Module Seven Notes
"Project Scheduling and Network Analysis"

Index to Module Seven Notes

7.1: Project Scheduling

7.2: Uncertain Activity Times

7.3: Time-Cost Tradeoffs

7.1: Project Scheduling: PERT/CPM

The Project Evaluation and Review Technique (PERT) and Critical Path Method (CPM) were developed by management scientists to help organizations with planning, scheduling and controlling large projects, such as building a new hospital or launching a new product. I first became familiar with the utility of project scheduling in my Air Force career when we used PERT/CPM to schedule activities associated with the construction of an air field in Spain. More recently, last semester I worked with an MBA student in applying the techniques to help in scheduling subcontractors to build car wash facilities throughout the state. I even wanted to use PERT/CPM to schedule our wedding activities, but when you are young, in love and/or elope you don't have time for analysis (just kidding - at least about the analysis part).

When I speak of large projects I mean an undertaking that has series of interdependent activities that take time to complete, require funds and resources, such as time and labor. Interdependence means that activities follow a given sequence or precedence relationship - some activities cannot start until others are completed. PERT is a scheduling technique that was specifically designed by the Navy in 1958 for projects with uncertain activity times. CPM was designed by Remington-Rand and DuPont in 1957 to address the time-cost tradeoff: if the project manager wishes to accelerate a project so that it is completed faster than originally planned, there is a cost tradeoff. Today, we generally speak of PERT/CPM as a single quantitative method with a number of analysis components. Further, today the method is as applicable to small projects as to large ones.

Project Scheduling can be broken down into twelve general steps. These are summarized below, and I will explain them in more detail with an example.

Step 1. Identify the activities

Step 2. Determine activity relationships (immediate predecessors of each activity)

Step 3. Estimate activity completion times and costs

Step 4. Construct an activity network

Step 5. Execute a forward pass to determine earliest start and earliest finish times for each activity, and project completion time

Step 6. Execute a backward pass to determine latest start and latest finish times for each activity

Step 7. Identify activity slack (length of time an activity can be delayed without delaying the project completion time)

Step 8. Find the activities with zero slack; these are critical activities and make up at least one critical path

Step 9. Use information from Steps 5 - 8 to develop the activity schedule for the project.

Step 10. Find project completion time variance and conduct probability analysis, such as the probability of meeting a customer target completion time, under the condition of uncertainty in activity times.

Step 11. Consider time-cost tradeoffs

Step 10. Implement , monitor and control the project

The example project involves the marketing of a new product. There are ten activities (Project Scheduling Step 1) with precedence relationships (Step 2) and completion times in weeks (Step 3) shown in Table 7.1.1 It is noted that Project Scheduling Step 3 also includes gathering activity cost data. That part of Step 3 will be addressed in Module 7.3 Notes.


Table 7.1.1

ACTIVITY	TITLE	IMMEDIATE PREDECESSOR	ACTIVITY TIME
A	Identify market need	-	3
B	Conduct R&D	A	60
C	Design Packaging	A	5
D	Select production site	A	15
E	Conduct product test	B	6
F	Install production process	D	40
G	Perform market analysis	C , E	10
H	Production startup	F	7
I	Make modifications	G	6
J	Market product	H , I	12

The first activity that begins this example is to identify market need for the new product. Please understand that this main activity may involve multiple sub-activities, each with their own precedence relationships, resource needs, time and cost. Thus, this is an example of a top-level project schedule. I am adding a letter to identify each activity, as The Management Scientist uses only letters for activity identification.

The next activity is to conduct project research and development. This activity takes 60 weeks and cannot begin until Activity A is completed...this is what is meant by precedence relationships... and so goes the list of project activities. Note that Activity G, Perform Market Analysis, cannot begin until both Activities C and E are completed.
Project Scheduling Step 4 involves construction of a graphical representation of the project, called an Activity Network. The activity network for this project is shown in Figure 7.1.1.


Figure 7.1.1.

