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P E R T

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P E R T P E R T Presentation Transcript

  • PERT
  • PERT
    • In 1957 the Critical Path Method (CPM) was developed as a network model for project management.
    • It is a deterministic method that uses a fixed time estimate for each activity.
    • While CPM is easy to understand and use, but does not consider uncertainty in activity time estimation.
    • Uncertainty such as weather, equipment failure, absenteeism can have a great impact on the completion time of a complex project.
  • PERT
    • The Program Evaluation and Review Technique (PERT) is a network model that allows for randomness in activity completion times.
    • Generally used when there is a risk of time associated with project.
      • R & D projects where correct time determinations cannot be made.
      • Example : project launching the spacecraft.
  • PERT
    • PERT was developed in the late 1950's for the U.S. Navy's Polaris ballistic missile system project having thousands of contractors.
    • This project was notable in that it finished 18 months ahead of schedule and within budget.
    • It has the potential to reduce both the time and cost required to complete a project.
  • PERT
    • This method uses statistical tools for Implication of uncertainties on project time
    • Or
    • Stochastic Modeling of Network
    • A distinguishing feature of PERT is its ability to deal with uncertainty in activity completion times. For each activity, the model usually includes three time estimates:
  • Three Time Estimates 1 2 3 4 5 2-5-12 4-7-16 1-6-23 3-7-20 2-5-10
  • Times
    • Optimistic time – Shortest possible time in which an activity can be completed under ideal conditions. This is denoted by t o
    • Pessimistic time - the longest time that an activity might require. If everything went wrong and abnormal situation prevails.however, it doesn't”t include highly unusual catastrophies such as earthquake, floods, fires. It is denoted by t p
    • Most likely time (Most Frequent-Mode)- the completion time having the highest probability. Normal condition prevails. It is denoted by t L
  • t 0 t p t m 1 2 3 4 5 2-5-12 4-7-16 1-6-23 3-7-20 2-5-10
  • Problem: 54 trenches of same dimensions by different parties Find :Optimistic, Pessimistic & Most Likely Times
  • Times
  • Most Likely Time Tallest peak of the curve- Most Likely time or Mode
  • Expected Time & Standard Deviation: Beta Distribution
    • Expected time  = ( Optimistic  +  4 x Most likely  +  Pessimistic ) / 6
    Expected time : Time corresponding 50 % probability of performance SD: How tightly a set of values is clustered around the mean. Standard Deviation: Sigma: measure of uncertainty = (b-a)/6
  • Calculate Expected Time & Standard Deviation: Write down their significance
  • Expected Time & Standard Deviation Comment on Standard Deviation: Second case measure of dispersion is higher Activity t o t m t p 1 4 7 16 2 1 6 23
  • A systematic and scientific method of finding critical path lies in the calculation of event time which is described by i) The Earliest Expected Occurance Time (T E ) ii) The Latest Allowable Occurance Time (T L ) The Earliest Expected Time (T E ) is the time when an event can be expected to occur earliest. The calculation of TE of an event is same as calculation of EOT of CPM network If more than one activity are directed to the event, maximum of the sum of T E 's along various path will give the expected mean time of the event. Expected mean time of the initial event is taken as zero and process is repeated for each succeeding event and ultimately to the final event. The method is usually called the forward pass. (T E )j = Max [(T E )i + t ij ] The Latest Allowable Occurence Time (T L ) : The latest time by which an event must occur to keep the project on schedule is called the latest allowable occurence time (T L ). The calculation of T L of an event is same as that LOT of CPM network by the method known as Backward Pass. ; (T L )i = min ((T L )j – t ij )
  • Scheduled Completion Time (T s ) Whenever a PERT network is taken in hand decision is made regarding the completion time of the project and the accepted figure is called the Scheduled Completion Time (T s ). Naturally. T s refers to the latest allowable occurence time (T L ) of the last event of the project, i.e. (Ts= T L )· SLACK . Time box having two compartments is made at each event. the value in the left compartment indicating the value of T E and that of in the right compartment indicating T L of that event. And the slack of the event is given by, Slack (S) = (T L – T E ) Thus the slack is difference between event times denoting the range within which an event time can vary. Thus, slack gives the idea of "time to spare". Slack means more time to work and less to worry about. It also spots which are potential trouble areas. Slack may be positive, zero or negative depending upon the value of T E and T L of that event.
