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A Unified Conceptual Framework for GeographicalOptimization Using Evolutionary Algorithms

Ningchuan Xiao

Department of Geography, The Ohio State University

During the last two decades, evolutionary algorithms (EAs) have been applied to a wide range of optimizationand decision-making problems. Work on EAs for geographical analysis, however, has been conducted in aproblem-specific manner, which prevents an EA designed for one type of problem from being used on others.In this article, a formal, conceptual framework is developed to unify the design and implementation of EAsfor many geographical optimization problems. The key element in this framework is a graph representationthat defines the spatial structure of a broad range of geographical problems. Based on this representation, fourtypes of geographical optimization problems are discussed and a set of algorithms is developed for problems ineach type. These algorithms can be used to support the design and implementation of EAs for geographicaloptimization. Knowledge specific to geographical optimization problems can also be incorporated into theframework. An example of solving political redistricting problems is used to demonstrate the application ofthis framework.Key Words: evolutionary algorithms, geographical optimization problems, locationallocation analysis,political redistricting, spatial representation.

Durante lasultimas dos decadas, se han aplicado algoritmos evolucionarios (evolutionary algorithms, EA) a unaamplia variedad de problemas de optimizacion y toma de decisiones. Sin embargo, el trabajo en los EA paraanalisis geograficos se ha realizado de manera especfica al problema, que evita que un EA disenado para un tipode problema se use para resolver otros problemas. En este art culo se desarrolla un marco conceptual formal paraunificar el diseno y la implementacion de EA para muchos problemas de optimizacion geografica. El elementoclave en este marco es una representacion grafica que define la estructura espacial de una amplia variedad deproblemas geograficos. Con base en esta representacion, se discutieron cuatro tipos de problemas de optimizaciongeografica y se desarrollo un conjunto de algoritmos para cada tipo de problemas. Estos algoritmos se pueden usarpara apoyar el diseno y la implementacion de EAs para optimizacion geografica. Tambien se puede incorporar enel marco conocimiento especfico a los problemas de optimizacion geografica. Se usa un ejemplo de resolucion dereasignacion de distritos polticos para demostrar la aplicacion de este marco.Palabras clave: algoritmos evolutivos,problemas de optimizacion geografica, analisis de ubicacion-asignacion, reasignacion de distritos pol ticos, representacion

espacial.

Geographers and researchers from relateddisciplines have devoted a significant amountof attention to the development of solution

methods for geographical optimization problems. Theseproblems can be found in the literature of locationalanalysis (Rushton 1988; Densham and Rushton 1996),

natural resource management (Murray and Church1995; Hof and Bevers 1998), nature reserve selection(Church, Stoms, and Davis 1996), regionalization andpolitical redistricting (Williams 1995), spatial datamining and exploratory analysis (Han, Kamber, andTung 2001), and spatial decision making in public

Annals of the Association of American Geographers, 98(4) 2008, pp. 795817 C2008 by Association of American GeographersInitial submission, November 2006; revised submission, July 2007; final acceptance, October 2007

Published by Taylor & Francis, LLC.


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A Unified Conceptual Framework for Geographical Optimization Using Evolutionary Algorithms 797

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Figure 1. An example evolutionary algorithm (EA). A binary representation is used for a problem that maximizes f =x 1+ 2x2 x23 , where

x1, x2, and x3 are integer variables between 0 and 7. A binary string of length 9 is used to represent a solution in which each three-bitsubstring encodes an integer variable. (A) An encoded solution ofx1 = 3, x2 = 1, and x3 = 3. (B) A one-point crossover operation forrecombination. (C) A mutation operation. (D) An example procedure of the EA where a shaded box is used to illustrate solutions in ageneration. The population size in this example is 6. Each individual is marked by a letter (AF) identifying the individual in the populationand a digit indicating the generation (02). Results of evolutionary operations performed during each iteration are enclosed in unshadedboxes. Individuals with a highlighted bit are manipulated by the mutation operation.


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V={1, 2, 3, 4, 5, 6, 7, 8, 9,10}E={(1,2), (1,7), (1,10), (2,3), (2,5),

V'={{1,2,5,7,8,10},

E' = {(1,2), (1,7), (1,10), (2,5),

(8,10)}

(3,4), (3,5), (3,6), (4,6), (4,9)(5,6), (5,7), (6,7), (6,8), (6,9)(7,8), (7,10), (8,9), (8,10)}

(3,4), (3,6), (4,6), (4,9)

(5,7), (6,9), (7,8), (7,10)

{3,4,6,9}}

V'={2, 6, 9}

V'={{1,2,4,5,6,8},{3,7,9,10}}

V'={3, 6, 7}E'={(3,6), (6,7)}

E' = *

E' = *

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Figure 2. A graph representation for geographical optimization problems. In each row, the left figure is a hypothetical map that shows thedistribution of the spatial units, and the figure in the middle is an illustration of the corresponding graph, which is formally denoted using thenotations on the right. (A) A graph with ten vertices and a set of edges. (B) An example selection problem without spatial constraints (threevertices are selected). (C) An example problem that selects a contiguous set of three vertices. (D) A partitioning problem of two subdivisions(shaded and unshaded) without spatial constraints. (E) A partitioning problem that requires two contiguous subdivisions.


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Ui is the i th subdivision (1 i p). The spatialrelationship between vertices in{Ui }is defined by E

,

which is often a subset ofE .It should be noted that asubdivision need not be spatially contiguous. Instead,in some cases, partitioning can refer to an aspatialclassification, in which each unit (represented as a

vertex) is assigned an integer that indicates a particularclass (see, for example, Armstrong, Xiao, and Bennett2003; Bennett, Xiao, and Armstrong 2004).

In summary, the input of a geographical optimiza-tion problem consists of sets G,D, and A. The goalof solving such a problem is to find a solution G that,without loss of generality, minimizes a set ofkobjectivefunctions: F =( f1, f2, . . . , fk), where each objectivefunction can be denoted as fi :G G

D A (1 i k ).

