Difference between aon and pert

Overview

1) Overview – The continued discussion of project implementation by covering various scheduling techniques.

2) Background – Per the text, “A schedule is the conversion of a project action plan into an operating timetable.” A schedule is important because each project is unique in its own way. The basic process is to identify all tasks and sequential relationships between them, that is, which tasks must precede or succeed others. There are a number of benefits to the creation and use of these networks. Some of them are as follows:

a) It is a consistent framework for planning, scheduling, and controlling the project.

b) It can be used to determine a start and end date for every project task.

c) It identifies so-called critical activities that, if delayed will delay the project completion.

3) Network Techniques: PERT (ADM) and CPM (PDM) – PERT and CPM are the most commonly used approaches to project scheduling. Both were introduced in the 1950s. PERT has been primarily associated with R&D projects, while CPM with construction projects. Today PERT is not used much since project management software generates CPM style networks. The primary difference between them is that PERT uses probabilistic techniques to determine task durations, while CPM relies on a single duration estimate for each task. Both techniques identify the critical path (tasks that cannot be delayed without delaying the project) and associated float or slack in the schedule. In 2005 the Project Management Institute (PMI) deemed it necessary to change the names of these techniques. According to PMI, PERT is called ADM/PERT (Arrow Diagram Method) and CPM is PDM/CPM (Precedence Diagramming Method).

a) Terminology – Following are the key terms associated with the development and use of networks:

i) Activity – A specific task or set of tasks that have a start and end, and consume resources.

ii) Event – The result of completing one or more activities. Events don’t use resources.

iii) Network – The arrangement of all activities and events in their logical sequence represented by arcs and nodes.

iv) Path – The series of connected activities between any two events in a network.

v) Critical – Activities, events or paths which, if delayed, will delay the completion of the project.

To construct a network the predecessors and successors of each activity must be identified. Activities that start the network will have no predecessor. Activities that end the network will have no successor. Regardless of the technique used, it is a good practice to link the activities with no other predecessor to a START milestone. Those without any successor should be linked to an END milestone. PDM/CPM networks identify the activities as nodes in the network, called the Activity on Node (AON) network. The arrows in between the nodes depict the predecessor/successor relationships among the activities. The ADM/PERT method, on the other hand, uses Activity on Arrow (AOA) networks. Here the nodes represent events and the arrows represent the actual activities.

b) Constructing the Network, AON Version – The text illustrates the development of a simple AON network. All the major project management software packages will generate this type of network.

c) Constructing the Network, AOA Version – The AOA network has some development rules that make it somewhat more difficult to construct than the AON network. The primary rule is that any given activity must have its source in one and only one node. As a result, some network relationships can only be depicted with the use of a “dummy” activity. This is an activity that has no duration and consumes no resources. Its sole purpose is to indicate a precedence relationship. The text uses various figures to illustrate the use of dummy activities.

d) Gantt (Bars) Charts and Microsoft® Project (MSP) – The most familiar tool for depicting project schedules is the Gantt chart, invented by Henry L. Gantt in 1917. The activities are depicted as horizontal bars with their length proportional to their duration. This method results in an easy to read graphical depiction of the project schedule. Gantt charts can be difficult to maintain if there are large changes in the project schedule. Tools like Microsoft® Project (MSP) will automatically draw the Gantt chart as a byproduct of the network entered by the user. The disadvantage of the Gantt chart is that it typically does not depict the network relationships.

e) An Important Aside on Estimating Activity Times – It is important that the duration of each of the tasks in a project is honestly estimated. The deadlines should be set as close to the actual deadline needed to ensure proper buy-in from everyone throughout the project. As is the norm with many project managers, the deadlines are set earlier than what is required and the workers overestimate the deadlines to ensure task completions and delivery, both of which lead to serious errors in the long run of a major project.

f) Solving the Network – The text illustrates the development and solving an AON network based on the project detailed in Table 8-1.

g) Calculating Activity Times – The sample project in the text has three duration estimates for each activity: optimistic (a), most likely (m) and pessimistic (b). Optimistic and pessimistic are defined as the durations that represent 99 percent certainty. In other words the actual duration of an activity will be less than the optimistic or greater than the pessimistic only one percent of the time. Then, the expected time (TE) is found with the formula:

TE = (a + 4m + b)/6

where:

a = optimistic time estimate

b = pessimistic time estimate

m = most likely time estimate, the mode

This formula is based on the beta statistical distribution. In spite of a flurry of discussion in the 1980s the assumptions used to derive this formula have stood the test of time. Along with TE, the variance of the durations can be calculated as:

and the standard deviation as:

h) Critical Path and Time – Using the example network, the text describes the concept of the critical path. For simple projects, the critical path can be found by determining the longest path through the network.

i) Slack (aka, Float) – In the previous section, the earliest possible dates for each activity were determined. By starting the analysis at the end of the network and working through it backwards, the latest possible dates for each activity can be determined. The difference between the early dates and the late dates is float or slack. Activities on the critical path have zero float.

j) Precedence Diagramming – The Precedence Diagram Method allows for additional relationships to be established between activities. They are:

i) Finish to Start – The successor activity cannot begin until the predecessor finishes. This is the most common relationship

ii) depicted in networks.

iii) Start to Start – The successor activity cannot begin until the predecessor begins.

iv) Finish to Finish – The successor activity cannot finish until the predecessor activity finishes.

v) Start to Finish – The successor activity cannot finish until the predecessor activity starts. This relationship is rarely used.

In addition to these relationships, PDM allows for leads and lags which is the introduction of a specific time period between the linked activities. For example, in a Start to Start relationship with five days of lag, the successor activity cannot begin until five days after the predecessor starts. The critical path and slack calculation resulting from these relationships can be complicated and counter intuitive.

k) Once again, Microsoft® Project – The text illustrates the use of MSP for calculating the most likely project duration using the PERT method.

l) Exhibits Available from Software, a Bit More MSP – The text illustrates the types of outputs available from MSP.

m) Uncertainty of Project Completion Time – The chance of completing a project within a given time period can be calculated. The project activities are assumed to be statistically independent and the variance of a set of activities is equal to the sum of the variances of the individual activities comprising the set. Then the chance of meeting a particular project duration can be calculated as:

where:

D = the desired project completion time

µ = the critical time of the project, the sum of the TEs for activities on the critical path

= the variance of the critical path, the sum of the variances of activities on the critical path
Z = the number of standard deviation of a normal distribution (the standard normal deviate)

The weakest element of this technique is that it is difficult to account for the possibility that other paths through the network may become critical due to variation in their duration. Simulation techniques using tools like Crystal Ball® or Risk +® can be used to better model this situation.

n) Toward Realistic Time Estimates – The traditional PERT method uses optimistic and pessimistic duration estimates at the 99% confidence level. The calculations can be adjusted for lower confidence levels that estimators may be more comfortable in predicting such as 90 or 95%.

4) Risk Analysis Using Simulation With Crystal Ball® – Tools like Crystal Ball® can be used to model the project and determine the likelihood of completion within a certain time. Since Crystal Ball® works with Microsoft® Excel, the project network must be modeled in the spreadsheet. This involves creating cells that calculate the early and late dates for each activity. Then a distribution (commonly triangular) with the appropriate parameters can be assigned to each duration. Finally, Crystal Ball® runs its simulation and the results are displayed.

a) Traditional Statistics or Simulation? – With the advent of inexpensive and easy to use tools, simulation is the recommended way to model uncertainty in project durations. Both methods require the development of three durations for each activity. The simulation method, however, does a much better job of handling the possibility that the critical path will shift due to variation in durations of activities. This issue can be analyzed with traditional statistics, but it takes considerable manual effort on the part of an analyst.

5) Using These Tools – The text gives an example of the use of the tools on a specific project.

Teaching Tips

The authors correctly applaud the advent of many user-friendly and powerful project management tools available for PCs. They have performed a great service by integrating the use of tools like Microsoft® Project into the subject matter. My experience in teaching these tools, however, reminds me of the story about IBM’s new programming language. The story goes that IBM, in their marketing campaign for their new language, said that the language was so easy to use that it would virtually eliminate the position of a programmer. Anybody would be able to use this language to easily create computer programs. The punch line is that the new language was FORTRAN, a 1960’s era product long ago supplanted by other “user-friendly” products. Even today, with all the marvelous tools we have available, the position of a programmer has not disappeared. The point of all this is that in the classroom students will have a wide range of skills and abilities that may or may not be applicable to Microsoft® Project and similar tools. My experience is that it is unreasonable to expect students to pick up MSP and use it successfully based solely on the printed examples in the text. Unfortunately, even students who claim to be experienced in the use of the tool often know how to draw a Gantt chart and little else. This is due to three reasons:

· MSP on the surface may look like a fancy spreadsheet, but “under the hood” it’s is a very complex tool. Many of its processes are dependent on complex algorithms, controlled by a seemingly endless series of settings with mysterious titles.

