Basics Of PERT/CPM PERT=Project Evaluation & Review Technique CPM = Critical Path Method Network • Network is a graphic representation of a project's operations and is composed of activities and events that must be completed to reach the end of the objective of a project, showing the planning sequence of their accomplishment , their dependence and inter-relationships. Basic Component • Activity is a task, or item of work to be done, that consume time, effort, money or other resources. Its lies between two events. An activity is presented by an arrow its head indicating the sequence in which the events are to occur • Events is the start (beginning) or completion (end) of some activity and as such it consume no time, no resources. It is known as NODE. i Activity j Starting Event Completion Event Predecessor Activity: an activity which must be completed before one or more other activities start is known as predecessor activity(i predecessor j ) Successor activity: an activity which started immediately after one or more of other activities are Completed is known as successor activity (j successor i) Dummy activity: an activity which does not consume either any resource and time is known as dummy activity (dotted line in the network) Local Sequencing • The entire project may be considered as a series of activities which being only after another activity completed • This types of relationship is called constrains which are represent by inequalities [eg. A<B (the activity A must be completed before the start of activity B)] Situations in network diagram B A A must finish before either B or C can start C A C both A and B must finish before C can start B A C both A and B must finish before either of C B or D can start D A B A must finish before B can start Dummy both A and C must finish before D can start C D Errors in Network Diagram • Looping : if an activity were represented as going back in time B A C • Close loop produced endless cycle in computer programmes without a built-in routine for detection of identification of the cycle Cycles •A cycle is any path of jobs that leads back into itself. •It represents logical error. It has to be removed before scheduling computation. •A job list with cycles cannot be put in topological order. Errors in Network Diagram • Dangling :No activity should end without being joined to maintain the continuity of the system. Such end – events other then the end of the projects as a whole are called dangling events B E A C F D Dangling Dummy The CPM Diagram • “Tasks” are Arrows • “Events” are Circles • “Critical Tasks” are Thick Arrows • “Dummy Tasks” are Dashed Arrows Rules for Network Construction 1. Each activity is represented by one and only one arrow 2. Each activity must be identify by its starting and end node –Two activities should not be identified by the same completion events –Activity must be represented by their symbol or corresponding order pair of starting and completion events –Nodes are numbered to identify an activity uniquely. 3. Between any pair of nodes, there should be one and only one activity 4. Arrow should be kept straight or not curved or bent. 5. The logical sequence between activities must follow following rules - An event cannot occur until all the incoming activities into it have been completed - An activity cannot start unless all the preceding activities on which it depends have been completed - Dummy activities should only be introduced if obviously necessary Numbering the Events • Event number should be unique. • Event numbering should be carried out on a sequential basis from left to right. • The initial event which has all outgoing arrows with no incoming arrows is numbered 0 or 1 • The head of an arrow should always bear a number higher than the one assigned at all tail of the arrow. • Gaps should be left in the sequence of event numbering to accommodate subsequent inclusion of activities, if necessary. Illustration - 1 • A television is manufactured in six steps, labeled A through F. Because of its size and complexity, the television is produced one at a time. The production control manager thinks that network scheduling techniques might be useful in planning future production. He recorded the following information • A is the first step and precedes B and C • C precedes D and E. • B follows D and precedes E. • D, E is successor of F. Activity on node diagram • A is the first step and precedes B and C (A < B, C) • C precedes D and E. • B follows D and precedes E. • D, E is successor of F. B A C Activity on node diagram • A is the first step and precedes B and C • C precedes D and E. (C < D, E) • B follows D and precedes E. • D, E is successor of F. B D A C E Activity on node diagram • A is the first step and precedes B and C • C precedes D and E. • B follows D and precedes E. (D < B > E) • D, E is successor of F. B D A C E Activity on node diagram • A is the first step and precedes B and C • C precedes D and E. • B follows D and precedes E. • D, E is successor of F. (F < D, E ) B D A F C E Illustration - 2 Construct the network diagram comprising activities B, C, …, Q and N such that the following constraints are satisfies B < E, F; C < G, L; E, G < H; L, H < I; L < M; H < N; H < J; I,J < P; P < Q. H I E 5 6 9 2 J 10 F B 3 P 1 C G 11 4 Q L M N 7 8 12 Illustration - 3 Construct the network diagram comprising activities A, B, C > NONE; A<D B, C < E A<F C<G H < D, E, F D<I G < J, K H, J < L K<M I, L < N Illustration - 3 Construct the network diagram comprising activities A, B, C > NONE; A<D B, C < E A<F D 2 5 H C<G F I 9 H < D, E, F A 7 D<I 1 C 4 G G < J, K 8 B H, J < L K<M E 3 6 I, L < N Illustration - 4 Network analysis of a minor redesign of a product and its associated packaging. The key question is: How long will it take to complete this project ? For clarity, this list is kept to a minimum by specifying only immediate relationships, that is relationships involving activities that "occur near to each other in time". Practice Example A social project manager is faced with a project with the following activities: Activity Description Duration Social work team to live in village 5w Social research team to do survey 12w Analyse results of survey 5w Establish mother & child health program 14w Establish rural credit programme 15w Carry out immunization of under fives 4w Draw network diagram and show the critical path. Calculate project duration. Practice problem Activity Description Duration 1-2 Social work team to live in village 5w 1-3 Social research team to do survey 12w 3-4 Analyse results of survey 5w 2-4 Establish mother & child health program 14w 3-5 Establish rural credit programme 15w 4-5 Carry out immunization of under fives 4w 4 2 1 5 3 CPM calculation • Path – A connected sequence of activities leading from the starting event to the ending event • Critical Path – The longest path (time); determines the project duration • Critical Activities – All of the activities that make up the critical path Critical Path Analysis • The objective of critical path analysis is to estimate the total project duration – Total duration needed for the completion of the project – The activities of the project as being critical or non-critical •An Activity in a network diagram is said to be critical is the delay in its start will further delay the project completion time. Critical Path Analysis • The following terms shall be used in the critical path calculation – Ei = Earliest occurrence time of event i – Lj = Latest occurrence time of event j – Tij = duration of the activity (i, j) Forward Pass • Earliest Start Time (ES) – earliest time an activity can start – ES = maximum EF of immediate predecessors • Earliest finish time (EF) – earliest time an activity can finish – earliest start time plus activity time EF= ES + t Backward Pass Latest Start Time (LS) Latest time an activity can start without delaying critical path time LS= LF - t Latest finish time (LF) latest time an activity can be completed without delaying critical path time LS = minimum LS of immediate predecessors CPM analysis • Draw the CPM network • Analyze the paths through the network • Determine the float for each activity – Compute the activity’s float float = LS - ES = LF - EF – Float is the maximum amount of time that this activity can be delay in its completion before it becomes a critical activity, i.e., delays completion of the project • Find the critical path is that the sequence of activities and events where there is no “slack” i.e.. Zero slack – Longest path through a network • Find the project duration is minimum project completion time CPM Example: • CPM Network f, 15 g, 17 h, 9 a, 6 i, 6 b, 8 d, 13 j, 12 c, 5 e, 9 CPM Example • ES and EF Times f, 15 g, 17 h, 9 a, 6 0 6 i, 6 b, 8 0 8 d, 13 j, 12 c, 5 0 5 e, 9 CPM Example • ES and EF Times f, 15 6 21 g, 17 h, 9 a, 6 0 6 6 23 i, 6 b, 8 0 8 d, 13 j, 12 8 21 c, 5 0 5 e, 9 5 14 CPM Example • ES and EF Times f, 15 6 21 g, 17 h, 9 a, 6 21 30 0 6 6 23 i, 6 23 29 b, 8 0 8 d, 13 j, 12 8 21 21 33 c, 5 0 5 e, 9 Project’s EF = 33 5 14 CPM Example • LS and LF Times f, 15 6 21 h, 9 21 30 a, 6 g, 17 24 33 0 6 6 23 i, 6 23 29 b, 8 27 33 0 8 d, 13 j, 12 8 21 21 33 c, 5 21 33 0 5 e, 9 5 14 CPM Example • LS and LF Times f, 15 6 21 h, 9 18 24 21 30 a, 6 g, 17 24 33 0 6 6 23 i, 6 4 10 10 27 23 29 b, 8 27 33 0 8 d, 13 j, 12 0 8 8 21 21 33 c, 5 8 21 21 33 0 5 e, 9 7 12 5 14 12 21 CPM Example • Float f, 15 3 6 21 h, 9 9 24 a, 6 g, 17 3 21 30 24 33 3 0 6 4 6 23 i, 6 3 9 10 27 4 23 29 b, 8 27 33 d, 13 0 0 8 j, 12 0 8 0 8 21 0 21 33 c, 5 8 21 21 33 e, 9 7 0 5 7 12 7 5 14 12 21 CPM Example • Critical Path f, 15 g, 17 h, 9 a, 6 i, 6 b, 8 d, 13 j, 12 c, 5 e, 9 Critical Path Analysis • A critical path consists that set of dependent tasks (each dependent on the preceding one), which together take the longest time to complete. • One way is to draw critical path tasks with a double line instead of a single line. • The critical path for any given method may shift as the project progresses; this can happen when tasks are completed either behind or ahead of schedule, causing other tasks which may still be on schedule to fall on the new critical path PERT • PERT is based on the assumption that an activity’s duration follows a probability distribution instead of being a single value • Three time estimates are required to compute the parameters