Introduction
CPM/PERT or Network Analysis as the technique is sometimes called, developed along two parallel streams, one industrial and the other military.
CPM (Critical Path Method) was the discovery of M.R.Walker of E.I.Du Pont de Nemours & Co. and J.E.Kelly of Remington Rand, circa 1957. The computation was designed for the UNIVAC-I computer. The first test was made in 1958, when CPM was applied to the construction of a new chemical plant. In March 1959, the method was applied to maintenance shut-down at the Du Pont works in Louisville, Kentucky. Unproductive time was reduced from 125 to 93 hours.
PERT (Project Evaluation and Review Technique) was devised in 1958 for the POLARIS missile program by the Program Evaluation Branch of the Special Projects office of the U.S.Navy, helped by the Lockheed Missile Systems division and the Consultant firm of Booz-Allen & Hamilton. The calculations were so arranged so that they could be carried out on the IBM Naval Ordinance Research Computer (NORC) at Dahlgren, Virginia.
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.
Engineering Professional Practice Chapter 2 Ethics and Professionalism Pokhar...Hari Krishna Shrestha
Lecture Notes related to Chapter 2 (Ethics and Professionalism) of Engineering Professional Practice, prepared by Prof. Dr. Hari Krishna Shrestha, Nepal Engineering College
The document discusses the Program Evaluation and Review Technique (PERT), a network model used for project management that accounts for uncertainty in activity times. PERT was developed in the late 1950s for large, complex projects like the Polaris missile system. It uses three time estimates - optimistic, pessimistic, and most likely - to calculate the expected time, variance, and probability of completing a project by a certain date using statistical methods like the beta distribution and normal distribution. PERT identifies the critical path that determines the project completion time and allows project managers to focus on critical activities.
Project Management: NETWORK ANALYSIS - CPM and PERTS.Vijaya Bhaskar
This document provides information about project management techniques including network analysis using Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT). It defines what a project and network are, and describes the basic steps and concepts involved in CPM and PERT. These include drawing networks, identifying critical paths, calculating expected durations and variances of activities, and determining the probability of completing projects within given timeframes. Examples are provided to illustrate key concepts like crashing project durations and calculating associated costs. The document is intended as a teaching aid for understanding fundamental aspects of project network analysis.
Internship Report on Building ConstructionEsmael Aragaw
K2N Architecture and Engineering Consultancy PLC is an Ethiopian consulting firm providing architectural, engineering, and project management services. The company aims to provide quality services that exceed client expectations while adhering to high technical and ethical standards. K2N has experience on various project types both in Ethiopia and internationally. The document provides details on K2N's vision, mission, values, organizational structure, management, areas of expertise, and example projects.
Suman Jyoti
Madan Ashrit Memorial Technical Collage. Gothatar, Kathmandu
The Report Includes:-
Building (Drawing, Theory and Estimation)
Highway ( Detailing, Theory and Calculation)
Sanitary ( ( Detailing, Theory and Calculation)
Irrigation (Theory only)
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.
L&T Inplant training at chennai metro reporthemanth kumar
Larsen and Toubro (L&T) was awarded contracts to complete two unfinished sections of the Chennai Metro rail project after the previous contractor abruptly pulled out. L&T is responsible for the design and construction of underground stations and tunnels between Government Estate and Saidapet. Extensive surveys and investigations were conducted before construction, including basic surveys, soil investigations, and subsurface investigations to understand the land, topography, and soil conditions. The Environmental Health and Safety department of L&T oversees safety procedures and training to ensure worker and public safety during the large-scale metro construction project.
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.
Engineering Professional Practice Chapter 2 Ethics and Professionalism Pokhar...Hari Krishna Shrestha
Lecture Notes related to Chapter 2 (Ethics and Professionalism) of Engineering Professional Practice, prepared by Prof. Dr. Hari Krishna Shrestha, Nepal Engineering College
The document discusses the Program Evaluation and Review Technique (PERT), a network model used for project management that accounts for uncertainty in activity times. PERT was developed in the late 1950s for large, complex projects like the Polaris missile system. It uses three time estimates - optimistic, pessimistic, and most likely - to calculate the expected time, variance, and probability of completing a project by a certain date using statistical methods like the beta distribution and normal distribution. PERT identifies the critical path that determines the project completion time and allows project managers to focus on critical activities.
Project Management: NETWORK ANALYSIS - CPM and PERTS.Vijaya Bhaskar
This document provides information about project management techniques including network analysis using Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT). It defines what a project and network are, and describes the basic steps and concepts involved in CPM and PERT. These include drawing networks, identifying critical paths, calculating expected durations and variances of activities, and determining the probability of completing projects within given timeframes. Examples are provided to illustrate key concepts like crashing project durations and calculating associated costs. The document is intended as a teaching aid for understanding fundamental aspects of project network analysis.
Internship Report on Building ConstructionEsmael Aragaw
K2N Architecture and Engineering Consultancy PLC is an Ethiopian consulting firm providing architectural, engineering, and project management services. The company aims to provide quality services that exceed client expectations while adhering to high technical and ethical standards. K2N has experience on various project types both in Ethiopia and internationally. The document provides details on K2N's vision, mission, values, organizational structure, management, areas of expertise, and example projects.
Suman Jyoti
Madan Ashrit Memorial Technical Collage. Gothatar, Kathmandu
The Report Includes:-
Building (Drawing, Theory and Estimation)
Highway ( Detailing, Theory and Calculation)
Sanitary ( ( Detailing, Theory and Calculation)
Irrigation (Theory only)
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.
L&T Inplant training at chennai metro reporthemanth kumar
Larsen and Toubro (L&T) was awarded contracts to complete two unfinished sections of the Chennai Metro rail project after the previous contractor abruptly pulled out. L&T is responsible for the design and construction of underground stations and tunnels between Government Estate and Saidapet. Extensive surveys and investigations were conducted before construction, including basic surveys, soil investigations, and subsurface investigations to understand the land, topography, and soil conditions. The Environmental Health and Safety department of L&T oversees safety procedures and training to ensure worker and public safety during the large-scale metro construction project.
