Introduction
Phases of CPM and PERT
Some Important Definitions
Project management or representation by a network diagram
Types of activities
Types of events
Common Errors
Rules of network construction
Numbering the events
Time analysis
Determination of Floats and Slack times
Critical activity and Critical path
2 Critical Path Method - CPM
3 Program Evaluation and Review Technique - PERT
Network problem cpm and pert BY MITESH KUMARMitesh Kumar
The document discusses project scheduling techniques like PERT and CPM. It begins by defining key concepts like activities, events, critical path analysis and float/slack times. It then provides examples to demonstrate how to construct a network diagram by numbering events and determining earliest and latest times for activities. The document aims to explain the basic concepts and procedures for using PERT/CPM techniques to schedule projects.
The document discusses network construction for planning and scheduling large projects. It defines key terms like activities, events, precedence relationships. It describes activity on arrow and activity on node diagrams and provides rules for constructing networks without errors. Examples show how to build networks from descriptions of activities and their relationships. Dummy activities are introduced to properly represent connections when needed. Events are numbered according to flow to uniquely identify activities.
The document discusses time-cost optimization in project management using the critical path method. It provides information on normal and crash time estimates, direct and indirect costs, and outlines the steps to determine the optimum duration that results in the lowest total project cost. These include establishing direct cost-time relationships, determining activity cost slopes, computing costs at normal durations, and crashing critical activities from lowest to highest slope while tracking total costs.
Selection Criteria of Equipment in Construction ProjectIRJET Journal
This document discusses the selection criteria for construction equipment on projects. It begins by defining the importance of selecting the proper equipment, as this decision can impact time, cost and productivity. The document then outlines various types of earthmoving, concrete placing and lifting equipment commonly used. It describes factors that influence equipment choice, such as economic considerations, site conditions, manufacturer preferences, and labor availability. The selection process and specific criteria for different equipment types are also examined. Overall, the document provides guidance on strategically matching equipment to tasks to maximize efficiency during construction.
This document summarizes network techniques for project planning and management. It discusses Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) which are used to evaluate the time required to complete projects. CPM is used for projects with deterministic activity times while PERT is used for probabilistic activity times. The key steps for both methods include describing the project, diagramming the network, estimating activity times, and monitoring progress. Critical paths and floats are identified to determine the minimum time needed to complete projects.
This document provides an introduction to the subject of estimating and costing for the 2nd year intermediate vocational course in construction technology. It discusses key definitions like estimating, costing, and procedures for estimating. The importance of estimation and costing is explained. The data required for preparing an estimate like drawings, specifications, and rates is described. The document also discusses complete estimates, lump sum items, and work charged establishment. Measurement units for different construction items like earthwork, concrete, masonry, woodwork, and finishing works are listed with the applicable unit of measurement and payment.
Lecture 7 transportation problem finding initial basic feasible solutionAathi Suku
This document describes three methods for finding an initial basic feasible solution to a transportation problem:
1. The North-West Corner Method which allocates values starting from the top-left cell and proceeds row-by-row, column-by-column.
2. The Least Cost Method which allocates to the cell with the lowest unit cost, exhausting that row or column before proceeding.
3. Vogel's Approximation Method which calculates penalties for each row/column and allocates to the row/column with the highest penalty.
It provides an example problem and shows that Vogel's Method results in the lowest total transportation cost compared to the other two methods.
Project Management Tools and Techniques (PERT- Project Evaluation and Review ...Zulfiquer Ahmed Amin
Project management involves planning, executing, and controlling projects to achieve specific goals within defined time and resource constraints. It utilizes tools like PERT (Program Evaluation and Review Technique) to estimate activity times, determine critical paths, and update schedules as projects progress. PERT uses three time estimates - optimistic, most likely, and pessimistic - to calculate expected activity times and identify float and slack. The critical path determines the minimum project duration, and crashing or fast tracking can potentially shorten schedules at increased cost. Project management tools help make projects more efficient and effective.
Network problem cpm and pert BY MITESH KUMARMitesh Kumar
The document discusses project scheduling techniques like PERT and CPM. It begins by defining key concepts like activities, events, critical path analysis and float/slack times. It then provides examples to demonstrate how to construct a network diagram by numbering events and determining earliest and latest times for activities. The document aims to explain the basic concepts and procedures for using PERT/CPM techniques to schedule projects.
The document discusses network construction for planning and scheduling large projects. It defines key terms like activities, events, precedence relationships. It describes activity on arrow and activity on node diagrams and provides rules for constructing networks without errors. Examples show how to build networks from descriptions of activities and their relationships. Dummy activities are introduced to properly represent connections when needed. Events are numbered according to flow to uniquely identify activities.
The document discusses time-cost optimization in project management using the critical path method. It provides information on normal and crash time estimates, direct and indirect costs, and outlines the steps to determine the optimum duration that results in the lowest total project cost. These include establishing direct cost-time relationships, determining activity cost slopes, computing costs at normal durations, and crashing critical activities from lowest to highest slope while tracking total costs.
Selection Criteria of Equipment in Construction ProjectIRJET Journal
This document discusses the selection criteria for construction equipment on projects. It begins by defining the importance of selecting the proper equipment, as this decision can impact time, cost and productivity. The document then outlines various types of earthmoving, concrete placing and lifting equipment commonly used. It describes factors that influence equipment choice, such as economic considerations, site conditions, manufacturer preferences, and labor availability. The selection process and specific criteria for different equipment types are also examined. Overall, the document provides guidance on strategically matching equipment to tasks to maximize efficiency during construction.
This document summarizes network techniques for project planning and management. It discusses Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) which are used to evaluate the time required to complete projects. CPM is used for projects with deterministic activity times while PERT is used for probabilistic activity times. The key steps for both methods include describing the project, diagramming the network, estimating activity times, and monitoring progress. Critical paths and floats are identified to determine the minimum time needed to complete projects.
