Application of Network Analysis to Project Management.

Transcription

1 Applicatio of Network Aalysis to Proect Maagemet. R.A. Adeleke, O.Y. Halid, O.D. Oguwale, ad A.O. Olubiyi Departmet of mathematical Scieces, Faculty of Sciece, Uiversity of Ado-Ekiti, Ekiti State, Nigeria. ABSTRACT This paper cosidered the activities ivolved i a costructio site as a etwork ad proposed a etwork aalytic techique amely the Critical Path Method (CPM) ad a traditioal method for all activities ivolved i the site to obtai the latest fiish time (completio time) of the proect. Results show that the CPM gave a shorter completio date tha the traditioal method sice it gave a lower latest fiish time ad cosequetly, lower ruig ad persoel costs. (Keywords: etwork, etwork aalytic techique, critical path maagemet, CPM, latest fiish time) INTRODUCTION Network aalysis is the geeral ame give to certai specific techiques which ca be used for plaig, maagemet ad cotrol of proect. It ofte acts as a etwork maagemet tool for breakig dow proects ito compoets or idividual activities ad recordig the result o a flow chart or etwork diagram. These results geerally reveal iformatio that is used to determie duratio, resource limitatios ad cost estimates associated with the proect. It offers isight ito what is occurrig at each critical poit of the etwork. Proect maagemet ad efficiet resource allocatio are two critical aspects of the productio ad operatios maagers resposibilities. Sice a proect is o-repetitive ad temporal i ature, the mode of maagemet differs from the usual ob shop or other related types of schedulig. A proect cosists of tasks with defiite startig ad ultimate edig poits ad hece a proect maager is saddled with the resposibilities of gettig ob doe o schedule withi allowable cost ad time costrait specified by the maagemet. Typically all proects ca be broke ito: Separate activities where each activity has a associated completio time (time from the start of the activity to its fiish). Precedece relatioships which gover order i which we may perform the activities. The mai problem is to brig all these activities together i a coheret fashio to complete the proect at a required time. Apart from the traditioal method of addig activity duratios, these exist two differet techiques for etwork aalysis amely the PERT Program Evaluatio ad Review Techique ad CPM Critical Path Maagemet. PERT has the ability to cope with ucertaity i activity completio times while CPM emphasized o the trade-off betwee cost of the proect ad its overall completio time. The CPM has the advatage of decreasig completio times by probably spedig more moey. Network aalysis eables us to take a systematic quatitative structural approach to the problem of maagig a proect through to successful completio. Also, sice it has a graphical represetatio, it ca be easily uderstood ad used by those with a less techical backgroud. MATERIALS AND METHODS The followig is the list of activities obtaied from a costructio firm ivolved i the costructio of a shoppig complex is sequetial The Pacific Joural of Sciece ad Techology Volume 1. Number 1. May 11 (Sprig)

