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open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"PERT" redirects here. For other uses, see <a href="/wiki/PERT_(disambiguation)" class="mw-redirect mw-disambig" title="PERT (disambiguation)">PERT (disambiguation)</a>.</div> <p class="mw-empty-elt"> </p> <p>The <b>program </b> <b>evaluation and review technique</b> (<b>PERT</b>) is a statistical tool used in <a href="/wiki/Project_management" title="Project management">project management</a>, which was designed to analyze and represent the <a href="/wiki/Task_(project_management)" title="Task (project management)">tasks</a> involved in completing a given <a href="/wiki/Project" title="Project">project</a>. </p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Pert_chart_colored.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Pert_chart_colored.svg/309px-Pert_chart_colored.svg.png" decoding="async" width="309" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Pert_chart_colored.svg/464px-Pert_chart_colored.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Pert_chart_colored.svg/618px-Pert_chart_colored.svg.png 2x" data-file-width="308" data-file-height="188"></a><figcaption>PERT network chart for a seven-month project with five <a href="/wiki/Milestone_(project_management)" title="Milestone (project management)">milestones</a> (10 through 50) and six activities (A through F).</figcaption></figure> <p>PERT was originally developed by Charles E. Clark for the <a href="/wiki/United_States_Navy" title="United States Navy">United States Navy</a> in 1958; it is commonly used in conjunction with the <a href="/wiki/Critical_Path_Method" class="mw-redirect" title="Critical Path Method">Critical Path Method</a> (CPM), which was also introduced in 1958.<sup id="cite_ref-origins_1-0" class="reference"><a href="#cite_note-origins-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Overview"><span class="tocnumber">1</span> <span class="toctext">Overview</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#History"><span class="tocnumber">2</span> <span class="toctext">History</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Terminology"><span class="tocnumber">3</span> <span class="toctext">Terminology</span></a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Events_and_activities"><span class="tocnumber">3.1</span> <span class="toctext">Events and activities</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Time"><span class="tocnumber">3.2</span> <span class="toctext">Time</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Management_tools"><span class="tocnumber">3.3</span> <span class="toctext">Management tools</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Implementation"><span class="tocnumber">4</span> <span class="toctext">Implementation</span></a> <ul> <li class="toclevel-2 tocsection-8"><a href="#Example"><span class="tocnumber">4.1</span> <span class="toctext">Example</span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Next_step,_creating_network_diagram_by_hand_or_by_using_diagram_software"><span class="tocnumber">4.2</span> <span class="toctext">Next step, creating network diagram by hand or by using diagram software</span></a></li> <li class="toclevel-2 tocsection-10"><a href="#Next_step,_determination_of_critical_path_and_possible_slack"><span class="tocnumber">4.3</span> <span class="toctext">Next step, determination of critical path and possible slack</span></a></li> <li class="toclevel-2 tocsection-11"><a href="#Avoiding_loops"><span class="tocnumber">4.4</span> <span class="toctext">Avoiding loops</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-12"><a href="#As_project_scheduling_tool"><span class="tocnumber">5</span> <span class="toctext">As project scheduling tool</span></a> <ul> <li class="toclevel-2 tocsection-13"><a href="#Advantages"><span class="tocnumber">5.1</span> <span class="toctext">Advantages</span></a></li> <li class="toclevel-2 tocsection-14"><a href="#Disadvantages"><span class="tocnumber">5.2</span> <span class="toctext">Disadvantages</span></a></li> <li class="toclevel-2 tocsection-15"><a href="#Uncertainty_in_project_scheduling"><span class="tocnumber">5.3</span> <span class="toctext">Uncertainty in project scheduling</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-16"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#Further_reading"><span class="tocnumber">8</span> <span class="toctext">Further reading</span></a></li> <li class="toclevel-1 tocsection-19"><a href="#External_links"><span class="tocnumber">9</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Overview">Overview</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=1" title="Edit section: Overview" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>PERT is a method of analyzing the tasks involved in completing a project, especially the time needed to complete each task, and to identify the minimum time needed to complete the total project. It incorporates uncertainty by making it possible to schedule a project while not knowing precisely the details and durations of all the activities. It is more event-oriented than start- and completion-oriented, and is used more for projects where time is the major constraint rather than cost. It is applied to very large-scale, one-time, complex, non-routine infrastructure projects, as well as <a href="/wiki/R%26D" class="mw-redirect" title="R&D">R&D</a> projects. </p><p>PERT offers a management tool,<sup id="cite_ref-FOOTNOTEKerzner2009_2-0" class="reference"><a href="#cite_note-FOOTNOTEKerzner2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 497">: 497 </span></sup> which relies "on arrow and node diagrams of <i>activities</i> and <i>events</i>: arrows represent the <i>activities</i> or work necessary to reach the <i>events</i> or nodes that indicate each completed phase of the total project."<sup id="cite_ref-MB_1968_3-0" class="reference"><a href="#cite_note-MB_1968-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>PERT and CPM are complementary tools, because "CPM employs one time estimation and one cost estimation for each activity; PERT may utilize three time estimates (optimistic, expected, and pessimistic) and no costs for each activity. Although these are distinct differences, the term PERT is applied increasingly to all critical path scheduling."<sup id="cite_ref-MB_1968_3-1" class="reference"><a href="#cite_note-MB_1968-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="History">History</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=2" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>PERT was developed primarily to simplify the planning and scheduling of large and complex projects. It was developed by the <a href="/wiki/United_States_Navy_Special_Projects_Office" title="United States Navy Special Projects Office">United States Navy Special Projects Office</a>, <a href="/wiki/Lockheed_Aircraft" class="mw-redirect" title="Lockheed Aircraft">Lockheed Aircraft</a>, and <a href="/wiki/Booz_Allen_Hamilton" title="Booz Allen Hamilton">Booz Allen Hamilton</a> to support the Navy's <a href="/wiki/Polaris_missile" class="mw-redirect" title="Polaris missile">Polaris missile</a> project.