![30
A = [76+4(86)+120]/6 = 90
A = (120 - 76)/6 = 7.33 A
2 = (7.33)2 = 53.78
Activity
A 90 7.33 53.78
B 15 1.00 1.00
C 5 0.33 0.11
D 20 3.00 9.00
E 21 1.00 1.00
F 25 2.33 5.44
G 14 2.00 4.00
H 28 1.33 1.78
I 30 4.67 21.78
J 45 3.67 13.44
2
KLONE COMPUTERS –
Finding activities’ mean and variance](https://image.slidesharecdn.com/unit3pert-240331181916-b7eb398b/85/Unit-3-PERT-pptx-in-related-to-supply-chain-30-320.jpg)
Unit 3 PERT.pptx in related to supply chain
- 2. 2 • A project is a collection of tasks that must be completed in minimum time or at minimal cost. • Objectives of Project Scheduling – Completing the project as early as possible by determining the earliest start and finish of each activity. – Calculating the likelihood a project will be completed within a certain time period. – Finding the minimum cost schedule needed to complete the project by a certain date. Introduction
- 3. 3 • A project is a collection of tasks that must be completed in minimum time or at minimal cost. • Objectives of Project Scheduling – Investigating the results of possible delays in activity’s completion time. – Progress control. – Smoothing out resource allocation over the duration of the project. Introduction
- 4. 4 • Tasks are called “activities.” – Estimated completion time (and sometimes costs) are associated with each activity. – Activity completion time is related to the amount of resources committed to it. – The degree of activity details depends on the application and the level of specificity of data. Task Designate
- 5. 5 Identifying the Activities of a Project • To determine optimal schedules we need to – Identify all the project’s activities. – Determine the precedence relations among activities. • Based on this information we can develop managerial tools for project control.
- 6. 6 Identifying Activities, Example KLONE COMPUTERS, INC. • KLONE Computers manufactures personal computers. • It is about to design, manufacture, and market the Klonepalm 2000 palmbook computer.
- 7. 7 • There are three major tasks to perform: – Manufacture the new computer. – Train staff and vendor representatives. – Advertise the new computer. • KLONE needs to develop a precedence relations chart. • The chart gives a concise set of tasks and their immediate predecessors. KLONE COMPUTERS, INC
- 8. 8 Activity Description A Prototype model design B Purchase of materials Manufacturing C Manufacture of prototype model activities D Revision of design E Initial production run F Staff training Training activities G Staff input on prototype models H Sales training Advertising activities I Pre-production advertising campaign J Post-redesign advertising campaign KLONE COMPUTERS, INC
- 9. 9 From the activity description chart, we can determine immediate predecessors for each activity. Activity A is an immediate predecessor of activity B, because it must be competed just prior to the commencement of B. A B KLONE COMPUTERS, INC
- 10. 10 Immediate Estimated Activity Predecessor Completion Time A None 90 B A 15 C B 5 D G 20 E D 21 F A 25 G C,F 14 H D 28 I A 30 J D,I 45 Precedence Relationships Chart KLONE COMPUTERS, INC
- 11. 11 3 The PERT/CPM Approach for Project Scheduling • The PERT/CPM approach to project scheduling uses network presentation of the project to – Reflect activity precedence relations – Activity completion time • PERT/CPM is used for scheduling activities such that the project’s completion time is minimized.
- 12. 12 KLONE COMPUTERS, INC. - Continued • Management at KLONE would like to schedule the activities so that the project is completed in minimal time. • Management wishes to know: – The earliest and latest start times for each activity which will not alter the earliest completion time of the project. – The earliest finish times for each activity which will not alter this date. – Activities with rigid schedule and activities that have slack in their schedules.
- 13. 13 Earliest Start Time / Earliest Finish Time • Make a forward pass through the network as follows: – Evaluate all the activities which have no immediate predecessors. • The earliest start for such an activity is zero ES = 0. • The earliest finish is the activity duration EF = Activity duration. – Evaluate the ES of all the nodes for which EF of all the immediate predecessor has been determined. • ES = Max EF of all its immediate predecessors. • EF = ES + Activity duration. – Repeat this process until all nodes have been evaluated • EF of the finish node is the earliest finish time of the project.
