|
"Project Scheduling: Pert/CPM" |
Index to Module Three Notes |
|
"Project Scheduling: Pert/CPM" |
Index to Module Three Notes |
3.1: 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 pathStep 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
Identify market need Conduct R&D Design Packaging Select production site Conduct product test Install production process Perform market analysis Production startup Make modifications Market product - -> --- --- -> --- --- -> --- --- -> --- --- -> | | ^ ^ | | | ---- ---| -> --- --- --- --- --- --- --- | | | | | | | | - -> --- --- -> --- --- -> --- --- --- --- --- --- --- - - -> -> -> -> -> 85 97 | | ^ ^ | | | -> --- --- --- --- --- | | | | | | | - -> -> -> --- --- --- --- --- -
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 3.1.1 It is noted that Project Scheduling Step 3 also
includes gathering activity cost data. That part of Step 3 will be
addressed in Module 3.3 Notes.
Table 3.1.1
PREDECESSOR
TIME
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 3.1.1.
Figure 3.1.1.
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 3.1.2 to add the activity analysis to
an expanded copy of the activity network.
Figure 3.1.2
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 3.1.2 provides the definition for
these numbers, as well as the latest start and finish times which I
will fill in and illustrate in Figure 3.1.3 in a few more
paragraphs.
Table 3.1.2
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'simmediate 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 3.1.3.
Figure 3.1.3
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 thatactivities 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 3.1.1
PROJECT SCHEDULING WITH PERT/CPM
********************************
*** PROJECT ACTIVITY LIST ***
PREDECESSORS
TIME
--------------------------------------------------
*** ACTIVITY SCHEDULE ***
-----------------------------------------------------------------------------
START
START
FINISH
FINISH
ACTIVITY
----------------------------------------------------------------------------
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 3.2 and 3.3 notes,
below.
3.2: 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 3.2.1 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 3.2.1
PREDECES
SORS
MISTIC
TIME
PROBABLE TIME
MISTIC
TIME
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 TimeA = (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:
VarianceA = [ (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 DeviationProject = 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, 10095% of Data is Within: Mean +/- 2 Std Dev =
97 + 2 (3) = 91, 103100% 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. 75 - 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 3.2.1
PROJECT SCHEDULING WITH
PERT/CPM
********************************
*** PROJECT ACTIVITY LIST ***
PREDECESSORS
TIME
PROBABLE TIME
TIME
-----------------------------------------------------------------------------
EXPECTED TIMES AND VARIANCES FOR ACTIVITIES
TIME
*** ACTIVITY SCHEDULE ***
START
START
FINISH
FINISH
ACTIVITY
----------------------------------------------------------------------------
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.
3.3: 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 3.3.1 illustrates cost information for the example problem.
Table 3.3.1
PREDECESSORS
TIME
TIME
COST
COST
COST/WEEK
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. The Management Scientist does
have linear programming capability which we will examine later on in
the course.
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 3, "Warehouse
Expansion," in the text, pp. 524-526. The case answers via e-mail and
The Management Scientist computer output file are due February
24, 2001. If you want a "free" review of your draft responses/output,
please forward as a draft by Tuesday, February 20, 2001.
About the
Course |
Module
Schedule |
WebBoard |
