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<title>Critical path method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Critical path method</span></span>

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<p>The <b>critical path method</b> (<b>CPM</b>), or <b>critical path analysis</b> (<b>CPA</b>), is an <a href="Algorithm" title="Algorithm">algorithm</a> for <a href="Schedule_(project_management)" title="Schedule (project management)">scheduling</a> a set of project activities.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A critical path is determined by identifying the longest stretch of dependent activities and measuring the time<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> required to complete them from start to finish. It is commonly used in conjunction with the <a href="Program_evaluation_and_review_technique" title="Program evaluation and review technique">program evaluation and review technique</a> (PERT).

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The CPM is a project-modeling technique developed in the late 1950s by Morgan R. Walker of <a href="DuPont" title="DuPont">DuPont</a> and James E. Kelley Jr. of <a href="Remington_Rand" title="Remington Rand">Remington Rand</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Kelley and Walker related their memories of the development of CPM in 1989.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Kelley attributed the term "critical path" to the developers of the PERT, which was developed at about the same time by <a href="Booz_Allen_Hamilton" title="Booz Allen Hamilton">Booz Allen Hamilton</a> and the <a href="U.S._Navy" class="mw-redirect" title="U.S. Navy">U.S. Navy</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The precursors of what came to be known as critical path were developed and put into practice by DuPont between 1940 and 1943 and contributed to the success of the <a href="Manhattan_Project" title="Manhattan Project">Manhattan Project</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>Critical path analysis is commonly used with all forms of projects, including construction, aerospace and defense, software development, research projects, product development, engineering, and plant maintenance, among others. Any project with interdependent activities can apply this method of mathematical analysis. CPM was used for the first time in 1966 for the major skyscraper development of constructing the former <a href="World_Trade_Center_Twin_Towers" class="mw-redirect" title="World Trade Center Twin Towers">World Trade Center Twin Towers</a> in New York City. Although the original CPM program and approach is no longer used,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> the term is generally applied to any approach used to analyze a project network logic diagram.

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<div class="mw-heading mw-heading2"><h2 id="Basic_techniques">Basic techniques</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Components">Components</h3></div>

<p>The essential technique for using CPM<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> is to construct a model of the project that includes:

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<ol><li>A list of all activities required to complete the project (typically categorized within a <a href="Work_breakdown_structure" title="Work breakdown structure">work breakdown structure</a>)</li>

<li>The time (duration) that each activity will take to complete</li>

<li>The dependencies between the activities</li>

<li>Logical end points such as milestones or <a href="Deliverable" title="Deliverable">deliverable</a> items</li></ol>

<p>Using these values, CPM calculates the <a href="Longest_path" class="mw-redirect" title="Longest path">longest path</a> of planned activities to logical end points or to the end of the project, and the earliest and latest that each activity can start and finish without making the project longer. This process determines which activities are "critical" (i.e., on the longest path) and which have no float/slack or "total float" zero (i.e., can not be delayed without making the project longer). In project management, a critical path is the sequence of project network activities that adds up to the longest overall duration, regardless of whether that longest duration has float or not. This determines the shortest time possible to complete the project. "Total float" (unused time) can occur within the critical path. For example, if a project is testing a solar panel and <a href="Task_(project_management)" title="Task (project management)">task</a> 'B' requires 'sunrise', a scheduling constraint on the testing activity could be that it would not start until the scheduled time for sunrise. This might insert dead time (total float) into the schedule on the activities on that path prior to the sunrise due to needing to wait for this event. This path, with the constraint-generated total float, would actually make the path longer, with total float being part of the shortest possible duration for the overall project. In other words, individual tasks on the critical path prior to the constraint might be able to be delayed without elongating the critical path; this is the total float of that task, but the time added to the project duration by the constraint is actually <a href="Critical_path_drag" title="Critical path drag">critical path drag</a>, the amount by which the project's duration is extended by each critical path activity and constraint.

</p><p>A project can have several, parallel, near-critical paths, and some or all of the tasks could have free float and/or total float. An additional parallel path through the network with the total durations shorter than the critical path is called a subcritical or noncritical path. Activities on subcritical paths have no drag, as they are not extending the project's duration.

</p><p>CPM analysis tools allow a user to select a logical end point in a project and quickly identify its longest series of dependent activities (its longest path). These tools can display the critical path (and near-critical path activities if desired) as a cascading waterfall that flows from the project's start (or current status date) to the selected logical end point.