This document provides information about estimating and costing for construction projects. It discusses the need for estimation to determine project feasibility and cost, outlines the estimating process, and lists the key data required which includes drawings, specifications, and rates. Measurement rules and units for different construction items like earthwork, concrete, masonry, woodwork, and finishing works are also covered. The document emphasizes accurate measurement of work quantities as the basis for preparation of estimates.
This document provides definitions and explanations of key concepts in reinforced concrete design. It defines reinforced concrete as a composite material made of concrete and steel reinforcement. The purpose of reinforcement is to improve the tensile strength of concrete. The Limit State Method of design considers both the strength limit state and serviceability limit state, making it a more realistic and economical approach compared to other methods like Working Stress Method and Ultimate Load Method. Key factors of safety in the Limit State Method include partial factors for concrete γc = 1.5, and for steel γs = 1.15.
The document is an internship report on Bharat Pride Park, a residential and commercial development project in Kalaburagi, India. It provides details about the project, including an overview of the consulting firm and the intern's responsibilities. The internship involved taking measurements to calculate quantities of materials, interpreting drawings, and assisting engineers and supervisors. The intern gained experience in construction processes and project documentation during the one-month internship.
The document provides an introduction to the concepts of estimation and quantity surveying in construction projects. It discusses the key steps and considerations in preparing different types of cost estimates, including preliminary, detailed, quantity, revised, and annual repair estimates. Various methods for preparing approximate estimates are described, such as the service unit method, plinth area method, floor area method, carpet area method, percentage method, and cubical content method. Requirements for preparing accurate estimates like drawings, specifications, and rate analysis are also outlined.
The document is a practical training report submitted by Sumit Kumar, a civil engineering student, documenting his six-week internship with PKS Buildmart Pvt. Ltd. It provides an overview of the company and details of the commercial building construction project he worked on, which involved tasks in both the office and on-site, including setting out, excavation, concrete work, and masonry. The report documents the materials, procedures, and Sumit's contributions on the project over the training period.
This document is a training report submitted by Sher Bahadur to Kurukshetra University for a degree in civil engineering. It provides an overview of building construction topics covered during the training period, including different types of buildings, loads, building components, foundations, materials used, and quality control tests. The training gave Sher Bahadur hands-on experience in building construction that supplemented his theoretical classroom knowledge and prepared him for a career in the field.
Chapter 3 Construction Planning and Scheduling lect. 3.pptssuserb4074f
This document provides an overview of construction planning and scheduling. It discusses the importance of construction planning, including the advantages of planning before, during, and after a project's completion. It also outlines the steps involved in construction planning, including determining activities, sequencing activities, estimating activity durations, and developing schedules. The document also discusses construction scheduling techniques like bar charts, network scheduling using critical path method (CPM) and program evaluation review technique (PERT), and how to calculate critical paths.
I will describe those things in the documents--
Types of Cost Estimates Used in Construction
Following are the different types of cost estimates used in construction,
Preliminary Cost Estimate
Plinth Area Cost Estimate
Cube Rate Cost Estimate
Approximate Quantity Method Cost Estimate
Detailed Cost Estimate
Revised Cost Estimate
Supplementary Cost Estimate
Annual Repair Cost Estimate
This document discusses Gantt charts, which are bar charts developed in the early 1900s to illustrate a project schedule. A Gantt chart uses two axes - horizontal for time and vertical for tasks. Tasks are represented as bars whose length indicates the time to complete that task. The document provides an example Gantt chart for a project with 5 tasks ranging from 5 to 30 days. It notes advantages like ease of creation and visual representation, but also limitations like lack of facilitating project control, monitoring, and updating for large projects.
minor project report on design of residential buildingtushar garg
This document is a minor project report submitted by Tushar Garg to Rajendra Kumar Khyalia for a Bachelor of Technology degree. It includes an acknowledgement, abstract, declaration, and table of contents sections. The content covers the aim of designing a residential building, including selecting a plot, surveying the site, requirements for residential buildings, building bye laws and regulations, room arrangements, and sanitation provisions. Drawings and photos are also included.
This document provides an overview of Coastal Projects Limited (CPL), an infrastructure construction company, and CPL's role in the Bangalore Metro Rail Project. It discusses CPL's vision, mission, values and quality assurance policies. It then describes the Majestic Metro Station project, including station layout details. CPL is constructing the underground Majestic Metro Station, which will be an interchange station for the Purple and Green metro lines. Construction is proceeding according to a grid-wise plan from the center of Zone E outwards in all directions.
The document discusses the future of civil engineering. It covers innovative technologies like 3D printing, swarm construction, smart buildings, smart cities, vertical cities, and intelligent transportation systems. It also discusses housing demand and infrastructure development in India, including the need for 51 million housing units to address shortages. Challenges for civil engineers include developing infrastructure like roads, ports, airports, railways and power while addressing this housing crisis.
Report on surveying and leveling during my summer Internship period.
The two-week program did not cover too much and it was done in the 4th semester so the report is of beginner level.
This document provides an overview of a project report on designing a multi-storied reinforced concrete building using ETABS software. The objectives are to analyze, design, and detail the structural components of the building. The methodology involves preparing CAD drawings, calculating loads, analyzing the structure, and designing and detailing structural elements. The building to be designed is a residential building with ground + 5 floors located in Chalikkavattom. Loads like dead, live, wind, and seismic loads will be calculated according to Indian codes and applied in the ETABS analysis model.
An internship report on a Construction CompanyOsama Yousaf
This document is an internship report submitted by Osama Yousaf to the Institute of Management Studies at the University of Peshawar. The report details Osama's internship at Maimaar Construction Company in Peshawar. It provides an overview of the construction sector in Pakistan, introduces Maimaar Construction, and analyzes the company's organizational structure, human resources practices, training programs, and activities. The report also includes a SWOT analysis and recommendations to improve the company's performance.
Presentation on The Use of automation in construction industryIEI GSC
Presented By Akshay Tejankar
at 31st National Convention of Civil Engineers organised by Gujarat State Center, The Institution of Engineers (India) at Ahmedabad
Summer Internship Report of Civil Engineering in Construction SiteRAVI KUMAR
The document provides details about the internship completed by Ravi Kumar at Fidesto Projects Private Limited from June 17 to July 30, 2019. It includes an acknowledgement, information about the organization and the proposed residential project in Pune on which Ravi Kumar worked. Safety protocols at construction sites like use of safety equipment and formwork are also summarized. The key steps involved in building construction are outlined.