<sup id="cite_ref-MRCW_1959_4-0" class="reference"><a href="#cite_note-MRCW_1959-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-byte198205_5-0" class="reference"><a href="#cite_note-byte198205-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It found applications throughout industry. An early example is the <a href="/wiki/1968_Winter_Olympics" title="1968 Winter Olympics">1968 Winter Olympics</a> in <a href="/wiki/Grenoble" title="Grenoble">Grenoble</a> which used PERT from 1965 until the opening of the 1968 Games.<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> This project model was the first of its kind, a revival for the <a href="/wiki/Scientific_management" title="Scientific management">scientific management</a> of Frederick Taylor and later refined by Henry Ford (<a href="/wiki/Fordism" title="Fordism">Fordism</a>). <a href="/wiki/DuPont" title="DuPont">DuPont</a>'s CPM was invented at roughly the same time as PERT. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PERT_Summary_Report_Phase_2,_1958.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/PERT_Summary_Report_Phase_2%2C_1958.jpg/170px-PERT_Summary_Report_Phase_2%2C_1958.jpg" decoding="async" width="170" height="218" class="mw-file-element" data-file-width="979" data-file-height="1253"></noscript><span class="lazy-image-placeholder" style="width: 170px;height: 218px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/PERT_Summary_Report_Phase_2%2C_1958.jpg/170px-PERT_Summary_Report_Phase_2%2C_1958.jpg" data-width="170" data-height="218" data-mw-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/PERT_Summary_Report_Phase_2%2C_1958.jpg/255px-PERT_Summary_Report_Phase_2%2C_1958.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/PERT_Summary_Report_Phase_2%2C_1958.jpg/340px-PERT_Summary_Report_Phase_2%2C_1958.jpg 2x" data-class="mw-file-element"> </span></a><figcaption><i>PERT Summary Report Phase 2</i>, 1958</figcaption></figure> <p>Initially PERT stood for <i>Program Evaluation Research Task,</i> but by 1959 was renamed.<sup id="cite_ref-MRCW_1959_4-1" class="reference"><a href="#cite_note-MRCW_1959-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It had been made public in 1958 in two publications of the U.S. Department of the Navy, entitled <i>Program Evaluation Research Task, Summary Report, Phase 1.</i><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> and <i>Phase 2.</i><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> both primarily written by Charles F. Clark.<sup id="cite_ref-origins_1-1" class="reference"><a href="#cite_note-origins-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In a 1959 article in <i><a href="/wiki/The_American_Statistician" title="The American Statistician">The American Statistician</a></i>, <a href="/wiki/Willard_Fazar" title="Willard Fazar">Willard Fazar</a>, Head of the Program Evaluation Branch, Special Projects Office, U.S. Navy, gave a detailed description of the main concepts of PERT. He explained: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Through an electronic computer, the PERT technique processes data representing the major, finite accomplishments (events) essential to achieve end-objectives; the inter-dependence of those events; and <a href="/wiki/Estimation_(project_management)" title="Estimation (project management)">estimates</a> of time and range of time necessary to complete each activity between two successive events. Such time expectations include estimates of "most likely time", "optimistic time", and "pessimistic time" for each activity. The technique is a management control tool that sizes up the outlook for meeting objectives on time; highlights danger signals requiring management decisions; reveals and defines both methodicalness and slack in the flow plan or the network of sequential activities that must be performed to meet objectives; compares current expectations with <a href="/wiki/Schedule_(project_management)" title="Schedule (project management)">scheduled</a> completion dates and computes the probability for meeting scheduled dates; and simulates the effects of options for decision— before decision.<sup id="cite_ref-SDFJWM_1959_9-0" class="reference"><a href="#cite_note-SDFJWM_1959-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></p></blockquote> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PERT_Guide_for_management_use,_June_1963.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/PERT_Guide_for_management_use%2C_June_1963.jpg/170px-PERT_Guide_for_management_use%2C_June_1963.jpg" decoding="async" width="170" height="224" class="mw-file-element" data-file-width="1020" data-file-height="1342"></noscript><span class="lazy-image-placeholder" style="width: 170px;height: 224px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/PERT_Guide_for_management_use%2C_June_1963.jpg/170px-PERT_Guide_for_management_use%2C_June_1963.jpg" data-width="170" data-height="224" data-mw-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/PERT_Guide_for_management_use%2C_June_1963.jpg/255px-PERT_Guide_for_management_use%2C_June_1963.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/PERT_Guide_for_management_use%2C_June_1963.jpg/340px-PERT_Guide_for_management_use%2C_June_1963.jpg 2x" data-class="mw-file-element"> </span></a><figcaption><i>PERT Guide for Management Use</i>, June 1963</figcaption></figure> <p>Ten years after the introduction of PERT, the American <a href="/wiki/Librarian" title="Librarian">librarian</a> Maribeth Brennan compiled a selected <a href="/wiki/Bibliography" title="Bibliography">bibliography</a> with about 150 publications on PERT and CPM, all published between 1958 and 1968.<sup id="cite_ref-MB_1968_3-2" class="reference"><a href="#cite_note-MB_1968-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For the subdivision of work units in PERT<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> another tool was developed: the <a href="/wiki/Work_Breakdown_Structure" class="mw-redirect" title="Work Breakdown Structure">Work Breakdown Structure</a>. The Work Breakdown Structure provides "a framework for complete networking, the Work Breakdown Structure was formally introduced as the first item of analysis in carrying out basic PERT/CPM."<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> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Terminology">Terminology</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=3" title="Edit section: Terminology" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"><h3 id="Events_and_activities">Events and activities</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=4" title="Edit section: Events and activities" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>In a PERT diagram, the main building block is the <i>event</i>, with connections to its known predecessor events and successor events. </p> <ul><li><i>PERT event</i>: a point that marks the start or completion of one or more activities. It consumes no time and uses no resources. When it marks the completion of one or more activities, it is not "reached" (does not occur) until <i>all</i> of the activities leading to that event have been completed.</li> <li><i>predecessor event</i>: an event that immediately precedes some other event without any other events intervening. An event can have multiple predecessor events and can be the predecessor of multiple events.</li> <li><i>successor event</i>: an event that immediately follows some other event without any other intervening events. An event can have multiple successor events and can be the successor of multiple events.