		-	->	B	---	---	->	E	---	---	->	G	---	---	->	I	---	---	->	J
		|		60				6				10				6				12
		|										^								^
		|										|								|
A	----	---|	->	C	---	---	---	---	---	---	---	|								|
3		|		5																|
		|																		|
		|																		|
		-	->	D	---	---	->	F	---	---	->	H	---	---	---	---	---	---	---	-
				15				40				7

One purpose of the activity network is to illustrate the precedence relationships by arrows. These arrows connect to nodes that represent the activities and their time duration. Another purpose is to illustrate the network paths which are formed by tracing activities and arrows from left to right through the network. This network shows three paths from the first activity to the last. The top path is A-B-G-E-I-J. The summation of the activity times on this path is 3+60+6+10+6+12 or 97 weeks. The middle path is A-C-G-I-J which takes 3+5+10+6+12 or 36 weeks. The bottom path is A-D-F-H-J which takes 3+15+40+7+12 or 77 weeks.

While all activities on all paths have to be completed, the longest path through the network is the top path which takes 97 weeks. This path is called the critical path. If any activity is delayed on the critical path, the entire project is delayed.

I used the "brute force" method of finding the critical path. Project Scheduling Step 5 consists of a mathematical algorithm for finding the critical path, and is used by The Management Scientist and other Project Management Software such as Microsoft Project. Specialized project management software has the advantage of producing graphical activity networks which The Management Scientist cannot do. However, the disadvantage is that these packages cost much more than The Management Scientist and are limited to one of the family of quantitative methods covered in this course.

While you will be using The Management Scientist for Steps 5 and 6, I will illustrate the algorithms so we can better understand the concept. We will use Figure 7.1.2 to add the activity analysis to an expanded copy of the activity network.


Figure 7.1.2

		-	->	B	3	63	->	E	63	69	->	G	69	79	->	I	79	85	->	J	85	97
		|		60				6				10				6				12
		|										^								^
		|										|								|
A	0	3	->	C	3	8	---	---	---	---	---	|								|
3				5																|
		|																		|
		|																		|
		-	->	D	3	18	->	F	18	58	->	H	58	65	---	---	---	---	---	-
				15				40				7

Note that I added some numbers to the right of the activity name (letter). The numbers represent earliest start and earliest finish times for each activity. Table 7.1.2 provides the definition for these numbers, as well as the latest start and finish times which I will fill in and illustrate in Figure 7.1.3 in a few more paragraphs.


Table 7.1.2

Activity Name or Letter	Earliest Start Time for Activity	Earliest Finish Time for Activity
Activity Time	Latest Start Time for Activity	Latest Finish Time for Activity

Under Project Scheduling Step 5, the first thing we do is indicate "0" as the earliest start (ES) time for all project starting activities (those activities having no immediate predecessors). There is only one activity that has no immediate predecessors for this example - Activity A, so it gets a "0" ES. Next, we compute the earliest finish (EF) time for Activity A using the general formula:

EF Time for an Activity = ES time for an Activity + Activity Time

For Activity A, this is 0 + 3 = 3, as shown on Figure 3.1.2. To compute the ES for all activities that have immediate predecessors, we use the formula:

ES for an Activity = Largest of the EF times for that activity's

immediate predecessors

Thus, the ES's for Activities B, C, and D are all 3, since 3 is the EF for Activity A, the immediate predecessor to B, C and D. Figure 3.1.2 shows the remaining forward pass computations. Note that Activity G has two immediate predecessors, C and E, with EF's of 69 for E and 8 for C. The earliest time Activity G can start is 69, since it can't start until both E and C are completed.

At the end of the forward pass, the maximum of the EF times for all terminal activities (activities with no successors) is the project duration. In this example there is just one terminal activity which has an EF of 97 weeks - that is the expected project completion time. To complete the activity time analysis, we follow Step 6 and execute a backward pass. The numbers for the backward pass are shown in Figure 7.1.3.


