research article a comparison of local search methods for...

Research ArticleA Comparison of Local Search Methods for the MulticriteriaPolice Districting Problem on Graph

F Liberatore1 and M Camacho-Collados23

1UC3M-BS Institute of Financial Big Data Charles III University of Madrid Getafe 28903 Madrid Spain2Deputy Directorate for Operations Spanish National Police Corps 28010 Madrid Spain3Statistics and Operations Research Department University of Granada 18071 Granada Spain

Correspondence should be addressed to F Liberatore fliberatinstuc3mes

Received 8 June 2015 Revised 29 January 2016 Accepted 16 February 2016

Academic Editor Francisco Chicano

Copyright copy 2016 F Liberatore and M Camacho-ColladosThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In the current economic climate law enforcement agencies are facing resource shortages The effective and efficient use of scarceresources is therefore of the utmost importance to provide a high standard public safety service Optimization models specificallytailored to the necessity of police agencies can help to ameliorate their useTheMulticriteria Police Districting Problem (MC-PDP)on a graph concerns the definition of sound patrolling sectors in a police districtThe objective of this problem is to partition a graphinto convex and continuous subsets while ensuring efficiency and workload balance among the subsets The model was originallyformulated in collaboration with the Spanish National Police Corps We propose for its solution three local search algorithms aSimple Hill Climbing a Steepest Descent Hill Climbing and a Tabu Search To improve their diversification capabilities all thealgorithms implement a multistart procedure initialized by randomized greedy solutions The algorithms are empirically testedon a case study on the Central District of Madrid Our experiments show that the solutions identified by the novel Tabu Searchoutperform the other algorithms Finally research guidelines for future developments on the MC-PDP are given

1 Introduction

The Police Districting Problem concerns the definition ofsoundpatrolling sectors in a police district An extensive liter-ature review on this family of problems is given by Camacho-Collados et al [1] The newest member of this family is theMulticriteria Police Districting Problem (MC-PDP) [1] Thenovelty of this model stands in that it evaluates the workloadassociated with a specific patrol sector according to multiplecriteria such as area crime risk diameter and isolationand that it finds a balance between global efficiency andworkload distribution among the agents according to thepreferences of a decision-maker (ie the service coordinatorin charge of the patrolling operations in a police district)TheMC-PDPwas originally formulated in collaborationwiththe Spanish National Police Corps (SNPC) and it was solvedby means of a fast heuristic algorithm that is capable ofrapidly generating patrolling configurations that are more

efficient than those adopted by the SNPC When combinedwith Predictive Policing methodologies [2] the MC-PDPallows designing patrolling configurations that focus thedistribution of resources on the most relevant locationswith a consequential improvement in the effectiveness ofpatrolling operations This is the rationale of the paperby Camacho-Collados and Liberatore [3] that presented aDecision Support System (DSS) for the implementation of aparadigm of Predictive Patrolling in the SNPC

The contributions of this paper are the following Inthis research we extend the applicability and the qualityof the solutions found by the MC-PDP with the objectiveof improving the performance of the DSS for PredictivePolicing In particular we tackle one of the major limitationsof the original formulation of the MC-PDP and proposeand compare new heuristic algorithms More specificallythe original MC-PDP was formulated to partition a grid Inthis paper we formulate the MC-PDP to generate patrolling

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3690474 13 pageshttpdxdoiorg10115520163690474

2 Mathematical Problems in Engineering

district on a generic graph without any assumption on itstopology This allows for the definition of patrolling configu-rations using census districts as the atomic unit of patrollingAs explained by Sarac et al [4] the use of a structure basedon census districts is desirable as it allows easy access todemographic data and at the same time it is suitable for useby other agencies Translating theMC-PDP to a generic graphrequires the definition of an efficient and practical conditionfor set convexity that we derive from the classical definitionof convexity in graphs In terms of solution methodologieswe propose three local search algorithms for the MC-PDPon a graph including a Tabu Search (TS) Thanks to itsability to escape from local optima and its versatility theTS has been successfully applied to a very wide breadth ofcontexts and problems such as parameter optimization [5]vehicle routing [6] hardwaresoftware partitioning [7] andjob shop scheduling [8] The MC-PDP is a variant of thegraph partitioning problem The first application of the TSto the graph partitioning problem is due to Rolland et al[9] In recent years the TS has been successfully applied tothis family of problems either individually or combined withother approaches [10ndash14]

The proposed algorithms are extensively tested on areal dataset based on a case study of the Central Districtof Madrid Their performances are then compared andanalyzed statistically Finally the best solutions found by thealgorithms are illustrated and operational insights are drawn

The remainder of the paper is structured as follows Thefollowing section presents a review of the most relevantcontributions to the literature In Section 3 we formulate theMC-PDP for a generic graph and propose a methodologyto deal with the problem of partitioning a generic graphinto convex blocks Next we present in detail the localsearch algorithms developed for the solution of the modelsIn Section 5 we explain the dataset and the computationalexperiments run to test the algorithms Also we analyzethe results and provide insights on the solutions obtainedFinally we conclude the paperwith some guidelines for futureresearch

2 Related Work

In his seminal work Mitchell [15] proposes a clusteringheuristic for the redesign of patrol beats in Anaheim Califor-nia The underlying optimization model considers both thetotal expected weighted distance to incidents and a workloadmeasure defined as the sum of the expected service and traveltime A different approach is presented by Bodily [16] thatadopts a utility theory model that takes into considerationthe preferences of the citizens the administrators and theservice personnel The problem is solved by means of a localsearch algorithm that explores the solution space by swappingelements between the sectors Benveniste [17] includes for thefirst time workload equalization criteriaThe final model wasnonlinear and stochastic in nature and was solved by meansof an approximation algorithm DrsquoAmico et al [18] proposea simulated annealing algorithm The solutions identified bythe search algorithm are evaluated by an external softwareprogram PCAM [19 20] that calculates sector workloads

The external routine is based on a queuing model thatcomputes relevant statistics regarding a sector including theoptimal number of cars to be allocated Equity in terms ofarea to be patrolled is enforced in the model by constrainingthe ratio of the size of the largest and the smallest sectorsA simpler approach is presented by Curtin et al [21] thattackle the problem of partitioning a police district by usinga covering model that maximizes the number of incidentsthat are close to the centers of the sectors Five years laterthe authors extend their approach [22] and include backupcoverage (eg multiple coverage of high priority locations)Finally Zhang and Brown [23] propose a heuristic algorithmfor the generation of districting evaluated using an agent-based simulation model The MC-PDP [1 3] differs fromthose proposed so far in the literature in a number of relevantaspects

(i) It focuses on crime prevention rather than detentionFor this reason it does not consider the emergencycalls but it is based on a crime risk estimation (Thecrime risk estimation can be obtained by a PredictivePolicing model as illustrated by the authors in [3])Previous approaches [18 23] optimize the reactiontime to crime incidents that is crimes that havealready happened

(ii) It optimizes at the same time attributes of area crimerisk compactness and support In particular mutualsupport is a novel attribute that differs from backupcoverage [22] in that the former regards the possibilityof receiving backing in any point of the patrol sectorfrom any other agent in the district while the latteronly concerns the overlapping areas between patrolsectors