This document provides an introduction to the subject of estimating and costing for the 2nd year intermediate vocational course in construction technology. It discusses key definitions like estimating, costing, and procedures for estimating. The importance of estimation and costing is explained. The data required for preparing an estimate like drawings, specifications, and rates is described. The document also discusses complete estimates, lump sum items, and work charged establishment. Measurement units for different construction items like earthwork, concrete, masonry, woodwork, and finishing works are listed with the applicable unit of measurement and payment.
Lecture 7 transportation problem finding initial basic feasible solutionAathi Suku
This document describes three methods for finding an initial basic feasible solution to a transportation problem:
1. The North-West Corner Method which allocates values starting from the top-left cell and proceeds row-by-row, column-by-column.
2. The Least Cost Method which allocates to the cell with the lowest unit cost, exhausting that row or column before proceeding.
3. Vogel's Approximation Method which calculates penalties for each row/column and allocates to the row/column with the highest penalty.
It provides an example problem and shows that Vogel's Method results in the lowest total transportation cost compared to the other two methods.
Project Management Tools and Techniques (PERT- Project Evaluation and Review ...Zulfiquer Ahmed Amin
Project management involves planning, executing, and controlling projects to achieve specific goals within defined time and resource constraints. It utilizes tools like PERT (Program Evaluation and Review Technique) to estimate activity times, determine critical paths, and update schedules as projects progress. PERT uses three time estimates - optimistic, most likely, and pessimistic - to calculate expected activity times and identify float and slack. The critical path determines the minimum project duration, and crashing or fast tracking can potentially shorten schedules at increased cost. Project management tools help make projects more efficient and effective.
The document summarizes the key types of demand guarantees used in construction and engineering projects, including tender guarantees, performance guarantees, advance payment guarantees, retention money guarantees, warranty guarantees, and others. It provides an overview of the history and development of the Uniform Rules for Demand Guarantees (URDG), which aim to establish uniform rules for cross-border guarantees. The URDG 758 are the most recent version and establish standards for demand guarantees to make the process more efficient and reduce rejection risks for beneficiaries.
The document discusses the history and concepts of Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT), two network analysis techniques used for project management. CPM was developed by DuPont in the 1950s for chemical plant projects, focusing on cost-time tradeoffs. PERT was developed by the US Navy for the Polaris missile program to minimize completion time given uncertain activity durations. Both techniques use network diagrams of nodes and arrows to represent activities and their dependencies. The critical path is the longest sequence of activities determining the minimum project duration.
This document discusses drip and sprinkler irrigation methods. It defines drip irrigation as a system that slowly delivers water directly to the soil through a network of pipes and emitters. The advantages of drip irrigation include maximum water use efficiency, reduced weeds and erosion, and lower costs. Disadvantages include potential clogging issues and higher initial costs compared to other methods. Sprinkler irrigation is described as applying water through pressurized pipes and sprinklers to break water into drops simulating rainfall. Advantages include suitability for various landscapes and adding fertilizers through the system, while limitations include water loss to evaporation and uneven distribution in windy conditions.
This document discusses system modeling and different types of models used in system modeling. It covers context models, interaction models, structural models, behavioral models, and model-driven engineering. Some key points include:
- System modeling involves developing abstract models of a system from different perspectives or views. Models are often developed using the Unified Modeling Language (UML).
- Common model types include use case diagrams, sequence diagrams, class diagrams, state diagrams, and activity diagrams.
- Structural models show the organization and structure of a system. Behavioral models show the system's dynamic behavior and responses to events.
- Model-driven engineering is an approach where models rather than code are the primary outputs and code is generated
Construction Planning and Scheduling of Residential Building by MS Project by...Shabaz Khan
This document provides a project report for the planning, scheduling, and estimation of a residential building project consisting of a ground floor and first floor. It includes the project objective, various drawings of the site plan, floor plans, and elevations. It also provides details on the construction planning principles, estimation process, and a title abstract estimate with quantities and costs for various construction items.
PERT (Program Evaluation and Review Technique) is a project management tool developed by the US Navy to analyze complex projects by breaking them into tasks, estimating activity times, and identifying the critical path of tasks. It facilitates decision making by determining the earliest and latest start/finish times and calculating slack time to reduce project time and costs. PERT uses a network diagram and beta distribution to model activity times and account for uncertainty in complex, non-routine projects.
This document provides an overview of activity diagrams and state chart diagrams. It describes the key elements of each, including:
- For activity diagrams: activities, actions, transitions, decisions, synchronization bars, start/end points. Activity diagrams are used to model business processes and workflow.
- For state chart diagrams: states, transitions, events, initial/final states. State chart diagrams are used to model the lifetime of an object and the different states it can be in.
The document defines each element, provides examples, and explains how they are graphically represented in UML diagrams. It also discusses concepts like concurrent states, history states, and swimlanes.
This document provides an introduction to the Critical Path Method (CPM) project scheduling technique. It defines CPM and explains that it was developed in the 1950s to assist with scheduling complex projects. The document outlines the key steps in CPM, including constructing a network diagram of tasks, calculating early and late start/finish times, and identifying the critical path. An example is provided to demonstrate how to determine the critical path of a project using CPM. Benefits and limitations of the technique are also summarized.
The document discusses network analysis techniques CPM and PERT. It outlines key benefits of CPM including scheduling, monitoring, and controlling projects while minimizing costs. Limitations of CPM include uncertainty in activity time estimates and inability to dynamically control changes. PERT differs from CPM in using probabilistic rather than deterministic estimates, with activity durations represented as ranges and probabilities rather than single values. Both techniques use network diagrams and share steps for planning activity schedules.
Project report of kota super thermal power plantHîmãńshu Mêęńä
This document provides a summary of a practical training report submitted by Himanshu Derwal at the Kota Super Thermal Power Station from June 1-30, 2013. The report describes the power station's layout and key components including the coal handling plant, ash handling plant, boiler, steam turbine, turbo generator, cooling system, water treatment plant, and control room. It provides technical details and specifications of the various units and aims to document the practical experience and knowledge gained during the training.
