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. 2
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. 3
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 Activity Predecessor Duration (Months) 1. Design house and - 3 obtain financing 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 7. Finish work 4, 6 1 7
Drawing the Project Network AOA Network for House Building Project 8
Drawing the Project Network 9
Drawing the Project Network 10
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 . 12
8.4. CPM and PERT To compute LTs, make a backward pass on the project network:
8.4. CPM and PERT 14
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. 15
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. 16
8.4. CPM and PERT 17
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
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. 19
8.4. CPM and PERT 20
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
8.4. CPM and PERT 22
8.4. CPM and PERT 23
■ 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
■ 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
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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. 28
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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 x ij 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? 40
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. 41
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. 42
8.5. Minimum Spanning Tree Problems 43
8.5. Minimum Spanning Tree Problems 44
8.5. Minimum Spanning Tree Problems Finding the MST:
8.5. Minimum Spanning Tree Problems Example: Step 1 Step 0 Step 3 Step 2 46
8.5. Minimum Spanning Tree Problems Example: Step 3 Step 4 Step 6: Total length = 14 Step 5 miles 47
The Minimum Spanning Tree Problem Example 2 Network of Possible Cable TV Paths – Connect all nodes so that 48 the cable length is minimized.
The Minimum Spanning Tree Problem Example 2 Partial Tree with nodes 1 and 3 49
The Minimum Spanning Tree Problem Example 2 Partial Tree with nodes 1, 3 and 4 50
The Minimum Spanning Tree Problem Example 2 Partial Tree with nodes 1, 3, 4 and 2 51
The Minimum Spanning Tree Problem Example 2 Partial Tree with nodes 1, 3, 4, 2 and 5 52
The Minimum Spanning Tree Problem Example 2 Partial Tree with nodes 1, 3, 4, 2, 5 and 7 53
The Minimum Spanning Tree Problem Example 2 Minimum Spanning Tree 54
The Minimum Spanning Tree Problem Example 3 55
The Minimum Spanning Tree Problem Example 3 56
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