Internship report
The report is prepared on request from Maaz and Shahrukh for their 4-week internship program.
The data has been provided by them.
This report includes theoretical, pictorial and mathematical material.
A beautiful power point presentation on land acquisition act- 1894 and the acquisition procedure in Kerala..Highly useful for revenue officers in KERALA.
AN INTERNSHIP REPORT ON RESIDENTIAL BUILDING CONSTRUCTIONAbhishek Singh
This document appears to be an internship report submitted by four students - Abhishek Singh, Naval Tej Singh Ahuja, Sahil Thakur, and Swapnil Singh - to their supervisor Mr. Kapil Bhardwaj at Universal Buildwell Pvt. Ltd. in Gurgaon, Haryana, India. The report provides details about a residential construction project called Universal Aura, including project specifications, building materials used, and work ongoing at the site during the students' summer internship from June 13 to July 13, 2016.
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.
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 provides information about estimating and costing for construction projects. It discusses the need for estimation to determine project feasibility and cost, outlines the estimating process, and lists the key data required which includes drawings, specifications, and rates. Measurement rules and units for different construction items like earthwork, concrete, masonry, woodwork, and finishing works are also covered. The document emphasizes accurate measurement of work quantities as the basis for preparation of estimates.
This document provides definitions and explanations of key concepts in reinforced concrete design. It defines reinforced concrete as a composite material made of concrete and steel reinforcement. The purpose of reinforcement is to improve the tensile strength of concrete. The Limit State Method of design considers both the strength limit state and serviceability limit state, making it a more realistic and economical approach compared to other methods like Working Stress Method and Ultimate Load Method. Key factors of safety in the Limit State Method include partial factors for concrete γc = 1.5, and for steel γs = 1.15.
The document is an internship report on Bharat Pride Park, a residential and commercial development project in Kalaburagi, India. It provides details about the project, including an overview of the consulting firm and the intern's responsibilities. The internship involved taking measurements to calculate quantities of materials, interpreting drawings, and assisting engineers and supervisors. The intern gained experience in construction processes and project documentation during the one-month internship.
The document provides an introduction to the concepts of estimation and quantity surveying in construction projects. It discusses the key steps and considerations in preparing different types of cost estimates, including preliminary, detailed, quantity, revised, and annual repair estimates. Various methods for preparing approximate estimates are described, such as the service unit method, plinth area method, floor area method, carpet area method, percentage method, and cubical content method. Requirements for preparing accurate estimates like drawings, specifications, and rate analysis are also outlined.
The document is a practical training report submitted by Sumit Kumar, a civil engineering student, documenting his six-week internship with PKS Buildmart Pvt. Ltd. It provides an overview of the company and details of the commercial building construction project he worked on, which involved tasks in both the office and on-site, including setting out, excavation, concrete work, and masonry. The report documents the materials, procedures, and Sumit's contributions on the project over the training period.
This document is a training report submitted by Sher Bahadur to Kurukshetra University for a degree in civil engineering. It provides an overview of building construction topics covered during the training period, including different types of buildings, loads, building components, foundations, materials used, and quality control tests. The training gave Sher Bahadur hands-on experience in building construction that supplemented his theoretical classroom knowledge and prepared him for a career in the field.
Chapter 3 Construction Planning and Scheduling lect. 3.pptssuserb4074f
This document provides an overview of construction planning and scheduling. It discusses the importance of construction planning, including the advantages of planning before, during, and after a project's completion. It also outlines the steps involved in construction planning, including determining activities, sequencing activities, estimating activity durations, and developing schedules. The document also discusses construction scheduling techniques like bar charts, network scheduling using critical path method (CPM) and program evaluation review technique (PERT), and how to calculate critical paths.
I will describe those things in the documents--
Types of Cost Estimates Used in Construction
Following are the different types of cost estimates used in construction,
Preliminary Cost Estimate
Plinth Area Cost Estimate
Cube Rate Cost Estimate
Approximate Quantity Method Cost Estimate
Detailed Cost Estimate
Revised Cost Estimate
Supplementary Cost Estimate
Annual Repair Cost Estimate
This document discusses Gantt charts, which are bar charts developed in the early 1900s to illustrate a project schedule. A Gantt chart uses two axes - horizontal for time and vertical for tasks. Tasks are represented as bars whose length indicates the time to complete that task. The document provides an example Gantt chart for a project with 5 tasks ranging from 5 to 30 days. It notes advantages like ease of creation and visual representation, but also limitations like lack of facilitating project control, monitoring, and updating for large projects.
minor project report on design of residential buildingtushar garg
This document is a minor project report submitted by Tushar Garg to Rajendra Kumar Khyalia for a Bachelor of Technology degree. It includes an acknowledgement, abstract, declaration, and table of contents sections. The content covers the aim of designing a residential building, including selecting a plot, surveying the site, requirements for residential buildings, building bye laws and regulations, room arrangements, and sanitation provisions. Drawings and photos are also included.
This document provides an overview of Coastal Projects Limited (CPL), an infrastructure construction company, and CPL's role in the Bangalore Metro Rail Project. It discusses CPL's vision, mission, values and quality assurance policies. It then describes the Majestic Metro Station project, including station layout details. CPL is constructing the underground Majestic Metro Station, which will be an interchange station for the Purple and Green metro lines. Construction is proceeding according to a grid-wise plan from the center of Zone E outwards in all directions.
The document discusses the future of civil engineering. It covers innovative technologies like 3D printing, swarm construction, smart buildings, smart cities, vertical cities, and intelligent transportation systems. It also discusses housing demand and infrastructure development in India, including the need for 51 million housing units to address shortages. Challenges for civil engineers include developing infrastructure like roads, ports, airports, railways and power while addressing this housing crisis.
Report on surveying and leveling during my summer Internship period.
The two-week program did not cover too much and it was done in the 4th semester so the report is of beginner level.
This document provides an overview of a project report on designing a multi-storied reinforced concrete building using ETABS software. The objectives are to analyze, design, and detail the structural components of the building. The methodology involves preparing CAD drawings, calculating loads, analyzing the structure, and designing and detailing structural elements. The building to be designed is a residential building with ground + 5 floors located in Chalikkavattom. Loads like dead, live, wind, and seismic loads will be calculated according to Indian codes and applied in the ETABS analysis model.