This document discusses the estimation and costing of soil and water conservation measures. It describes the different types of estimates needed including preliminary estimates, detailed estimates, revised estimates, and supplementary estimates. It provides details on how estimates are prepared including using data books, schedules of rates, and determining conveyance and lead/lift costs for different items. The key information included are the types of estimates needed for soil and water conservation projects, how rates are determined, and what components like labor, materials, and transportation are included in cost estimates.
The document discusses functional dependencies and database normalization. It provides examples of functional dependencies and explains key concepts like:
- Functional dependencies define relationships between attributes in a relation.
- Armstrong's axioms are properties used to derive functional dependencies.
- Decomposition aims to eliminate redundancy and anomalies by breaking relations into smaller, normalized relations while preserving information and dependencies.
- A decomposition is lossless if it does not lose any information, and dependency preserving if the original dependencies can be maintained on the decomposed relations.
This document discusses the process of updating a project network diagram based on actual project progress and changes. It provides definitions and explanations of updating, describes the data and steps required for the updating process, and gives an illustrative example of updating the network for a project after 12 days where some activities finished earlier than planned while others were delayed. The example network is updated by assigning completion times, remaining durations and revised estimates based on new information, and recalculating the critical path.
The document discusses various behavioral diagrams in UML including use case diagrams, sequence diagrams, collaboration diagrams, state machine diagrams and activity diagrams. It provides details on how to model scenarios and interactions using these diagrams, including modeling conditional paths, asynchronous messages, procedural vs flat sequencing, and how interaction diagrams relate to use cases at different levels of abstraction.
This document provides an overview of Programme Evaluation and Review Technique (PERT) and Critical Path Method (CPM), which are network analysis techniques used for project scheduling. PERT is used for projects with uncertainty, uses probabilistic time estimates, and focuses on scheduling and monitoring. CPM is used for projects with deterministic time estimates, focuses on time/cost tradeoffs, and allows expediting activities for extra cost. Both techniques identify the critical path and activities with slack. The document then discusses network terminology, provides an example project, and demonstrates how to construct a PERT network and perform calculations for expected activity times, earliest/latest event times, and slack.
This document provides an overview of program evaluation and review technique (PERT) and critical path method (CPM) for project scheduling. It defines key concepts like activities, events, critical path, floats and provides examples of how to draw network diagrams and calculate event and activity times. The examples demonstrate how to identify the critical path, calculate earliest and latest start/finish times, and use PERT to determine the probability of completing a project by a certain deadline while accounting for uncertainty in activity durations.
The document summarizes the key types of demand guarantees used in construction and engineering projects, including tender guarantees, performance guarantees, advance payment guarantees, retention money guarantees, warranty guarantees, and others. It provides an overview of the history and development of the Uniform Rules for Demand Guarantees (URDG), which aim to establish uniform rules for cross-border guarantees. The URDG 758 are the most recent version and establish standards for demand guarantees to make the process more efficient and reduce rejection risks for beneficiaries.
The document discusses the history and concepts of Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT), two network analysis techniques used for project management. CPM was developed by DuPont in the 1950s for chemical plant projects, focusing on cost-time tradeoffs. PERT was developed by the US Navy for the Polaris missile program to minimize completion time given uncertain activity durations. Both techniques use network diagrams of nodes and arrows to represent activities and their dependencies. The critical path is the longest sequence of activities determining the minimum project duration.
This document discusses drip and sprinkler irrigation methods. It defines drip irrigation as a system that slowly delivers water directly to the soil through a network of pipes and emitters. The advantages of drip irrigation include maximum water use efficiency, reduced weeds and erosion, and lower costs. Disadvantages include potential clogging issues and higher initial costs compared to other methods. Sprinkler irrigation is described as applying water through pressurized pipes and sprinklers to break water into drops simulating rainfall. Advantages include suitability for various landscapes and adding fertilizers through the system, while limitations include water loss to evaporation and uneven distribution in windy conditions.
This document discusses system modeling and different types of models used in system modeling. It covers context models, interaction models, structural models, behavioral models, and model-driven engineering. Some key points include:
- System modeling involves developing abstract models of a system from different perspectives or views. Models are often developed using the Unified Modeling Language (UML).
- Common model types include use case diagrams, sequence diagrams, class diagrams, state diagrams, and activity diagrams.
- Structural models show the organization and structure of a system. Behavioral models show the system's dynamic behavior and responses to events.
- Model-driven engineering is an approach where models rather than code are the primary outputs and code is generated
Construction Planning and Scheduling of Residential Building by MS Project by...Shabaz Khan
This document provides a project report for the planning, scheduling, and estimation of a residential building project consisting of a ground floor and first floor. It includes the project objective, various drawings of the site plan, floor plans, and elevations. It also provides details on the construction planning principles, estimation process, and a title abstract estimate with quantities and costs for various construction items.
PERT (Program Evaluation and Review Technique) is a project management tool developed by the US Navy to analyze complex projects by breaking them into tasks, estimating activity times, and identifying the critical path of tasks. It facilitates decision making by determining the earliest and latest start/finish times and calculating slack time to reduce project time and costs. PERT uses a network diagram and beta distribution to model activity times and account for uncertainty in complex, non-routine projects.
This document provides an overview of activity diagrams and state chart diagrams. It describes the key elements of each, including:
- For activity diagrams: activities, actions, transitions, decisions, synchronization bars, start/end points. Activity diagrams are used to model business processes and workflow.
- For state chart diagrams: states, transitions, events, initial/final states. State chart diagrams are used to model the lifetime of an object and the different states it can be in.
The document defines each element, provides examples, and explains how they are graphically represented in UML diagrams. It also discusses concepts like concurrent states, history states, and swimlanes.
This document provides an introduction to the Critical Path Method (CPM) project scheduling technique. It defines CPM and explains that it was developed in the 1950s to assist with scheduling complex projects. The document outlines the key steps in CPM, including constructing a network diagram of tasks, calculating early and late start/finish times, and identifying the critical path. An example is provided to demonstrate how to determine the critical path of a project using CPM. Benefits and limitations of the technique are also summarized.
