research article an improved ant colony algorithm and its...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 785383 9 pageshttpdxdoiorg1011552013785383

Research ArticleAn Improved Ant Colony Algorithm and Its Application inVehicle Routing Problem

Min Huang and Ping Ding

School of Software Engineering South China University of Technology Guangzhou Guangdong 510006 China

Correspondence should be addressed to Min Huang minhscuteducn

Received 6 August 2013 Revised 16 October 2013 Accepted 18 October 2013

Academic Editor Ali Davoudi

Copyright copy 2013 M Huang and P Ding This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Optimal path planning is an important issue in vehicle routing problem This paper proposes a new vehicle routing path planningmethod which adds path weight matrix and save matrix The method uses a new transition probability function adding the anglefactor function and visibility function while setting penalty function in a new pheromone updatingmodel to improve the accuracyof the route searching Finally after each cycle we use 3-opt method to update the optimal solution to optimize the path lengthThe results of comparison also confirm that this method is better than the traditional ant colony algorithm for vehicle routing pathplanning methodThe result of computer simulation confirms that the method can plan a more rational rescue path focused on thereal traffic situation

1 Introduction

Vehicle routing problem (VRP) is one of combinatorialoptimization problems and VRP can be turned into otherapplication problems for example Chen and Wang haveturned VRP into tour planning problem [1] A typical VRPcan be described as follows a warehouse that offers servicesto different positions of customers at the lowest cost pathplanning It has real economic significance in many fieldssuch as transportation scheduling routing railway trans-portation and other practical problems Currently heuristicalgorithm is the main method for solving vehicle routingproblem Heuristic algorithm can be divided into simpleheuristic algorithm two-phase heuristic algorithm and arti-ficial intelligence methods

Vehicle routing problem has successfully applied in manyareas such as Li et al [2] and Malekly et al [3] They solvethe uncertain and ambiguous problem in vehicle routingproblem using fuzzy set theories Toth and Vigo [4] con-sidered VRP as a significant part in logistic handing andgrouped the methods that have been found to solve theproblem of VRP Kim et al [5] divided the waste collectionbusiness into different areas and proposed methods to solvethese problems in VRP Many researchers have proposed

heuristics or metaheuristics algorithm for efficiently solvingthe vehicle routing problem Semet and Taillard [6] designedthe taboo algorithm considering time window and differenttypes of vehicles which mainly used methods to generatethe initial solution then optimize the initial solution withtaboo search algorithm It is a local neighborhood searchfor an extension Cordeau and Laporte [7] studied simulatedannealing algorithm for the VRP they proposed a simulatedannealing method which is suitable for solving the vehiclerouting problem and shows the advantage of accuracy andthe speed of search convergence Genetic algorithm is goodfor solving combinatorial optimization problems Niazy andBadr [8] and Zulvia et al [9] use genetic algorithm (GA)encoding to solve VRP problem Niazy and Badr solved thecomplexity of CVRP with the goal of minimizing the totaldistances using the cellular genetic algorithm (CGA) Mul-tiple ant colony algorithms are proposed by Gambardella etal [10] and Bell and McMullen [11] to solve the time windowof the multiobjective VRP problem Ant colony optimization[12ndash14] (ACO) was successfully applied to emergency rescue[15 16] which is an ideal heuristic algorithm and capable ofintelligent search and global optimization

In this paper we apply ant colony optimization algorithmto the path planning of emergency rescue vehicle This paper

2 Mathematical Problems in Engineering

Table 1 Distance map and delivery of CVRP example

Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119909 0 3 minus2 minus2 0 minus3 3 minus4 3 minus3 2 1 minus1 minus2 2 4119910 0 2 1 minus2 3 minus3 minus1 minus1 4 0 1 3 minus1 3 minus2 3Delivery 0 18 08 15 25 02 24 05 06 09 13 2 06 02 14 09

will focus on the shortcomings of ant colony algorithm Ourproposed algorithm adds the angle factor pathweightmatrixand save matrix in the transfer rules and optimizes visibilityfunction and penalty function thus it avoids falling into localoptimum and speeds up the convergence of the algorithm Atthe same time pheromone update rules with local update andglobal update update the global optimal solution only usethe best ant to improve the algorithmrsquos global search Finallyafter each cycle update the optimal solution in accordancewith 3-opt method [17 18] to shorten the length of the rescuepath Use benchmark test [19] to compare the results with theliterature [20] in the algorithm to verify the effectiveness ofthe algorithm

2 Problem Description andMathematical Model

Disasters such as earthquakes and floods are sudden andunpredictable which cause serious sufferings for peopleHowto find an effective and short rescue path for the affectednodes in the limited time after the disasters become thefocus thing Before making vehicle routing problem pathplanning we first establish reasonable free space modelCapacitated vehicle routing problem [21 22] (CVRP) wasdefined as a transportation and a distribution problem withthe constrained of vehicle load travel time or distanceResearch on this model is the longest and the results areachieved the most Meanwhile there are a large numberof heuristic algorithms for solving this problem Thus weestablished CVRP model for disaster emergency response

Vehicle routing problem can be modeled by describingas the following There is a rescue center 119886

0to deliver

goods to 119896 disaster areas We suppose that the 119894th area sentby the rescue centerrsquos 119898 vehicle where each vehicle cancarry 119882 ton relief supplies demands 119892

119894(119894 = 1 2 119896)

The goods are sent to each disaster area and finally backto the rescue center If we know 119892

