Project Scheduling

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Project Scheduling
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What is Project Scheduling?

• The task of planning timetables and the
  establishment of dates during which resources
  such as equipment and personnel, will perform the
  activities required to complete the project.

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The Project Schedule Integrates the Following
Information

• Estimated duration of activities
• The technical precedence relations among
  activities
• Constraints imposed by the availability of
  resources and the budget
• Due-date requirements

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What questions should by answered by the project
schedule?

• If each of the activities goes according to plan,
  when will the project be completed?
• Which tasks are most critical to ensure the
  timely completion of the project?
• Which tasks can be delayed, if necessary, without
  delaying the project completion, and by how much?
• At what times should each activity begin and end?
• What is the range of dollars that should have
  been spent at any given time during the project?
• Is it worthwhile to incur extra costs to
  accelerate some of the activities?

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Schedule Presentation

• Schedules can be represented in several different
  ways to match the needs of the user. For
  example
• WBS
• Gantt chart
• Master schedule all schedules should relate to
  the master schedule, which gives a time-phased
  picture of the principle activities and
  highlights the major milestones.
• Is a detailed schedule the best schedule?
• Depends on the needs of the user!
• If the users needs an overview but has to work
  with a very detailed schedule, then important
  communication could be delayed and divert
  attention from critical activities.

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WBS Example Develop a New MBA ProgramFour
Level Divide the work content according to the
year in the program

• 1. Development of an MBA curriculum

1.1 First-year Courses
1.2 Second-year Courses
1.1.1 Courses in Finance
1.6.1 Courses in Accounting
1.2.6 Courses in Accounting
1.2.1 Courses in Finance
1.1.1 Introduction to Finance
1.2.1.2 Corporate Finance
1.2.6.3 Corporate Accounting
1.1.6.4 Managerial Accounting
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What are the benefits to using a schedule?

• Communications tool about the project
• Coordination tool to link resources, time and
  activities
• Tool for planning
• Tool for monitoring progress
• Tool for justifying corrective action

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How do you start a schedule?

• Top down - define major tasks, (sometimes
  referred to as milestones or phases) and then
  decompose each milestone/phase into more detail
• Bottom up - list all the activities in a project
  in any order. Group the list into
  phases/milestones based on the required sequence,
  constraints, and assumptions

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Milestones

• A milestone can be defined as an important event
  in the project life cycle. Examples include
• The fabrication of a prototype
• The start of a new phase
• A status review
• A test
• First shipment

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How do you start a schedule? Top Down

• Place milestones on a timeline, which becomes the
  skeleton for the master schedule
• Should be defined for all major phases of the
  project

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Milestones

• The completion of the milestone should be easily
  verifiable, but in reality, this may not be the
  case.
• Why?
• Design, testing and review tend to run together.
  There is always a desire to do a little more work
  correct superficial flaws or to extract a
  marginal improvement in performance. These
  activities tend to blur the milestones and make
  project control much more difficult.
• When setting milestones, you must balance the
  number for the project. Too many may be
  considered over control by those involved,
  whereas too few can lead to continuity problems.

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Estimating the Duration of Project Activities

• Guidelines
• The length of each activity should be
  approximately in the range of .5 to 2 of the
  length of the project. Example If the project
  takes about 1 year, each activity should be
  between a day and a week.
• Critical activities that fall below this range
  should be included. Examples Tests, reviews and
  other significant activities that may be short in
  duration, but important to the project.
• If the number of activities is very large (above
  250), the project should be divided into
  subprojects by function, products, or other
  logical division.
• Schedules with too many activities quickly become
  unwieldy and are difficult to monitor and
  control.

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Stochastic Approach to Estimating the Duration of
a Project

• How often do we know the exact duration of an
  activity?
• Almost never!
• How do we estimate the duration of activities
  whose durations cannot be known exactly?
• One method is to analyze historical data from
  similar projects
• Data analysis usually starts with representing
  the data in some visual form such as a table or
  chart like Figure 7-3 on p.309 of your text.
• We can describe this set of historical data with
  statistical measures of centrality (such as the
  mean, mode, and the median) and measures of the
  distribution of the data (such as variance,
  standard deviation and the interquartile range).
• When we have historical data, we need to fit a
  continuous distribution to facilitate analysis.

