1. 1 Chapter 13 Mathematical models of networks give us algorithms so computationally efficient that we can employ them to evaluate problems too big to be solved any other way. Network Models

2. 2 The Structure of Network Problems  Networks have features of the following: arcs (e.g., roads), nodes (e.g., cities), and arc values representing distance or flow.

3. 3 Types of Networks and Applications  Networks have been used in the following.  There are four basic network models.  Shortest route problems.  Minimal spanning trees.  Maximum flow.  Minimum-cost maximum flow.

4. 4 The Shortest Route Problem  This problem uses the network as prop. We find the shortest route from A to G.  The START node is evaluated first (shaded). Then all direct links between evaluated and unevaluated nodes are identified and the distances back to START are computed along connecting arcs.  The node with smallest distance, written above it, becomes the next evaluated one and is shaded.  An arrow points from there along the connecting arc.

5. 5 The Shortest Route Problem  Node C joins the evaluated set. An arrow is added pointing back to A, and the cumulative distance, 3, back to START from C is entered above.  The process continues.  Node B is next to join the evaluated set, at a distance of 4 back to START.

6. 6 The Shortest Route Problem  The process continues.  Node D is next to join the evaluated set with a distance of 5 back to START.

7. 7 The Shortest Route Problem  The process continues.  Node E is next to join the evaluated set with a distance of 7 back to START.

8. 8 The Shortest Route Problem  The process continues.  Two nodes, F and G, are next to join the evaluated set, each with with a distance of 9 back to START.

9. 9 The Shortest Route Problem  All nodes are evaluated. The shortest route is found by tracing back from FINISH following the arrows.  The shortest route from A to G is A-B-D-E-G for a distance of C = 9.

10. 10 The Minimal Spanning Tree  A tree is a set of arcs connecting nodes in such a way that only one route involving those arcs connects any two nodes.  Imagine an ant on a real tree. It has just one way to walk from any leaf to another.  A spanning tree connects with all nodes.  It is like railroad tracks connecting all cities, but with only one routing between any two.  A minimal spanning tree has the smallest sum of its arc distances C (tree size).  In connecting all circuit-board solder points with gold wire, it would use the least gold.  It would have the least tracks for a railway.

11. 11 Finding the Minimal Spanning Tree  As first connected node pick any (here A). Find all arcs directly joining connected to unconnected. Join the shortest arc to tree.  Connected nodes are shaded. A-C joins the tree.

12. 12 Finding the Minimal Spanning Tree  As new arcs join the tree, more nodes become connected. We consider only arcs directly joining connected to unconnected nodes.  Arc A-B joins the tree.

13. 13 Finding the Minimal Spanning Tree  The process continues.  Arc C-F joins the tree.

14. 14 Finding the Minimal Spanning Tree  The process continues.  Arcs B-E, D-E, F-J, H-I, and I-J join tree.

15. 15 Finding the Minimal Spanning Tree  The process continues.  Arc G-H joins the tree. Since all nodes are connected, the tree has been found. The sum of the arc lengths gives its size C.

16. 16 Maximizing Flow  Arcs in a maximum flow problem are directed and have upper bounds. Flow moves one way.  A node is designated as the SOURCE and another as the SINK.  Flow-augmenting paths from SOURCE to SINK are found and flows sent over the arcs. If no path can be found, flow is maximized.  A flow-augmenting path ordinarily involves arcs directed away from the SOURCE toward the SINK.  But an arc can point in the opposite direction if some of its current flow would be reduced and be redirected to another arc.  It doesn’t matter which path is used. The possibilities shrink as more are found.  Flow into an interior node must equal the flow out.

17. 17 Maximizing Flow  The bottleneck arc on a path has the lowest remaining capacity. Here it is H-J.

18. 18 Maximizing Flow  Arc H-J is saturated. Flow over saturated arcs may be decreased only.  The next path’s bottleneck arc is E-I.

19. 19 Maximizing Flow  The next path has two bottleneck arcs: C-F and K-L.

20. 20 Maximizing Flow  The next path has two bottleneck arcs: D-F and I-J.

21. 21 Maximizing Flow  This path goes against the direction of J-K flow. Some J-K flow is redirected over J-L.  The bottlenecks are B-D and I-K.

22. 22 Maximizing Flow  There are no more flow-augmenting paths. The optimal solution has been found with maximum flow (sum into SINK ) of C = 14.

23. 23 Minimum-Cost Maximum Flow  Transportation problems are special cases of minimum-cost maximum flow problems.  The general problem has bounded arcs (routes) and is represented as a network.  It may be solved by an elaborate procedure, the out-of-kilter algorithm, involving shortest routes and maximum flows.  However, it is best solved on the computer.  QuickQuant may be used for this purpose.  It can perform the out-of-kilter algorithm.  The problem can also be solved as a general linear program (with QuickQuant or Excel).

24. 24 Solving with QuickQuant  The following first iteration involves a smaller version of the problem in the text.