Figure 7.1.3

		-	->	B	3	63	->	E	63	69	->	G	69	79	->	I	79	85	->	J	85	97
		|		60	3	63		6	63	69		10	69	79		6	79	85		12	85	97
		|										^								^
		|										|								|
A	0	3	->	C	3	8	---	---	---	---	---	|								|
3	0	3		5	64	69														|
		|																		|
		|																		|
		-	->	D	3	18	->	F	18	58	->	H	58	65	---	---	---	---	---	-
				15	23	38		40	38	78		7	78	85

Project Scheduling Step 6, the backward pass, which results in finding the latest start (LS) and latest finish (LF) times for each activity, begins at the terminal activities. We make the LF for Activity J the same as the EF for that activity since it is the maximum of the terminal activity earliest finish times. In doing so, the LF time can be thought of as the latest time an activity can finish without delaying the project. The LS for Activity J is found by the formula:

LS for an Activity = LF for the Activity - Activity Time

So for Activity J, the LS is 97 - 12 = 85. The latest start time is also the latest time an activity can start without delaying the project. We next compute the LF for the immediate predecessors by the formula:

LF for an Activity = Smallest of the LS times for that

activities immediate successors

Looking at Activity I, J is the only successor activity so the LF time for Activity I would be the LS for Activity J, which is 85. The LS for Activity I is 85 - 6 = 79. The Backward pass continues moving right to left through the network until we get to Activity A, the single starting activity. To find the LF for Activity A, we compare the LS its three successors: 3 for Activity B, 64 for Activity C, and 23 for Activity D. Since 3 is the minimum, 3 becomes the LF time for Activity A. The LS for Activity A is 3 - 0 = 0. At least one starting activity must have 0 for the LS time, or there was an error in the backward pass calculations.

Pause and Reflect
We have just completed the mechanics of finding, for each activity, earliest start, earliest finish, latest start, and latest finish times, as well as the project completion time. This tactical information is critical to project managers to help ensure successful projects (on time projects). I know that several corporations in Southwest Florida, GE Client Business Services for one, expects their MBAs to be successful at project management - understanding basic scheduling metrics is part of that success.

At this point we have finished the measurements for activity analysis. The project manager knows when each activity should start and stop in order to keep projects on schedule. We can use this information to determine which activities have slack, which is Project Scheduling Step 7.

Activity slack is computed by the following formula:

Slack = LS - ES = LF - EF

For example, the slack for Activity G is "0", since LS - ES = 69 - 69 = 0 and LF - EF = 79 - 79 = 0. This means that Activity G must start at week 69 and finish by week 79: any delay will delay the project, thus we say it has no slack. Suppose there was a problem in getting data for market analysis (Activity G) and the activity takes 11 weeks instead of 10. What is the new project completion time? That's right: 98 weeks, reflecting the one extra week of activity time.

Now let's look at Activity D. You should note that Activity D has slack of 20 weeks (LS - ES = 23 - 3 = 20). This means that Activity D can start at week 3, but it doesn't have to. It could start at week 4, or 5, or at any delay up to week 23, the latest start time (a 20 week delay) - it just can't start later than week 23. If it does, the project will be delayed. Let's say Activity D takes 40 weeks instead of the original estimate of 15. The new project completion time will be 102 weeks. Did you get that? The added activity time is 40 - 15 or 25 weeks, but that's not the delay for the project since there is an offset by the 20 weeks of slack. So, 25 - 20 gives a net delay of 5 weeks, making the project completion time 97 + 5 = 102.

We can proceed through the network and compute the slack for each activity. By observation I note that Activities A, B, E, G, I and J all have no slack. These activities are said to be critical activities because they have no slack. Note that these activities are connected on one path. There will always be at least one path through a network containing only critical activities. This path is called the critical path (hence the title, Critical Path Method). From a scheduling standpoint, the activities on the critical path require careful monitoring since they have no slack. The identification of the critical path(s) in the network is Project Scheduling Step 8. This example has just one critical path, but there could be more than one.

When we have finished Step 8, we know activity earliest and latest start and stop times, activity slack, project completion time and the critical path(s). Putting this all together provides an activity schedule, Project Scheduling Step 9. One way to show the schedule is to show the activity start and stop times on the network diagram, and somehow indicate the critical path (I put the activity letters in bold in Figure 3.1.3). The better alternative is to rely on a project management software program to provide the activity schedule.