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Stochastic Approach to Estimating the Duration of
a Project
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What distributions will we use?

• Normal and beta.
• While the normal is easy to work with, it has a
  long left-hand tail which could imply a negative
  performance time.
• The left-hand tail of the beta distribution does
  not cross the zero duration point and is not
  symmetric.

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How do we incorporate probabilistic
considerations into project scheduling?

• By Assuming that the estimate for each activity
  can be derived from three different values
• a optimistic time, which will be required if
  execution goes extremely well
• m most likely time, which will be required if
  execution is normal
• b pessimistic time, which will be required if
  execution goes extremely bad
• a and b represent the upper and lower bounds on
  the frequency distribution.
• To convert m, a and b into estimates of the
  expected value d and variance v, of the elapsed
  time required by the activity, two assumptions
  are made

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How do we incorporate probabilistic
considerations into project scheduling?

• Assumption 1
• The standard deviation, s (square root of the
  variance), equals 1/6 the range of possible
  outcomes.
•  
• s (b-a)/6
•  
• Assumption 2
• The duration of the activities follows a beta
  distribution with its unimodal point occurring at
  m and its endpoints at and b.
• The expected value of the activity duration is
  given by
•  
• d 1/32m 1/2(a b) (a 4m b)/6

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Deterministic Approach to Estimating the
Duration of a Project

• How do you estimate the duration of tasks when
  you do not have data from past projects to use as
  a baseline?
• You can use on of 3 techniques
• Modular
• Benchmark job
• Parametric

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Modular Technique for Estimating the Duration of
a Project

• The Modular Technique decomposes each activity
  into subactivities (or modules), estimates the
  performance time of each module, and then totals
  the results to get an approximate performance for
  the activity.

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Benchmark Job Technique for Estimating the
Duration of a Project

• The benchmark job technique uses elapsed time
  data from previous jobs to estimate the time
  required to complete tasks in a project.
• This approach is the same concept as using
  standard data in time studies.
• Time data is stored in a company database for
  routine tasks such as lifting boxes, loading
  trucks, or assembling a product.

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When do you use the benchmark job technique?

• The benchmark job technique is used when a
  project contains many repetitions of standard
  activities whose execution time is additive. If
  the nature of the work does not support the
  additivity assumption, then the parametric
  technique should be used.
• The extent to which this technique can be used
  depends on the quality of the database of common
  activities maintained by the company.

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Parametric Technique for Estimating the Duration
of a Project

• Parametric technique is the same as using
  Regression Analysis to solve a problem in
  Statistics.
• The parametric technique attempts to predict the
  time required to complete a task by modeling the
  independent variables that impact the time
  required in a mathematical equation.
• Regression analysis is used to predict the value
  of one variable on the basis of other variables.
• The technique involves developing a mathematical
  equation that describes the relationship between
  the variable to be forecast, which is called the
  dependent variable, and variables that the
  statistician believes are related to the
  dependent variable.

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Parametric Technique for Estimating the Duration
of a Project

• The dependent variable is denoted by "y", while
  the related variables are called independent
  variables and are denoted x1, x2, , xk (where
  "k" is the number of independent variables).
• Equations such as
• E mc2
• F ma
• Are deterministic models because with exception
  of small errors, these equations allow us to
  determine the value of the independent variable
  (on the left side of the equation) from the value
  of the independent variables.
• These equations do not represent the random
  nature of real life. Equations that contain some
  measure of randomness are called probabilistic
  models.

