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# Applications of Tabu Search - PowerPoint PPT Presentation

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

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### Applications of Tabu Search

OPIM 950

Gary Chen

9/29/03

• 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

• Bioengineering

• Finance

• Manufacturing

• Scheduling

• Political Districting

Many of the applications of Tabu Search are very similar to Simulated Annealing

• 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.

• 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

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Implementation - Part 1

• Construct student timetable without considering section enrollments

• Model course sections as undirected graphs

• 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

• 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

• 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.

• 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

• For course selection, class size is strategically oscillated.

• Problem: Partition a territory into voting districts. Political influence problems.

• Solution: Using tabu search for deciding districts will result in a fair, unbiased answer

• 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

• Clustering approach

• First pick several pre-determined centroid districts.

• “Grow” districts outward.

• Previous attempts

• Branch-and-bound trees (NP-hard)

• Simulated annealing

• 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

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

• Rj(x) = length of jth district boundary

• R = perimeter of entire territory

• Sj(x) = standard deviation of income in district j.

• = average income in entire territory

• Oj(x) = largest overlay of district j and similar district in new solution

• A = Entire territory area

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

• Pj(x) = total population in district j.

• “Grow” district by merging it with adjacent units until reached or no adjacent unit are available.

• 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.

• 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.