Graph TheoryBased Formal Problem Types

The four problem types already discussed can be for-mally denoted based on the relation between V andVand the characteristics ofE .Two kinds of relationsbetweenVandV can be recognized:

V V. V is a subset ofV.Selection problems be-long to this group (see Figures 2B and 2C, wherethree vertices are selected).

| Ui | = |V|. In this case, all vertices inVare usedto construct a feasible solution and therefore V hasthe same size as V, although vertices in V may beassigned to different subdivisions or categories (de-noted asUi ). Partitioning problems have this char-acteristic (see Figures 2D and 2E, where two subdi-visions are created).

For the relations between E and E , it is knownthat E defines the spatial relations among all verticesin V, and that, equivalently, E confines the spatialrelations among vertices in a solution. One can identifythe following two relations between E andE :

E = . For problems without spatial constraints, spa-tial relations among vertices in a solution are not

needed and it is unnecessary to specify the explicitcontents of E . Note that E = means that E isnot needed in the solution representation (or we donot care), but it does not imply that a spatial relationamong vertices inV does not physically or logicallyexist (see Figures 2B and 2D).

E E. Selection or partitioning problems withspatial constraints belong to this category. In theexample of Figure 2C, the selected three vertices are

Table 1. Relations betweenVandV andbetweenE and E

Relation

Code Meaning Vand V E and E

Subset V V E E

E Equal size | Ui | = |V| Do not care E

contiguous. In the example of Figure 2E, two con-tiguous subdivisions are created.

Table 1 lists a set of Greek letters that can be used torepresent the relation types discussed here. Using thisnotation, each type of problem can be identified by acombination of two conditions: the relation between Vand V, and between E and E . Each combination isdenoted by a string of two letters delimited by a slash (/):

/: selection problems without spatial constraints /: selection problems with spatial constraints E /: partitioning problems without spatial con-

straints E /: partitioning problems with spatial constraints

Because all geographical optimization problems re-quire the arrangement of spatial units, it is unrealisticto have a type ofV = . Additionally, ifE is identicaltoE , the solution will have exactly the same structureof the input, which makes the problem trivial to solve.

Therefore, problem type of E

= E is not included inthe typology. Finally, if a dot ()isusedtodenoteallpos-sible conditions, one can specify the following generalproblem types:

/: selection problems E /: partitioning problems /: problems with spatial constraints /: problems without spatial constraints /: all geographical optimization problem types

EA Design for Geographical OptimizationProblems

Using the graph representation developed in the pre-vioussection,asetofprinciplesforthedesignofEAscanbe specified for each type of geographical optimizationproblem. The purpose here is not to develop universaloperations that can be immediately plugged in to solveall problems. Instead, the focus is placed on the generalprinciples that can be applied (and possibly extended)


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to specific problems. An example of using a subset ofthese principles is presented in the next section.

Encoding Strategies

Although the use of a binary representation is useful

for many optimization problems (Goldberg 1989), it hasbecome common for researchers to choose or design arepresentation technique that is natural to the prob-lem being addressed (Falkenauer 1994; Michalewicz1996; D. B. Fogel and Angeline 1997). Geographicaloptimization problems can be conveniently encodedusing graphs.

For selection problems (/), where V is a subsetofV,it is not necessary to record the location of eachvertex in a solution. Instead, the unique identificationnumber of each vertex in an individual solution canbe directly used. In a p-median problem, for example,

each individual contains an array of p integers thatrepresent the facility nodes. For partitioning problems(E/), because | V| = |V|,astringofn integers can beused to represent feasible solutions; the value of thei thelement of the string indicates the subdivision assignedto the corresponding spatial entity.

For problems with spatial constraints (/), edge in-formation must also be stored so that the spatial con-straints can be effectively formulated and maintained.There are different approaches to storing edge infor-mation. In some previous studies, edge information isexplicitly recorded for each vertex in an individual EA

solution (Xiao, Bennett, and Armstrong 2002). Thisstrategy, however, may be inefficient because whenevera solution is changed (e.g., some vertices in the solutionare modified), edge information for all vertices in thatsolution must be accordingly updated. This issue canbe addressed by making edge information, or E,avail-able to the entire EA as a global variable. While E isavailable, it is unnecessary to store edge information foreach individual vertex redundantly. Consequently, thedata structure for /remain in the form of a string ofintegers (as identification numbers).

Spatial Constraint Handling

Three general strategies of constraint handling canbe identified from the EA literature. The first approachuses a specifically designed encoding method such thatinfeasible solutions will not occur during a solution pro-cess. A decoding method is needed to translate the en-coded information to a feasible solution. This methodhas been used to solve the traveling salesperson problem

(Grefenstette et al. 1985). Although this represents anelegant way of handling constraints, it is impractical todesign a general encoding method that can be used formany geographical optimization problems.

The second type of constraint handling method isbased on a penalty function that can be used to decrease

the fitness values for infeasible individual solutions suchthat they are unlikely to be included in the next gener-ation (Michalewicz 1996). This approach has been es-pecially effective for numerical optimization problems.Many geographical optimization problems, however, of-ten have a large number of infeasible solutions in an EApopulation, which makes it ineffective to use penaltyfunctions to promote feasibility that only exhibit in asmall number of individuals (see Bergey, Ragsdale, andHoskote 2003).

The third type of constraint handling method re-lies on various algorithms that will only create fea-

sible solutions, sometimes through a repair mecha-nism that converts infeasible solutions to feasible ones(Michalewicz 1996). Previous research has found thismethod to be flexible and effective when applied togeographical problems (Xiao, Bennett, and Armstrong2002; Bergey, Ragsdale, and Hoskot 2003; Xiao 2006).Although the implementation of this approach can beproblem dependent, with the graph representation de-fined it is possible to design algorithms for classes ofproblems that share common properties. The EA frame-work for solving geographical optimization problemsdiscussed in this article is based on this approach.

Design of Initialization Strategies

For selection problems (/), an initialization oper-ation (called Algorithm I 1) can be designed based onthe use of an accretion procedure in which a feasiblesolution is constructed from a set of seed vertices. Inthis algorithm, and others that follow, variablep is usedto denote the number of vertices to be selected, or thenumber of subdivisions to be partitioned, and set V isalways used to denote the set of vertices in a feasiblesolution.