Using The Management Scientist Software Package
We will be using The Management Scientist PERT/CPM Module to do the actual activity, project completion time and critical path analyses. To illustrate the package for the first example, click Windows Start/Programs/The Management Scientist/The Management Scientist Icon/Continue/Select Module 7 PERT/CPM/OK/File/New and you are ready to load the example problem. The next dialog screen asks you to enter Known or Uncertain Activity times. For this example, select Known Activity Times, then enter 10 for Number of Activities, then click OK, and start entering your data. Note that the left input dialogue screen requires the selection of an activity by clicking it , then selecting the activity's predecessor(s) by clicking from the available list in the center input screen. The predecessor list will build as you continue selecting activities. Be sure to enter the activity time in the right input box. After data entry, select Solution, then Solve and you should get the following solution:

Printout 7.1.1

PROJECT SCHEDULING WITH PERT/CPM

********************************

*** PROJECT ACTIVITY LIST ***

ACTIVITY	IMMEDIATE PREDECESSORS	EXPECTED TIME
A	-	3
B	A	60
C	A	5
D	A	15
E	B	6
F	D	40
G	C , E	10
H	F	7
I	G	6
J	H , I	12

--------------------------------------------------

*** ACTIVITY SCHEDULE ***

-----------------------------------------------------------------------------

ACTIVITY	EARLIEST START	LATEST START	EARLIEST FINISH	LATEST FINISH	SLACK	CRITICAL ACTIVITY
A	0	0	3	3	0	YES
B	3	3	63	63	0	YES
C	3	64	8	69	61
D	3	23	18	38	20
E	63	63	69	69	0	YES
F	18	38	58	78	20
G	69	69	79	79	0	YES
H	58	78	65	85	20
I	79	79	85	85	0	YES
J	85	85	97	97	0	YES

----------------------------------------------------------------------------

CRITICAL PATH: A-B-E-G-I-J

PROJECT COMPLETION TIME = 97

Please note that I added the table boarder to address the spacing problem that I experience in going from The Management Scientist OUT file to Word to the Web page. You do not have to worry about that since you are not translating your OUT file to a Web page: all you have to do is insert the OUT file in an e-mail or in a Word Document.

The activity schedule part of the output provides all of the necessary start, finish, and slack times, as well as the identification of the critical path(s) and the critical activities.
That finishes our introduction to project scheduling. There are two extensions that address common attributes to projects: uncertain activity times and time-cost tradeoffs. These will be covered in Module 7.2 and 7.3 notes, below.


7.2: Uncertain Activity Times (Project Scheduling Step 10)

Anecdotal evidence suggests that activity times often vary, especially for new or unique projects. For many years project managers have accepted the Beta Distribution to capture the variance. Of course, project managers don't go around talking about Beta Distributions at construction sites - if they did, someone may call a doctor (MD, not Ph.D.)!

But they do go around talking about most likely, worst case and best case activity times. Statistically, the best distribution to model uncertainty involving three time estimates (most likely, pessimistic and optimistic) is the Beta Distribution. It is like the symmetric bell-shaped distribution we call the Normal Distribution, but the mean is given a heavier weighting.

Table 7.2.1 illustrates this new condition of uncertainty in the activity times by showing the Optimistic, Most Probable, and Pessimistic Times. These times are not computed - they are inputs based upon historical experience, project team estimation, and so forth.

Table 7.2.1

ACTIVITY	IMMEDIATE PREDECES SORS	OPTI MISTIC TIME	MOST PROBABLE TIME	PESSI MISTIC TIME	EXPECTED TIME	VARIANCE
A	-	1	2	9	3	1.78
B	A	57	58	71	60	5.44
C	A	3	4	11	5	1.78
D	A	5	15	25	15	11.11
E	B	5	6	7	6	0.11
F	D	37	39	47	40	2.78
G	C , E	9	10	11	10	0.11
H	F	3	7	11	7	1.78
I	G	4	5	12	6	1.78
J	H , I	10	12	14	12	0.44

The last two columns are computations necessary to analyze the uncertainty and to conduct Project Scheduling Step 10 Probability Analysis.

The Expected Time for an activity is its weighted average. The formula is:

Expected Time = (Optimistic + 4*Most Likely + Pessimistic) / 6

Note that we divide by six to get the average since we weight the most likely value by 4, resulting in six numbers in the numerator. The expected time for Activity A is:

Expected Time_A = (1 + 4*2 + 9) / 6 = 3

To compute the variance, we use the formula:

Variance = [ ( Pessimistic - Optimistic) / 6 ] ²

Why do we divide the numerator by six? Well, the Beta Distribution has many of the same properties as the Normal Distribution. Recall from statistics that about all of the data in a distribution lies in the area of the mean +/- 3 standard deviations. That means there are six standard deviations from the lowest to the highest number - thus we divide the range by six to estimate the standard deviation, then square it to get the variance. Slick, huh!!

