Research ArticleAn Improved Ant Colony Algorithm for Solving the PathPlanning Problem of the Omnidirectional Mobile Vehicle

Jiang Zhao Dingding Cheng and Chongqing Hao

School of Electrical Engineering Hebei University of Science and Technology Hebei Shijiazhuang 050018 China

Correspondence should be addressed to Dingding Cheng chengdingding0306126com

Received 20 May 2016 Accepted 21 August 2016

Academic Editor Cheng-Tang Wu

Copyright copy 2016 Jiang Zhao et al This 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

This paper presents an improved ant colony algorithm for the path planning of the omnidirectional mobile vehicle The purposeof the improved ant colony algorithm is to design an appropriate route to connect the starting point and ending point of theenvironment with obstacles Ant colony algorithm which is used to solve the path planning problem is improved according to thecharacteristics of the omnidirectional mobile vehicle And in the improved algorithm the nonuniform distribution of the initialpheromone and the selection strategy with direction play a very positive role in the path searchThe coverage and updating strategyof pheromone is introduced to avoid repeated search reducing the effect of the number of ants on the performance of the algorithmIn addition the pheromone evaporation coefficient is segmented and adjusted which can effectively balance the convergence speedand search ability Finally this paper provides a theoretical basis for the improved ant colony algorithm by strict mathematicalderivation and some numerical simulations are also given to illustrate the effectiveness of the theoretical results

1 Introduction

Path planning which is one of the hot topics in motioncontrol research requires the control object to determine thepath avoid obstacles and achieve the goal autonomously [1]As a special kind ofmobile robot the omnidirectionalmobilerobot has wide application in the field of industrial trans-portation With the increase of production cost and workefficiency the path planning problem of the omnidirectionalmobile vehicle needs to be resolved as soon as possible Thepath planning problem of mobile robot was proposed in the1960s With the research of Lozano-Perez andWesley [2] theproblem has aroused the interest of many scholars

Simulated annealing algorithm [3] potential functiontheory [4] genetic algorithm [5] particle swarm algorithm[6] and ant colony algorithm [7] are representative methodsof path planning which can find the shortest path Inaddition the research on path planning has made greatprogress in recent years The idea of Mo and Xu [8] solvedthe path planning by PSOwith the position updating strategyand biogeography particle swarm optimization algorithm toincrease the diversity of population and optimize the paths

in a static environment Precup et al [9] proposed that theadaptive charged system search algorithm was applied to theoptimal path planning problem of multiple mobile robots instatic environment Several modifications and improvementsof A star algorithm were introduced by Ducho et al [10] andGuruji et al [11] considering a static or dynamic environmentAnd it was the first time that MOVNS [12] was proposed todeal with the path planning problem of mobile robots aboutthe path safety length and smoothness In addition manyother findings also made contributions to the related pathplanning problem [13ndash15]

Many scholars applied ant colony algorithm [16] to solvethe path planning problem because it can represent obstaclesflexibly and easily by using a tabu list On this basis thefurther research was carried out Later according to thecharacteristics of the above algorithms some methods com-bined ant colony algorithm and other algorithms which wereproposed and hadmore advantages than one algorithm suchas ant colony particle swarm algorithm [2] and ant colonygenetic algorithm [17] However the application of ant colonyalgorithm ismuchmore than that As discussed byTiwari andVidyarthi [18] the ant colony optimization algorithm solved

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7672839 10 pageshttpdxdoiorg10115520167672839

2 Mathematical Problems in Engineering

Figure 1 The omnidirectional mobile vehicle

the scheduling problem effectively through a kind of specialants that remain alive till the fitter lazy ants are generatedin the successive generations Wang et al [19] proposed amodified ACS scheme that can identify the true attack patheven without the entire network routing information Jianget al [20] presented a coevolutionary improved multiantcolony optimization algorithm to design appropriate piperoutes in the layout space under various kinds of constraintsWhat is more Saidi-Mehrabad et al [21] also made someimprovements to the ant colony algorithm to solve theproblem Therefore solving the problem in accordance withthe feature of the control object environment and existingmethods is the best way

