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Chapter 8
Network Models – Part 2
Assoc. Prof. Dr. Arslan M. ÖRNEK
8.4. CPM and PERT
■ Network representation is useful for project analysis.
■ Project examples: Construction of a building, organization of
a conference, installation of a computer system etc.
■ Networks show how project activities are organized and are
used to determine time duration of projects.
■ Network techniques used are:
▪ CPM (Critical Path Method)
▪ PERT (Project Evaluation and Review Technique)
■ Developed independently during late 1950’s.
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8.4. CPM and PERT
■ All activities (steps) of the project should be identified.
■ The sequential relationships of the activities (which activity
comes first, which follows, etc.) is identified by precedence
relationships.
■ For each activity, there is a set of activities (called the
predecessors of the activity) that must be completed before
the activity begins.
■ Steps of project planning:
■ Make time estimates for activities, determine project
completion time.
■ Compare project schedule objectives, determine resource
requirements.
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8.4. CPM and PERT
Activity-on-Arc (AOA) Network
■ An arc reflects an activity of a project.
■ A node represents the beginning and end of activities, referred
to as events.
■ Arcs in the network indicate precedence relationships.
■ When an activity is completed at a node, it has been realized.
8.4. CPM and PERT
■ Time duration of activities are shown on arcs.
■ Activities can occur at the same time (concurrently).
■ Node 1: Start node, last node: finish node of the project.
■ A dummy activity shows a precedence relationship but reflects
no passage of time.
■ Two or more activities cannot share the same start and end
nodes.
8.4. CPM and PERT
The Project Network
House Building Project Data
No. Activity
1. Design house and
obtain financing
Activity Predecessor
Duration (Months)
-
3
2. Lay foundation
1
2
3. Order Materials
1
1
4. Build house
2, 3
3
5. Select paint
2, 3
1
6. Select carpet
5
1
4, 6
1
7. Finish work
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Drawing the Project Network
AOA Network for House Building Project
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Drawing the Project Network
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Drawing the Project Network
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8.4. CPM and PERT
To compute ETs, make a forward pass on the project network:
8.4. CPM and PERT
It can be shown that ET(i)
is the length of the
longest path in the
project network from
node 1 to node i.
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8.4. CPM and PERT
To compute LTs, make a backward pass on the project network:
8.4. CPM and PERT
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8.4. CPM and PERT
If LT(i) = ET(i), any delay in the occurrence of node i will delay the
completion of the project.
For example, because LT(4) = ET(4), any delay in the occurrence
of node 4 will delay the completion of the project.
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8.4. CPM and PERT
The total float of an activity is the amount by which the duration
of the activity can be increased without delaying the completion
of the project.
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8.4. CPM and PERT
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8.4. CPM and PERT
A critical path in any project network is the longest path from
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the start node to the finish node.
8.4. CPM and PERT
The free float of an activity is the amount by which the duration
of the activity can be increased without delaying the start of any
other activity.
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8.4. CPM and PERT
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8.4. CPM and PERT
In many situations, the project manager must complete the project
in a time that is less than the length of the critical path.
This is called crashing the project.
Suppose that Widgetco must complete the project within 25 days.
By allocating additional resources to an activity, Widgetco can
reduce the duration of any activity by as many as 5 days. The cost
per day of reducing the duration of an activity is shown.
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8.4. CPM and PERT
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8.4. CPM and PERT
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■ Activity time estimates usually cannot be made with certainty.
■ PERT is used for probabilistic activity times.
■ In PERT, three time estimates are used: most likely time (m), the optimistic
time (a), and the pessimistic time (b).
■ These provide an estimate of the mean and variance of a beta distribution:
variance of activity (i,j):
mean (expected time) of activity (i,j) :
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■ Total time of the critical path is normally distributed (by the
Central Limit Theorem):
With this assumption, we can answer questions concerning
the probability that the project will be completed by a given
date.
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What is the probability that the project will be completed within 35 days?
To answer this question, we must also make the following assumption: No matter
what the durations of the project’s activities turn out to be, 1–2–3–4–5–6 will be a
critical path.
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8.5. Minimum Cost Network Flow Problems
The transportation, assignment, transshipment,
shortest-path, maximum flow, and CPM problems are
all special cases of the minimum cost network flow
problem (MCNFP).