2 order. Clearig of site Settig out Excavatio Cocrete of colum base Colum reiforcemet Castig of strips Form works Castig of colums Settig of block wall sub-structure Earth fillig Hard core Electrical fixig I Plumbig fixig I Castig of groud floor slab Reiforcemet of groud floor colums/litel Electrical fixig II Plumbig fixig II Form work of groud floor colums/litel Cast of groud floor colums/litel Form work for first beam/first floor slab Settig of groud floor block wall Electrical fixig III Plumbig fixig III First floor slab reiforcemet Castig of first floor slab Reiforcemet of the first floor colums /litel Electrical fixig IV Plumbig fixig IV Form work first floor colums/litel Castig of first floor colums/litel Form work for roof beam/reiforcemet Castig roof beam/reiforcemet Settig of first floor block wall Electrical fixig V Plumbig fixig V Roof/ceilig Door/widows Plasterig Fial electrical fixig Fiishig Paitig/decoratio Exteral works (Electrical/Plumbig) The activities are tabulated i the succeedig tables. Liear Programmig Formulatio To esure the formulatio, we assume that our route is to be determied betwee odes where also equals the total umber of etwork modes. It will also be assumed that a sigle brach coects ode to ode (i.e. distaces ad braches associated with odes that are ot directly coected are igored). Also, i the evet of a directed brach, we do ot cosider flows i oppositio to the directio. Thus, we wish to fid the set mi imize d X i 1 1 k subect to X 1 k 1 1 X k, ik, i, k k, i 1 1 i, i, 1 1 to: X X for k,..., 1 Xik,,1 for all i, whered i, dis ta ce from ode i to ( for directly coected odes) 1 if the brach from ode i to is coected X i, otherwise The first two costraits assure us that exactly oe brach is take from the first ode whereas exactly oe brach is take ito the fial ode. The third costraits simply force the umber of braches ito a itermediate ode to equal those out of that ode. The Critical Path Compositio All activities have bee elisted i the path to completio. The CPM will be used to choose the path with the shortest completio time. The Pacific Joural of Sciece ad Techology 6 Volume 1. Number 1. May 11 (Sprig)

3 Table 1: List of Activities with Estimated Duratio ad Immediate Predecessor. S/N Duratio (Days) Activity Immediate Predecessor 1 1 Clearig of site - 1 Settig out 1 9 Excavatio 4 Cocrete of colum base Colum reiforcemet 6 Castig of strip, Form work 6 8 Castig of colums 6,7 9 Settig of block wall sub-structure 8 1 Earth fillig Hard core Electrical fixig I 1 1 Plumbig fixig I 1 14 Castig of groud floor slab 11, 1, Reiforcemet of groud floor colums/litel Electrical fixig II 1 17 Plumbig fixig II 1 18 Form work of groud floor colums/litel 16, Cast of groud floor colums/litel 18 Form work for first beam/first floor slab Settig of groud floor block wall Electrical fixig III 19 Plumbig fixig III First floor slab reiforcemet 1 Castig of first floor slab 4 6 Reiforcemet of the first floor colums /litel 7 Electrical fixig IV 8 Plumbig fixig IV 6, Form work first floor colums/litel 8 4 Castig of first floor colums/litel 9 1 Form work for roof beam/reiforcemet 9 8 Castig roof beam/reiforcemet 1 6 Settig of first floor block wall 4 8 Electrical fixig V 1 Plumbig fixig V, Roof/ceilig, Door/widows, Plasterig Fid electrical fixig Fiishig Paitig/decoratio Exteral works Exteral works electrical 8, 9, Exteral works plumbig 8, 9, 4 The Pacific Joural of Sciece ad Techology 7 Volume 1. Number 1. May 11 (Sprig)

4 Activity Duratio (Days) Immediate Predecessor [1, ] 1 - [, ] 1 [1, ] [, 4] 9 [, ] [4, ] [, 4] [4, 6] [, 4] [, 6] [4, ] [6, 7] [4, ], [, 6] [7, 8] 4 [6, 7] [8, 9] [7, 8] [9, 1] [8, 9] [1, 11] [9, 1] [11, 1] 4 [1, 11] [11, 1] 1 [1, 11] [11, 14] [1, 11] [1, 14] [11, 1] [1, 14] [11, 1] [14, 1] [1, 14], [11, 14], [1, 14] [1, 16] 4 [14, 1] [16, 17] [1, 16] [16, 18] [1, 16] [17, 18] [16, 17] [18, 19] [16, 18], [17, 18] [19, ] [18, 19] [, 1] [19, ] [, ] [19, ] [, ] [19, ] [1, 4] 1 [, 1] [1, ] 4 [, 1] [, 4] [, ] [, 4] [, ] [4, 4] [1, 4] [, 6] 1 [1, ] [6, 7] [, 6] [6, 8] [, 6] [7, 8] [6, 7] [8, 9] [6, 8], [7, 8] [9, ] [8, 9] [, 1] 4 [9, ] [1, ] [, 1] [, ] 8 [1, ] [, ] 8 [, ] [4, ] [, 4] [, 6] 1 [, ], [4, ] [, 7] 4 [, ], [4, ] [6, 7] [, 6] [7, 8] 1 [, 7], [6, 7] [7, 9] 1 [, 7], [6, 7] [8, 4] [7, 8] [9, 4] 7 [7, 9] [9, 41] 14 [7, 9] [9, 4] 7 [7, 9] [9, 4] 9 [7, 9] [4, 4] [9, 4] [41, 4] [9, 41] [4, 4] [9, 4] [4, 44] 8 [9, 4], [4, 4], [4, 4] [4, 4] 1 [9, 4], [4, 4], [4, 4] [44, 4] [4, 44] The Pacific Joural of Sciece ad Techology 8 Table : Activity Duratios with Immediate Predecessor Activities. Volume 1. Number 1. May 11 (Sprig)