The variance for Activity A is:

Variance_A = [ (9 - 1 ) / 6 ]² = 1.78

The expected time for the project is the sum of the expected times for activities on the critical path, in this case, 97 weeks. The variance of the project completion time is the sum of the variances of the activities on the critical path. Thus, the variance for this project completion time is the sum of the variances for Activities A, B, E, G, I and J = 1.78 + 5.44 + 1.78 + 11.11 + 0.11 + 2.78 + 0.11 + 1.78 + 1.78 + 0.44 = 9.67. Note carefully that we do not include the variances for activities that have slack as those activities do not impact the project completion time (more on this a bit later). To get the standard deviation for project completion time, we take the square root of the variance:

Standard Deviation_Project = Square Root (9.67) =

3.1 or about 3 weeks

It can be shown (not in this course but a mathematically based statistics course) that the distribution of project completion times (not activity completion times), which is made up of beta distributed activity times, is normal.

Remember from statistics that if we know the mean (expected project completion time) and the standard deviation from a Normal Distribution, we can describe the data with The Empirical Rule:

68% of Data is within: Mean +/- 1 Std Dev =

97 + 1 (3) = 94, 100

95% of Data is Within: Mean +/- 2 Std Dev =

97 + 2 (3) = 91, 103

100% of Data is Within: Mean +/- 3 Std Dev =

97 + 3 (3) = 88, 106

This is very valuable information for the project manager. Data in this case means expected project completion times. The project manager can expect most projects with these activity times will be completed within 91 to 103 weeks (95%) - that is good information for planning and budgeting purposes. The 88 to 106 range can be used to detect an "outlier" project completion time. If a project takes more than 106 weeks, something out of the ordinary or unexpected happened.

Knowing the mean and standard deviation is also very useful for probability analysis. Remember that 100% of the distribution has to be accounted for. Thus, what is the probability that we will have a project completion time less than 91 weeks? Note that 95% of the project completion times are between 91 and 103 weeks. That means that 5% of the times are less than 91 and greater than 103. Assuming approximate symmetry to the distribution, 2.5% of the times are below 91, and 2.5% are above 103. So the answer is: there is a 2.5% chance or probability that the completion time will be less than 91 weeks.

What if the boss wants to know the probability that the project completion time is less than 92 weeks. This number isn't at the 1, 2 or 3 standard deviation intervals so I can't use the Empirical Rule to give the boss a fairly close estimate of the probability. So, what do we do?

Well, you can use the material on the normal probability distribution from the text (pp. 78 - 84). We would convert 92 to a z-score and look up its equivalent probability in the z-table on page 78. The other alternative (and my choice here) is to use the normal distribution function in Excel. In an active cell of an Excel Spreadsheet, type the following formula:

= NORMDIST(92,97,3,TRUE)

where 92 is the target number, 97 is the mean, 3 is the standard deviation, and TRUE returns the commutative probability up to the number 92. The result from this formula is 0.048 so we would say that there is a 4.8% chance of finishing this project in 92 weeks. By the way, what's the probability that it will take longer than 92 days to complete the project? That's right, 1.0 - 0.048 = .952 or 95.2%

We can go the other way as well. What if the boss wants to know what project completion time is associated with a high probability of occurring (say, 75%). Type the following formula in an active cell of the Excel Spreadsheet:

= NORMINV(.75,97,3)

Excel returns 99 (99 weeks). That is, there is a 75% chance of finishing this project in 99 weeks. Please note that NORMDIST and NORMINV are called functions in Excel. To find a function for which you do not know the input parameters, go to the top tool bar in an open Excel Workbook, select Insert, then Function, then Statistical, then scroll for the function you want. The dialog screen will ask you for the information needed (such as the number, the mean, the standard deviation).

That wraps up our examination of Project Scheduling Step 9.