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Parametric Technique for Estimating the Duration
of a Project

• To build a probabilistic model, we start with a
  deterministic model that approximates the
  relationship we want to model. We then add a
  random term that measures the error of the
  deterministic component.
• Example
• Suppose that the cost of building a new house is
  about 75 per square foot and that most lots sell
  for about 25,000. The approximate selling price
  would be
• y 25000 75x
• Where y selling price and x Size of the house
  in square feet
• Thus a house of 2000 square feet would be
  estimated to sell for
• y 25000 75(2000) 175,000

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Parametric Technique for Estimating the Duration
of a Project

• We know the price is not exactly 175,000, but
  between 150,000 - 200,000. To represent this
  situation properly, we should use the
  probabilistic model
• y 25000 75x Î
• where " Î " ( the Greek letter epsilon)
  represents the random term (Called the error
  variable). Î is the difference between the actual
  selling price and the estimated price based on
  the size of the house. Thus the random term
  accounts for all the variables, both measurable
  and immeasurable, that are not part of the model
  such as number of bedrooms and location.
• The value of Î will vary from house to house
  depending on location, number of bedrooms, etc

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First-order Linear Model or Simple Linear
Regression Model

• First-order Linear Model
• y b 0 b 1x Î
• Where
• y dependent variable
• x independent variable
• b 0 y-intercept
• b 1 slope of the line (defined as the ratio
  rise/run or change in y / change in x)
• Î error variable

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Least Squares Method

• The problem addressed by the simple linear
  regression model is to analyze the relationship
  between two variables, x and y, (both which must
  be quantitative).
• To do this we must know b 0 and b 1.
• These coefficients are population parameters,
  which are almost always unknown.
• How do we estimate b 0 and b 1?
• We draw a random sample from the populations of
  interest and calculate the statistics we need
  just like we have throughout this course.
• The estimators of b 0 and b 1 are based on
  drawing a straight line through sample data.

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First-order Linear Model or Simple Linear
Regression Model
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First-order Linear Model or Simple Linear
Regression Model
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First-order Linear Model or Simple Linear
Regression Model

• Our task is to draw the straight line that
  provides the best possible fit.
• Our best line will have some points above and
  below. Meaning we will have differences that will
  be positive (above the line) and negative (below
  the line).
• To eliminate the positive and negative
  differences, we draw the line that minimizes the
  squared differences. That is, we want to
  determine the line that minimize

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First-order Linear Model or Simple Linear
Regression Model

• Where yi represents the observed value y and
  represents the value of y calculated from the
  equation of the line. That is

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First-order Linear Model or Simple Linear
Regression Model

• The technique that produces this line is called
  the least squares method.
• The line itself is called the least square line,
  the fitted line or the regression line.
• The "hats" on the coefficients remind us that
  they are estimators of the parameters b 0 and b 1.

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First-order Linear Model or Simple Linear
Regression Model

• Calculations

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First-order Linear Model or Simple Linear
Regression Model
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First-order Linear Model or Simple Linear
Regression Model

• Shortcut Formulas for SSx and SSxy

SSx sum of the squared difference between the
observations of x and their mean SSxy is not a
sum of squares (note that nothing is squared)
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First-order Linear Model or Simple Linear
Regression Model

• Sum of x
• Sum of y
• Sum of x-squared
• Sum of x times y

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First-order Linear Model or Simple Linear
Regression Model
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First-order Linear Model or Simple Linear
Regression Model

• Thus the least squares regression line is

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First-order Linear Model or Simple Linear
Regression Model

• Example
• Car dealers are interested in the influence of
  the number of miles a used car has on the selling
  price of the car.
• To examine the issue, a used car dealer randomly
  selected 100 three year old Ford Tauruses that
  were sold at auction during the past month.
• Each car was in top condition and equipped with
  the same features (automatic transmission, AM/FM
  cassette tape player and air conditioning).
• The dealer recorded the price and the number of
  miles on the odometer.
• These data are stored in the file XM19-02. The
  dealer wants to find the regression line.

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First-order Linear Model or Simple Linear
Regression Model
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First-order Linear Model or Simple Linear
Regression Model

• What do we know?
• Dependent variable y selling price
• Independent variable x odometer reading (miles
  driven)
• What do we want to know?
• How the odometer reading affects the selling
  price, thus a simple linear regression.

