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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.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/ence-603-management-science-applications-in-project-management-lectures-5-7-project-management-lp-models-in-scheduling-integer-programming/656168/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/ence-603-management-science-applications-in-project-management-lectures-5-7-project-management-lp-models-in-scheduling-integer-programming/656168.png","ImageObject",300,407,{"name":92,"@type":93},"Putri","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-25",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the main purposes of CPM in project scheduling?","Question",{"text":112,"@type":113},"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.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do AON and AOA network models differ?",{"text":117,"@type":113},"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.",{"name":119,"@type":110,"acceptedAnswer":120},"How are earliest and latest times computed in network scheduling?",{"text":121,"@type":113},"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.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},656168,1793073881,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":21,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},962085571259,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","ENCE 603  \nManagement Science Applications in Project  \nManagement  \nLectures 5-7  \nProject Management LP Models  \nin Scheduling, Integer Programming  \nSpring 2009  \nInstructor: Dr. Steven A. Gabriel [www.eng.umd.edu/](www.eng.umd.edu/)~sgabriel  \n1  \nCopyright 2008, Dr. Steven A. Gabriel  \nOutline  \n• Project Scheduling  \n• Critical Path Method (CPM)  \n• AON and AOA methods  \n• Project Crashing  \n• Precedence Diagramming Method (PDM)  \n• Gantt Charts  \n2  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \n• Project activities described by a network  \n• Can use the activity-on-node (AON) model  \n• Nodes are activities, arrows (arcs) indicate the precedence relationships  \n• Could also consider the activity-on-arc (AOA) model which has arcs for activities with nodes being the starting and ending points  \n• AON used frequently in practical, non-optimization situations, AOA is used in optimization settings  \n• First AON, then AOA  \n• Main idea for both is to determine the critical path (e.g., tasks whose delay will cause a delay for the whole project)  \n3  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \n• Sample project network (AON) (read left to right)  \n• Dashed lines indicate dummy activities  \n• Key: Activity, Duration (days)  \n4  \nCopyright 2008, Dr. Steven A. Gabriel  \nNetwork Analysis  \n• Network Scheduling:  \n• Main purpose of CPM is to determine the “critical path”  \n• Critical path determines the minimum completion time for a project  \n• Use forward pass and backward pass routines to analyze the project network  \n• Network Control:  \n• Monitor progress of a project on the basis of the network schedule  \n• Take correction action when required  \n• “Crashing” the project  \n• Penalty/reward approach  \n5  \nCopyright 2008, Dr. Steven A. Gabriel  \nActivity on Node (AON) Representation of Project  \nNetworks  \n6  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \nA: Activity identification (node) ES: Earliest starting time  \nEC: Earliest completion time  \nLS: Latest starting time  \nLC: Latest completion time  \nt: Activity duration  \nP(A): set of predecessor nodes to node A  \nS(A): set of successor nodes to node A  \n7  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \n• In tabular form  \nActivity Predecessor Duration A n/a 2  \nB n/a 6  \nC n/a 4  \nD A 3  \nE C 5  \nF A 4  \nG B,D,E 2  \nSample Computations  \nES(A) =Max{EC(j), j in P(A)}=EC(start)=0  \nEC(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.  \nstart  \nA,2  \nB,6  \nC,4  \nD,3  \nE,5  \nF,4  \nG,2  \nend  \n8  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \n• Notation: Above node ES(i), EC(i), below node LS(i),LC(i)  \n• Zero project slack convention in force  \n0,2 2,6  \nA,2  \n4,6 0,0  \nstart  \n0,4 0,0  \nC,4  \n0,4  \nF,4  \n0,6  \nB,6  \n3,9  \n2,5 D,3  \n6,9  \n4,9  \nE,5  \n7,11 9,11  \nG,2  \n9,11  \n11,11  \nend 11,11  \n4,9  \nSample Computations  \nLC(F) =Min{LS(i), i in S(F))}=11 LS(F)=LC(F)-tF=11-4=7  \netc.  \n9  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \n• During the forward pass, it is assumed that each activity will begin at its earliest starting time  \n• An activity can begin as soon as the last of its predecessors has finished  \nC must wait for both A and B to finish before it can start  \nC  \nCompletion of the forward pass determines the earliest completion time of the project  \n• During the backward pass, it is assumed that each activity begins at its latest completion time  \n• Each activity ends at the latest starting time of the first activity in the project network  \n10  \nCopyright 2008, Dr. Steven A. Gabriel  \nProject Networks  \n• Note:  \n1=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  \n• P(3)= {1,2}  \n• S(3)= {4,5}  \nRule 1: ES(1)=0 (unless otherwise stated)  \nRule 2: ES(i)=Max j in P(i) {EC(j)}  \n• Why do we use “max” of the predecessor EC’s in rule 2?  \n","cbCaieJgIeJGPIzI","https://ap.wps.com/l/cbCaieJgIeJGPIzI","pdf",441599,"English","# Outline\n## Project Scheduling and Critical Path Method (CPM)\n## Project Networks: AON and AOA Models\n## Network Analysis: Scheduling and Control\n## Activity Time Notation and Sample Computations\n## Forward Pass and Backward Pass Rules\n## Slack, Critical Activities, and CPM Steps","[{\"question\":\"What are the main purposes of CPM in project scheduling?\",\"answer\":\"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.\"},{\"question\":\"How do AON and AOA network models differ?\",\"answer\":\"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.\"},{\"question\":\"How are earliest and latest times computed in network scheduling?\",\"answer\":\"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.\"}]","ENCE 603 - Management Science Applications in Project Management - Lectures 5-7 - Project Management LP Models in Scheduling, Integer Programming | PDF",1792966827,176]