1. Welcome Unit 6 Seminar MM305 Wednesday 8:00 PM ET Quantitative Analysis for Management Delfina Isaac

2. Characteristics of Network Models A node is a specific location An arc connects 2 nodes Arcs can be 1-way or 2-way

3. Network Models with QM for Windows Appendix 6.1 Minimal-spanning tree technique – cost or distance Go to module -> networks -> file -> new -> Minimum spanning tree Maximal flow technique – flow and reverse flow capacity. Go to module-> networks -> file-> new-> Maximal flow Shortest-route technique – distance associated with each branch. Go to module-> networks, file-> new-> Shortest route

4. Network Models with QM for Windows Appendix 6.1 In order to use QM for Windows to solve network problems, the network must be drawn and labeled Data about this network is input to QM Branches (arcs) are identified by the starting and ending nodes for each leg of the network The following information must also be input: Minimal-spanning tree technique – cost or distance Maximal flow technique – flow and reverse flow capacity Shortest-route technique – distance associated with each branch

5. Steps for Solving Minimal Spanning Tree 1.Selecting any node in the network. 2.Connect this node to the nearest node minimizing the total distance. 3. Find and connect the nearest unconnected node.  If there is a tie for the nearest node, one can be selected arbitrarily.  A tie suggests that there may be more than one optimal solution. 4.Repeat the third step until all nodes are connected. Decision: Which arcs to choose to connect all nodes? Objective: Minimize the total distance of the arcs chosen

6. Minimal-Spanning Tree Technique Network for Lauderdale Construction (Figure 6.2) 1st Iteration

7. Minimal-Spanning Tree Technique Network for Lauderdale Construction (Figure 6.3) 2nd and 3rd Iterations

8. Minimal-Spanning Tree Technique Network for Lauderdale Construction (Figure 6.4) 4th and 5th Iterations

9. Minimal-Spanning Tree Technique Network for Lauderdale Construction (Figure 6.5) 6th and 7 th (Final) Iterations

10. Minimal-Spanning Tree Technique Network for Lauderdale Construction

12. Steps for the Maximal-Flow Technique 1. Pick any path from the start (source) to the finish (sink) with some flow.  If no path with flow exists, then the optimal solution has been found. 2. Find the arc on this path with the smallest flow capacity available.  Call this capacity C.  This represents the maximum additional capacity that can be allocated to this route. 3. For each node on this path, decrease the flow capacity in the direction of flow by the amount C.  For each node on this path, increase the flow capacity in the reverse direction by the amount C. 4. Repeat these steps until an increase in flow is no longer possible. Decision: How much flow on each arc? Objective: Maximize flow through the network from an origin to a destination

13. Maximal-Flow Technique Road Network for Waukesha (Figure 6.6)

14. Maximal-Flow Technique Road Network for Waukesha (Figure 6.7) Capacity Adjustment for path 1–2–6 1 st iteration

15. Maximal-Flow Technique Road Network for Waukesha (Figure 6.7) New Path

16. Maximal-Flow Technique Road Network for Waukesha Repeat the Process

17. Maximal-Flow Technique Road Network for Waukesha (Figure 6.8) Capacity adjustment for path 1–2–4-6 iteration 2

18. Maximal-Flow Technique Road Network for Waukesha (Figure 6.8) New Network

19. Maximal-Flow Technique Road Network for Waukesha (Figure 6.9) 3rd Iteration

20. Maximal-Flow Technique Road Network for Waukesha (Figure 6.9) 4 th and Final Iteration

21. Maximal-Flow Technique Road Network for Waukesha (Figure 6.9) There are no more paths from nodes 1 to 6 with unused capacity so this represents a final iteration The maximum flow through this network is 500 cars PATHFLOW (CARS PER HOUR) 1–2–61–2–6200 1–2–4–61–2–4–6100 1–3–5–61–3–5–6200 Total500 Final Network Flow

22. Maximal-Flow Technique Road Network for Waukesha

24. Steps of the Shortest-Route Technique 1.Find the nearest node to the origin (plant). 2.Find the next-nearest node to the origin and put the distance in a box by the node. Several paths may have to be checked to find the nearest node. 3.Repeat this process until you have gone through the entire network. Decision: Which arcs to travel on? Objective: Minimize the distance (or time) from the origin to the destination

25. Shortest-Route Technique: Figure 6.10 Roads from Ray’s Plant to Warehouse

26. Shortest-Route Technique: Figure 6.11 Ray’s Design 1 st Iteration

27. Shortest-Route Technique: Figure 6.12 Ray’s Design 2nd Iteration

28. Shortest-Route Technique: Figure 6.13 Ray’s Design 3rd Iteration

29. Shortest-Route Technique: Figure 6.14 Ray’s Design 4 th and Final Iteration

30. Shortest-Route Technique Roads from Ray’s Plant to Warehouse