5 Figure 1: Network Diagram of Activities ad Duratios i the Site. This is divided ito: Forward Pass through the etwork, Backward Pass through etwork, Float time computatio ad Critical path determiatio. Forward Pass through the Network We compute the earliest occurrece of evet as: ES max[ ES D ] for all i, i i i, where i evet of the 1,,..., idicates the precedig activities that termiates at Also, the earliest fiish/completio time of evet is: EFi ESi D i, Backward Pass through the Network The pass determie the latest occurrece time for each evet. This is give by: LF ES at si k ad LF mi[ LF D ] for all i, i, i, The latest occurrece time of evet is: ES the earliest occurrece time of evet that i whe i 1, ES i so LFi LF D i i of the 1,,..., idicates the successor evets activities. The Pacific Joural of Sciece ad Techology 9 Volume 1. Number 1. May 11 (Sprig)

6 Determiatio of Fial Time ad Critical Path LF EF i Float time is activity ca be delayed from its ES time without disturbig the critical activities that follow it. It is equal to LS mius ES or LF mius EF for activity that is: Fi LSi ESi or The float time for evet is the differece betwee the earliest ad latest occurrece time for the evet. Note that all activities with zero float costitute the critical path. RESULTS Results are show i Table. The activities with zero float are: [1,], [,], [,4], [4,6], [6,7], [7,8], [8,9], [9,1], [1,11], [11,1], [1,14], [14,1], [1,16], [16,18], [18,19], [19,], [,1], [1,], [,6], [6,8], [8,9], [9,], [,1], [1,], [,], [,], [6,7], [7,9], [9,4] ad [4,4]. Hece the critical path is: With total duratio: = 149 Days Tora optimizatio system was also used to aalyze the data yieldig the followig output (Table 4): The Pacific Joural of Sciece ad Techology 1 Volume 1. Number 1. May 11 (Sprig)

7 Activity D i Table : Results ES EF i i LS i LF Float [1, ] [, ] [, 4] [4, ] [4, 6] [, 6] [6, 7] [7, 8] [8, 9] 8 8 [9, 1] 8 8 [1, 11] 8 8 [11, 1] [11, 1] [11, 14] [1, 14] [1, 14] [14, 1] [1, 16] [16, 17] [16, 18] [17, 18] [18, 19] [19, ] 8 8 [, 1] [, ] [, ] [1, 4] [1, ] [, 4] [, 4] [4, 4] [, 6] [6, 7] [6, 8] [7, 8] [8, 9] [9, ] [, 1] [1, ] [, ] [, 4] [, ] [4, ] [, 6] [, 7] [6, 7] [7, 8] [7, 9] [8, 4] [9, 4] [9, 41] [9, 4] [9, 4] [4, 4] [41, 4] [4, 4] [4, 44] [4, 4] [44,4] The Pacific Joural of Sciece ad Techology 11 Volume 1. Number 1. May 11 (Sprig)