(iii) It considers the decision-makerrsquos preferences in theobjective function In the formulation proposed byDrsquoAmico et al [18] the users can specify their prefer-ences only by adjusting the righthand side coefficientsin the constraints while in the models presented byCurtin et al [21 22] no user preference is considered

(iv) It requires a limited amount of data to functionwhile all the approaches previously presented in theliterature require specific information such as thetime location and service time of incidents andemergency calls whichmight not be available For thesame reason these methodologies do not take intoconsideration and hence they cannot be extendedto all the nonviolent crimes that are not reported byemergency calls such as pickpockets theft of vehiclesor property damage

In this paper we extend the first formulation of theMC-PDP [1 3] solving the problem on a generic graphrather than on a grid This improvement allows for thedefinition of patrolling configurations using census districtsas the atomic unit of patrolling which results in an increasedoperationality of the patrolling configurations designed bythe problem a simpler access to demographic data and favorscommunication with other agencies To translate the MC-PDP into a generic graph we devised a definition of set

Mathematical Problems in Engineering 3

convexity specifically tailored to the structure of the problemIn terms of methodology we propose and compare threelocal search algorithms that are capable of generating goodpatrolling configurations in a short time In fact the MC-PDP has been designed to be included in a Decision SupportSystem Therefore the ability of generating good patrollingconfigurations in a short time is of primary importance InSection 5 we show that the solutions obtained by two of thenew algorithms outperform those of the former approachAll these improvements greatly enhance the realism appli-cability and effectiveness of the MC-PDP compared to theprevious formulation and solution methodology [1 3]

The MC-PDP is also related to another family of prob-lems namely the Convex 119901-Partition Problem The Convex119901-Partition Problem concerns the partition of a graph into119901 convex subgraphs Research on the decision form of theproblem is extremely recent and the first contributions onthe topic are due to Artigas et al In two subsequent articles[24 25] the authors prove that the problem of deciding if agraph can be partitioned into 119901 convex subgraphs is NP-complete in general and polynomial for cographs In recentyears it has been shown that this problem is also polynomialfor bipartite graphs for all 119901 ge 2 [26] and for planar graphswhen119901 = 2 [27] However determiningwhether the problemis polynomial for planar graph and 119901 ge 3 (which is thepremise of the MC-PDP) is still an open question

So far the research on the problem has focused primarilyon its decision form (meaning establishing whether a graphcan be partitioned into convex subsets) To the best of theauthorsrsquo knowledge the literature has a lack of models thattackle the problem of optimizing convex partitions that ispartitioning a graph into convex subgraphs while optimizingan objective function or a set of criteria In fact optimiza-tion of convex partitions is a prerogative of the districtingproblem a special case of the graph partitioning problemto which the MC-PDP belongs and most of the researchin the area focused on proposing metrics of convexity fordistricting problems (see eg [28ndash30]) With respect to thePolice Districting Problem the only model that includesa measure of graph convexity in the optimization processapart from the MC-PDP is that proposed by DrsquoAmico et al[18] However the authors recognize that their definition ofconvexity ldquois somewhat unclearrdquo In fact instead of relying onthe formal definition of convexity as we do for the MC-PDPthey consider a set of feasibility constraints that accordingto the authors should ensure convexity in the final solutionNonetheless the constraints seem to be rather arbitrary andno formal proof is given

In the following section we formulate the MC-PDP for ageneric graph and propose a methodology to deal with theproblem of partitioning a generic graph into convex subsets

3 The Multicriteria Police DistrictingProblem on Graph

The MC-PDP concerns the design of patrol sector config-urations that are efficient and that distribute the workloadhomogeneously among the police officers A solution to

the MC-PDP defined on graph 119866 = (119873 119864) is partition119875 of set of nodes 119873 Each block 119860 isin 119875 of the partitionis a connected subset of the node set and represents apatrol sector Therefore from this point onward the termsldquopartition blockrdquo ldquopatrol sectorrdquo and ldquosectorrdquo will be usedinterchangeably The MC-PDP requires the partition blocksto be convex This condition has been introduced to ensurethat all the patrol sector would be intrinsically efficient thatis the agent can move within the sector always following theshortest path Finally the number of subsets in the partitionmust be exactly 119901 The formal elements of the model arepresented in the following

31 Data and Properties Wedefine theMC-PDP on a genericgraph119866 = (119873 119864) with119873 being the set of nodes and 119864 the setof edges For each node 119894 isin 119873 the following data is required

(i) 119886119894isin Rge0 total length of the streets to be patrolled at

node 119894 isin 119873(ii) 119903119894isin Rge0 risk of crime at node 119894 isin 119873

Also each edge (119894 119895) isin 119864 is characterized by the following(i) 119897119894119895isin Rge0 length of edge (119894 119895) isin 119864

Finally 119901 isin Nge2

is the number of patrolling sectors to bedefined

Additionally on the set of nodes 119873 and all of its subsets1198731015840sube 119873 we define the following operations(i) 119889119894119895(1198731015840) shortest path distance between nodes 119894 119895 isin

1198731015840 computed using only the nodes in 119873

1015840 Thisdistance is calculated considering the length of theedges in the path

(ii) 1198891119894119895(1198731015840) shortest edge distance between nodes 119894 119895 isin


1015840 Thisdistance is calculated considering exclusively thenumber of edges in the path

Given a node subset1198731015840 sube 119873 the shortest distances betweenall the nodes are obtained using the Floyd-Warshall algorithm[31 32] The algorithm is initialized with 119897

119894119895for 119889119894119895(1198731015840)

and with the adjacency matrix for 1198891119894119895(1198731015840) Other relevant

properties defined on the set of nodes119873 and all of its subsets1198731015840sube 119873 are as follows(i) ⊘1198731015840 diameter of subset 119873

1015840 The diameter is themaximum distance between two nodes belonging to1198731015840 that is ⊘

1198731015840 = max

119894119895isin1198731015840119889119894119895(1198731015840)

(ii) 1198881198731015840 center of subset 1198731015840 We define the center of a

subset of nodes1198731015840 sube 119873 as the node belonging to thesubset that minimizes the maximum risk-weighteddistance to all the other nodes in the subset Incase of ties the node that minimizes the sum of therisk-weighted distances is chosen In summary 119888

1198731015840 =

arg Lexmin119894isin1198731015840(max

119895isin1198731015840119903119895119889119894119895(1198731015840) sum119895isin1198731015840 119903119895119889119894119895(119873

1015840))

where Lex stands for lexicographic optimization (iehierarchical optimization) of the two objectives Weconsider risk-weighted distances as we assume thatthe agents should spend more time patrolling thenodes having greater risk


32 Patrol Sector Attributes and Workload The MC-PDPevaluates the patrol sectors defined by a partition accordingto fourmain attributes area isolation demand and diameterAll the attributes explained in the following are expressed asdimensionless ratios so as to be comparable

(i) Area 120572119860 This attribute is a measure of the size of theterritory that an agent should patrol It is expressed as theratio of the area encompassed by patrol sector119860 to the wholedistrict area

120572119860=

sum119894isin119860

119886119894

sum119894isin119873

119886119894

(1)