AlgorithmI 1.{Accretion for selection}Input:V,E ,pOutput:V

1. i := 0,V :=, V1:=V2. repeat untili = p3. Randomly select a vertex (v) fromV14. Addv intoV

5. UpdateV1so that it only contains eligible vertices6. i := i + 1


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V1 = {1,3,4,8,9,11,12,13,14,15,16}V' = {6} V' = {6,8}

V1 = {1,3,6,11,13,14,15,16} V1 = {1,3,9,11,16}V' = {6,8,14}

V' = {6,7,10}V1 = {1,2,3,5,8,9,11,14}

V' = {6,7}V1={1,2,3,5,8,9,10,11}

V'={6}V1 = {2,5,7,10}

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Figure 3. Initialization for (A) a constraint that requires all selected vertices to be contiguous and (B) a requirement that does not allowselected entities to be adjacent. For both (A) and (B), the process starts from the left figure and ends at the right, as shown by the arrows. Ablack circle represents a selected vertex, and a gray circle represents an eligible vertex maintained in V1.

In Algorithm I 1, a vertex is called eligible if it can

be added into V

without violating spatial constraints.Here, a set V1 is maintained to contain only eligi-ble vertices. Figure 3 illustrates example initializationstrategies for two kinds of spatial constraints. When nospatial constraint is required, every unselected vertexis eligible and V1 will contain all unselected verticesinV.

A second initialization operation (Algorithm I 2)can be designed to generate initial feasible solutionsto partitioning problems (E /). Step 3 of Algorithm I 2will yieldV = {Ui }, where Ui (i =1, . . . ,p ) containsone and only one unique vertex that serves as the seed

of the i th subdivision. The specific procedure used instep 4 may vary for different problems, although thegeneral principle of the algorithm still holds and theconcept of eligible vertices can also be applied. Figure4 shows an example of this procedure for problems suchas political redistricting. When no spatial constraint isspecified, step 4 can be implemented in a random fash-ion with each vertex randomly assigned to a subdivision(see Bennett, Xiao, and Armstrong 2004).

AlgorithmI 2.{Accretion for partitioning}

Input:V,E ,pOutput:V

1. i := 0,V:= 2. Randomly selectp vertices fromV3. Add each of the p vertices to a subdivision (Ui ) in V

4. Assign each of the remainingn p vertices inVto asubdivision

Design of Recombination Operations

The recombination operations reported in the EAliterature share a similar behavior: The components

from two selected parent individuals are used to createnew child individual solutions by exchanging theircomponents (cf. Figure 1B). For geographical optimiza-tion problems, an overlay and repair approach can beused to recombine two individual solutionsand to createnew ones. In this approach, an overlay operation is con-ducted first to combine the vertices of two solutions intoa temporary set, which normally will contain more ver-tices than a feasible solution (see examples that follow).


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Figure 4. Initialization for a partitioning problem that requires all subdivisions (two in this example) to be contiguous. The four figures showthe progress of the process (indicated by the arrows) starting from two seed vertices (6 and 11), each assigned to a different subdivision.

A repair procedure is therefore needed to form a feasiblesolution.

For selection problems (/), the overlay operationwill result in a superset that contains at least one feasiblesolution. The repair process can be developed by iter-atively selecting vertices from the superset and addingthem to a partial solution until a feasible solution iscreated. A recombination operation, called AlgorithmR1, based on this mechanism is outlined as follows:

AlgorithmR 1.{Overlay and repair: selection}Input:V,E , V1, V

2,p

Output:V

1. V :=, V3 :=V

1 V

2

2. repeat until |V

| = p3. Randomly select a vertex (v) fromV34. Create a partial solution usingv andV

5. ifthe partial solution does not violate spatialconstraintsthen

6. V :=V v

7. Removev fromV3

For some selection problems, however, the overlayand repair procedure may not always yield solutions

that are different from the parent solutions. For exam-ple, in cases when the two parent solutions V1 and V

2

do not have overlapping vertices (i.e., V

1 V

2 =),the repair procedure will return either V1 or V

2. To

address this issue, Xiao (2006) developed a local searchapproach in which new solutions are created based ona single (instead of two) individual solution. Althoughthe mechanism of such an operation seems to be similarto a mutation method (described in the next section),it performs in a different fashion. First, a local searchis conducted based on the fitness values of individuals.That is, individuals with high fitness values have a highchance to be modified using the local search method.Mutation operations, however, are often performed

regardless of fitness values. Second, a local search isoften designed to improve individual solutions in termsof their objective function values, whereas mutationoperations tend to introduce randomness into anexisting solution that may not necessarily be improved.This type of operation is also called an asexual crossoveror transposition in which new (better) solutions arecreated based on a single solution (see Simoes and Costa2000).


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(A) (B) (C) (D)

Figure 5. Overlay and repair for a partitioning problem (p = 3) that requires contiguity of each subdivision. A cell here represents a vertexor a spatial unit and an edge exists between two cells if they are adjacent and on the same row or column. Numbers are used to uniquelyidentify each cell. (A) A hypothetical parent solution; (B) another hypothetical parent solution; (C) the result of overlay that creates fivesubdivisions; (D) a possible result of a repair procedure.

For partitioning problems with spatial constraints(E /), a set of new distinct subdivisions will begenerated after the overlay operation. Figure 5, forexample, shows the result of overlaying two solutionsin (A) and (B) to yield an intermediate solution asshown in (C), which can be repaired as shown in Figure5D. This operation is outlined in Algorithm R2. Forthe particular example in Figure 5, step 1 will generatea set V3 ={{1,2},{3,5,6},{4},{7,8},{9}}and themerging process (steps 46) merges the subdivisions of{1,2}and{4}, and of{7,8}and{9}.

AlgorithmR2.{Overlay and repair based on merging}Input:V, E,V1, V

2,p

Output:V

1. V3: ={Ui | Ui is a subdivision after overlaying V1and

V2}2. V: = 3. if |V3 | = p ,thenV

:=V3 andstop4. repeat until |V3 | = p

5. Randomly select two subdivisions,Ui andUj , fromV36. MergeUi andUj if doing so does not violate the

constraints7. V:=V3

Algorithm R 2 also can be used to solve partitioningproblems without spatial constraints (E /) by simplymerging (Ui ) and (Uj ) in step 6 without checkingconstraint violation. A more flexible recombinationapproach, called Algorithm R3, for this type ofproblem can be designed to assign a vertex to one ofthe subdivisions in its parent solutions.

