gis-basednichehybridbatalgorithmforsolvingoptimal spatialsearch · 2020. 5. 11. ·...

Research ArticleGIS-Based Niche Hybrid Bat Algorithm for Solving OptimalSpatial Search

Guoming Du 1 Yangbo Chen 1 and Wei Sun 12

1School of Geography and Planning Sun Yat-sen University 135 West Xingang RD Guangzhou 510275 China2Southern Marine Science and Engineering Guangdong Laboratory Zhuhai 519082 China

Correspondence should be addressed to Yangbo Chen eescybmailsysueducn

Received 29 December 2019 Revised 19 March 2020 Accepted 8 April 2020 Published 11 May 2020

Academic Editor Ricardo Aguilar-Lopez

Copyright copy 2020 Guoming Du et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Complex nonlinear optimization problems are involved in optimal spatial search such as location allocation problems that occur inmultidimensional geographic space Such search problems are generally difficult to solve by using traditional methods )e bat algorithm(BA) is an effectivemethod for solving optimization problems However the solution of the standard BA is easily trapped at one of its localoptimum values )e main cause of premature convergence is the loss of diversity in the population )e niche technique is an effectivemethod tomaintain the population diversity to enhance the exploration of the new search domains and to avoid premature convergenceIn this paper a geographic information system- (GIS-) based niche hybrid bat algorithm (NHBA) is proposed for solving the optimal spatialsearch )e NHBA is able to avoid the premature convergence and obtain the global optimal values )e GIS technique provides robustsupport for processing a substantial amount of geographical data A case in Fangcun District Guangzhou City China is used to test theNHBA)e comparative experiments illustrate that the BA GA FA PSO andNHBA algorithms outperform the brute-force algorithm interms of computational efficiency and the optimal solutions are more easily obtained with NHBA than with BA GA FA and PSOMoreover the precision of NHBA is higher and the convergence of NHBA is faster than those of the other algorithms under thesame conditions

1 Introduction

)e general goal of the optimal spatial search is to find a setof contiguous places that meet specific optimization ob-jectives such as minimizing the total cost and maximizingthe proximity to certain facilities [1] Optimal spatial searchproblems often exhibit considerable complexity especiallywhen they are involved in large multiple objective locationallocation problems many of which contain combinatorialoptimization under complicated highly nonlinear con-straints and thus they are often NP-hard [2 3] )e tra-ditional solutions include the fixed charge location modelthe covering model in noncompetitive location theory andthe medianoid and centroid models from competitive lo-cation theory Because of the exponential increase in thesearch space with problem size the combinatorial optimalproblems where the reliability goal is achieved by discretechoices made from available parts encounter computationaldifficulties )erefore the methods are difficult to run in the

acceptable computational time Consequently many re-search efforts have been devoted to improving or findingefficient algorithms Metaheuristic algorithms derived fromthe behavior of biological or physical systems found innature have become powerful methods to solve the toughspatial optimization problems [4] Inspired by this idea thegenetic algorithm (GA) [5] ant colony optimization (ACO)[6] particle swarm optimization (PSO) [7] artificial fishswarm algorithm (AFSA) [8] cuckoo search (CS) [9]monkey algorithm (MA) [10] and firefly algorithm (FA)[11] have been proposed and applied widely Every algo-rithm has its own advantages and disadvantages and is alsosuited for solving different problems)e bat algorithm (BA)is also a metaheuristic algorithm inspired by a propertyknown as echolocation which is a type of sonar that guidesbats in their flying and hunting behavior [12] Not only canthe bats move but also they can distinguish different types ofinsects even in complete darkness thanks to such capabil-ities [12 13] )e BA can solve optimization problems

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2843436 11 pageshttpsdoiorg10115520202843436

completely stronger than the PSO GA and harmony searchalgorithms [14] )e main reason is that the BA uses anappropriate combination of significant advantages of thementioned algorithms

However the solution of the BA is sometimes trapped atone of its local optima because of its premature convergenceand weak exploitation capabilities )erefore it cannot al-ways obtain the ideal results when BA is used alone [15] )emain cause of premature convergence is the loss of diversityin the population )e niche techniques are regarded as aneffective method to maintain the population diversity toenhance the exploration of the new search domains [16])eniche techniques have been proven particularly useful inproblems that require multiple solutions to be found by asearch algorithm such as multimodal and multiobjectiveoptimization problems [17] As a consequence the aim ofthis study is to propose the niche hybrid bat algorithm(NHBA) to enhance the performance of BA on nonlinearoptimization problems avoid becoming trapped in a locallyoptimal value with the subsequent premature convergenceproblem of BA and solve the complex problems of optimalspatial search more efficiently precisely and reliably

)e main contributions of this study are as follows

(1) It provides a robust solution by hybridizing the nichetechniques with BA in combination with GIStechniques

(2) It contributes to maintaining the diversity of thepopulation properly improving the global searchability and overcoming the deficiency in the pre-mature convergence on only one solution

(3) It provides an effective method for solving optimalspatial search and multiobjective nonlinear optimi-zation problems

)e rest of the paper is structured as follows Section 2gives the literature review of the related works Section 3introduces the basic concepts theory and procedure of BASection 4 proposes the GIS-based niche hybrid bat algorithm(NHBA) Section 5 analyzes and discusses the experimentalresults by implementing comparative studies Section 6concludes the paper

2 Literature Review

Optimal spatial search is involved in combinatorial opti-mization problems such as location allocation )e geneticalgorithm (GA) which searches over regions was firstproposed to identify a final feasible optimal or near-op-timal solution to a relaxed version of the redundancy al-location problem [18] Because GA involves extensivecoding and decoding it will take an excessive amount oftime to compute Simulated annealing was used to solvehigh-dimensional nonlinear optimization problems formultisite land use allocation (MLUA) problems [2] Simu-lated annealing can almost guarantee finding the optimalsolution when the cooling process is slow enough and thesimulation is running long enough However the fine ad-justment of parameters affects the convergence rate of the

optimization process )e multiobjective harmony searchalgorithm was used to optimize the multiserver locationallocation problem in congested systems [19] A modifiedfruit fly optimization algorithm (MFOA) was proposed tosolve a location allocation-inventory problem in a two-echelon supply chain network [20] Ant colony optimization(ACO) with stochastic demand was used to solve the sto-chastic uncapacitated location allocation problem with anunknown number of facilities [21] A metaheuristic algo-rithm based on the harmony search algorithm was presentedfor the location allocation and routing of temporary healthcenters in rural areas in a crisis situation [22] An improvedartificial bee colony was proposed for the facility locationallocation problem of the end-of-life vehicles recoverynetwork [23] A self-learning particle swarm optimization(SLPSO) algorithm was developed to solve the multiecheloncapacitated location-allocation-inventory problem [24] It isdifficult to avoid becoming stuck at the local minimum valuewhen using the single algorithm Furthermore these algo-rithms lack the support of GIS techniques