The document discusses network analysis techniques CPM and PERT. It outlines key benefits of CPM including scheduling, monitoring, and controlling projects while minimizing costs. Limitations of CPM include uncertainty in activity time estimates and inability to dynamically control changes. PERT differs from CPM in using probabilistic rather than deterministic estimates, with activity durations represented as ranges and probabilities rather than single values. Both techniques use network diagrams and share steps for planning activity schedules.
Project report of kota super thermal power plantHîmãńshu Mêęńä
This document provides a summary of a practical training report submitted by Himanshu Derwal at the Kota Super Thermal Power Station from June 1-30, 2013. The report describes the power station's layout and key components including the coal handling plant, ash handling plant, boiler, steam turbine, turbo generator, cooling system, water treatment plant, and control room. It provides technical details and specifications of the various units and aims to document the practical experience and knowledge gained during the training.
This document discusses the estimation and costing of soil and water conservation measures. It describes the different types of estimates needed including preliminary estimates, detailed estimates, revised estimates, and supplementary estimates. It provides details on how estimates are prepared including using data books, schedules of rates, and determining conveyance and lead/lift costs for different items. The key information included are the types of estimates needed for soil and water conservation projects, how rates are determined, and what components like labor, materials, and transportation are included in cost estimates.
The document discusses functional dependencies and database normalization. It provides examples of functional dependencies and explains key concepts like:
- Functional dependencies define relationships between attributes in a relation.
- Armstrong's axioms are properties used to derive functional dependencies.
- Decomposition aims to eliminate redundancy and anomalies by breaking relations into smaller, normalized relations while preserving information and dependencies.
- A decomposition is lossless if it does not lose any information, and dependency preserving if the original dependencies can be maintained on the decomposed relations.
This document discusses the process of updating a project network diagram based on actual project progress and changes. It provides definitions and explanations of updating, describes the data and steps required for the updating process, and gives an illustrative example of updating the network for a project after 12 days where some activities finished earlier than planned while others were delayed. The example network is updated by assigning completion times, remaining durations and revised estimates based on new information, and recalculating the critical path.
The document discusses various behavioral diagrams in UML including use case diagrams, sequence diagrams, collaboration diagrams, state machine diagrams and activity diagrams. It provides details on how to model scenarios and interactions using these diagrams, including modeling conditional paths, asynchronous messages, procedural vs flat sequencing, and how interaction diagrams relate to use cases at different levels of abstraction.
This document provides an overview of Programme Evaluation and Review Technique (PERT) and Critical Path Method (CPM), which are network analysis techniques used for project scheduling. PERT is used for projects with uncertainty, uses probabilistic time estimates, and focuses on scheduling and monitoring. CPM is used for projects with deterministic time estimates, focuses on time/cost tradeoffs, and allows expediting activities for extra cost. Both techniques identify the critical path and activities with slack. The document then discusses network terminology, provides an example project, and demonstrates how to construct a PERT network and perform calculations for expected activity times, earliest/latest event times, and slack.
This document provides an overview of program evaluation and review technique (PERT) and critical path method (CPM) for project scheduling. It defines key concepts like activities, events, critical path, floats and provides examples of how to draw network diagrams and calculate event and activity times. The examples demonstrate how to identify the critical path, calculate earliest and latest start/finish times, and use PERT to determine the probability of completing a project by a certain deadline while accounting for uncertainty in activity durations.
Critical Path Explained With 7 Q&As and a Free Excel TemplateSHAZEBALIKHAN1
The critical path method is the most used project management technique. Learn the CPM with 7 FAQs ranging from definition to method, process, tests. The attached excel sheet (hyperlinked in the article) explains the implementation of CPM in a project schedule.
This document discusses project management techniques like PERT and CPM. It begins by explaining that PERT and CPM help managers plan, schedule, and control projects using limited resources like time, money, staff etc. It then defines key PERT and CPM terms like activities, events, dummy activities and explains how to construct a network diagram. The document also covers topics like calculating earliest and latest times, determining float and slack, identifying the critical path, and using probability in PERT to estimate project duration.
(PROGRAM EVALUATION AND REVIE TECHNIQUE)hannahpepino1
The Program Evaluation and Review Technique (PERT) is a project management tool used to calculate the time needed to complete a project. It involves identifying all activities, determining the order they must be completed in, estimating activity times, and finding the critical path - the longest sequence of activities that determines the project's duration. PERT assigns three time estimates to each activity - optimistic, pessimistic, and most likely - and uses these to determine expected activity times and variances.
The document discusses the network diagram, which is a tool used in project planning and management. It graphically depicts the sequence and interdependencies of project activities. The network diagram is developed from the work breakdown structure and provides the basis for scheduling resources, enhancing communication, estimating duration, and highlighting critical activities. It involves identifying each activity, precedence relationships between activities, and performing forward and backward passes to determine early and late start/finish dates. The critical path is the longest path through the network with the least amount of slack.
The document discusses scheduling a housing construction project using Critical Path Method/Program Evaluation and Review Technique (CPM/PERT). It contains an activity list and precedence relationships for the project. It then summarizes how CPM/PERT can be used to schedule the project by determining the critical path, earliest and latest start/finish times for each activity, and identifying slack. CPM/PERT provides a way to visually display the project, calculate the total time needed, and determine which activities are on the critical path and how delays will impact the schedule.
The document discusses various project scheduling methods including:
1. Gantt charts which show activity start/duration/completion in a bar chart format.
2. Resource leveling which adjusts work plans to fit staffing constraints.
3. Crashing which aims to accelerate schedules by adding people, increasing productivity, overlapping tasks, removing scope, or innovating approaches.
PERT/CPM networks are also covered as tools to plan, schedule, and monitor complex projects using activities, events, time estimates, critical paths, and other techniques.
This document discusses project planning techniques like PERT and CPM. It explains that PERT and CPM are important for project timeline planning and control. PERT was developed by the US Navy in 1958 and allows for probabilistic time estimates, while CPM was developed earlier in 1957 and assumes deterministic time estimates. The document provides details on key aspects of PERT including identifying activities and events, establishing activity order, creating network diagrams, estimating activity durations, and identifying critical paths. It also discusses advantages and limitations of both PERT and CPM for project management.