25. 25 Solving with QuickQuant  The initial solution is infeasible. A series of iterations yields the optimal solution.

26. 26 NetworkTemplates  shortest route  maximum flow  minimum cost maximum flow

27. 27 Shortest Route for Yellow Jacket Freightways (Figure 13-11) 2. Enter the distances above the diagonal in the table B9:H15. They will automatically be entered below the diagonal. This is the upper portion of Figure 13-11. The lower portion is shown next. 1. Enter the problem name in B3. 3. If 1000 is not large enough to denote the impossibility of going between two cities, use a larger number.

28. 28 Shortest Route for Yellow Jacket Freightways (Figure 13-11) This is the lower portion of Figure 13-11. The length of the shortest route is in cell E9. Here it is 9. The shortest route is found from the table in cells A20:H27. Here it is A-B-D- E-G.

29. 29 Shortest Route for Yellow Jacket Freightways (Figure 13-11) This is the lower portion of Figure 13-11. 4. Click on Tools and use Solver to find the shortest route. The Solver Parameters dialog box is shown on the next slide. 5. The starting point is assumed to be A and the ending point G. If this is different adjust the required flow in cells B31:H31 accordingly. 6. For problems with more than 7 cities, expand the distance and path tables and check to make sure that all the formulas have the proper ranges.

30. 30 Solver Parameters Dialog Box (Figure 13-12) 1. Enter the value of the objective function, E17, in the Target Cell line, either with or without the $ signs. 2. The Target Cell is to be minimized so click on Min in the Equal To line. 3. Enter the decision variables in the By Changing Cells line, B21:H27. 4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner. NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

31. 31 The Add Constraint Dialog Box 1. Enter the net flows B30:H30 (or $B$30:$H$30) in the Cell Reference line. 2. Enter = as the sign because the net flow must be equal to the required flow, given next in Step 3. 3. Enter the required flow B31:H31 in the Constraint line (or =$B$31:$H$31). 4. Click the OK button. Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint. If you need to change a constraint, the Change Constraint dialog box functions just like this one.

32. 32 Maximum Flow for Lulliput Telephone Company (Figure 13-31) This is the upper portion of Figure 13-31. The lower portion is shown next. 2. (a) Enter the capacities in the table B9:M20. 1. Enter the problem name in B3. 2. (b) A big number is entered for the upper limit on the return flow from L to A.

33. 33 Maximum Flow for Lulliput Telephone Company (Figure 13-31) This is the lower portion of Figure 13-31. The maximum flow is in cell E22. Here it is 14. The flow along each arc is found from the table in cells A20:H27. For example, cell C26 has a 1 in it. This means one unit of flow goes from A to B.

34. 34 Maximum Flow for Lulliput Telephone Company (Figure 13-31) This is the lower portion of Figure 13-31. 3.Click on Tools and use Solver to find the maximum flow. The Solver Parameters dialog box is shown on the next slide. 4. The starting point is the first node and the ending point the last one. 5. For problems with more than 12 nodes, expand the capacities and flows tables and check to make sure that all the formulas have the proper ranges.

35. 35 Solver Parameters Dialog Box (Figure 13-32) 1. Enter the value of the objective function, E22, in the Target Cell line, either with or without the $ signs. 2. The Target Cell is to be maximized so click on Max in the Equal To line. 3. Enter the decision variables in the By Changing Cells line, B26:M37. 4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box (obtained by clicking on the Add button) as was done for the shortest route template. If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner. NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

36. 36 Minimum Cost Maximum Flow for BigCo (Figure 13-40) This is the upper portion of Figure 13-40. The lower portion is shown next. 1. Enter the problem name in B3. 2. Enter the costs and capacities in the table B8:G10 and the corresponding From and To names in cells A8:A9 and B7:F7. 3. Enter the minimum quantities in the table B15:F16. 4. Enter the maximum quantities in the table B21:F22.

37. 37 Minimum Cost Maximum Flow for BigCo (Figure 13-40) This is the lower portion of Figure 13-40. The minimum cost is in cell E24. Here it is $12,110. The optimal shipping schedule is given in the table in cells A28:G29.

38. 38 Minimum Cost Maximum Flow for BigCo (Figure 13-40) This is the lower portion of Figure 13-40. 5.Click on Tools and use Solver to find the optimal solution. The Solver Parameters dialog box is shown on the next slide. 6. For other problems insert (or delete) the appropriate number of rows or columns and check to make sure that all the formulas have the proper ranges.

39. 39 Solver Parameters Dialog Box (Figure 13-41) 1. Enter the value of the objective function, E24, in the Target Cell line, either with or without the $ signs. 2. The Target Cell is to be minimized so click on Min in the Equal To line. 3. Enter the decision variables in the By Changing Cells line, B27:F28. 4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box (obtained by clicking on the Add button) as was done for the shortest route template. If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner. NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.