)e BA was proposed by Yang in 2010 )ere are manyvariants of BA in the related literature A multiobjective bat-inspired algorithm was proposed for solving nonlinearglobal optimization problems [14] Simulation results sug-gested that the proposed algorithm worked efficientlyYangrsquos original work was expanded to solve complicatedconstrained nonlinear optimization problems [25] Amodified optimization methodology called opposition-based BA (OBA) was proposed for the infinite impulseresponse system identification problem [26] )e simulationresults revealed OBA to be a more competent candidate thanother evolutionary algorithms (EA) such as real-coded GAdifferential evolution (DE) and PSO in terms of accuracyand convergence speed )e discrete real BA (RBA) wasproposed for route search optimization of a graph-basedroad network [27] Compared with ACO and the intelligentwater drops (IWD) algorithm RBA achieved a betterconvergence rate A novel BA was developed in order toimplement the loading pattern optimization of a nuclearreactor core [28] )e test results showed that BA was a verypromising algorithm for LPO problems and has the potentialto be used in other nuclear engineering optimizationproblems A new modified BA was proposed to solve theoptimal management of the multiobjective reconfigurationproblem [29] )is algorithm was tested on the 32-bus IEEEradial distribution system and demonstrated its feasibilityand effectiveness )e bat-inspired algorithm was combinedwith differential evolution (BA-DE) into a new hybrid al-gorithm to solve constrained optimization problems [15]Comparisons showed that the BA-DE outperformed themost advanced methods in terms of the final solutionrsquosquality A new multiobjective optimization based on themodified BA and Pareto front was developed for solvingpassive power filter design problems [30] Local and globalsearch characteristics of BA (EBA) through three differentmethods were enhanced for solving optimization problems[31] It was proven that EBA was more effective than otheralgorithms such as the DE EA FA artificial immune systemand charged system search BA was used to solve the

2 Mathematical Problems in Engineering

reliability redundancy allocation problem (RAP) andshowed that the BA was competitive with the best knownheuristics for redundancy allocation [4] BA was used todesign and optimize the concurrent tolerance in mechanicalassemblies [32] It was found that the BA produced betterresults than other methods (such as EA and GA) in the initialgenerations of concurrent tolerance problems )e multi-directional bat algorithm (MDBAT) which hybridized thebat algorithm with the multidirectional search algorithmwas proposed for solving unconstrained global optimizationproblems [33] It could accelerate the convergence to theregion of the optimal response )e multiobjective batalgorithm was proposed for association rule mining [34] Anovel multiobjective BA was proposed for communitydetection on dynamic social networks [35] )e algorithmused the mean shift algorithm to generate the new solutionsand avoid the random process by defining a new mutationoperator A combination of bat and scaled conjugategradient algorithms is proposed to improve neural networklearning capability [36] )e bat algorithm was firstpractically implemented in swarm robotics [37] An au-tonomous binary version of the bat algorithm was used tosolve the patient bed assignment problem [38] )e swarmbat algorithm with improved search was introduced in [39])e adaptive multiswarm bat algorithm (AMBA) was su-perior to the others over 20 benchmark functions in acomparison with six algorithms [40] A hybrid bat algo-rithm with a genetic crossover operation and smart inertiaweight (SGA-BA) was used to solve multilevel imagesegmentation [41] A novel bat algorithm with doublemutation operators was applied to the low-velocity impactlocalization problem [42]

3 Standard Bat Algorithm

)e standard BA is a bioinspired algorithm based on theecholocation behavior of bats [12] Bats emit a very loudsound pulse and listen for the echo that bounces back fromthe surroundings Bats fly randomly using frequency ve-locity and position to search for prey [43 44] In the BA thefrequency velocity and position of each bat in the pop-ulation are updated for further movements)is algorithm isformulated to imitate the behavior of bats finding their preysuch that it serves to solve both single objective and mul-tiobjective optimization problems in the continuous solu-tion domain [14] )e implementation of the standard BA isas follows [12ndash14]

(1) All bats use echolocation to sense distance and theyalso ldquoknowrdquo the difference between prey and back-ground barriers in an uncanny way

(2) With a varied frequency fi varied wavelength λ andloudness A0 bats fly randomly with velocity vi atposition xi to search for prey )ey can automaticallyadjust the wavelength (or frequency) according tothe distance from the prey

(3) Although the loudness can vary in different ways it isassumed that the loudness varies from large A0 to aminimum value Amin

In the BA each bat is defined by its position xti velocity

vti frequency fi loudness At

i and the emission pulse rate rti

at time t in a D-dimensional search space )e new solutionat time t is given by

xti x

tminus1i + v

ti (1)

)e velocity is updated by

vti v

tminus1i + x

tminus1i minus x

tgbest1113872 1113873fi (2)

where xtgbest is the current global best location found by all

the bats in the past generations )e pulse frequency fi isdominated by

fi fmin + fmax minus fmin( 1113857β (3)

where S is a random number drawn from a uniform dis-tribution )e two parameters fmin and fmax are the mini-mum and maximum frequency values respectively

Whenever a solution is selected as the current optimalsolution (xgbest) in the local search areas the new solution(xi) is generated by a random walk as follows

xi xgbest + randlowastAt (4)

where rand is a random number uniformly distributedwithin [minus1 1] and controls the direction and power of therandom walk At is the average loudness of all the bats

With the ith bat approaching the prey its loudness Aiwill decrease and its rate ri of pulse emission will increaseSuch a change can be achieved by the following formulas

At+1i αA

ti (5)

rt+1i r

0i [1 minus exp(minusct)] (6)

where c and c are constants For any 0lt αlt 1 and cgt 0 wehave

At+1i ⟶ 0 r

ti⟶ r

0i as t⟶infin (7)

)e pseudocode of the original BA is given inAlgorithm 1

4 GIS-Based Niche Hybrid BatAlgorithm (NHBA)

41 Brief Description of NHBA )e standard BA has apremature convergence rate and is easily trapped in a localoptimum To overcome these problems NHBA is proposedby means of the niche technique hybridizing BA)e idea ofNHBA is as follows

First the Euclidean distance dij is calculated between thearbitrary two bats Bati and Batj among the population If theEuclidean distance dij denoted as || Bati minusBatj || is less thanthe niche distance L which is a threshold value and is givenpreviously based on experience it signifies that the bats Biand Bj have higher similarity )e fitness values of twoindividuals are compared )en the lower fitness value isgiven a penalty factor in order to decrease its fitness valueRegarding the two individuals within the niche distance Lthe individuals with lower fitness values will become worse

Mathematical Problems in Engineering 3

and be eliminated by a large probability in the next gen-erations It means that there exist only elite individualswithin the niche distance L )erefore the population di-versity is maintained Meanwhile the two different indi-viduals maintain a certain distance and all the individualsare scattered over the whole constrained space Finally theniche hybrid bat algorithm is implemented in this way

