ppendix 7.1
PERT and PERT Simulation
LEARNING OBJECTIVES
After reading this appendix you should be able to:
- LO A7-1
Calculate basic Pert Simulation projections.
PERT—PROGRAM EVALUATION AND REVIEW TECHNIQUE
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:
te = a + 4m + b6 | (7.1) |
where
- t_{e}=weighted average activity time
- a=optimistic activity time (1 chance in 100 of completing the activity earlier under normal conditions)
- b=pessimistic activity time (1 chance in 100 of completing the activity later under normal conditions)
- m=most likely activity time
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.
σte = (b − a6) | (7.2) |
σTE = ∑σte2 | (7.3) |
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.
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Knowing 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.
Z = TS − TE∑σte2 | (7.4) |
where
- T_{E}=critical path duration
- T_{S}=scheduled project duration
- Z=probability (of meeting scheduled duration) found in statistical Table A7.2
A HYPOTHETICAL EXAMPLE USING THE PERT TECHNIQUE
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
Activity | a | m | b | t_{e} | [(b – a)/6]^{2} |
1–2 | 17 | 29 | 47 | 30 | 25 |
2–3 | 6 | 12 | 24 | 13 | 9 |
2–4 | 16 | 19 | 28 | 20 | 4 |
3–5 | 13 | 16 | 19 | 16 | 1 |
4–5 | 2 | 5 | 14 | 6 | 4 |
5–6 | 2 | 5 | 8 | 5 | 1 |
FIGURE A7.2 Hypothetical Network
The expected project duration (T_{E}) 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 (T_{S}) 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:
Z=TS − TE∑σte2=67 − 6425 + 9 + 1 + 1=+336=+ 0.50P=0.69
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.
TABLE A7.2 Z Values and Probabilities
Z Value | Probability | Z Value | Probability |
−3.0 | .001 | +0.0 | .500 |
−2.8 | .003 | +0.2 | .579 |
−2.6 | .005 | +0.4 | .655 |
−2.4 | .008 | +0.6 | .726 |
−2.2 | .014 | +0.8 | .788 |
−2.0 | .023 | +1.0 | .841 |
−1.8 | .036 | +1.2 | .885 |
−1.6 | .055 | +1.4 | .919 |
−1.4 | .081 | +1.6 | .945 |
−1.2 | .115 | +1.8 | .964 |
−1.0 | .159 | +2.0 | .977 |
−0.8 | .212 | +2.2 | .986 |
−0.6 | .274 | +2.4 | .992 |
−0.4 | .345 | +2.6 | .995 |
−0.2 | .421 | +2.8 | .997 |
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Conversely, the probability of completing the project by time period 60 is computed as follows:
Z=60 − 6425 + 9 + 1 + 1=− 436=− 0.67P≈0.26
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From Table A7.2, a Z value of −0.67 gives an approximate probability of 0.26, which is interpreted to mean there is about a 26 percent chance of completing the project on or before 60 time units. Note that this same type of calculation can be made for any path or segment of a path in the network.
When such probabilities are available to management, trade-off decisions can be made to accept or reduce the risk associated with a particular project duration. For example, if the project manager wishes to improve the chances of completing the project by 64 time units, at least two choices are available. First, management can spend money up front to change conditions that will reduce the duration of one or more activities on the critical path. A more prudent, second alternative would be to allocate money to a contingency fund and wait to see how the project is progressing as it is implemented.
EXERCISES
- Given the project information below, what is the probability of completing the National Holiday Toy project in 93 time units?
Act. ID | Description | Predecessor | Optm. (a) | Most likely (m) | Pess. (b) | Act time t_{e} | Variance [(b − a)/6]^{2} | Critical |
1 | Design package | None | 6 | 12 | 24 | |||
2 | Design product | 1 | 16 | 19 | 28 | |||
3 | Build package | 1 | 4 | 7 | 10 | |||
4 | Secure patent | 2 | 24 | 27 | 36 | |||
5 | Build product | 2 | 17 | 29 | 47 | |||
6 | Paint | 3, 4, 5 | 4 | 7 | 10 | |||
7 | Test market | 6 | 13 | 16 | 19 |
- The Global Tea and Organic Juice companies have merged.