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<div class="mw-heading mw-heading3"><h3 id="Visualizing_critical_path_schedule">Visualizing critical path schedule</h3></div>

<p>Although the activity-on-arrow diagram (PERT chart) is still used in a few places, it has generally been superseded by the activity-on-node diagram, where each activity is shown as a box or node and the arrows represent the logical relationships going from predecessor to successor as shown here in the "Activity-on-node diagram".

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<p>In this diagram, Activities A, B, C, D, and E comprise the critical or longest path, while Activities F, G, and H are off the critical path with floats of 15 days, 5 days, and 20 days respectively. Whereas activities that are off the critical path have float and are therefore not delaying completion of the project, those on the critical path will usually have critical path drag, i.e., they delay project completion. The drag of a critical path activity can be computed using the following formula:

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<ol><li>If a critical path activity has nothing in parallel, its drag is equal to its duration. Thus A and E have drags of 10 days and 20 days respectively.</li>

<li>If a critical path activity has another activity in parallel, its drag is equal to whichever is less: its duration or the total float of the parallel activity with the least total float. Thus since B and C are both parallel to F (float of 15) and H (float of 20), B has a duration of 20 and drag of 15 (equal to F's float), while C has a duration of only 5 days and thus drag of only 5. Activity D, with a duration of 10 days, is parallel to G (float of 5) and H (float of 20) and therefore its drag is equal to 5, the float of G.</li></ol>

<p>These results, including the drag computations, allow managers to prioritize activities for the effective management of project, and to shorten the planned critical path of a project by pruning critical path activities, by "fast tracking" (i.e., performing more activities in parallel), and/or by "crashing the critical path" (i.e., shortening the durations of critical path activities by adding <a href="Resource_(project_management)" title="Resource (project management)">resources</a>).

</p><p>Critical path drag analysis has also been used to optimize schedules in processes outside of strict project-oriented contexts, such as to increase manufacturing throughput by using the technique and metrics to identify and alleviate delaying factors and thus reduce assembly lead time.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Crash_duration">Crash duration</h3></div>

<p>"Crash duration" is a term referring to the shortest possible time for which an activity can be scheduled.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It can be achieved by shifting more resources towards the completion of that activity, resulting in decreased time spent and often a reduced quality of work, as the premium is set on speed.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

Crash duration is typically modeled as a linear relationship between cost and activity duration, but in many cases, a <a href="Convex_function" title="Convex function">convex function</a> or a <a href="Step_function" title="Step function">step function</a> is more applicable.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Expansion">Expansion</h3></div>

<p>Originally, the critical path method considered only logical <a href="Dependency_(project_management)" title="Dependency (project management)">dependencies</a> between terminal elements. Since then, it has been expanded to allow for the inclusion of resources related to each activity, through processes called activity-based resource assignments and resource optimization techniques such as <a href="Resource_Leveling" class="mw-redirect" title="Resource Leveling">Resource Leveling</a> and <a href="Resource_smoothing" title="Resource smoothing">Resource smoothing</a>. A resource-leveled schedule may include delays due to resource bottlenecks (i.e., unavailability of a resource at the required time), and may cause a previously shorter path to become the longest or most "resource critical" path while a resource-smoothed schedule avoids impacting the critical path by using only free and total float.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> A related concept is called the <a href="Critical_chain" class="mw-redirect" title="Critical chain">critical chain</a>, which attempts to protect activity and project durations from unforeseen delays due to resource constraints.

</p><p>Since project schedules change on a regular basis, CPM allows continuous monitoring of the schedule, which allows the <a href="Project_manager" title="Project manager">project manager</a> to track the critical activities, and alerts the project manager to the possibility that non-critical activities may be delayed beyond their total float, thus creating a new critical path and delaying project completion. In addition, the method can easily incorporate the concepts of stochastic predictions, using the PERT and <a href="Event_chain_methodology" title="Event chain methodology">event chain methodology</a>.

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<div class="mw-heading mw-heading3"><h3 id="Flexibility">Flexibility</h3></div>

<p>A schedule generated using the critical path techniques often is not realized precisely, as <a href="Estimation_(project_management)" title="Estimation (project management)">estimations</a> are used to calculate times: if one mistake is made, the results of the analysis may change. This could cause an upset in the implementation of a project if the estimates are blindly believed, and if changes are not addressed promptly. However, the structure of critical path analysis is such that the variance from the original schedule caused by any change can be measured, and its impact either <a href="https://en.wiktionary.org/wiki/ameliorated" class="extiw external" title="wikt:ameliorated">ameliorated</a> or adjusted for. Indeed, an important element of project postmortem analysis is the 'as built critical path' (ABCP), which analyzes the specific causes and impacts of changes between the planned schedule and eventual schedule as actually implemented.