(ii) Isolation 120573119860 In the MC-PDP two patrol sectors supporteach other if the distance between their centers is less than orequal to a defined constant119870The value of119870 can be providedby an expert Alternatively for the MC-PDP on graph werecommend the following

119870 =⊘119873

2radic119901 (2)

that is we suggest 119870 to be set equal to the total diameter ofthe graph divided by twice the square root of the number ofsubsets to be defined The support received by a patrol sectorcan be calculated by

119887119860=10038161003816100381610038161003816119861 isin 119875 | 119889

119888119860119888119861

(119873) le 119870 119860 = 11986110038161003816100381610038161003816 (3)

that is the support 119887119860 is equal to the number of sectors whosecenters are within a distance of 119870 from the center of thecurrently considered subsetTherefore the isolation of sector119860 is computed as

120573119860=119901 minus 1 minus 119887

119860

119901 minus 1 (4)

(iii) Risk 120574119860 This attribute is a measure of the total riskassociated with the sector that an agent patrols It is expressedas the ratio of the total risk of sector 119860 to the whole districtrisk

120574119860=

sum119894isin119860

119903119894

sum119894isin119873

119903119894

(5)

(iv) Diameter 120575119860 The diameter has been introduced in theMC-PDP as an efficiency measure In fact the diametercan be interpreted as the maximum distance that the agentassociated with the sector would have to travel in case of anemergency call Therefore a small diameter results in a lowresponse timeThediametermeasure used to evaluate a patrolsector is the ratio of the subset diameter to the diameter of thegraph that is the maximum possible diameter

120575119860=

⊘119860

⊘119873

(6)

The decision-maker can express their preference on eachattribute by defining a normalized vector of weights w isin R4By linearly combining the attributes with preference weightsw we can compute a measure of workload 119882

119860 of a sector 119860as

119882119860= 119908120572sdot 120572119860+ 119908120573sdot 120573119860+ 119908120574sdot 120574119860+ 119908120575sdot 120575119860 (7)

33 Objective Function The objective of the MC-PDP is togenerate patrolling configurations that are efficient and atthe same time that distribute the workload homogeneouslyamong the patrol sectors The objective function of the MC-PDP takes into consideration the preferences of the decision-maker for these factors by introducing coefficient 120582 isin R0 le 120582 le 1 that expresses the decision-makerrsquos preferencebetween optimization and workload balance

obj (119875) = 120582 sdotmax119860isin119875

119882119860 + (1 minus 120582) sdot

sum119860isin119875

119882119860

119901 (8)

The term max119860isin119875

119882119860 represents the worst workload while

the term sum119860isin119875

119882119860119901 is the average workload This objective

function allows the decision-maker to examine the trade-offbetween optimization and balance by a parametric analysisIn fact by varying 120582 the model gives a range from optimiza-tion (120582 = 0) to balance (120582 = 1)

34 Problem Formulation We can now present a mathemat-ical formulation for the MC-PDP

min obj (119875) (9)

st 0 notin 119875 (10)

⋃

119860isin119875

119860 = 119873 (11)

119860 cap 119861 = 0 forall119860 119861 isin 119875 | 119860 = 119861 (12)

|119875| = 119901 (13)

Conn (119860) = 1 forall119860 isin 119875 (14)

Conv (119860) = 1 forall119860 isin 119875 (15)

The model optimizes the objective function (8) Con-straints (10)ndash(12) represent the conditions held by partition 119875defined on119873 that is119875 should not contain the empty set 0 (10)and the partition blocks cover119873 (11) and are pairwise disjoint(12) Restriction (13) concerns the partition cardinality andenforces the number of partition blocks to be exactly 119901Conditions (14) and (15) regard the geometry of the patrolsectors In fact Conn(119860) is an indicator function that equals1 when 119860 is connected and zero otherwise and Conv(119860) isan indicator function that equals 1 when 119860 is convex andzero otherwise The model establishes that only connectedpartition blocks are feasible This condition implies that anagent cannot be assigned to patrol sectors composed of two ormore separate areas of the city Furthermore all the partitionblocks are required to be convex When a subset is convex


procedure 119871119900119888119886119897119878119890119886119903119888ℎ(1198750)

119875larr 1198750 Initialize the best solution found to the initial solution

119905 larr 0while not119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() do119875119905+1

larr 119878119890119897119890119888119905119873119890119894119892ℎ119887119900119903(119875119905) Select a neighboring solution

if 119875119905+1

better than 119875 then

119875larr 119875119905+1

Save the best solution found so farend if119905 larr 119905 + 1 Increase the iteration counter

end whilereturn 119875

end procedure

Algorithm 1 Local search algorithm pseudocode

it is also optimally efficient in terms of distance betweenthe points In fact in a convex subset there is a minimalshortest path connecting any pair of points Therefore thiscondition allows for the generation of patrol sectors that aremore efficient in terms of movement inside of the area In thefollowing we illustrate more in detail the concept of graphconvexity

35 A Note on Graph Convexity and on Convex GraphPartitioning Let 119866 = (119873 119864) be a finite simple graph Let119860 sube 119873 its closed interval 119868[119860] being the set of all nodes lyingon shortest paths between any pair of nodes of 119860 The set 119860is convex if 119868[119860] = 119860 In this work the following equivalentcondition is applied

1198891

119894119895(119860) = 119889

1

119894119895(119873) forall119894 119895 isin 119860 lArrrArr Conv (119860) = 1 (16)

Lemma 1 Equation (16) is a proper condition for set convexity

Proof Let119860 be a nonconvex set It follows from the definitionthat 119868[119860] = 119860 Let us consider nodes 119894 119895 isin 119860 and node 119896 isin 119873

such that 119896 isin 119868[119860] and 119896 notin 119860 It follows that 1198891119894119895(119860) gt 119889

1

119894119895(119873)

In fact if it were that 1198891119894119895(119860) = 119889

1

119894119895(119873) then 119896 would need to

belong to 119860 Now let 119860 be a convex set It follows from thedefinition that 119868[119860] = 119860 More specifically all the nodes lyingon the shortest path in 119894 119895 isin 119860 also belong to 119860 It followsthat necessarily 1198891

119894119895(119860) = 119889

1

119894119895(119873)

Artigas et al [25] prove that the problem of decidingif a graph can be partitioned into 119901 ge 2 convex setsis NP-complete As we do not make any assumption ongraph 119866 convexity for all the patrol sectors could not alwaysbe possible In order to always obtain a solution we relaxconstraint (15) and penalize its violation in the objectivefunction by means of a Lagrange multiplier The resultingprogram is

min obj (119875) = obj (119875) + 120583sum

119860isin119875

(1 minus Conv (119860)) (17)

st (10) (11) (12) (13) and (14) (18)

Coefficient 120583 is the Lagrange multiplier associated withthe convexity constraint (15)We suggest setting120583 gt 1 In fact

as obj(119875) le 1 setting 120583 gt 1 translates into always preferringa convex graph partition over a nonconvex one regardless ofthe value of obj(119875)