</li></ul> <p>Besides events, PERT also tracks activities and sub-activities: </p> <ul><li><i>PERT activity</i>: the actual performance of a task which consumes time and requires resources (such as labor, materials, space, machinery). It can be understood as representing the time, effort, and resources required to move from one event to another. A PERT activity cannot be performed until the predecessor event has occurred.</li> <li><i>PERT sub-activity</i>: a PERT activity can be further decomposed into a set of sub-activities. For example, activity A1 can be decomposed into A1.1, A1.2 and A1.3. Sub-activities have all the properties of activities; in particular, a sub-activity has predecessor or successor events just like an activity. A sub-activity can be decomposed again into finer-grained sub-activities.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Time">Time</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=5" title="Edit section: Time" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>PERT defines four types of time required to accomplish an activity: </p> <ul><li><i>optimistic time</i>: the minimum possible time required to accomplish an activity (o) or a path (O), assuming everything proceeds better than is normally expected<sup id="cite_ref-FOOTNOTEKerzner2009_2-1" class="reference"><a href="#cite_note-FOOTNOTEKerzner2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 512">: 512 </span></sup></li> <li><i>pessimistic time</i>: the maximum possible time required to accomplish an activity (p) or a path (P), assuming everything goes wrong (but excluding major catastrophes).<sup id="cite_ref-FOOTNOTEKerzner2009_2-2" class="reference"><a href="#cite_note-FOOTNOTEKerzner2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 512">: 512 </span></sup></li> <li><i>most likely time</i>: the best estimate of the time required to accomplish an activity (m) or a path (M), assuming everything proceeds as normal.<sup id="cite_ref-FOOTNOTEKerzner2009_2-3" class="reference"><a href="#cite_note-FOOTNOTEKerzner2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 512">: 512 </span></sup></li> <li><i>expected time</i>: the best estimate of the time required to accomplish an activity (te) or a path (TE), accounting for the fact that things don't always proceed as normal (the implication being that the expected time is the average time the task would require if the task were repeated on a number of occasions over an extended period of time).<sup id="cite_ref-FOOTNOTEKerzner2009_2-4" class="reference"><a href="#cite_note-FOOTNOTEKerzner2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 512–513">: 512–513 </span></sup></li></ul> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle te={\frac {o+4m+p}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>o</mi> <mo>+</mo> <mn>4</mn> <mi>m</mi> <mo>+</mo> <mi>p</mi> </mrow> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle te={\frac {o+4m+p}{6}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10574ccabea7758e49d2ed6ded06ef7149f89e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.038ex; height:5.343ex;" alt="{\displaystyle te={\frac {o+4m+p}{6}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.038ex;height: 5.343ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10574ccabea7758e49d2ed6ded06ef7149f89e40" data-alt="{\displaystyle te={\frac {o+4m+p}{6}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TE=\sum _{i=1}^{n}te_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>E</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>t</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TE=\sum _{i=1}^{n}te_{i}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d70c1e8845ce3c3a73f7fd1888f42b19ba4eb4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.975ex; height:6.843ex;" alt="{\displaystyle TE=\sum _{i=1}^{n}te_{i}}"></noscript><span class="lazy-image-placeholder" style="width: 12.975ex;height: 6.843ex;vertical-align: -3.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d70c1e8845ce3c3a73f7fd1888f42b19ba4eb4e" data-alt="{\displaystyle TE=\sum _{i=1}^{n}te_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl></dd></dl></dd></dl> <ul><li><i>standard deviation of time</i> : the variability of the time for accomplishing an activity (σ<sub>te</sub>) or a path (σ<sub>TE</sub>)</li></ul> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\sigma _{te}={\frac {p-o}{6}}\\[8pt]&\sigma _{TE}={\sqrt {\sum _{i=1}^{n}{\sigma _{te_{i}}}^{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <mi>o</mi> </mrow> <mn>6</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\sigma _{te}={\frac {p-o}{6}}\\[8pt]&\sigma _{TE}={\sqrt {\sum _{i=1}^{n}{\sigma _{te_{i}}}^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8f0591db32c23b563de325fbda071969eefed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:18.487ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}&\sigma _{te}={\frac {p-o}{6}}\\[8pt]&\sigma _{TE}={\sqrt {\sum _{i=1}^{n}{\sigma _{te_{i}}}^{2}}}\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 18.487ex;height: 14.843ex;vertical-align: -6.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8f0591db32c23b563de325fbda071969eefed4" data-alt="{\displaystyle {\begin{aligned}&\sigma _{te}={\frac {p-o}{6}}\\[8pt]&\sigma _{TE}={\sqrt {\sum _{i=1}^{n}{\sigma _{te_{i}}}^{2}}}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Management_tools">Management tools</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=6" title="Edit section: Management tools" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>PERT supplies a number of tools for management with determination of concepts, such as: </p> <ul><li><i><a href="/wiki/Float_(project_management)" title="Float (project management)">float</a> or <a href="/wiki/Float_(project_management)" title="Float (project management)">slack</a></i> is a measure of the excess time and resources available to complete a task. It is the amount of time that a project task can be delayed without causing a delay in any subsequent tasks (<i>free float</i>) or the whole project (<i>total float</i>). Positive slack would indicate <i>ahead of schedule</i>; negative slack would indicate <i>behind schedule</i>; and zero slack would indicate <i>on schedule</i>.</li> <li><i><a href="/wiki/Critical_path_method" title="Critical path method">critical path</a></i>: the longest possible continuous pathway taken from the initial event to the terminal event. It determines the total calendar time required for the project; and, therefore, any time delays along the critical path will delay the reaching of the terminal event by at least the same amount.</li> <li><i>critical activity</i>: An activity that has total float equal to zero. An activity with zero free float is not necessarily on the critical path since its path may not be the longest.</li> <li><i><a href="https://en.wiktionary.org/wiki/lead#Verb_2" class="extiw" title="wikt:lead">lead</a> time</i>: the time by which a <i>predecessor event</i> must be completed in order to allow sufficient time for the activities that must elapse before a specific PERT event reaches completion.</li> <li><i>lag time</i>: the earliest time by which a <i>successor event</i> can follow a specific PERT event.