119894le 119882119898 then we look

for the shortest length that meet the demand of the affectednode It can be described as an undirected graph 119866 =

(119881 119864) where 119866 represents the layout of affected area after theoccurrence of disasters 119881 is the vertices set correspondingwith affected spot and rescue center and 119864 stands for anyedge of two nodes

In this paper we define several constraints as follows

(a) the start and the end of each route are at the rescuecenter 119886

0

Rescuecenter

1

23

8

910

14 13

12

11

4

5

67

15

Figure 1 A real example of CVRP

(b) the total amount of relief supplies loaded by vehiclemust not exceed the maximum affected node ofvehicle load

(c) each affected node is visited once by only one vehiclein the limited time

Figure 1 shows awork environment instance of the vehiclepath planning In this example a rescue center 119886

0 the

number of vehicles is 5 (119887 = 5 119887 means the number ofvehicles) and the affected node is 15 (V = 15) Table 1shows the affected nodesrsquo coordinates and each arearsquos reliefsupplies demands of Figure 1 and the rescue center coor-dinates are (0 0) The units of axes are meters 119909 representsthe 119909-coordinate and 119910 represents the 119910-coordinate Thenumbers 0 1 2 15 sign each area

When choosing the right path themore spacious smoothroad is more likely to be selected thus road grade trafficcongestion levels traffic and other factors make up how totry to meet various requirements to plan a trip with a morereasonable path problem When disaster occurs how timelyand reasonable given the rescue path is essential Then thevehicle path planning is necessary to consider the followingaspects path length road conditions traffic smoothnesssingle time constraint that means each area can withstand themaximum rescue time and weather conditions Pheromonessubstance such as more than one rescue center and total timeand length constraints needs to be pointed out in the paperAccording to the [16] proposed method to calculate weights

Mathematical Problems in Engineering 3

matrix based on AHP as is shown in Table 2 the number 0node represents rescue center

3 Brief Introduction of Ant ColonyOptimization Algorithm

31 Ant Colony Optimization Algorithm Ant colony opti-mization algorithm and the traditional ant colony algorithmare different in these three aspects as followsThat traditional

ant colony algorithm is of slow convergence and is easy tofall into local optimum and stagnation in the vehicle pathplanning applications while improved ant colony algorithmsolves these problems in the following aspects

(1) State transition probability function the transfer rulesthat ants select the next affected node are modifiedwhich is calculated as follows

119901119896

119894119895(119905) =

[120591119894119895(119905)]

119886

lowast 119863120573lowast [119908119894119895(119905)]

120574

lowast [119880119894119895(119905)]

120574

lowast 120577119894119895(119895)

sum119896isinallowed119896

[120591119894119896(119905)]119886lowast 119863120573lowast [119908119894119896(119905)]120574lowast [119880119894119896(119905)]120574lowast 120577119894119896(119896)

if 119895 isin allowed119896

0 otherwise

(1)

where 120591119894119895represents pheromones substance on the path

edge (119894 119895) 119863 stands for visibility 119901119896119894119895is transition probabili-

ties of the 119896th ant from the affected node 119894 to affected node 119895the user parameters 120572 and 120573 are adjustable parameters thatdetermine the relative influence of the pheromone andthe visibility in the transition probabilities 120574 means therelative importance of the path weights in the transitionprobability and allowed

119896represents the neighbor affected

node of affected node 119894 that 119896th ant has not yet visited Hereintroducing the path weight value 119908

119894119895that stands for the

weight value between affected nodes and the rescue center119863 = 1(119889

119894119895+ 1198891198951198860) replaces the traditional visibility function

that is 120578119894119895= 1119889

119894119895 where the value 119889

119894119895means the distance

between 119894 and 119895 and the value 1198891198951198860

is the distance betweenthe affected node 119895 and the rescue centre node To modifyvisibility function is to enhance the antrsquos perception of rescuecentre node in the current affected node which will guide theants to move and therefore avoid falling into local optimum

search results and reduce the search time Where 119880119894119895stands

for the save distances between nodes both the affected nodesand the rescue centre The save matrix calculated by theformula

After adding this factor it may avoid out the connectionof a direct connection with the affected nodes withoutthrough disaster rescue center node meanwhile shorteningthe length of the path Shown in Figure 2 120577

119894119895(119895) is

an angle factor which is the angle between thevector [

997888997888997888997888997888997888997888997888997888997888997888rarr

Node1198941Node

119894

997888997888997888997888997888997888997888997888997888997888997888rarr

Node119894Node119895] Node

1198941 stands for the

previous node before visited affected node 119894 thevalue Node

119894that the black point represents the current

visiting affected node 119894 Node119895represents the next visiting

affected node 119895 red arrow direction equals with theopposite

997888997888997888997888997888997888997888997888997888997888997888rarr

Node1198941Node

119894 yellow arrow direction equals

with997888997888997888997888997888997888997888997888997888997888997888rarr

Node119894Node119895 and 120577

119894119895(119895) is calculated as (2) When 120579 is

between [0 1205872] 120577119894119895(119895) is a constant 000001 which is for

escape when an ant is faced with a dead-end street Consider

120577119894119895(119895) =

120579

120587

if 1205872

le 120579 le 120587

000001 if 0 le 120579 le 1205872

(2)

119895 =

argmax119906isin119869119869119896

119894

120591119894119906(119905)120572lowast 119863120573lowast [119908119894119906(119905)]120574lowast [119880119894119906(119905)]120574 if 119902 le 119902