· The training provided by most organizations ranges between non-existent and abominable. If there is training, it’s usually administered by someone who has never managed a project and doesn’t understand much more than the student he/she is teaching. One of my very computer literate colleagues will probably become homicidal, if he is forced to go to yet another training session that concentrates on issues vital to the PM like changing the color of the fonts. In my twenty years of both managing projects and teaching project management, I have encountered just one person who is both an accomplished user of MSP on actual projects and a capable instructor.

· MSP training, if it exists, is usually done out of context. It is taught as a standalone computer tool, without any of the concepts of project management to put its use in the proper perspective.

The opportunity then, for any project management instructor, is to provide both the concept and meaningful tools training in one package. Ideally, lectures on concept should be alternated with lab sessions using the tool. If this is not possible then, at a minimum, the instructor must set aside class time to demonstrate the key elements of MSP needed to complete the homework problems. Then, a week later, be prepared for the questions and frustrations that will erupt from the class.

A good reference case for this chapter follows:

Ivey cases:

9A97D001 Note on Introduction to Project Management. An introduction to projects with a simple AON problem.

9A97D002 Gadget Toy Company. A simple AON problem that also introduces Microsoft Project (MSP).

9A95D015 H.M.S. Pinafore. A moderately longer AOA problem.

9A98D020 Procter & Gamble Canada: Dayquil Sampling Operations. A realistic problem based on an actual summer intern’s experience involving a quick decision on a new product line that requires a number of tasks to be executed.

University of Virginia Darden case: UVA-OM-0803 Tastee Snax Cookie Company. A straightforward, but more involved network problem.

Material Review Questions

Question 1:

Activity: Activities have a start and end, and consume resources.

Event: Events do not consume resources. Typically, events designate the start or the completion of an activity or a path

Path: A path traces the predecessor-successor relationships that exist among a set of interconnected activities and events. The boundaries of a path use a starting event and a completion event to designate the path’s endpoints.

Dummy Activity: Dummy activities are tasks of zero duration that have no resources assigned to them. They are used to maintain predecessor/successor relationships in AOA networks.

Question 2:

Activities along the critical path cannot be delayed without delaying the completion of the project.

Question 3:

Gantt Chart: The Gantt chart compares planned and actual progress for the detailed tasks in a project.

Master Schedule: The Gantt chart format (bars to represent progress over time) may be used to display data regarding the master project schedule, but the master schedule is oriented towards overall management of the project and will only focus on the major project tasks. For example, the Gantt view in MS-Project can be filtered to only show summary tasks at a particular level of the WBS hierarchy.

Question 4:

Total slack vs. free slack:

Total slack is the difference between the calculated earliest finish time of the very last activity and the project’s required completion time.

Free slack is the time an activity can be delayed without affecting the start time of any successor activity.

So, the total slack deals with the relationship between the current activity and the total project completion time, while free slack relates to the next activity.

Question 5:

The authors of the text have suggested that PERT and CPM are very similar. Therefore, the terms PERT and CPM have been used interchangeably in the textbook, when basic educational concepts of a project schedule is explained. The following guidelines are suggested with regard to when to use each type of scheduling technique discussed in this chapter.

1) PDM/CPM should be used when the control of costs associated with expediting work is an important concern. PDM networks should be used when the project requires the use of leads and lags between activities. PDM is easier to draw than ADM, is used in most project software applications, and tends to be preferred when CPM is used.

2) ADM/PERT should be used when the activity times are estimated using probability distributions in order to evaluate the range of uncertainty around the expected project duration. ADM networks should be used, when it is desirable to show completion events as a part of the scheduling network, though nothing prevents the use of START and FINISH events in a PDM/CPM network.

3) The Gantt chart is a useful tool for displaying the schedule regardless of what method is used to derive it. The Gantt chart can be used directly to develop small project schedules. A less-known approach: GERT should be used when the project plan is complex enough to require loop backs and/or the use of multiple probability distributions associated with branching options in the relationships between activities.

Question 6:

AON (activity on node) places the activities or tasks on a rectangle (node), whereas the AOA places the activities on arrows connecting nodes. Typically the AON provides more information per activity in the diagram itself because more information can be placed on the node itself (start time, finish time, etc).