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<div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2></div>

<p>Critical path techniques are widely used in planning, managing and controlling the delivery of <a href="Construction" title="Construction">construction</a> projects. A technique known as "as-built critical path analysis" can also be used to assess the causes of a <a href="Construction_delay" title="Construction delay">delay in completing a project</a>, especially where there may have been more than one delaying factor and liability needs to be established for compensation and <a href="Damages" title="Damages">damages</a> purposes. However, the use of as-built CPA in a legal context has been criticised, for example in the <a href="Courts_of_Scotland" title="Courts of Scotland">Scottish court</a> case of <i>City Inn Ltd. v Shepherd Construction</i> (2007).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>Currently, there are several <a href="Software" title="Software">software</a> solutions available in industry which use the CPM method of scheduling; see <a href="List_of_project_management_software" class="mw-redirect" title="List of project management software">list of project management software</a>. The method currently used by most project management software is based on a manual calculation approach developed by Fondahl of Stanford University.

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<div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2></div>

<ul><li>In <i><a href="Odds_On" title="Odds On">Odds On</a></i>, the first novel by <a href="Michael_Crichton" title="Michael Crichton">Michael Crichton</a>, robbers use a critical path computer program to help plan a heist.</li>

<li><i><a href="The_Nome_Trilogy" title="The Nome Trilogy">The Nome Trilogy</a></i> (in the first book, <i>Truckers</i>) by <a href="Terry_Pratchett" title="Terry Pratchett">Terry Pratchett</a> mentions "something called <i>critical path analysis</i>" and says that it means "there's always something you should have done first."</li></ul>

<ul><li><a href="Critical_chain_project_management" title="Critical chain project management">Critical chain project management</a> – Method of planning and managing projects</li>

<li><a href="Gantt_chart" title="Gantt chart">Gantt chart</a> – Type of chart to show a project schedule</li>

<li><a href="Graphical_Evaluation_and_Review_Technique" title="Graphical Evaluation and Review Technique">Graphical Evaluation and Review Technique (GERT)</a> – Project management tool for the probabilistic treatment of activity duration and project structure</li>

<li><a href="Liebig's_law_of_the_minimum" title="Liebig's law of the minimum">Liebig's law of the minimum</a> – Growth is limited by the scarcest resource</li>

<li><a href="List_of_project_management_software" class="mw-redirect" title="List of project management software">List of project management software</a></li>

<li><a href="List_of_project_management_topics" class="mw-redirect" title="List of project management topics">List of project management topics</a></li>

<li><a href="Main_path_analysis" title="Main path analysis">Main path analysis</a> – Mathematical tool</li>

<li><a href="Program_evaluation_and_review_technique" title="Program evaluation and review technique">Program evaluation and review technique (PERT)</a> – Statistical tool used in project management</li>

<li><a href="Project_management" title="Project management">Project management</a> – Practice of leading the work of a team to achieve goals and criteria at a specified time</li>

<li><a href="Project_planning" title="Project planning">Project planning</a> – Aspect of project management which uses schedules to plan and report progress</li>

