Project Scheduling Models

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Example 5.10

• Project Scheduling Models

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Background Information

• Tom Lingley, an independent contractor, has
  agreed to build a new room on an existing house.
• He plans to begin work on Monday morning, June 1.
• The main question is when will he complete his
  work, given that he works only on weekdays.
• The owner of the house is particularly hopeful
  that the room will be ready in 15 or fewer
  working days that is, by the end of Friday,
  June 19.
• The work proceeds in stages, labeled A through J,
  as summarized in the table on the next slide.

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Background Information -- continued

• Three of these activities, E, F, and G, will be
  done by separate independent subcontractors.

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Background Information -- continued

• Lingley wants to know how long the project will
  take, given the activity times (durations) in the
  table.
• He also wants to know the critical activities.

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Solution

• The project network appears here. The activity
  time for each activity is shown on its arc.

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Solution -- continued

• We suggest that you verify rule 3 from our
  general discussion if an activitys arc leads
  out of a node, then all of this activitys
  predecessors should have arcs leading into the
  node.
• The key to the solution is finding event times
  for each node in the network, where the event
  time for node i is the earliest time we could
  reach that point in the network.
• Denote the event time for node i by Ei. We begin
  by setting E10. Node 1 is the start node, so its
  event time is 0 right away.

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Solution -- continued

• Also, if node n is the finish node, then the
  earliest the entire project can be completed is
  at time En. Therefore, the total project time is
  E8 for our example. In general, let i and j be
  any nodes joined by an arc with activity time
  tij.
• Then we must have Ej ? Ei tij. The reasoning is
  that the event at node j cannot occur until at
  least time tij after the event at node i.
• Actually, we can do better than this. The event
  time Ej is equal to the maximum of the quantities
  Ei tij, where the maximum is taken over all arcs
  that lead into node j.

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Solution -- continued

• We could use this relationship to find the event
  times. In fact, we will do this when we revisit
  this example in Chapter 12.
• For right now, however, we will not use maximums.
• Instead, we will exploit inequalities.

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Developing the Model -- continued

• The completed spreadsheet model appears on the
  next slide.
• It can be developed with the following steps.
• Enter data. The data for this model include the
  predecessors and activity times (durations) in
  the range C5D14.
• Event times. We will eventually use the Solver to
  find the event times that satisfy inequalities.
  For now, enter any event times in the EventTimes
  range. The EventTime range will be the changing
  cells range.

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Developing the Model -- continued

• Network Information. Enter the information in the
  range D18F28. This comes directly from the
  network shown earlier.
• Inequalities. Now we implement inequalities in
  the range G18I28. We could enter the formulas
  one cell at a time, but it is more convenient to
  use VLOOKUP functions. To do so, enter the
  formula VLOOKUP(F18,LTable2,2) in cell G18 and
  copy it down. Each of these values corresponds to
  Ei tij. As usual, make sure you understand
  exactly how these formulas work. Lookup functions
  can be intimidating, but they can save a lot of
  work!
• Project time. Enter the formula B25 in cell B30.
  This creates a link to the event time for the
  final node in cell B25 that is, to the time to
  complete the project.

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Using Solver

• We want to choose the event times so as to
  minimize the project time and satisfy
  inequalities. Therefore, set up the Solver as
  shown here.

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Using Solver -- continued

• When we click on Solve and see the dialog box
  indicating that the Solver has found the optimal
  solution, we now request a sensitivity report,
  which is shown on the next slide.
• We stated in Chapter 3 that these sensitivity
  reports can sometimes be misleading, but they
  provide useful information in this example.
• Specifically, consider the Shadow Price column of
  the sensitivity report.

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Using Solver -- continued

• Each row of this table corresponds to one of the
  inequalities in the model that is, to one of
  the activities in the project.
• In general, a shadow price indicates the change
  in the target cell if the right side of a
  constraint increases by 1.
• Because our constraints include activity times on
  their right sides, each shadow price indicates
  how much the total project time will increase if
  the corresponding activity time increases by 1.

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Using Solver -- continued

• We see that some activity time increases have no
  effect on the project time, whereas others have a
  positive effect.
• This is how we find the critical path. It
  includes exactly those activities with positive
  shadow prices. These are indicated by asterisks.
• The activities that are on the critical path are
  A, B, D, E, H, and J.
• If the activity times for activities not on the
  critical path could increase, at least a little,
  with no effect on the total project time.