Example

PROBLEM

Recently, you were assigned to manage a project to remodel the seminar room for your company. You have constructed a network diagram depicting the various activities in the project (shown below to the right). In addition, you have asked your team to estimate the amount of time that they would expect each of the activities to take. Their responses are shown in the following table: Time Estimates (days) Optimistic Most Likely Pessimistic Activity | Start Finish a. What is the expected completion time of the project? First, calculate the expected time, te, for each activity. To calculate the expected time, te for the activity, use the formula below, where a is the optimistic time estimate, b is the pessimistic time estimate, and m is the most likely time estimate. te = a +4m + b Beginning with activity A, substitute known values into the equation and evaluate. te = a + 4m + b 6+4(9)+10 = 8.67 Repeat the process using the values for activity B. te = a + 4m + b 5+4(7) +11 7.33 Find the expected time for activity C. te = a + 4m + b 6+4(7) +9 = 7.17 Repeat the process using the values for activity D. a + 4m + b te = 2 +4(5)+7 = 4.83 Finally, find the expected time for activity E. te = a + 4m + b 5+46) + 10 = 6.50 To find the expected completion time of the project, determine the project's critical path. The critical path is the sequence of activities between a project's start and finish that takes the longest time to complete. Thus, the activities along the critical path determine the completion time of the project. Look at each path from start to finish and determine the time it takes to complete the path, using the expected time values found above. Path A-C A-D-E B-E Expected Time (days) 8.67 +7.17 = 15.84 8.67 +4.83 +6.50 = 20 7.33+6.50 = 13.83 The critical path is path A-D-E and the expected completion time is 20 days. b. What is the probability of completing the project in 22 days? Use the normal distribution and the z-transformation formula shown below. Note that Te is the sum of the expected activity times on the critical path, o is the sum of the variances of activities on the critical path, and T is the due date of the project. T-TE ZEE In this case, T = 22. Also, note that Te is equal to the expected completion time found in part (a). Thus, TE = 20. To calculate the variance, 62, for an activity, use the formula below, where a is the optimistic time estimate, and b is the pessimistic time estimate. Beginning with activity A, substitute known values into the equation and evaluate. 02=1072) = 0.44 Repeat the process using the values for activity D. = 0.69 Finally, find the variance for activity E. 02=443) = 0.69 Compute z. T-TE 22-20 10.44 +0.69 +0.69) = 1.48 Use technology or a standard normal distribution table to find the probability of completing the project in 22 days, which is equal to the probability that z is less than 1.48. Refer to the standard normal table This probability is 0.9306 c. What is the probability of completing the project in 18 days? Note 18 and the value of T from part (b), 22, are equidistant from the value of TE, 20. Because the normal distribution is symmetrical, the probability the project can be completed in 18 days can be found by subtracting the probability found in part (b) from 1. Find the probability of completing the project in 18 days 1 -0.9306 = 0.0694 Problem 21 Question Help Recently, you were assigned to manage a project for your company. You have constructed a network diagram depicting the various activities in the project (shown below to the right). In addition, you have asked your team to estimate the amount of time that they would expect each of the activities to take. Their responses are shown in the following table: Activity Time Estimates (days) Optimistic Most Likely Pessimistic 10 W NAUDU Start Finish 0 Finish cowo a. What is the expected completion time of the project? The expected completion time is days. (Enter your response as an integer.)