Any MCNFP can be solved by a generalization of the
transportation simplex called the network simplex.
8.5. Minimum Cost Network Flow Problems
Flow balance equations
8.5. Minimum Cost Network Flow Problems
8.5. Minimum Cost Network Flow Problems
8.5. Minimum Cost Network Flow Problems
The flow balance equations in any MCNFP have the following
important property:
Each variable xij has a coefficient of 1 in the node i flow balance
equation, a coefficient of -1 in the node j flow balance equation,
and a coefficient of 0 in all other flow balance equations.
An MCNFP can be solved by a generalization of the
transportation simplex known as the network simplex algorithm.
If the problem parameters are integers, the optimal solution will
also be integer.
The network simplex is efficient and easy to use, so it is
important to formulate an LP, if at all possible, as an MCNFP.
8.5. Minimum Cost Network Flow Problems
Example: Each hour, an average of 900 cars enter the
network below at node 1 and seek to travel to node 6.
The time it takes a car to traverse each arc is shown in
the table. The number above each arc is the arc
capacity. Formulate an MCNFP that minimizes the
total time required for all cars to travel from node 1
to node 6.
8.5. Minimum Cost Network Flow Problems
Constraints:
8.5. Minimum Spanning Tree Problems
• Suppose that each arc (i, j) in a network has a length.
• Arc (i, j) represents a way of connecting node i to node j.
• For example, if each node in a network represents a computer in
our university, then arc (i, j) might represent an underground cable
that connects computer i with computer j.
• We want to determine the set of arcs in a network that connect
all nodes such that the sum of the length of the arcs is minimized.
• Such a group of arcs should contain no loop. (A loop is often
called a closed path or cycle.) For example, the sequence of arcs
(1, 2)–(2, 3)–(3, 1) is a loop.
8.5. Minimum Spanning Tree Problems
8.5. Minimum Spanning Tree Problems
– A cycle (loop): A path beginning and ending at the same node.
– The loop DE-ED is a directed loop
– The loop AB-BC-AC is an undirected loop because A-B-C-A is
an undirected path (a chain).
– Two nodes are connected if there is at least one chain
between them.
– A connected network is a network where every pair of nodes
is connected. (Which arcs should be removed to make the
below network unconnected?
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8.5. Minimum Spanning Tree Problems
– Consider a network with n nodes where all arcs are deleted.
– A tree can be “grown” by adding one arc (or “branch”) at a
time to connect all nodes.
• Each arc should be added such that it is between one node
already in the tree, and one not in the tree, and no cycles
will be formed.
• Total number of arcs = n – 1.
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8.5. Minimum Spanning Tree Problems
– The resulting tree connects all nodes, which is called
a spanning tree (: a connected network for all n
nodes that contains no undirected cycles).
– Every spanning tree has n-1 arcs.
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8.5. Minimum Spanning Tree Problems
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8.5. Minimum Spanning Tree Problems
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8.5. Minimum Spanning Tree Problems
Finding the MST:
8.5. Minimum Spanning Tree Problems
Example:
Step 0
Step 3
Step 1
Step 2
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8.5. Minimum Spanning Tree Problems
Example:
Step 3
Step 6:
Total length = 14
miles
Step 4
Step 5
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The Minimum Spanning Tree Problem
Example 2
Network of Possible Cable TV Paths – Connect all nodes so that
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the cable length is minimized.
The Minimum Spanning Tree Problem
Example 2
Partial Tree with nodes 1 and 3
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The Minimum Spanning Tree Problem
Example 2
Partial Tree with nodes 1, 3 and 4
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The Minimum Spanning Tree Problem
Example 2
Partial Tree with nodes 1, 3, 4 and 2
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The Minimum Spanning Tree Problem
Example 2
Partial Tree with nodes 1, 3, 4, 2 and 5
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The Minimum Spanning Tree Problem
Example 2
Partial Tree with nodes 1, 3, 4, 2, 5 and 7
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The Minimum Spanning Tree Problem
Example 2
Minimum Spanning Tree
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The Minimum Spanning Tree Problem
Example 3
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The Minimum Spanning Tree Problem
Example 3
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