Applications of Tabu Search

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

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 • fpop(x) = population equality function • fcomp(x) = compactness function • fsoc(x) = socio-economic homogeneity function • fsim(x) = similarity to previous districting function • fint() = integrity of communities function

16. Population Equality Pj(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 • Rj(x) = length of jth district boundary • R = perimeter of entire territory

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

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

20. Example

21. Integrity of Communities • Gj(x) = largest population of a given community (Chinese, latino, etc) in district j. • Pj(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.