SLIDE 1 Chapter 8 Network Models – Part 2
- Assoc. Prof. Dr. Arslan M. ÖRNEK
SLIDE 2 ■ 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.
8.4. CPM and PERT
2
SLIDE 3 ■ 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.
8.4. CPM and PERT
3
SLIDE 4
■ 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. Activity-on-Arc (AOA) Network
8.4. CPM and PERT
SLIDE 5
■ 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
SLIDE 6
8.4. CPM and PERT
SLIDE 7 The Project Network House Building Project Data
Activity Predecessor Duration (Months)
- 1. Design house and
- 3
- btain financing
- 2. Lay foundation
1 2
1 1
2, 3 3
2, 3 1
5 1
4, 6 1
7
SLIDE 8 Drawing the Project Network AOA Network for House Building Project
8
SLIDE 9 Drawing the Project Network
9
SLIDE 10 Drawing the Project Network
10
SLIDE 11
8.4. CPM and PERT
To compute ETs, make a forward pass on the project network:
SLIDE 12 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.
12
SLIDE 13
8.4. CPM and PERT
To compute LTs, make a backward pass on the project network:
SLIDE 14 8.4. CPM and PERT
14
SLIDE 15 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
- f node 4 will delay the completion of the project.
15
SLIDE 16 8.4. CPM and PERT
The total float of an activity is the amount by which the duration
- f the activity can be increased without delaying the completion
- f the project.
16
SLIDE 17 8.4. CPM and PERT
17
SLIDE 18 8.4. CPM and PERT
A critical path in any project network is the longest path from the start node to the finish node.
18
SLIDE 19 8.4. CPM and PERT
The free float of an activity is the amount by which the duration
- f the activity can be increased without delaying the start of any
- ther activity.
19
SLIDE 20 8.4. CPM and PERT
20
SLIDE 21 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.
21
SLIDE 22 8.4. CPM and PERT
22
SLIDE 23 8.4. CPM and PERT
23
SLIDE 24 ■ 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) :
24
SLIDE 25 ■ 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.
25
SLIDE 28 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.
28
SLIDE 30
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.
SLIDE 31
8.5. Minimum Cost Network Flow Problems
Flow balance equations
SLIDE 32
8.5. Minimum Cost Network Flow Problems
SLIDE 33
8.5. Minimum Cost Network Flow Problems
SLIDE 34
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.
SLIDE 35 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.
SLIDE 36
8.5. Minimum Cost Network Flow Problems
SLIDE 37
Constraints:
SLIDE 38 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
- ur 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.
SLIDE 39
8.5. Minimum Spanning Tree Problems
SLIDE 40 – 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?
8.5. Minimum Spanning Tree Problems
40
SLIDE 41 – 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.
8.5. Minimum Spanning Tree Problems
41
SLIDE 42 – 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.
8.5. Minimum Spanning Tree Problems
42
SLIDE 43 8.5. Minimum Spanning Tree Problems
43
SLIDE 44 8.5. Minimum Spanning Tree Problems
44
SLIDE 45
8.5. Minimum Spanning Tree Problems
Finding the MST:
SLIDE 46 Step 0 Step 1 Step 2 Step 3
8.5. Minimum Spanning Tree Problems Example:
46
SLIDE 47 Step 3 Step 4 Step 5 Step 6: Total length = 14 miles
8.5. Minimum Spanning Tree Problems Example:
47
SLIDE 48 Network of Possible Cable TV Paths – Connect all nodes so that the cable length is minimized.
The Minimum Spanning Tree Problem Example 2
48
SLIDE 49 Partial Tree with nodes 1 and 3
The Minimum Spanning Tree Problem Example 2
49
SLIDE 50 Partial Tree with nodes 1, 3 and 4
The Minimum Spanning Tree Problem Example 2
50
SLIDE 51 Partial Tree with nodes 1, 3, 4 and 2
The Minimum Spanning Tree Problem Example 2
51
SLIDE 52 Partial Tree with nodes 1, 3, 4, 2 and 5
The Minimum Spanning Tree Problem Example 2
52
SLIDE 53 Partial Tree with nodes 1, 3, 4, 2, 5 and 7
The Minimum Spanning Tree Problem Example 2
53
SLIDE 54 Minimum Spanning Tree
The Minimum Spanning Tree Problem Example 2
54
SLIDE 55 The Minimum Spanning Tree Problem Example 3
55
SLIDE 56 The Minimum Spanning Tree Problem Example 3
56