Question 7:

Simulation requires the project schedule to be modeled mathematically, which happens to be a by-product of any of the network scheduling techniques. Once the model is established, simulation involves inputting appropriately distributed random numbers into the independent variables and analyzing the resulting distribution of the dependent variables (those calculated by the model). To make the result meaningful, hundreds if not thousands of trials are run, to build a statistically significant output distribution. Once the output distribution is established, probabilities of various outcomes can be calculated.

Question 8:

Networks are drawn from the left to the right. Arrowheads indicate the direction of flow in the network. The flow designates the precedence relationships between activities in the network.

Question 9:

Late start time: Given the precedence relationships in a network, this is the latest time that an activity can begin without extending the time required to complete the entire project.

Early start time: Given the precedence relationships in a network, this is the earliest time that an activity can begin. In order to begin, all predecessor constraints must have been satisfied.

Early finish time: Given the precedence relationships in a network and the activities duration, this is the earliest time that an activity can be completed if all predecessor constraints are satisfied.

Question 10:

The critical path is determined by performing the “forward pass” and “backward pass” calculations. Float is calculated by subtracting the early dates from the late dates, specifically the early start from the late start or the early finish from the late finish. If an activity has zero float, then it is on the critical path. Any delay in an activity along the critical path would extend the project’s completion date.

Question 11:

Slack is important for two reasons: 1) Slack tells us that we can be a bit more forgiving about delays on paths with slack, whereas our primary attention should be focused on the critical path. 2) If we need additional resources for some reason (such as a delay on the critical path), the first place to look is at the resources on paths with slack in case they might be available for use.

Class Discussion Questions

Question 12:

The network diagram could serve as a rough process flowchart showing the steps in a manufacturing cycle. The direct and indirect costs for each step could be identified and scheduled for each iteration of the total manufacturing cycle. However, the basic networks are not sophisticated enough to capture costs associated with variables such as production yields, rework loop backs, and branching logic commonly associated with control points to assure quality.

Question 13:

1) Benefits:

a) Illustrates task interdependencies

b) Establishes the sequence of activities (precedence)

c) Highlights critical and near-critical paths and their tasks

d) Highlights activities that contain float

2) Disadvantages:

a) Emphasizes time at the expense of other dimensions of project success

b) Large networks are difficult to print in a convenient format and they may require significant wall space to view the entire network

c) As the network technique becomes more complex, its effectiveness as a control tool is reduced

Question 14:

This is a good question to kick off a lively class discussion. There are no black and white answers to this question, but here are a couple thoughts:

· It’s easy to become obsessed by the use of even more sophisticated tools and lose sight of the big picture. Project management tools are only useful if they help projects achieve their cost, schedule, and scope goals. Just because a tool is more sophisticated doesn’t mean that it will yield a better result for the business.

· Organizations have to clearly articulate the goals of a project, put together some kind of a plan, and then meticulously monitor its execution. Many organizations gloss over the monitoring part because they believe it smacks of micro-management. In spite of what Dilbert thinks, managers must have a mechanism for knowing where their project is every day. This allows corrective action to be taken before the problem grows beyond recovery. This attention to detail is boring and repetitive, but it’s far more fundamental to the success of the project than the sophistication of the simulation tools used to model the plan.

Question 15:

Both methods are of significant value because they force the PM to consider the relationships among the project activities. Then using these relationships, both methods produce a schedule for those activities. In addition, both methods can be used for analysis of variances and problems when the schedule is executed.

Question 16:

There are many ways to deal with uncertainty. The most common in the scheduling process are:

1) Adding buffer or padding to the duration of each activity

2) Adding buffer to the overall project schedule

3) Developing schedules based on a range of activity durations

4) Calculating probabilities of completion using statistical or simulation techniques

5) Taking specific actions to reduce the uncertainty in duration for some or all the activities

Question 17:

The “free slack” as it is called, is the slack along a path in the project and is the minimum of all the slacks on that path. Thus, if the path of interest is A-B-C and the slacks on A and B are 3 each, while the slack on C is 2, the free slack on the path is 2.

Question 18:

Activity times are generally estimated in a manner similar to budgets. For example, they can be individually estimated by the participants or calculated based on time estimates from earlier projects.