An internship report on a Construction CompanyOsama Yousaf
This document is an internship report submitted by Osama Yousaf to the Institute of Management Studies at the University of Peshawar. The report details Osama's internship at Maimaar Construction Company in Peshawar. It provides an overview of the construction sector in Pakistan, introduces Maimaar Construction, and analyzes the company's organizational structure, human resources practices, training programs, and activities. The report also includes a SWOT analysis and recommendations to improve the company's performance.
Presentation on The Use of automation in construction industryIEI GSC
Presented By Akshay Tejankar
at 31st National Convention of Civil Engineers organised by Gujarat State Center, The Institution of Engineers (India) at Ahmedabad
Summer Internship Report of Civil Engineering in Construction SiteRAVI KUMAR
The document provides details about the internship completed by Ravi Kumar at Fidesto Projects Private Limited from June 17 to July 30, 2019. It includes an acknowledgement, information about the organization and the proposed residential project in Pune on which Ravi Kumar worked. Safety protocols at construction sites like use of safety equipment and formwork are also summarized. The key steps involved in building construction are outlined.
Internship report
The report is prepared on request from Maaz and Shahrukh for their 4-week internship program.
The data has been provided by them.
This report includes theoretical, pictorial and mathematical material.
A beautiful power point presentation on land acquisition act- 1894 and the acquisition procedure in Kerala..Highly useful for revenue officers in KERALA.
AN INTERNSHIP REPORT ON RESIDENTIAL BUILDING CONSTRUCTIONAbhishek Singh
This document appears to be an internship report submitted by four students - Abhishek Singh, Naval Tej Singh Ahuja, Sahil Thakur, and Swapnil Singh - to their supervisor Mr. Kapil Bhardwaj at Universal Buildwell Pvt. Ltd. in Gurgaon, Haryana, India. The report provides details about a residential construction project called Universal Aura, including project specifications, building materials used, and work ongoing at the site during the students' summer internship from June 13 to July 13, 2016.
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.
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.
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.
Network analysis is an important project management tool. It helps with planning, scheduling, and controlling projects through representing activities as a project network. It identifies the interrelationships between activities and helps minimize time, costs, and idle resources while avoiding delays. PERT and CPM are two common network analysis techniques. PERT is useful for complex projects with uncertain activity times, while CPM focuses on cost optimization for projects with known activity times. Both aim to identify the critical path that determines the minimum project duration.
The document provides an overview of critical path method (CPM) and program evaluation and review technique (PERT) network analysis techniques. It defines CPM and PERT, explaining that CPM is used for projects with known activity times and deterministic models, while PERT is for uncertain activity times and probabilistic models. The key aspects of developing a network are described, including defining the project, determining activity relationships and time estimates, and identifying the critical path which determines the project duration. The differences between CPM and PERT are that PERT is for non-repetitive first projects while CPM is for repetitive projects with prior experience to estimate activity times.
The document discusses various network scheduling techniques used for project management including Gantt charts, PERT, CPM, and precedence diagramming. It focuses on describing the Program Evaluation and Review Technique (PERT) which is a management tool used to analyze time and resource requirements of projects. PERT involves identifying all tasks, determining their sequence and interdependencies, and estimating time durations to identify the critical path and assess project deadline risks. Key aspects of PERT include event-based networks, three-point estimates for activity durations, and computation of slack times.
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.
The key steps in the PERT planning process are:
1) Identifying the specific activities and milestones of the project.
2) Determining the proper sequence of activities by constructing a network diagram that shows the interdependencies.
3) Estimating the time required to complete each activity.
4) Identifying the critical path which is the longest sequence of activities that determines the minimum time to complete the project.
(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 three examples of project managers and their responsibilities on different projects:
1) Construction of a retail development with 26 units and a supermarket. Responsible for coordinating contractors to ensure on-time and on-budget completion.
2) Directing trials of a new analgesic drug. Responsible for designing experiments and ensuring proper scientific and legal procedures are followed.
3) Introducing multimedia resources at a teacher training college in New Delhi. Responsible for purchasing and developing resources as well as encouraging acceptance by lecturers and students.
The document discusses organizing procurement processes and activities through project management techniques. It explains that procuring big projects requires well-organized activities prior to execution. Project management involves planning, scheduling, and controlling phases. Planning includes defining goals and teams. Scheduling sequences activities and allocates time. Controlling monitors resources, costs, quality and budgets to revise plans if needed. Popular techniques like Gantt charts, PERT and CPM help with planning, scheduling and controlling. These network approaches involve defining relationships between activities to identify critical paths and ensure on-time completion.
This document provides an introduction to project management techniques PERT and CPM. It defines key concepts like activities, events, nodes, dummy activities and paths in a network diagram. It explains the stages of project management including planning, appraisal, implementation and review/control. The document outlines the steps to determine critical path in CPM and describes crashing a project to reduce duration. It compares PERT and CPM, noting PERT uses 3 time estimates and is probabilistic while CPM uses one estimate and focuses on tradeoffs between time and cost.
The document discusses project management techniques CPM and PERT. It provides:
1) A brief history of CPM and PERT, including that CPM was developed by DuPont for chemical plants, while PERT was developed by the US Navy for planning the Polaris missile program.
2) An overview of the key differences between CPM and PERT, such as CPM using deterministic activity times while PERT uses probabilistic times to account for uncertainty.
3) An example application of network analysis to determine the minimum time needed to complete a product redesign and packaging project. The analysis identifies a critical path of 24 weeks.
The document discusses project scheduling and some key related concepts. It defines project scheduling as the process of converting a general project plan into a time-based graphic presentation given information on available resources and time constraints. It outlines the basic principles of project scheduling such as compartmentalizing tasks, determining interdependencies, allocating time, and defining responsibilities, outcomes and milestones. It also defines some basic terminology like tasks, activities, work products, and events. It then explains common scheduling techniques like network diagrams, bar charts, Gantt charts and milestones charts. Finally, it discusses objectives and applications of network analysis in project scheduling.