Algorithm R3.{Recombination: partitioning without spatialconstraints}Input:V, E,V1, V

2,p

Output:V

1. V:= {Ui |Ui =, 1 i p}2. foreach vertexv in V3. Randomly seti to be one of vs subdivision in V1

andV24. Addv toUi inV

Design of Mutation Operations

A simple, straightforward way of introducing ran-domness into a current EA population is to replacean existing solution with a new one created using aninitialization operator. This method, called AlgorithmM1, is described next.

AlgorithmM1.{Mutation: reinitialization}Input:V, E, V

Output:V

1. Create a new solutionV using Algorithm I 1 or I 2

A more complicated mutation approach is to createa new solution based on an existing one (see Algo-rithm M2). For this type of algorithm, a mechanism isdesigned to modify the morphology of an existing so-lution by exchanging some of its vertices either withunselected vertices (for selection problems) or amongdifferent subdivisions (for partitioning problems). Herea setV2is maintained to contain all moveable vertices

in V. A vertex is moveable if it can be removed fromV without creating a partial solution that violates spa-tial constraints, if any. A repair procedure is neededsubsequently to create a feasible solution. For problemswithout spatial constraints, all vertices in V are move-able andV2is identical toV

.

AlgorithmM2.{Mutation: morphing}Input:V,E ,V

Output:V

1. Find all moveable vertices inV and put them inV22. Randomly select a vertex (v) fromV23. UPDATE(V, v)

Step 1 of Algorithm M2 yields a set of all moveablevertices inV. In step 3, function UPDATEis given in ageneral form without further specifications; this is be-cause each specific problem may have different require-ment and it is impossible to have a one-size-fits-all al-gorithm. In essence, however, function UPDATE(V, v)repairs V by replacing v with a new vertex (selec-tion problems) or assigning v to a new subdivision


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806 Xiao

(partitioning problems); this function also guaranteesthat the spatial constraint, if any, not be violated.Let us consider the solution in Figure 3A (the figureto the right) as an example. Here, because we haveV2 = {7, 10}, we can remove either vertex 7 or 10and the resulting partial solution, {6, 10} or {6, 7},

respectively, still satisfies the contiguity constraint. Ifvertex 7 is removed, vertices 2, 3, 5, 9, 10, 11, and 14become eligible vertices, each of which can be addedinto the partial solution (caused by the removal ofvertex 7) to form a feasible solution.

Algorithm M2 can be tailored for problems with orwithout spatial constraints. Moreover, it is possible todevise randomized versions of Algorithm M2 specif-ically for problems without spatial constraints (/).These algorithms, M3 for selection problems and M4for partitioning problems, as outlined next, swap thememberships of two randomly selected vertices.

Algorithm M3. {Mutation for selection without spatialconstraints}Input:V, E, V

Output:V

1. Randomly select a vertex (v) fromV, v / V

2. Randomly select a vertex (v) fromV

3. Removev fromV

4. Addv intoV

Algorithm M4. {Mutation for partitioning without spatialconstraints}Input:V, E, V

Output:V

1. Randomly select two subdivisionsUi andUj2. Randomly select a vertex (vi ) fromUi3. Randomly select a vertex (vj ) fromUj4. Assignvi to subdivisionUj5. Assignvj to subdivisionUi

Incorporating Problem-Specific Knowledge

Knowledge about solutions to geographical op-timization problems has proven to be useful in thedevelopment of traditional heuristic methods (seeDensham and Rushton 1991, 1992). Such knowledge

can also be incorporated into the design of EAs for solv-ing geographical optimization problems. A variety ofapproaches can be used in all stages of EA development(representation, initialization, and evolutionary oper-ations). More specifically, in this article, I categorizepotential incorporation strategies into two types.

The first type of incorporation approach can be im-plemented on a macro level in which knowledgeabout the entire problem and its solutions is used in an

EA. An example of using macro-level problem-specificknowledge is the design of representation strategy insolving the p -median problem with EAs (or genetic al-gorithms in this specific case). An early application ofgenetic algorithms to the p -median problem by Hosageand Goodchild (1986) showed relatively poor perfor-

mance, partly due to the representation strategy of us-ing a binary string with a length equal to the numberof all demand nodes. Since then, significant improve-ments were made by researchers when they changedthe representation of the problem by using an integerstring with the length equal to the number of facili-ties to locate (Bianchi and Church 1993; Dibble andDensham 1993). Studies along this research line havedemonstrated that EAs can be successfully used to findhigh-quality solutions to p -median (Estivill-Castro andTorres-Velazquez 1999; Alp, Erkut, and Drezner 2003;Mladenovic et al. 2007).

Another example of incorporating macro-levelproblem-specific knowledge into EAs is the use ofexisting solutions to a problem. Bennett, Xiao, andArmstrong (2004), for example, found that solutionsgenerated by other methods can be used as part of aninitial EA population and can improve the overall EAperformance.

The second type of approach to incorporatingproblem-specific knowledge is often implemented ona micro level because of the use of existing heuris-tic methods in the design of evolutionary operations inEAs. This idea follows the trend of hybridization be-

tween EAs and other search algorithms that are oftenderived from classic problem-specific algorithms (Fox1993; Anderson 1996; Preux and Talbi 1999). Previ-ous research has shown that the use of a hybridizationstrategy in EAs can greatly improve EA performance(Krzanowski and Raper 1999; Ruiz-Andino et al. 2000;Lin, Hwang, and Wang 2001).

In the algorithms designed in this article, many stepsare executed in a random fashion. For example, step3 of Algorithm R1 specifies that a vertex is randomlyselected from a set. This operation, however, can bedesigned in a more heuristic way by utilizing problem-

specific knowledge. Many researchers have imposed agreedy operation such that this process will only im-prove the solution being considered. The next sectiondiscusses the implementation of some knowledge-basedEA operations.

Although the use of problem-specific knowledgein EA design has shown generally positive results,hybridization must be executed with care, especiallyfor large-size problems with a great number of local


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optimal solutions. The randomness used in EAs under-lines a sense of emergence, a hope that high-qualitysolutions will emerge from a random start (Holland1975, 1998). Excessive use of problem-specific knowl-edge may decrease the ability of the hybrid algorithmin escaping local optima.