This document discusses project scheduling techniques like PERT and CPM. It begins by explaining that project management involves planning, scheduling, and controlling activities with limited resources. PERT was developed for complex projects like missile development, using three time estimates per activity to determine expected duration probabilistically. CPM uses a single time estimate.
The document then covers the three phases of project management - planning, scheduling, and controlling. It describes how to construct a network diagram showing activities and their logical relationships. Key steps like event numbering, float calculation, and critical path analysis are explained. The document concludes by providing an example project network to demonstrate these concepts.
The document provides an overview of project management techniques including network diagrams, determining the critical path, and project scheduling. It discusses fundamentals of project management including defining a project and project management. The document also provides examples of network diagrams and calculating critical paths for projects.
This document discusses project management techniques PERT and CPM. It begins by outlining the basic steps of PERT/CPM, which include planning, scheduling, allocating resources, and controlling projects. It then discusses network diagrams and how they are used to represent projects, define activities and their relationships. The document provides examples of network diagrams and common errors in drawing them. It also discusses critical paths and how crashing can be used to reduce project duration at additional cost. The framework for PERT and CPM involves 6 steps including defining activities, relationships, and time/cost estimates.
This document discusses project scheduling techniques such as bar charts, milestone charts, and network techniques like PERT and CPM. It explains how these techniques are used for work breakdown, structuring activities, and allocating resources and monitoring projects in construction management. The document also provides details on early start times, early finish times, late start times, late finish times, total float, free float, and independent float in critical path method scheduling. It includes an example of calculating the expected time of activities and identifying the critical path in a sample project network diagram.
This document provides an overview of CPM (Critical Path Method) and PERT (Project Evaluation and Review Technique) techniques for project scheduling. Both methods use network diagrams to visually map out the sequence and dependencies of project activities. The key difference is that CPM uses single time estimates while PERT uses three time estimates and probability theory. The document outlines the basic steps and concepts for developing network diagrams and calculating the critical path of activities that determine the minimum project duration.
The document discusses technical aspects of project management. It provides an example of a housing project with specifications including building 3 bedroom homes for 50 families within 3 years. It then discusses key project elements like stakeholders, life cycle, work breakdown structure, Gantt charts, critical path analysis, and PERT/CPM techniques. Project management concepts like activities, events, precedence, floats, critical paths, and time estimates are explained with examples.
This document discusses key concepts in project management including:
1) How to construct a project network diagram representing activities and their dependencies.
2) How to determine a project's critical path which indicates its minimum duration.
3) How to calculate activity floats which provide flexibility in scheduling.
This document discusses project management using the network model known as PERT/CPM. It uses the example of planning seminars to illustrate key concepts. The document outlines the two key pieces of information needed for project planning and control: 1) the length of time to complete each task and 2) which tasks must be completed before another can begin. It presents this information for the seminar planning project in a table. The document then explains how to represent the information as a network diagram with activities as nodes and sequencing as arrows. It defines the critical path as the longest path of connected activities, which determines the shortest time to complete the entire project. The document outlines how to calculate early start/finish times and late start/finish times
This document discusses project management techniques like PERT and CPM. It provides definitions and explanations of key concepts in project management including:
- PERT was developed by the US Navy and handles uncertain activity times, while CPM was developed by DuPont for industrial projects with known activity times.
- Project networks use nodes and arrows to represent events and activities. Forward and backward passes are used to calculate earliest and latest start/finish times.
- Critical paths are those where activities must be completed on schedule or the project will be delayed. Slack time indicates how long an activity can be delayed.
This document discusses project scheduling techniques like PERT and CPM. It begins by explaining that project management involves planning, scheduling, and controlling limited resources like time, money, and personnel. It then defines PERT and CPM, noting that PERT uses three time estimates while CPM uses a single estimate. The document outlines the three phases of project management: planning, scheduling, and controlling. It also explains key components of PERT/CPM networks like activities, events, precedence, and dummy activities. The document provides examples of constructing PERT/CPM networks.
Critical Path Method: Activities And Events, Float/SlacksPrasad Thanthratey
A presentation on Critical Path Method: Activities And Events, Float/Slacks course of Construction Management from students of 10th Semester Architecture at VNIT, Nagpur (January- April 2018)
Introduction to LPP
Components of Linear Programming Problem
Basic Assumption in LPP
Examples of LPP
2 Formulation of LPP
Steps for Mathematical Formulation of LPP’s
Examples on Formulation of LPP
3 Basic Definitions
4 Graphical Method for solving LPP
5 Examples on Graphical method for solving LPP
1 Introduction
2 Types of events
3 Classical definition of probability
4 Examples on probability
5 Conditional probability
6 Bayes theorem
7 Random variables and Probability distributions
This document discusses number systems, including the decimal, binary, and octal systems. It begins by introducing positional and non-positional number systems. The decimal system uses base 10 with digits 0-9, where the place value of each digit depends on its position. The binary system uses base 2 with digits 0-1. Conversions between decimal, binary, and octal systems are demonstrated through examples such as decimal to binary conversion by repeated division. Fractions are also converted between number systems. Finally, the octal system is introduced, which uses base 8 with digits 0-7.
This document provides an introduction to error analysis in numerical techniques. It discusses approximate vs exact numbers, significant figures, rounding off numbers, different types of errors including absolute, relative and percentage errors. It also covers error in arithmetic operations due to inherent, truncation and rounding errors. A general error formula is presented to calculate the error in a function with multiple variables based on the errors in each variable. An example is given to calculate the maximum relative error in a numerical computation.
This document outlines lecture material on sampling techniques from Dr. Tushar Bhatt of Saurashtra University. It discusses various probability and non-probability sampling methods including simple random sampling, stratified sampling, cluster sampling, systematic sampling, and PPS sampling. For each method, it provides definitions, formulas, steps for implementation, and examples. The document is intended as teaching material, covering core concepts in sampling and how to select samples from different populations.
