6/7/2018. Overview PERT / CPM PERT/CPM. Project Scheduling PERT/CPM PERT/CPM

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/7/018 PERT / CPM BSAD 0 Dave Novak Summer 018 Overview Introduce PERT/CPM Discuss what a critical path is Discuss critical path algorithm Example Source: Anderson et al.

Loucks 01 Cengage Learning Project Scheduling Projects often consist of many different tasks that are performed by a number of different individuals and departments It is not possible for managers to

1 /7/018 PERT / CPM BSAD 0 Dave Novak Summer 018 Overview Introduce PERT/CPM Discuss what a critical path is Discuss critical path algorithm Example Source: Anderson et al., 01 Quantitative Methods for Business 1 th edition some slides are directly from 1 J. Loucks 01 Cengage Learning Project Scheduling Projects often consist of many different tasks that are performed by a number of different individuals and departments It is not possible for managers to remember and account for all the information related to project planning and scheduling There are quantitative techniques to help managers with the planning process PERT and CPM are two well-known techniques PERT/CPM PERT Program Evaluation and Review Technique Developed by U.S. Navy for Polaris missile project to handle uncertainty associated with activity times CPM Critical Path Method Developed by DuPont & Remington Rand Developed for industrial projects for which activity times generally were known PERT/CPM CPM assumes that activity times are known with certainty (variation in the time it takes to complete a particular activity is very small and is not a concern) Activity times can be reduced by adding more workers or resources to the activity Identifies trade-offs between time and cost for the activities of the project Reduce activity time at an increased cost when is it worthwhile, and for which activities? PERT/CPM Modern project management software combines features from both approaches PERT and CPM techniques are used to plan, schedule, and control a wide variety of projects such as: R&D of new products and processes Construction of buildings and highways Maintenance of large and complex equipment Design and installation of new systems 1

2 /7/018 7 PERT/CPM PERT/CPM is used to plan the scheduling of individual activities or tasks that make up a specific project Large projects may have several thousand activities A complicating factor in scheduling and carrying out all the activities in a project is that some activities depend on the completion of other activities before they can be started 8 PERT/CPM Used to answer questions such as: What is the total time to complete the project? What are the scheduled start and finish dates for each specific activity? Which activities are critical and must be completed exactly as scheduled to keep the project on schedule? How long can noncritical activities be delayed before they cause an increase in completion time for the entire project? Network model Begin by constructing a model of the project network to observe sequencing of activities Nodes of the network represent the activities The arcs or links between the nodes of the network reflect the precedence (or timebased, pre-requisite sequencing) of the relationships of the activities A critical path for the network is a path consisting of activities with zero slack Float building example Frank s Floats is in the business of building elaborate parade floats. Frank s crew has a new float to build and want to use PERT/CPM to help them manage the project. The table on the next slide shows the activities that comprise the project as well as each activity s estimated completion time (in days) and each activities immediate predecessors (previous activities). Frank wants to know: 1) the total time to complete the project, ) which activities are critical, and ) the earliest and latest start and finish dates for each activity Steps to critical path algorithm 1) List all activities ) ID immediate predecessors for each activity ) Estimate completion time for each activity ) Draw project network Step Initial Paperwork B. Build Body D. Body Final Paperwork G. Mount Body on Frame ) Forward pass ) Backward pass 7) Calculate slack for each activity 8) Find activities with ZERO slack Build Frame E. Frame H. Install Skirt on Frame 9) ID critical path 11 1

3 /7/018 Network representation Depending on the network being modeled, a node can represent many different things (abstract or real, at different scales) People in a social network Transfer stations and holding facilities in a supply network Buildings, entire organizations, countries etc. Units within an organization Cultural or performance attributes Network representation The arcs or links connecting the nodes can also represent many different things (abstract or real, at different scales) Physical or wireless links in a communications network Roadways in a transportation network Strength of relationships between people or organizations Measures of organizational knowledge, etc. 1 1 What is a path? In network analysis (most any network transportation, telecommunication, social, etc.) a PATH is a sequence of nodes connected by arcs or links i.e., the specific set and sequence of links and nodes that are followed to get from the start to the finish What is a path in our example? In project management a path the is sequence of activity or task nodes that connects the entire project from to the Some path examples for our project network: 1 1 What is a critical path? In the project management case, all paths must be traversed for the project to be completed If we find the path that takes the MOST time, this represents our total time to complete the project, and essentially represents a binding time constraint if all other paths take less time, then those paths are not the ones that are most critical to completing the project What is a critical path? If activities along the longest path are delayed, then the entire project is delayed If there is slack (excess time) in other paths, those paths can be delayed (up to some point) before the entire project is delayed Here, the longest path = the critical path 17 18