Applications

The algorithms already described provide a guidelinefor the design and implementation of EAs to solve thefour types of geographical optimization problems. Ingeneral, an EA designed using this framework can bedenoted as a 4-tuple:

GEA =C, I,R,M,

whereCis /, /,E /, orE / referring to the rep-

resentation and encoding strategy, I is a set of initial-ization algorithms,Ris a set of recombination methods,and M is a set of mutation methods. Table 2 summa-rizes these algorithms and lists example EAs for eachproblem type. For each of the evolutionary operations,it is possible to have multiple implementations. For ex-ample, both M1 and M2 can exist in an EA, but onlyone of them will be (randomly) chosen at one time.

A variety of applications can be used to demonstratethe use of the framework set forth in this article. Anexample of using this framework to solve partitioningproblems without spatial constraints is illustrated in the

work of Bennett, Xiao, and Armstrong (2004), wherethe combination of land use types is sought to satisfy ob-jectives such as maximizing environmental benefit andminimizing public investment. Xiao (2006) employedsome major concepts of this framework (e.g., graph rep-resentation, recombination, and mutation) for selectionproblems with spatial constraints in a case study of sitesearch problems where a contiguous set of land parcelsmust be identified. For selection problems without spa-tial constraints, recent EA implementations in solvingthe p-median problem employed representation strate-

gies and evolutionary operations that are similar to whathas been discussed here.

In this section, the use of the EA framework isdemonstrated by applying it to a partitioning problemwith spatial constraints ( E/). More specifically, po-litical redistricting problems are addressed. Redistrict-

ing problems are critical in the political system of theUnited States and solving them requires subdivision ofa region (e.g., a state) into a number of districts thatare as equal in population as possible. Two fundamen-tal criteria (i.e., contiguity and population equality) arerequired by law, although many states may have addi-tional requirements such as compact districts. A feasibleredistricting plan must be contiguous, meaning that thespatial units of a district form an entire region and onecan walk between any two points in the district with-out leaving it. Each district, of course, also must haveat least one spatial unit.

Political redistricting problems have been studiedfrom various perspectives in the geography and relatedliterature (Morrill 1976, 1981; Williams 1995; Eagles,Katz, and Mark 2000). This type of problem has beenconsidered as a typical NP-complete problem (Altman1998a);5 exact solution approaches are generally ineffi-cient and many heuristic methods have been developed(Williams 1995). In practice, redistricting plans are of-ten created using an interactive computer program suchas a geographical information system (GIS) that allowsusers to modify the membership of each spatial unitand monitor the demographic impact when a change

to a plan is made (Altman, MacDonald, and McDonald2005).6 Although these tools are useful, they typicallyrely on manual, interactive processes that may not pro-vide a sufficient guide to practitioners with high-qualitysolutions.

The focus of this section is the application and eval-uation of the EA framework in solving a particular typeof problem. To keep the discussion relatively simple,the application concentrates on population equality,the primary concern of political redistricting, but dis-trict compactness is not explicitly considered.7 Being

Table 2. EA operations for different geographical optimization problem types

Problem type Initialization Recombination Mutation Example EAs

/ I 1 R1 M1, M2 /,I 1,R1, {M1,M2}/ I 1 R1 M1,M2, M3 /,I 1,R1, M3E/ I 2 R2 M1, M2 E/,I 1,R1, M2E/ I 2 R2, R 3 M1,M2, M4 E/, I 1, R1, {M2, M4}

Note:EA = evolutionary algorithm.


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aware of the variety of existing solution approaches inthe literature (see Morrill 1981; Williams 1995), I notethat a full discussion of using EAs in this area warrantsa longer and more detailed article, especially in a mul-tiobjective context where EAs have been widely used(Deb 2001; Xiao, Bennett, and Armstrong 2007). An

EA that can be used to find high-quality redistrictingplans can also be extended to include more criteria suchas compactness and minority representation.

The problem has population equality as the singleobjective function:

miny =100 1

P

r

j =1

|pj P|,

where P is the total population, r is the number ofdistricts to be created, P is the ideal population forall districts (computed as the rounded integer value ofP /r ), andp j is the population size of the j th district. Inother words, the goal of solving the problem is to min-imize the overall population deviation of each districtfrom the ideal population.

EA Implementation for Political RedistrictingProblems

The EA used to solve political redistricting problemscan be denoted as E /, I 2, R2, (M1, M2). Theprocedure outlined in Algorithm I 2 (illustrated in

Figure 4) is used to create initial redistricting plans. Forrecombination operations, two versions are created,both based on Algorithm R2 (see also Figure 5).The first version of recombination operation is astraightforward application of Algorithm R2 wherestep 5 is randomly executed. In the second version,problem-specific knowledge is incorporated in step 5in Algorithm R2. Here, instead of randomly choosingtwo subdivisions after overlay to merge, the algorithm(called R2K,details provided later) randomly choosesone (step 5.1) and then chooses another subdivisionthat has the smallest population (step 5.2). In this way,

the merging process (step 6) is encouraged to producenew subdivisions with low population deviation fromeach other to pursue the goal of population equality.

AlgorithmR2K. {Overlay and repair based on merging, withknowledge}Input:V, E,V1,V

2, p

Output:V

1. V3: = {Ui |Ui is a subdivision after overlaying V1 and

V2}

2. V := 3. if |V3 | = p ,thenV

:=V3 andstop4. repeat until |V3 | = p

5.1 Randomly select a subdivisions,Ui fromV

3

5.2 Choose the subdivision with smallest population(Uj ) from the neighboring subdivisions ofUi

6. MergeUi andUj

7. V: =V3

The mutation operations used here also have twoversions: random and knowledge-based. The randomversion is designed using the guideline specified in Al-gorithmM2 and is calledM2R.

AlgorithmM2R.{(Random) mutation for redistricting}Input:V, E, V

Output:V

1. Find all moveable vertices inV and put them inV22. Randomly select a vertex (v) fromV2

3. Randomly assignv

to one of its neighboring districts

The knowledge-based mutation operation is calledM2K, also designed based on Algorithm M2. Algo-rithmM2Kdiffers fromM2Rin terms of how a move-able vertex is chosen in step 2. Here, the moveablevertex with the smallest population is chosen. In thisway, the overall population of each district will be leastdisturbed when a new solution with a different spatialconfiguration is created.

