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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

Applications of Tabu Search

OPIM 950

Gary Chen

9/29/03


Basic tabu search overview
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


Applications
Applications

  • Bioengineering

  • Finance

  • Manufacturing

  • Scheduling

  • Political Districting

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


Application 1 student course scheduling
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.


Objectives and constraints
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


Implementation part 1

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Implementation - Part 1

  • Construct student timetable without considering section enrollments

  • Model course sections as undirected graphs


Maximum cardinality independent sets
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


Implementation part 2
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


Tabu search
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.


Strategic oscillation
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


Strategic oscillation cont
Strategic Oscillation (Cont)

  • For course selection, class size is strategically oscillated.


Application 2 tabu search for political districting
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


Constraints
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


General solution strategy
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


Problem formulation
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


Population equality
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


Compactness
Compactness

  • Rj(x) = length of jth district boundary

  • R = perimeter of entire territory


Socio economic homogeneity
Socio-Economic Homogeneity

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

  • = average income in entire territory


Similarity to previous districting
Similarity to Previous Districting

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

  • A = Entire territory area



Integrity of communities
Integrity of Communities

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

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


Tabu search1
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.


Tabu search cont
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.



References
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.


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