Question 19:

Yes and no. Critical path activities deserve closer scrutiny. If critical activities run late the project is sure to be late. In a situation where scant resources have to be allocated to help late activities recover, the critical path activities would get the resources before the noncritical path activities.

This, however, does not absolve the PM from monitoring the noncritical path items. Items off the noncritical path may feed the critical path, so if they are late they could delay the project indirectly. Also if noncritical path activities get late enough then the critical path may shift to them, again delaying the entire project.

Question 20:

I’m not aware of any network relationship that can’t be built through some combination of the PDM relationships with leads and lags. As the text points out, the relationships can become quite complicated, which leads to anomalies in the critical path.

Questions for Project Management in Practice

Replacing the Atigun Section of the TransAlaska Pipeline

Question 21:

Probably work was done underground in order to avoid wildlife like grizzly bears.

Question 22:

Petroleum engineers built a bypass system, which was used to divert the oil flow temporarily, for repairs without interrupting it.

Question 23:

The environment for this project was very hostile. In addition to limited sunlight (3 hours per day), temperatures were as low as –60 degrees during winter. Unless robots could be used, shifts were likely to be limited to the time one could withstand the temperatures and still avoid frostbites.

Designing and Delivering a Rush Vehicle for War

Question 24:

For the first billion dollar contract, the army paid around $450,000 per vehicle and about $590,000 per vehicle for the second order. The high price can be attributed to the ultra-efficient schedule followed by the manufacturing company. The company maintained tight schedules and efficient problem-solution relay mechanisms to provide a quality solution within the short time-frame for the army. Producing a car is never such a critical problem. A highly efficient and tight schedule is not necessary to produce a car. Producing a rush vehicle for a war is entirely different than producing a car for, say, the general public. The situation and its relative importance definitely demand a premium price.

Question 25:

Need is the mother of invention. Never before the military encountered such a situation where they needed a new type of vehicle specialized for the situation. The tight schedule demanded a different approach of parallel production to development, which is not a norm with the military equipment industry and to which Oshkosh adopted seamlessly.

Question 26:

This new approach may hinder the development of new components. The time-frame of such projects is so small that rarely a firm would be able to develop and test new components. Another weakness could be the inefficient testing that could result from fast prototypes to productions, which in turn would increase the price of the end-product.

Hosting the Annual Project Management Institute Symposium

Question 27:

One unique aspect of this project is its length. The Gantt chart shows that planning for the symposium began more than four years before the event and continued for a year after. This means that several symposia are continuously in the planning process throughout the United States.

Question 28:

The symposium took place in September 1992 and the supporting project completed in April of 1993.

Question 29:

The activities after completion of the symposium are tasks associated with project closeout work. This would include tasks related to contract closeout and administrative closeout, and creation of a project archive and a summary report of lessons learned. For this project, closeout lasted approximately 6 months. This question points out the importance of establishing a common understanding of when any project is actually done. The answer may not be obvious, and it can come back to bite the PM when he/she least expects it.

Election Returns within Three Hours

Question 30:

While each of the two thrusts had its own difficulties, I believe that the integration of each citizen’s records with their respective biometric data was a more difficult task. The primary reason for this is that the capturing of pertinent information of each and every citizen involved is a humungous task, which is only possible with a dedicated team of professionals since it is a continuing effort spanning many years. The development and integration of tools needed for this task and their subsequent successful use is highly commendable.

Question 31:

It is indeed difficult but not impossible for other countries to replicate this system. The only problems these countries would encounter would be the efficient handling of such a long project, that will span years since counting and integrating each and every citizen on the face of the country is a monstrous task, and can only be accomplished through thoughtful leadership and thorough planning. I believe that sooner or later every developed nation would think of building and customizing a system like this as a necessity.

Question 32:

The developed nations could easily ignore the technological aspects of this establishment. Many developed nations already have the necessary technological infrastructure that is used in this system. Other nations could ignore the biometric fingerprint integration of each and every citizen and proceed with simple establishment of a database through valid birth and death records, which are mostly present in every country now. This would be very important in achieving faster returns through fast development.

Problems
NOTE: Many of the AON graphics in this solutions set depict the start day of the successor activity to be the same day as the completion of the predecessor. This is consistent with the presentation in the text. It is not consistent with the result that would be obtained using Microsoft® Project, where the start day of the successor is always the next working day after the completion of the predecessor.

