ENCE 603 - Management Science Applications in Project Management - Lectures 5-7 - Project Management LP Models in Scheduling, Integer Programming

Oct 25, 2026

Upload by Putri

Lecture material for ENCE 603 covering project network scheduling using activity-on-node (AON) and activity-on-arc (AOA) models. Focuses on critical path determination via CPM, network analysis through forward and backward passes, and project control concepts including monitoring progress, correction actions, and project crashing using penalty/reward ideas. Introduces key notations (ES, EC, LS, LC, slack) and shows sample computations for earliest/latest times and critical activities.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages70

Reading Time2 h 56 min

AccessFree to download

Outline

Project Scheduling and Critical Path Method (CPM)

Project Networks: AON and AOA Models

Network Analysis: Scheduling and Control

Activity Time Notation and Sample Computations

Forward Pass and Backward Pass Rules

Slack, Critical Activities, and CPM Steps

ENCE 603 Management Science Applications in Project Management Lectures 5-7 Project Management LP Models in Scheduling, Integer Programming Spring 2009 Instructor: Dr. Steven A. Gabriel [www.eng.umd.edu/](www.eng.umd.edu/)~sgabriel 1 Copyright 2008, Dr. Steven A. Gabriel Outline • Project Scheduling • Critical Path Method (CPM) • AON and AOA methods • Project Crashing • Precedence Diagramming Method (PDM) • Gantt Charts 2 Copyright 2008, Dr. Steven A. Gabriel Project Networks • Project activities described by a network • Can use the activity-on-node (AON) model • Nodes are activities, arrows (arcs) indicate the precedence relationships • Could also consider the activity-on-arc (AOA) model which has arcs for activities with nodes being the starting and ending points • AON used frequently in practical, non-optimization situations, AOA is used in optimization settings • First AON, then AOA • Main idea for both is to determine the critical path (e.g., tasks whose delay will cause a delay for the whole project) 3 Copyright 2008, Dr. Steven A. Gabriel Project Networks • Sample project network (AON) (read left to right) • Dashed lines indicate dummy activities • Key: Activity, Duration (days) 4 Copyright 2008, Dr. Steven A. Gabriel Network Analysis • Network Scheduling: • Main purpose of CPM is to determine the “critical path” • Critical path determines the minimum completion time for a project • Use forward pass and backward pass routines to analyze the project network • Network Control: • Monitor progress of a project on the basis of the network schedule • Take correction action when required • “Crashing” the project • Penalty/reward approach 5 Copyright 2008, Dr. Steven A. Gabriel Activity on Node (AON) Representation of Project Networks 6 Copyright 2008, Dr. Steven A. Gabriel Project Networks A: Activity identification (node) ES: Earliest starting time EC: Earliest completion time LS: Latest starting time LC: Latest completion time t: Activity duration P(A): set of predecessor nodes to node A S(A): set of successor nodes to node A 7 Copyright 2008, Dr. Steven A. Gabriel Project Networks • In tabular form Activity Predecessor Duration A n/a 2 B n/a 6 C n/a 4 D A 3 E C 5 F A 4 G B,D,E 2 Sample Computations ES(A) =Max{EC(j), j in P(A)}=EC(start)=0 EC(A)=ES(A)+tA=0+2=2 ES(B)= EC(start)=0 EC(B)=ES(B)+ tB=0+6=6 ES(F)= EC(A)=2 EC(F)= ES(F)+4=6, etc. start A,2 B,6 C,4 D,3 E,5 F,4 G,2 end 8 Copyright 2008, Dr. Steven A. Gabriel Project Networks • Notation: Above node ES(i), EC(i), below node LS(i),LC(i) • Zero project slack convention in force 0,2 2,6 A,2 4,6 0,0 start 0,4 0,0 C,4 0,4 F,4 0,6 B,6 3,9 2,5 D,3 6,9 4,9 E,5 7,11 9,11 G,2 9,11 11,11 end 11,11 4,9 Sample Computations LC(F) =Min{LS(i), i in S(F))}=11 LS(F)=LC(F)-tF=11-4=7 etc. 9 Copyright 2008, Dr. Steven A. Gabriel Project Networks • During the forward pass, it is assumed that each activity will begin at its earliest starting time • An activity can begin as soon as the last of its predecessors has finished C must wait for both A and B to finish before it can start C Completion of the forward pass determines the earliest completion time of the project • During the backward pass, it is assumed that each activity begins at its latest completion time • Each activity ends at the latest starting time of the first activity in the project network 10 Copyright 2008, Dr. Steven A. Gabriel Project Networks • Note: 1=first node (activity),n=last node,i,j=arbitrary nodes, P(i)= immediate predecessors of node i, S(j)= immediate successors of node j, Tp=project deadline time • P(3)= {1,2} • S(3)= {4,5} Rule 1: ES(1)=0 (unless otherwise stated) Rule 2: ES(i)=Max j in P(i) {EC(j)} • Why do we use “max” of the predecessor EC’s in rule 2?

FAQs

What are the main purposes of CPM in project scheduling?

CPM determines the project’s critical path and the minimum completion time. It uses forward-pass and backward-pass routines to analyze the project network.

How do AON and AOA network models differ?

In AON, nodes represent activities and arrows represent precedence relationships. In AOA, activities are represented by arcs while nodes represent the start and end points.

How are earliest and latest times computed in network scheduling?

During the forward pass, each activity starts at its earliest starting time and waits for predecessor completion, using ES and EC rules. During the backward pass, each activity is analyzed from latest completion backwards using LC and LS rules.

Oct 25, 2026

Upload by Putri