The following information has been collected for the “Consolidation Project.”
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Activity | Description | Predecessor | a opt | m ml | b pess |
1 | Codify accounts | None | 16 | 19 | 28 |
2 | File articles of unification | None | 30 | 30 | 30 |
3 | Unify price and credit policy | None | 60 | 72 | 90 |
4 | Unify personnel policies | None | 18 | 27 | 30 |
5 | Unify data processing | 1 | 17 | 29 | 47 |
6 | Train accounting staff | 1 | 4 | 7 | 10 |
7 | Pilot run data processing | 5 | 12 | 15 | 18 |
8 | Calculate P & L and balance sheet | 6, 7 | 6 | 12 | 24 |
9 | Transfer real property | 2 | 18 | 27 | 30 |
10 | Train salesforce | 3 | 20 | 35 | 50 |
11 | Negotiate with unions | 4 | 40 | 55 | 100 |
12 | Determine capital needs | 8 | 11 | 20 | 29 |
13 | Explain personnel policies | 11 | 14 | 23 | 26 |
14 | Secure line of credit | 9, 12 | 13 | 16 | 19 |
15 | End | 10, 13, 14 | 0 | 0 | 0 |
- Compute the expected time for each activity.
- Compute the variance for each activity.
- Compute the expected project duration.
- What is the probability of completing the project by day 112? Within 116 days?
- What is the probability of completing “Negotiate with Unions” by day 90?
- The expected times and variances for the project activities are given below. What is the probability of completing the project in 25 periods?
ID | Description | Predecessor | t_{e} | Variance [(b − a)/6]^{2} |
1 | Pilot production | None | 6 | 3 |
2 | Select channels of distrib. | None | 7 | 4 |
3 | Develop mktg. program | None | 4 | 2 |
4 | Test market | 1 | 4 | 2 |
5 | Patent | 1 | 10 | 5 |
6 | Full production | 4 | 16 | 10 |
7 | Ad promotion | 3 | 3 | 2 |
8 | Release | 2, 5, 6, 7 | 2 | 1 |
Case A7.1
International Capital, Inc.—Part A
International Capital, Inc. (IC), is a small investment banking firm that specializes in securing funds for small- to medium-sized firms. IC is able to use a standardized project format for each engagement. Only activity times and unusual circumstances change the standard network. Beth Brown has been assigned to this Page 246client as project manager partner and has compiled the network information and activity times for the latest client as follows:
Activity | Description | Immediate Predecessor | ||
A | Start story draft using template | — | ||
B | Research client firm | — | ||
C | Create “due diligence” rough draft | A, B | ||
D | Coordinate needs proposal with client | C | ||
E | Estimate future demand and cash flows | C | ||
F | Draft future plans for client company | E | ||
G | Create and approve legal documents | C | ||
H | Integrate all drafts into first-draft proposal | D, F, G | ||
I | Line up potential sources of capital | G, F | ||
J | Check, approve, and print final legal proposal | H | ||
K | Sign contracts and transfer funds | I, J | ||
Time in Workdays | ||||
Activity | Optimistic | Most Likely | Pessimistic | |
A | 4 | 7 | 10 | |
B | 2 | 4 | 8 | |
C | 2 | 5 | 8 | |
D | 16 | 19 | 28 | |
E | 6 | 9 | 24 | |
F | 1 | 7 | 13 | |
G | 4 | 10 | 28 | |
H | 2 | 5 | 14 | |
I | 5 | 8 | 17 | |
J | 2 | 5 | 8 | |
K | 17 | 29 | 45 |
MANAGERIAL REPORT
Brown and other broker partners have a policy of passing their plan through a project review committee of colleagues. This committee traditionally checks that all details are covered, times are realistic, and resources are available. Brown wishes you to develop a report that presents a planned schedule and expected project completion time in workdays. Include a project network in your report. The average duration for a sourcing capital project is 70 workdays. IC partners have agreed it is good business to set up projects with a 95 percent chance of attaining the plan. How does this project stack up with the average project? What would the average have to be to ensure a 95 percent chance of completing the project in 70 workdays?