<li><a href="Work_breakdown_structure" title="Work breakdown structure">Work breakdown structure</a> – Deliverable-orientated breakdown of a project into smaller components</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKelley" class="citation book cs1">Kelley, James. <i>Critical Path Planning</i>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSantiago2009" class="citation web cs1">Santiago, Jesse (February 4, 2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181024032348/http://web.stanford.edu/class/cee320/CEE320B/CPM.pdf">"Critical Path Method"</a> <span class="cs1-format">(PDF)</span>. <i>Stanford</i>. Archived from <a rel="nofollow" class="external text" href="https://web.stanford.edu/class/cee320/CEE320B/CPM.pdf">the original</a> <span class="cs1-format">(PDF)</span> on October 24, 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">October 24,</span> 2018</span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFKelleyWalker1959" class="citation book cs1">Kelley, James E.; Walker, Morgan R. (1959). "Critical-path planning and scheduling". <i>Papers presented at the December 1-3, 1959, eastern joint IRE-AIEE-ACM computer conference on - IRE-AIEE-ACM '59 (Eastern)</i>. pp. <span class="nowrap">160–</span>173. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1460299.1460318">10.1145/1460299.1460318</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4503-7868-0</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFKelleyWalker1989" class="citation journal cs1">Kelley, James; Walker, Morgan (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jS9aAAAAYAAJ">"The Origins of CPM: A Personal History"</a>. <i>The PM Net Work</i>. <b>3</b> (2): <span class="nowrap">7–</span>22.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFNewellGrashina2004" class="citation book cs1">Newell, Michael W.; Grashina, Marina N. (2004). <i>The Project Management Question and Answer Book</i>. AMACOM, American Management Association. p. 98. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8144-7164-7</bdi>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFThayer1996" class="citation book cs1">Thayer, Harry (1996). <i>Management of the Hanford Engineer Works in World War II</i>. pp. <span class="nowrap">66–</span>67. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2F9780784401606">10.1061/9780784401606</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7844-0160-6</bdi>.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">A Brief History of Scheduling: <a rel="nofollow" class="external text" href="http://www.mosaicprojects.com.au/PDF_Papers/P042_History%20of%20Scheduing.pdf">mosaic projects.com.au</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150518071346/http://www.mosaicprojects.com.au/PDF_Papers/P042_History%20of%20Scheduing.pdf">Archived</a> May 18, 2015, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Samuel L. Baker, Ph.D. <a rel="nofollow" class="external text" href="http://hspm.sph.sc.edu/COURSES/J716/CPM/CPM.html">"Critical Path Method (CPM)"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100612142236/http://hspm.sph.sc.edu/COURSES/J716/CPM/CPM.html">Archived</a> June 12, 2010, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <i>University of South Carolina</i>, Health Services Policy and Management Courses</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFArmstrong-Wright,_MICE1969" class="citation book cs1">Armstrong-Wright, MICE, A. T. (1969). <i>Critical Path Method: Introduction and Practice</i>. London: Longman Group LTD. p. 5ff.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFSedore2014" class="citation thesis cs1">Sedore, Blake William Clark (2014). <i>Assembly lead time reduction in a semiconductor capital equipment plant through constraint based scheduling</i> (Thesis). <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F93851">1721.1/93851</a></span>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFHendricksonTung2008" class="citation book cs1"><a href="Chris_T._Hendrickson" class="mw-redirect" title="Chris T. Hendrickson">Hendrickson, Chris</a>; Tung, Au (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170324031153/http://pmbook.ce.cmu.edu/">"11. Advanced Scheduling Techniques"</a>. <a rel="nofollow" class="external text" href="http://pmbook.ce.cmu.edu/"><i>Project Management for Construction</i></a> (2.2 ed.). Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-731266-5</bdi>. Archived from <a rel="nofollow" class="external text" href="http://pmbook.ce.cmu.edu/11_Advanced_Scheduling_Techniques.html">the original</a> on March 24, 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">October 27,</span> 2011</span>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrooks1975" class="citation book cs1">Brooks, F.P. (1975). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mythicalmanmonth00broo"><i>The Mythical Man-Month</i></a></span>. Reading, MA: Addison Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-00650-6</bdi>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFHendricksonB.N._Janson1984" class="citation journal cs1">Hendrickson, C.; B.N. Janson (1984). "A Common Network Flow Formulation for Several Civil Engineering Problems". <i>Civil Engineering Systems</i>. 4. <b>1</b> (4): <span class="nowrap">195–</span>203. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984CEngS...1..195H">1984CEngS...1..195H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02630258408970343">10.1080/02630258408970343</a>.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite class="citation book cs1">"6.5.2.3 Resource Optimization". <i>A Guide to the Project Management Body of Knowledge (PMBOK® Guide)</i> (6th ed.). <a href="Project_Management_Institute" title="Project Management Institute">Project Management Institute</a>. 2017. p. 720. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-62825-382-5</bdi>.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Pease, I., <a rel="nofollow" class="external text" href="http://www.primafacie.biz/resources/articles/080121-plj202-p11-14.pdf">Delaying tactics</a>, <i>Property Law Journal</i>, published on 21 January 2008, accessed on 5 December 2024</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li><cite id="CITEREFEmmanuel_Asanga2013" class="citation book cs1">Emmanuel Asanga (2013). <i>A Guide To The Project Management Body Of Knowledge</i> (5th ed.). Project Management Institute. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-935589-67-9</bdi>.</cite></li>

