22,.11-19, (1972). 7. E.D. Dvid, PERT nd simultion, in Proc. of the 10th conference on Winter simultion (Mimi Bech, FL, 1978),.89-98.