42 GIS-Based NHBA for Optimal Spatial Search )e GIS-based NHBA has the advantage of being applied to optimalspatial search with combined GIS techniques during a se-quence of iterations Optimal spatial search is involved in theresource or location allocation of facilities which includehospitals supermarkets schools cinemas and fire stationsHence optimal spatial search plays an important role ingeographic information science )is paper uses the optimallocation allocation of facilities as an example )is type ofproblem is involved in complex nonlinear optimizationscenarios especially in multiobjective location allocation Ifwe use the brute-force algorithm ie list all the combina-torial results to obtain the optimal value the computationalvolume is so enormous that it is impossible to finish thecomputing work in an acceptable time )e GIS-basedNHBA is an effective algorithm that can easily optimize thecomplex spatial search )e performance procedure is de-scribed as follows

421 Define the Objective Function )e objective functionis strongly associated with the population density trafficaccessibility and competitive factors of facilities

(1) Population Density Population density is an importantfactor affecting the spatial distribution of facilities When afacility is located in a position with a large populationdensity it will undoubtedly increase the passenger volume)e influence of the facilities on the population rapidlyattenuates with the increase in distance Suppose the central

position of the facility is P(x y) the formula of which has aneffect on a certain point Pj(xj

prime yjprime) in the 2-dimensional

geographical space is expressed by the following equation

Dj eminus k

xminus xjprime( 1113857

2+ yminus yj

prime( 11138572

1113969

(8)

where k is the decay coefficient )e greater the value k thefaster the decay is and vice versa When the facility scale islarge its influence is also large and when the value k isrelatively low it means there is a wider scope of influence Inaddition traffic accessibility has a greater impact on thevalue k When the transportation is developed the value k isrelatively lower and the influence scope of the facilities isbroader In this paper we suppose that every facility scale isfixed which means the value k is constant )e facility with agood position should try to cover the maximum populationdensity ie the formula max Dj

(2) Traffic Accessibility Traffic accessibility is another im-portant factor affecting the spatial distribution of facilitiesConvenient traffic can attract a larger passenger volume Asa consequence the population number will be reducedaccordingly with the increasing space distance Suppose thedistance d(x y) between an arbitrary point Pj(xj

prime yjprime) and the

central position of a certain facility P(x y) is described by theEuclidean distance and expressed by the following equation

d(x y)

x minus xjprime1113872 1113873

2+ y minus yj

prime1113872 11138732

1113970

(9)

Traffic accessibility means that the optimal locations ofthe facilities meet the minimum sum of the distances be-tween the population and the facilities ie the formulamin1113936dj(x y)

(3) Competitive Factor Competition exists between thedifferent facilities so that it is necessary to keep a certaindistance between the facilities If the factor of the facilityscale is not considered the proximity principle will be

Objective function F(x) x (x1x2xd)T

Initialize the bat population xi (i 12 n) and vi

Define the pulse frequency fi at xiInitialize the pulse rates ri and the loudness AiWhile (tlt Imax) (where Imax is the max number of iterations)

Generate new solutions by adjusting the frequency and updating velocities and locations using equations (1)ndash(3)If (rand1gt ri)Select a solution among the best solutionsGenerate a local solution around the selected best solution using equation (4)

End ifGenerate a new solution by flying randomlyIf (rand2ltAi and f(xi)lt f(xgbest))Accept the new solutionsIncrease ri and reduce Ai using equations (5) and (6)

End ifRank the bats and find the current best xgbest

End while

ALGORITHM 1 )e pseudocode of the original BA


followed by people Suppose the minimum distance djminbetween the n facilities P1(x1 y1) P2(x2 y2) Pn(xn yn)and the jth population cell Pj(xj

prime yjprime) exists and is given as

follows

djmin min dj xj yj1113872 1113873 min

xi minus xjprime1113872 1113873

2+ yi minus yj

prime1113872 11138732

1113970

(10)

where i 1 2 n j 1 2 h)e parameters n and h arethe number of facilities and the number of population cellsrespectively

422 Objective Function According to the above defini-tions the problem should be how to locate the facilities in acertain place when the number of facilities is constant Firstthe objective function is defined to solve the problem )einfluence of the facilities will gradually attenuate with thedistance )us the objective function based on the vectordata is calculated as follows

F C 1113936

ni1 1113936

hj1 DjAje

minus kdij

1113936hj1 DjAjdjmin

(11)

where F is the fitness value of the objective function C is aconstant Dj is the population density of the jth populationcell Aj is the area of the jth population cell wherej 12 h and k is the decay parameter )e parameter dijis equal to

(xi minus xj)

2 + (yi minus yj)2

1113969 where dij is the distance

between point (xi yi) and point (xj yj) )e point (xi yi)represents the coordinates of the ith facility and the point(xj yj) represents the coordinates of the jth population cell

)e combinatorial optimization problem can be for-mulated as computing the maximum value )erefore thepurpose of the research is to obtain the coordinates of fa-cilities Pg(xgbest ygbest) under the constraint condition of themaximization of F

423 Introduction to NHBA )e NHBA treats each solu-tion as a bat searching in D-dimensional hyperspace Dif-ferent from other optimal problems the optimal spatialsearch is in a 2-dimensional geographic space in which eachpoint includes X and Y coordinates )e concept of geo-graphic multidimensional space is different from that of D-dimensional hyperspace D is equal to 2n where n is thenumber of facilities Consequently the position vector of theith bat is Bi (xi1 yi1 xi2 yi2 xin yin) Each bat flies over thesearch space and its velocity vector isVI(vxi1 vyi1 vxi2 vyi2 vxin vyin) )e bats can adjusttheir positions and velocities according to the current op-timal value B(j)Best and global optimal value Bgbest )eupdating formula of the position is expressed by

xi(t + 1) xi(t) + vxi(t + 1)

yi(t + 1) yi(t) + vyi(t + 1)

⎧⎨

⎩ (12)


vxi(t + 1) vxi(t) + xi(t) minus xgbest(t)1113872 1113873fxi

vyi(t + 1) vyi(t) + yi(t) minus ygbest(t)1113872 1113873fyi

⎧⎪⎨

⎪⎩(13)

where t is the iteration )e parameters xgbest(t) andygbest(t) are the global optimal positions of X and Y axisdirections found by all the bats in the tth iteration )efrequency f is dominated by

fxi fxmin + fxmax minus fxmin( 1113857βxi

fyi fymin + fymax minus fymin1113872 1113873βyi

⎧⎨

⎩ (14)

where βxi and βyi are random vectors subjected to a uniformdistribution in the range of [0 1] Typically fxmin fymin 0fxmax fymax 100 and the initial frequencies of each batalong the X and Y axis directions are uniformly distributedrandom numbers in the range of [fxmin fxmax] and [fyminfymax] respectively

xinew(t + 1) xiold(t) + εxi(t) middot Axi(t)

yinew(t + 1) yiold(t) + εyi(t) middot Ayi(t)