0

119869119869 if 119902 gt 1199020

(3)

Ants choose the next affected node 119895 in accordance with(3)

Where 119869119869 is determined by (1) 119902 is a random variablenumber between 0 and 1 119902

0is an adjustable parameter which

is between 0 and 1 and here we define 1199020as 005 When (3)

satisfied the condition 119902 gt 1199020 it chooses the next access node

in accordance with (1)There are two rules in global update One is the local

optimal iteration and the other is the global optimal iteration

(2) Update the local pheromone substances in accor-dance with the following

120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (

119905) + (1 minus 120588) 120591119894119895 (0) (4)

In the progress of searching paths according to statetransition rules ants apply (4) to choose route whosepheromone should be updated After all ants move a steplocal pheromones update once Where 120591

119894119895(0) represents the

initial value of pheromone of edge (119894 119895) usually a small


Table 2 The path weight matrix

119882119894119895

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 0 14 9 15 20 10 5 13 16 12 13 7 8 2 10 51 14 0 8 12 9 7 15 4 11 15 8 10 12 3 5 122 9 8 0 7 19 14 5 17 12 16 1 2 13 10 11 63 15 12 7 0 9 15 12 10 7 14 9 8 14 11 16 24 20 9 19 9 0 8 11 14 17 8 17 13 21 4 5 85 10 7 14 15 8 0 9 13 10 6 10 11 12 21 7 146 5 15 5 12 11 9 0 7 18 21 5 7 9 3 16 237 13 4 17 10 14 13 7 0 15 10 9 12 15 21 9 98 16 11 12 7 17 10 18 15 0 8 3 4 12 19 21 109 12 15 16 14 8 6 21 10 8 0 7 23 16 10 9 1310 13 8 1 9 17 10 5 9 3 7 0 11 12 21 7 311 7 10 2 8 13 11 7 12 4 23 11 0 9 3 16 212 8 12 13 14 21 12 9 15 12 16 12 9 0 21 9 1713 2 3 10 11 4 21 3 21 19 10 21 3 21 0 21 814 10 5 11 16 5 7 16 9 21 9 7 16 9 21 0 1615 5 12 6 2 8 14 23 9 10 13 3 2 17 8 16 0

positive number to prevent ant colony from falling into localoptimal solution prematurely 120588 is a user-defined coefficientwhich stands for the degree of global pheromone evaporation

(3) Update the Global pheromone substances with thefollowing

120591119894119895(119905 + 119899) = 120588 sdot 120591

119894119895(119905) + (1 minus 120588) Δ120591

119894119895 (5)

Δ120591119894119895=

1

119871

if edge (119894 119895) is part of the bestantrsquos route at this iteration

0 otherwise

(6)

where 120591119894119895(119905 + 119899) represents pheromone of the best

path (119894 119895) from time 119905 to time 119905 + 119899 after all of the antscomplete a trip which is unlike traditional ant colonyalgorithm In (6) 119871 represents length of the best movingpath corresponding to ants crawling at this iteration Afteriteration update the global pheromone by (5)

Meanwhile to take advantage of convergence of thealgorithm this paper adds incentive factor 120574 in pheromoneglobal updating It is calculated by the following

120574 =

119871+minus 119871

119879

(7)

119879 is a positive constant whose value can be up to thesame order of magnitude as current working space In thispaper set 119879 = 1000 Thus it reduced the complexity of thesearch and improved the accuracy and convergence speed ofthe route search 119871+ is the shortest path length for movingtill previous cycle If the path length of optimal solution inthis cycle is shorter than optimal solution having been foundthen 120574 gt 0 otherwise 120574 lt 0 which avoids falling into localoptimum

32 The Steps of Path Optimization

Step 1 Initialize phase 119898 the number of ants NC maxthe maximum number of iterations the number ofiterations NC = 0 best len the current optimum pathlength and set concentration of pheromones on eachnode at initial time (0001) all the ants at the rescuecentre 119886

0 119882 indicates the vehicle load and weigh[ ] stands

for the demand of affected nodes Set each ant a tabu listarray tabu[ ] to store the city that the ants have alreadyvisited that is to say the cities will not be visited in the nextchoice the path length passed ant[119896]path length arrayROUTES[119896][119899] stored the 119896th antrsquos route where the path hasalready been visited in the 119899th iteration and add 119886

0to each

antrsquos tabu list (119896 = 1 119898 119899 = 1 NC max)

Step 2 Set variable 119894 = 1

Step 3 Calculate the next affected node the ant can chooseand exclude the affected node in tabu list or the obstacleaffected node Array to visit[ ] and Len to visit recordedseparately the affected nodersquos mark the ants can visit and thenumber of the affected node while Len to visit ge 1 and 119882 geweigh[ ] go to Step 4 otherwise go to Step 7

Step 4 Use (3) and roulette method to select an affected nodeto visit for each ant 119896 (119896 = 1 to 119898) And move ant 119896 tothis node Meanwhile update the ant krsquos length of pathroute ROUTES[119896][119899] = add[ROUTES[119896][119899] to visit] theant moves the next affected node to visit put to visit in antkrsquos tabu list array tabu[ ] and 119882 = 119882 minus weigh [to visit]

Step 5 Perform a local updating rule according to (4)

