INCORPORATING GENETIC ALGORITHM INTOSIMULATED ANNEALING BASED REDISTRICTING

Sim Kwan Hua

A thesis submittedIn full fulfillment of the requirements for the degree of

Master of Sciencein Information Technology

Faculty of Information Technology

UNIVERSITI MALAYSIA SARAWAK

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Acknowledgements

I would like to express my deepest appreciation to my supervisor, Dr. Wang Yin Chai, forhis encouraging words and for allowing me freedom to explore. His persistent guidance andpatient has led to the success of this dissertation. For specialized knowledge and advice, Iwould like to convey my special thank to Dr. Stan Open&w and his team members fortheir valuable conference proceedings. Thanks also to Dr. Kenneth Been for many Fruitfulemail discussion on the implementation of Simulated Annealing. On a more personal note, I.would like to thank my family for their unwavering encouragement and support throughoutmy research. Their love and prayers have me endured through all the difficult times. I amalso feel very thankful to my great fellow colleagues for making my time doing this researchboth an enjoyable and rewarding experience.

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2.6.2.1 Selection .................................................................................................... ..I 6

2.6.2.2 Crossover.. .................................................................................................. .16

2.6.2.3 Mutation .................................................................................................... ..18

2.6.3 Genetic Algorithms in Solving NP-hard Problem ..................... ..i ...................... 1 8

2.7 Summary ................................................................................................................... 1 9

CH/$‘TER3 SOLIJTIONDESIGN ................................................................... .,-. . . ............ .203.1 Introduction.. ............................................................................................................. 20

3.2 Extended Analysis of Simulated Annealing .............................................................. 20

3.2 . l Analogy of Simulated Annealing ....................................................................... 20

3.2.2 Outline of Simulated Annealing ....................................................................... ..2 1

3.3 Specification of Simulated Annealing In Redistricting Process ................................ 22

3.3.1 The Basic AZP Algorithm ................................................................................ ..2 2

3.3.2 A Simulated Annealing AZP Algorithm.......................................................... ..2 3

3.4 Algorithm Design of The Proposed Redistricting Algorithm .., ................................ .25

3.4.1 ProposedModelofIncorporating Genetic Algorithm.. .................................... ..2 5

3.4.2 Incorporated Redistricting Algorithm.. ............................................................... 28

3.4.3 Reproduction Effect of Genetic Algorithm ........................................................ 29

3.4.4 Evolve Through a Set of Potential Candidate Solutions ..................................... 30

3.4.5 Annealing Schedule.. .......................................................................................... 3 1

3.4.5.1 Initial Temperature - 3 2......................................................................................

3.4.5.2 Length of Iteration ....................................................................................... 32

3.4.5.3 Temperature Decrement Ratio.. ..................................... a........; .................... 33

3.4.6 Selection Method ................................................................................................ 34

3.5 summary ................................................................................................................... 36

CHAPTER 4 IMPLEMENTATION ..................................................................................... 37

4.1 Introduction ............................................................................................................... -37

4.2 ImplementationEnvironment .................................................................................... 37

4.3 Test Data.. .................................................................................................................. 37

4.4 Simulation Development ........................................................................................... 38

4.4.1 Initialization ........................................................................................................ 38

4.4.1 .l Definition of Objective Function ................................................................. 39

4.4.1.1.1 Population Equality .................................................................................. 39

4.4.1.1.2 Compactness ............................................................................................. 40

4.4.1.2 Preliminary Experiment ...................... . ...................................................... ..4 1

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Appendix III .......................................................................................................................... 96Incorporating Genetic Algorithm into Simulated Annealing Based Redistricting Algorithm............................................................................................................................................... 96

Abstract ............................................................................................................................. 96

Keywords....................................................................................................................... 96

1 .O Introduction .............................................................................................................. 97

2.0 Redistricting ..................................................................................... ..~.....~~.~~......~.~ ... 97

3.0 Simulated Annealing 98..................................................................................................