- 14. 14 Earliest Start / Earliest Finish – Forward Pass A 90 B 15 C 5 F 25 I 30 G 14 D 20 E 21 H 28 J 45 90,105 90,115 90,120 105,110 110,124 115,129 129,149 149,170 149,177 120,165 149,194 170 194 A 0,90 B I F C G D E H J 177 194 EARLIEST FINISH
- 15. 15 Latest start time / Latest finish time • Make a backward pass through the network as follows: – Evaluate all the activities that immediately precede the finish node. • The latest finish for such an activity is LF = minimal project completion time. • The latest start for such an activity is LS = LF - activity duration. – Evaluate the LF of all the nodes for which LS of all the immediate successors has been determined. • LF = Min LS of all its immediate successors. • LS = LF - Activity duration. – Repeat this process backward until all nodes have been evaluated.
- 16. 16 B F C A I E D G H H 28 166,194 J J 45 149,194 E 21 173,194 90,105 90,115 90,120 105,110 115,129 129,149 149,170 149,177 149,194 153,173 146,166 194 129,149 0,90 129,149 D 20 129,149 129,149 129,149 129,149 129,149 129,149 129,149 G 14 115,129 I 30 119,149 29,119 C 5 110,115 B 15 95,110 5,95 F 25 90, 115 0,90 A 90 Latest Start / Latest Finish – Backward Pass
- 17. 17 • Activity start time and completion time may be delayed by planned reasons as well as by unforeseen reasons. • Some of these delays may affect the overall completion date. • To learn about the effects of these delays, we calculate the slack time, and form the critical path. Slack Times
- 18. 18 – Slack time is the amount of time an activity can be delayed without delaying the project completion date, assuming no other delays are taking place in the project. Slack Time = LS - ES = LF - EF Slack Times
- 19. 19 Critical activities must be rigidly scheduled Activity LS - ES Slack A 0 -0 0 B 95 - 90 5 C 110 - 105 5 D 119 - 119 0 E 173 - 149 24 F 90 - 90 0 G 115 - 115 0 H 166 - 149 17 I 119 - 90 29 J 149 - 149 0 Slack time in the Klonepalm 2000 Project
- 20. 20 • The critical path is a set of activities that have no slack, connecting the START node with the FINISH node. • The critical activities (activities with 0 slack) form at least one critical path in the network. • A critical path is the longest path in the network. • The sum of the completion times for the activities on the critical path is the minimal completion time of the project. The Critical Path
- 21. 21 B F C A I E D G H H 28 166,194 J J 45 149,194 E 21 173,194 90,105 90,115 90,120 105,110 115,129 129,149 149,170 149,177 149,194 D 20 0,90 129,149 G 14 115,129 I 30 119,149 A 90 C 5 110,115 B 15 95,110 F 25 90, 115 0,90 The Critical Path
- 22. 22 The Probability Approach to Project Scheduling • Activity completion times are seldom known with 100% accuracy. • PERT is a technique that treats activity completion times as random variables. • Completion time estimates are obtained by the Three Time Estimate approach
- 23. 23 • The Three Time Estimate approach provides completion time estimate for each activity. • We use the notation: a = an optimistic time to perform the activity. m = the most likely time to perform the activity. b = a pessimistic time to perform the activity. The Probability Approach – Three Time Estimates
- 24. 24 = t h e m e a n c o m p l e t i o n t i m e = a + 4 m + b 6 = t h e s t a n d a r d d e v i a t i o n = b - a 6 Approximations for the mean and the standard deviation of activity completion time are based on the Beta distribution. The Distribution, Mean, and Standard Deviation of an Activity
- 25. 25 To calculate the mean and standard deviation of the project completion time we make some simplifying assumptions. The Project Completion Time Distribution - Assumptions
- 26. 26 • Assumption 2 – The time to complete one activity is independent of the time to complete any other activity. • Assumption 3 – There are enough activities on the critical path so that the distribution of the overall project completion time can be approximated by the normal distribution. The Project Completion Time Distribution - Assumptions • Assumption 1 – A critical path can be determined by using the mean completion times for the activities. – The project mean completion time is determined solely by the completion time of the activities on the critical path.