</li> <li><i><a href="/wiki/Critical_path_method" title="Critical path method">fast tracking</a></i>: performing more critical activities in parallel</li> <li><i><a href="/wiki/Critical_path_method" title="Critical path method">crashing critical path</a></i>: Shortening duration of critical activities</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Implementation">Implementation</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=7" title="Edit section: Implementation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <p>The first step for scheduling the project is to determine the tasks that the project requires and the order in which they must be completed. The order may be easy to record for some tasks (e.g., when building a house, the land must be graded before the foundation can be laid) while difficult for others (there are two areas that need to be graded, but there are only enough bulldozers to do one). Additionally, the time estimates usually reflect the normal, non-rushed time. Many times, the time required to execute the task can be reduced for an additional cost or a reduction in the quality. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=8" title="Edit section: Example" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>In the following example there are seven tasks, labeled <i>A</i> through <i>G</i>. Some tasks can be done concurrently (<i>A</i> and <i>B</i>) while others cannot be done until their predecessor task is complete (<i>C</i> cannot begin until <i>A</i> is complete). Additionally, each task has three time estimates: the optimistic time estimate (<i>o</i>), the most likely or normal time estimate (<i>m</i>), and the pessimistic time estimate (<i>p</i>). The expected time (<i>te</i>) is computed using the formula (<i>o</i> + 4<i>m</i> + <i>p</i>) ÷ 6.<sup id="cite_ref-FOOTNOTEKerzner2009_2-5" class="reference"><a href="#cite_note-FOOTNOTEKerzner2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 512–513">: 512–513 </span></sup> </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Activity </th> <th rowspan="2">Predecessor </th> <th colspan="3">Time estimates </th> <th rowspan="2">Expected time </th></tr> <tr> <th>Opt. (<i>o</i>) </th> <th>Normal (<i>m</i>) </th> <th>Pess. (<i>p</i>) </th></tr> <tr> <td><i>A</i> </td> <td>— </td> <td>2 </td> <td>4 </td> <td>6 </td> <td>4.00 </td></tr> <tr> <td><i>B</i> </td> <td>— </td> <td>3 </td> <td>5 </td> <td>9 </td> <td>5.33 </td></tr> <tr> <td><i>C</i> </td> <td><i>A</i> </td> <td>4 </td> <td>5 </td> <td>7 </td> <td>5.17 </td></tr> <tr> <td><i>D</i> </td> <td><i>A</i> </td> <td>4 </td> <td>6 </td> <td>10 </td> <td>6.33 </td></tr> <tr> <td><i>E</i> </td> <td><i>B</i>, <i>C</i> </td> <td>4 </td> <td>5 </td> <td>7 </td> <td>5.17 </td></tr> <tr> <td><i>F</i> </td> <td><i>D</i> </td> <td>3 </td> <td>4 </td> <td>8 </td> <td>4.50 </td></tr> <tr> <td><i>G</i> </td> <td><i>E</i> </td> <td>3 </td> <td>5 </td> <td>8 </td> <td>5.17 </td></tr></tbody></table> <p>Once this step is complete, one can draw a <a href="/wiki/Gantt_chart" title="Gantt chart">Gantt chart</a> or a network diagram. </p> <dl><dd><figure class="mw-halign-none" typeof="mw:File/Frame"><a href="/wiki/File:Pert_example_gantt_chart.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/7/73/Pert_example_gantt_chart.gif" decoding="async" width="774" height="178" class="mw-file-element" data-file-width="774" data-file-height="178"></noscript><span class="lazy-image-placeholder" style="width: 774px;height: 178px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/7/73/Pert_example_gantt_chart.gif" data-width="774" data-height="178" data-class="mw-file-element"> </span></a><figcaption>A Gantt chart created using <a href="/wiki/Microsoft_Project" title="Microsoft Project">Microsoft Project</a> (MSP). Note (1) the <a href="/wiki/Critical_path_method" title="Critical path method">critical path</a> is in red, (2) the <a href="/wiki/Float_(project_management)" title="Float (project management)">slack</a> is the black lines connected to non-critical activities, (3) since Saturday and Sunday are not work days and are thus excluded from the schedule, some bars on the Gantt chart are longer if they cut through a weekend.</figcaption></figure></dd> <dd><figure class="mw-halign-none" typeof="mw:File/Frame"><a href="/wiki/File:Pert_example_gantt_chart.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/en/9/9a/Pert_example_gantt_chart.png" decoding="async" width="643" height="317" class="mw-file-element" data-file-width="643" data-file-height="317"></noscript><span class="lazy-image-placeholder" style="width: 643px;height: 317px;" data-mw-src="//upload.wikimedia.org/wikipedia/en/9/9a/Pert_example_gantt_chart.png" data-width="643" data-height="317" data-class="mw-file-element"> </span></a><figcaption>A Gantt chart created using <a href="/wiki/OmniPlan" title="OmniPlan">OmniPlan</a>. Note (1) the <a href="/wiki/Critical_path_method" title="Critical path method">critical path</a> is highlighted, (2) the <a href="/wiki/Float_(project_management)" title="Float (project management)">slack</a> is not specifically indicated on task 5 (d), though it can be observed on tasks 3 and 7 (b and f), (3) since weekends are indicated by a thin vertical line, and take up no additional space on the work calendar, bars on the Gantt chart are not longer or shorter when they do or don't carry over a weekend.</figcaption></figure></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Next_step,_creating_network_diagram_by_hand_or_by_using_diagram_software"><span id="Next_step.2C_creating_network_diagram_by_hand_or_by_using_diagram_software"></span>Next step, creating network diagram by hand or by using diagram software</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=9" title="Edit section: Next step, creating network diagram by hand or by using diagram software" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>A network diagram can be created by hand or by using diagram software. There are two types of network diagrams, activity on arrow (<a href="/wiki/Arrow_Diagramming_Method" class="mw-redirect" title="Arrow Diagramming Method">AOA</a>) and activity on node (<a href="/wiki/Precedence_Diagram_Method" class="mw-redirect" title="Precedence Diagram Method">AON</a>). Activity on node diagrams are generally easier to create and interpret. To create an AON diagram, it is recommended (but not required) to start with a node named <i>start</i>. This "activity" has a duration of zero (0). Then you draw each activity that does not have a predecessor activity (<i>a</i> and <i>b</i> in this example) and connect them with an arrow from start to each node. Next, since both <i>c</i> and <i>d</i> list <i>a</i> as a predecessor activity, their nodes are drawn with arrows coming from <i>a</i>. Activity <i>e</i> is listed with <i>b</i> and <i>c</i> as predecessor activities, so node <i>e</i> is drawn with arrows coming from both <i>b</i> and <i>c</i>, signifying that <i>e</i> cannot begin until both <i>b</i> and <i>c</i> have been completed. Activity <i>f</i> has <i>d</i> as a predecessor activity, so an arrow is drawn connecting the activities. Likewise, an arrow is drawn from <i>e</i> to <i>g</i>. Since there are no activities that come after <i>f</i> or <i>g</i>, it is recommended (but again not required) to connect them to a node labeled <i>finish</i>. </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Pert_example_network_diagram.