1. Notes 6IE3121 Knapsack Model Intuitive idea: what is the most valuable collection of items that can be fit into a backpack?

2. Notes 6IE3122 Race Car Features Budget of $35,000 Which features should be added?

3. Notes 6IE3123 Decision variables ILP Formulation

4. Notes 6IE3124 LINGO Formulation MODEL: SETS: FEATURES /F1,F2,F3,F4,F5,F6/: INCLUDE,SPEED_INC,COST; ENDSETS DATA: SPEED_INC = 8 3 15 7 10 12; COST = 10.2 6.0 23.0 11.1 9.8 31.6; BUDGET = 35; ENDDATA MAX = @SUM( FEATURES: SPEED_INC * INCLUDE); @SUM( FEATURES: COST * INCLUDE) <= BUDGET; @FOR( FEATURES: @BIN( INCLUDE)); END Specify index sets All the constants Objective Constraints Variables indexed by this set Decision variables are binary Note ; to end command : to begin an environment

5. Notes 6IE3125 Solve using Branch & Bound Solution? Candidate Problem Relaxed Problem

6. Notes 6IE3126 What is the Relative Worth? Want to add this feature first Want to add this feature second

7. Notes 6IE3127 Solve Relaxed Problem Solution: Relaxed Problem Objective  24.8

8. Notes 6IE3128 Now the other node … Relaxed Problem Solution: Objective  27.8

9. Notes 6IE3129 Next Step? Objective  24.8 Objective  27.8

10. Notes 6IE31210 Rule of Thumb: Better Value Obj  24.8 Relaxed Problem Solution: Relaxed Problem Obj.  26.4Obj.  27.8 Solution:

11. Notes 6IE31211 Next Level Obj  24.8 Obj.  26.4 Infeasible Candidate Problem Solution: Obj. = 25 (This turns out to be true.) Now What?

12. Notes 6IE31212 Next Steps … Obj  24.8 Obj.  26.4 InfeasibleObj. = 25 Obj  25 Still need to continue branching here. Finally we will have accounted for every solution!

13. Notes 6IE31213 Capital Budgeting Multidimensional knapsack problems are often called capital budgeting problems Idea: select collection of projects, investments, etc, so that the value is maximized (subject to some resource constraints)

14. Notes 6IE31214 NASA Capital Budgeting

15. Notes 6IE31215 Formulation Decision variables Budget constraints

16. Notes 6IE31216 Formulation Mutually exclusive choices Dependencies

17. Notes 6IE31217 Set Packing, Covering, and Partitioning

18. Notes 6IE31218 Select Locations

19. Notes 6IE31219 Ways of Splitting the Set Set covering constraints Set packing constraints Set partitioning constraints

20. Notes 6IE31220 Example: Choosing OR Software Formulate a set covering problem to acquire the minimum cost software with LP, IP, and NLP capabilities. Formulate set partitioning and set packing problems. What goals do they meet?

21. Notes 6IE31221 Maximum Coverage Perhaps the budget only allows $9000 What can we then do  Maximum coverage How do we now formulate the problem? Need new variables

22. Notes 6IE31222 Travelling Salesman Problem (TSP) Ames Fort Dodge Boone Carroll Marshalltown West Des Moines Waterloo What is the shortest route, starting in Ames, that visits each city exactly ones?

23. Notes 6IE31223 TSP Solution Ames Fort Dodge Boone Carroll Marshalltown West Des Moines Waterloo

24. Notes 6IE31224 Not a TSP Solution Ames Fort Dodge Boone Carroll Marshalltown West Des Moines Waterloo

25. Notes 6IE31225 Applications Routing of vehicles (planes, trucks, etc.) Routing of postal workers Drilling holes on printed circuit boards Routing robots through a warehouse, etc.

26. Notes 6IE31226 Formulating TSP A TSP is symmetric if you can go both ways on every arc

27. Notes 6IE31227 Example 1 5 2 6 34 10 1 1 1 1 1 1 Formulate a TSP

28. Notes 6IE31228 Subtours It is not sufficient to have two arcs connected to each node Why? Must eliminate all subtours Every subset of points must be exited

29. Notes 6IE31229 How do we eliminate subtours? 1 5 2 6 34 10 1 1 1 1 1 1

30. Notes 6IE31230 Asymmetric TSP Now we have decision variables Constraints

31. Notes 6IE31231 Asymmetric TSP (cont.) Each tour must enter and leave every subset of points Along with all variables being 0 or 1, this is a complete formulation

32. Notes 6IE31232 Example 1 5 2 6 34 10 1 1 1 1 1 1 Assume a two unit penalty for passing from a high to lower numbered node. This is now an asymmetric TSP. Why?

33. Notes 6IE31233 Subtour Elimination Making sure there are no subtours involves a very large number of constraints Can obtain simpler constraints if we go with a nonlinear objective function

34. Notes 6IE31234 Quadratic Assignment Formulation

35. Notes 6IE31235 Example: reformulate 1 5 2 6 34 10 1 1 1 1 1 1