11/8/2018. Overview. PERT / CPM Part 2

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Amos Phillips
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for project Using the z-table to estimate project completion probabilities Activities with little variation in completion time Last class, assumed that time required to complete each activity (t i )

1 /8/08 PERT / CPM Part BSAD 0 Dave Novak Fall 08 Source: Anderson et al., 0 Quantitative Methods for Business th edition some slides are directly from J. Loucks 0 Cengage Learning Overview Last class introduce basic CPM project management approach Today introduce uncertainty into PERT/CPM approach and the Beta distribution Estimating mean and variance for project Using the z-table to estimate project completion probabilities Activities with little variation in completion time Last class, assumed that time required to complete each activity (t i ) was known with certainty (deterministic) Common assumption used in certain industrial production planning applications, and useful in demonstrating CPM Uncertain activity times In reality, completion times associated with many project tasks are uncertain (stochastic) PERT is a project scheduling technique that treats the completion time of each activity as a random variable A common approach for estimating activity completion times is the Three Time Estimate Approach Node / activity notation Node / Activity A Time required to complete activity A A t A ES A LS A EF A LF A Instead of a fixed, known value, t A is a random variable Completion time for each activity Activity ES EF G. 8 8 E. H t LS LF

2 /8/08 approach rationale Recall that PERT was developed to manage the Polaris missile system project Most of the tasks / activities required in this project had never been performed before The time required to complete these new tasks was unknown and subject to variation approach rationale PERT focuses on estimating the probability distribution for total project time based on the uncertainty associated with the time to complete each activity in the project The three time estimate approach Three different time estimates are required for each project task / activity. a = an optimistic time to perform the activity (this is the minimum time assuming everything progresses ideally). m = most probable, average or expected activity time under normal conditions. b = a pessimistic time to perform the activity (this is the maximum time, assuming substantial delays) 0 approach We are not going to estimate these parameters (a, m, and b are our parameters) in this class they will be given In practice, where would activity time estimates for a, m, and b come from? Quick refresher A parameter estimate (or sample statistic) is a descriptive measure of a population Because we generally can t measure an entire population (i.e. the height of all males between - years old in the U.S.), we take a random sample from the population We estimate population parameters (based on the sample) along with the error associated with these estimates Quick refresher Normal distribution characterized by two parameters: mean (µ) and standard deviation (σ) Exponential distribution characterized by one parameter: mean ( ) In the PERT application, the Beta distribution is characterized by three parameters: optimal (a), expected (m), and pessimistic (b) NOTE: this is a slightly different interpretation of the Beta parameters than you will find in a statistics source

3 /8/08 approach The three time activity completion estimates can be approximated using a Beta distribution The Beta distribution has different shapes approach The fact that the Beta distribution is defined on a finite interval and can be symmetric or skewed (right or left) makes it more realistic and more flexible than the normal distribution Why? a m b a m b a m b Source: slides from unknown author Cal State Fullerton, Project Scheduling with PERT Estimating the mean and variance for activities The mean activity completion time can be estimated using a weighted average Weights are /, /, and / on a, m, and b respectively Weight optimistic estimate (a) / (or 0.7) Weight expected estimate (m) / (or 0.7) Weight pessimistic estimate (b) / (or 0.7) Weights sum to Estimating the mean and variance for activities Mean μ = a+m+b Variance σ = b a Why are we using as the denominator? The area we are concerned with is within the interval range (b a) This area is within standard deviations on either side of the mean ( standard deviations total) Assumption : We can find the critical path using the mean, or the expected completion times for each activity This implies that the expected completion time for the entire project is based only on the average completion time of the activities / tasks on the critical path Assumption : There are enough activities on the critical path so that the overall project completion time can be approximated using the normal distribution 7 8