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/en/4/40/Pert_example_network_diagram.gif" decoding="async" width="1000" height="350" class="mw-file-element" data-file-width="1000" data-file-height="350"></noscript><span class="lazy-image-placeholder" style="width: 1000px;height: 350px;" data-mw-src="//upload.wikimedia.org/wikipedia/en/4/40/Pert_example_network_diagram.gif" data-width="1000" data-height="350" data-class="mw-file-element"> </span></a><figcaption>A network diagram created using <a href="/wiki/Microsoft_Project" title="Microsoft Project">Microsoft Project</a> (MSP). Note the <a href="/wiki/Critical_path_method" title="Critical path method">critical path</a> is in red.</figcaption></figure> <div class="thumb tright" style=""><div class="thumbinner" style="width:222px"><div class="thumbimage noresize" style="width:220px;text-align:center"> <table class="wikitable" style="text-align:center;"> <tbody><tr> <td style="width:70px;">Early<br>Start </td> <td style="width:70px;">Duration </td> <td style="width:70px;">Early<br>finish </td></tr> <tr> <td colspan="3">Task Name </td></tr> <tr> <td>Late<br>Start </td> <td>Slack </td> <td>Late<br>finish </td></tr></tbody></table></div><div class="thumbcaption">A node like this one can be used to display the activity name, duration, ES, EF, LS, LF, and slack.</div></div></div> <p>By itself, the network diagram pictured above does not give much more information than a Gantt chart; however, it can be expanded to display more information. The most common information shown is: </p> <ol><li>The activity name</li> <li>The expected duration time</li> <li>The early start time (ES)</li> <li>The early finish time (EF)</li> <li>The late start time (LS)</li> <li>The late finish time (LF)</li> <li>The <a href="/wiki/Float_(project_management)" title="Float (project management)">slack</a></li></ol> <p>In order to determine this information it is assumed that the activities and normal duration times are given. The first step is to determine the ES and EF. The ES is defined as the maximum EF of all predecessor activities, unless the activity in question is the first activity, for which the ES is zero (0). The EF is the ES plus the task duration (EF = ES + duration). </p> <ul><li>The ES for <i>start</i> is zero since it is the first activity. Since the duration is zero, the EF is also zero. This EF is used as the ES for <i>a</i> and <i>b</i>.</li> <li>The ES for <i>a</i> is zero. The duration (4 work days) is added to the ES to get an EF of four. This EF is used as the ES for <i>c</i> and <i>d</i>.</li> <li>The ES for <i>b</i> is zero. The duration (5.33 work days) is added to the ES to get an EF of 5.33.</li> <li>The ES for <i>c</i> is four. The duration (5.17 work days) is added to the ES to get an EF of 9.17.</li> <li>The ES for <i>d</i> is four. The duration (6.33 work days) is added to the ES to get an EF of 10.33. This EF is used as the ES for <i>f</i>.</li> <li>The ES for <i>e</i> is the greatest EF of its predecessor activities (<i>b</i> and <i>c</i>). Since <i>b</i> has an EF of 5.33 and <i>c</i> has an EF of 9.17, the ES of <i>e</i> is 9.17. The duration (5.17 work days) is added to the ES to get an EF of 14.34. This EF is used as the ES for <i>g</i>.</li> <li>The ES for <i>f</i> is 10.33. The duration (4.5 work days) is added to the ES to get an EF of 14.83.</li> <li>The ES for <i>g</i> is 14.34. The duration (5.17 work days) is added to the ES to get an EF of 19.51.</li> <li>The ES for <i>finish</i> is the greatest EF of its predecessor activities (<i>f</i> and <i>g</i>). Since <i>f</i> has an EF of 14.83 and <i>g</i> has an EF of 19.51, the ES of <i>finish</i> is 19.51. <i>Finish</i> is a milestone (and therefore has a duration of zero), so the EF is also 19.51.</li></ul> <p>Barring any <a href="/wiki/Theory_of_Constraints" class="mw-redirect" title="Theory of Constraints">unforeseen events</a>, the project should take 19.51 work days to complete. The next step is to determine the late start (LS) and late finish (LF) of each activity. This will eventually show if there are activities that have <a href="/wiki/Float_(project_management)" title="Float (project management)">slack</a>. The LF is defined as the minimum LS of all successor activities, unless the activity is the last activity, for which the LF equals the EF. The LS is the LF minus the task duration (LS = LF − duration). </p> <ul><li>The LF for <i>finish</i> is equal to the EF (19.51 work days) since it is the last activity in the project. Since the duration is zero, the LS is also 19.51 work days. This will be used as the LF for <i>f</i> and <i>g</i>.</li> <li>The LF for <i>g</i> is 19.51 work days. The duration (5.17 work days) is subtracted from the LF to get an LS of 14.34 work days. This will be used as the LF for <i>e</i>.</li> <li>The LF for <i>f</i> is 19.51 work days. The duration (4.5 work days) is subtracted from the LF to get an LS of 15.01 work days. This will be used as the LF for <i>d</i>.</li> <li>The LF for <i>e</i> is 14.34 work days. The duration (5.17 work days) is subtracted from the LF to get an LS of 9.17 work days. This will be used as the LF for <i>b</i> and <i>c</i>.</li> <li>The LF for <i>d</i> is 15.01 work days. The duration (6.33 work days) is subtracted from the LF to get an LS of 8.68 work days.</li> <li>The LF for <i>c</i> is 9.17 work days. The duration (5.17 work days) is subtracted from the LF to get an LS of 4 work days.</li> <li>The LF for <i>b</i> is 9.17 work days. The duration (5.33 work days) is subtracted from the LF to get an LS of 3.84 work days.</li> <li>The LF for <i>a</i> is the minimum LS of its successor activities. Since <i>c</i> has an LS of 4 work days and <i>d</i> has an LS of 8.68 work days, the LF for <i>a</i> is 4 work days. The duration (4 work days) is subtracted from the LF to get an LS of 0 work days.</li> <li>The LF for <i>start</i> is the minimum LS of its successor activities. Since <i>a</i> has an LS of 0 work days and <i>b</i> has an LS of 3.84 work days, the LS is 0 work days.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Next_step,_determination_of_critical_path_and_possible_slack"><span id="Next_step.2C_determination_of_critical_path_and_possible_slack"></span>Next step, determination of critical path and possible slack</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=10" title="Edit section: Next step, determination of critical path and possible slack" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The next step is to determine the <a href="/wiki/Critical_path_method" title="Critical path method">critical path</a> and if any activities have <a href="/wiki/Float_(project_management)" title="Float (project management)">slack</a>. The critical path is the path that takes the <b>longest</b> to complete. To determine the path times, add the task durations for all available paths. Activities that have slack can be delayed without changing the overall time of the project. Slack is computed in one of two ways, slack = LF − EF <i>or</i> slack = LS − ES. Activities that are on the critical path have a slack of zero (0). </p> <ul><li>The duration of path <i>adf</i> is 14.83 work days.</li> <li>The duration of path <i>aceg</i> is 19.51 work days.</li> <li>The duration of path <i>beg</i> is 15.67 work days.