So accordingly ant colony algorithm is a key pointbecause it is the basis of the above methods Therefore basedon the motion characteristics of omnidirectional mobilevehicle this paper proposes a kind of improved ant colonyalgorithmThe initial pheromone distribution is nonuniformwhich helps reduce the blindness of the search In thestrategy of selection heuristic information with direction isintroduced to improve the probability of finding the opti-mal solution And piecewise adjusting pheromone behaviorhelps avoid falling into local optimal solution Finally thepheromone coverage and updating strategy can ensure thatthe algorithm can find the optimal path strictly in theory

2 Problem Statement

21 Control Object Aswe all know solving the path planningproblem in accordance with the feature of the control objectis the best way Different from mobile robot the omnidi-rectional mobile vehicle as shown in Figure 1 is equippedwith chassis and drive which can contribute to moving in anydirection In terms of its control system the omnidirectionalmobile vehicle has 8 basic motion units upper upper rightright lower right lower lower left left and upper left asshown in Figure 2Therefore its form of motion which is theoutput of the motion path can be composed of a number ofbasic motion units

1

2

3

4

5

6

7

8

Figure 2 Basic motion directions

22 Problem Description Without considering the omnidi-rectional mobile vehicle height the working environmentcan be considered as a two-dimensional plane So the pathplanning needs to solve three problems

(1) In a nonchanging work environment find a path toconnect the starting point and the ending point

(2) Find the shortest path with avoiding all obstacles(3) The algorithm should have certain simplicity low

complexity and good stability

3 Mathematical Model

31 Environment Model In practical applications the elec-tronic map is often needed to represent the working environ-ment This paper describes the work environment using thegrid method in which the grid coordinates are replaced bythe position of each center point So working environment isdivided into119872lowast119873 squares And the obstacles are representedby gray grids (less than one by one count) distinguishingthem from available parts The area is numbered from topto bottom and from left to right in the literature [22] whilethe squares can be represented by the set of numbered 119861 =

1 2 119887 119872 lowast 119873 which is shown in Figure 3 In orderto identify obstacles the white grid cell is represented by 0and the gray grid unit is represented by 1 Thus the workingenvironment is represented by119872 lowast 119873 binary matrix whichis denoted by 119866

32 Path Representation It can be known from the char-acteristics of the omnidirectional mobile vehicle and theenvironment representation that the resolution of the systemis related to the number of grids Finally the path is obtainedby adding the components of the solution step by step toobtain the solution The path of the vehicle can be expressedas 119871 = (119909

1 1199101) (119909

119894 119910119894) (119909

119905minus1 119910119905minus1) (119909119905 119910119905) the

starting point is (1199091 1199101) denoted as 119878 and the ending point

coordinate is (119909119905 119910119905) denoted as 119864 From the environmental

model we can know that the path can be expressed by the

Mathematical Problems in Engineering 3

[[[[[[[[[[

0 0 0 0 0

0 0 1

Start

End

6 7 8 9

1 2 3 4 5

1 0

0 0 1 1 0

0 0 0 0 0

0 0 0 0 0

]]]]]]]]]]

Figure 3 Mathematical model of working environment

grid number 119871 = 1198971 119897119894 119897119905minus1 119897119905 and (119909

119894 119910119894) is replaced

by 119897119894 119894 = 1 119905 minus 1 119905

4 Improved Ant Colony Algorithm

41 Improvement Aspects

(1)NonuniformDistribution of Initial Pheromone In the initialphase of traditional ant colony algorithm the pheromonedistribution is described by119872lowast119873 dimensionalmatrix wherethe element 120591

119887isin 119879 represents the initial value of pheromone

in 119887 grid As shown below

120591119887= 1198880

1198880is a constant (1)

To solve the problems of lack of initial pheromone andlow speed of search in literatures [10 11] particle swarmoptimization algorithm and genetic algorithm are proposedto generate an initial path which can be transformed into theinitial pheromone distribution so as to reduce the blindnessof the ant colony search In order to improve the efficiencyand simplicity of the algorithm the nonuniform distributionof initial pheromone is proposed in this paper So thepheromone distribution is described by matrix 119879 whereelement 120591