Step 6 Record the current iteration of the shortest routeupdate best length with the best antrsquos path and record


the optimal path G best route[ ] while update the optimalsolution in accordance with 3-opt method

Step 7 Ant 119896 comes back to affected centre updatebest length and route ROUTES[119896][119899] = add[ROUTES[119896][119899] 119886

0] Set 119894 = 119894 + 1 If 119894 le 119889 switch to Step 4 otherwise

go to Step 8

Step 8 Update global concentration of pheromones of eachnode by optimal ant in accordance with (5) (6) and (7)

Step 9 Set NClarrNC + 1 If NC ltNC max and the entire antcolony has not converged to take the same path then set allthe ants at the rescue node 119886

0again and go to Step 3 If NC lt

NC max and the entire ant colony has converged to take thesame path or NC =NC max then finish the cycle and outputthe best route

4 Experiment of Computer Simulation andAnalysis of Results

In this part we use the established model shown in Part 2 todo the simulation experiment of the optimal algorithm Theexperiment runs onWindows XP Professional SP3 (symmet-ric multiprocessor system) and the compiler environment isMATLAB R2009a Each parameter is set to NC max = 5 120588 =085 120572 = 10 120573 = 10 120574 = 17 119876 = 1000 and 119898 = 20All of the parameters are under the optimal experiment inthe literature [15] The results of the simulation are shown inFigures 3ndash8

In order to illustrate the superiority of the proposedmethod we use the example above to compare the resultsof three methods including traditional ant colony algorithmthe literature [15] and improved ant colony algorithmFigures 3 4 and 5 shows the results that each polygon meansone route From Figure 3 we can see the path planning oftraditional ant colony algorithm the best length is 5852mthe route is

0rarr 10rarr 4rarr 13rarr 12rarr 00rarr 3rarr 5rarr 7rarr 9rarr 2rarr 00rarr 8rarr 6rarr 14rarr 0 0rarr 1rarr 15rarr 11rarr 0

When using the weight matrix we define that if the tworoads are not connected the weight value is 0 then thehigher the weight value the better the clearer roadThe searchresult is shown as Figure 4 the best length is 4849m and theoptimal path is

0rarr 1rarr 15rarr 8rarr 00rarr 13rarr 2rarr 3rarr 5rarr 7rarr 9rarr 12rarr 00rarr 14rarr 6rarr 0 0rarr 11rarr 4rarr 0 0rarr 10rarr 0

This paper introduced angle factor in the transfer rulebecause in an emergency we need to avoid obstacles to theroad to gainmore timeThus we choose the next node wherethe offset angle is small and closer to the direction of thetransfer which is more likely to avoid obstacles in the roadsThe save matrix is to compensate for the optimal path length

The direction ofnext movement

The oppositedirection of

current movement

120579

Figure 2 A view of added angle factor

minus4 minus2 0 2 4minus3

minus2

minus1

0

1

2

3

4

X (m)

Y(m

)

Figure 3 A result of ant colony algorithm path planning

increasing brought by a perspective to factors Meanwhileafter each cycle update pheromone by reward function anduse 3-opt algorithm to the optimal solutionmethod to updatethe global optimal path length Results shown in Figure 5 thebest length is 4161m and the optimal path is

0rarr 13rarr 4rarr 11rarr 0 0rarr 6rarr 14rarr 00rarr 8rarr 15rarr 1rarr 10rarr 00rarr 12rarr 3rarr 5rarr 7rarr 9rarr 2rarr 0

In this instance the literature [15] and this paperrsquosproposed algorithmrsquos optimal path length are best among thethree methods After adding the angle the vehiclersquos routeis more concise and clear while the new transfer rules andpheromone update rule this algorithm only sent four rescuevehicles to meet relief needs which is more reasonable

Convergence property where Figures 6 7 and 8 showtheir convergence property Blue line shows the optimal


minus4 minus3 minus2 minus1 0 1 2 3 4minus3

minus2

minus1

0

1

2

3

4

X (m)

Y(m

)

Figure 4 A result of the literature [15] path planning


minus2

minus1

0

1

2

3

4

X (m)

Y(m

)

Figure 5 A result of improved ant colony algorithm path planningin this paper

solution of the traditional ant colony algorithm convergenceprocess the black line is the literature [15] in the convergenceprocess and the red line is this paperrsquos improved ant colonyalgorithm convergence process As can be seen from thefigure in the same five iterations the curve of traditional antcolony algorithm fluctuates the optimal solution is 5852mthe literature [15] proposed optimization algorithm and thealgorithm this paper proposed converges within 5 iterationsFrom the view of convergence improved ant colony algo-rithm overcomes the shortcomings of slow convergence andthe optimal solution (4161m) that is superior to the literature[15] algorithm is the optimal solution (4849m)

In this part the simulation instances from benchmark[10] were tested by the above algorithms We labeled theexample of examples that data points were non-decreasingorder as I1 I2 I14 where the conditions for instances I1ndashI5 are similar to the instances I9ndashI14 while instances I6ndashI10and I11-I12 involve limited routing path length and service

1 2 3 4 50

10

20

30

40

50

60

70

Iterations

Path

leng

th

Figure 6 The convergence curve of ant colony algorithm pathplanning

1 2 3Iterations

Path

leng

th

4 50

5

10

15

20

25

30

35

40

45

50

Figure 7The convergence curve of the literature [15] path planning

time We conducted five experiments the first group wasthe traditional GA the second group was the traditionalACS the third group was the literature [15] the fourthgroup was the literature [20] and the last group was forthis paper proposed optimization algorithm All parameterswere under experimental environment of the optimum algo-rithm shown in Table 3 Here parameters 119875