</li></ul> <p>The critical path is <i>aceg</i> and the critical time is 19.51 work days. It is important to note that there can be more than one critical path (in a project more complex than this example) or that the critical path can change. For example, let's say that activities <i>d</i> and <i>f</i> take their pessimistic (b) times to complete instead of their expected (T<sub>E</sub>) times. The critical path is now <i>adf</i> and the critical time is 22 work days. On the other hand, if activity <i>c</i> can be reduced to one work day, the path time for <i>aceg</i> is reduced to 15.34 work days, which is slightly less than the time of the new critical path, <i>beg</i> (15.67 work days). </p><p>Assuming these scenarios do not happen, the slack for each activity can now be determined. </p> <ul><li><i>Start</i> and <i>finish</i> are milestones and by definition have no duration, therefore they can have no slack (0 work days).</li> <li>The activities on the critical path by definition have a slack of zero; however, it is always a good idea to check the math anyway when drawing by hand. <ul><li>LF<sub>a</sub> – EF<sub>a</sub> = 4 − 4 = 0</li> <li>LF<sub>c</sub> – EF<sub>c</sub> = 9.17 − 9.17 = 0</li> <li>LF<sub>e</sub> – EF<sub>e</sub> = 14.34 − 14.34 = 0</li> <li>LF<sub>g</sub> – EF<sub>g</sub> = 19.51 − 19.51 = 0</li></ul></li> <li>Activity <i>b</i> has an LF of 9.17 and an EF of 5.33, so the slack is 3.84 work days.</li> <li>Activity <i>d</i> has an LF of 15.01 and an EF of 10.33, so the slack is 4.68 work days.</li> <li>Activity <i>f</i> has an LF of 19.51 and an EF of 14.83, so the slack is 4.68 work days.</li></ul> <p>Therefore, activity <i>b</i> can be delayed almost 4 work days without delaying the project. Likewise, activity <i>d</i> <b>or</b> activity <i>f</i> can be delayed 4.68 work days without delaying the project (alternatively, <i>d</i> and <i>f</i> can be delayed 2.34 work days each). </p> <figure class="mw-halign-center" typeof="mw:File/Frame"><a href="/wiki/File:Pert_example_network_diagram_visio.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/4/46/Pert_example_network_diagram_visio.gif" decoding="async" width="1000" height="750" class="mw-file-element" data-file-width="1000" data-file-height="750"></noscript><span class="lazy-image-placeholder" style="width: 1000px;height: 750px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/4/46/Pert_example_network_diagram_visio.gif" data-width="1000" data-height="750" data-class="mw-file-element"> </span></a><figcaption>A completed network diagram created using <a href="/wiki/Microsoft_Visio" title="Microsoft Visio">Microsoft Visio</a>. Note the <a href="/wiki/Critical_path_method" title="Critical path method">critical path</a> is in red.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Avoiding_loops">Avoiding loops</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=11" title="Edit section: Avoiding loops" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Depending upon the capabilities of the data input phase of the critical path algorithm, it may be possible to create a loop, such as A -> B -> C -> A. This can cause simple algorithms to loop indefinitely. Although it is possible to "mark" nodes that have been visited, then clear the "marks" upon completion of the process, a far simpler mechanism involves computing the total of all activity durations. If an EF of more than the total is found, the computation should be terminated. It is worth saving the identities of the most recently visited dozen or so nodes to help identify the problem link. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="As_project_scheduling_tool">As project scheduling tool</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=12" title="Edit section: As project scheduling tool" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <div class="mw-heading mw-heading3"><h3 id="Advantages">Advantages</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=13" title="Edit section: Advantages" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li>PERT chart explicitly defines and makes visible <a href="/wiki/Dependency_(project_management)" title="Dependency (project management)">dependencies</a> (precedence relationships) between the work breakdown structure (commonly <a href="/wiki/Work_breakdown_structure" title="Work breakdown structure">WBS</a>) elements.</li> <li>PERT facilitates identification of the critical path and makes this visible.</li> <li>PERT facilitates identification of early start, late start, and slack for each activity.</li> <li>PERT provides for potentially reduced project duration due to better understanding of dependencies leading to improved overlapping of activities and tasks where feasible.</li> <li>The large amount of project data can be organized and presented in diagram for use in decision making.</li> <li>PERT can provide a probability of completing before a given time.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Disadvantages">Disadvantages</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=14" title="Edit section: Disadvantages" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <ul><li>There can be potentially hundreds or thousands of activities and individual dependency relationships.</li> <li>PERT is not easy to scale down for smaller projects.</li> <li>The network charts tend to be large and unwieldy, requiring several pages to print and requiring specially-sized paper.</li> <li>The lack of a timeframe on most PERT/CPM charts makes it harder to show status, although colours can help, <i>e.g.</i>, specific colour for completed nodes.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Uncertainty_in_project_scheduling">Uncertainty in project scheduling</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=15" title="Edit section: Uncertainty in project scheduling" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>During project execution a real-life project will never execute exactly as it was planned due to uncertainty. This can be due to ambiguity resulting from subjective estimates that are prone to human errors or can be the result of variability arising from unexpected events or risks. The main reason that PERT may provide inaccurate information about the project completion time is due to this schedule uncertainty. This inaccuracy may be large enough to render such estimates as not helpful. </p><p>One possible method to maximize solution robustness is to include safety in the baseline schedule in order to absorb disruptions. This is called <i>proactive scheduling</i>, however, allowing for every possible disruption would be very slow and couldn't be accommodated by the baseline schedule. A second approach, termed <i>reactive scheduling</i>, defines a procedure to react to disruptions that cannot be absorbed by the baseline schedule. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=16" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Activity_diagram" title="Activity diagram">Activity diagram</a></li> <li><a href="/wiki/Arrow_diagramming_method" title="Arrow diagramming method">Arrow diagramming method</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT distribution</a></li> <li><a href="/wiki/Critical_chain_project_management" title="Critical chain project management">Critical chain project management</a></li> <li><a href="/wiki/Float_(project_management)" title="Float (project management)">Float (project management)</a></li> <li><a href="/wiki/Gantt_chart" title="Gantt chart">Gantt chart</a></li> <li><a href="/wiki/GERT" class="mw-redirect" title="GERT">GERT</a></li> <li><a href="/wiki/Precedence_diagram_method" title="Precedence diagram method">Precedence diagram method</a></li> <li><a href="/wiki/Project_network" title="Project network">Project network</a></li> <li><a href="/wiki/Project_management" title="Project management">Project management</a></li> <li><a href="/wiki/Project_planning" title="Project planning">Project planning</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular distribution</a></li> <li><a href="/wiki/PRINCE2" title="PRINCE2">PRINCE2</a></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=17" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-origins-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-origins_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-origins_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKelleyWalkerSayer1989" class="citation journal cs1">Kelley, James E.; Walker, Morgan R.; Sayer, John S. (February 1989). <a rel="nofollow" class="external text" href="https://www.pmi.org/learning/library/origins-cpm-personal-history-3762">"The Origins of CPM: a personal history"</a>. <i>Project Management</i>. <b>3</b> (2). Project Management Institute: 18<span class="reference-accessdate">. Retrieved <span class="nowrap">20 March</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Project+Management&rft.atitle=The+Origins+of+CPM%3A+a+personal+history&rft.volume=3&rft.issue=2&rft.pages=18&rft.date=1989-02&rft.aulast=Kelley&rft.aufirst=James+E.&rft.au=Walker%2C+Morgan+R.&rft.au=Sayer%2C+John+S.&rft_id=https%3A%2F%2Fwww.pmi.org%2Flearning%2Flibrary%2Forigins-cpm-personal-history-3762&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEKerzner2009-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEKerzner2009_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKerzner2009_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKerzner2009_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKerzner2009_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKerzner2009_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKerzner2009_2-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKerzner2009">Kerzner 2009</a>.</span> </li> <li id="cite_note-MB_1968-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-MB_1968_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MB_1968_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-MB_1968_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Brennan, Maribeth; <i>PERT and CPM: a selected bibliography</i>, Council of Planning Librarians, Monticello (IL), 1968, p. 1</span> </li> <li id="cite_note-MRCW_1959-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-MRCW_1959_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MRCW_1959_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Malcolm, Donald G.; Roseboom, John H.; Clark, Charles E.; <a href="/wiki/Willard_Fazar" title="Willard Fazar">Fazar, Willard</a>; "Application of a Technique for Research and Development Program Evaluation", <i>Operations Research</i>, vol. 7, no. 5, September–October 1959, pp. 646–669</span> </li> <li id="cite_note-byte198205-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-byte198205_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZimmermanConrad1982" class="citation magazine cs1">Zimmerman, Steve; Conrad, Leo M. (May 1982). <a rel="nofollow" class="external text" href="https://archive.org/details/eu_BYTE-1982-05_OCR/page/n466/mode/1up?view=theater">"Programming PERT in BASIC"</a>. <i>BYTE</i>. pp. <span class="nowrap">465–</span>478<span class="reference-accessdate">. Retrieved <span class="nowrap">29 December</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BYTE&rft.atitle=Programming+PERT+in+BASIC&rft.pages=%3Cspan+class%3D%22nowrap%22%3E465-%3C%2Fspan%3E478&rft.date=1982-05&rft.aulast=Zimmerman&rft.aufirst=Steve&rft.au=Conrad%2C+Leo+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feu_BYTE-1982-05_OCR%2Fpage%2Fn466%2Fmode%2F1up%3Fview%3Dtheater&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://www.la84foundation.org/6oic/OfficialReports/1968/or1968.pdf">1968 Winter Olympics official report</a>, p. 49. Accessed 1 November 2010. <span class="languageicon">(in English and French)</span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">U.S. Department of the Navy, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151112203807/http://www.dtic.mil/dtic/tr/fulltext/u2/735902.pdf"><i>Program Evaluation Research Task, Summary Report, Phase 1</i></a>, Government Printing Office, Washington (DC), 1958</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">U.S. Department of the Navy, <a rel="nofollow" class="external text" href="https://catalog.hathitrust.org/Record/100954569"><i>Program Evaluation Research Task, Summary Report, Phase 2</i></a>, Government Printing Office, Washington (DC), 1958</span> </li> <li id="cite_note-SDFJWM_1959-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-SDFJWM_1959_9-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Willard_Fazar" title="Willard Fazar">Willard Fazar</a> cited in: Stauber, B. Ralph; Douty, Harry M.; Fazar, Willard; Jordan, Richard H.; Weinfeld, William; and Manvel, Allen D.; <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2682310">"Federal Statistical Activities"</a>, <i>The American Statistician</i>, 13(2): 9–12 (April 1959), pp. 9–12</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">Cook, Desmond L.; <i>Program Evaluation and Review Technique</i>, 1966, p. 12</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"><a href="/wiki/Harold_Bright_Maynard" title="Harold Bright Maynard">Maynard, Harold Bright</a>, <i>Handbook of Business Administration,</i> 1967, p. 17</span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Program_evaluation_and_review_technique&action=edit&section=18" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 30em;"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProject_Management_Institute2013" class="citation book cs1">Project Management Institute (2013). <i>A Guide to the Project Management Body of Knowledge</i> (5th ed.). Project Management Institute. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-935589-67-9" title="Special:BookSources/978-1-935589-67-9"><bdi>978-1-935589-67-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+the+Project+Management+Body+of+Knowledge&rft.edition=5th&rft.pub=Project+Management+Institute&rft.date=2013&rft.isbn=978-1-935589-67-9&rft.au=Project+Management+Institute&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlastorin2003" class="citation book cs1">Klastorin, Ted (2003). <i>Project Management: Tools and Trade-offs</i> (3rd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-41384-4" title="Special:BookSources/978-0-471-41384-4"><bdi>978-0-471-41384-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Project+Management%3A+Tools+and+Trade-offs&rft.edition=3rd&rft.pub=Wiley&rft.date=2003&rft.isbn=978-0-471-41384-4&rft.aulast=Klastorin&rft.aufirst=Ted&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKerzner2009" class="citation book cs1"><a href="/wiki/Harold_Kerzner" title="Harold Kerzner">Kerzner, Harold</a> (2009). <i>Project Management: A Systems Approach to Planning, Scheduling, and Controlling</i> (10th ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-27870-3" title="Special:BookSources/978-0-470-27870-3"><bdi>978-0-470-27870-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Project+Management%3A+A+Systems+Approach+to+Planning%2C+Scheduling%2C+and+Controlling&rft.edition=10th&rft.pub=Wiley&rft.date=2009&rft.isbn=978-0-470-27870-3&rft.aulast=Kerzner&rft.aufirst=Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-20822-8" title="Special:BookSources/978-0-471-20822-8"><bdi>978-0-471-20822-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Project+Management+ToolBox%3A+Tools+and+Techniques+for+the+Practicing+Project+Manager&rft.pub=Wiley&rft.date=2003&rft.isbn=978-0-471-20822-8&rft.aulast=Milosevic&rft.aufirst=Dragan+Z.