Case A7.2
Advantage Energy Technology Data Center Migration—Part B
In Chapter 6, Brian Smith, network administrator at Advanced Energy Technology (AET), was given the responsibility of implementing the migration of a large data center to a new office location.Page 247
Time in Workdays | ||||||
Task Name | Optimistic Dur. |
Most Likely Dur. |
Pessimistic Dur. |
Immediate Predecessor |
Critical Path | |
1 | AET DATA CENTER MIGRATION | 54 | 68 | 92 | ||
2 | Team meeting | 0.5 | 1 | 1.5 | ✓ | |
3 | Hire contractors | 6 | 7 | 8 | 2 | |
4 | Network design | 12 | 14 | 16 | 2 | |
5 | Ventilation system | — | — | — | — | |
6 | Order ventilation system | 18 | 21 | 30 | 2 | |
7 | Install ventilation system | 5 | 7 | 9 | 6 | |
8 | New racks | — | — | — | — | |
9 | Order new racks | 13 | 14 | 21 | 2 | ✓ |
10 | Install racks | 17 | 21 | 25 | 9 | ✓ |
11 | Power supplies and cables | — | — | — | — | |
12 | Order power supplies & cables | 6 | 7 | 8 | 2 | |
13 | Install power supplies | 5 | 5 | 11 | 12, 16 | |
14 | Install cables | 6 | 8 | 10 | 12, 16 | ✓ |
15 | Renovation of data center | 19 | 20 | 27 | 3, 4 | |
16 | City inspection | 1 | 2 | 3 | 3, 7, 10 | ✓ |
17 | Switchover Meetings | — | — | — | — | |
18 | Facilities | 7 | 8 | 9 | 14 | |
19 | Operations/systems | 5 | 7 | 9 | 14 | |
20 | Operations/telecommunications | 6 | 7 | 8 | 14 | |
21 | Systems & applications | 7 | 7 | 13 | 14 | |
22 | Customer service | 5 | 6 | 13 | 14 | ✓ |
23 | Power check | 0.5 | 1 | 1.5 | 13, 14, 15 | ✓ |
24 | Install test servers | 5 | 7 | 9 | 18, 19, 20, 21, 22, 23 | ✓ |
25 | Management safety check | 1 | 2 | 3 | 7, 23, 24 | ✓ |
26 | Primary systems check | 1.5 | 2 | 2.5 | 25 | ✓ |
27 | Set date for move | 1 | 1 | 1 | 26 | ✓ |
28 | Complete move | 1 | 2 | 3 | 27 | ✓ |
Careful planning was needed because AET operates in the highly competitive petroleum industry. AET is one of five national software companies that provide an accounting and business management package for oil jobbers and gasoline distributors. A few years ago, AET jumped into the “application service provider” world. Their large data center provides clients with remote access to AET’s complete suite of application software systems. Traditionally, one of AET’s primary competitive advantages has been the company’s trademark IT reliability. Due to the complexity of this project, the Executive Committee insisted that preliminary analysis of the anticipated completion date be conducted.
Brian compiled the following information, in preparation for some PERT analysis:
- Based on these estimates and the resultant expected project duration of 69 days, the executive committee wants to know what the probability is of completing the project before a scheduled time (T_{S}) of 68 days.
- The significance of this project has the executive committee very concerned. The committee has decided that more analysis of the duration of each activity is needed. Prior to conducting that effort, they asked Brian to calculate what the expected project duration would have to be to ensure a 93 percent chance of completion within 68 days.
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ADVANTAGE ENERGY TECHNOLOGY (AET)—ACCOUNTS PAYABLE SYSTEM
The AET sales department has been concerned about a new start-up company that is about to release an accounts payable system. Their investigation indicates that this new package will provide features which will seriously compete with AET’s current Accounts Payable system and in some cases exceed what AET offers.