/7/018 19 Steps 1 presented together Immediate Completion Activity Description Predecessors Time (days) A Initial Paperwork --- B Build Body A C Build Frame A D Body B E Frame C 7 F Final Paperwork

4 /7/ Steps 1 presented together Immediate Completion Activity Description Predecessors Time (days) A Initial Paperwork --- B Build Body A C Build Frame A D Body B E Frame C 7 F Final Paperwork B,C G Mount Body to Frame D,E H Install Skirt on Frame C 0 Node / activity notation Node / Activity A Time required to complete activity A A t A ES A LS A EF A LF A Node / activity notation Node / activity notation Node / Activity A Earliest feasible time activity A can start (ES) Node / Activity A START times go first A ES A EF A Earliest feasible time activity A can finish (EF) A ES A EF A EARLY start and finish times go on top t A LS A LF A t A LS A LF A LATE start and finish times go on bottom Latest time activity A can start to stay on critical path (LS) Latest time activity A can finish to stay on critical path (LF) FINISH times go last 1 Step add completion time B. D. G. E. H. Step forward pass ID earliest start and finish times In our project network, we have 8 activities (i = A, B, C, ) Beginning at the node compute: Earliest Time = the maximum of the earliest finish times of all activities immediately preceding activity i This is 0 for an activity with no predecessors

5 /7/018 Step forward pass Beginning at the node compute: Earliest Time = (Earliest Time) + (Time to complete activity i ) This is 0 for an activity with no predecessors The project completion time is the maximum of the Earliest Times at the node Step forward pass EF i = ES i + t i where EF i = earliest finish for activity (i) ES i = earliest start for activity (i) t i = time to complete activity (i) 7 The forward pass operations Objective is to calculate ES and EF Time required to complete activity A A t A Earliest feasible time activity A can start (ES) ES A EF A Earliest feasible time activity A can finish (EF) Logically, if the activity begins at ES, then the earliest the activity can possibly finish is EF because the activity requires t time to complete EF = ES + t 8 Step forward pass ES EF Activity A A Step forward pass ES EF Activity B B Step forward pass ES EF Activity D D 9 0

6 /7/018 Step forward pass Step forward pass 1 0 B. D. 9 9 G E. 1 H. 7 7 What do you do, when there is more than one preceding activity? Logically, the EARLIEST start time, has to be the latest finish time of the preceding activities (i.e. F can t start until B is finished and G can t start until E is finished The expected completion time of project is the largest of the earliest finish time values for the nodes directly connected to the F, G, and H are last they are directly connected to the node 9 days for F 7 days for H 18 days for G Expected completion time for the project is 18 days Step backward pass ID latest start and finish times Beginning at the node work backwards through network: Set Latest time for ALL of the nodes that are directly linked to the finish (F, G, H) equal to the expected project completion time and work backwards Set LF for all nodes connected to finish = 18 Step backward pass ID latest start and finish times Beginning at the node work backwards through network: For the node this is the project completion time Latest Time = (Latest Time) - (Time to complete activity i) Step backward pass LS i = LF i - t i where LF i = latest finish for activity (i) LS i = latest start for activity (i) t i = time to complete activity (i) The backwards pass operations Objective is to calculate LS and LF Time required to complete activity A A t A ES A LS A EF A LF A Latest time activity A can be finished if we want to stay on critical path and not delay the project completion time (LF) Latest time activity A can start to stay on critical path (LS). If activity starts after LS, we know that LF increases, and all activities following activity A are impacted If the activity must be finished at LF, then the latest the activity can start without impacting the project completion time is LS because the activity requires t time to complete LS = LF t

7 /7/018 Step backward pass Activity G Step backward pass Activity H G 1 18 H 7 LS LF LS LF 7 8 Step backward pass Step backward pass Activity D D 9 LS LF B. D G E. 1 H What do you do, when there is more than one following activity? 9 0 Set LF C = minimum LS of the three activities following LS H = 1, LS E =, and LS F = 1. The smallest of these values is LS E =. LF C = LS E = 1 Step backward pass Activity C C LS C precedes E, F, and H So, LF C is the minimum of the LS values for E, F, and H LS E =, LS F = 1, LS H = 1 LF C = LF Step 7 Calculate slack The slack of an activity is the amount of time the activity can be delayed without increasing the project completion time For each activity i, calculate: Slack = (Latest ) - (Earliest ), or (Latest ) - (Earliest ) Slack i = LS i - ES i = LF i - EF i 7