Pause and Reflect
Uncertainty in activity completion times can be addressed and analyzed by determining activity and project means and standard deviations. Knowing these, we can conduct probability analysis on the project completion time. The assumption needed to do this is that activity times follow the Beta Distribution, and project times follow the Normal Distribution. One limitation of the probability analysis is that it only addresses critical path activities. If an activity with uncertain completion times has relatively little slack and high variance, the chance that it could become critical is very high - project management software ignore this limitation (unless they incorporate a simulation module) and we should be aware of that.

Well, what if the boss really wanted to finish this project in 92 weeks. You would tell the boss there is only a 4.8% chance of that happening (then duck!). Actually, this happens all the time in project management. It just happened at FGCU. The University is pressing the contractor to hurry up the project schedule to get the Whitaker Science and Technology Building open by start of class next year. We call this accelerating or crashing the project. That is our last topic in Project Scheduling. Before we get there, let's look at The Management Scientist routine that will take our optimistic, most likely, and pessimistic inputs and find expected times and variances, as well as perform the activity analysis.


Using The Management Scientist Software Package
To incorporate uncertain activity time in using the The Management Scientist PERT/CPM Module to do the actual activity, project completion time and critical path analyses, start as before: click Windows Start/Programs/The Management Scientist/The Management Scientist Icon/Continue/Select Module 7 PERT/CPM/OK/File/New and you are ready to load the example problem. On the next dialog screen select Uncertain Activity Times, enter 10 for the Number of Activities, then click OK. The data input screen is similar to the screen for Known Activity Times, but note the addition of the need to enter Optimistic, Most Probable and Pessimistic Activity Times. After data entry, select Solution, and then Solve to get the following:


Printout 7.2.1

PROJECT SCHEDULING WITH PERT/CPM

********************************

*** PROJECT ACTIVITY LIST ***

ACTIVITY	IMMEDIATE PREDECESSORS	OPTIMISTIC TIME	MOST PROBABLE TIME	PESSIMISTIC TIME
A	-	1	2	9
B	A	57	58	71
C	A	3	4	11
D	A	5	15	25
E	B	5	6	7
F	D	37	39	47
G	C , E	9	10	11
H	F	3	7	11
I	G	4	5	12
J	H , I	10	12	14

-----------------------------------------------------------------------------

EXPECTED TIMES AND VARIANCES FOR ACTIVITIES

ACTIVITY	EXPECTED TIME	VARIANCE
A	3	1.78
B	60	5.44
C	5	1.78
D	15	11.11
E	6	0.11
F	40	2.78
G	10	0.11
H	7	1.78
I	6	1.78
J	12	0.44

*** ACTIVITY SCHEDULE ***

ACTIVITY	EARLIEST START	LATEST START	EARLIEST FINISH	LATEST FINISH	SLACK	CRITICAL ACTIVITY
A	0	0	3	3	0	YES
B	3	3	63	63	0	YES
C	3	64	8	69	61
D	3	23	18	38	20
E	63	63	69	69	0	YES
F	18	38	58	78	20
G	69	69	79	79	0	YES
H	58	78	65	85	20
I	79	79	85	85	0	YES
J	85	85	97	97	0	YES

----------------------------------------------------------------------------

CRITICAL PATH: A-B-E-G-I-J

EXPECTED PROJECT COMPLETION TIME = 97

VARIANCE OF PROJECT COMPLETION TIME = 9.67

Note carefully that the software reports the variance, not the standard deviation. You have to take the square root on your calculator (or watch, or in Excel, or with paper and pencil - yeah, right!). Now it's off to the time-cost tradeoff.


7.3: Time-Cost Tradeoffs (Project Scheduling Step 10)

Precedence relationships, start and finish times, slack, critical activities, and probabilities of completion are not the only considerations in project scheduling. Activities cost money to complete and often require labor and other resources. This section examines the activity cost resource consideration, and the tradeoff between cost and time.

Table 7.3.1 illustrates cost information for the example problem.


