Engineering Finance
Lessons



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 add-in.

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Final Three Weeks/Final Exam
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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 add-ins, 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.

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

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Chapter 9: Project Scheduling
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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 add-in.

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Elements of the Model
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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.

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Critical Path Formulas
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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.

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Critical Path for Example
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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 add-in, 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 add-in.

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 add-in.

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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Scheduling Summary
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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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