Knapsack Model

1. Knapsack Model • Intuitive idea: what is the most valuable collection of items that can be fit into a backpack? IE 312

2. Race Car Features Budget of $35,000 Which features should be added? IE 312

3. Formulation • Decision variables • ILP IE 312

4. 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 Variables indexed by this set Specify index sets Note ; to end command : to begin an environment All the constants Objective Constraints Decision variables are binary IE 312

5. Solve using Branch & Bound Candidate Problem Relaxed Problem Solution? IE 312

6. What is the Relative Worth? Want to add this feature second Want to add this feature first IE 312

7. Solve Relaxed Problem Relaxed Problem Solution: Objective <= 24.8 IE 312

8. Now the other node… Relaxed Problem Solution: Objective <= 27.8 IE 312

9. Next Step? Objective <= 24.8 Objective <= 27.8 IE 312

10. Relaxed Problem Solution: Obj. <=26.4 Obj. <= 27.8 Relaxed Problem Solution: Rule of Thumb: Better Value Obj <= 24.8 IE 312

11. Next Level Obj <= 24.8 Obj. <=26.4 Obj. = 25 Infeasible Now What? IE 312

12. Next Steps … Still need to continue branching here. Obj <= 24.8 Obj. <= 26.4 Finally we will have accounted for every solution! Obj. = 25 Infeasible IE 312

13. 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) IE 312

14. NASA Capital Budgeting IE 312

15. Formulation • Decision variables • Budget constraints IE 312

16. Formulation • Mutually exclusive choices • Dependencies IE 312

17. Assignment Problems • Assignment problems deal with optimal pairing or matching of objects in two distinct sets • Decision variable • Let A be the set of allowed assignments and cij be the cost of assigning i to j. IE 312

18. Formulation IE 312

19. Example • We must determine how jobs should be assigned to machines to minimize setup times, which are given below: IE 312

20. Hungarian Algorithm • Step 1: (a) Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from each row. (b) Find the minimum cost in each column of the new matrix, and subtract this from each column. This is the reduced cost matrix. IE 312

21. Example: Step 1(a) IE 312

22. Example: Step 1(b) IE 312

23. Hungarian Algorithm • Step 2: Draw the minimum number of lines that are needed to cover all the zeros in the reduced cost matrix. If m lines are required, then an optimal solution is available among the covered zeros in the matrix. Otherwise, continue to Step 3. IE 312

24. Example: Step 2 We need 3<4 lines, so continue to Step 3 IE 312

25. Hungarian Algorithm • Step 3: Find the smallest nonzero element (say, k) in the reduced cost matrix that is uncovered by the lines. Subtract k from each uncovered element, and add k to each element that is covered by two lines. Return to Step 2. IE 312

26. Example: Step 3 IE 312

27. Example: Step 2 (again) Need 4 lines, so we have the optimal assignment and we stop IE 312

28. Example: Final Solution Optimal assignment IE 312

29. Travelling Salesman Problem (TSP) Fort Dodge Waterloo Boone Ames Carroll Marshalltown What is the shortest route, starting in Ames, that visits each city exactly ones? West Des Moines IE 312

30. TSP Solution Fort Dodge Waterloo Boone Ames Carroll Marshalltown West Des Moines IE 312

31. Not a TSP Solution Fort Dodge Waterloo Boone Ames Carroll Marshalltown West Des Moines IE 312

32. Applications • Routing of vehicles (planes, trucks, etc.) • Routing of postal workers • Drilling holes on printed circuit boards • Routing robots through a warehouse, etc. IE 312

33. Formulating TSP • A TSP is symmetric if you can go both ways on every arc IE 312

34. 10 1 2 1 1 1 10 1 3 4 1 1 10 5 6 Example Formulate a TSP IE 312

35. 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 IE 312

36. 10 1 2 1 1 1 10 1 3 4 1 1 10 5 6 How do we eliminate subtours? IE 312

37. Asymmetric TSP • Now we have decision variables • Constraints IE 312

38. 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 IE 312

39. 10 1 2 1 1 1 10 1 3 4 1 1 10 5 6 Example Assume a two unit penalty for passing from a high to lower numbered node. This is now an asymmetric TSP. Why? IE 312

40. 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 IE 312

41. Quadratic Assignment Formulation IE 312

42. 10 1 2 1 1 1 10 1 3 4 1 1 10 5 6 Example: reformulate IE 312

43. Solving TSP • We can use branch-and-bound to get an exact solution to the TSP problem • As always, the key to implementing branch-and-bound is to relax the problem so that we can easily solve the relaxed problem, but we still get good bounds • How can we relax the TSP? IE 312

44. Reduces to the assignment problem Relaxing the TSP IE 312

45. Branching Which cities should be selected? What is the most important variable? IE 312

46. Example: Solution to Assignment Problem Ames CR IC DSM IE 312

47. Branching Always branch to split up the subtours! Solving the assignment problem will hopefully yield a feasible solution soon. IE 312

48. Solution to New Assignment Problem (Left Branch) This makes both subtours impossible, so we get Ames CR IC DSM Thus, optimal solution is found by solving a total of three assignment problems! IE 312

49. Poor Branching Examples IE 312

50. Set Packing, Covering, and Partitioning IE 312