119888and 119875

119898are for

Crossover probability and mutation probability in the GAalgorithm respectively Each experiment was performed 20times respectively and compared with each optimal and


1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

35

40

45

Iterations

Path

leng

th

Figure 8The convergence curve of improved ant colony algorithmpath planning in this paper

0

200

400

600

800

1000

1200

1400

1600

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14

GAACS Literature (Jin and Zhang 2010)

Literature (Chen etal 2008)

This paper proposed

Figure 9 The simulation results of best length

average path length The experimental results are shownin Figures 9 and 10 To illustrate our proposed algorithmdrawing better performance we record CPU processing timeunder the experiment above and then compare their averagetime shown as Figure 11

Figure 11 shows the average running time of the fivealgorithms in various numerical examples from I1 to I14 Tocompare against GA and ACS our proposed algorithm hashigher overhead time However it can still be able to obtaingood solutions in a short time and all the solving time iswithin 100 secondsAt the same time our proposed algorithmperforms better than [15] and [20] From Figure 11 we cansee that from I1 to I8 [15 20] and our proposed algorithmhave the almost same computation speed however from I9

0

200

400

600

800

1000

1200

1400

1600

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14

GAACS

Literature (Jin and Zhang 2010)Literature (Chen etal 2008)

This paper proposed

Figure 10 The simulation results of average length

0

20

40

60

80

100

120

I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14

GAACSLiterature (Jin and Zhang 2010)Literature (Chen etal 2008)This paper proposed

Figure 11 The simulation results of average time

to I14 our proposed algorithm takes less time than [15 20]significantly

As is seen from the experimental results the optimaland the average results of the paper proposed algorithmare superior to other four methods clearly comparison interms of simulation time our proposed algorithm also hasadvantage of average running time as the problem sizeincreases that is to say from I1 to I14 the size of examplesincrease this advantage has become increasingly apparentTaken all together our proposed algorithm draws the best


Table 3 Parameter settings

120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898

GA sim sim sim 08 008 50 100 5ACS l 2 sim 07 08 50 100 5Literature [15] 01 2 15 01 09 50 100 5Literature [20] 1 1 sim 02 09 50 100 5This paper 1 1 17 015 005 50 100 5

performance and in the future it can be applied inmuchmorecomplex environments as well

5 Conclusions

In this paper we established a new model of disaster emer-gency response which adopts new transfer rules adds thepath weight matrix save matrix angle factor functions andnewvisibility functions at the same time updates pheromonemodel with reward function overcoming the shortcomingsthat are of slow convergence and are easy to fall into localoptimum and stagnation of ant colony algorithm in thevehicle applications thus reducing the complexity of thesearch process We use 3-opt method to update the optimalsolution to shorten the length of rescue route Finally thecomputer simulation experiments confirm the correctnessand validity of this method and due to considering the actualroad conditions and other factors such as the angle which ismore reasonable when applies to solve real emergency rescue

Conflict of Interests

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

Acknowledgments

This work was supported in part by a Grant from twoGuangdong Province Production Education and ScientificStudy Programs of China (no 2011B090400056 and no2012B091100490)

References

[1] R-M Chen and C-M Wang ldquoVehicle routing problem appli-cation in tour planning multimedia system with metaheuristicdesignedrdquo Advances in Information Sciences and Service Sci-ences vol 3 no 2 pp 1ndash11 2011

[2] X Li P Tian and S C H Leung ldquoVehicle routing problemswith time windows and stochastic travel and service timesmodels and algorithmrdquo International Journal of ProductionEconomics vol 125 no 1 pp 137ndash145 2010

[3] H Malekly B Haddadi and T-M Reza ldquoA fuzzy randomvehicle routing problem the case of Iranrdquo in Proceedingsof the International Conference on Computers and IndustrialEngineering (CIE rsquo09) pp 1070ndash1075 Troyes France July 2009

[4] P Toth and D VigoThe Vehicle Routing Problem SIAMMono-graphs on Discrete Mathematics and Applications Philadel-phia Pa USA 2002

[5] B-I Kim S Kim and S Sahoo ldquoWaste collection vehicle rout-ing problem with time windowsrdquo Computers and OperationsResearch vol 33 no 12 pp 3624ndash3642 2006

[6] F Semet and E Taillard ldquoSolving real-life vehicle routingproblems efficiently using tabu searchrdquo Annals of OperationsResearch vol 41 no 4 pp 469ndash488 1993

[7] J F Cordeau and G Laporte ldquoTabu search heuristics forthe vehicle routing problemrdquo Metaheuristic Optimization ViaMemory and Evolution vol 30 pp 145ndash163 2005

[8] N S Niazy and A Badr ldquoComplexity of capacitated vehiclesrouting problem using cellular genetic algorithmsrdquo Interna-tional Journal of Computer Science and Network Security vol12 no 2 p 5 2012

[9] E F Zulvia R J Kuo and T-L Hu ldquoSolving CVRP with timewindow fuzzy travel time and demand via a hybrid ant colonyoptimization and genetic algorithmrdquo in Proceedings of the IEEEWorld Congress on Computational Intelligence (WCCI rsquo2012)Brisbane Australia

