Network analysis is a vital technique in Operations Research (OR) used for planning, managing, and optimizing complex projects and systems. It involves representing problems as a network of interconnected activities or tasks and analyzing them to improve efficiency. This approach is widely used in project management, transportation, logistics, and supply chain optimization. Network analysis helps organizations minimize project completion time, allocate resources effectively, and identify critical tasks that influence overall efficiency.
Network analysis is a mathematical and graphical representation of interrelated activities in a system. It involves constructing a network diagram consisting of nodes (events) and edges (activities) to model dependencies and constraints within a project. By analyzing these networks, decision-makers can determine the most efficient way to execute tasks, identify critical paths, and optimize resource utilization. Network analysis is based on graph theory, where a system is represented as a network of nodes (representing tasks or events) and arcs (representing dependencies or relationships). Two primary techniques used in network analysis are:
Critical Path Method (CPM) – Used for deterministic project scheduling, CPM identifies the longest path of dependent tasks in a project, determining the minimum time required for completion. It helps in resource allocation and project planning.
Program Evaluation and Review Technique (PERT) – Used for probabilistic project scheduling, PERT incorporates uncertainty by considering multiple time estimates for each activity (optimistic, pessimistic, and most likely). It helps in estimating the probability of completing a project within a specified timeframe.
Objectives of Network Analysis
Network Analysis is a highly effective technique used in Operations Research for planning, monitoring, and controlling complex projects involving multiple interdependent activities. It provides a structured approach to managing large-scale projects efficiently. The key objectives of network analysis are:
To Minimize Project Cost – By analyzing the sequence of activities and optimizing resource allocation, network analysis helps reduce overall project expenses. Identifying critical and non-critical tasks allows decision-makers to allocate resources where they are most needed, preventing cost overruns.
To Minimize Project Time – Network analysis techniques like the Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) help determine the shortest possible time required to complete a project. By identifying the critical path, project managers can focus on crucial tasks that directly affect project completion.
To Ensure Optimum Utilization of Human and Other Resources – Effective resource management is essential for successful project execution. Network analysis helps in scheduling tasks efficiently, ensuring that manpower, equipment, and materials are used effectively, reducing wastage and idle time.
To Ensure Minimum Conflicts and Unnecessary Delays – By providing a clear timeline of activities and dependencies, network analysis helps avoid scheduling conflicts, bottlenecks, and delays. It enhances coordination among teams, ensuring smooth execution of the project without unnecessary interruptions.
Basic key concepts in network analysis
Activity
An activity refers to the physically identifiable part of the project which consumes time and cost. An activity is represented by an arrow. The activity may have a predecessor activity and a successor activity. An activity is a task or a set of tasks required to complete a project. It consumes time and resources. Activities are represented by arrows in network diagrams (especially in the PERT method). An activity is any task, operation, or work that needs to be completed as part of a project. It requires time, resources, and effort to accomplish. In network analysis, activities are usually represented by arrows in diagrams, with the length of the arrow sometimes indicating the duration of the task. Activities can be dependent on one another, meaning some tasks must be completed before others can begin. Activities are classified into critical activities, which directly impact project completion time, and non-critical activities, which can have some flexibility. For example, in a construction project, "laying the foundation" is an activity that must be completed before "building walls" can begin.
How to represent an activity?
Activity should be represented by an arrow connecting 2 nodes.
All activities should be in forward direction, it may be upward or downward direction but it can never be in reverse direction.
Activities should be either represented by numbers or alphabet.
All alphabets and numbers should be in ascending order.
Event (Node)
An event is a milestone or point in time that marks the start or end of an activity. It does not consume time or resources. Events are represented by circles (nodes) in network diagrams. An event, also called a node, represents a specific point in time within a project where an activity either starts or finishes. Unlike activities, events do not consume time or resources; they serve as milestones in the workflow. Events help in tracking project progress and determining dependencies between tasks. In network diagrams, events are typically represented by circles or rectangles, with numbers or labels indicating their sequence. For instance, in software development, an event could be "Approval of Design Documents," marking the completion of the planning phase and the start of the coding phase.
Example:
The completion of "Building a foundation" is an event that signals the start of the next activity.