Table 7.3.1

ACTIVITY	IMMEDIATE PREDECESSORS	NORMAL TIME	CRASH TIME	NORMAL COST	CRASH COST	CRASH COST/WEEK
A	-	3	1	$ 5,000	$ 5,600	$ 300
B	A	60	40	15,000	25,000	500
C	A	5	3	5,000	7,000
D	A	15	13	7,000	11,000
E	B	6	6	6,000	6,000	n/a
F	D	40	36	12,000	14,000
G	C , E	10	9	8,000	8,300	300
H	F	7	5	7,000	9,000
I	G	6	4	5,000	9,000	500
J	H , I	12	11	9,000	11,000	2000

All of the columns represent data input except the last column which is a computation. The normal activity time is considered the activity expected time as before. If we complete Activity A in the normal time of 3 weeks, it is estimated to cost $ 5,000. On the other hand, we can accelerate Activity A and do it in 1 week, but the cost increases to $5,600. Perhaps we can add one more telemarketer to help with the telephone survey part of determining market need. The benefit is time reduction, but it comes at a higher cost. In project scheduling, we call the accelerated time the crash time, and the associated cost is the crash cost. While the normal time is considered an expected time, the crash time represents the technological minimum time, at least in the short run.

The last column reflects the computation of the slope of the linear relationship between the change in cost related to a unit change in time. The computation is straightforward:

Cost/Time Unit = Change in Cost / Change in Time
Cost/Time Unit = (Normal Cost - Crash Cost) /

(Crash Time - Normal time)

For Activity A, the computations are:

Cost/Week = (5,600 - 5,000 ) / ( 3 - 1 ) = $300 per week

If we crash Activity A by one week, the cost will be $300 more or $ 5,300; and if we crash Activity A by two weeks, the cost will be $5,600.

The remaining time-cost computations are shown in Table 3.3.1, with two exceptions. The first exception is to note that crash cost per week is only computed for activities on the critical path. We crash activities to shorten project time. If you crash a non-critical activity, all you do is reduce slack - there is no savings in project duration. The other thing to note is that there is no cost per week for Activity E, even though it is on the critical path. Note that the crash and normal times and costs are the same for this activity - this suggests that the activity cannot be further accelerated.

The major purpose in preparing the crash cost per time unit computations is to create a measure to make an economic decision concerning which activity to accelerate to reduce project duration. The economic criteria is to choose that activity on the critical path that has the minimum crash cost per time unit. Crash that activity one time unit.

For this example, both Activities A and G have the same minimum crash cost per week, so it would be a toss-up as to which activity to accelerate one week based on the economic criteria. However, we could break the tie by selecting the earliest activity, Activity A, as the activity to crash one week. The reason for this is that by crashing Activity A now, we leave the option open to crash Activity G later. If we decide to crash Activity G and not Activity A, by the time we get to Activity G we obviously would have foregone the option to crash Activity A.

If the project manager wanted to continue crashing activities, then the next step would be to redo the activity analysis. The reason for this is that as activities are crashed, they lose slack and may join the list of critical path activities. Then repeat the economic analysis and select that activity that now has the minimum crash cost per time unit as the next activity to crash one time unit. This process continues until we reach a target completion time or exhausted all of the crash opportunities.

This is one area where a software program really comes in handy. While the PERT/CPM module does not have the time-cost tradeoff evaluation feature, this can be done by the quantitative method described as linear programming. We can demonstrate the use of linear programming for network analysis through a simple example with activities indicated in Table 7.3.2

Table 7.3.2

ACTIVITY	IMMEDIATE PREDECESSORS	NORMAL TIME	CRASH TIME	NORMAL COST	CRASH COST	CRASH COST/WEEK
A	-	3 weeks	1 week	$ 5,000	$ 5,600	$ 300
B	A	6	4	15,000	25,000	5000
C	A	5	3	5,000	7,000	1000
D	B,C	4	3	5,000	6,000	1000

The network is represented in figure 7.3.1 below.

Using the Management Scientist with certain times, we solved this network with the solution presented in table 7.3.3 below:

TABLE 7.3.3

PROJECT ACTIVITY LIST
	IMMEDIATE	EXPECTED
ACTIVITY	PREDECESSORS	TIME
A	-	3
B	A	6
C	A	5
D	B,C	4

***ACTIVITY SCHEDULE ****

	EARLIEST	LATEST	EARLIEST	LATEST		CRITICAL
ACTIVITY	START	START	FINISH	FINISH	SLACK	ACTIVITY
A	0	0	3	3	0	YES
B	3	3	9	9	0	YES
C	3	4	8	9	1
D	9	9	13	13	0	YES

CRITICAL PATH = A-B-D

PROJECT COMPLETION TIME = 13 Weeks

In a simple network like this one we can arrive at the crashing decision without problems. The activity we would like to crash is in the critical path (A-B-D) and it should have the lowest crash cost per week (A). In this example, activity A should be selected to be crashed if we want to reduce the project completion time by one week (the expected project duration now is 12 weeks), and the corresponding cost to complete the project goes up by $300. The total cost is now $30,300.00. However, in a more complex network, crashing decisions are not obvious and linear programming can be used to solve the decision problem. If we desire to complete the project is 12 weeks (that is, crash the original project by one week), using time and cost information from Table 7.3.2 we can formulate a cost minimization model to make crashing decisions.