., 4.0 Simulated Annealing BasedRedistricting Algorithm .......................................... JO 0

5.0 Genetic Algorithm ................................................................................................. 102

6.0 Development of Proposed Redistricting Algorithm .............................................. 1 0 3

7.0 Result and Discussion............................................................................................ 108

8.0 Conclusion ............................................................................................................. 111

References. ................................................................................................................... 1 1 1

List of Tables

Table 2. 1 Redistricting problems that are at least NP-hard (Al~nan, 1998) . . . . . . . . . . . . . . . . . . . . . . . . . 11

Table 3. 1 Selection probability of individual solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Table 4. 1 Euclidean Measurement ....................................................................................... 42

Table 4.2 Parameters Setting For Simulated Annealing Features ........................................ 42

Table 4.3 Parameters Setting For Genetic Algorithm Features ............................................ 43

Table 5 . 1 Summary of redistricting simulation for modest districting plan . . . . . . . . . . . . . . . . . . . . . . . . . 54

Table 5 . 2 Summary of redistricting simulation for modest number of areas . . . . . . . . . . . . . . . . . . . . . . . 60

Table 5. 3 Summary of this redistricting simulation for greater number of areas . . . . . . . . . . . . . . . . .63

Table 5.4 Summary of this redistricting simulation for modest area size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66I,

1;:Table 5.5 Summary of this redistricting simulation for greater area size *............................ 68

/ Table 5.6 Summary of this redistricting simulation for odd shape districting plan..............70‘,I’ Table 5.7 Objective function value of output districting plan . . . . . . . . . . . . . . . . . . ..*.........*.*............. 7 1”

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List of Figures,

Figure 2. 1 More compact district shape ................................................................................. 7

Figure 2.2 Less compact district shape ................................................................................... 8

Figure 2.3 Transformation of shapes. (Atlrnan, 1998) ............ . .............................................. 8

Figure 2.4 Geographical concern in redistricting ,,................................................................ . 1 0

Figure 2. 5 Fitness landscape................................................................ ..4........;.~..; ...... .......... :.16;. .

Figure 3. 1 AZP redistricting algorithm -23................................................................................

Figure 3. 2 Simulated annealing AZP algorithm ................................................................... 24

Figure 3.3 The concept of incorporating GA into SA .......................................................... ‘26

Figure 3.4 Architecture Design of Proposed Redistricting Model ....................................... 27

Figure 3. 5 Transition of area into adjacent district ............................................................... 27

Figure 3.6 Incorporated redistricting algorithm .................................................................... 28

Figure 4. 1 Diagram 0 of the Data Flow Diagram ................................................................ 38

Figure 4.2 Level 1 Data Flow Diagram for Process 1 .O ....................................................... 39

Figure 4.3 Shape compactness of district ............................................................................. 41

Figure 4.4 Level 1 Data Flow Diagram for Process 2.0 ,i4,................................................. ..... .

Figure 4.5 Command to select polygons .............................................................................. &I* ‘2’/ ‘,

Figure 4.6 Command to select adjacent polygons ................................................................ 45\“,

Figure 4.7 Command to perform loop task on all the district ............................................... 45

Figure 4.8 Level 1 Data Flow Diagram for Process 3.0 ....................................................... 46

Figure 4.9 Subcommand in the STATISTIC command......................................................... 46

Figure 4. 10 Syntax for calculating idea population size ....................................................... 47

Figure 4.11 Command to obtain sum of population ............................................................. 47

Figure 4. 12 Command to dissolve polygon ................................................... ..t .................... 47

Figure 4. 13 Command to Join Info Table............................................................................. 47

Figure 4. 14 Obtain standard deviation for population .......................................................... 48

Figure 4. 15 Syntax for calculating compactness value ......................................................... 48

Figure 4. 16 Syntax for calculation objective function value ................................................ 48

Figure 4. 17 Checking for contiguity of district .................................................................... 49

Figure 4. 18 Command to pass arguments ............................................................................ 50

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Figure 5. 1 Statistical data of the sample districting plan ...................................................... 53

Figure 5 . 2 Final output of districting plan obtained from enhanced algorithm.. ................. .54

Pigure 5.3 Final output of districting plan obtained by SA algorithm ................................... 54

Figure 5.4 Objective function value during redistricting simulation.................................... 55

Figure 5. 5 Number of iteration by SA algorithm ................................................................. 56

Figure 5.6 Number of iteration by enhanced algorithm ........................................................ 56

Figure 5.7 Simulation time for SA algorithm ....................................................................... 57

Figure 5. 8 Simulation time for enhanced algorithm ............................................................. 57

Figure 5.9 Simulation time to achieve objective function value 1.72 by SA algorithm.......5 8

Figure 5. 10 Simulation time to achieve objective diction value 1.72 by enhanced algorithm....................................................................................................................................... 58

Figure 5. 11 Sample data contains 55 areas ........................................................................... 59