Path
A path is a sequence of connected activities that leads from the start of a project to the end. The length of a path is determined by the sum of the durations of its activities. A path in network analysis is a sequence of interconnected activities that lead from the starting event to the ending event of a project. Each path has a duration, calculated by summing up the time required for all the activities in that path. A project can have multiple paths, but only the longest one is considered the critical path, as it determines the project's minimum completion time. If any activity along this critical path is delayed, the entire project is delayed. For example, in an IT project, one path may include "Requirement Gathering → UI Design → Coding → Testing → Deployment," while another might be "Requirement Gathering → Database Setup → API Development → Integration → Deployment.
Example:
In software development, a path might be "Requirement Gathering → Design → Development → Testing → Deployment."
Network Diagram
A network diagram is a visual representation of all activities, events, and dependencies within a project. It helps managers and teams understand the project structure, identify bottlenecks, and optimize schedules. Network diagrams are widely used in PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) for project scheduling and planning.
Rules for construction of network diagram
Each activity is represented by one and only one arrow in the network
No two activities can be identified by the same head and tail events
Except for the nodes at the beginning and at the end every node must have at least one activity preceding it and at least one following it
Only one activity may connect any two nodes.
Dummy Activity
A dummy activity is an artificial task that does not consume any time or resources but is used in network diagrams to maintain logical relationships between tasks. Dummy activities are particularly useful in PERT charts, where multiple activities share common dependencies. They are represented by dashed arrows to indicate that they do not contribute to the project's duration. For example, if two different tasks must be completed before a third one begins, a dummy activity may be used to distinguish them without affecting the sequence.
Critical Path
The critical path is the longest sequence of dependent activities in a project that determines the shortest possible project completion time. Any delay in an activity on the critical path will directly impact the final completion date. Identifying the critical path helps project managers focus on high-priority tasks that require strict adherence to schedules. For example, in an aircraft manufacturing project, the critical path might include "Blueprint Design → Material Procurement → Assembly → Testing → Final Approval." If the assembly phase is delayed, the testing phase will also be pushed back, leading to an overall delay in project completion.
Slack (Float)
Slack, also known as float, is the amount of time an activity can be delayed without delaying the project’s overall completion. Slack helps managers understand which tasks have scheduling flexibility and which do not. There are two main types of slack:
Total Slack – The amount of time an activity can be delayed without affecting the project's completion date.
Free Slack – The amount of time an activity can be delayed without affecting the start of the next dependent activity.
For example, if a testing phase is scheduled to start on the 15th day but has a total slack of 3 days, it can be delayed until the 18th day without affecting the overall project deadline.
Sequence of activities of a House construction Project
Solution
Interpretation
Network diagram representing house construction project.
The network diagram in the above figure shows the procedure relationship between the activities.
Activity A (preparation of house plan), has a start event 1 as well as an ending event 2. Activity B (Construction of house) begins at event 2 and ends at event 3. The activity B cannot start until activity A has been completed. Activities C and D cannot begin until activity B has been completed, but they can be performed simultaneously. Similarly, activities E and F can start only after completion of activities C and D respectively. Both activities E and F finish at the end of event 6.
Uses of Network Techniques for management
Here are some key applications of network techniques in project and business management:
1. Project Planning & Scheduling
Network techniques such as PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) help managers plan project tasks, determine dependencies, and establish realistic schedules. These methods identify the critical path, ensuring that key activities are completed on time without project delays.
2. Resource Allocation & Optimization
Helps in efficiently distributing resources such as manpower, machinery, and materials. Reduces bottlenecks by highlighting which activities require additional attention or redistribution of resources.
3. Time Management
Identifies slack (float) in tasks, allowing managers to delay non-critical tasks without affecting the overall project deadline. Enables estimation of earliest start (ES), earliest finish (EF), latest start (LS), and latest finish (LF) for activities.
4. Cost Control & Budgeting
By determining the most time-efficient path, network techniques help in reducing costs related to project delays. Cost-based CPM variations assist in finding the lowest-cost method to complete a project within a deadline.
5. Risk Analysis & Decision Making
PERT incorporates uncertainty in task durations, allowing managers to estimate risks and create contingency plans. Helps in decision-making by providing alternative paths in case of unforeseen delays.