⎧⎨

⎩ (15)

wherein εxi(t) and εyi(t) are random numbers in the range of[minus1 1] and Axi(t) and Ayi(t) are the average loudness of allthe bats along the X and Y axis directions in the tth iteration

Axi(t + 1) α middot Axi(t)

Ayi(t + 1) α middot Ayi(t)

⎧⎨

⎩ (16)

)e parameter α is the adjustment coefficient and takesthe appropriate value according to the specific problem

Axi(t) 11139362nminus1

j135

rxij(t)

n

Ayi(t) 11139362n

j246

rxij(t)

n

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

rxi(t) rxi(0) middot 1 minus eminus ct( 1113857

ryi(t) ryi(0) middot 1 minus eminus ct( 11138571113896 (18)

)e parameters rxi(0) and ryi(0) are the initial pulserates rxi(t) and ryi(t) are the tth iteration of the ith bat andc is the adjustment coefficient Because the x and y coor-dinates pair up j 1 3 5 2nminus1 and j 2 4 6 2nalong the X- and Y-axis directions respectively

424 Implementing the Procedure of NHBA for OptimalSpatial Search Based on the above concepts the imple-mentation procedure of NHBA is described as follows

(1) Define the objective function F(Bi) based onequation (11)

(2) Initialize the parameters for each bat the pop-ulation size M position x velocity v loudness Aand pulse rate r

(3) Set the arithmometer that is used to count thenumber of bats Generate M bats from the initialpopulation with random positions and velocitiesand calculate the fitness value F(Bi) of each bat of


the population based on the objective function instep 1

(4) Evaluate each bat and find the current optimal batin each population and save it as B(j)Best

(5) Begin to iterate from each bat in the population

(51) Calculate the frequency velocity and positionfor the ith bat based on the equations(12)ndash(14)

(52) Judge whether the positions xi and yi of all thebats exceed the boundaries If so let xi and yibe equal to the critical values of boundaries

(53) If randgt rxi or randgt ryi generate a new so-lution for each bat based on equation (15) Arandom walk procedure has been used togenerate a new solution for each bat to im-prove the variability of the possible solutionwhere the parameter rand is a randomnumber and rand isin [-1 1]

(54) Adjust the loudness A and pulse rates r of allthe bats in the tth iteration At each iterationof the algorithm the loudness Ax(t) andAy(t) and the pulse rates rxi(t) and ryi(t) areupdated based on the equations (16)ndash(18) ifthe new solutions are improved It means thatthese bats are moving toward the optimalsolution

(55) A solution is accepted if a number drawn atrandom is less than the loudness Ai and meetsthe relation F(Bi) gtBgbest where Bgbest is theglobal optimum value

(56) Update the frequency velocity positionloudness pulse rate and fitness values of eachbat because the values of the parameterschange

(6) Calculate the Euclidean distance dij between twoarbitrary bats Bi and Bj in the population where dij

Bi minus Bj 1113936

nk1[(xik minus xjk)2 + (yik minus yjk)2]

1113969

(7) Implement the elimination algorithm of theniche When the relation dij lt L is met comparethe fitness values of two bats Bi and Bj L is theniche distance It is an empirical value and isgiven based on experience in the following ex-periment (see Tables 1 and 2) Impose thestronger penalty factor on the bat with the lowerfitness value ie min(F(Bi) F(Bj)) = Penalty sothat the diversity of the population is maintained)e Penalty is a very small number For examplewe let Penalty = 10minus6 By doing so it can help thepopulation improve its average fitness value andaccelerate the speed of obtaining the optimalvalue

(8) )e new population is sorted in descending orderby the new fitness values )e bats of which thefitness values are less than the elimination factorelm are divided into the population V )e othersare divided into the population W and memorized

)e bats of population V will be eliminated andreinitialized

(9) Judge whether it meets the termination condition Ifit does not meet the termination condition updatethe arithmometer t= t+ 1 )e former W individ-uals produced in step 8 are treated as the newpopulation of the next iteration )en go to step 5If the termination condition is met ie |Ft+1minusFt|lt ε(a critical value) or the maximum iterations areexceeded end the algorithm

(10) Obtain the best solution including the best posi-tions and best fitness values

(11) Memorize and output the results

5 Experiment and Discussion

51 Data andMethods )e spatial search needs to combinethe location data with the attribute data Regarding theproblems of locating the facilities in a certain place GIS canprovide robust support Some important information whichincludes the centroid coordinates (xi yi) and area of eachpopulation cell and the width and height of the search spaceneeds to be extracted from the map of the population densitydistribution by means of GIS Python a common languagetool is used in this research We can import and manipulatethe input data and thus present the generated map

)e digitalized population density map is a polygonvector map of Fangcun District Guangzhou City ChinaEach polygon represents a population cell which is theminimal unit of the geographic area )e fields should in-clude OID and Population in the attribute database of GIS)e field OID represents the object identity code which isunique )e field Population represents the populationdensity of each population cell which can be extracted from

Table 1 List of parameters when n 1

Parameters Meaning Selected valuesM )e number of bats 80C Constant 100000K Decay coefficient 00004c1 c2 Constant 2E min Minimum error 10ndash6

α )e adjustment coefficient 090c Constant cgt 0 096L )e niche distance 500

Table 2 List of parameters when ngt 1

Parameters Meaning ValuesM )e number of bats 108C Constant 100000K Decay coefficient 00004Emin Minimum error 10ndash6

α )e adjustment coefficient 086c Constant cgt0 095L )e niche distance 2000


the attribute database of GIS Compared with the raster datathe precision of the vector data we use in the paper is high

)e map of the population density in Fangcun DistrictGuangzhou City China is shown in Figure 1

52 Case for Single Facility (n 1) )is is the simplest casewhen n 1 because there is no competition from otherfacilities )e parameters are listed in Table 1

)e optimal value is Fmax 220968 after 6 iterationsbased on NHBA )e results are shown in Figure 2

We compare NHBA with BA GA FA PSO and thebrute-force algorithm under the same parameters (see Ta-ble 1) by our programming in C language )e comparisonresults are shown in Tables 3 and 4

)e optimal value of BA is Fmax 220967 after 7 iter-ations NHBA has fewer iterations and a higher precisionthan BA under the same parameters GA achieves a similaroptimal value with the same iterations compared with BA)e optimal value of PSO is Fmax 220968 after 24 itera-tions )e results are similar among PSO BA and NHBAHowever PSO needs more iterations FA obtains a smalleroptimal value with more iterations It means that FA is theworst algorithm and NHBA is the best algorithm among thefive algorithms

)e brute-force algorithm is used in order to validate theresults )e time complexities of the algorithm and theaccuracies of the solutions are closely related to the spatialsampling interval )e results of the fitness values timecomplexities and x and y coordinates are shown in Table 4