4 Local Search Methods for the MC-PDP

Local search algorithms move from solution to solutionin the space of candidate solutions (the search space) byapplying local changes until certain termination criteria aresatisfied for example a solution deemed optimal is found ora time bound is elapsed One of the main advantages of localsearch algorithms is that they are anytime algorithms whichmeans that they can return a valid solution even if they areinterrupted at any time before they end For this reason theyare often used to tackle hard optimization problems in a real-time environment such as the MC-PDP All the local searchalgorithms proposedmake use of the same solution structure

119875 = 1198891 1198892 119889

|119873| (19)

where 119889119894isin [1 119901] forall119894 = 1 119873 In summary a solution

is a vector of |119873| elements one for each node in the graphthat can take any value from one to 119901 that is the numberof patrolling sectors A generic pseudocode for a local searchalgorithm is presented in Algorithm 1

The procedure starts the search from a given initialsolution 119875

0and it iteratively moves to a solution belonging

to the neighborhood of the incumbent one until certaintermination criteria are met The neighborhood of a solutionis the set of solutions that can be obtained from the currentone by changing it slightly In this research we considerall the solutions that can be obtained by removing a nodefrom a patrol sector and assigning it to another one withoutviolating constraints (10)ndash(14) Different implementationsof 119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() and 119878119890119897119890119888119905119873119890119894119892ℎ119887119900119903() result indifferent local search algorithms The characteristics of thealgorithms developed in this research are presented in thefollowing

Simple Hill Climbing At each iteration the Simple HillClimbing (SHC) algorithm [33] explores the neighborhoodof the incumbent solution to find a better one In our SHC119878119890119897119890119888119905119873119890119894119892ℎ119887119900119903(119875

119905) procedure explores the neighborhood


procedure 119872119906119897119905119894119878119905119886119903119905()

while not119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() do119875 larr 119868119899119894119905119894119886119897119878119900119897119906119905119894119900119899() Generate an initial solution1198751015840larr 119871119900119888119886119897119878119890119886119903119888ℎ(119875) Improve the current solution

if 1198751015840 better than 119875 then

119875larr 1198751015840 Save the best solution found so far

end ifend whilereturn 119875

end procedure

Algorithm 2 Multistart pseudocode

of partition 119875119905in a random fashion and returns the first

improving solution foundThe algorithm terminateswhen noimproving solution is found or the time limit is exceeded

Steepest Descent Hill Climbing The Steepest Descent HillClimbing (SDHC) algorithm [33] is a variant of the SHC thatexplores the whole neighborhood of the incumbent solutionand chooses the best solution belonging to itThis is the samealgorithm originally proposed for the solution of the MC-PDP [1]

Tabu Search Similarly to the SDHC the Tabu Search (TS)algorithm [34 35] explores the whole neighborhood of theincumbent solution However the TS chooses for the nextiteration the best solution found that is not tabu Also the TSdoes not terminate if an improving solution is not foundThisallows the algorithm to escape local optima The criterionthat is used to declare a certain point of the neighborhoodas tabu is based on a short-term memory At each iterationthe TS presented in this paper stores the current solution inthe short-termmemorywith an associated expiration counterinitially set to 119879 During the exploration of a neighborhoodall the solutions found that are already included in theshort-term memory are marked as tabu and their expirationcounter is reset to 119879 Finally at the end of the iteration all theexpiration counters are decreased by one and the solutionswhose counters have reached zero are removed from theshort-term memory

The algorithm terminateswhen the time limit is exceededwhen no nontabu solution is found in the current neighbor-hood or after a fixed number 119868 of nonimproving iterationsWe suggest setting parameters 119879 and 119868 to the cardinality ofthe node set that is 119879 = 119868 = |119873|

41 Multistart Local Search Algorithms Local search meth-ods are very good at exploring certain zones of the solutionspace but they generally end up in local optima Multistartis a very simple and general diversification method In orderto better explore distant portions of the solution space thesearch is started more than one time from different pointsThe pseudocode of a multistart procedure is illustrated inAlgorithm 2

The procedure alternates a solution generation procedurewith a local search step until the time limit is exceeded

Begin

Terminationcriteria

No

InitialSolution

LocalSearch

Solutionimprovement

test

Yes End

SHC

SDHC

TS

Figure 1 Algorithm flow-chart

Generating an Initial Solution To generate an initial solutionat each iteration of the multistart algorithm we use therandom greedy algorithm proposed in Camacho-Colladoset al [1] adapted to work on a generic graph In summarythe algorithm generates a solution by randomly choosing thefirst node of each sector and then expanding the sectors in agreedy fashion while preserving their connectivity Initiallythe partition blocks are empty In the first phase of thealgorithm each block is initialized with a randomly chosennode Subsequently at each iteration of the second phase thealgorithm extends the initial solution by assigning a nodeto a single sector The algorithm chooses the combinationof node and sector that results in the best feasible solutionThe algorithm ends when all the points have been assigned tosubsets It is important to notice that in the current version ofthe algorithm the solutions are evaluated by using the relaxedobjective function (17)

Complete Algorithm Structure Figure 1 illustrates the flow-chart of the complete multistart algorithm First an initialsolution is generated using the aforementioned randomgreedy algorithm Second the solution is improved by meansof local search The local search algorithm could be eitherSHC SDHC or TS Finally the incumbent solution iscomparedwith the best found that is updated if requiredThissequence of steps is repeated until the termination criteria aremet the algorithm terminates and the best solution found isreturned

5 Results and Discussion

51 Dataset We test our algorithms on the Central Districtof Madrid dataset presented in Camacho-Collados et al [1]However in this research the data is aggregated with respect


Figure 2 Census districts in theCentral District ofMadrid (in gray)and the corresponding graph (in black)

to the census district rather than a grid As reported by Saracet al [4] the use of a structure based on census districtsis preferable as it allows easy access to demographic dataand is suitable for use by other agencies Figure 2 showsthe subdivision of the territory and the associated graphThe borders of the census districts are plotted in gray Thenodes of the graph identified by black bullets correspondto the centroids of the census districts Finally black linesrepresent the edges of the graph that connect neighboringcensus districts Overall the graph is comprised of 111 nodesand 277 edges The total length of the streets at each node119886119894 is obtained by summing the length of the parts of street

contained within the borders of each census district Thelength of each edge 119897

119894119895 is computed as the great-circle

distance between the nodes In terms of the risk of crime ateach node 119903

119894 we consider the thefts that occurred during the

following shifts

(i) SATT3 Saturday 10132012 night shift (10 PMndash8AM)

(ii) SUNT1 Sunday 10142012 morning shift (8AMndash3 PM)

(iii) MONT2 Monday 10152012 afternoon shift (3 PMndash10 PM)

These shifts have been identified by a service coordinator incharge of the patrolling operations of the Central Districtof Madrid as typical scenarios representing different crimeactivity patterns as illustrated in Figure 3 In the SATT3shift the district is characterized by a high level of nightlifetherefore thefts are committed in almost all the territory withthe highest levels distributed around popular meeting placesin the center and in the northeast of the district SUNT1 hasa low level of criminality mostly concentrated in the south ofthe district where a popular flea market is held every Sundaymorning Finally MONT2 presents the characteristics of anormal business day with criminal activity spread in the

central area of the territory which is where the commercialactivities are located