Control is a system for measuring and checking or inspecting a phenomenon. It suggests when to inspect, how often to inspect and how much to inspect. Control ascertains quality characteristics of an item, compares the same with prescribed quality characteristics of an item, compares the same with prescribed quality standards and separates defective item from non-defective ones.
Statistical Quality Control (SQC) is the term used to describe the set of statistical tools used by quality professionals.
SQC is used to analyze the quality problems and solve them. Statistical quality control refers to the use of statistical methods in the monitoring and maintaining of the quality of products and services.
Variation in manufactured products is inevitable; it is a fact of nature and industrial life. Even when a production process is well designed or carefully maintained, no two products are identical.
The difference between any two products could be very large, moderate, very small or even undetectable depending on the sources of variation.
For example, the weight of a particular model of automobile varies from unit to unit, the weight of packets of milk may differ very slightly from each other, and the length of refills of ball pens, the diameter of cricket balls may also be different and so on.
The existence of variation in products affects quality. So the aim of SQC is to trace the sources of such variation and try to eliminate them as far as possible.
The Statistical Inference is the process of drawing conclusions about on underlying population based on a sample or subset of the data.
In most cases, it is not practical to obtain all the measurements in a given population.
The statistical inference is deals with decision problems. There are two types of decision problems as mentioned below:
(i) Problems of estimation and
(ii) Test of hypotheses
In the problem of estimation, we must determine the value of parameter(s), while in test of hypothesis we must decide whether to accept or reject a specific value(s) of a parameter(s).
Decision theory as the name would imply is concerned with the process of making decisions. The extension to statistical decision theory includes decision making in the presence of statistical knowledge which provides some information where there is uncertainty. The elements of decision theory are quite logical and even perhaps intuitive. The classical approach to decision theory facilitates the use of sample information in making inferences about the unknown quantities. Other relevant information includes that of the possible consequences which is quantified by loss and the prior information which arises from statistical investigation. The use of Bayesian analysis in statistical decision theory is natural. Their unification provides a foundational framework for building and solving decision problems. The basic ideas of decision theory and of decision theoretic methods lend themselves to a variety of applications and computational and analytic advances.
The purpose of the book is to present the current techniques of operations research in such a way that they can be readily comprehended by the average business student taking an introductory course in operations research. Several OR teachers and teachers from management schools suggested that we should bring out a separate volume on OR with a view to meet the requirements of OR courses, which can also be used by the practising managers. The book can be used for one semester/term introductory course in operations research. Instructors may like to decide the appropriate sequencing of major topics covered.
This book will be useful to the students of management, OR, industrial and production engineering, computer sciences, chartered and cost-accountancy, economics and commerce. The approach taken here is to illustrate the practical use of OR techniques and therefore, at places complicated mathematical proofs have been avoided. To enhance the understanding of the application of OR techniques, illustrations have been drawn from real life situations. The problems given at the end of each chapter have been designed to strengthen the student's understanding of the subject matter. Our long teaching experience indicates that an individual's comprehension of the various quantitative methods is improved immeasurably by working through and understanding the solutions to the problems.
It is not possible for us to thank individually all those who have contributed to the case histories. Our colleagues and many people have contributed to these studies and we gratefully acknowledge their help. Without their support and cooperation this book could not have been brought out. Our special thanks are due to Dr. K. H. Atkotiya who have assisted me in editing the case studies. we wish to express my sincere thanks to Mr. Chandraprakash Shah making available all facilities needed for this job. We express my gratitude to my parents who have been a constant source of Inspiration.
We Strongly believe that the road to improvement is never-ending. Suggestions and criticism of the books will be very much appreciated and most gratefully acknowledged.
THIS PRESENTATION COVERED FOLLOWING TOPICS IN MATRIX ALGEBRA
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(c) Central Differences
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(a) Newton Forward and Backward Interpolation Formula
(b) Gauss Forward and Backward Interpolation Formula
(c)Stirling’s Interpolation Formula
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The presentation on Numerical Methods covered the following topics :
1. Introduction
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How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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Network_Analysis.pdf
1. Unit - V: Network Analysis
Dr. Tushar Bhatt
Ph. D, Mathematics
Assistant professor
Department of Science and Humanities
Faculty of Engineering and Technology
Atmiya University
Rajkot - 360005
March 3, 2023
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 1 / 45
2. Table of Contents
1 Introduction
Phases of CPM and PERT
Some Important Definitions
Project management or representation by a network diagram
Types of activities
Types of events
Common Errors
Rules of network construction
Numbering the events
Time analysis
Determination of Floats and Slack times
Critical activity and Critical path
2 Critical Path Method - CPM
3 Program Evaluation and Review Technique - PERT
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 2 / 45
3. 1. Introduction
The techniques of operations research used for planning, scheduling
and controlling large and complex projects are often referred as
network analysis. All these techniques are based on representation of
the project as a network of activities.
A network is a graphical plan consisting of a certain configuration of
arrows and nodes for showing the logical sequence of various activities
to be performed to achieve project objectives.
PERT (Program Evaluation and Review Technique) and CPM
(Critical Path Method) are two of the well known techniques belong
to the family of network analysis.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 3 / 45
4. 1.1 Phases of CPM and PERT
Project management by CPM and PERT can be based on the following
three phases.
1 Planning
In this phase, the whole project is broken down into smaller projects
and further into activities.
The time estimates for these activities are then determined.
Network diagram is constructed.
Study the network diagram in detail and incorporate any modification
at the initial or execution stage.
2 Scheduling
Scheduling phase involves time related activities.
A time chart is constructed showing the start and finish times of each
activity.
The chart also shows or relates one activity to other activities of the
project.
Identify non critical activities and show the amount of slack or float
times.This is very essential to take the advantages of delay in execution
of activities or using limited available resources.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 4 / 45
5. 1.1 Phases of CPM and PERT
3 Control
Controlling includes careful supervision of progress of the project with
the help of network diagram and time chart.