8 /7/018 Step 7 calculate slack Step 7 Calculate slack Activity t ES EF LS LF B. D G E. 1 H Activity A B C D E F G H Earliest 0 1 Earliest Latest Latest Slack Critical (?) 0 Y N 0 Y N 0 Y 9 N 0 Y 11 N Step 7 Calculate slack Step 8 ID activities with ZERO slack Step 9 ID critical path Critical path is a path of activities from to node, with ZERO slack times What is the critical path here? Step 9 ID critical path Project completion time = the maximum of the earliest finish times for all the activities included in the critical path What is the project completion time? 7 8 8

9 /7/018 Step 9 critical path Summary B. D G E. 1 H Introduce PERT/CPM Define nodes, links, path Discuss what a critical path is Discuss critical path algorithm Example Critical Path: A C E G minute break PERT / CPM Part BSAD 0 Dave Novak Summer Source: Anderson et al., 01 Quantitative Methods for Business 1 th edition some slides are directly from J. Loucks 01 Cengage Learning Overview Last class introduce basic CPM project management approach Today introduce uncertainty into PERT/CPM Three time estimate 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 9

10 /7/018 Uncertain activity times In reality, the completion times associated with many project tasks is 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 0 0 B. D G E. 1 H Three time estimate 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 7 t LS LF 8 9 Three time estimate 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 0 The three time estimate approach Three different time estimates are required for each project task / activity 1. 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) 10

11 /7/018 Three time estimate 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 1 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 Three time estimate approach The three time activity completion estimates can be approximated using a Beta distribution The Beta distribution has different shapes a m b a m b a m b Source: slides from unknown author Cal State Fullerton, Project Scheduling with PERT Three time estimate 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? Estimating the mean and variance for activities The mean activity completion time can be estimated using a weighted average, applying the weights 1/, /, and 1/ on a, m, and b respectively Weight optimistic estimate (a) 1/ (or 0.17) Weight expected estimate (m) / (or 0.7) Weight pessimistic estimate (b) 1/ (or 0.17) Weights sum to 1 11

12 /7/018 7 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) 8 Assumptions for PERT Assumption 1: 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 expected completion time of the activities / tasks on the critical path Assumptions for PERT Assumption : There are enough activities on the critical path so that the overall project completion time can be approximated using the normal distribution Assumptions for PERT Assumption : use of Beta and Normal (Gaussian) distributions 9 70 Assumptions for PERT 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 Assumptions for PERT Together, these assumptions imply that the overall project completion time follows a normal distribution with: Expected project completion time = the sum of all the expected activity times along the critical path Variance of project completion time = sum of the variances of all activities along the critical path

13 /7/018 7 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 F B, C G B, C 1 1. H E, F 7 I E, F 8 J D, H..7. K G, I 7 7 Estimate expected (or mean) completion time for each activity (µ i ) 7 Estimate expected completion time for each activity (µ i ) 7 Activity Expected Time (a + m + b) µ = A B C D E 1 F G H I J K D. E. 1 B. These are the expected or mean times that we just calculated G. H. I. J. 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

14 /7/018 : Calculate ES and EF D. H. E. 1 J. D. 0 E H J. 19 I. 9 1 I B. G. K. B. 0 G K Calculating ES and EF At this point, we know the estimated project completion time 8 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 ) D. 0 E H I J. 19 : 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 B. 0 G K

15 /7/018 0 D. E H J D E H J I I K. 18 K. 18 B. 0 G B. 0 9 G Calculating LS and LF 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 = 1 and LF I = 1), so LF F = D E B. 0 9 H I G J K Calculating slack Calculating slack 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 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

/7/018 91 : Calculate slack (LF EF) Activity ES EF LS LF Slack A 0 0 0 * B 0 9 C 9 9 0 * D 11 1 0 9 E 7 1 1 F 9 1 9 1 0 * G 9 11 1 18 7 H 1 19 1 0 1 I 1 18 1 18 0 * J 19 0 1 K 18 18 0 * 9 : Determine

16 /7/ : Calculate slack (LF EF) Activity ES EF LS LF Slack A * B 0 9 C * D E F * G H I * J K * 9 : Determine the critical path The critical path is a path of activities from to node, with ZERO slack times The project completion time equals the maximum of all activity EF times 0 0 D E H J Estimate variance in completion time (σ ) for each activity on the critical path 9 1 I B. 0 G K 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? 9 9 1

17 /7/018 : 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? Summary Introduce uncertainty into PERT/CPM Three time estimate approach and the Beta distribution Estimating mean and variance Using the z-table to estimate project completion probabilities

11/1/2018. Overview PERT / CPM. Network representation. Network representation. Project Scheduling. What is a path?

11/1/2018. Overview PERT / CPM. Network representation. Network representation. Project Scheduling. What is a path? PERT / CPM BSD Dave Novak Fall Overview Introduce Discuss what a critical path is Discuss critical path algorithm Example Source: nderson et al., 1 Quantitative Methods for Business 1 th edition some slides