8 Table 4: Solutio Steps FORWARD PASS BACKWARD PASS STEP NODE EARLIEST TIME STEP NODE LATEST TIME At sik, LF = ES = 149.days after the 4 th activity. The Pacific Joural of Sciece ad Techology 1 Volume 1. Number 1. May 11 (Sprig)

9 DISCUSSION The etwork diagram (Figure 1) shows that activity [1, ] has to wait for days before activity [, ] ca cotiue to [, 4]. The latest start time for activity [1, ] is zero ad its latest time is 1 day. For total float, activity [, ] ca be delayed without affectig the total proect duratio ad so is activity [, 4]. Activity [7, 8] ca also be delayed for 4 days without affectig the overall duratio while activity [4, ] ca be delayed for days whe all succeedig activities are completed as early as possible. The etwork diagram also reveals that the proect will be completed after the 4 th evet. Table o float time was obtaied from the etwork diagram. It reveals the critical path (the additio of all activities with zero float) with total a duratio of 149 days. Also iputtig the data (Table 1) ito the Tora system, the output gave a critical path with total duratio of 149 days which coicided with the maual computatio obtaied earlier. CONCLUSION AND RECOMMENDATION A total of 149 days was obtaied as a completio time for the proect usig the CPM techique whereas a time of 199 days would have bee required if the traditioal method of additio was adopted. We coclude that 149 days is the required completio time for the proect. This will save a period of days i the costructio process. It could also save ruig, persoel, ad other resource costs. We therefore recommed that persoel relevat to the field of etwork aalysis should be ivolved i plaig of a proect prior to its commecemet. This will i tur give a idea of the proect s shortest completio period. Maagemet should also seriously cosider ad implemet the results of such persoel to avoid udue delay i the completio of ay proect. Lastly, iefficiet utilizatio of resources ad icrease i cost of utilities beyod reasoable proportios could lead to the abadomet of proect works. I such cases, we recommed that additioal resources such as overtime paymets, bouses ad icreased labour iput should be icorporated so as to re-adust the critical path. REFERENCES 1. Taha, H.A Operatios Research A Itroductio (Sixth Editio). Pretice-Hall: New York, NY.. Hillier, F.S. ad Lieberma, G.J.. A Itroductio to Operatios Research. McGraw Hill Higher Educatio: New York, NY.. Igizio J.P Liear Programmig i Sigle ad Multiple-Obective Systems. Pretice-Hall: New York, NY. ABOUT THE AUTHORS R.A. Adekee, holds a Ph.D. degree i Statistics. He is curretly a Seior Lecturer i the Departmet of Mathematical Scieces, Uiversity of Ado-Ekiti, Nigeria. His area of research is probability theory ad stochastic processes. O.Y. Halid, holds a Master s Degree i Statistics. He is curretly a Assistat Lecturer i the Departmet of Mathematical Scieces, Uiversity of Ado-Ekiti, Nigeria. His area of research is probability distributio theory ad its applicatios O.D. Oguwale, holds a Master s Degree i Statistics. He is curretly a Assistat Lecturer i the Departmet of Mathematical Scieces, Uiversity of Ado-Ekiti, Nigeria. His area of research is probability theory ad stochastic processes A.O. Olubiyi, holds a Master s Degree i Statistics. She is curretly a Assistat Lecturer i the Departmet of Mathematical Scieces, Uiversity of Ado-Ekiti, Nigeria. Her area of research is evirometal statistics. SUGGESTED CITATION Adeleke, R.A., O.Y. Halid, O.D. Oguwale, ad A.O. Olubiyi. 11. Applicatio of Network Aalysis to Proect Maagemet. Pacific Joural of Sciece ad Techology. 1(1): - 1. Pacific Joural of Sciece ad Techology The Pacific Joural of Sciece ad Techology 1 Volume 1. Number 1. May 11 (Sprig)