119887isin 119879 as shown below

120591119887=

1198880times 1205791

119887 isin 119860

1198880

otherwise(2)

where 119860 is the collection of all numbers between number1198971and 119897

119905 that is to say 119860 = min119897

1 119897119905min119897

1 119897119905 +

1 max1198971 119897119905 1205791is a constant greater than 1 Formula (2)

indicates that the number of pieces of information betweenthe starting point and the ending point is slightly higher thanothers Similar to the principle of the zero point theorem inview of the characteristics of the path the probability of anarbitrary path set between the starting point and the endingpoint is 1 Therefore the simple initial distribution strategyhas advantages to reduce the blindness of ant colony searchshorten the path search time And at the same time it doesnot increase the complexity of the algorithm

(2) Heuristic Strategy with Direction Information In the tradi-tional ant colony algorithm the probability of the next nodeis selected by rotating the roulette wheel method as follows

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1 119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(3)

where119901119897119894 119897119894+1

is the probability of the next node of 119897119894 120591119897119894119897119894+1

is thepheromone of the path from 119897

119894to 119897119894+1

and 120572 is the pheromonecoefficient Even 120578

119897119894 119897119894+1is the heuristic information of 119897

119894to 119897119894+1

and 120573 is the heuristic information parameter where 119889

119897119894+1 119897119905is

the distance between node 119897119894+1

and end node 119897119905 (119909119897119894+1 119910119897119894+1)

and (119909119897119905 119910119897119905) are the coordinates of 119897

119894+1and 119897119905

The direction information is proposed as the heuristicinformation of the node transfer strategy by Wang et al [19]to solve the path planning problem without obstacle

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120578119897119894 119897119894+1

=

1205792

if state is toward the goal

1 otherwise

(4)

where 120578119897119894 119897119894+1

is the direction of information and 1205792is a constant

little more than 1 In order to reduce the blindness of thesearch the paper puts forward the strategy of the directioninformation and the probability formula is

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

119897119894+1

isin the next node

0 otherwise

(5)

4 Mathematical Problems in Engineering

End

11

16

21 22

12 13

18

23

Figure 4 Directional information representation

where

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(6)

The direction information 120583119894119894+1

is expressed as follows

120583119897119894 119897119894+1

= 120579120574

3 (7)

where 120574 is the number of the same directions of node 119897119894to

next 119897119894+1

and node 119897119894to end 119897

119905 120574 = 0 1 2 And 120579

3is a constant

slightly greater than 1 As shown in Figure 4 the next node of17 may be 11 12 13 16 18 21 22 and 23 where node 13 hastwo same directions as the end so the direction informationis 12057923 And as such the direction information of nodes 11 12

18 and 23 is 12057913 and for 16 21 and 22 it is 1205790

2

(3) Coverage and Updating Strategy In the traditional antcolony algorithm the next node position is decided by theroulette wheel method and repeated until the target point isobtained The pheromones of all nodes are updated by thefollowing rules after every ant 120585 searching

120591119897119894119897119894+1

(119896 120585 + 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 120585) + Δ120591119897119894119897119894+1

(119896 120585)

120591119897119894 119897119894+1

(119896 + 1 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 119877) + Δ120591119897119894119897119894+1

(119896 119877)

Δ120591119897119894119897119894+1

(119896 120585) =

1

PL119896120585

through 119897119894to 119897119894+1

0 otherwise

(8)

where 120588 is the evaporation rate of pheromone where thefunction is to avoid the pheromone accumulation 0 lt 120588 lt 1119877 is the number of the ants starting from the starting point inevery round Δ120591

119897119894119897119894+1(119896 120585) is the addition of the 119896th round 120585th

ant And PL119896120585

is the path length of the 120585th ant which can getto the end point in the 119896th round

In the improved ant colony algorithm the next nodeposition is decided by the roulette wheel method until the

target point is obtained And in a cycle the pheromone ofall the nodes is updated by following the rules in each roundof ants

1205911015840

119897119894119897119894+1(119896 + 1) = (1 minus 120588 (119896)) 120591

119897119894 119897119894+1(119896) + Δ120591

119897119894119897119894+1(119896)

12059110158401015840

119897119894119897119894+1(119896 + 1) =

1205794120591119897119894119897119894+1

(0)

ln (119896 + 1)+ Δ120591119897119894119897119894+1

(119896)