)e smaller the spatial sampling interval is the greaterthe time complexities are and the higher the accuracies ofthe solutions are When the spatial sampling interval is 1times 1meters the fitness value is 22096775 which is roughlysimilar to most algorithms However the time complexity ofthe brute-force algorithm is O (86times107) BA needs 7 it-erations and the time complexity is O (122times106) NHBAonly needs 6 iterations and the time complexity is O(105times106) Consequently NHBA achieves a faster con-vergence speed

)e 3D perspective map regarding the fitness values andgeographical positions in Fangcun District is shown inFigure 3

)e map is obtained by means of developing techniquesbased on Python and R languages First evenly generate100lowast100 original sample points in the 2D space within theFangcun District boundary)en calculate the fitness valuesas zi coordinates corresponding to every point (xi yi) withinthe boundary Finally create the 3D perspective map basedon every point (xi yi zi) in R language

In Figure 3 the x and y coordinates represent the po-sition of a facility the z coordinate represents its fitnessvalue We can roughly judge the optimal position in which afacility is located based on Figure 3

53 Case for Two and More Facilities (ngt 1) When thenumber of facilities n is more than 1 it is considered acombinatorial optimization problem which is computa-tionally difficult For example if the brute-force algorithm is

GF FacilityPopulation density

0 ndash 2186421865 ndash 5161751618 ndash 89388


GF

0 1600 3200800 meters

Figure 2 Map of the population density and facility distribution(n 1) based on NHBA in Fangcun District

Population density0 ndash 2186421865 ndash 5161751618 ndash 89388


0 1600 3200800 meters

Figure 1 Map of population density in Fangcun District


used the time complexity is O (765times1058) when the spatialsampling interval is 1meter and n 8 Because the com-puting time is unacceptable the brute-force algorithm has tobe abandoned

)e parameters are listed in Table 2 )e optimal valueFmax 43694 after 13 iterations (GENNHBA 13) based onNHBA )e results are shown in Figure 4

In general the parameterm namely the population sizeor number of bats is given based on the researcherrsquos ex-perience In this paper the parameter m is obtained byconducting experiments

In Figure 5 the larger the parameter m is the larger thefitness value is (that means that the results are better) whenm is lower than 108 However the fitness value becomeslower when m is equal to 110 In addition the computationtime will increase with the increasing m value As a result

the parameterm is 108 when the number of facilities is 8)eadjustment coefficients are α 086 and c 095 in thisexperiment)e comparative results among the BA GA FAPSO and NHBA algorithms based on the same parametersare as follows

)e fitness values are calculated by the objective functionF(Bi) based on equation (1) )e larger the fitness values are

428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m

Figure 5)e relationship between the parameterm and the fitnessvalues

Table 3 )e comparison results of NHBA with BA GA FA andPSO

Algorithms Optimal values Iterations Time complexitiesBA 220967 7 122times106

GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106

Table 4 Results of the brute-force algorithm based on the spatialsampling interval

Spatial samplinginterval

Fitnessvalues X Y Time

complexities100times100 21188921 345795 249204 871210times10 22095361 340795 250204 8636251times1 22096775 340665 250384 86times107

x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20

Figure 3 )e 3D perspective map of the fitness values and geo-graphical positions in Fangcun District

0 1600 3200800 meters

FacilityPopulation density



Figure 4 Map of the population density and facilities distribution(n 8) based on NHBA in Fangcun District


the better the results are In Figure 6 the fitness values of BAGA FA PSO and NHBA increase with the iterations )efitness values of NHBA are larger than those of the otheralgorithms under the same iterations )e fitness values ofNHBA tend to be stable under 13 iterations(GENNHBA 13) while BA needs 15 iterations GA needs 17iterations FA needs 68 iterations and PSO needs 54 iter-ations )e maximum fitness values of BA GA FA PSOand NHBA are 41342 32480 36332 43085 and 43694respectively It is easier for NHBA to obtain the optimalvalues than it is for the other algorithms )is finding meansthat the NHBA approach has the ability to converge tooptimal solutions more quickly than other algorithms

)e time complexities are defined as O (GENlowastmlowast2nlowasth)Here GEN is the number of iterations m is the populationsize (m 108) n is the number of facilities (n 8) and h isthe number of population cells (h 1093) Consequently thetime complexities of BA GA FA PSO and NHBA areapproximately O (283times107) O (321times 107) O (128times108)O (102times108) and O (246times107) in this experiment re-spectively (Table 5) As mentioned above the time com-plexity of the brute-force algorithm is O (765times1058) underthe same condition In conclusion the convergence ofNHBA is faster than that of the other algorithms Fur-thermore the computation time of BA GA FA PSO orNHBA is significantly less than that of the brute-forcealgorithm

54 Experiment Based on the Expanded Shaffer Function)e NHBA has random elements and the statistics of onlyone experimental result are not convincing )e expandedShaffer function (equations (19) and (20)) is used to test theabove algorithms

g(x y) 05 +sin2

x2 + y2

1113968minus 05

10 + 0001 x2 + y2( 11138571113858 11138592 st minus100ltx ylt 100

(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)

When x(x1 x2 xD) locates in the position (0 0 0)the minimum value of the expanded Shaffer function is equalto 0 After every algorithm is run 50 times in D-dimensionalspace (D 50) the statistical means the best optimal valuesthe worst optimal values and the variances of experimentalresults are obtained and listed in Table 6

In Table 6 from the means we can see that the mean ofNHBA is 1314 which is the smallest )e second smallestmean is achieved by FA (1528) and the others in ascendingorder are as follows PSO (1759) BA (2147) and GA(2378) )e order of the best optimal values is the same asthat of the means However the smallest value of the worstoptimal values is obtained by FA instead of NHBA )esmallest variance is obtained by PSO (130) It means that thenumerical change range of PSO is the smallest Because themeans can better reflect the overall level it means thatNHBA surpasses other methods and GA is worse than otheralgorithms

6 Conclusions

Complicated combinatorial optimization problems are in-volved in the optimal spatial search and are difficult to besolved by traditional algorithms such as the brute-forcealgorithm In this paper a GIS-based NHBA is proposed forsolving the optimal spatial search It is characterized asfollows

(1) )e GIS-based NHBA is able to maintain the pop-ulation diversity enhance the exploration of the newsearch domain and overcome the deficiency that

Table 5 Comparison results of BA GA FA PSO and NHBA

Methods Fitness values Iterations Time complexitiesBA 41342 15 283times107

GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA

Figure 6 )e comparative results among the BA GA FA PSOand NHBA algorithms with the fitness values and iterations

Table 6 Comparison results of NHBA and other algorithms basedon the expanded Shaffer function after 50 runs (D 50)

Methods Means Best optimalvalues

Worst optimalvalues Variances

BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278


occurs in the premature convergence to only onesolution of optimal spatial search

(2) )e spatial search occurs in multidimensional geo-graphical space Optimal spatial search involvesmore complicated optimization problems )e GIS-based NHBA provides robust support for this work