  • POSITIVE SLACK When T L is more than T E . positive slack is obtained. It indicates the project is ahead of schedule meaning thereby the excess resources. ZERO SLACK When T L is equal to T E zero slack' is obtained. It indicates that the project is going on schedule meaning thereby adequate resources. NEGATIVE SLACK When the scheduled completion time Ts (and hence T L ) is less than T E negative slack is obtained. It indicates the project is behind schedule meaning thereby the lack of resources. CRITICAL EVENT The event having the least slack value is known as a critical event CRITICAL PATH The path joining the critical events is called a critical path of the PERT network. The critical path may be one or more than one. Time wise. the critical path is the longest path connecting the initial event to the final event. A critical path is distinctly marked in the network. usually by a thick line or double lines.
  • Determine the Expected time for Each Path & Find the critical Path
  • Critical Path
  • Probability of Meeting The Schedule Date
  • Normal Distribution Function
    • Sum of all expected time of all activities along critical path is equal to the expected time of last event= 50 % time of completion of project
    • Though individual activities assume random( beta distribution) but T E of the project as a whole assume normal distribution
  • Normal Distribution Function
  • Normal Distribution Function
  • Normal Deviate (x): Distance from the mean expressed in terms of sigma 1. Normal Deviate = 0, it is the expected time, probability of completion = 50 % 2. Normal Deviate = 1, probability of completion = 84 %. 3. Normal Deviate = -1, probability of completion = 16 %
  • Normal Deviate
    • If Ts is the schedules time of completion
    • & Te is the expected time of completion
    • Z = Ts-Te/sigma
    • Sigma = (Sum of variances along critical path) 0.5
    • Variance = (tp-to/6) 2
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  • Exp. For the given PERT network, determine a) Expected time, Standard deviation and variance of the PROJECT and show the critical path also. b) Probability of completion of project in 35 days. c) Time duration that will provide 90% probability of its completion in time. The three time estimates of each activity. are mentioned on the network.
  • Expected mean time of activity t e = (t a + 4t m + t b )/6 Standard deviation of activity  t = (t b - t a )/6 Variance of activity vt = (standard deviation) 2 . Earliest Expected Mean Time (T E ) and Latest allowable occurrence time (T L ) are marked in time box at each event. Slack (S) = (T L - T E ) is also mentioned on the network. Since scheduled completion time of project is not mentioned, for the last event (8), T L = T E has been taken.
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  • Least slack value = 0 :: All the events having zero slack are critical. CRITICAL PATH-I = 1- 2- 3 - 6-7 - 8 CRITICAL PATH-II = 1- 2-4 - 6-7 – 8 Expected Mean Time of Project (  T ) = 31 days. Variance of project along critical path I (VT I) = 1 + 7.1 + 5.44 +1.78 + 0.44 = 15.76 Variance along critical path II (VrII ) = 1 + 4 + 1 + 1.78 + 0.44 = 2.86 :. Variance of the project (V T ) = 15.76 Standard Deviation of the project (  T ) = sqrt(15.76) = 3.97 b) Probability factor (z) corresponding to x = 35 days z = (x-  T )/  t = (35-31)/3.97 = 1.007 = 1.0 probability % corresponding to z = 1.0 (from table) pr= 84.13% c) for 90% probability, the value of z = 1.32 (from table ) 1.32 = (x- 31 )/3.97 So x = 36.24 days.
  • Four activities to be undertaken in series for the completion of II project are as follows,
    • Estimate the time required at
    • 95% probability, and
    • 5% probability to complete the work.
    • Also which of the above four activities has the most reliable time estimates?
  •  
  • Problem:
    • Expected Project Length is 50 weeks
    • Variance 16
    • How many weeks required to complete the project to complete with
      • 95 % Probability
      • 75 % probability
      • 40 % Probability
    57 weeks 53 weeks 49 weeks
  • Find The probability of completion within 35 days 10 9 9 7 11 5 8 Critical path 1-2-4-5, Te= 30 Variance 1-2= (18-6/6) 2 =4, + 9 + 9 = 22 SD= 4.69
  • Benefits of PERT
    • PERT is useful because it provides the following information:
      • Expected project completion time.
      • Probability of completion before a specified date.
      • The critical path activities that directly impact the completion time.
      • The activities that have slack time and that can lend resources to critical path activities.
      • Activity start and end dates.
  • Limitations
    • The activity time estimates are somewhat subjective and depend on judgement. In cases where there is little experience in performing an activity, the numbers may be only a guess.
    • Even if the activity times are well-estimated, PERT assumes a beta distribution for these time estimates, but the actual distribution may be different.
    • Even if the beta distribution assumption holds, PERT assumes that the probability distribution of the project completion time is the same as the that of the critical path. Because other paths can become the critical path if their associated activities are delayed, PERT consistently underestimates the expected project completion time.