52 Computational Experiments We now present and ana-lyze the solution values obtained by the heuristic algorithmspresented It is important to notice that all the algorithms(ie SHC SDHC and TS) implement the multistart methodthat is the local search algorithms are repeatedly run startingfrom a different randomly generated solution until the globaltermination criteria are met For the sake of consistencythe experiments have been run using the same parametersadopted in previous researches on the subject [1 3]

(i) Decision-maker preference weights and balance coef-ficient (119908

120572 119908120573 119908120574 119908120575) = (045 005 045 005) and

120582 = 01 These values have been provided by a servicecoordinator in charge of the patrolling operations ofthe Central District of Madrid as their preference

(ii) Number of patrol sectors 119901 = 2 6 On an ldquoaveragedayrdquo the Central District of Madrid is either splitinto two big sectors or partitioned according to its sixneighborhoods

(iii) The parameters of the TS algorithm have been set as119879 = 119868 = 111 as the graph is comprised of 111 nodes

Given the random nature of the algorithms proposed we raneach combination of algorithm shift and number of patrolsectors 50 times Each run had a time limit of 60 seconds tosimulate the real-time environment of DSSThe experimentswere run on a computer with an Intel Core i5-2500K CPUhaving four cores at 330GHz and 4GBRAM memory andthe algorithms were programmed in C++

Tables 1(a)ndash1(f) show the average relaxed objective func-tion value obj(119875) and the corresponding standard deviationfor each group In the tables the rows correspond to thealgorithm and the best average solution value is highlightedin bold Please note that a solution value that is less thanone indicates that the solution is feasible with respect to theconvexity constraints (15) From the tables we can observethat on average the TS algorithm finds the best solution infour out of six groups and the SDHC in the remaining twogroups

For the sake of reproducibility and comparison withfuture contributions to the subject the average number ofmultistart iterations (The number of solutions explored ina full local search run depends exclusively on the initialsolution and in the case of the SHC on the neighborhoodexploration order which is random Therefore the onlymetric that fully depends on the capabilities of themachine isthe number ofmultistart iterations Reporting its value allowsfor comparisons between runs on different machines) andthe corresponding standard deviations are provided in Tables2(a)ndash2(f)

53 Statistical Analysis As a first step we verify that thesolution values are normally distributed by using the Shapiro-Wilk test The results are shown in Table 3 As all the 119901

values are greater than the chosen alpha level 005 the null


(a) SATT3 (b) SUNT1

(c) MONT2

Figure 3 Maps of the number of thefts reported in the Central District of Madrid The red shade represents a high crime level while thewhite shade represents no criminal activity

hypothesis that the solution values are normally distributedcannot be rejected To understand if the differences in themeans are statistically significant we can now run one-wayANOVA tests The results are illustrated in Table 4

We highlighted in bold the rows of the groups where asignificant difference was detected We can immediately seethat there is no significant difference in the groups wherethe SDHC algorithm was the best We run post hoc Tukeyrsquostests to understand more in detail which algorithm performsbetter for the solution of theMC-PDP Tukeyrsquos test is a single-step multiple comparison procedure used to find means thatare significantly different from each other and that is moresuitable for multiple comparisons than doing a number of119905-tests would be The results are illustrated in Tables 5(a)ndash5(f) In the tables the rows are associated with the pairs of

algorithms being tested We highlighted in bold the rowsshowing a significant difference From the results of thestatistical tests we can draw the following conclusions

(i) The performances of the SHC and the SDHC in termsof solutionsrsquo objective function values are alwaysidentical except for shift SATT3 with six patrolsectors where the SDHC produces solutions that aresignificantly better than those of the SHC

(ii) The TS produces on average better solutions in fourout of six groups and its performances are notworse than those of the other two algorithms in theremaining two groups Therefore we can claim thatit is preferable to use the TS over the SHC and theSDHC


Table 1 Average relaxed objective function value obj(119875) andstandard deviation for each group

(a) Shift SATT3 119901 = 2

Algorithm Avg St DevSHC 050109 000435SDHC 049997 000413TS 053567 019831

(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6


54 Solution Analysis Figures 4(a)ndash4(f) illustrate the bestsolutions found for each shift and number of patrol sectorsin terms of relaxed objective function value All the solutionshave been identified by the TS In the figures the borders ofthe census districts have been plotted in black the streets havebeen plotted in gray and each patrol sector is represented bya different color By observing the patrolling configurationssome insights can be drawn

(i) SATT3 police activity is focusedmostly on the centeras well as on the northeast part of the district wheremost of the crimes are committed The reason forthat is that those areas are very busy nightlife meetingplaces

(ii) SUNT1 the patrolling configurations concentrate onthe southern part of the territory where most ofthe thefts happen on Sunday morning because of

Table 2 Average number ofmultistart iterations and correspondingstandard deviation for each group

(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6


the popular flea market In the six patrol sectorsrsquoconfiguration we can see that one sector is dedicatedexclusively to the area with the highest concentrationof crimes that corresponds exactly to the location ofthe flea market

(iii) MONT2 the district is uniformly partitionedbetween northeast and southwest The configurationwith six patrol sectors assigns higher importance tothe central-western part of the district correspondingto the commercial area

6 Conclusions

In this paper we extended the MC-PDP to generate efficientconvex partitions on generic graphs which increases thepractical usefulness and applicability of the model Also wepropose and compare three local search algorithms and test


Table 3 Values of Shapiro-Wilk test statistics and corresponding 119901value for each group

(a) Shift SATT3 119901 = 2

Algorithm Statistic119882 119901 valueSHC 097584 006277SDHC 098543 03412TS 098595 03707

(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6


Table 4 Results of the one-wayANOVA tests on the solution values

Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212

SATT3 6 262 185e minus 10SUNT1 2 3744 719e minus 14SUNT1 6 2816 443e minus 11MONT2 2 0754 0472

MONT2 6 2212 401e minus 09

them on real crime data from the Central District of MadridThe results of the computational experiments show that theTS presented in this paper produces solutions that are onaverage better than those identified by the SDHC algorithmproposed in a previous research [1]

Table 5 Results of Tukeyrsquos test for each group

(a) Shift SATT3 119901 = 2

Pair 119901 valueSHC-SDHC 099867TS-SDHC 026697TS-SHC 028945

(b) Shift SATT3 119901 = 6

Pair 119901 valueSHC-SDHC 001698TS-SDHC 609e minus 5TS-SHC lt1e minus 7

(c) Shift SUNT1 119901 = 2

Pair 119901 valueSHC-SDHC 057937TS-SDHC lt1e minus 7TS-SHC lt1e minus 7

(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6

Pair 119901 valueSHC-SDHC 099018TS-SDHC 2e minus 7TS-SHC 1e minus 7

This research offers new interesting lines to be pursued Interms ofmodeling solving theMC-PDPon a graph simplifiesthe inclusion of demographic data in the model such as theracial composition of a census district A future work mightexplore the impact of the minimization of racial profiling onthe performance of the resulting patrolling configurationsFurthermore it would be interesting to research optimalsolution algorithms for the MC-PDP Given its intrinsicallynonlinear structure and the interdependence between thepatrol sectors to compute the isolation ratio we believe thata simultaneous column-and-row generation algorithm [36]should be used Although this methodology is rather timeconsuming it could still generate good heuristic solutionswithin the allowed time limit The quality of these solutionscould then be compared with that of the TS proposed in thispaper As explained in Introduction theMC-PDP is a variantof the graph partitioning problems It could be interesting


