Industrial Engineering : PERT and CPM

Project Network diagram: A network in project management displays the duration of project activities and the dependencies between activities graphically or as a table. In a network, nodes (rectangles) represent activities and events. Arrows connect nodes with each other. Arrows represent the dependency between the activities or events.

Rules for network construction:

• An activity can only be started when all the activities earlier to it are completed.
• No two or more activities may have the same head and tail event.
• In this condition to represent the same logic dummy activity will be required.

Terminology in Network analysis:

Activity: It is the recognizable part of a project which consumes time and resources for its completion and it may involve physical or mental work when all the activities are executed, then only a project gets completed. 

Event: It denotes the point of time or accomplishment occurring at a moment and is used to denote the starting and endpoint of an activity, event neither consumes any time nor resources for its completion.

Dummy Activity: An activity which use to show the logic, dependency or relationship of one activity over the other but doesn’t consume any time or resources for its completion it termed a dummy activity it is represented by a dotted line arrow. Dummy activities only are used when it is necessary but there is no restriction on the no. of dummy activity used. There should be no looping & dangling on the network diagram.

Dangling: It is the condition when activity other than the final activity doesn’t have any successor activity. Such activity should be connected directly to the last event of network diagram.

PERT(Programme Evaluation and Review Technique)

• PERT is event oriented.
• PERT is a probabilistic model i.e. it takes into account uncertainties involved in the estimation of time of a job or an activity.
• It uses three estimates of the activity time - optimistic, Pessimistic and Most Likely.
  • Optimistic time (t₀ or a): It is the minimum time required to complete an activity when everything goes acc. to the plan.
  • Pessimistic time (t_p or b): It is the maximum time required to complete an activity when everything goes against the plan.
  • Most likely time (t_m or m): It is a time required to complete an activity when executed under normal working condition.
• PERT is generally used for those projects, which are not of repetitive in nature (e.g. Research and Development Projects) and where time required to complete Various activities are not known in advance.
• Avg or expected time to complete an activity is given byProbability of completing project within scheduled time: If T_E is expected project completion time, s is std. deviation along the critical path then the prob. of completing project within a scheduled time (T_S) is given by

CPM (Critical Path Method):

• CPM is a technique which is used for the projects where the time needed for completion of project is already known.
• A network is a graphical representation of a project, depicting the flow as well as the sequence of well-defined activities and events. A network path consists of a set of activities that connects the networking beginning event to the network terminal event.
• The longest path through the network is called the critical path and its length determines the minimum duration in which the said project can be completed.
• An activity in a network diagram is said to be critical, if the delay in its start will further delay the project completion time. In this activity times are known with certainity.
• Critical path is the largest path in project management which always provide minimum time taken for completion of project.

Earliest and Latest Time: 

• Earliest Start Time (EST) of Activity: It is earliest event time of tail and event.
• Earliest Finish Time (EFT) of Activity: It is the sum of earliest start time and activity time.

                               EFT = EST + activity time

• Latest Start Time (LST) of Activity: It is the latest possible time by which an activity can start without delaying the date of completion of the project.
• Latest Finish Time (LFT) of Activity: It is latest possible time by which the activity must be completed so that the scheduled date for the completion of the project may not be delayed.

                             LFT = LST + activity time

• Earliest Event Time (E_i): It is the maximum of earliest finish times of all activities ending into that event.

                            E_j = max. of all {E_i + t_Eij}

• Latest Event Time (L_i): It is the minimum of the latest start time of all activities originating from that event.

                                 _Li= min. of all {Lj - t_Eij}

Float or Slack: Float or slack means a margin of extra time over and above its duration which a non-critical activity can consume without delaying the date of completion of the project.

• Slack term used for PERT (event), It is amount of time by which particular event can be delayed without delaying the project schedule.

                                S_i = L_i - E_i

• Float term used for CPM (activity)
• Total Float: It is extra time available for an activity without delaying the project schedule.

                            TF = LFT - EFT = LST - EST

• Free Float: It is that part of total float which can be used without affecting the float of succeeding activity.

                           FF = TF - Head event slack

• Independent Float: It is the amount of time which can be used without affecting either the head or the tail event.

                          IF = FF - Tail event slack

• TF ≥ FF ≥ IF
• Interfering Float: It can be defined as that part of the total float which causes a reduction in the float of the successor activities. It can be defined as the difference between the latest finish time of the activity under consideration and earliest start time of the following activity or zero whichever is larger.

Crashing or Time Cost Model: It is an extension of critical path method that consider a compromise between the time and cost required to complete a project the total cost of any project consists of direct and indirect cost involves in its execution.

The objective of crashing a network is to determine optimum project duration corresponding to minimum cost of project and the steps involved are:

1. In the critical path select the critical activity having minimum cost slope.
2. Reduce the duration of this activity by one-time unit.
3. Revise the network diagram by adjusting the time & cost of crashed activity again find critical path, project duration and total cost of project.
4. If the optimum project duration is obtained then stop otherwise repeat from step i.
   
   

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