[10] L M Gambardella E Taillard and G AgazziMACS-VRPTWAMultiple Ant Colony SystemForVehicle Routing ProblemsWithTime Windows in New ideAs in optimization 1999 Edited by DCorne

[11] J E Bell and P R McMullen ldquoAnt colony optimization tech-niques for the vehicle routing problemrdquo Advanced EngineeringInformatics vol 18 no 1 pp 41ndash48 2004

[12] B-B Jiang H-M Chen L-N Ma and L Deng ldquoTime-dependent pheromones and electric-field model a new ACOalgorithm for dynamic traffic routingrdquo International Journal ofModelling Identification and Control vol 12 no 1-2 pp 29ndash352011

[13] R Zhou H P Lee and A Y C Nee ldquoApplying Ant ColonyOptimisation (ACO) algorithm to dynamic job shop schedulingproblemsrdquo International Journal ofManufacturing Research vol3 no 3 pp 301ndash320 2008

[14] M Lopez-Ibanez and C Blum ldquoBeam-ACO for the travellingsalesman problem with time windowsrdquo Computers amp Opera-tions Research vol 37 no 9 pp 1570ndash1583 2010

[15] B Jin and L Zhang ldquoAn improved ant colony algorithm for pathoptimization in emergency rescuerdquo in Proceedings of the 2ndInternational Workshop on Intelligent Systems and Applications(ISA rsquo10) May 2010

[16] X C Bian C Zhu and Y X He ldquoApplication of path weightsant colony algorithm in emergency rescuerdquo Industrial Safety andEnvironmental Protection vol 36 no 11 pp 60ndash62 2010

[17] H Wang and C Cao ldquoResearch on an intelligent ant algorithmin conjuction with local O3-opt optimizationrdquo Computer Appli-cations and Software vol 10 pp 89ndash91 2010

[18] R Kamil and S Reiji ldquoAccelerating 2-opt and 3-opt localsearch using GPU in the travelling salesman problemrdquo IEEEConference Publications pp 489ndash495 2012


[19] N Christofides A Mingozzi and P Toth ldquoThe vehicle routingproblemrdquo in Combinatorial Optimization pp 315ndash338 WileyChichester UK 1979

[20] P Chen H-K Huang and X-Y Dong ldquoHybrid heuristicalgorithm for the vehicle routing problem with simultaneousdelivery and pickuprdquo Chinese Journal of Computers vol 31 no4 pp 565ndash573 2008

[21] M Y Khoshbakht and M Sedighpour ldquoAn optimization algo-rithm for the capacitated vehicle routing problem based on antcolony systemrdquoAustralian Journal of Basic and Applied Sciencesvol 5 no 12 pp 2729ndash2737 2011

[22] B Lyamine H Amir and K Abder ldquoA hybrid heuristicapproach to solve the capacitated vehicle routing problemrdquoJournal of Artificial Intelligence vol 1 no 1 p 31 2010

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

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Table 1 Distance map and delivery of CVRP example

Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119909 0 3 minus2 minus2 0 minus3 3 minus4 3 minus3 2 1 minus1 minus2 2 4119910 0 2 1 minus2 3 minus3 minus1 minus1 4 0 1 3 minus1 3 minus2 3Delivery 0 18 08 15 25 02 24 05 06 09 13 2 06 02 14 09

will focus on the shortcomings of ant colony algorithm Ourproposed algorithm adds the angle factor pathweightmatrixand save matrix in the transfer rules and optimizes visibilityfunction and penalty function thus it avoids falling into localoptimum and speeds up the convergence of the algorithm Atthe same time pheromone update rules with local update andglobal update update the global optimal solution only usethe best ant to improve the algorithmrsquos global search Finallyafter each cycle update the optimal solution in accordancewith 3-opt method [17 18] to shorten the length of the rescuepath Use benchmark test [19] to compare the results with theliterature [20] in the algorithm to verify the effectiveness ofthe algorithm

2 Problem Description andMathematical Model

Disasters such as earthquakes and floods are sudden andunpredictable which cause serious sufferings for peopleHowto find an effective and short rescue path for the affectednodes in the limited time after the disasters become thefocus thing Before making vehicle routing problem pathplanning we first establish reasonable free space modelCapacitated vehicle routing problem [21 22] (CVRP) wasdefined as a transportation and a distribution problem withthe constrained of vehicle load travel time or distanceResearch on this model is the longest and the results areachieved the most Meanwhile there are a large numberof heuristic algorithms for solving this problem Thus weestablished CVRP model for disaster emergency response

Vehicle routing problem can be modeled by describingas the following There is a rescue center 119886

0to deliver

goods to 119896 disaster areas We suppose that the 119894th area sentby the rescue centerrsquos 119898 vehicle where each vehicle cancarry 119882 ton relief supplies demands 119892

119894(119894 = 1 2 119896)

The goods are sent to each disaster area and finally backto the rescue center If we know 119892

119894le 119882119898 then we look

for the shortest length that meet the demand of the affectednode It can be described as an undirected graph 119866 =

(119881 119864) where 119866 represents the layout of affected area after theoccurrence of disasters 119881 is the vertices set correspondingwith affected spot and rescue center and 119864 stands for anyedge of two nodes

In this paper we define several constraints as follows

(a) the start and the end of each route are at the rescuecenter 119886

0

Rescuecenter

1

23

8

910

14 13

12

11

4

5

67

15

Figure 1 A real example of CVRP

(b) the total amount of relief supplies loaded by vehiclemust not exceed the maximum affected node ofvehicle load