(a) Shift SATT3 119901 = 2 (b) Shift SATT3 119901 = 6

(c) Shift SUNT1 119901 = 2 (d) Shift SUNT1 119901 = 6

(e) Shift MONT2 119901 = 2 (f) Shift MONT2 119901 = 6

Figure 4 Best solutions found


to compare our TS with adaptations of solution techniquesproposed in the literature for this family of problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Spanish National PoliceCorps personnel of the SEYCO (Statistics and OperativeControl Section) the CPD (Data Processing Center) theSala 091 of Madrid (Police Emergency Call Center) theCentral District of Madrid and the Central District ofGranada for their help and collaboration The investigationof Camacho-Collados was partially financed by the SpanishPolice Foundation research grant for civil servants of theSpanish National Police Corps and by the Fulbright ProgramThe research of Liberatore was supported by the SpanishGovernment Grant TIN2012-32482 All support is gratefullyacknowledged

References

[1] M Camacho-Collados F Liberatore and J M Angulo ldquoAmulti-criteria police districting problem for the efficient andeffective design of patrol sectorrdquo European Journal of Opera-tional Research vol 246 no 2 pp 674ndash684 2015

[2] W L Perry B McInnis C C Price S C Smith and J SHollywood Predictive Policing The Role of Crime Forecastingin Law Enforcement Operations RAND Corporation SantaMonica Calif USA 2013

[3] M Camacho-Collados and F Liberatore ldquoA decision supportsystem for predictive police patrollingrdquo Decision Support Sys-tems vol 75 pp 25ndash37 2015

[4] A Sarac R Batta J Bhadury and C Rump ldquoReconfiguringpolice reporting districts in the city of Buffalordquo OR Insight vol12 no 3 pp 16ndash24 1999

[5] D He and Y L Hong ldquoAn improved tabu search algorithmbased on grid search used in the antenna parameters optimiza-tionrdquo Mathematical Problems in Engineering vol 2015 ArticleID 947021 8 pages 2015

[6] X H Zhang S Q Zhong Y L Liu and X L Wang ldquoA framinglink based tabu search algorithm for large-scale multidepotvehicle routing problemsrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 152494 13 pages 2014

[7] G Lin W Zhu and M M Ali ldquoA tabu search-based memeticalgorithm for hardwaresoftware partitioningrdquo MathematicalProblems in Engineering vol 2014 Article ID 103059 15 pages2014

[8] Y Z Yang and X S Gu ldquoCultural-based genetic tabu algorithmformultiobjective job shop schedulingrdquoMathematical Problemsin Engineering vol 2014 Article ID 230719 14 pages 2014

[9] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 pp 209ndash232 1996

[10] B Ucar C Aykanat K Kaya and M Ikinci ldquoTask assignmentin heterogeneous computing systemsrdquo Journal of Parallel andDistributed Computing vol 66 no 1 pp 32ndash46 2006

[11] U Benlic and J-K Hao ldquoAn effective multilevel memeticalgorithm for balanced graph partitioningrdquo in Proceedingsof the 22nd International Conference on Tools with ArtificialIntelligence (ICTAI rsquo10) pp 121ndash128 Arras France October2010

[12] U Benlic and J-K Hao ldquoA multilevel memetic approach forimproving graph k-partitionsrdquo IEEE Transactions on Evolution-ary Computation vol 15 no 5 pp 624ndash642 2011

[13] U Benlic and J-K Hao ldquoAn effective multilevel tabu searchapproach for balanced graph partitioningrdquo Computers andOperations Research vol 38 no 7 pp 1066ndash1075 2011

[14] P Galinier Z Boujbel and M Coutinho Fernandes ldquoAn effi-cient memetic algorithm for the graph partitioning problemrdquoAnnals of Operations Research vol 191 no 1 pp 1ndash22 2011

[15] P S Mitchell ldquoOptimal selection of police patrol beatsrdquo TheJournal of Criminal Law Criminology and Police Science vol63 no 4 article 577 1972

[16] S E Bodily ldquoPolice sector design incorporating preferences ofinterest groups for equality and efficiencyrdquoManagement Sciencevol 24 no 12 pp 1301ndash1313 1978

[17] R Benveniste ldquoSolving the combined zoning and locationproblem for several emergency unitsrdquo Journal of the OperationalResearch Society vol 36 no 5 pp 433ndash450 1985

[18] S J DrsquoAmico S-J Wang R Batta and C M Rump ldquoA simu-lated annealing approach to police district designrdquo Computersamp Operations Research vol 29 no 6 pp 667ndash684 2002

[19] J M Chaiken and P Dormont ldquoA patrol car allocation modelbackgroundrdquo Management Science vol 24 no 12 pp 1280ndash1290 1978

[20] J M Chaiken and P Dormont ldquoA patrol car allocation modelcapabilities and algorithmsrdquo Management Science vol 24 no12 pp 1291ndash1300 1978

[21] K Curtin F Qui K Hayslett-McCall and T Bray ldquoGeographicinformation systems and crime analysisrdquo in Integrating GIS andMaximal Coverage Models to Determine Optimal Police PatrolAreas pp 214ndash235 Idea Group 2005

[22] K M Curtin K Hayslett-McCall and F Qiu ldquoDeterminingoptimal police patrol areas with maximal covering and backupcovering locationmodelsrdquoNetworks and Spatial Economics vol10 no 1 pp 125ndash145 2010

[23] Y Zhang andD E Brown ldquoPolice patrol districtingmethod andsimulation evaluation using agent-basedmodel ampGISrdquo SecurityInformatics vol 2 article 7 2013

[24] D Artigas M C Dourado and J L Szwarcfiter ldquoConvexpartitions of graphsrdquo Electronic Notes in Discrete Mathematicsvol 29 pp 147ndash151 2007

[25] D Artigas S Dantas M C Dourado and J L SzwarcfiterldquoPartitioning a graph into convex setsrdquo Discrete Mathematicsvol 311 no 17 pp 1968ndash1977 2011

[26] L N GrippoMMatamala M D Safe andM J Stein ldquoConvexp-partitions of bipartite graphsrdquo Theoretical Computer Sciencevol 609 part 2 pp 511ndash514 2016

[27] R Glantz and H Meyerhenke ldquoFinding all convex cuts ofa plane graph in cubic timerdquo in Algorithms and Complexityvol 7878 of Lecture Notes in Computer Science pp 246ndash263Springer 2013

[28] J Bozeman L Pyrik and JTheoret ldquoNearly convex sets and theshape of legislative districtsrdquo International Journal of Pure andApplied Mathematics vol 49 no 3 pp 341ndash347 2008

[29] C P Chambers and A D Miller ldquoA measure of bizarrenessrdquoQuarterly Journal of Political Science vol 5 no 1 pp 27ndash442010