Continuously analyze and update all activities involved and if necessary
reschedule the project for the remaining portion of the project.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 5 / 45
6. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
7. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
2 Event: Beginning and end points of an activity are called events or
nodes.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
8. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
2 Event: Beginning and end points of an activity are called events or
nodes.
3 Critical activity: An activity is said to be critical if any delay in its
start further delays the completion of the whole work.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
9. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
2 Event: Beginning and end points of an activity are called events or
nodes.
3 Critical activity: An activity is said to be critical if any delay in its
start further delays the completion of the whole work.
4 Amount of Slack or Float time: The difference between the
earliest time and the latest time is called as activity slack.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
10. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
2 Event: Beginning and end points of an activity are called events or
nodes.
3 Critical activity: An activity is said to be critical if any delay in its
start further delays the completion of the whole work.
4 Amount of Slack or Float time: The difference between the
earliest time and the latest time is called as activity slack.
5 Total amount of slack or total float time: Total slack is the
difference of latest finish time and the duration.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
11. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
2 Event: Beginning and end points of an activity are called events or
nodes.
3 Critical activity: An activity is said to be critical if any delay in its
start further delays the completion of the whole work.
4 Amount of Slack or Float time: The difference between the
earliest time and the latest time is called as activity slack.
5 Total amount of slack or total float time: Total slack is the
difference of latest finish time and the duration.
6 Free slack or free float time: Free slack is the amount of time a
job can be delayed without affecting the early start of any other job.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
12. 1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.
2 Event: Beginning and end points of an activity are called events or
nodes.
3 Critical activity: An activity is said to be critical if any delay in its
start further delays the completion of the whole work.
4 Amount of Slack or Float time: The difference between the
earliest time and the latest time is called as activity slack.
5 Total amount of slack or total float time: Total slack is the
difference of latest finish time and the duration.
6 Free slack or free float time: Free slack is the amount of time a
job can be delayed without affecting the early start of any other job.
7 Independent slack: Independent slack = Free slack −(Lf − Es)
where Lf the amount of latest finish time of a job is and Es is the
amount of earlier start of a job.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 6 / 45
13. 1.3 Project management or representation by a network
diagram
Network diagram is the graphical representation of logically and
sequentially connected arrows and node, representing activities and events
respectively in a project.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 7 / 45
14. 1.3 Project management or representation by a network
diagram
Network diagram is the graphical representation of logically and
sequentially connected arrows and node, representing activities and events
respectively in a project.
Figure: Events and activity
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 7 / 45
15. 1.3 Project management or representation by a network
diagram
Now observe the following figure i and j indicates event represented by the
nodes in which node i known as tail and j is known as head moreover the
symbol A means an arrow in between i and j indicates an activity.
Figure: Network diagram
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 8 / 45
16. 1.4 Types of activities
1 Preceding activity:
Activity that must be accomplished before a given event can occur.
2 Succeeding activity:
Activity that cannot be accomplished until an event has occurred.
3 Concurrent activity:
Activity taking place at same time or in the same location.
4 Dummy activity:
Activity which neither consumes time nor resources but is used simply
to represent a connection or a link between the events is known as
dummies. It is shown in network by a dotted line.
5 In above mentioned figure, A is a preceding activity of B, B is a
preceding activity of C, D is a preceding activity of E and E is a
preceding activity of F.
6 Similarly B is a succeeding activity of A, C is a succeeding activity of
B, E is a succeeding activity of C and F is a succeeding activity of E.
7 Moreover the activities A and D arise from even 1 at same time of
same location therefore A and D are becomes concurrent activities.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 9 / 45
17. 1.4 Types of activities
Now for understanding the dummy activity observe the following figure:
Figure: Network diagram and dummy activity
Here according to the rule of network theory, there is only one starting
point and one ending point. But in above mentioned figure, we observe
that there are two ending points node 4 and node 5. So it is necessary to
add a dummy activity between node 3 and node 4 because after that
procedure event 4 is not an end of the network. Therefore the activity E in
above figure is called dummy activity and it is denoted by a dotted line.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 10 / 45
18. 1.5 Types of events
1. Merge Event
An event is said to be Merge event if two or more than two activities are
end at the same node (event).
Figure: Merge Event
In above figure an event i is Merge event.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 11 / 45
19. 1.5 Types of events
2. Burst Event
An event is said to be Burst event if it is a starting point of two or more
than two activities.
Figure: Burst Event
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 12 / 45
20. 1.6 Common Errors
1. Looping (Cycling)
Drawing an endless loop in a network is known as error of looping.
Figure: Loop or Cycle
In above figure double arrow for activity B create an endless loop in
between the events 2, 3 and 4.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 13 / 45
21. 1.6 Common Errors
2. Dangling
To disconnect an activity before the complication of all activities called
error of dangling.
Figure: Dangling
In above figure an activity F is disconnected before completion of
remaining all activities.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 14 / 45
22. 1.6 Common Errors
3. Redundancy
Unnecessarily inserting the dummy activity in a network diagram is known
as error of redundancy.
Figure: Redundancy
In above figure an activity F is a dummy activity which is unnecessary
because there is a path between events 2 to 4 is already exists. Therefore
it creates an error namely redundancy.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 15 / 45
23. 1.7 Rules of network construction
Try to avoid the arrow the cross each other.
Use straight arrow.
No event can occur until every activity preceding it has been
completed.
An event can’t occur twice.
Dummies should be introduced only if it is extremely necessary.
Network has only one entry point called start event and one end point
(emergence).
Use arrow left to right. Avoid mixing two directions.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 16 / 45
24. 1.8 Numbering the events
Number must be unique.
Number should be carried out on a sequential basis from left to right
and top to bottom.
Initial event must be numbered as 1.
Number of all new starts events 2, 3 and so on.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 17 / 45
25. 1.9 Time analysis
1 Tij = tij = Estimate completion time of activity (i, j).
2 ESij = Earliest starting time of activity (i, j).
3 EFij = Earliest finishing time of activity (i, j).
4 LSij = Latest starting time of activity (i, j).
5 LFij = Latest finishing time of activity (i, j).
Forward pass computation (Left to Right)
Zero is the starting time for the project.