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First-order Linear Model or Simple Linear
Regression Model
4,309,340,160
-134,269,296
Using the Sums of squares, we find the slope
coefficient
-134,269,296/4,309,340,160 -.0311577
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First-order Linear Model or Simple Linear
Regression Model

• To determine the intercept, we need to find x-bar
  and y-bar

541,141/100 5,411.41
3,600,945/100 36,009.45
Thus
411.41 - (-.0311577)( 36,009.45) 6,533.38
The sample regression line is
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First-order Linear Model or Simple Linear
Regression Model

• Using Minitab
• Type or import the data into 2 columns
• Click Stat, Regression, and Regression
• Type the name of the dependent variable (Response
  - Price or C2)
• Hit tab, and type the name of the independent
  variable (Predictors - Odometer or C1)
• Click O.K.
• Click Stat, Regression, and Fitted Line Plot
• Type the name of the dependent variable (Response
  - Y - Price or C2)
• Hit tab, and type the name of the independent
  variable (Predictors - X - Odometer or C1)
• Click O.K.

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First-order Linear Model or Simple Linear
Regression Model

• Interpreting the Results
• -.0312 which means that for each additional
  mile on the odometer, the price decreases by an
  average of .0312 or 3.12 cents.
• 6533 since this is the y - intercept, we might
  think that a car with 0 miles would sell for
  6533. In this case however, the intercept is
  probably meaningless. Because our sample did not
  include any cars with 0 miles on the odometer, we
  have no basis for interpreting . As a general
  rule, we cannot determine the value of y for a
  value of x that is far outside the range of
  sample values of x. In this sample, x ranged from
  19075 to 49223.

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Multiple Regression

• With simple linear regression, we had one
  dependent and one independent variable.
• These models were simple, but of limited value.
• Now we will consider models were we have more
  than one independent variables.
• Multiple regression models generally fit the data
  better than simple (single) regression models.
• In general we will include as many independent
  variables as can be shown to significantly affect
  the dependent variable.

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Multiple Regression

• We now assume that k independent variables are
  potentially related to the dependent variable

Where y dependent variable x1, x2, , xk
independent variables b 1, , b k
coefficients ? error variable
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Multiple Regression

• The independent variables may actually be
  functions of other variables
• x2 x12
• x5 x3 x4
• x7 log(x6)
• We will not discuss how and under what
  circumstances such functions can be used in
  regression analysis (thus we will not use these
  "functions of other variables").
• When we have more than one independent variable
  in regression analysis, we refer to the graphical
  depiction of the equation as a response surface
  rather than a straight line.
• When k2, the regression equation creates a
  plane.
• When k equals something greater than 2 we can
  only imagine the response surface, we cannot draw
  it.

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Multiple Regression

• Example
• The president of a chain of video stores is
  deciding where to locate a new store. She plans
  to use a regression model to help select a
  location for a new store. She decides to use
  annual gross revenue as a measure of success,
  which is the dependent variable. The president
  believes that determinants of success include the
  following variables
• Number of people living within one mile of the
  store (People)
• Mean income of households within one mile of the
  store (Income)
• Number of competitors within one mile of the
  store (Computers)
• Rental price of a newly released movie (Price)

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Multiple Regression

• The president randomly selects 50 video stores
  and records the values of each of the variables
  listed above plus annual gross revenue (Revenue).
  These data are stored in file XM20-01.
• She proposes the following multiple regression
  model
• Revenue b 0 b 1(People) b 2(Income) b
  3(Computers) b 4(Price) ?

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Multiple Regression

• Menu Commands
• Type or import the data
• Click Stat, Regression, and Regression
• Type the name of the dependent variable
  (Response) (Revenue)
• Hit tab, and type the names of the independent
  variables (Predictors) (People, Income,
  Computers, and Price)
• Click O.K.
• From the computer results we can define the
  multiple linear regression model as
• Revenue -20297 6.44(People) 7.27(Income) -
  6709(Computers) 15969(Price)

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Multiple Regression

• So how do we develop a regression equation to
  predict the time required to complete a task?
• Identify the independent variables that affect
  activity duration
• Collect data on past performance time of the
  activity for different values of the independent
  variable.
• Check the correlation between the variables. If
  necessary, use transformations and only then
  generate the regression equation using step-wise
  regression analysis.