The document discusses project scheduling and some key concepts related to it. It defines project scheduling as the process of converting a general project plan into a time-based graphic presentation given information on available resources and time constraints. It outlines the basic principles of project scheduling such as compartmentalizing tasks, determining interdependencies, allocating time, and defining responsibilities, outcomes and milestones. It also discusses some basic scheduling terminology like tasks, activities, work products, and events. Finally, it explains common scheduling techniques like network diagrams and bar charts, as well as objectives and applications of network analysis in project scheduling.
This document provides an overview of project management techniques network analysis, CPM, and PERT. It discusses that CPM and PERT were developed in the 1950s, with CPM by DuPont for chemical plants and PERT by the US Navy for missile programs. CPM uses deterministic activity times and a single estimate, while PERT uses probabilistic times and multiple estimates. Both techniques use networks of interconnected activities to plan and schedule projects. The critical path is the longest path through the network and determines the minimum project time. Network analysis helps answer questions about project schedules, costs, and critical activities.
1) PERT and CPM are network analysis techniques used to schedule, coordinate, and control complex projects. They analyze the tasks involved in a project and their interdependencies to estimate project duration and identify the critical path.
2) The document discusses key aspects of PERT and CPM including activity networks, time estimates, critical path identification, and slack calculations. It also provides an example of applying PERT/CPM to schedule a construction project.
3) PERT and CPM both help answer questions about project completion date, start/finish times of tasks, critical tasks, and float or slack times. The example project demonstrates how its network diagram can be used to determine a 44-week schedule with identified
This document provides an overview of project management techniques including network analysis, CPM, and PERT. It defines key terms like activities, events, paths, and networks. Network analysis is used to plan, schedule, and control complex projects by representing relationships between activities. CPM uses deterministic activity times while PERT accounts for uncertainty with probabilistic estimates. The critical path consists of activities that directly determine the project duration. Project management involves planning, scheduling, and controlling a project over its stages.
Most construction sites that run into trouble do so for reasons related to managerial factors rather than because of technical problems. The site-based management can make significant improvements in the cost and time savings during the construction process without involving a mass of additional work. The role of site managers is to control and maintain work performance and then taking actions to rectify situations where performance is unsatisfactory.
Concepts and Formulas of Construction Site Layout Planning Elements:
A well-planned site including all temporary facilities and utilities lead to: 1) increasing productivity and safety, 2) reducing area(s) needed for temporary construction, and 3) maximizing utilization. The following points should be considered in good site layout
Site layout planning can affect productivity and is crucial to project success. However, as construction is heterogeneous in the nature of its organizations, project designs, time constraints, environmental effects, etc., site layout planning for each project becomes unique. Affected by many uncertainties variables! And variations, site layout planning is a typical multi objective problem.
Problem solving requires representing the problem in a language that problem solvers can understand. However, solutions of most construction problems rely on empirical Knowledge about the site layout that can be as a site space allocation for material storage, working areas, units of accommodation, plant positions, general circulation areas, and also access and egress for deliveries and emergency services. Furthermore, conflicting objectives and the uniqueness of construction projects like bad site layout make the problems difficult to conceptualize and define.
1) The document discusses project management tools including network analysis techniques like CPM and PERT.
2) CPM and PERT are used to plan and schedule complex projects using network diagrams that show the logical sequence and relationships of tasks.
3) PERT uses three time estimates for each activity - optimistic, most likely, and pessimistic - to calculate the expected duration using probability.
Introduction 1
Network is a technique used for planning and scheduling of large projects in the fields of construction, maintenance, fabrication, purchasing, computer system instantiation, research and development planning etc. There is multitude of operations research situations that can be modeled and solved as network. Some recent surveys reports that as much as 70% of the real-world mathematical programming problems can be represented by network related models. Network analysis is known by many names _PERT (Programme Evaluation and Review Technique), CPM (Critical Path Method), PEP (Programme Evaluation Procedure), LCES (Least Cost Estimating and Scheduling), SCANS (Scheduling and Control by Automated Network System), etc
This chapter will present three of algorithms.
1. PERT & CPM
2. Shortest- route algorithms
3. Maximum-flow algorithms
ASSIGNMENT III
B Tech 8th Semester
Submit it on or before 20th April 2020
01. Construct a network for each of the activities and their precedence relationships are given below:
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Design of steel structure as per is 800(2007)ahsanrabbani
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Network analysis cpm module3
1. DEPARTMENT OF CIVIL ENGINEERING
B TECH 8th
SEMESTER
01 1827 CONSTRUCTION PLANNING AND MANAGEMENT
L-T-P: 3-0-0 Credit: 3
Network analysis, PERT: Leveling of Resources
Total number of Lecture: 8
Introduction
CPM/PERT or Network Analysis as the technique is sometimes called,
developed along two parallel streams, one industrial and the other military.
CPM (Critical Path Method) was the discovery of M.R.Walker of E.I.Du
Pont de Nemours & Co. and J.E.Kelly of Remington Rand, circa 1957. The
computation was designed for the UNIVAC-I computer. The first test was
made in 1958, when CPM was applied to the construction of a new chemical
plant. In March 1959, the method was applied to maintenance shut-down at the
Du Pont works in Louisville, Kentucky. Unproductive time was reduced from
125 to 93 hours.
PERT (Project Evaluation and Review Technique) was devised in 1958 for
the POLARIS missile program by the Program Evaluation Branch of the
Special Projects office of the U.S.Navy, helped by the Lockheed Missile
Systems division and the Consultant firm of Booz-Allen & Hamilton. The
calculations were so arranged so that they could be carried out on the IBM
Naval Ordinance Research Computer (NORC) at Dahlgren, Virginia.
The methods are essentially network-oriented techniques using the same
principle. PERT and CPM are basically time-oriented methods in the sense that
they both lead to determination of a time schedule for the project. The
significant difference between two approaches is that the time estimates for the
different activities in CPM were assumed to be deterministic while in PERT
these are described probabilistically. These techniques are referred as project
scheduling techniques.
In CPM activities are shown as a network of precedence relationships using
activity-on- node network construction
– Single estimate of activity time
– Deterministic activity times
2. USED IN: Production management - for the jobs of repetitive in nature
where the activity time estimates can be predicted with considerable certainty
due to the existence of past experience.
