6/10/01network problems: djk1 network problems chapters 9 and 10
TRANSCRIPT
6/10/01 Network Problems: DJK 1
Network Problems
Chapters 9 and 10
6/10/01 Network Problems: DJK 2
General
• Networks consist of nodes and arcs
• A node is either an origin or a destination
• An arc connects two nodes and may represent distances, costs or flows
6/10/01 Network Problems: DJK 3
Types of Problems
• Shortest distance
• Minimum spanning tree
• Maximum flow
• Project management
6/10/01 Network Problems: DJK 4
Shortest Distance
• Start by drawing the network as shown on page 410
• Then label the nodes closest to the start with tentative distances and most recent node, e.g. node 2 [15,1] and node 3 [10,1]
• The closest node (#3) now becomes a “solved node”
6/10/01 Network Problems: DJK 5
Continued
• We now repeat the process, using nodes 1 and 3.
• This gives us temporary values for nodes 2 and 5 [13,3] and [14,3]
• By continuing this process, we get nodes 2 and 5 as solved nodes and tentative values for 4, 6 and 7 [18,5],[16,5] and [22,6] which is an optimum. See p417 for package output
6/10/01 Network Problems: DJK 6
1 4
3
2
7
6
5
15
3
10
17
6
4
5 6
4
2
6/10/01 Network Problems: DJK 7
1 4
2
7
6
5
15
3
10
17
6
4
5 6
4
2 3
[15,1]
[10,1]
6/10/01 Network Problems: DJK 8
1 4
3
2
7
6
5
15
3
10
17
6
4
5 6
4
2
[22,6]
[16,5]
[14,3]
N4-[18,5]
[10,1]
[13,3]
6/10/01 Network Problems: DJK 9
Minimum Spanning Tree
• Start at any node
• Find the closest node
• Connect them
• Repeat, using either node
• Continue until all nodes are connected
• Useful for network layout
6/10/01 Network Problems: DJK 10
1 4
3
2
7
6
5
15
3
10
17
6
4
5 6
4
2
Fir st 2 st eps, bl ue, t h en r ed
6/10/01 Network Problems: DJK 11
Maximum Flow
• Useful for message transmission, detours, etc.
• Stepwise procedure described on page 422
• Basis for calculation is bi-directional flow information between two nodes[N1]-7--------------0-[N2] changed to
[N1]-1--------------6-[N2]
6/10/01 Network Problems: DJK 12
Algorithm
• Find a path from source to destination with positive flow capacities for all arcs
• Increase flow on that path as much as you can
• Repeat first two until no paths exist which have positive flows in desired direction
• Keep track of paths and flows
6/10/01 Network Problems: DJK 13
Project Management
• There are two basic network methods for managing a complex project, CPM and PERT
• CPM, Critical Path Method, is based on a single time estimate for each activity
• PERT, Project Evaluation Reporting Technique, uses 3 estimates-optimistic, realistic and pessimistic
6/10/01 Network Problems: DJK 14
Functions
• Both methods will develop the “critical path”, the sequence of activities which will delay the entire project if they are delayed
• We need predecessor activities for each• CPM is basically easier to do, so we will
concentrate on it• “Crash costing” considers ways to reduce
completion time by applying money
6/10/01 Network Problems: DJK 15
Critical Path
• To find the critical path, we need to do a forward pass and a backward pass through the network
• Of course, that means we need to draw the network first
• Having drawn the network, the forward pass will develop ES and EF times for each activity
6/10/01 Network Problems: DJK 16
Forward Pass
• If t is the duration of a task, and ES is the earliest start, EF the earliest finishES= maximum of EF’s for all predecessorsEF=ES+t
• ES for any starting activity will be zero
6/10/01 Network Problems: DJK 17
Backward Pass
• Using similar notation (LF and LS are latest finish and start times) with LF for the last activity or activities being the final completion time (the EF for those activities)LS = LF-t
• The LF’s for any predecessors will be the LS for the successor
6/10/01 Network Problems: DJK 18
Critical Path
• Calculate “slack” for each activity. Either LS-ES or LF-EF
• Any activity with zero slack is on the critical path
6/10/01 Network Problems: DJK 19
Crash Costing
• Crash costing: shorten the critical path by using money to shorten times of activities on the critical path
• Watch outs– Not all can be reduced– Cost of reduction varies– As one path is reduced, another may become
critical
6/10/01 Network Problems: DJK 20
Manpower Smoothing
• Use activities with slack time to balance out the need for critical skills