[30] J K Hodge E Marshall and G Patterson ldquoGerrymanderingand convexityrdquo The College Mathematics Journal vol 41 no 4pp 312ndash324 2010

[31] R W Floyd ldquoAlgorithm 97 shortest pathrdquo Communications ofthe ACM vol 5 no 6 artilce 345 1962

[32] S Warshall ldquoA theorem on boolean matricesrdquo Journal of theAssociation for Computing Machinery vol 9 pp 11ndash12 1962

[33] S Russell and P Norvig Artificial Intelligence A ModernApproach Pearson 3rd edition 2009

[34] F Glover ldquoTabu searchmdashpart Irdquo ORSA Journal on Computingvol 1 no 3 pp 190ndash206 1989

[35] F Glover ldquoTabu searchmdashpart IIrdquo ORSA Journal on Computingvol 2 no 1 pp 4ndash32 1990

[36] I Muter S Ilker Birbil and K Bulbul ldquoSimultaneous column-and-row generation for large-scale linear programs withcolumn-dependent-rowsrdquoMathematical Programming vol 142no 1-2 pp 47ndash82 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


district on a generic graph without any assumption on itstopology This allows for the definition of patrolling configu-rations using census districts as the atomic unit of patrollingAs explained by Sarac et al [4] the use of a structure basedon census districts is desirable as it allows easy access todemographic data and at the same time it is suitable for useby other agencies Translating theMC-PDP to a generic graphrequires the definition of an efficient and practical conditionfor set convexity that we derive from the classical definitionof convexity in graphs In terms of solution methodologieswe propose three local search algorithms for the MC-PDPon a graph including a Tabu Search (TS) Thanks to itsability to escape from local optima and its versatility theTS has been successfully applied to a very wide breadth ofcontexts and problems such as parameter optimization [5]vehicle routing [6] hardwaresoftware partitioning [7] andjob shop scheduling [8] The MC-PDP is a variant of thegraph partitioning problem The first application of the TSto the graph partitioning problem is due to Rolland et al[9] In recent years the TS has been successfully applied tothis family of problems either individually or combined withother approaches [10ndash14]

The proposed algorithms are extensively tested on areal dataset based on a case study of the Central Districtof Madrid Their performances are then compared andanalyzed statistically Finally the best solutions found by thealgorithms are illustrated and operational insights are drawn

The remainder of the paper is structured as follows Thefollowing section presents a review of the most relevantcontributions to the literature In Section 3 we formulate theMC-PDP for a generic graph and propose a methodologyto deal with the problem of partitioning a generic graphinto convex blocks Next we present in detail the localsearch algorithms developed for the solution of the modelsIn Section 5 we explain the dataset and the computationalexperiments run to test the algorithms Also we analyzethe results and provide insights on the solutions obtainedFinally we conclude the paperwith some guidelines for futureresearch

2 Related Work

In his seminal work Mitchell [15] proposes a clusteringheuristic for the redesign of patrol beats in Anaheim Califor-nia The underlying optimization model considers both thetotal expected weighted distance to incidents and a workloadmeasure defined as the sum of the expected service and traveltime A different approach is presented by Bodily [16] thatadopts a utility theory model that takes into considerationthe preferences of the citizens the administrators and theservice personnel The problem is solved by means of a localsearch algorithm that explores the solution space by swappingelements between the sectors Benveniste [17] includes for thefirst time workload equalization criteriaThe final model wasnonlinear and stochastic in nature and was solved by meansof an approximation algorithm DrsquoAmico et al [18] proposea simulated annealing algorithm The solutions identified bythe search algorithm are evaluated by an external softwareprogram PCAM [19 20] that calculates sector workloads

The external routine is based on a queuing model thatcomputes relevant statistics regarding a sector including theoptimal number of cars to be allocated Equity in terms ofarea to be patrolled is enforced in the model by constrainingthe ratio of the size of the largest and the smallest sectorsA simpler approach is presented by Curtin et al [21] thattackle the problem of partitioning a police district by usinga covering model that maximizes the number of incidentsthat are close to the centers of the sectors Five years laterthe authors extend their approach [22] and include backupcoverage (eg multiple coverage of high priority locations)Finally Zhang and Brown [23] propose a heuristic algorithmfor the generation of districting evaluated using an agent-based simulation model The MC-PDP [1 3] differs fromthose proposed so far in the literature in a number of relevantaspects

(i) It focuses on crime prevention rather than detentionFor this reason it does not consider the emergencycalls but it is based on a crime risk estimation (Thecrime risk estimation can be obtained by a PredictivePolicing model as illustrated by the authors in [3])Previous approaches [18 23] optimize the reactiontime to crime incidents that is crimes that havealready happened

(ii) It optimizes at the same time attributes of area crimerisk compactness and support In particular mutualsupport is a novel attribute that differs from backupcoverage [22] in that the former regards the possibilityof receiving backing in any point of the patrol sectorfrom any other agent in the district while the latteronly concerns the overlapping areas between patrolsectors

(iii) It considers the decision-makerrsquos preferences in theobjective function In the formulation proposed byDrsquoAmico et al [18] the users can specify their prefer-ences only by adjusting the righthand side coefficientsin the constraints while in the models presented byCurtin et al [21 22] no user preference is considered

(iv) It requires a limited amount of data to functionwhile all the approaches previously presented in theliterature require specific information such as thetime location and service time of incidents andemergency calls whichmight not be available For thesame reason these methodologies do not take intoconsideration and hence they cannot be extendedto all the nonviolent crimes that are not reported byemergency calls such as pickpockets theft of vehiclesor property damage

In this paper we extend the first formulation of theMC-PDP [1 3] solving the problem on a generic graphrather than on a grid This improvement allows for thedefinition of patrolling configurations using census districtsas the atomic unit of patrolling which results in an increasedoperationality of the patrolling configurations designed bythe problem a simpler access to demographic data and favorscommunication with other agencies To translate the MC-PDP into a generic graph we devised a definition of set






















119894119895for 119889119894119895(1198731015840)




1198731015840 = max

119894119895isin1198731015840119889119894119895(1198731015840)



1198731015840 =


119895isin1198731015840119903119895119889119894119895(1198731015840) sum119895isin1198731015840 119903119895119889119894119895(119873

1015840))





120572119860=

sum119894isin119860

119886119894

sum119894isin119873

119886119894

(1)


119870 =⊘119873

2radic119901 (2)


119887119860=10038161003816100381610038161003816119861 isin 119875 | 119889

119888119860119888119861

(119873) le 119870 119860 = 11986110038161003816100381610038161003816 (3)


120573119860=119901 minus 1 minus 119887

119860

119901 minus 1 (4)


120574119860=

sum119894isin119860

119903119894

sum119894isin119873

119903119894

(5)


120575119860=

⊘119860

⊘119873

(6)





obj (119875) = 120582 sdotmax119860isin119875

119882119860 + (1 minus 120582) sdot

sum119860isin119875

119882119860

119901 (8)







min obj (119875) (9)

st 0 notin 119875 (10)

⋃

119860isin119875

119860 = 119873 (11)

119860 cap 119861 = 0 forall119860 119861 isin 119875 | 119860 = 119861 (12)

|119875| = 119901 (13)