(3) )e parameter population size m is traditionallyobtained by the researcherrsquos experience It is derivedby the experiments conducted in this paper in orderto improve the precision of the results

(4) )e BA GA FA PSO and NHBA outperform thebrute-force algorithm in terms of computationalefficiency

(5) )e optimal value is easier to obtain by employingNHBA than it is by using BA GA FA and PSOFurthermore the precision of NHBA is higher andthe convergence of NHBA is faster than that of BAGA FA and PSO under the same conditions

Data Availability

)e data used to support the findings of this study weresupplied under license and so cannot be made freelyavailable Requests for access to these data should be made toGuoming Du via eesdgmmailsysueducn

Conflicts of Interest

)e authors declare that there are no conflicts of interest

Acknowledgments

)is work was supported by the National Program on KeyResearch Projects of China (no 2017YFC1502706) It wasalso partially and financially supported by the SouthernMarine Science and Engineering Guangdong Laboratory(Zhuhai) (no 99147-42080011) and the Hundred TalentsProgram of Sun Yat-Sen University (37000-18841201)

References

[1] T A Arentze AW J Borgers and H J P Timmermans ldquoAnefficient search strategy for site-selection decisions in anexpert systemrdquo Geographical Analysis vol 28 no 2pp 126ndash146 1996

[2] J C J H Aerts and G B M Heuvelink ldquoUsing simulatedannealing for resource allocationrdquo International Journal ofGeographical Information Science vol 16 no 6 pp 571ndash5872002

[3] X S Yang and A Hossein Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[4] T P Talafuse and E A Pohl ldquoA bat algorithm for the re-dundancy allocation problemrdquo Engineering Optimizationvol 48 no 5 pp 900ndash910 2016

[5] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Ardor Michigan 1975

[6] A Colorni M Dorigo V Maniezzo and M Trubian ldquoAntsystem for job-shop schedulingrdquo Belgian Journal of

Operations Research Statistics and Computer Science vol 34no 1 pp 39ndash54 1994

[7] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of ICNNrsquo95-International Conference on NeuralNetworks Perth Australia December 1995

[8] X L Li Z J Shao and J X Qian ldquosrdquo System EngineeringIeory and Practice vol 22 no 11 pp 32ndash38 2002 inChinese

[9] X S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings Of the 2009 World Congress on Nature amp Bio-logically Inspired Computing (NaBIC) pp 210ndash214 IEEECoimbatore India December 2009

[10] R Q Zhao and W S Tang ldquoMonkey algorithm for globalnumerical optimizationrdquo Journal of Uncertain Systems vol 2no 3 pp 165ndash176 2008

[11] X S Yang Nature-inspired Metaheuristic Algorithmspp 83ndash96 Luniver Press London UK 2008

[12] X S Yang ldquoA new metaheuristic bat-inspired algorithmrdquoNature Inspired Cooperative Strategies for Optimization(NICSO 2010) Vol 284 of Studies in Computational Intel-ligence pp 65ndash74 Springer Berlin Heidelberg 2010

[13] M Zineddine ldquoOptimizing security and quality of service in areal-time operating system using multi-objective Bat algo-rithmrdquo Future Generation Computer Systems vol 87pp 102ndash114 2018

[14] X S Yang ldquoBat algorithm for multi-objective optimisationrdquoInternational Journal of Bio-Inspired Computation vol 3no 5 pp 267ndash274 2011

[15] S Y Pei A Ouyang and L Tong ldquoA hybrid algorithm basedon bat-inspired algorithm and differential evolution forconstrained optimization problemsrdquo International Journal ofPattern Recognition and Artificial Intelligence vol 29 no 4Article ID 1559007 2015

[16] L Wei and M Zhao ldquoA niche hybrid genetic algorithm forglobal optimization of continuous multimodal functionsrdquoApplied Mathematics and Computation vol 160 no 3pp 649ndash661 2005

[17] X M Zhang and L R Wang ldquoTwo-parameter inversion offluid-saturated porous medium with niche ant colony algo-rithmrdquo Mathematical Problems in Engineering vol 2014Article ID 164932 6 pages 2014

[18] DW Coit and A E Smith ldquoReliability optimization of series-parallel systems using a genetic algorithmrdquo IEEE Transactionson Reliability vol 45 no 2 pp 254ndash260 1996

[19] V Hajipour S H A Rahmati S H R Pasandideh andS T A Niaki ldquoA multi-objective harmony search algorithmto optimize multi-server location-allocation problem incongested systemsrdquo Computers amp Industrial Engineeringvol 72 pp 187ndash197 2014

[20] S M Mousavi N Alikar S T A Niaki and A BahreininejadldquoOptimizing a location allocation-inventory problem in atwo-echelon supply chain network a modified fruit fly op-timization algorithmrdquo Computers amp Industrial Engineeringvol 87 pp 543ndash560 2015

[21] J-P Arnaout G Arnaout and J E Khoury ldquoSimulation andoptimization of ant colony optimization algorithm for thestochastic uncapacitated location-allocation problemrdquo Jour-nal of Industrial andManagement Optimization vol 12 no 4pp 1215ndash1225 2016

[22] M Alinaghian and A Goli ldquoLocation allocation and routingof temporary health centers in rural areas in crisis solved byimproved harmony search algorithmrdquo International Journalof Computational Intelligence Systems vol 10 no 1pp 894ndash913 2017


[23] Y Lin H Jia Y Yang G Tian F Tao and L Ling ldquoAnimproved artificial bee colony for facility location allocationproblem of end-of-life vehicles recovery networkrdquo Journal ofCleaner Production vol 205 pp 134ndash144 2018

[24] E B Tirkolaee J Mahmoodkhani M R Bourani andR Tavakkoli-Moghaddam ldquoA self-learning particle swarmoptimization for robust multi-echelon capacitated location-allocation-inventory problemrdquo Journal of AdvancedManufacturing Systems vol 18 no 04 pp 677ndash694 2019

[25] A H Gandomi X S Yang A H Alavi and S Talatahari BatAlgorithm for Constrained Optimization Tasks Springer-Verlag London UK 2012

[26] S K Saha R Kar D Mandal S P Ghoshal andV Mukherjee ldquoA new design method using opposition-basedbat algorithm for IIR system identification problemrdquo Inter-national Journal of Bio-Inspired Computation vol 5 no 2pp 99ndash132 2013

[27] C Sur and A Shukla ldquoAdaptive amp discrete real bat algorithmsfor route search optimization of graph based road networkrdquoin Proceedings of the 2013 International Conference on Ma-chine Intelligence and Research Advancement (ICMIRA 2013)pp 120ndash124 IEEE Katra India December 2013

[28] S Kashi A Minuchehr N Poursalehi and A Zolfaghari ldquoBatalgorithm for the fuel arrangement optimization of reactorcorerdquo Annals of Nuclear Energy vol 64 pp 144ndash151 2014