Figure 5. 12 Districting plan produced by SA algorithm ...................................................... 59

Figure 5.13 Districting plan produced by enhanced algorithm............................................. 60

Figure 5.14 Simulation result on sample data contains 55 areas .......................................... 60

Fi&e 5. 1 SSimulation time by SA algorithm ....................................................................... 6 1

Figure 5. 16 Simulation time by enhanced algorithm............................................................ 6 1

Figure 5.17 Sample data contains 395 areas ......................................................................... 62

Figure 5. 18 Districting plan produced by SA algorithm ...................................................... 62

Figure 5. 19 Districting plan produced by enhanced algorithm............................................. 62

Figure 5.20 Simulation result on sample data contains 395 areas ........................................ 63

Figure 5.21 Simulation time requires for SA algorithm ....................................................... 64

Figure 5.22 Simulation time require for enhanced algorithm............................................... 64

Figure 5.23 Sample data with modest area size.................................................................... 65

Figure 5.24 Final output fkom SA algorithm ........................................................................ 65

Figure 5.25 Final output by enhanced algorithm .................................................................. 66

Figure 5.26 Simulation result for modest area size .............................................................. 66

Figure 5.27 Sample data with greater average size of area .................................................. 67

Figure 5.28 Output by SA algorithm .................................................................................... 67

Figure 5.29 Output by enhanced algorithm .......................................................................... 67

Figure 5.30 Simulation result on greater average area size .................................................. 68

Figure 5.3 1 Sample data with odd shape area ...................................................................... 69

Figure 5.32 Output districting plan by SA algorithm ........................................................... 69

:Rgure. 5.33 Output districting plan ~,ymhmmad~a~go&i~ i~,.,.,..,..............,..,.,.,....,.,.,,...,,..,. 6 9

Figwre 5.34 Simulation result on odd shape districting plan . . . ..i.......................................... 70.

Abstract

Redistricting is a process of drawing lines as a boundary, it plays an important role in theprocess of decision making on space and spatial allocation, In redistricting, the mainproblem to be solved is to find the districting plan that maximizes the value function

involved. Thus, redistricting process can be characterized as a combinatorial optimizationproblem and considered to be computationally intractable or NP-hard problem, so gettingtrapped in local optimal and difficulty in obtaining the most optimal solution have be&ethe great challenge of redistricting problem. Latest approach of solving redistricting problemusing simulated annealing has shown a significant improvement with its ability to escapefrom local optimal. However, simulated annealing does not guarantee outstandingperformance in escaping from local optimal though it posses that special capability.Therefore, this research will incorporate the advantages of genetic algorithm to overcomethe deficiency of simulated annealing in the process of redistricting. The evolutionaryfeature of genetic algorithm that have potential to overcome some of the dilemmas ofredistricting process are adapted to enhance the efficiency and performance of simulatedannealing redistricting algorithm. At the same time, the strengths of simulated annealing willstill being retained. Thus, the proposed redistricting algorithm has been enhanced by thereproduction effect of genetic algorithm in approaching more optimal districting plan withbetter objective function value during redistricting process. The implementation ofsimulation on the proposed model shows that genetic algorithm can be used to incorporatewith simulated annealing in solving hard spatial optimization problem to achieve optimalresult as a final output of districting plan. The outcome will contribute significantly tovarious applications and systems including political redistricting, business territorymanagement and allocation planning. There are also several recommendations .for futurework as an extension of this research,