6. Workflow Optimization
Helps improve the efficiency of business operations, production planning, and logistics by mapping interdependencies. Reduces idle time and enhances productivity.
7. Performance Evaluation
Provides a clear timeline and checkpoints to assess whether a project is on track. Helps in analyzing completed projects to improve future project execution strategies.
Critical Path Method (CPM)
The Critical Path Method (CPM) is one of the most widely used project management techniques that helps in planning, scheduling, and controlling complex projects efficiently. It was developed in the late 1950s by Morgan R. Walker of DuPont and James E. Kelley of Remington Rand to address the challenges of scheduling large-scale industrial projects. CPM is a deterministic model, meaning it assumes that the duration of each task is known and does not change, unlike the Program Evaluation and Review Technique (PERT), which deals with uncertainty. The fundamental principle of CPM is that every project consists of a sequence of interdependent activities, and among these, there exists a longest path of tasks that dictates the minimum completion time for the entire project—this is known as the critical path. The method involves identifying all activities required to complete a project, determining the dependencies between these tasks, estimating their durations, and constructing a network diagram to visualize their sequence. By performing a forward pass and backward pass analysis, CPM helps determine key scheduling parameters such as earliest start (ES), latest start (LS), earliest finish (EF), and latest finish (LF) times, as well as identifying float (slack), which indicates the flexibility available for non-critical tasks.
The main benefit of CPM is its ability to pinpoint activities that are most crucial to the project timeline, allowing project managers to allocate resources more efficiently, prevent bottlenecks, and avoid unnecessary delays. This method is particularly valuable in construction, manufacturing, engineering, research and development, and software project management, where strict deadlines and resource optimization are critical. CPM not only provides a realistic project schedule but also aids in cost control and risk assessment, making it an essential tool in modern project management practices. However, despite its advantages, CPM has limitations, such as not accounting for uncertainty in activity durations and becoming overly complex for large-scale projects with numerous interdependencies. Nevertheless, it remains a foundational technique for ensuring structured and efficient project execution, particularly in industries where meeting deadlines and optimizing resources are of utmost importance.
Steps involved in critical path method
Draw a network diagram
Find the Earliest event time(TE) and latest event time (TL) of each event and show in network diagram
Calculate Earliest start time, Earliest finish time, Latest start time and Latest finish time for each activity
Determine float for each activity
Identify critical activities (having zero floats)
Draw double line in the network diagram passing through critical activities.
The double line shows the critical path
Calculate the total project duration which is the sum of durations of critical activities
Terms in CPM
Earliest and Latest event times
Before critical path in a network is determined it is necessary to find the earliest and latest time of each event to know the earliest expected time(TE) at which the activities originating from the event can be started and to know the latest allowable time(TL) at which activities terminating at the event can be completed.
Forward pass rule (Earliest start time rule)
Earliest Event Time(TE): it is the earliest time at which as event can occur.
1. Earliest occurrence of initial event=0, as there is no predecessor event.
ie: E1 = 0
2. Earliest occurrence of an event when there is only one immediate predecessor activity= Earliest occurrence of predecessor event + duration of the predecessor activity.
4 2
E1=0
E2=0+4=4
E3= E2+duration of(2-3)
=4+2=6
3. Earliest occurrence of an event when there are many predecessor activities= Maximum value selected from earliest occurrences of all the predecessor events+duration of each of the corresponding predecessor activity
E1=0
E2=0+2=2
E3=max(E2+duration of activity (2-3), E1+ duration of (1-3)
=max(2+3, 0+4)
= max(5,4)=5
E3=5
Backward pass (latest finish time rule)
Latest event time (TL): It is the latest time by which an event must occur to keep the project on schedule. Latest occurrence of an event say 2 is denoted by L2.
For eg:
1. Latest occurrence of terminal event= Earliest occurrence of the terminal event. L10=E10
2. Latest occurrence of an event when there is only one immediate successor= Latest occurrence of the successor event- duration of the successor activity:
L9=L10 -duration of activity 9-10
3. Latest occurrence of an event when there are many successor activities= Minimum chosen from latest occurrences of all the successor events- duration of each corresponding successor activity.