[29] M Golmaryami S Saleh A B Ardekani and F Kavousi-Fard ldquoA new modified bat algorithm to solve optimalmanagement of multi-objective reconfiguration problemrdquoJournal of Intelligent amp Fuzzy Systems vol 27 no 3pp 1567ndash1573 2014

[30] N-C Yang and M-D Le ldquoOptimal design of passive powerfilters based on multi-objective bat algorithm and paretofrontrdquo Applied Soft Computing vol 35 pp 257ndash266 2015

[31] S Yilmaz and E U Kucuksille ldquoA newmodification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015

[32] L R Kumar K P Padmanaban S G Kumar andC Balamurugan ldquoDesign and optimization of concurrenttolerance in mechanical assemblies using bat algorithmrdquoJournal of Mechanical Science and Technology vol 30 no 6pp 2601ndash2614 2016

[33] M A Tawhid and A F Ali ldquoMulti-directional bat algorithmfor solving unconstrained optimization problemsrdquo Opsearchvol 54 no 4 pp 684ndash705 2017

[34] K E Heraguemi N Kamel andH Drias ldquoMulti-objective batalgorithm for mining numerical association rulesrdquo Interna-tional Journal of Bio-Inspired Computation vol 11 no 4pp 239ndash248 2018

[35] I Messaoudi and N Kamel ldquoA multi-objective bat algorithmfor community detection on dynamic social networksrdquo Ap-plied Intelligence vol 49 no 6 pp 2119ndash2136 2019

[36] P M R Bento J A N Pombo M R A Calado andS J P S Mariano ldquoOptimization of neural network withwavelet transform and improved data selection using batalgorithm for short-term load forecastingrdquo Neurocomputingvol 358 pp 53ndash71 2019

[37] P Suarez A Iglesias and A Galvez ldquoMake robots be batsspecializing robotic swarms to the Bat algorithmrdquo Swarm andEvolutionary Computation vol 44 pp 113ndash129 2019

[38] C Taramasco R Olivares R Munoz R Soto M Villar andV H C Albuquerque ldquo)e patient bed assignment problemsolved by autonomous bat algorithmrdquo Applied Soft Com-puting vol 81 2019

[39] R Chaudhary and H Banati ldquoSwarm bat algorithm withimproved search (SBAIS)rdquo Soft Computing vol 23 no 22pp 11461ndash11491 2019

[40] R Chaudhary and H Banati ldquoAdaptive multi-swarm batalgorithm (AMBA)rdquo in Advances in Intelligent Systems andComputing K Das J Bansal K Deep A NagarP Pathipooranam and R Naidu Eds vol 1048 pp 805ndash8212020

[41] X Yue and H Zhang ldquoModified hybrid bat algorithm withgenetic crossover operation and smart inertia weight formultilevel image segmentationrdquo Applied Soft Computingvol 90 2020

[42] Q Liu J Li L Wu F Wang and W Xiao ldquoA novel batalgorithm with double mutation operators and its applicationto low-velocity impact localization problemrdquo EngineeringApplications of Artificial Intelligence vol 90 2020

[43] J D Altringham Bats Biology and Behavior Oxford Uni-versity Press Oxford UK 1996

[44] P Richardson Bats Natural History Museum London UK2008


completely stronger than the PSO GA and harmony searchalgorithms [14] )e main reason is that the BA uses anappropriate combination of significant advantages of thementioned algorithms

However the solution of the BA is sometimes trapped atone of its local optima because of its premature convergenceand weak exploitation capabilities )erefore it cannot al-ways obtain the ideal results when BA is used alone [15] )emain cause of premature convergence is the loss of diversityin the population )e niche techniques are regarded as aneffective method to maintain the population diversity toenhance the exploration of the new search domains [16])eniche techniques have been proven particularly useful inproblems that require multiple solutions to be found by asearch algorithm such as multimodal and multiobjectiveoptimization problems [17] As a consequence the aim ofthis study is to propose the niche hybrid bat algorithm(NHBA) to enhance the performance of BA on nonlinearoptimization problems avoid becoming trapped in a locallyoptimal value with the subsequent premature convergenceproblem of BA and solve the complex problems of optimalspatial search more efficiently precisely and reliably

)e main contributions of this study are as follows

(1) It provides a robust solution by hybridizing the nichetechniques with BA in combination with GIStechniques

(2) It contributes to maintaining the diversity of thepopulation properly improving the global searchability and overcoming the deficiency in the pre-mature convergence on only one solution

(3) It provides an effective method for solving optimalspatial search and multiobjective nonlinear optimi-zation problems

)e rest of the paper is structured as follows Section 2gives the literature review of the related works Section 3introduces the basic concepts theory and procedure of BASection 4 proposes the GIS-based niche hybrid bat algorithm(NHBA) Section 5 analyzes and discusses the experimentalresults by implementing comparative studies Section 6concludes the paper

2 Literature Review

Optimal spatial search is involved in combinatorial opti-mization problems such as location allocation )e geneticalgorithm (GA) which searches over regions was firstproposed to identify a final feasible optimal or near-op-timal solution to a relaxed version of the redundancy al-location problem [18] Because GA involves extensivecoding and decoding it will take an excessive amount oftime to compute Simulated annealing was used to solvehigh-dimensional nonlinear optimization problems formultisite land use allocation (MLUA) problems [2] Simu-lated annealing can almost guarantee finding the optimalsolution when the cooling process is slow enough and thesimulation is running long enough However the fine ad-justment of parameters affects the convergence rate of the

optimization process )e multiobjective harmony searchalgorithm was used to optimize the multiserver locationallocation problem in congested systems [19] A modifiedfruit fly optimization algorithm (MFOA) was proposed tosolve a location allocation-inventory problem in a two-echelon supply chain network [20] Ant colony optimization(ACO) with stochastic demand was used to solve the sto-chastic uncapacitated location allocation problem with anunknown number of facilities [21] A metaheuristic algo-rithm based on the harmony search algorithm was presentedfor the location allocation and routing of temporary healthcenters in rural areas in a crisis situation [22] An improvedartificial bee colony was proposed for the facility locationallocation problem of the end-of-life vehicles recoverynetwork [23] A self-learning particle swarm optimization(SLPSO) algorithm was developed to solve the multiecheloncapacitated location-allocation-inventory problem [24] It isdifficult to avoid becoming stuck at the local minimum valuewhen using the single algorithm Furthermore these algo-rithms lack the support of GIS techniques