4 /8/08 Assumption : use of Beta and Normal (Gaussian) distributions Assumption : The time required to complete each activity is independent of the time required to complete any other activity This does not imply that the sequencing of the activities within the project is independent Only that the time it takes to complete activity B, is not affected by the time it takes to complete activity A 9 0 Together, these assumptions imply that the overall project completion time follows a normal distribution with: Expected or average project completion time = the sum of all the expected / average activity times along the critical path Variance of project completion time = sum of the variances of all activities along the critical path Activity Immediate Predecessor The parameters for Beta distribution a Optimistic Time (hours) m Most Likely Time (hours) b Pessimistic Time (hours) A B C A D A E A 0.. F B, C G B, C. H E, F 7 I E, F 8 J D, H..7. K G, I 7 Estimate expected (or mean) completion time for each activity (µ i ) Estimate expected completion time for each activity (µ i )

5 /8/08 Activity Expected Time (a + m + b) µ = These are the expected or mean times that we just calculated H. J. A B C D E F G H I J K E. G. I. K. : Calculate ES and EF Early (ES) and Early (EF) are calculated on TOP ROW of each node by moving forward through the network the forward pass by setting ES = 0 for all nodes directly connected to The LARGEST EF time of the nodes connected to the FINISH (J and K) represents the project completion time Logically, the project can t be completed until the activity that is finished last is completed : Calculate ES and EF H. E. I. G. J. K. 7 8 Calculating ES and EF 0 E. 7 H. 9 J. 9 At this point, we know the estimated project completion time 9 I G. 9 K

6 /8/08 Calculating ES and EF Review: moving forward through the network calculating ES and EF If an activity has more than one precedent (i.e activities F, G, H, I, J, and K all have more than one precedent), the ES of that activity MUST be set to the MAXIMUM EF of ALL preceding activities For example, I cannot start activity I until BOTH E and F are completed. Therefore, ES I is constrained by EF F (not EF E ) 0 E. H G. 9 I. 8 J. 9 K. 8 : Calculate LS and LF Late (LS) and Late (LF) are calculated on the BOTTOM ROW by moving backwards through the network Begin the backward pass by setting the EF for all nodes directly connected to the finish equal to the estimated project completion time 0 E H. 9 I. 8 J. 9 K. 8 0 G. 9 Calculating LS and LF E H I G. 9 8 J. 9 0 K. 8 8 Review: moving backwards through the network calculating LS and LF If an activity precedes more than one activity (i.e activities F, E, C, and A) precede multiple activities, the LF of that activity MUST be set to the MINIMUM LS value of ALL activities it precedes For example, F precedes BOTH H and I. Therefore, LF F is set to the smallest value (LF H = and LF I = ), so LF F =

/8/08 Calculating slack 0 0 0 E. 7 9 9 H. 9 0 I. 8 8 J.

7 /8/08 Calculating slack E H. 9 0 I. 8 8 J. 9 0 Slack measures how much leeway or extra time each activity has between the earliest it can start and the latest it can start (or the earliest the activity can finish and the latest it can finish) without impacting the entire project G. 9 8 K. 8 8 slack = LF i EF i = LS i ES i Activities with NO SLACK are critical there is no extra time between the earliest and latest times that the activity can start (or finish) without impacting the project completion time 7 8 Calculating slack : Calculate slack (LF EF) 9 0 Activity ES EF LS LF Slack A * B 0 9 C * D 0 9 E 7 F * G H 9 0 I * J 9 0 K * : Determine the critical path The critical path is a path of activities from to node, with ZERO slack times 0 0 E. 7 0 H. 9 0 J. 9 0 The project completion time equals the maximum of all activity EF times I G K

8 /8/08 Estimate variance in completion time (σ ) for each activity on the critical path Estimate variance in completion time (σ ) for each activity on the critical path : Uncertainty calculations What is the probability that the project will be completed within hours? : Uncertainty calculations What is the probability that the project will be completed within hours? : Uncertainty calculations What is the probability that the project will be completed within hours? : Uncertainty calculations What is the probability that the project will be take more than hours to complete? 7 8 8

9 /8/08 Summary Introduce uncertainty into PERT/CPM approach and the Beta distribution Estimating mean and variance Using the z-table to estimate project completion probabilities 9 9

A scheme developed by Du Pont to figure out

CPM Project Management scheme. A scheme developed by Du Pont to figure out Length of a normal project schedule given task durations and their precedence in a network type layout (or Gantt chart) Two examples