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiller1963" class="citation book cs1">Miller, Robert W. (1963). <i>Schedule, Cost, and Profit Control with PERT - A Comprehensive Guide for Program Management</i>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780070419940" title="Special:BookSources/9780070419940"><bdi>9780070419940</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schedule%2C+Cost%2C+and+Profit+Control+with+PERT+-+A+Comprehensive+Guide+for+Program+Management&rft.pub=McGraw-Hill&rft.date=1963&rft.isbn=9780070419940&rft.aulast=Miller&rft.aufirst=Robert+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSapolsky1971" class="citation book cs1">Sapolsky, Harvey M. (1971). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/polarissystemdev0000sapo"><i>The Polaris System Development: Bureaucratic and Programmatic Success in Government</i></a></span>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0674682254" title="Special:BookSources/0674682254"><bdi>0674682254</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Polaris+System+Development%3A+Bureaucratic+and+Programmatic+Success+in+Government&rft.pub=Harvard+University+Press&rft.date=1971&rft.isbn=0674682254&rft.aulast=Sapolsky&rft.aufirst=Harvey+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpolarissystemdev0000sapo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProgram+evaluation+and+review+technique" class="Z3988"></span></li></ul> </div> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" 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class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Liz" data-user-gender="unknown" data-timestamp="1742443266"> <span>Last edited on 20 March 2025, at 04:01</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D8%B3%D9%84%D9%88%D8%A8_%D8%AA%D9%82%D9%8A%D9%8A%D9%85_%D9%88%D9%85%D8%B1%D8%A7%D8%AC%D8%B9%D8%A9_%D8%A7%D9%84%D9%85%D8%B4%D8%B1%D9%88%D8%B9" title="أسلوب تقييم ومراجعة المشروع – Arabic" lang="ar" hreflang="ar" data-title="أسلوب تقييم ومراجعة المشروع" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%95%D0%A0%D0%A2" title="ПЕРТ – Bulgarian" lang="bg" hreflang="bg" data-title="ПЕРТ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/T%C3%A8cnica_de_revisi%C3%B3_i_avaluaci%C3%B3_de_programes" title="Tècnica de revisió i avaluació de programes – Catalan" lang="ca" hreflang="ca" data-title="Tècnica de revisió i avaluació de programes" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Program_evaluation_and_review_technique" title="Program evaluation and review technique – German" lang="de" hreflang="de" data-title="Program evaluation and review technique" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/PERTi_diagramm" title="PERTi diagramm – Estonian" lang="et" hreflang="et" data-title="PERTi diagramm" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/T%C3%A9cnica_de_revisi%C3%B3n_y_evaluaci%C3%B3n_de_programas" title="Técnica de revisión y evaluación de programas – Spanish" lang="es" hreflang="es" data-title="Técnica de revisión y evaluación de programas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D9%88%D8%B4_%D8%A8%D8%B1%D8%B1%D8%B3%DB%8C_%D9%88_%D8%A7%D8%B1%D8%B2%DB%8C%D8%A7%D8%A8%DB%8C_%D8%A8%D8%B1%D9%86%D8%A7%D9%85%D9%87" title="روش بررسی و ارزیابی برنامه – Persian" lang="fa" hreflang="fa" data-title="روش بررسی و ارزیابی برنامه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/PERT" title="PERT – French" lang="fr" hreflang="fr" data-title="PERT" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8D%BC%ED%8A%B8" title="퍼트 – Korean" lang="ko" hreflang="ko" data-title="퍼트" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/PERT/CPM" title="PERT/CPM – Italian" lang="it" hreflang="it" data-title="PERT/CPM" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/PERT" title="PERT – Dutch" lang="nl" hreflang="nl" data-title="PERT" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/PERT" title="PERT – Japanese" lang="ja" hreflang="ja" data-title="PERT" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/PERT" title="PERT – Occitan" lang="oc" hreflang="oc" data-title="PERT" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/PERT" title="PERT – Polish" lang="pl" hreflang="pl" data-title="PERT" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/PERT" title="PERT – Portuguese" lang="pt" hreflang="pt" data-title="PERT" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Tehnic%C4%83_de_evaluare_%C8%99i_analiz%C4%83_a_programelor" title="Tehnică de evaluare și analiză a programelor – Romanian" lang="ro" hreflang="ro" data-title="Tehnică de evaluare și analiză a programelor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/PERT" title="PERT – Russian" lang="ru" hreflang="ru" data-title="PERT" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Program_Evaluation_and_Review_Technique" title="Program Evaluation and Review Technique – Sundanese" lang="su" hreflang="su" data-title="Program Evaluation and Review Technique" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/PERT" title="PERT – Finnish" lang="fi" hreflang="fi" data-title="PERT" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Program_Evaluation_and_Review_Technique" title="Program Evaluation and Review Technique – Swedish" lang="sv" hreflang="sv" data-title="Program Evaluation and Review Technique" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Program_Evaluation_and_Review_Technique" title="Program Evaluation and Review Technique – Tagalog" lang="tl" hreflang="tl" data-title="Program Evaluation and Review Technique" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B9%80%E0%B8%A1%E0%B8%B4%E0%B8%99%E0%B9%81%E0%B8%A5%E0%B8%B0%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%95%E0%B8%A3%E0%B8%A7%E0%B8%88%E0%B8%AA%E0%B8%AD%E0%B8%9A%E0%B9%82%E0%B8%9B%E0%B8%A3%E0%B9%81%E0%B8%81%E0%B8%A3%E0%B8%A1" title="การประเมินและการตรวจสอบโปรแกรม – Thai" lang="th" hreflang="th" data-title="การประเมินและการตรวจสอบโปรแกรม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%85%D0%BD%D1%96%D0%BA%D0%B0_%D0%BE%D1%86%D1%96%D0%BD%D1%8E%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F_%D1%82%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7%D1%83_%D0%BF%D1%80%D0%BE%D0%B3%D1%80%D0%B0%D0%BC" title="Техніка оцінювання та аналізу програм – Ukrainian" lang="uk" hreflang="uk" data-title="Техніка оцінювання та аналізу програм" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/K%E1%BB%B9_thu%E1%BA%ADt_%C6%AF%E1%BB%9Bc_l%C6%B0%E1%BB%A3ng_v%C3%A0_%C4%90%C3%A1nh_gi%C3%A1_Ch%C6%B0%C6%A1ng_tr%C3%ACnh" title="Kỹ thuật Ước lượng và Đánh giá Chương trình – Vietnamese" lang="vi" hreflang="vi" data-title="Kỹ thuật Ước lượng và Đánh giá Chương trình" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A8%88%E7%95%AB%E8%A9%95%E6%A0%B8%E8%A1%93" title="計畫評核術 – Chinese" lang="zh" hreflang="zh" data-title="計畫評核術" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> <ul id="footer-icons" class="footer-icons"> <li id="footer-copyrightico"><a href="https://www.wikimedia.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" 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