120591119897119894119897119894+1

(119896 + 1) = max 1205911015840119897119894 119897119894+1

(119896 + 1) 12059110158401015840

119897119894119897119894+1(119896 + 1)

(9)

Among them Δ120591119897119894119897119894+1

(119896) is the pheromone update part of119897119894to 119897119894+1

in the 119896th round 1205794is a constant and the maximum

value of the pheromone in the iterative process is the initialset value 120591max = 120591(0) 120588(119896) is the pheromone evaporationcoefficient 0 lt 120588(119896) lt 1 In the entire search space

Δ120591119897119894119897119894+1

(119896) = 119900 (Δ120591119897119894119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 120585)

Δ120591119897119894119897119894+1

(119896 119877 minus 1) Δ120591119897119894 119897119894+1

(119896 119877))

(10)

where 119900(Δ120591119897119894 119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 119877)) is a nonzero pherom-one space covering operation For example

119860 (1) =[[

[

1 2 4

0 0 1

3 0 0

]]

]

119860 (2) =[[

[

2 3 5

1 0 0

2 0 0

]]

]

so 119900 (119860 (1) 119860 (2)) =[[

[

2 3 5

1 0 1

2 0 0

]]

]

(11)

In formula (10)

Δ120591119897119894119897119894+1

(119896 120585) =

119876

PL119896120585

120591119897119894119897119894+1

(119896 120585) through 119897119894to 119897119894+1

0 otherwise(12)

Among them119876 is a constant It can be seen that if the antcan get to the end point the added part of the pheromone isin inverse proportion with the length and if the ant cannotreach the ending point it is recorded as 0 The rule canguide the search for the shortest path improve the speedof convergence and avoid the possibility of the pheromoneaccumulation caused by repeated search meanwhile theeffect on the performance of the algorithm caused by thequantity of the ants is reduced

(4) Evaporation Coefficient Segment When the problemscale is relatively large due to the presence of pheromoneevaporation the pheromone of some nodes will be reducedgreatly even close to zero which reduces the search ability ofthe algorithm When 120588 is large the search ability is affected

Mathematical Problems in Engineering 5

by repeated selection Meanwhile when 120588 is small the searchability of the algorithm is enhanced but the convergencespeed is decreased Therefore the heuristic informationcoefficient is adjusted as follows

120588 (119896 + 1) =

1205795120588 (119896) 120588 (119896) ge 120588min

120588min 120588 (119896) lt 120588min(13)

where 1205795is a constant less than 1 119896 is the number of

search rounds 119896 = 0 1 119870 120588(0) = 120588max 120588max and120588min are the maximum and minimum of the coefficientof evaporation The pheromone evaporation coefficient offragmentation can enhance the search ability in the initialstage of search increase the convergence speed later andimprove the performance of ant colony algorithm

42 Algorithm Steps The improved ant colony algorithm isto find the optimal path according to the following steps

Step 1 The nonuniform distribution of initial pheromone isproposed and pheromone matrix 119879 is constructed accordingto formula (2)

Step 2 Send 119870 rounds and each round of 119877 ants which areplaced at the starting point

Step 3 Send a round of ants and each ant to select the nodeaccording to the roulette wheel method with the probabilityof each point calculated by formulas (5) (6) and (7)

Step 4 After a round of ant search the pheromone wascalculated by formulas (10) and (12) the evaporation coeffi-cient was calculated by (13) and the pheromone was updatedaccording to formulas (9)

Step 5 Send the next round of ants and repeat above Steps 3and 4 until the end of the iteration

Step 6 Record the shortest path of each round then draw thecurve of the shortest path length of each round

Step 7 Compare the shortest paths of all the ants Finallyoutput the current global optimal path

43 Proof of Convergence The proof depends on a necessaryassumption there is a path between the starting point and theending at least According to formulas (10) and (12) the addedpart of pheromonemust be greater than or equal to 0That is

Δ120591119897119894119897119894+1

(119896) ge 0 (14)

Formulas (9) show that the pheromone can be expressedas follows

120591119897119894119897119894+1

(119896) ge max((1 minus 120588 (119896))119896 120591 (0) 1205794120591 (0)

ln (119896 + 1)) (15)