In PERT activities are shown as a network of precedence relationships using
activity-on- arrow network construction
– Multiple time estimates
– Probabilistic activity times
USED IN: Project management - for non-repetitive jobs (research and
development work), where the time and cost estimates tend to be quite
uncertain. This technique uses probabilistic time estimates.
Benefits of PERT/CPM
Useful at many stages of project management
Mathematically simple
Give critical path and slack time
Provide project documentation
Useful in monitoring costs
Limitations of PERT/CPM
Clearly defined, independent and stable activities
Specified precedence relationships
Over emphasis on critical paths
Applications of CPM / PERT
These methods have been applied to a wide variety of problems in industries
and have found acceptance even in government organizations. These include
Construction of a dam or a canal system in a region
Construction of a building or highway
Maintenance or overhaul of airplanes or oil refinery
Space flight
Cost control of a project using PERT / COST
Designing a prototype of a machine
3. Development of supersonic planes
Basic Steps in PERT / CPM
Project scheduling by PERT / CPM consists of four main steps
1. Planning
The planning phase is started by splitting the total project in to small
projects. These smaller projects in turn are divided into activities and are
analyzed by the department or section.
The relationship of each activity with respect to other activities are defined
and established and the corresponding responsibilities and the authority are
also stated.
Thus the possibility of overlooking any task necessary for the completion of
the project is reduced substantially.
2. Scheduling
The ultimate objective of the scheduling phase is to prepare a time chart
showing the start and finish times for each activity as well as its relationship
to other activities of the project.
Moreover the schedule must pinpoint the critical path activities which
require special attention if the project is to be completed in time.
For non-critical activities, the schedule must show the amount of slack or
float times which can be used advantageously when such activities are
delayed or when limited resources are to be utilized effectively.
3. Allocation of resources
Allocation of resources is performed to achieve the desired objective. A
resource is a physical variable such as labour, finance, equipment and space
which will impose a limitation on time for the project.
When resources are limited and conflicting, demands are made for the same
type of resources a systematic method for allocation of resources become
essential.
Resource allocation usually incurs a compromise and the choice of this
compromise depends on the judgment of managers.
4. Controlling
4. The final phase in project management is controlling. Critical path methods
facilitate the application of the principle of management by expectation to
identify areas that are critical to the completion of the project.
By having progress reports from time to time and updating the network
continuously, a better financial as well as technical control over the project
is exercised.
Arrow diagrams and time charts are used for making periodic progress
reports. If required, a new course of action is determined for the remaining
portion of the project.
The Framework for PERT and CPM
Essentially, there are six steps which are common to both the techniques. The
procedure is listed below:
I. Define the Project and all of its significant activities or tasks. The Project (made
up of several tasks) should have only a single start activity and a single finish
activity.
II. Develop the relationships among the activities. Decide which
activities must precede and which must follow others.
III. Draw the "Network" connecting all the activities. Each Activity should
have unique event numbers. Dummy arrows are used where required to
avoid giving the same numbering to two activities.
IV. Assign time and/or cost estimates to each activity
V. Compute the longest time path through the network. This is called the
critical path.
VI. Use the Network to help plan, schedule, and monitor and control the project.
The Key Concept used by CPM/PERT is that a small set of activities, which
make up the longest path through the activity network control the entire
project. If these "critical" activities could be identified and assigned to
responsible persons, management resources could be optimally used by
concentrating on the few activities which determine the fate of the entire
project.
Non-critical activities can be replanned, rescheduled and resources for them
can be reallocated flexibly, without affecting the whole project. Five useful
questions to ask when preparing an activity network are:
Is this a Start Activity?
5. Is this a Finish Activity?
What Activity Precedes this?
What Activity Follows this?
What Activity is Concurrent with this?
Network Diagram Representation
In a network representation of a project certain definitions are used
1. Activity
Any individual operation which utilizes resources and has an end and a
beginning is called activity. An arrow is commonly used to represent an
activity with its head indicating the direction of progress in the project. These
are classified into four categories
1. Predecessor activity – Activities that must be completed immediately prior
to the start of another activity are called predecessor activities.
2. Successor activity – Activities that cannot be started until one or more of
other activities are completed but immediately succeed them are called
successor activities.
3. Concurrent activity – Activities which can be accomplished concurrently
are known as concurrent activities. It may be noted that an activity can be a
predecessor or a successor to an event or it may be concurrent with one or
more of other activities.
4. Dummy activity – An activity which does not consume any kind of
resource but merely depicts the technological dependence is called a
dummy activity.
The dummy activity is inserted in the network to clarify the activity pattern
in the following two situations
To make activities with common starting and finishing points distinguishable
To identify and maintain the proper precedence relationship between
activities that is not connected by events.
For example, consider a situation where A and B are concurrent activities. C
is dependent on A and D is dependent on A and B both. Such a situation can
be handled by using a dummy activity as shown in the figure.
6. 2. Event
An event represents a point in time signifying the completion of some activities
and the beginning of new ones. This is usually represented by a circle in a
network which is also called a node or connector.
The events are classified in to three categories
1. Merge event – When more than one activity comes and joins an event such
an event is known as merge event.
2. Burst event – When more than one activity leaves an event such an event is
known as burst event.
3. Merge and Burst event – An activity may be merge and burst event at the
same time as with respect to some activities it can be a merge event and
with respect to some other activities it may be a burst event.
3. Sequencing
The first prerequisite in the development of network is to maintain the
precedence relationships. In order to make a network, the following points
should be taken into considerations
What job or jobs precede it?
What job or jobs could run concurrently?
What job or jobs follow it?
What controls the start and finish of a job?
Since all further calculations are based on the network, it is necessary that a
network be drawn with full care.
Rules for Drawing Network Diagram
Rule 1
Each activity is represented by one and only one arrow in the network
7. Rule 2
No two activities can be identified by the same end events
Rule 3
In order to ensure the correct precedence relationship in the arrow diagram,
following questions must be checked whenever any activity is added to the
network
What activity must be completed immediately before this activity can start?
What activities must follow this activity?
What activities must occur simultaneously with this activity?
In case of large network, it is essential that certain good habits be practiced to draw an
easy to follow network
Try to avoid arrows which cross each other
Use straight arrows
Do not attempt to represent duration of activity by its arrow length
Use arrows from left to right. Avoid mixing two directions, vertical
and standing arrows may be used if necessary.
