Review
This is a review of the Project Scheduling
portion of the final exam. This review cannot teach all
the material that are included in this part of the course.
It's purpose is to summarize the contents of the individual
lessons. The student should also study the event and homework
keys on the class Canvas site. The link below opens a page
with links to the scheduling lessons as well as to links in
that open pages in the ORMM site that deal with the Project
Management addin.


Final
Three Weeks/Final Exam 


In addition to the scheduling lessons, the final
exam will also cover lessons 18 and 19. The exam may also include
materials covered in the earlier lessons. 

Exam Format 

The final exam will either be scheduled separately for each
section at various times during the final exam period or at one time for all students. If the exams
are scheduled separately, they will be administered in the discussion room. The final exam is comprehensive
in that questions may deal with any aspect of the course; however,
it will stress material studied since the second exam. The exam
will be entirely open website and open textbook. You may use
the class website, calculators available on the site, the Excel
addins, the ORMM website, and the textbook during the entire
exam. No cell phones or other materials may be used. You may not access the
solutions to past homework assignments, events, or exams or bring printed
material to the exam. You may not open personal computer files
during the exam except those used for the exam. You may not communicate
with any other persons via the internet or cell phone.
The exam will be presented on paper and space will be provided
for answers on the paper. To receive credit you must answer
the questions on the paper form. You will upload files used
for the exam to the Canvas class site. You may not share
computer files with other persons in the class. 

Goals 

Each lesson in the section has a collection
of goals. Most of them describe an activity that you should understand
and be able to perform. The paragraphs below describe the purpose
of each lesson. Click the bush icon to see the goals of the individual
lessons.


20. The
Model
This lesson describes the model we use for project
scheduling. The model includes a list of activities
that comprise the project, a list of the immediate
predecessors for each activity, and a list of estimates
of the time required for each activity. The model
may also describe the resources and cash flows associated
with the activities.

21. Critical
Path
This lesson describes methods to find the critical
time, the minimum time required to complete the
project. We also discover the critical path,
the set of activities that must be performed in strict
sequence to finish the project as soon as possible.
The process of finding the critical time and path determines
two schedules  the earliest schedule and the latest
schedule.

22. Uncertainty
There are many reasons for uncertainty
with regard to the project scheduling model. This lesson
recognizes uncertainty in activity times by describing
the times with continuous probability distributions.
Using several approximations with regard to the critical
path, the normal distribution is used to provide probability
estimates about the duration of the project. Simulation
is used when the normality assumption is not appropriate.




Text 

Most of the materials in this section
are in chapter 9 of the text.


Chapter
9:
Project Scheduling 




The model for project scheduling must include
all the physical tasks necessary to complete the project. The
work breakdown schedule, WBS, describes a project
by a collection of work packages (WP). Necessary characteristics
of the WBS is that WPs cannot overlap in function and that they
must collectively describe everything that must be done to complete
the project. When discussing project scheduling we usually use
the term activities to
describe the work contents of a project.

The set of activities
describes all of the steps necessary to complete
a project. Activities cannot overlap in function.


The sequence of activities is described by listing
the immediate predecessors of each activity.

In our model, all an activity's
predecessors must be complete before the activity
may begin.


The linked document shows the elements of the model
as they are described by the Project Management addin.


The goal of this lesson is to find the critical
time,
the minimum time required to complete the project. We also discover
the critical
path, the set of activities that must be performed in strict
sequence to finish the project a soon as possible. The process
of finding the critical time and path determines two schedules  the earliest schedule and the latest schedule.
The early schedule starts each job as early as
possible while fulfilling the following requirement.

An activity cannot begin until
all of its predecessors are finished.


The late schedule starts each job as late as possible while
fulfilling the following requirement.

An activity must finish before
any of its successors may begin.


The equations to compute earliest start and finish
times and latest start and finish times for the activities are
in the document linked below.
See the three short movies to learn how to find the critical
path on the project network.


Solving
for the Earliest Times 




Solving
for the Latest Times 




Construction
of the Critical Path 


Click the icon to open a document that shows the
critical path as well as the early and late schedules.


Critical
Path for Example 




This lesson
considers the uncertainty in the estimates of the time duration
of activities. We model the effects of activity time uncertainty
using continuous random variables with known probability distributions.
With the approximation that the project duration has a normal
distribution, probability statements can be made about the project
duration. When the normality assumption is not invoked, simulation
provides more accurate estimates for the statistics of the project
duration.
Our models provide three methods for estimating
uncertain activity times. With the help of the Random Variables addin,
any of the available distributions can be used.
Traditional PERT 
PERT reflects
uncertainty by allowing three estimates: the minimum time, a,
the most likely (or mode) time, m, and the maximum
time, b. Based
on these parameters, empirical formulas approximate the
mean and variance of the activity duration. The
traditional method is convenient for hand or spreadsheet calculations.

Triangular Estimate 
The triangular estimate also uses the three
parameters: the minimum time, a, the most likely
time, m, and the maximum time, b, but
here we use the parameters to define a triangular probability
distribution. The mean and variance can be calculated using
the Random Variables addin. 
Beta Estimate 
The Beta distribution is defined by its
lower and upper bounds, as well as two parameters defining
its shape. A variety of shapes can be defined. The mode
is a function of the parameters. The mean and variance can
be calculated using the Random Variables addin. 
The simplest approach for describing the uncertainty
of the project duration uses the critical path method to find
a single path based on the mean value of the activity times.
The approach assumes that this path remains critical for all
activity time realizations. It also assumes that the project
duration is normally distributed. Of course these assumptions
are not really true, but the results of the approximation may
be useful.
Use the mean values for the activity
times. 
Find the critical path using the mean
values. The path is identified by the critical activities. 
Find the mean duration by summing the
mean times on the critical path. 
Find the variance of the duration by
summing the activity variances on the critical path. The
standard deviation is the square root of the variance. 
Assume the duration has a normal distribution.
Make probability statements regarding the duration using
the cumulative standard normal distribution with the computed
mean and standard deviation. 
Monte Carlo simulation provides statistical results
that don't depend on the normality assumption.
The simulation accumulates statistics on the project duration
and the number of times each activity appears in the critical
path.
Do the following NSIM (the number of
simulated observations) times. 
 Simulate the activity times using the Monte Carlo
method.
 Find the critical path. Compute the duration for
the path and store it in an array of simulated values.
 Observe the activities on the critical path and increase
the count for each critical activity.

Compute the mean, variance and standard
deviation of the simulated durations. 
Compute the proportion of the observations
when each activity was identified as critical. 
Report the results. 
When probability statements are desired,
assume the duration has a normal distribution with the
mean and standard deviation determined by the simulation.
Make probability statements regarding the duration using
the cumulative standard normal distribution. 


Summary 


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