1. 6/10/01Network Problems: DJK1 Network Problems Chapters 9 and 10

2. 6/10/01Network Problems: DJK2 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

3. 6/10/01Network Problems: DJK3 Types of Problems Shortest distance Minimum spanning tree Maximum flow Project management

4. 6/10/01Network Problems: DJK4 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”

5. 6/10/01Network Problems: DJK5 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. 6/10/01Network Problems: DJK6

7. 6/10/01Network Problems: DJK7

8. 6/10/01Network Problems: DJK8

9. 6/10/01Network Problems: DJK9 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

10. 6/10/01Network Problems: DJK10

11. 6/10/01Network Problems: DJK11 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]

12. 6/10/01Network Problems: DJK12 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

13. 6/10/01Network Problems: DJK13 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

14. 6/10/01Network Problems: DJK14 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

15. 6/10/01Network Problems: DJK15 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

16. 6/10/01Network Problems: DJK16 Forward Pass If t is the duration of a task, and ES is the earliest start, EF the earliest finish ES= maximum of EF’s for all predecessors EF=ES+t ES for any starting activity will be zero

17. 6/10/01Network Problems: DJK17 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

18. 6/10/01Network Problems: DJK18 Critical Path Calculate “slack” for each activity. Either LS-ES or LF-EF Any activity with zero slack is on the critical path

19. 6/10/01Network Problems: DJK19 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

20. 6/10/01Network Problems: DJK20 Manpower Smoothing Use activities with slack time to balance out the need for critical skills