Let x_i = the completion time for activity i i=A, B, C, D

y_i = the amount of time activity i is crashed i=A, B, C, D

Min Z = 300 y_A + 5000 y_B + 1000 y_C + 1000 y_D

S.T.

1) x_A + y_A ≥ 3

2) x_B + y_B – x_A ≥ 6

3) x_C + y_C – x_A ≥ 5

4) x_D + y_D – x_B ≥ 4

5) x_D + y_D – x_C ≥ 4

6) x_D ≤ 12

7) y_A ≤ 2

8) y_B ≤ 2

9) y_C ≤ 2

10) y_D ≤ 1

All x_i,y_i ≥ 0

Let's review this formulation. Because the total project cost for a normal completion time is fixed at $30,000 (see Table 7.3.2), we minimize the total project cost by minimizing total crashing costs (note that total project cost is given by normal cost plus crashing cost). The coefficients 300, 5000, 1000, and 1000, are the crash cost per week (also called cost slope) for each of the activities and are computed in Table 7.3.2. When using software, the complete objective function should include the completion time for each activity (x_i) as well with a coefficient of zero. The formulation is then written as:

Min Z = 300 y_A + 5000 y_B + 1000 y_C + 1000 y_D + 0 x_A + 0 x_B + 0 x_C + 0 x_D

Constraints 1 through 5 represent the following logic: Completion time = Earliest start time + Activity time. But since activities may have slack time, then we can write:

Completion time ≥ Earliest start time + Activity time.

Note that 'Activity time' is computed as (Normal duration time - amount of time the activity is crashed). Using our notation for Activity A:

Completion time_A ≥ Earliest start time_A + Activity time_A.

x_A ≥ 0 + (3 - y_A), or moving y_A to the left,

x_A + y_A ≥ 3

Moving forward in the network, we see that the earliest time for activity B is x_A, the finish time for activity A. thus, the constraint corresponding to the finish time for activity B is:

x_B ≥ x_A + (6 - y_B) or

x_B + y_B – x_A ≥ 6

Similar formulation is made for activities C and D. Equation 6 represents management requirement to complete the project in 12 weeks, therefore the completion time for activity D should be no later than 12, or

x_D ≤ 12

And finally, equations 7 through 10 represent the maximum allowable crashing time for each activity. Note this is the difference between Normal Time and Crash Time in Table 7.3.2. Thus to determine the optimal crashing for each of the activities, we must solve an 8-variable, 10-constraint linear programming model.

Table 7.3.4 contains the solution for the crashing problem. It indicates that activity A should be crashed by one week (YA variable = 1) and the project cost is now the original cost $30,000 plus $300 (the result of the optimization problem) to crash this activity. Activity A now has duration of 2 weeks.

Table 7.3.4

Activity	Duration
A	2 (crashed by 1 week)
B	6
C	5
D	4

This LP formulation problem can also be reviewed in Chapter 12, pp. 539, ‘Linear Programming Model for Crashing’.

Pause and Reflect
Project managers worry about being on time and at (or below) cost. I hope this introduction to the metrics side of the subject gives you some appreciation that project completion time can be accelerated, but only at a cost. Analysis is required to evaluate the tradeoff. Obviously, other considerations such as availability and substitutability of subcontractors, need to be addressed, even if only by incremental additions to the cost component. This classic time-cost tradeoff analysis makes one important assumption: the relationship between time and cost is linear. That may not be true, especially when the activity time approaches its minimum and costs increase at an increasing rate to gain incremental time savings.

You should be ready to tackle the assignment for Module 7, "R.C. Coleman," pp. 548-549 in the text.

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