Algorithm M2K. {(Knowledge-based) mutation for redis-tricting}Input:V, E, V

Output:V

1. Find all moveable vertices inV and put them inV22. Select from V2 the vertex (v

) that has the smallestpopulation

3. Randomly assignv to one of its neighboring districts

Based on the preceding implementations, the EA forpolitical redistricting, namedGEAPR,is outlined next.The output ofGEAPRis a set of solutions{(V,E )},one of which is the best solution found by the EA thatexhibits the lowest objective function (population de-

viation) value. GEAPR outlined next can be imple-mented in two versions: random and knowledge based.The difference is in steps 5.2 and 5.3. For the randomversion, Algorithm R2 is used in step 5.2 for recombi-nation and the mutation operation in step 5.3 is exe-cuted by randomly choosing between Algorithms M1and M2R. The knowledge-based version uses Algo-rithm R2Kin step 5.2 and randomly chooses betweenM1 andM2Kin step 5.3.


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AlgorithmGEAPR{EA for a selection problem with spatialconstraints}Input:V,EOutput:{(V, E)}1. t := 02. Initialize population P (t) using Algorithm I 1 based

on theE /encoding

3. repeat until a user-specified termination criterion ismet

4. Evaluate each individual inP (t)5. Generate offspring ofP (t) by

5.1 Selecting parent solutions fromP (t)5.2 Using a recombinationoperationtocreatenew

individuals5.3 Applying a mutation operation

6. Copy the new individuals toP (t)7. t := t + 1

In step 4 ofGEAPR,the objective function value iscalculated for each solution and then the fitness value is

computed as ymaxyymaxymin , whereymaxandyminare the max-imum and minimum objective function values of allsolutions in the current EA population, respectively,and y is the objective function value of the individualsolution being evaluated. In this application, a propor-tional selection approach (Goldberg 1989) is utilized instep 5.1 where the probability of the i th individual to beselected is calculated as fiN

j =1 fj, with the denominator

being the sum of fitness values of all solutions in a pop-ulation and fi the fitness of the i th solution. The actualselection process can be described as throwing a ball

onto a roulette wheel, where solutions with high fitnessvalues occupy a big sector and thus have a high chanceof being selected. Each time, the algorithm selects twoindividuals (step 5.1) and, by a high chance (95 per-cent here), a recombination operation is used to createa new individual (step 5.2). Then, the new individualwill have a small chance (5 percent here) of undergoinga mutation operation (step 5.3). This process continuesuntil the number of new solutions equals the numberof solutions in the current generation. At the end ofeach iteration, the newly created population is used toreplace the current population and the process repeats

(steps 6 and 7).

Computational Experiments

Two types of data sets are used to test GEAPR.Thefirst type of data set includes three population gridscalled Grid1, Grid2, and Grid3 of different sizes (seethe first two columns of Table 3); each cell in a gridrepresents a hypothetical spatial unit (a county or cen-

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(C)Figure 6. Redistricting of twenty-five spatial units in data set Grid1into two (A), three (B), and four (C) districts. The population ofeach spatial unitis shown. Thick black lines indicate the boundariesbetween districts.

sus block, for example). The population of each cellis randomly determined and is shown in the followingmaps. The total population for each data set is shown inthe third column of Table 3. Each data set is used to de-fine three redistricting problems with different numbersof districts (r ), as listed in the fourth column of Table

3. In the same table, the population size and numberof iterations used in GEAPRare listed in the fifth andsixth columns, respectively. These are often problem-specific EA parameters and must be determined basedon a method of trial and error. The values adoptedhere have been determined to be reasonable in terms oftheir impact on EA performance during numerous priorexperiments.

To evaluate the performance ofGEAPR,it is neces-sary to know the closeness of the best solution found tothe global optimal solution. Although it is difficult todisplay the spatial configuration of the global optimal

solution, it is possible to derive the objective functionvalue of thetheoreticalglobal optimal solutions.8 Thesevalues are listed in the seventh column of Table 3. TheCPU time reported in the table shows that knowledge-based approaches are slightly more efficient for most ofthe cases. This can be explained by examining algo-rithmsR2 andR2K.In Algorithm R2 (for the randomversion), the repair process (steps 4, 5, and 6) does notguarantee that merging two randomly selected subdi-visions will not violate the contiguity constraint, andthe algorithm may go through a large number of theseoperations until a feasible solution can be created. Al-

gorithmR2K,however, does not have this problem aseach subdivision picked in step 5.1 will have at least oneneighboring subdivision and merging can be conducted.

The random and knowledge-based versions ofGEAPR are executed ten times for each redistrictingproblem, and the minimum, median, and maximumobjective function values found during the ten runsare reported in Table 3. The maps of the best solu-tions found are shown in Figures 6, 7, and 8. The


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(C)

Figure 7. Redistricting of 100 spa-tial units in data set Grid2 into three(A), four (B), and five (C) districts.The population of each spatial unit isshown. Thick black lines indicate theboundaries between districts.

results clearly demonstrate that GEAPR can be usedto find high-quality solutions to the experimental prob-lems used here. Knowledge-based approaches can pos-sibly find good solutions more quickly (see the columnsmarked Last in Table 3) and, over ten independentruns, have a better chance to find good (if not optimal)

solutions. Note that the theoretical optimal objectivefunction values were not reached for the two problems(Grid3 with r = 10 and 20), although we should re-alize that these theoretical optimal solutions may notexist.

Figure 9 shows the performance ofGEAPRfor Grid1with four districts (r = 4). The general trend of decreas-ing minimal, median, and maximal objective functionvalues of each iteration suggests the effectiveness of theEA in improving current solutions. The irregular curvemarked Max in the figure indicates the introduction of

new, random solutions to the population in each itera-tion using the mutation operations.