Conn (119860) = 1 forall119860 isin 119875 (14)

Conv (119860) = 1 forall119860 isin 119875 (15)



procedure 119871119900119888119886119897119878119890119886119903119888ℎ(1198750)


119905 larr 0while not119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() do119875119905+1


if 119875119905+1


119875larr 119875119905+1



end procedure




1198891

119894119895(119860) = 119889

1





1

119894119895(119873)


1



119894119895(119860) = 119889

1

119894119895(119873)


min obj (119875) = obj (119875) + 120583sum

119860isin119875

(1 minus Conv (119860)) (17)

st (10) (11) (12) (13) and (14) (18)





119875 = 1198891 1198892 119889

|119873| (19)









procedure 119872119906119897119905119894119878119905119886119903119905()





end procedure









Begin

Terminationcriteria

No

InitialSolution

LocalSearch

Solutionimprovement

test

Yes End

SHC

SDHC

TS













following shifts








120572 119908120573 119908120574 119908120575) = (045 005 045 005) and










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























119894119895for 119889119894119895(1198731015840)




1198731015840 = max

119894119895isin1198731015840119889119894119895(1198731015840)



1198731015840 =


119895isin1198731015840119903119895119889119894119895(1198731015840) sum119895isin1198731015840 119903119895119889119894119895(119873

1015840))





120572119860=

sum119894isin119860

119886119894

sum119894isin119873

119886119894

(1)


119870 =⊘119873

2radic119901 (2)


119887119860=10038161003816100381610038161003816119861 isin 119875 | 119889

119888119860119888119861

(119873) le 119870 119860 = 11986110038161003816100381610038161003816 (3)


120573119860=119901 minus 1 minus 119887

119860

119901 minus 1 (4)


120574119860=

sum119894isin119860

119903119894

sum119894isin119873

119903119894

(5)


120575119860=

⊘119860

⊘119873

(6)





obj (119875) = 120582 sdotmax119860isin119875

119882119860 + (1 minus 120582) sdot

sum119860isin119875

119882119860

119901 (8)







min obj (119875) (9)

st 0 notin 119875 (10)

⋃

119860isin119875

119860 = 119873 (11)

119860 cap 119861 = 0 forall119860 119861 isin 119875 | 119860 = 119861 (12)

|119875| = 119901 (13)

Conn (119860) = 1 forall119860 isin 119875 (14)

Conv (119860) = 1 forall119860 isin 119875 (15)



procedure 119871119900119888119886119897119878119890119886119903119888ℎ(1198750)


119905 larr 0while not119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() do119875119905+1


if 119875119905+1


119875larr 119875119905+1



end procedure




1198891

119894119895(119860) = 119889

1





1

119894119895(119873)


1



119894119895(119860) = 119889

1

119894119895(119873)


min obj (119875) = obj (119875) + 120583sum

119860isin119875

(1 minus Conv (119860)) (17)

st (10) (11) (12) (13) and (14) (18)





119875 = 1198891 1198892 119889

|119873| (19)









procedure 119872119906119897119905119894119878119905119886119903119905()





end procedure









Begin

Terminationcriteria

No

InitialSolution

LocalSearch

Solutionimprovement

test

Yes End

SHC

SDHC

TS













following shifts








120572 119908120573 119908120574 119908120575) = (045 005 045 005) and










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120572119860=

sum119894isin119860

119886119894

sum119894isin119873

119886119894

(1)


119870 =⊘119873

2radic119901 (2)


119887119860=10038161003816100381610038161003816119861 isin 119875 | 119889

119888119860119888119861

(119873) le 119870 119860 = 11986110038161003816100381610038161003816 (3)


120573119860=119901 minus 1 minus 119887

119860

119901 minus 1 (4)


120574119860=

sum119894isin119860

119903119894

sum119894isin119873

119903119894

(5)


120575119860=

⊘119860

⊘119873

(6)





obj (119875) = 120582 sdotmax119860isin119875

119882119860 + (1 minus 120582) sdot

sum119860isin119875

119882119860

119901 (8)







min obj (119875) (9)

st 0 notin 119875 (10)

⋃

119860isin119875

119860 = 119873 (11)

119860 cap 119861 = 0 forall119860 119861 isin 119875 | 119860 = 119861 (12)

|119875| = 119901 (13)

Conn (119860) = 1 forall119860 isin 119875 (14)

Conv (119860) = 1 forall119860 isin 119875 (15)



procedure 119871119900119888119886119897119878119890119886119903119888ℎ(1198750)


119905 larr 0while not119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() do119875119905+1


if 119875119905+1


119875larr 119875119905+1



end procedure




1198891

119894119895(119860) = 119889

1





1

119894119895(119873)


1



119894119895(119860) = 119889

1

119894119895(119873)


min obj (119875) = obj (119875) + 120583sum

119860isin119875

(1 minus Conv (119860)) (17)

st (10) (11) (12) (13) and (14) (18)





119875 = 1198891 1198892 119889

|119873| (19)









procedure 119872119906119897119905119894119878119905119886119903119905()





end procedure









Begin

Terminationcriteria

No

InitialSolution

LocalSearch

Solutionimprovement

test

Yes End

SHC

SDHC

TS













following shifts








120572 119908120573 119908120574 119908120575) = (045 005 045 005) and










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










procedure 119871119900119888119886119897119878119890119886119903119888ℎ(1198750)


119905 larr 0while not119879119890119903119898119894119899119886119905119894119900119899119862119903119894119905119890119903119894119886() do119875119905+1


if 119875119905+1


119875larr 119875119905+1



end procedure




1198891

119894119895(119860) = 119889

1





1

119894119895(119873)


1



119894119895(119860) = 119889

1

119894119895(119873)


min obj (119875) = obj (119875) + 120583sum

119860isin119875

(1 minus Conv (119860)) (17)

st (10) (11) (12) (13) and (14) (18)





119875 = 1198891 1198892 119889

|119873| (19)









procedure 119872119906119897119905119894119878119905119886119903119905()





end procedure









Begin

Terminationcriteria

No

InitialSolution

LocalSearch

Solutionimprovement

test

Yes End

SHC

SDHC

TS













following shifts








120572 119908120573 119908120574 119908120575) = (045 005 045 005) and










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










procedure 119872119906119897119905119894119878119905119886119903119905()





end procedure









Begin

Terminationcriteria

No

InitialSolution

LocalSearch

Solutionimprovement

test

Yes End

SHC

SDHC

TS













following shifts








120572 119908120573 119908120574 119908120575) = (045 005 045 005) and










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















following shifts








120572 119908120573 119908120574 119908120575) = (045 005 045 005) and










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) SATT3 (b) SUNT1

(c) MONT2









(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6






(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6




6 Conclusions




(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6



Shift 119901 119865(2 147) Pr(gt119865)SATT3 2 157 0212


MONT2 6 2212 401e minus 09



(a) Shift SATT3 119901 = 2


(b) Shift SATT3 119901 = 6


(c) Shift SUNT1 119901 = 2


(d) Shift SUNT1 119901 = 6


(e) Shift MONT2 119901 = 2


(f) Shift MONT2 119901 = 6












Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a comparison of local search methods for...

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