Problem 1:

q8-1a

Problem 2:

Problem 3:

1) The arrows cannot form a loop such as the one shown between nodes 2, 3, and 5.

2) The dummy arrow between nodes 6 and 7 is not required because 6 precedes 5 and 5 precedes 7.

3) Nodes 8 and 9 do not have successors, so it appears that this network has two final termination nodes. This is not a conventional diagramming technique. An arrow from 8 should point to 9.

Problem 4:

a) The critical path is B-E-G.

b) 23 work periods.

Problem 5:q8-5a

Initial PDM Diagram

q8-5bAdjusted PDM Diagram

Problem 6:

q8-6a PDM Diagram 6a

PDM Diagram 6b

q8-6b

Problem 7:

q8-7Figure 7a is the AOA diagram.

Figure 7b is the AON diagram.

Problem 8:

q8-8Please see note about network depiction preceding Problem 1

a) The critical path activities are A, C, E, and G.

b) The project’s duration is 22 days.

c) Yes. Activity B can be delayed by one day without delaying the completion of the project.

Problem 9:

Task

a

m

b

Expected

Variance

Std Dev.

A

6.5

7.5

14.5

8.5

1.78

1.33

B

8.5

10.5

12.5

10.5

0.44

0.67

C

2.5

3.5

4.5

3.5

0.11

0.33

D

6.5

7.5

8.5

7.5

0.11

0.33

E

5.5

5.5

9.5

6.2

0.44

0.67

F

5.5

7.5

9.5

7.5

0.44

0.67

G

4.5

6.5

8.5

6.5

0.44

0.67

H

2.5

3.5

3.5

3.3

0.03

0.17

Desired Duration

Expected Project Duration

Sum of Variances Critical Path

Z

Probability

21

24.7

2.77

–2.2

a) 1.4%

22

24.7

2.77

–1.6

b) 5.5%

25

24.7

2.77

0.2

c) 57.9%

Problem 10:

a) The critical path is AC CB BE EF.

b) The only event with slack is “D” at 3 days.

c) If “D” were the final event in the network, then the critical path would be AC CB BD.

d) The following spreadsheet excerpt illustrates the calculation of the probability of completion in 14 days:

Task

a

m

b

Expected

Variance

Std Dev.

AB

3

6

9

6.0

1.00

1.00

AC

1

4

7

4.0

1.00

1.00

CB

0

3

6

3.0

1.00

1.00

CD

3

3

3

3.0

0.00

0.00

CE

2

2

8

3.0

1.00

1.00

BD

0

0

6

1.0

1.00

1.00

BE

2

5

8

5.0

1.00

1.00

DF

4

4

10

5.0

1.00

1.00

DE

1

1

1

1.0

0.00

0.00

EF

1

4

7

4.0

1.00

1.00

Desired Duration

Expected Project Duration

Sum of Variances Critical Path

Z

Probability

14

16.0

4.00

–1

15.9%

e) If CD slips to six days the critical path is unchanged, but slack on D is reduced. If CD slips to seven days, then there are two critical paths: AC CB BE EF and AC CD DF. If CD slips to eight days then the critical path shifts to AC CD DF and the project duration extends to 17 days.

Problem 11:

Figure 8.11 shows duration on the arrow in matching the “(i,j)” notation used to define the problem’s source data.

b) The critical path is A, D, C, E, F, G, H, and J.

c) The completion time is 43 days.

Problem 12:

q8-12 Figure 8.12a shows the PDM network for the data from Table A of Problem 8-12 assuming that the data were applied as shown in Figure 8.12b.

Please see note about network depiction preceding Problem 1

1) The critical path is 2, 3, 4, 5, 7, 8, and 9.

2) The slack for activity 1 is 11.7 days. The slack for activity 6 is 4 days.

2) The following table shows the calculation of the expected completion time:

Activity

a

m

b

Expected

1

8

10

13

10.2

2

5

6

8

6.2

3

13

15

21

15.7

4

10

12

14

12.0

5

11

20

30

20.2

6

4

5

8

5.3

7

2

3

4

3.0

8

4

6

10

6.3

9

2

3

4

3.0

Expected Project Duration

66.4

Problem 13: q8-13

Figure 8.13 shows the network for problem 13.

1) The critical path is A, B, E, I, L, M, N, and P.

2) The completion time is 75 months.

Problem 14:

q8-14

Figure 8.14a shows the original network diagram for problem 14.

Please see note about network depiction preceding Problem 1