Use dummies freely in rough draft but final network should not have
any redundant dummies.
The network has only one entry point called start event and one point
of emergence called the end event.
Common Errors in Drawing Networks
The three types of errors are most commonly observed in drawing network diagrams
1. Dangling
To disconnect an activity before the completion of all activities in a network
diagram is known as dangling. As shown in the figure activities (5 – 10) and (6
– 7) are not the last activities in the network. So the diagram is wrong and
8. indicates the error of dangling
2. Looping or Cycling
Looping error is also known as cycling error in a network diagram. Drawing an
endless loop in a network is known as error of looping as shown in the
following figure.
3. Redundancy
Unnecessarily inserting the dummy activity in network logic is known as the
error of redundancy as shown in the following diagram
Advantages and Disadvantages
PERT/CPM has the following advantages
A PERT/CPM chart explicitly defines and makes visible dependencies
9. (precedence relationships) between the elements,
PERT/CPM facilitates identification of the critical path and makes this visible,
PERT/CPM facilitates identification of early start, late start, and slack for
each activity,
PERT/CPM provides for potentially reduced project duration due to better
understanding of dependencies leading to improved overlapping of
activities and tasks where feasible.
PERT/CPM has the following disadvantages:
There can be potentially hundreds or thousands of activities and individual
dependency relationships,
The network charts tend to be large and unwieldy requiring several pages to
print and requiring special size paper,
The lack of a timeframe on most PERT/CPM charts makes it harder to show
status although colours can help (e.g., specific colour for completed nodes),
When the PERT/CPM charts become unwieldy, they are no longer used to
manage the project.
Critical Path in Network Analysis
Basic Scheduling Computations
The notations used are
(i, j) = Activity with tail event i and head event j
Ei = Earliest occurrence time of event i
Lj = Latest allowable occurrence time of event j
Dij = Estimated completion time of activity (i, j)
(Es)ij = Earliest starting time of activity (i, j)
(Ef)ij = Earliest finishing time of activity (i, j)
(Ls)ij = Latest starting time of activity (i, j)
(Lf)ij = Latest finishing time of activity (i, j)
The procedure is as follows
1. Determination of Earliest time (Ej): Forward Pass computation
Step 1
The computation begins from the start node and move towards the end node.
For easiness, the forward pass computation starts by assuming the earliest
occurrence time of zero for the initial project event.
10. Step 2
i. Earliest starting time of activity (i, j) is the earliest event time of the tail
end event i.e. (Es)ij = Ei
ii. Earliest finish time of activity (i, j) is the earliest starting time + the
activity time i.e (Ef)ij = (Es)ij + Dij or (Ef)ij = Ei +Dij
iii. Earliest event time for event j is the maximum of the earliest finish times
of all activities ending in to that event i.e. Ej = max [(Ef)ij for all immediate
predecessor of (i, j)] or Ej =max [Ei + Dij]
2. Backward Pass computation (for latest allowable time)
Step 1
For ending event assume E = L. Remember that all E’s have been computed by
forward pass computations.
Step 2
Latest finish time for activity (i, j) is equal to the latest event time of event j i.e.
(Lf)ij = Lj
Step 3
Latest starting time of activity (i, j) = the latest completion time of (i, j) – the
activity time or (Ls)ij =(Lf)ij - Dij or (Ls)ij = Lj - Dij
Step 4
Latest event time for event ‘i’ is the minimum of the latest start time of all
activities originating from that event i.e. Li = min [(Ls)ij for all immediate
successor of (i, j)] = min [(Lf)ij - Dij] = min [Lj - Dij]
3. Determination of floats and slack times
There are three kinds of floats
Total float – The amount of time by which the completion of an activity
could be delayed beyond the earliest expected completion time without
affecting the overall project duration time.
Mathematically
(Tf)ij = (Latest start – Earliest start) for activity ( i – j)
(Tf)ij = (Ls)ij - (Es)ij or (Tf)ij = (Lj - Dij) - Ei
Free float – The time by which the completion of an activity can be delayed
11. beyond the earliest finish time without affecting the earliest start of a
subsequent activity.
Mathematically
(Ff)ij = (Earliest time for event j – Earliest time for event i) – Activity time for ( i,j)
(Ff)ij = (Ej - Ei) - Dij
Independent float – The amount of time by which the start of an activity
can be delayed without effecting the earliest start time of any immediately
following activities, assuming that the preceding activity has finished at its
latest finish time.
Mathematically
(If)ij = (Ej - Li) - Dij
The negative independent float is always taken as zero.
Event slack - It is defined as the difference between the latest event and
earliest event times.
Mathematically
Head event slack = Lj – Ej, Tail event slack = Li - Ei
4. Determination of critical path
Critical event – The events with zero slack times are called critical events.
In other words the event i is said to be critical if Ei = Li
Critical activity – The activities with zero total float are known as critical
activities. In other words an activity is said to be critical if a delay in its start
will cause a further delay in the completion date of the entire project.
Critical path – The sequence of critical activities in a network is called
critical path. The critical path is the longest path in the network from the
starting event to ending event and defines the minimum time required to
complete the project.
Worked Examples on CPM
Example 1
Determine the early start and late start in respect of all node points and
identify critical path for the following network.