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Abstrak

Pembahagian semula kawasan merupakan satu proses pengarisan semula sempadan, iamemainkan peranan yang penting dalam proses membuat keputusan berkenaan denganruang dan pengagihan kawasan. Dalam pembahagian semula kawasan, masalah utama yanghendak diatasi adalah untuk mencari plan pembahagian semula kawasan yangmemaksimumkan nilai fungsi yang terlibat. Oleh yang demikian, proses pembahagiansemula kawasan boleh digolongkan sebagai masalah pengoptimalan kombinasi dan bolehdianggap sebagai masalah N.-hard, jadi terperangkap dalam optima tempatan dan kesulitanuntuk mendapatkan penyelesaian yang paling optima merupakan cabaran yang paling getirdalam masalah pembahagian semula kawasan. Pendekatan terbaru dalam menyelesaikanmasalah pembahagian semula kawasan ‘menggunakan simulated annealing telahmenunjukkan kemajuan yang menonjol dengan keupayaannya untuk melepaskan diridaripada terperangkap dalam optima tempatan. Walau bagaimanapun, simulated annealingtidak menjanjikan prestasi yang baik &lam melepaskan diri daripada terperangkap dalamoptima tempatan walaupun ia mempunyai keupayaan tersebut. Oleh yang demikian,penyelidikan ini akan mengadunkan kelebihan genetic algorithm untuk mengatasikekurangan yang terdapat pada simulated annealing &lam berdepan dengan prosespembahagian semula kawasan. Ciri evolusi daripada genetic algorithm yang mempunyaipotensi untuk mengatasi sebahagian daripada dilema proses pembahagian semula kawasantelah diadun untuk meningkatkan keberkesanan dan prestasi simulated annealing dalampembahagian semula kawasan. Pada masa yang sama, kekuatan yang terdapat pada ciri-cirisimulated annealing masih akan tents dikekalkan. Oleh itu, algoritma pembahagian semulakawasan yang dicadangkan telah diperhebatkan dengan kesan reproduksi semula daripadagenetic algorithm untuk menjejaki plan pembahagian semula kawasan yang lebih optimayang mempunyai nilai fungsi objektif yang lebih baik semasa process pembahagian semulakawasan. Implementasi simulated annealing pada model yang dicadangkan menunjukkangenetic algorithm berupayaan untuk diserapkan ke dalam simulated annealing &lammenyelraikan masalah optimasasi ruang yang sukar dengan mencapai keputusan yangop~t&,na r~baba&i output akhir plan pembahagian semula kawasan. Hasil penyelidikan akan

sumbangan yang penting kepada pelbagai aplikasi dan sistem termasuk‘simula kawasan politik, pengurusan kawasan perniagaan dan perancangan

petigqIih~$r. Terdapat juga beberapa penyesoran untuk kerja-kerja masa depan sebagaikesinambungan kepada penyelidikan ini.

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CHAPTER 1 INTRODUCTION

Redistricting can be characterized as a combinatorial optimization problem and it is acomplicated process. The complexity will increase when it also involves other factor suchas political, demographic, geographical and environmental. A complete solution forredistricting seems to require an interdisciplinary effort. Many different methods have beenused or suggested for finding optimal districts. However, redistricting process by itself is aNP-hard problem, current redistricting algorithms and conventional programming methodsare not efficient, especially on implementing spatial redistricting (Altman, 1998). Thus, thisresearch aims to incorporate the advantages of genetic algorithm to overcome the deficiencyof existing redistricting algorithm, enhancement will be made on existing algorithm forredistricting process in order to increase the chances of obtaining near to optimalredistricting plan.

1.1 Background

Redistricting is a process of drawing lines as a boundary. It plays an important role in theprocess of decision making on space and spatial allocation, According to Agnew (1994), thespaces within which political, social, cultural, and economic processes are not only staticbackdrops or location referents for human events, but also the products of distinct territorialstructures, identities, and ambitions, and they are deeply complicated by social and politicalchange. Therefore, redistricting process plays an important role to redraw the lines on adistrict, based on the social and natural changes from times to times with the purpose of wellspatial representation.

Thus, redistricting is extremely important in many different fields including ‘electionboundary, school boundary, law enforcement, business territory management and evenforest planning. Although each different redistricting process posses its own issues; such asgerrymanders for electoral boundary, some of the issues are consistent regardless of theredistricting domain; one of the biggest issue falls into this category is optimization in,redistricting process.

GIS technology available today has offered a great chance for more consistent analysis ofdata and the ability to make more credible redistricting decisions in redistricting process(Hayes, 1996). Generally, optimization is one of the key factors in producing optimal andcredible redistricting plan, but optimization could not be achieved completely with theimprovement of technology. However, algorithms used for redistricting process can beimproved in order to achieve the aim of optimization in redistricting.

1.2 General Problem Statement

In redistricting, the main problem to be solved is to find the districting plan that optimizesthe value function involved. Formally, it is to find the optimal partition of the districtingplan.

Redistricting process possess special difficulties because the size of the solution set can beenormous. Practically, it is impossible to solve the problem by a brute force search throughall possible districting arrangements. As a result, redistricting process is considered to becomputationally intractable or NP-hard problem A problem will be said to becomputationally intractable (also “computationally complex,” or “computationally hard”) if

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the optimal solution method for solving the problem cannot solve all instances in polynomialtime.