(c) each affected node is visited once by only one vehiclein the limited time

Figure 1 shows awork environment instance of the vehiclepath planning In this example a rescue center 119886

0 the

number of vehicles is 5 (119887 = 5 119887 means the number ofvehicles) and the affected node is 15 (V = 15) Table 1shows the affected nodesrsquo coordinates and each arearsquos reliefsupplies demands of Figure 1 and the rescue center coor-dinates are (0 0) The units of axes are meters 119909 representsthe 119909-coordinate and 119910 represents the 119910-coordinate Thenumbers 0 1 2 15 sign each area

When choosing the right path themore spacious smoothroad is more likely to be selected thus road grade trafficcongestion levels traffic and other factors make up how totry to meet various requirements to plan a trip with a morereasonable path problem When disaster occurs how timelyand reasonable given the rescue path is essential Then thevehicle path planning is necessary to consider the followingaspects path length road conditions traffic smoothnesssingle time constraint that means each area can withstand themaximum rescue time and weather conditions Pheromonessubstance such as more than one rescue center and total timeand length constraints needs to be pointed out in the paperAccording to the [16] proposed method to calculate weights







119901119896

119894119895(119905) =

[120591119894119895(119905)]

119886

lowast 119863120573lowast [119908119894119895(119905)]

120574

lowast [119880119894119895(119905)]

120574

lowast 120577119894119895(119895)




0 otherwise

(1)









that is 120578119894119895= 1119889

119894119895 where the value 119889







119894119895(119895) is


997888997888997888997888997888997888997888997888997888997888997888rarr

Node1198941Node

119894

997888997888997888997888997888997888997888997888997888997888997888rarr







997888997888997888997888997888997888997888997888997888997888997888rarr

Node1198941Node


with997888997888997888997888997888997888997888997888997888997888997888rarr

Node119894Node119895 and 120577




120577119894119895(119895) =

120579

120587

if 1205872

le 120579 le 120587

000001 if 0 le 120579 le 1205872

(2)

119895 =

argmax119906isin119869119869119896

119894


0

119869119869 if 119902 gt 1199020

(3)









120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (

119905) + (1 minus 120588) 120591119894119895 (0) (4)






119882119894119895

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 0 14 9 15 20 10 5 13 16 12 13 7 8 2 10 51 14 0 8 12 9 7 15 4 11 15 8 10 12 3 5 122 9 8 0 7 19 14 5 17 12 16 1 2 13 10 11 63 15 12 7 0 9 15 12 10 7 14 9 8 14 11 16 24 20 9 19 9 0 8 11 14 17 8 17 13 21 4 5 85 10 7 14 15 8 0 9 13 10 6 10 11 12 21 7 146 5 15 5 12 11 9 0 7 18 21 5 7 9 3 16 237 13 4 17 10 14 13 7 0 15 10 9 12 15 21 9 98 16 11 12 7 17 10 18 15 0 8 3 4 12 19 21 109 12 15 16 14 8 6 21 10 8 0 7 23 16 10 9 1310 13 8 1 9 17 10 5 9 3 7 0 11 12 21 7 311 7 10 2 8 13 11 7 12 4 23 11 0 9 3 16 212 8 12 13 14 21 12 9 15 12 16 12 9 0 21 9 1713 2 3 10 11 4 21 3 21 19 10 21 3 21 0 21 814 10 5 11 16 5 7 16 9 21 9 7 16 9 21 0 1615 5 12 6 2 8 14 23 9 10 13 3 2 17 8 16 0



120591119894119895(119905 + 119899) = 120588 sdot 120591

119894119895(119905) + (1 minus 120588) Δ120591

119894119895 (5)

Δ120591119894119895=

1

119871


0 otherwise

(6)




120574 =

119871+minus 119871

119879

(7)






0to each











go to Step 8














current movement

120579



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)








minus2

minus1

0

1

2

3

4

X (m)

Y(m

)



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)




1 2 3 4 50

10

20

30

40

50

60

70

Iterations

Path

leng

th


1 2 3Iterations

Path

leng

th

4 50

5

10

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25

30

35

40

45

50



119888and 119875

119898are for



1 15 2 25 3 35 4 45 50

5

10

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30

35

40

45

Iterations

Path

leng

th


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This paper proposed




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GAACS


This paper proposed


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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14







120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119901119896

119894119895(119905) =

[120591119894119895(119905)]

119886

lowast 119863120573lowast [119908119894119895(119905)]

120574

lowast [119880119894119895(119905)]

120574

lowast 120577119894119895(119895)




0 otherwise

(1)









that is 120578119894119895= 1119889

119894119895 where the value 119889







119894119895(119895) is


997888997888997888997888997888997888997888997888997888997888997888rarr

Node1198941Node

119894

997888997888997888997888997888997888997888997888997888997888997888rarr







997888997888997888997888997888997888997888997888997888997888997888rarr

Node1198941Node


with997888997888997888997888997888997888997888997888997888997888997888rarr

Node119894Node119895 and 120577




120577119894119895(119895) =

120579

120587

if 1205872

le 120579 le 120587

000001 if 0 le 120579 le 1205872

(2)

119895 =

argmax119906isin119869119869119896

119894


0

119869119869 if 119902 gt 1199020

(3)









120591119894119895 (119905 + 1) = 120588 sdot 120591119894119895 (

119905) + (1 minus 120588) 120591119894119895 (0) (4)