12. Solution
Calculation of E and L for each node is shown in the network
Activity
(i, j)
Normal
Time
(Dij)
Earliest Time Latest Time Float Time
(Li - Dij ) - Ei
Start
(Ei)
Finish
(Ei + Dij )
Start
(Li - Dij )
Finish
(Li)
(1, 2) 10 0 10 0 10 0
(1, 3) 8 0 8 1 9 1
(1, 4) 9 0 9 1 10 1
(2, 5) 8 10 18 10 18 0
(4, 6) 7 9 16 10 17 1
(3, 7) 16 8 24 9 25 1
(5, 7) 7 18 25 18 25 0
(6, 7) 7 16 23 18 25 2
13. (5, 8) 6 18 24 18 24 0
(6, 9) 5 16 21 17 22 1
(7, 10) 12 25 37 25 37 0
(8, 10) 13 24 37 24 37 0
(9, 10) 15 21 36 22 37 1
Network Analysis Table
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
Example 2
Find the critical path and calculate the slack time for the following network
Solution
The earliest time and the latest time are obtained below
Activity
(i, j)
Normal
Time
(Dij)
Earliest Time Latest Time
Float Time
(Li - Dij ) - Ei
Start
(Ei)
Finish
(Ei + Dij )
Start
(Li - Dij )
Finish
(Li)
(1, 2) 2 0 2 5 7 5
(1, 3) 2 0 2 0 2 0
(1, 4) 1 0 1 6 7 6
14. (2, 6) 4 2 6 7 11 5
(3, 7) 5 2 7 3 8 1
(3, 5) 8 2 10 2 10 0
(4, 5) 3 1 4 7 10 6
(5, 9) 5 10 15 10 15 0
(6, 8) 1 6 7 11 12 5
(7, 8) 4 7 11 8 12 1
(8, 9) 3 11 14 12 15 1
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
Example 3
A project has the following times schedule
Activity Time (weeks) Activity Time (weeks) Activity Time (weeks)
(1 – 2) 4
(4 – 9) 5 (8 – 9)
1
(1 – 3)
1
(5 – 6)
4
(8 – 10)
8
(2 – 4)
1
(5 – 7) 8
(9 – 10) 7
(3 – 4)
1
(6 – 8) 1
Construct the network and compute
1. TE and TL for each event
16. (8 – 9) 1 17
18 0
(8 – 10) 8 17
25 0
(9 – 10)
7 18 25 0
The resultant network shows the critical path
The two critical paths are
i. 1 → 3 → 5 →7 → 8 → 9 →10
ii. 1 → 3 → 5 → 7 → 8 →10
Project Evaluation and Review Technique (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
Optimistic time – It is the shortest possible time in which the activity can be
finished. It assumes that every thing goes very well. This is denoted by t0.
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.
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.
17. This is denoted by tp.
In PERT calculation, all values are used to obtain the percent expected value.
1. 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 + 4 tm + tp ) / 6
2. The variance for the activity is given by
σ2
= [(tp – to) / 6] 2
Worked Examples
Example 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
Most likely time: 5 7 11 3 7 9 12 6 5 8 9
Find the earliest and latest expected time to each event and also
critical path in the network.
Solution
18. Task Least time (t0)
Greatest
time (tp)
Most likely
time (tm)
Expected time
(to + tp + 4tm)/6
A 4 8 5 5.33
B 5 10 7 7.17
C 8 12 11 10.67
D 2 7 3 3.5
E 4 10 7 7
F 6 15 9 9.5
G 8 16 12 12
H 5 9 6 6.33
I 3 7 5 5
J 5 11 8 8
K 6 13 9 9.17
`
Task
Expected
time (te)
Start Finish
Total float
Earliest Latest Earliest Latest
A 5.33 0 0 5.33 5.33 0
B 7.17 0 8.83 7.17 16 8.83
C 10.67 5.33 5.33 16 16 0
D 3.5 0 10 3.5 13.5 10
E 7 16 16 23 23 0
F 9.5 3.5 13.5 13 23 10
G 12 3.5 18.5 15.5 30.5 15
H 6.33 23 23 29.33 29.33 0
I 5 23 25.5 28 30.5 2.5
J 8 28 30.5 36 38.5 2.5
K 9.17 29.33 29.33 31.5 38.5 0
The network is
The critical path is A →C →E → H → K
19. Example 2
A project has the following characteristics
Activity
Most optimistic time (a) Most pessimistic time (b) Most likely time (m)
(1 – 2)
(2 – 3)
(2 – 4)
(3 – 5)
(4 – 5)
(4 – 6)
(5 – 7)
(6 – 7)
(7 – 8)
(7 – 9)
(8 – 10)
(9 – 10)
1
1
1
3
2
3
4
6
2
5
1
3
5
3
5
5
4
7
6
8
6
8
3
7
1.5
2
3
4
3
5
5
7
4
6
2
5
20. Construct a PERT network. Find the critical path and variance for each event.
Solution
Activity (a) (b) (m) (4m)
te
(a + b + 4m)/6
V
[(b – a) / 6]2
(1 – 2)
(2 – 3)
(2 – 4)
(3 – 5)
(4 – 5)
(4 – 6)
(5 – 7)
(6 – 7)
(7 – 8)
(7 – 9)
(8 – 10)
(9 – 10)
1
1
1
3
2
3
4
6
2
5
1
3
5
3
5
5
4
7
6
8
6
8
3
7
1.5
2
3
4
3
5
5
7
4
6
2
5
6
8
12
16
12
20
20
28
16
24
8
20
2
2
3
4
3
5
5
7
4
6.17
2
5
4/9
1/9
4/9
1/9
1/9
4/9
1/9
1/9
4/9
1/4
1/9
4/9
The network is constructed as shown below
The critical path = 1 → 2 → 4 → 6 → 7 →9 →10
Example 3
Calculate the variance and the expected time for each activity
Solution
24. 3. A project schedule has the following characteristics
Activity Most optimistic time Most likely time Most pessimistic
time
(1 – 2)
(2 – 3)
(2 – 4)
(3 – 5)
(4 – 5)
(4 – 6)
(5 – 7)
(6 – 7)
(7 – 8)
(7 – 9)
(8 – 10)
(9 – 10)
1
1
1
3
2
3
4
6
2
4
1
3
2
2
3
4
5
5
5
7
4
6
2
5
3
3
5
5
4
7
6
8
6
8
3
7
Construct a PERT network and find
out
a. The earliest possible time
b. Latest allowable time
c. Slack values
d. Critical path
4. Explain the following terms
a. optimistic time
b. Most likely time
25. c. Pessimistic time
d. Expected time
e. Variance
5. Calculate the variance and the expected time for each activity
Exercise
1. What is PERT and CPM?
2. What are the advantages of using PERT/CPM?
3. Mention the applications of PERT/CPM
4. Explain the following terms
a. Earliest time
b. Latest time
c. Total activity slack
d. Event slack
e. Critical path
5. Explain the CPM in network analysis.
6. What are the rules for drawing network diagram? Also mention the
common errors that occur in drawing networks.
7. What is the difference between PERT and CPM/
8. What are the uses of PERT and CPM?
9. Explain the basic steps in PERT/CPM techniques.
10. Write the framework of PERT/CPM.