<li><cite id="CITEREFDevaux2014" class="citation book cs1">Devaux, Stephen A. (2014). <i>Managing Projects as Investments: Earned Value to Business Value</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4822-1270-9</bdi>.</cite></li>

<li><cite id="CITEREFDevaux2015" class="citation book cs1">Devaux, Stephen A. (2015). <i>Total Project Control (2nd Edition): A Practitioner's Guide to Managing Projects as Investments</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4987-0677-3</bdi>.</cite></li>

<li><cite id="CITEREFHeerkens2001" class="citation book cs1">Heerkens, Gary (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/projectmanagemen00gary"><i>Project Management (The Briefcase Book Series)</i></a></span>. McGraw–Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-137952-5</bdi>.</cite></li>

<li><cite id="CITEREFKerzner2003" class="citation book cs1"><a href="Harold_Kerzner" title="Harold Kerzner">Kerzner, Harold</a> (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/projectmanagemen0000kerz_h0w8"><i>Project Management: A Systems Approach to Planning, Scheduling, and Controlling</i></a></span> (8th ed.). Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-22577-0</bdi>.</cite></li>

<li><cite id="CITEREFAtali2020" class="citation book cs1">Atali, Ozhan (2020). <i>Data-Driven Project Management: Spreadsheets and Finance</i>. Wremia Project Management. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-578-67030-0</bdi>.</cite></li>

<li><cite id="CITEREFKlastorin2003" class="citation book cs1">Klastorin, Ted (2003). <i>Project Management: Tools and Trade-offs</i> (3rd ed.). Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-41384-4</bdi>.</cite></li>

<li><cite id="CITEREFLewis2002" class="citation book cs1">Lewis, James (2002). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofpr00lewi_1"><i>Fundamentals of Project Management</i></a></span> (2nd ed.). American Management Association. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8144-7132-3</bdi>.</cite></li>

<li><cite id="CITEREFMalakooti2013" class="citation book cs1">Malakooti, B (2013). <i>Operations and Production Systems with Multiple Objectives</i>. John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-58537-5</bdi>.</cite></li>

<li><cite id="CITEREFMilosevic2003" class="citation book cs1">Milosevic, Dragan Z. (2003). <i>Project Management ToolBox: Tools and Techniques for the Practicing Project Manager</i>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-20822-8</bdi>.</cite></li>

<li><cite id="CITEREFO'BrienPlotnick2010" class="citation book cs1">O'Brien, James J.; Plotnick, Fredric L. (2010). <i>CPM in Construction Management, Seventh Edition</i>. McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-163664-3</bdi>.</cite></li>

<li><cite id="CITEREFTraunerManginelliLoweNagata2009" class="citation book cs1">Trauner; Manginelli; Lowe; Nagata; Furniss (2009). <i>Construction Delays, 2nd Ed.: Understanding Them Clearly, Analyzing Them Correctly</i>. Burlington, MA: Elsevier. p. 266. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-85617-677-4</bdi>.</cite></li>

<li><cite id="CITEREFWoolf2012" class="citation book cs1">Woolf, Murray B. (2012). <i>CPM Mechanics: The Critical Path Method of Modeling Project Execution Strategy</i>. ICS-Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-9854091-0-4</bdi>.</cite></li>

<li><cite id="CITEREFWoolf2007" class="citation book cs1">Woolf, Murray B. (2007). <i>Faster Construction Projects with CPM Scheduling</i>. McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-148660-6</bdi>.</cite></li></ul>

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</style><div id="Schedule_network_analysis29" style="font-size:114%;margin:0 4em"><a href="Schedule_network_analysis" title="Schedule network analysis">Schedule network analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Scenario_analysis" class="mw-redirect" title="Scenario analysis">Scenario analysis</a></li>

<li><a href="Critical_chain_project_management" title="Critical chain project management">Critical chain project management</a></li>

<li><a href="Resource_leveling" title="Resource leveling">Resource leveling</a></li>

<li><a href="Resource_smoothing" title="Resource smoothing">Resource smoothing</a></li>

<li><a href="Schedule_compression" title="Schedule compression">Schedule compression</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="noviewer" typeof="mw:File"></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:CPM_diagrams" class="extiw external" title="commons:Category:CPM diagrams">CPM diagrams</a> at Wikimedia Commons</li></ul>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q825377#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata726" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q825377#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata726" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85034138">United States</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00576449">Japan</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007533519505171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/7ed7a45c-00ab-4956-86ee-9fc5c98efc55">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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