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What is the probability of completing the project by day 112? Within 116 days?

Answer the following

The Global Tea and Organic Juice companies have merged.

The following information has been collected for the “Consolidation Project.”

1. Compute the expected time for each activity.

2. Compute the variance for each activity.

3. Compute the expected project duration.

4. What is the probability of completing the project by day 112? Within 116 days?

5. What is the probability of completing “Negotiate with Unions” by day 90

Appendix 7.1

PERT and PERT Simulation

Learning Objectives

· After reading this appendix you should be able to:

· A7-1 Calculate basic Pert Simulation projections.


LO A7-1

Calculate basic Pert Simulation projections.

In 1958 the Special Office of the Navy and the Booze, Allen, and Hamilton consulting firm developed PERT (program evaluation and review technique) to schedule the more than 3,300 contractors of the Polaris submarine project and to cover uncertainty of activity time estimates.

PERT is almost identical to the critical path method (CPM) technique except it assumes each activity duration has a range that follows a statistical distribution. PERT uses three time estimates for each activity. Basically, this means each activity duration can range from an optimistic time to a pessimistic time, and a weighted average can be computed for each activity. Because project activities usually represent work, and because work tends to stay behind once it gets behind, the PERT developers chose an approximation of the beta distribution to represent activity durations. This distribution is known to be flexible and can accommodate empirical data that do not follow a normal distribution. The activity durations can be skewed more toward the high or low end of the data range.  Figure A7.1A  depicts a beta distribution for activity durations that is skewed toward the right and is representative of work that tends to stay late once it is behind. The distribution for the project duration is represented by a normal Page 241(symmetrical) distribution shown in  Figure A7.1B . The project distribution represents the sum of the weighted averages of the activities on the critical path(s).

FIGURE A7.1 Activity and Project Frequency Distributions

Knowing the weighted average and variances for each activity allows the project planner to compute the probability of meeting different project durations. Follow the steps described in the hypothetical example given next. (The jargon is difficult for those not familiar with statistics, but the process is relatively simple after working through a couple of examples.)

The weighted average activity time is computed by the following formula:

When the three time estimates have been specified, this equation is used to compute the weighted average duration for each activity. The average (deterministic) value is placed on the project network as in the CPM method and the early, late, slack, and project completion times are computed as they are in the CPM method.

The variability in the activity time estimates is approximated by the following equations: Equation 7.2 represents the standard deviation for the activity. Equation 7.3 represents the standard deviation for the project. Note the standard deviation of the activity is squared in this equation; this is also called variance. This sum includes only activities on the critical path(s) or path being reviewed.

Finally, the average project duration (TE) is the sum of all the average activity times along the critical path (sum of te), and it follows a normal distribution.

Page 242Knowing the average project duration and the variances of activities allows the probability of completing the project (or segment of the project) by a specific time to be computed using standard statistical tables. The equation below (Equation 7.4) is used to compute the “Z” value found in statistical tables (Z = number of standard deviations from the mean), which, in turn, tells the probability of completing the project in the time specified.


TS = scheduled project duration

Z = probability (of meeting scheduled duration) found in statistical Table A7.2

TABLE A7.2 Values and Probabilities

The activity times and variances are given in Table A7.1. The project network is presented in  Figure A7.2 . This figure shows the project network as AOA and AON. The AON network is presented as a reminder that PERT can use AON networks as well as AOA.

TABLE A7.1 Activity Times and Variances
FIGURE A7.2 Hypothetical Network

The expected project duration (TE) is 64 time units; the critical path is 1-2-3-5-6. With this information, the probability of completing the project by a specific date can easily be computed using standard statistical methods. For example, what is the probability the project will be completed before a scheduled time (TS) of 67? The normal curve for the project would appear as shown in  Figure A7.3 .

FIGURE A7.3 Possible Project Durations

Using the formula for the Z value, the probability can be computed as follows:

Reading from Table A7.2, a Z value of +0.5 gives a probability of 0.69, which is interpreted to mean there is a 69 percent chance of completing the project on or before 67 time units.
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