# Project Scheduling -- Probabilistic PERT

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<p>Project SchedulingProbabilistic PERT</p> <p>PERT Probability Approach to Project Scheduling Activity completion times are seldom known with cetainty. PERT is a technique that treats activity completion times as random variables. Completion time estimates can be estimated using the Three Time Estimate approach. In this approach, three time estimates are required for each activity:a = an optimistic time to perform the activity P(Finish &lt; a) &lt; .01 m = the most likely time to perform the activity (mode) b = a pessimistic time to perform the activity P(Finish &gt; b) &lt; .01 Results from statistical studies Subjective best estimates</p> <p>3-Time Estimate Approach Probability Distribution With three time estimates, the activity completion time can be approximated by a Beta distribution. Beta distributions can come in a variety of shapes:</p> <p>a</p> <p>m</p> <p>b</p> <p>a</p> <p>m</p> <p>b</p> <p>a</p> <p>m</p> <p>b</p> <p>Mean and Standard Deviation for Activity Completion Times The best estimate for the mean is a weighted average of the three time estimates with weights 1/6, 4/6, and 1/6 respectively on a, m, and b. Since most of the area is with the range from a to b (b-a), and since most of the area lies 3 standard deviations on either side of the mean (6 standard deviations total), then the standard deviation is approximated by Range/6.a + 4m + b = the mean completion time = 6 b-a = the standard deviation = 6</p> <p>PERT Assumptions Assumption 1 A critical path can be determined by using the mean completion times for the activities. The project mean completion time is determined solely by the completion time of the activities on the critical path.</p> <p> Assumption 2 There are enough activities on the critical path so that the distribution of the overall project completion time can be approximated by the normal distribution.</p> <p> Assumption 3 The time to complete one activity is independent of the completion time of any other activity.</p> <p>The Project Completion Time DistributionThe three assumptions imply that the overall project completion time is normally</p> <p>distributed, with: = Sum of the s on the critical path 2 = Sum of the 2 s on the critical path</p> <p>(76 + 4(86) +120)/6</p> <p>The Probability Approach(120-76)/6 (7.33)2</p> <p>Activity Optimistic Most Likely Pessimistic A 76 86 120 B 12 15 18 C 4 5 6 D 15 18 33 E 18 21 24 F 16 26 30 G 10 13 22 H 24 28 32 I 22 27 50 J 38 43 60</p> <p> 90 15 5 20 20 21 21 25 25 14 14 28 28 28 30 30 30 45 45 45</p> <p> 7.33 1.00 0.33 3.00 3.00 1.00 1.00 2.33 2.33 2.00 2.00 1.33 1.33 1.33 4.67 4.67 4.67 3.67 3.67 3.67</p> <p>222 53.73 1.00 1.00 0.11 0.11 9.00 9.00 1.00 1.00 1.00 5.43 5.43 5.43 4.00 4.00 4.00 1.77 1.77 1.77 21.81 21.81 21.81 13.47 13.47 13.47</p> <p>Distribution For Klone Computers The project has a normal distribution. The critical path is A-F-G-D-J. A F G D J 90 25 14 20 45 1942 2 2 2 G 2 2 A F D J</p> <p> 53.78 5.44 4 9 13.44 </p> <p>85.66</p> <p> 2 85.66 9.255</p> <p>Standard Probability Questions1.</p> <p>What is the probability the project will be finished within 194 days?P(X &lt; 194)</p> <p>2.</p> <p>Give an interval within which we are 95% sure of completing the project.X values, xL, the lower confidnce limit, and xU, the upper confidnce limit, such that P(XxU) = .025</p> <p>3.</p> <p>What is the probability the project will be completed within 180 days?P(X &lt; 180)</p> <p>4.</p> <p>What is the probability the project will take longer than 210 days.P(X &gt; 210)</p> <p>5.</p> <p>By what time are we 99% sure of completing the project?X value such that P(X &lt; x) = .99</p> <p>Excel SolutionsNORMDIST(194, 194, 9.255, TRUE) NORMINV(.025, 194, 9.255) NORMINV(.975, 194, 9.255) NORMDIST(180, 194, 9.255, TRUE)</p> <p>1 - NORMDIST(210, 194, 9.255, TRUE)</p> <p>NORMINV(.99, 194, 9.255)</p> <p>Using the PERT-CPM Template for Probabilistic Models Instead of calculating and by hand, the Excel template may be used. Instead of entering data in the and columns, input the estimates for a, m , and b into columns C, D, and E. The template does all the required calculations After the problem has been solved, probability analyses may be performed.</p> <p>Enter a, m, b instead of </p> <p>Call Solver Click Solve Go to PERT OUTPUT worksheet</p> <p>Call Solver Click Solve</p> <p>To get a cumulative probability, enter a number here</p> <p>P(Project is completed in less than 180 days)</p> <p>Cost Analysis Using the Expected Value Approach Spending extra money, in general should decrease project duration.</p> <p> But is this operation cost effective? The expected value criterion can be used as a guide for answering this question.</p> <p>Cost Analyses Using ProbabilitiesSuppose an analysis of the competition indicated: If the project is completed within 180 days, this would yields an additional profit of \$1 million. If the project is completed in 180 days to 200 days, this would yield an additional profit of \$400,000.</p> <p>KLONE COMPUTERS Cost analysis using probabilities Completion time reduction can be achieved by</p> <p>additional training. Two possible activities are being considered. Sales personnel training: (Activity H) Cost \$200,000; New time estimates are a = 19, m= 21, and b = 23 days.</p> <p> Technical staff training: (Activity F) Cost \$250,000; New time estimates are a = 12, m = 14, and b = 16.</p> <p> Which, if either option, should be pursued?</p> <p>Analysis of Additional Sales Personnel TrainingSales personnel training (Activity H) is not a critical activity. Thus any reduction in Activity H will not affect the critical path and hence the distribution of the project completion time.</p> <p>This option should not be pursued at any cost.</p> <p>Analysis of Additional Technical Staff Training Technical Staff Training (Activity F) is on the critical path so this option should be analyzed. One of three things will happen: The project will finish within 180 days: Klonepalm will net an additional \$1 million</p> <p> The project will finish in the period from 180 to 200 days Klonepalm will net an additional \$400,000</p> <p> The project will take longer than 200 days Klonepalm will not make any additional profit.</p> <p>The Expected Value Approach Find the P(X &lt; 180), P(180 &lt; X &lt; 200), and P(X &gt; 200) under the scenarios that No additional staff training is done Additional staff is done</p> <p> For each scenario find the expected profit: Expected Additional Profit1000000(P(X</p>