The second type of experiment is based on the Iowacongressional redistricting case using the 2000 censusdata. The Iowa constitution provides that the coun-ties shall not be split.9 Subsequently, the ninety-nine

counties are used to create five congressional districtsto minimize the deviation from the ideal population. Aparameter setting ofGEAPR similar to that of Grid2with r = 5 is adopted here. In this particular case,the total population is 2,926,324 and the theoreticalglobal optimal solution should have an objective func-tion value of 0.0003. However, it is difficult to provethat such a solution would exist with the real data. Fig-ure 10A shows the best solution found by the EA usingthe knowledge-based version, and Figure 10B shows theofficially adopted redistricting plan. It can be noted that

Table 3. Test data and results

Random EA Knowledge-based EA

Data n Populationa r Popsizeb Iterationsc Optd Mine Medianf Maxg Lasth Timei Min Median Max Last Time

Grid1 5 5 384 2 50 500 0 0 0 0 5 7 0 0 0 4 7

3 50 500 0 0 0 1.04 84 7 0 0 1.56 33 6

4 50 500 0 0 1.56 4.16 141 7 0 0.52 1.56 73 6

Grid2 10 10 2,952 3 50 1,000 0 0 0 0.07 210 91 0 0 0 207 73

4 50 1,000 0 0 0.07 0.20 429 83 0 0 0.07 290 66

5 50 1,000 0.07 0.07 0.18 0.41 519 81 0.07 0.11 0.14 618 64

Grid3 25 40 74,663 5 50 1,500 0.0027 0.0027 0.0039 0.0230 1,039 21,506 0.0027 0.0043 0.0113 862 1 9,496

10 50 1,500 0.0040 0.0228 0.0790 2.4952 1,319 20,195 0.0174 0.0308 0.0440 1,185 18,901

20 50 1,500 0.0040 0.1594 0.4861 4.8605 1,387 19,755 0.0710 0.0978 0.2404 1,292 19,832

Note:EA = evolutionary algorithm.aThe total population of all spatial units in a data set.bThe number of individual solutions used inGEAPR.cThe total number of iterations ofGEAPR.dThe theoretically possible optimal objective function value.eThe objective function value of the best solution found in the ten runs.fThe median of the best objective function values found in the ten runs.gThe objective function value of the worst solution found in the ten runs.hAverage last iteration when a better solution is found.iAverage CPU time used for each run (in seconds).


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56 78 58 91 88 72 84 95 98 52 79 87 93 90 78 91 95 67 98 83 56 68 53 72 97 68 83 58 53 62 52 76 93 63 85 90 75 58 65

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65 71 91 70 80 68 95 93 99 81 53 89 90 84 60 76 97 95 98 91 81 76 59 52 67 85 87 73 90 82 71 55 62 96 57 52 66 58 99 87

60 91 72 87 62 88 99 83 64 67 84 82 95 73 67 52 74 67 60 67 99 55 61 95 87 65 60 74 67 62 69 83 96 87 61 62 58 99 69 81

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Figure 8. Redistricting of 1,000 spa-tial units in data set Grid3 into five(A), ten (B), and twenty (C) districts.The population of each spatial unit isshown. Thick black lines indicate theboundaries between districts.


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Objectivefunction

value

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Figure 9. The performance of the random version ofGEAPR whenused to solve the problem with twenty-five spatial units and fourdistricts. The minimal, median, and maximal objective functionvalues in each iteration are marked as Min, Median, and Max,respectively. The optimal solution is found at iteration 67.

the solution found by the EA has a higher populationequality, although the shape may be considered to beless compact. The point here is not to suggest a newredistricting plan. However, it can be observed throughthis application that EAs are useful in generating inter-esting, high-quality alternative plans for considerationin the decision-making process.

Discussion and Conclusions

The rapid development of computing techniquesduring the past several decades has encouraged an op-timistic view toward the computational issues that facegeographers (see, for example, Dobson 1983). However,researchers must bear in mind that the ultimate limita-tion on computation is the inherent complexity of theproblem to be solved, rather than the speed of computersystems (Garey and Johnson 1979; Armstrong 1993).Developing new methods to overcome the shortcom-

ings of existing algorithms has always been a motivationthat leads to progress in optimization research.

This research echoes geographers recent interestsin computational science (see Openshaw 1994;Fotheringham 1998; Armstrong 2000). This articleaddresses issues that are fundamental to geographicaloptimization in particular, and GIScience in general,including conceptualization of geographical problems,spatial representation, and algorithm analysis. In

a broader view, this research represents a startingpoint on a quest to establish a unified framework forgeographical optimization problems. Although thefocus is placed on the construction of a framework, thealgorithm guidelines discussed here are useful for manyparticular geographical problems and an application of

this framework is also discussed.It is also worthwhile to note possible limitations ofthe approach discussed here. The typology does notdistinguish between problems with a fixed number ofspatial entities to locate (e.g., the p-median problem)and those that may require a variable number of spatialentities (e.g., the set-covering problem). The latter isespecially important for problems such as spatial clus-ter detection in point data sets (Openshaw and Perree1996). Nevertheless, the graph representation discussedhere is flexible and can be applied to guide the design ofnew types of spatial EAs not discussed in the previous

sections. To incorporate such new encoding types, how-ever, additional algorithms may be needed for initial-ization, recombination, and mutation operations. As afuture research topic, techniquesused in variable-lengthEAs (see Srikanth, George, and Warsi 1995; Wu andBanzhaf 1999) can be used to implement EAs for geo-graphical optimization problems that require solutionswith variable lengths.

EAs do not come without drawbacks, however. Amajor issue of EAs is their computational load. Be-cause EAs are population-based computer programs, si-multaneously manipulating individual solutions across

multiple generations requires a significant amount ofcomputing resources. A straightforward way to speedup EA performance is to exploit the implicit paral-lel nature of EAs; a variety of parallel models havebeen developed in the literature (see, for example,Cantu-Paz 2000; Xiao and Armstrong 2003). Xu et al.(2003) suggested five speed-up strategies that utilize anew representation scheme and a set of efficient se-lection operations, which can be used to reduce theoverall search space and therefore expedite the searchprocess.

