1. Applications of Tabu Search OPIM 950 Gary Chen 9/29/03

2. Basic Tabu Search Overview Pick an arbitrary point and evaluate an initial solution Compute next set of solutions within neighborhood of current solution Pick best solution from the set. If solution is on Tabu (or forbidden) list, pick next best solution. Repeat until you come across solution not on Tabu list. After solution is chosen, repeat from step 2 until optima is reached. Parameters for tuning: Number of iterations, penalty points, size of Taboo list

3. Applications Bioengineering Finance Manufacturing Scheduling Political Districting Many of the applications of Tabu Search are very similar to Simulated Annealing

4. Application 1: Student Course Scheduling Problem: Registering for classes required students waiting in long queues. Solution: Allow course registration over the internet and using OR techniques (tabu search), give student satisfactory time schedule as well as balance section loads.

5. Objectives and Constraints Main Objective: Find conflict-free time schedule for each student Secondary Objectives: Balance number of classes per day Minimize gaps between classes Respect language preferences Student course selections must be respected Section enrollment must be balanced Section maximum capacity cannot be exceeded

6. Implementation - Part 1 Construct student timetable without considering section enrollments Model course sections as undirected graphs 1 5 2 3 1 2 4 5 3 2 6 1 1 6 6

7. Maximum Cardinality Independent Sets Objective: Find sets that contain one section of each course. Algorithm Find all cliques in the graphs. Pick one node or no nodes from each clique. Check if it ’ s a valid schedule. If it is retain as a possible solution set. repeat

8. Implementation - Part 2 Balance out section enrollment Each student has a set of possible time schedules. “ Optimal ” time schedule for a student adheres to following criteria: Balance number of classes per day Minimize gaps between classes Respect language preferences

9. Tabu Search Objective: Find satisfactory course schedule. “ Satisfactory ” being a solution no more than a threshold cost distance from the “ optimal ” course scheduling. Tabu list contains previously tried student course schedules. Tabu search combined with strategic oscillation used.

10. Strategic Oscillation Perform moves until hitting a boundary. Modify objective constraints or extend neighborhood function to allow crossing over to infeasible region. Proceed beyond boundary for a set depth Turn around to enforce feasibility

11. Strategic Oscillation (Cont) For course selection, class size is strategically oscillated.

12. Application 2: Tabu Search for Political Districting Problem: Partition a territory into voting districts. Political influence problems. Solution: Using tabu search for deciding districts will result in a fair, unbiased answer

13. Constraints Districts should be contiguous Voting population should be close to evenly divided among the districts Natural boundaries should be respected Existing political subdivisions, such as townships, should be respected Socio-economic homogeneity Integrity of communities should be respected

14. General Solution Strategy Clustering approach First pick several pre-determined centroid districts. “ Grow ” districts outward. Previous attempts Branch-and-bound trees (NP-hard) Simulated annealing

15. Problem Formulation minimize  i are user-supplied multiplers f pop (x) = population equality function f comp (x) = compactness function f soc (x) = socio-economic homogeneity function f sim (x) = similarity to previous districting function f int () = integrity of communities function

16. Population Equality P j (x) – represents population for each j district - represents total population/#districts  - represents user-defined constant fraction, 0    1 Require population in each district [(1-  ), (1+  ) ] Should equal 0 if each district lies in interval. Otherwise, will take a positive value

17. Compactness R j (x) = length of j th district boundary R = perimeter of entire territory

18. Socio-Economic Homogeneity S j (x) = standard deviation of income in district j. = average income in entire territory

19. Similarity to Previous Districting O j (x) = largest overlay of district j and similar district in new solution A = Entire territory area

20. Example

21. Integrity of Communities G j (x) = largest population of a given community (Chinese, latino, etc) in district j. P j (x) = total population in district j.

22. Tabu Search Start with initial solution Start with a seed unit for initializing a district. “ Grow ” district by merging it with adjacent units until reached or no adjacent unit are available.

23. Tabu Search (cont) After initial solution created, two possible moves. Give – give a basic unit from one district to another Swap – swap basic units along boundary of two adjacent districts Any basic units swapped or given are placed on a tabu list. Algorithm stops when value of current best solution has no improvements from previously known best solution.

24. Example

25. References Alvarez-Valdes, R. et al. Assigning students to course sections using tabu search. Annals of Operations Research. Vol. 96 (2000) p. 1-16 Bozkaya, Burcin. A tabu search heuristic and adaptive memory procedure for political districting. European Journal of Operational Research. Vol. 144 (2003) p. 12-26.