Eg: L5= Min( L6-duration of N, L7- duration of N)
EST, LFT, EFT LST of activities
EST of an activity = Earliest of its tail event
LFT of an activity = latest occurrence of the head event of that activity
EFT of an activity = EST of the activity+ activity duration
LST of an activity= LFT of the activity – activity duration.
Float= LST-EST
PROBLEMS
1. Draw a network diagram to the following activities
ActivityTime(days)
1-22
1-34
1-43
2-51
3-56
4-65
5-67
Solution
2. Draw a networking diagram
Solution
3. Draw a networking diagram
Program Evaluation and Review Technique (PERT)
The Program Evaluation and Review Technique (PERT) is a powerful project management tool used for planning, scheduling, and controlling large, complex projects, especially those with uncertain activity durations. Developed in the 1950s by the U.S. Navy for the Polaris missile project, PERT was designed to manage projects where time estimation is uncertain and involves a probabilistic approach rather than a deterministic one, as in the Critical Path Method (CPM). Unlike CPM, which assumes fixed activity durations, PERT incorporates variability by using three-time estimates for each activity: Optimistic time (O), Most likely time (M), and Pessimistic time (P). PERT system is preferred for those projects or operations which are of non repetitive nature or for those projects in which precise time determination for various activities cannot be made. In such projects management cannot be guided by the past experience. For example: the project a launching a space craft involves the work never done before. For such research and development projects, the time estimates made for use may be little more than guesses. PERT system is best suited for such projects.
Time Estimates
Here three time estimates are made for each activity;
Optimistic time(to)- shortest possible time in which an activity can be completed
Pessimistic time(tp)- it is the maximum time that would be required to complete an activity
Most likely time(tm)- it is the time which the activity will take most frequently
The expected(average) time based upon these three estimates is worked out by the formula:
te= to+4tm+tp 6
Steps involved in PERT calculation
Identify the events and activities and prepare a suitable network diagram
Events are numbered in ascending order from left to right
Obtain the various time estimates for each activity. They are optimistic time(to), pessimistic time(tp) and the most likely time(tm)
Compute the expected time(te) for each activity
Compute the float associated with each activity. Determine the critical path
Find the total expected duration time by adding the time estimates for various activities on the critical path.
CLICK HERE TO FOLLOW Table of Contents 1. Operations Research - Introduction-Click below link https://academicboosthub.blogspot.com/p/operations-research-introduction.html 2. Network Analysis- Click below link https://academicboosthub.blogspot.com/p/network-analysis.html
Program Evaluation and Review Technique (PERT) Introduction In today’s dynamic business environment, project management is crucial for achieving organizational goals efficiently. The Program Evaluation and Review Technique (PERT) is a powerful project management tool that helps managers plan, schedule, and control projects with uncertain durations. Developed by the U.S. Navy in the 1950s for the Polaris missile project , PERT has since been widely used in research & development, construction, and large-scale infrastructure projects. PERT is a probabilistic project scheduling technique that focuses on estimating the time required to complete each task while considering uncertainty. Unlike traditional project management methods, PERT provides a scientific approach to determining the expected project duration and identifying the critical path , which is essential for timely project completion. Features of PERT: Network-...
CLICK HERE TO FOLLOW Practice Problems Draw a network diagram for the following activities: Activity Time (days) 1-2 3 1-3 2 2-4 4 3-4 5 4-5 6 Construct a network diagram and determine the critical path for the following activities: Activity Time (days) 1-2 2 1-3 3 2-4 5 3-5 6 4-6 2 5-6 3 Develop a network diagram and find the earliest and latest start and finish times for each activity: Activity Time (days) 1-2 4 1-3 3 2-5 7 3-4 6 4-5 2 5-6 8 Draw the network diagram and calculate the total project duration and slack time for each activity: Activity Time (days) 1-2 3 1-3 2 2-4 6 3-5 5 4-6 4 5-6 7 Construct a network diagram for the following project and determine the float for each activity: Activity Time (days) 1-2 5 1-3 6 2-4 3 3-5 7 4-6 2 5-6 4 Develop a network diagram and use the PERT method to estimate the expected project completion time. Use the formula: T E = ( O + 4 M + P ) 6 TE = \frac{(O + 4M + P)}{6} TE = 6 ( O + 4 M + P ) Given optimistic ...
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