- 27. 27 Mean = Sum of mean completion times along the critical path. The three assumptions imply that the overall project completion time is normally distributed, the following parameters: The Project Completion Time Distribution Variance = Sum of completion time variances along the critical path. Standard deviation = Variance
- 28. 28 A c tiv ity O p tim is tic M o s t L ik e ly P e s s im is tic A 7 6 8 6 1 2 0 B 1 2 1 5 1 8 C 4 5 6 D 1 5 1 8 3 3 E 1 8 2 1 2 4 F 1 6 2 6 3 0 G 1 0 1 3 2 2 H 2 4 1 8 3 2 I 2 2 2 7 5 0 J 3 8 4 3 6 0 The Probability Approach – KLONE COMPUTERS
- 29. 29 • Management at KLONE is interested in information regarding the completion time of the project. • The probabilistic nature of the completion time must be considered. The Probability Approach – KLONE COMPUTERS
- 30. 30 A = [76+4(86)+120]/6 = 90 A = (120 - 76)/6 = 7.33 A 2 = (7.33)2 = 53.78 Activity A 90 7.33 53.78 B 15 1.00 1.00 C 5 0.33 0.11 D 20 3.00 9.00 E 21 1.00 1.00 F 25 2.33 5.44 G 14 2.00 4.00 H 28 1.33 1.78 I 30 4.67 21.78 J 45 3.67 13.44 2 KLONE COMPUTERS – Finding activities’ mean and variance
- 31. 31 • The mean times are the same as in the CPM problem, previously solved for KLONE. • Thus, the critical path is A - F- G - D – J. – Expected completion time = A +F +G +D +J=194. – The project variance =A 2 +F 2 +G 2 +D 2 +J 2 = 85.66 – The standard deviation = = 9.255 2 KLONE COMPUTERS – Finding mean and variance for the critical path
- 32. Steps in PERT (From the exam perspective) • Step1: Calculate µi and σi for each activity i • Step 2: Draw Network Diagram • Step 3: Calculate ES/EF by forward pass for each activity i • Step 4: Calculate LS/LF by backward pass for each activity i • Step 5: Calculate Slack Time Si for each activity i • Step 6: Identify critical activities and critical path • Step 7: Calculate µ (Expected project completion time). 2 (Project variance) and σ (Standard Deviation) for entire project 32
- 33. 33 Gantt Charts • Gantt charts are used as a tool to monitor and control the project progress. • A Gantt Chart is a graphical presentation that displays activities as follows: – Time is measured on the horizontal axis. A horizontal bar is drawn proportionately to an activity’ s expected completion time. – Each activity is listed on the vertical axis. • In an earliest time Gantt chart each bar begins and ends at the earliest start/finish the activity can take place.
- 34. 34 Here‘s how we build an Earliest Time Gantt Chart for KLONEPALM 2000
- 35. 35 Im m e d ia te E s tim a te d A c tiv ity P r e d e c e s s o r C o m p le tio n T im e A N o n e 9 0 B A 1 5 C B 5 D G 2 0 E D 2 1 F A 2 5 G C ,F 1 4 H D 2 8 I A 3 0 J D ,I 4 5 Im m e d ia te E s tim a te d A c tiv ity P r e d e c e s s o r C o m p le tio n T im e A N o n e 9 0 B A 1 5 C B 5 D G 2 0 E D 2 1 F A 2 5 G C ,F 1 4 H D 2 8 I A 3 0 J D ,I 4 5 A 90 90 B 15 F 25 I 30 105 C 5 115 G 14 129 D 20 149 E 21 H 28 J 45 194 194
- 36. 36 • Gantt chart can be used as a visual aid for tracking the progress of project activities. • Appropriate percentage of a bar is shaded to document the completed work. • The manager can easily see if the project is progressing on schedule (with respect to the earliest possible completion times). Gantt Charts- Monitoring Project Progress
- 37. 37 A 90 B 15 F 25 I 30 C 5 G 14 D 20 E 21 H 28 J 45 194 194 135 Monitoring Project Progress The shaded bars represent completed work BY DAY 135. Do not conclude that the project is behind schedule. Activity “I” has a slack and therefore can be delayed!!!
- 38. 38 • Advantages. – Easy to construct – Gives earliest completion date. – Provides a schedule of earliest possible start and finish times of activities. • Disadvantages – Gives only one possible schedule (earliest). – Does not show whether the project is behind schedule. – Does not demonstrate the effects of delays in any one activity on the start of another activity, thus on the project completion time. Gantt Charts – Advantages and Disadvantages