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Generally, current redistricting algorithms are still facing limited function and unable tosatisfy spatial needs (Ngu, 1998). Besides, some of the current redistricting methods onlyapproach to small size redistricting problems and others implement guessing procedures toget to near optimal solution. In solving bigger size of redistricting plan, the searchingalgorithms very often still not being able to approach near optimal solution and some ofthem will very likely to get trapped in local optimal.

Therefore, conventional programming methods for spatial redistricting algorithm not onlyhas limitation on the implementation process but also facing inflexibility because of spatialcomplexity (A&man, 1998).

1.3 Objective

The research will focus on following objectives:

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(III)

(Iv)

W)

Identify existing algorithms on redistricting and their weaknesses and problems insolving redistricting problem.

Study the requirement and specification of redistricting algorithm in redistrictingprocess.

Investigate suitable features of genetic algorithm and their potential to enhance theredistricting algorithm that will increase the optimality of the obtained districtingp l a n .

Design a redistricting algorithm by incorporating the evolutionary feature ofgenetic algorithms into existing simulated annealing based redistricting.

Develop a redistricting simulation for the proposed algorithm.

1.4 Scope of Study

This research focuses on enhancing the current redistricting algorithm by incorporating theadvantages of genetic algorithm. District boundary and compactness in redistricting processwill also be the focus of the research in assessing the performance of the proposedalgorithm.

Although there are numerous redistricting algorithms available currently, this research willonly concentrate on simulated annealing based redistricting algorithm. The research willfocus on the derivation of various components of the improved redistricting algorithm. Thepotential features of genetic algorithm are investigated in order to fmd the best way that theycan be incorporated into simulated annealing based redistricting algorithm, and eventuallyincrease the chances of obtaining district plan with better optimality.

Therefore, another main focus of this research will be on the genetic algorithm (GA)concept. Genetic algorithm uses process that analogous to natural evolutionary to generatesolutions for an optimization problem. Thus, GA seems to have many characteristics thatc’an be taken as an ideal solution on redistricting process to ensure better efficiency.

For redistricting simulation that will be ,in$emented based on proposed model, latestGeographical Information System (GIS) technology will be used to assist the manipulationof spatial and attribute data during the implementation process. This is based on the

capability of current GIS technology in managing and handling spatial relationship besidespowerful features and functions that allow spatial manipulation in the scope of study to bedone.

1.5 Research Methodology

This research will be conducted in a systematic and orderly manner. There are five mainphases in this research methodology, namely literature review, requirement specification,design, implementation and testing. This research methodology is to ensure that the researchwill achieve its objective systematically. The tasks of each stage in the methodology will bediscussed as below:

1.5.1 Literature Review

Reviews will be done on current redistricting algorithms with extensive review on thesimulated annealing based redistricting algorithm. At the same time, important redistrictingcriteria will also be identified. Genetic algorithms will also be studied and evaluated in orderto understand its characteristic as well as its ability in solving optimization problems todiscover the benefits of genetic algorithm on redistricting.

1.5.2 Requirement Specification & Analysis

Existing redistricting algorithm especially simulated annealing based redistricting algorithmwill be investigated to identify the requirement and specifications that are needed to yieldmodification and improvement. Requirements of proposed algorithm will also be specified.Serious concern will also be given concerning the feasibility of the new technique to ensurethat it will optimize the redistricting criteria. i. : .

1.5.3 Designing Proposed Algorithm

Based on the specification identified in previous stage, the most appropriate Latures ofgenetic algorithm will be incorporated into redistricting process to derive an enhancedredistricting algorithm, which will improve the capability of the redistricting algorithm inorder to produce a more optimal district plan. Therefore, the detail specification and designof the proposed algorithm will be performed, and the proposed technique should fulfill the .requirements and specifications of redistricting algorithm obtained from previous stage.

1.5.4 Implementation

A proposed redistricting algorithm will be implemented according to the specificationrequirements. A conceptual model on the proposed algorithm will be developed as a meansto manipulate the proposed algorithm in redistricting process. In this stage, programmingwill be done in order to put the developed algorithm into implementation. A redistrictingsimulation of the proposed algorithm will also be developed so that it can be provenpractically.