(EF)ij = (ES)ij + tij
Ej = maxi (Ej + tij )
Backward pass computation (Right to Left)
For ending event assume E = L.
(LFij ) = (LF)ij − tij .
Lj = minj (Lj − tij ).
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 18 / 45
26. 1.10 Determination of Floats and Slack times
1. Float
It is defined as the difference between the latest and earliest activity time.
There are three types of floats.
(a) Total Float
(TF)ij = (LS)ij − (ES)ij OR (Li − Ei ) − tij
Where Ei = Earliest time for tail event
Li =Latest time for tail event
tij = Normal time for activity (i, j).
(b) Free Float
FFij = (Ej − Ei ) − tij
Where Ei = Earliest time for tail event
tij =Normal time for activity (i, j).
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 19 / 45
27. 1.10 Determination of Floats and Slack times
(c). Independent Float
IFij = (Ej − Li ) − tij
Where Ei = Earliest time for tail event
Li = Latest time for tail event
tij = Normal time for activity (i, j).
2. Slack
It is defined as the difference between latest and earliest event time.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 20 / 45
28. 1.11 Critical activity and Critical path
An activity (i, j) is said to be critical, if it has total float TFij = 0.
A path throughout the network is said to be critical if it satisfies the
following three conditions.
1 ESi = LFi
2 ESj = LFj
3 ESj = ESi = LFj − LFi = tij
Where ESi = Earliest starting time of initial event.
LFi = Latest finishing time of initial event.
ESj = Earliest starting time of final event.
LFj = Latest finishing time of final event.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 21 / 45
29. 2. CPM
Ex-1: Construct a network for a project whose activities and precedence
relationships are as given below:
Activity A B C D E F G H I J K
Predecessor - - - A B B C D E H, I F, G
Figure: Given Network
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 22 / 45
30. 2. CPM - Ex - 1: Solu...
Figure: Required Network
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 23 / 45
31. 2. CPM - Ex - 2
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 24 / 45
32. 2. CPM - Ex - 2: Solu...
Solution:
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 25 / 45
33. 2. CPM - Ex - 3
Ex-3: A project schedule has the following properties. According to that
(1) Construct a network.
(2) Compute the earliest event time and latest event time.
(3) Determine the critical path and total project duration.
(4) Compute total and free float for each activity.
Activity 1-2 1-3 2-4 3-4
Time (days) 14 1 1 1
Solution:
(i) Network diagram
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 26 / 45
34. 2. CPM - Ex - 3: Solu...
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 27 / 45
35. 2. CPM - Ex - 3: Solu...
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 28 / 45
36. 2. CPM - Ex - 3: Solu...
The critical path = 1 − 2 − 4
Total project duration = 4 + 1 = 5 days
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 29 / 45
37. 2. CPM - EX - 4
Ex - 4: Determine the early start and late start in respect of all node
points and identify critical path for the following network
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 30 / 45
38. 2. CPM - Ex - 4: Solu...
Solution : Calculation of E and L for each node is shown in the network
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 31 / 45
39. 2. CPM - Ex - 4: Solu...
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 32 / 45
40. 2. CPM - Ex - 4: Solu...
From the table, the critical nodes are
(1, 2), (2, 5), (5, 7), (5, 8), (7, 10)and(8, 10).
From the table, there are two possible critical paths
(i)1 → 2 → 5 → 8 → 10
(ii)1 → 2 → 5 → 7 → 10
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 33 / 45
41. 2. CPM - Ex - 5
Ex - 5: Find the critical path and calculate the slack time for the
following network
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 34 / 45
42. 2. CPM - Ex - 5: Solu...
Solution: The earliest time and the latest time are obtained below
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 35 / 45
43. 2. CPM - Ex - 5: Solu...
Solution: From the above table, the critical nodes are the activities
(1, 3), (3, 5)and(5, 9)
The critical path is 1 → 3 → 5 → 9.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 36 / 45
44. 3. PERT
The main objective in the analysis through PERT is to find out the
completion for a particular event within specified date. The PERT
approach takes into account the uncertainties. The three time values are
associated with each activity.
1 Optimistic time: It is the shortest possible time in which the activity
can be finished. It assumes that everything goes very well. This is
denoted by t0.
2 Most likely time: It is the estimate of the normal time the activity
would take. This assumes normal delays. If a graph is plotted in the
time of completion and the frequency of completion in that time
period, then most likely time will represent the highest frequency of
occurrence. This is denoted by tm.
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 37 / 45
45. 3. PERT
3 Pessimistic time: It represents the longest time the activity could
take if everything goes wrong. As in optimistic estimate, this value
may be such that only one in hundred or one in twenty will take time
longer than this value. This is denoted by tp.
4 Expected time: It is the average time an activity will take if it were
to be repeated on large number of times and is based on the
assumption that the activity time follows Beta distribution, this is
given by
te =
(t0 + 4tm + tp)
6
5 The variance for the activity is given by
σ2
=
tp − t0
6
2
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 38 / 45
46. 3. PERT - Ex - 1
Ex - 1: For the project
Task A B C D E F G H I J K
Least time 4 5 8 2 4 6 8 5 3 5 6
Greatest time 8 10 12 7 10 15 16 9 7 11 13
Find the earliest and latest expected time to each event and also critical
path in the network.
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 39 / 45
47. 3. PERT - Ex - 1: Solu...
Solution:
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 40 / 45
48. 3. PERT - Ex - 1: Solu...
Solution:
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 41 / 45
49. 3. PERT - Ex - 1: Solu...
Solution:
The critical path is A → C → E → H → K.
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 42 / 45
50. 3. PERT - Ex - 2
Ex - 2: A project has the following characteristics
Construct a PERT network. Find the critical path and variance for each
event.
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 43 / 45
51. 3. PERT - Ex - 2: Solu...
Solution:
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 44 / 45
52. 3. PERT - Ex - 2: Solu...
Solution: The network is constructed as shown below
The critical path: 1 → 2 → 4 → 6 → 7 → 9 → 10.
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 45 / 45