119882119894119895

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 0 14 9 15 20 10 5 13 16 12 13 7 8 2 10 51 14 0 8 12 9 7 15 4 11 15 8 10 12 3 5 122 9 8 0 7 19 14 5 17 12 16 1 2 13 10 11 63 15 12 7 0 9 15 12 10 7 14 9 8 14 11 16 24 20 9 19 9 0 8 11 14 17 8 17 13 21 4 5 85 10 7 14 15 8 0 9 13 10 6 10 11 12 21 7 146 5 15 5 12 11 9 0 7 18 21 5 7 9 3 16 237 13 4 17 10 14 13 7 0 15 10 9 12 15 21 9 98 16 11 12 7 17 10 18 15 0 8 3 4 12 19 21 109 12 15 16 14 8 6 21 10 8 0 7 23 16 10 9 1310 13 8 1 9 17 10 5 9 3 7 0 11 12 21 7 311 7 10 2 8 13 11 7 12 4 23 11 0 9 3 16 212 8 12 13 14 21 12 9 15 12 16 12 9 0 21 9 1713 2 3 10 11 4 21 3 21 19 10 21 3 21 0 21 814 10 5 11 16 5 7 16 9 21 9 7 16 9 21 0 1615 5 12 6 2 8 14 23 9 10 13 3 2 17 8 16 0



120591119894119895(119905 + 119899) = 120588 sdot 120591

119894119895(119905) + (1 minus 120588) Δ120591

119894119895 (5)

Δ120591119894119895=

1

119871


0 otherwise

(6)




120574 =

119871+minus 119871

119879

(7)






0to each











go to Step 8














current movement

120579



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)








minus2

minus1

0

1

2

3

4

X (m)

Y(m

)



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)




1 2 3 4 50

10

20

30

40

50

60

70

Iterations

Path

leng

th


1 2 3Iterations

Path

leng

th

4 50

5

10

15

20

25

30

35

40

45

50



119888and 119875

119898are for



1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

35

40

45

Iterations

Path

leng

th


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This paper proposed


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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14







120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119882119894119895

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 0 14 9 15 20 10 5 13 16 12 13 7 8 2 10 51 14 0 8 12 9 7 15 4 11 15 8 10 12 3 5 122 9 8 0 7 19 14 5 17 12 16 1 2 13 10 11 63 15 12 7 0 9 15 12 10 7 14 9 8 14 11 16 24 20 9 19 9 0 8 11 14 17 8 17 13 21 4 5 85 10 7 14 15 8 0 9 13 10 6 10 11 12 21 7 146 5 15 5 12 11 9 0 7 18 21 5 7 9 3 16 237 13 4 17 10 14 13 7 0 15 10 9 12 15 21 9 98 16 11 12 7 17 10 18 15 0 8 3 4 12 19 21 109 12 15 16 14 8 6 21 10 8 0 7 23 16 10 9 1310 13 8 1 9 17 10 5 9 3 7 0 11 12 21 7 311 7 10 2 8 13 11 7 12 4 23 11 0 9 3 16 212 8 12 13 14 21 12 9 15 12 16 12 9 0 21 9 1713 2 3 10 11 4 21 3 21 19 10 21 3 21 0 21 814 10 5 11 16 5 7 16 9 21 9 7 16 9 21 0 1615 5 12 6 2 8 14 23 9 10 13 3 2 17 8 16 0



120591119894119895(119905 + 119899) = 120588 sdot 120591

119894119895(119905) + (1 minus 120588) Δ120591

119894119895 (5)

Δ120591119894119895=

1

119871


0 otherwise

(6)




120574 =

119871+minus 119871

119879

(7)






0to each











go to Step 8














current movement

120579



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)








minus2

minus1

0

1

2

3

4

X (m)

Y(m

)



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)




1 2 3 4 50

10

20

30

40

50

60

70

Iterations

Path

leng

th


1 2 3Iterations

Path

leng

th

4 50

5

10

15

20

25

30

35

40

45

50



119888and 119875

119898are for



1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

35

40

45

Iterations

Path

leng

th


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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14



This paper proposed




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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14

GAACS


This paper proposed


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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14







120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













go to Step 8














current movement

120579



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)








minus2

minus1

0

1

2

3

4

X (m)

Y(m

)



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)




1 2 3 4 50

10

20

30

40

50

60

70

Iterations

Path

leng

th


1 2 3Iterations

Path

leng

th

4 50

5

10

15

20

25

30

35

40

45

50



119888and 119875

119898are for



1 15 2 25 3 35 4 45 50

5

10

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35

40

45

Iterations

Path

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th


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This paper proposed




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GAACS


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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14







120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus2

minus1

0

1

2

3

4

X (m)

Y(m

)



minus2

minus1

0

1

2

3

4

X (m)

Y(m

)




1 2 3 4 50

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Iterations

Path

leng

th


1 2 3Iterations

Path

leng

th

4 50

5

10

15

20

25

30

35

40

45

50



119888and 119875

119898are for



1 15 2 25 3 35 4 45 50

5

10

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25

30

35

40

45

Iterations

Path

leng

th


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This paper proposed


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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14







120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 15 2 25 3 35 4 45 50

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I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 I13 I14







120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572 120573 120574 120588(or 119875119888) 119902

0(or 119875119898) NC max 119876 119898



5 Conclusions




Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an improved ant colony algorithm and its...

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