of an activity’s duration distribution: – pessimistic time (tp ) - the time the activity would take if things did not go well – most likely time (tm ) - the consensus best estimate of the activity’s duration – optimistic time (to ) - the time the activity would take if things did go well tp + 4 t m + to Mean (expected time): te = 6 2 tp - to Variance: Vt =2 = 6 PERT analysis • Draw the network. • Analyze the paths through the network and find the critical path. • The length of the critical path is the mean of the project duration probability distribution which is assumed to be normal • The standard deviation of the project duration probability distribution is computed by adding the variances of the critical activities (all of the activities that make up the critical path) and taking the square root of that sum • Probability computations can now be made using the normal distribution table. Probability computation Determine probability that project is completed within specified time x- Z= where = tp = project mean time = project standard mean time x = (proposed ) specified time PERT Example Immed. Optimistic Most Likely Pessimistic Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.) A -- 4 6 8 B -- 1 4.5 5 C A 3 3 3 D A 4 5 6 E A 0.5 1 1.5 F B,C 3 4 5 G B,C 1 1.5 5 H E,F 5 6 7 I E,F 2 5 8 J D,H 2.5 2.75 4.5 K G,I 3 5 7 PERT Example PERT Network D A E H J C B I K F G PERT Example Activity Expected Time Variance A 6 4/9 B 4 4/9 C 3 0 D 5 1/9 E 1 1/36 F 4 1/9 G 2 4/9 H 6 1/9 I 5 1 J 3 1/9 K 5 4/9 PERT Example Activity ES EF LS LF Slack A 0 6 0 6 0 *critical B 0 4 5 9 5 C 6 9 6 9 0* D 6 11 15 20 9 E 6 7 12 13 6 F 9 13 9 13 0* G 9 11 16 18 7 H 13 19 14 20 1 I 13 18 13 18 0* J 19 22 20 23 1 K 18 23 18 23 0* PERT Example Vpath = VA + VC + VF + VI + VK = 4/9 + 0 + 1/9 + 1 + 4/9 = 2 path = 1.414 z = (24 - 23)/(24-23)/1.414 = .71 From the Standard Normal Distribution table: P(z < .71) = .5 + .2612 = .7612 (a) Draw the activity network of the project (b) Find total float for each activity. Using above information crash the activity step by step until all path are critical. Activity Normal time Crash time 1-2 20 17 1-3 25 25 2-3 10 8 2-4 12 6 3-4 5 2 4-5 10 5 4-6 5 3 5-7 10 5 6-7 8 3 We will use PERT/CPM Analysis to determine Task Secondary properties: • Tail Event and Head Event • Earliest Start, Earliest Complete • Latest Start, Latest Complete • Critical / Non-Critical Status • Total Float, Free Float • Scheduled Start, Scheduled Complete • Actual Staffing, Duration, and Variable Costs We will then use Task Secondary Properties to generate Project Management Tools: • Gantt Chart (Project Schedule) • Manpower Chart • Expenditure Curves • Project Completion (PC) Generate Initial CPM Diagram • Must strictly enforce all prerequisite relationships. • Number of events is initially unknown • Critical path is initially unknown • Iterative Process • Try to minimize number of Dummy Tasks CPM Hint #1 • Add or remove events at your pleasure. • Do not number events until last. CPM Hint #2 • The initial event is the Tail Event for all tasks which have empty prerequisite sets (Initial Tasks). • The Final Event is the Head Event for all tasks which are not members of any prerequisite set (Final Tasks). CPM Hint #3 • Tasks which have identical prerequisite sets have the same Tail Event CPM Hint #4 • Starting with the Final Tasks, work backwards, enforcing the smallest prerequisite sets first. • Use Dummy Tasks to enforce any prerequisites in large sets which have already been enforced in a smaller set. Finish CPM Diagram • Remove all redundant Dummy Tasks • Remove all redundant Events • Number all remaining events • Not really finished . . haven’t identified critical tasks yet. Generate PERT Chart: Enter Data for Each Task • Task Symbol • Tail Event • Head Event • Task Duration (TD) Forward Pass: Determine Earliest Start (ES) and Earliest Complete (EC) for each Task • For all Initial Tasks, ES = 0 • Once ES is Determined, EC equals ES plus TD. • The ES for all tasks with tail [i] is equal to the largest value of EC for all tasks with head [i]. • PC is the largest value of EC for all Final Tasks. Backward Pass: Determine Latest Start (LS) and Latest Complete (LC) for each Task • For all Final Tasks, LC = PC • Once LC is Determined, LS equals LC minus TD. • The LC for all tasks with head [j], is equal to the smallest value of LS for all tasks with tail [j]. • At least one Initial Task must have LS = 0; none may be negative. Determine Total Float (TF): Allowable delay in start of task which will not delay Project Completion • For task with tail [i] and head [j], TF[i,j] = (LC[j] – ES[i]) – TD[i,j] • ES[i] is earliest start for all tasks with tail [i]. • LC[j] is latest complete for all tasks with head [j]. Determine Free Float (FF): Allowable delay in start of task which will not delay start of any other task. • For task with tail [i] and head [j], FF[i,j] = ES[j] - ES[i] - TD[i, j] = ES[j] - EC[i,j] • If [j] is the final event, use PC for ES[j] Determine Critical Path • All Tasks with zero Total Float are Critical. • Any delay in these Tasks will delay Project Completion. • Darken these Tasks to finish CPM Diagram.