Another type of approach to addressing the compu-

tational burden is to hybridize EAs with other heuristicsthat can help find high-quality solutions quickly. Re-searchers have discussed possible ways to apply othermetaheuristic approaches in EAs. For example, Bergey,Ragsdale, and Hoskot (2003) discussed the use ofsimulated annealing in a genetic algorithm to expeditethe search for optimal solutions to an electrical powerredistricting problem. More comprehensive approachesto incorporating a variety of metaheuristic methods


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(B)

Figure 10. Redistricting for Iowa where ninety-nine counties are subdivided into five congressional districts. (A) The best solution found by

GEAPRwith an objective function value of 0.0045, and the total absolute deviation from the ideal population is 131 persons. (B) The officialplan adopted by Iowa in 2000, with an objective function value of 0.0080, and the total absolute deviation from the ideal population is 235persons. Numbers printed on the maps show the population of each county.

have also been discussed (see, for example, Anderson1996; Preux and Talbi 1999). The algorithms devel-oped in this article have their roots in the heuristicoptimization literature. For example, the essence ofthe add, drop, and interchange algorithms developed

to solve the locationallocation problems (Kuehn andHamburger 1963; Feldman, Lehrer, and Ray 1966; Teitzand Bart 1968) can be found in most of the algorithmsdeveloped in this article. These traditional approachesare placed in an EA context as a way to create solutions


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814 Xiao

and handle spatial constraints. This hybrid approachhas been common in solving geographical optimizationproblems using EAs (see, for example, Reeves 1997;Estivill-Castro and Torres-Velazquez 1999; Krzanowskiand Raper 1999; Nalle, Arthur, and Sessions2002).

As a final note, although the literature has been gen-erally supportive about EA applications, a number ofauthors have also cautioned about overoptimistic viewsof EAs (Dowsland 1996; Ross 1997). To fine-tune anEA, for example, the user and designer must make aseries of decisions about EA parameters (such as thepopulation size and probability of recombination andmutation operations). In addition, the success of EAsoften relies on the use of diversification methods sothat individual solutions do not concentrate on a fewgood solutions and therefore avoid being trapped in lo-cal optima; these methods are often sensitive to their

parameters (Goldberg 1989). These issues have beendiscussed in the literature with respect to EA perfor-mance (De Jong 2006); the understanding of their im-pact on EA performance for spatial problems, however,is limited (Xiao 2006) and needs to be further stud-ied. Although the focus of this article is placed on themore geographical aspects of EA design, being awareof implementation issues will help researchers developa clear picture about the problem-solving landscape asthey continue to design better solution strategies forgeographical optimization problems.

Acknowledgments

I thank Marc P. Armstrong and Iris Hui for theirvaluable comments. Comments from Mei-Po Kwan andanonymous reviewers are also acknowledged.

Notes

1. The term geographical optimization problem is looselyused in this article. In general, such a problem requiressearch for the configuration of a set of discrete spatialentities. In other words, a solution to this type of prob-

lem exhibits aspatial patternthat can be displayed on amap. A solution to a p-median problem (Hakimi 1965),for example, represents a spatial configuration where thedemand in each geographical unit is served by its nearestfacility and a service area map can be drawn accordingly.Some optimization problems with spatial components(see, for example, Leung, Li, and Xu 1998) may not nec-essarily have this characteristic and therefore are notdirectly considered in this article, although research in-corporating these problems will be an interesting futuretopic.

2. If a problem can be solved in a time frame that is poly-nomial with respect to input size, the problem is placedinto class P and is typically regarded to be easy tosolve. For problems in class NP-complete (NP standsfor nondeterministic polynomial), however, polynomialtime algorithms have not been developed and it is likelythat such algorithms do not exist. See Garey and John-son (1979) and Cormen et al. (2001, ch. 34) for a more

formal discussion of this issue.3. Geographical optimization problems often contain im-

portant social, economic, and political factors that aredifficult, if not impossible, to place in a mathematicalformulation. Optimal solutions to mathematically well-formulated problems will become nonoptimal when theunmodeled factors or objectives are taken into account.Therefore, near optimal (or second best) solutions to aproblem may be favorable to decision makers (Simon1960; Brill 1979; Hopkins 1984).

4. It is important to distinguish spatial constraints andother constraints that may have spatial implications.In this article, spatial constraints refer to explicit re-quirements of the topological configuration of spatial

units. For many applications, some constraints may havespatial implications but they do not require the con-figuration (or pattern) of spatial units. For example, inthe work of Bennett, Xiao, and Armstrong (2004), it isrequired that the total area of particular land use typesshould not exceed 25 percent of the overall size of thestudy area. This requirement, however, does not confinethe topological relationship between spatial units andtherefore is not considered as a spatial constraint inthe context of this research.

5. For a redistricting problem of partitioning n spatialunits into r districts, the exact number of possibleredistricting plans is difficult to compute. However,we know that the upper bound of this number is aStirling number of the second kind that is defined

as S(n, r )= 1r !

ri =0(1)r [ r !(r i )!r ! ] (r i )n (see, Even1973, 60), which occurs when each unit is adjacent to allother units. The lower bound of the number of possibleplans occurs when the units are least connected, mean-ing units arranged as a line and, except for the unitsat both ends, each unit has only two adjacent units; in

this case, there areS1(n, r )= (n1)!

(nr )!(r 1)!possible plans.

For the example of Iowa redistricting, where n = 99 andr = 5, the total number of possible redistricting plans isbetween 3.6 106 and 1.3 1067.

6. Interactive redistricting software tools are availablein many GIS packages such as an ArcView extension(http://www.esri.com/software/arcview/extensions/distr-icting) and a commercial package called Maptitude for

Redistricting (http://www.caliper.com/mtredist.htm).Both URLs were last accessed on 18 December 2007.

7. Although compactness is a critical factor, studies didnot show that there is no necessary relation betweenthe shape of redistricting plans and gerrymandering, aprocess intended to favor a particular political party orinterest group (Taylor and Johnston 1979), and differentcompactness measures may lead to different conclusions(Altman 1998b).

8. If we assume all spatial units are adjacent to eachother, the optimal solution occurs when the population


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difference between any two districts is not greater than1. Accordingly, the optimal objective function value canbe calculated as 100 1

P |P r P |. Whenthe to-

tal population can be evenly divided by r, the theoreticaloptimal solution should have an objectivefunction valueof zero. It is important to stress the assumption used herebecause such a theoretical optimal objective functionvalue may not exist in the spatial connectivity of a realdata set.

9. Iowa Code 42.4(1)(b).

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