1.5.5 Testing and Evaluation

This is a simultaneous process with implementation. In this process, implemented solutionswill be evaluated. The performance of the proposed algorithm will be tested and analyzed.Different sets of input data are used to evaluate the model in term of its effectiveness,

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reliability, and performances. In the meantime, implementation alternatives, strength andweakness of the technique will also be identified and analyzed

1.6 Significance of the research

No one can deny that the redistricting process plays an important role in making redistrictingdecision involving political, social and even business prospect. Howevqr, currentredistricting algorithms and conversational programming methods seem to be notsatisfactory. Thus, incorporating genetic algorithms is one of the ways to improve theoptimality of the redistricting plan. This research should be benefiting all kinds ofredistricting process and application. In short, this research should formulate a redistrictingtechnique that makes redistricting tasks to be more effective in increasing the chances offinding near to optimal solution of redistricting plan.

1.7 Outline of the Dissertation

Every stage of the research will be documented. The outline gives a brief introduction ofeach chapter in this dissertation.

Chapter 2 describes the literature study of this research. It discusses the various elevation ofredistricting algorithms, definition of redistricting, ‘problems for the overall redistrictingprocess, advantages and disadvantages of existing redistricting algorithms, possibleapproach to improve current methodology, and prospect of genetic algorithm in solving NP-hard problem will be reviewed.

Chapter 3 discusses the design of the proposed algorithm. It explains the development. methodology. It contains an explanation on the solution and shows the general structure ofthe proposed algorithm. Various issues and factors are discussed throughout the designingprocess of the proposed redistricting algorithm.

Chapter 4 discusses the implementation of the proposed algorithm. It describes in details onthe techniques of implementation. This section discusses the development framework of theimplementation process. It describes all methods and techniques of implementation indetails.

Chapter 5 discusses the results obtained from the implementation of the proposed algorithmwith analysis from various aspects will also be carried. It explains the detail evaluation ofthe proposed algorithm by focusing on the incorporation of genetic algorithm into thealgorithm of redistricting.

Chapter 6 discusses on the findings, accomplishments, limitations and future enhancementof the research. The achievement shows the contribution of the developed redistrictingalgorithm in improving redistricting process.

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction

This chapter aims to present the detailed literature review on redistricting from differentperspectives related to this research. Redistricting algorithms are discussed thoroughly sinceit is one of the most crucial parts of redistricting process. This literature review tends tostudies thoroughly existing redistricting algorithms, and identifies the requirements andspecifications of the redistricting process. Among others, the review looks into, differentkinds of algorithms and techniques in terms of their advantages and disadvantages,strengths, limitations, and the lacking in the redistricting process. These problems shouldstimulate the proposed domain in the following chapter as a solution to the problem The.existing algorithm will be incorporated with genetic algorithm, hence the potential featuresand suitability of genetic algorithm will also be reviewed. In short, the outcomes of thischapter are to conduct reviews and analysis on all aspects of redistricting, and later, toident@ the specific problems of redistricting algorithm, and the potential of incorporatinggenetic algorithm for a better solution with better optimality district plan.

2.2 Redistricting

Redistricting process can be considered as an effort to explore the data for spatialrelationships and representation. Although’redistricting can be categorized into variouskinds of categories based on the nature of data and their respective objective function of theredistricting process, this research tend to focus on redistricting that involves spatial data re-aggregation which is similar to Modifiable Aria1 Unit Problem (MAUP) defined byOpenshaw and Taylor (1980), and viewed by Openshaw as one of the major redistricting

problem

One of the major task of redistricting process mentioned above starts with a set of data atone scale and then re-aggregate it again in order to create a new set of district plan suitablefor a specific purpose. It involves the aggregation of spatial data for N areas into M districts,where M is less than N and is usually a small fraction of N in size. It is very important tonote that all the districts should be internally connected and contiguous.

However, it is obvious that these activities will face problem due to the likely sensitivity ofthe result to the aggregation of the data being processed.

Expressed in mathematical notation, it looks like any other mathematical optimizationproblem trying to optimize F(Z), where F(Z) is a general function of Z, except Z is not a

simple set of linear or non linear parameters, but defines an aggregation of N initial areasinto M districts as output, and subject to constraints on Z to ensure the M district areinternally connected, F(Z) is any function defined on data for the M districts in Z, and Z isthe allocation of each of N areas to one of M districts such that each area is assigned to onlyone district.

In other words, there are implicit constraints on the function Z in such a way that each of theoriginal N areas only can to be assigned to exactly one output district. Moreover, all themembers from the same output districts have to be connected to form an individual districtso that when the internal boundaries are dissolved, they will still be able form a singlepolygon.

For example, electoral redistricting can be considered as one of the important variant ofredistricting, and Openshaw (1984) argued that it is a special case of a more general zone

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