)e BA was proposed by Yang in 2010 )ere are manyvariants of BA in the related literature A multiobjective bat-inspired algorithm was proposed for solving nonlinearglobal optimization problems [14] Simulation results sug-gested that the proposed algorithm worked efficientlyYangrsquos original work was expanded to solve complicatedconstrained nonlinear optimization problems [25] Amodified optimization methodology called opposition-based BA (OBA) was proposed for the infinite impulseresponse system identification problem [26] )e simulationresults revealed OBA to be a more competent candidate thanother evolutionary algorithms (EA) such as real-coded GAdifferential evolution (DE) and PSO in terms of accuracyand convergence speed )e discrete real BA (RBA) wasproposed for route search optimization of a graph-basedroad network [27] Compared with ACO and the intelligentwater drops (IWD) algorithm RBA achieved a betterconvergence rate A novel BA was developed in order toimplement the loading pattern optimization of a nuclearreactor core [28] )e test results showed that BA was a verypromising algorithm for LPO problems and has the potentialto be used in other nuclear engineering optimizationproblems A new modified BA was proposed to solve theoptimal management of the multiobjective reconfigurationproblem [29] )is algorithm was tested on the 32-bus IEEEradial distribution system and demonstrated its feasibilityand effectiveness )e bat-inspired algorithm was combinedwith differential evolution (BA-DE) into a new hybrid al-gorithm to solve constrained optimization problems [15]Comparisons showed that the BA-DE outperformed themost advanced methods in terms of the final solutionrsquosquality A new multiobjective optimization based on themodified BA and Pareto front was developed for solvingpassive power filter design problems [30] Local and globalsearch characteristics of BA (EBA) through three differentmethods were enhanced for solving optimization problems[31] It was proven that EBA was more effective than otheralgorithms such as the DE EA FA artificial immune systemand charged system search BA was used to solve the












xti x

tminus1i + v

ti (1)


vti v

tminus1i + x

tminus1i minus x

tgbest1113872 1113873fi (2)









At+1i αA

ti (5)

rt+1i r



At+1i ⟶ 0 r

ti⟶ r

0i as t⟶infin (7)













Dj eminus k


2+ yminus yj

prime( 11138572

1113969

(8)





d(x y)


2+ y minus yj

prime1113872 11138732

1113970

(9)









End while





follows



2+ yi minus yj

prime1113872 11138732

1113970

(10)



F C 1113936

ni1 1113936

hj1 DjAje

minus kdij


(11)


(xi minus xj)

2 + (yi minus yj)2





xi(t + 1) xi(t) + vxi(t + 1)

yi(t + 1) yi(t) + vyi(t + 1)

⎧⎨

⎩ (12)




⎧⎪⎨

⎪⎩(13)




⎧⎨

⎩ (14)




⎧⎨

⎩ (15)




⎧⎨

⎩ (16)



j135

rxij(t)

n

Ayi(t) 11139362n

j246

rxij(t)

n

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



















Bi minus Bj 1113936


1113969
































GF

0 1600 3200800 meters




0 1600 3200800 meters









428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References


























































xti x

tminus1i + v

ti (1)


vti v

tminus1i + x

tminus1i minus x

tgbest1113872 1113873fi (2)









At+1i αA

ti (5)

rt+1i r



At+1i ⟶ 0 r

ti⟶ r

0i as t⟶infin (7)













Dj eminus k


2+ yminus yj

prime( 11138572

1113969

(8)





d(x y)


2+ y minus yj

prime1113872 11138732

1113970

(9)









End while





follows



2+ yi minus yj

prime1113872 11138732

1113970

(10)



F C 1113936

ni1 1113936

hj1 DjAje

minus kdij


(11)


(xi minus xj)

2 + (yi minus yj)2





xi(t + 1) xi(t) + vxi(t + 1)

yi(t + 1) yi(t) + vyi(t + 1)

⎧⎨

⎩ (12)




⎧⎪⎨

⎪⎩(13)




⎧⎨

⎩ (14)




⎧⎨

⎩ (15)




⎧⎨

⎩ (16)



j135

rxij(t)

n

Ayi(t) 11139362n

j246

rxij(t)

n

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



















Bi minus Bj 1113936


1113969
































GF

0 1600 3200800 meters




0 1600 3200800 meters









428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References























































Dj eminus k


2+ yminus yj

prime( 11138572

1113969

(8)





d(x y)


2+ y minus yj

prime1113872 11138732

1113970

(9)









End while





follows



2+ yi minus yj

prime1113872 11138732

1113970

(10)



F C 1113936

ni1 1113936

hj1 DjAje

minus kdij


(11)


(xi minus xj)

2 + (yi minus yj)2





xi(t + 1) xi(t) + vxi(t + 1)

yi(t + 1) yi(t) + vyi(t + 1)

⎧⎨

⎩ (12)




⎧⎪⎨

⎪⎩(13)




⎧⎨

⎩ (14)




⎧⎨

⎩ (15)




⎧⎨

⎩ (16)



j135

rxij(t)

n

Ayi(t) 11139362n

j246

rxij(t)

n

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



















Bi minus Bj 1113936


1113969
































GF

0 1600 3200800 meters




0 1600 3200800 meters









428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References


















































follows



2+ yi minus yj

prime1113872 11138732

1113970

(10)



F C 1113936

ni1 1113936

hj1 DjAje

minus kdij


(11)


(xi minus xj)

2 + (yi minus yj)2





xi(t + 1) xi(t) + vxi(t + 1)

yi(t + 1) yi(t) + vyi(t + 1)

⎧⎨

⎩ (12)




⎧⎪⎨

⎪⎩(13)




⎧⎨

⎩ (14)




⎧⎨

⎩ (15)




⎧⎨

⎩ (16)



j135

rxij(t)

n

Ayi(t) 11139362n

j246

rxij(t)

n

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



















Bi minus Bj 1113936


1113969
































GF

0 1600 3200800 meters




0 1600 3200800 meters









428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References


























































Bi minus Bj 1113936


1113969
































GF

0 1600 3200800 meters




0 1600 3200800 meters









428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References































































GF

0 1600 3200800 meters




0 1600 3200800 meters









428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References






















































428

430

432

434

436

Fitn

ess v

alue

60 70 80 90 100 11050m




GA 220965 7 122times106

FA 210478 95 166times107

PSO 220968 24 420times106

NHBA 220968 6 105times106





x

3000032000

3400036000

38000

y

20000

22000

24000

26000

28000Fitness

05

1015

20


0 1600 3200800 meters









g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References



















































g(x y) 05 +sin2

x2 + y2

1113968minus 05


(19)

f(x) g x1 x2( 1113857 + g x2 x3( 1113857 + g x3 x4( 1113857 +

+ g xDminus1 xD( 1113857 + g xD x1( 1113857(20)



6 Conclusions





GA 32480 17 321times 107

FA 36332 68 128times108

PSO 43085 54 102times108

NHBA 43694 13 246times107

0 10 20 30 40 50 60 70

400

350

300

250

200

Iteration

Fitn

ess v

alue

BAGAFA

PSONHBA





BA 2147 1958 2279 46091GA 2378 2322 2438 56549FA 1528 1325 1698 23350PSO 1759 1460 2006 130NHBA 1314 920 1803 17278







Data Availability




Acknowledgments


References





















































Data Availability




Acknowledgments


References
















































gis-basednichehybridbatalgorithmforsolvingoptimal spatialsearch · 2020. 5. 11. ·...

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