Tom Wright, senior applications developer at AET, has been given the responsibility of analyzing, designing, developing, and delivering a new accounts payable system (A/P) for AET customers.
Complicating the issue is the concern of the sales department about AET’s recent inability to meet promised delivery dates. They have convinced CEO (Larry Martain) that a significant marketing effort will have to be expended to convince the clients they should wait for the AET product rather than jump to a package provided by a new entry to the petroleum software business. Companion to this effort is the importance of the performance of the software development group.
Consequently, Tom has decided to take the following action: tighten up the estimating effort by his developers; incorporate some new estimating procedures; and use some PERT techniques to generate probabilities associated with his delivery dates.
Tom’s planning team made a first-cut at the set of activities and associated durations:
Time in Workdays | ||||||
Task Name | Optimistic Dur. |
Most Likely Dur. |
Pessimistic Dur. |
Immediate Predecessor |
Critical Path | |
1 | ACCOUNTS PAYABLE SYSTEM | |||||
2 | Planning meeting | 1 | 1 | 2 | ✓ | |
3 | Team assignments | 3 | 4 | 5 | 2 | ✓ |
4 | Program specification | |||||
5 | Customer requirements | 8 | 10 | 12 | 3 | ✓ |
6 | Feasibility study | 3 | 5 | 7 | 5 | |
7 | Systems analysis | 6 | 8 | 10 | 5 | ✓ |
8 | Prelim budget & schedule | 1 | 2 | 3 | 7 | ✓ |
9 | Functional specification | 3 | 5 | 7 | 7 | ✓ |
10 | Prelim design | 10 | 12 | 14 | 9 | ✓ |
11 | Configuration & perf needs | 3 | 4 | 5 | 10 | ✓ |
12 | Hardware requirements | 4 | 6 | 8 | 11 | ✓ |
13 | System specification | 5 | 7 | 9 | 10 | |
14 | Detailed design | 12 | 14 | 16 | 12, 13 | ✓ |
15 | Program specification | 8 | 10 | 12 | 14 | ✓ |
16 | Programming—first phase | 27 | 32 | 37 | 15 | ✓ |
17 | Documentation | 14 | 16 | 18 | 10 | |
18 | Prototype | |||||
19 | Development | 5 | 7 | 9 | 16 | ✓ |
20 | User testing & feedback | 12 | 14 | 16 | 19 | ✓ |
21 | Programming—second phase | 10 | 12 | 14 | 16 | |
22 | Beta testing | 18 | 20 | 22 | 21 | |
23 | Final documentation pkg | 9 | 10 | 11 | 17, 20 | ✓ |
24 | Training pkg | 4 | 5 | 6 | 21SS, 23 | ✓ |
25 | Product release | 3 | 5 | 7 | 22, 23, 24 | ✓ |
SS = Start to Start lag
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- Based on these estimates and the critical path, the project duration is estimated at 149 days. But an AET salesperson in the Southeast Region has discovered that the competing A/P package (with significant improvements) is scheduled for delivery in approximately 145 days. The sales force is very anxious to beat that delivery time. The executive committee asks Tom for an estimated probability of reducing his expected project duration by two days.
- The executive committee is advised by Tom that after all the estimating was completed, he determined that one of his two critical systems analysts might have to move out of the area for critical family reasons. Tom is still very confident that with some staff rearrangements, assistance from a subcontractor, and some “hands on” activities on his part he can still meet the original delivery date, based on 149 days.
This news is very disconcerting to the committee and the sales staff. At this point, the committee decides that based on the most recent delivery performance of AET, a modified, comfortable delivery date should be communicated to AET clients—one that Tom and his staff are very likely to meet. Consequently, Tom is asked to calculate what the expected project duration would have to be to ensure a 98 percent chance of completion within 160 days—that is a “published, drop dead date” that can be communicated to the clients.
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