Event 119864119896means that the optimal solution is obtained for

the first time in the 119896th iteration Therefore event ⋀119870119896=1

119864119896

denotes that the algorithm can find the optimal solution forthe first time for 119870 iterations Then probability 119875(⋀119870

119896=1119864119896) is

satisfied

lim119896rarrinfin

119875(

119870

⋀

119896=1

119864119896) = 1 (16)

Proof 119897lowast119894+1

is the choice of the 119894th optimal solution and theprobability of finding the optimal solution for the 120585th ant inthe 119896th round 119901(119896 120585) can be obtained by formula (5) becausethe node selection is an independent event

1 ge 119901 (119896 120585) =

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

(120578119897119894 119897lowast

119894+1

)120573

120583119897119894 119897lowast

119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

=

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

120578119897119894 119897lowast

119894+1

)120573

(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

)

(17)

Define 120595(119897119894+1) = (120578

119897119894 119897119894+1120578119897119894 119897lowast

119894+1

)120573(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

) and 120595max =

max120595(119897119894+1) 119894 = 1 119905minus1 So formula (17) can be expressed

as

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

120595max (18)

Because of formula (12) it is known that the minimumvalue of pheromone in the 119896th iteration of the ant colonyalgorithm is

120591min (119896) ge1205794120591 (0)

ln (119896 + 1) forall119896 ge 1 (19)

Further the maximum value of the pheromone is theinitial value by setting some parameters So

120591max = 120591 (0) (20)

And the maximum number of 119873119888(119896 120585 (119897

119894 119897119894+1)) can be

expressed as

119873119888= max 119873

119888(119896 120577 (119897

119894 119897119894+1)) 119894 = 1 119905 minus 1 (21)

where119873119888(119896 120585 (119897

119894 119897119894+1)) is the number of options By (19) (20)

and (21) we can know that formula (18) meets the followingrelationship

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(120591119897119894 119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

120595max

gt ((1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

)

119905minus1

(22)

6 Mathematical Problems in Engineering

Record

119889 = ((1205794times 120591 (0))

120572

(120591 (0))120572times 119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

= ((1205794)120572

119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

(23)

So formula (22) is expressed as

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

=119889

(ln (119896 + 1))(119905minus1)120572

(24)

From the above we can know that

1 gt119889

(ln (119896 + 1))(119905minus1)120572gt 0 (25)

The probability that the optimal solution cannot be foundby any ants in the 119896th round is 119901(119896)

119901 (119896) = (1 minus 119901 (119896 120585))119877

(26)

Probability 119875(⋁119870119896=1

119864119896) that cannot find the optimal solu-

tion for the119870 rounds is given as follows

119875(

119870

⋁

119896=1

119864119896) =

119870

prod

119896=1

119901 (119896) =

119870

prod

119896=1

(1 minus 119901 (119896 120585))119877

lt

119870

prod

119896=1

(1 minus119889

(ln (119896 + 1))(119905minus1)120572)

119877

(27)

After the logarithm to the above formula

ln119875(119870

⋁

119896=1

119864119896) lt

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

119877

= 119877

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

le minus119877 times 119889

119870

sum

119896=1

(1

ln (119896 + 1)(119905minus1)120572)

le minus119877 times 119889

119870

sum

119896=1

1

119896 + 1

lim119870rarrinfin

ln119875(119870

⋁

119896=1

119864119896) lt lim119870rarrinfin

ln(minus119877 times 119889119870

sum

119896=1

1

119896 + 1)

= minusinfin

(28)

That is 119875(⋁119870119896=1

119864119896) = 0 So lim

119896rarrinfin119875(⋀119870

119896=1119864119896) = 1

5 Numerical Simulations

The experiments are made to demonstrate the effectivenessof the proposed algorithm The algorithm is compiled inMATLAB software Experiments were conducted using acomparative method to be more persuasive along with thesame experimental conditions

51 Simulation Experiments The experiment was dividedinto three parts with two algorithms the traditional antcolony algorithm and the improved ant colony algorithm Inorder to compare the effects of the two algorithms they areused in the same environment

(1) Example 1 In this example set the simulation environ-ment as 20times20 grids and the length of each unit is 1The start-ing point is the upper left corner in the grid (05 195) and theend point is the lower right corner (195 05) (see Figure 5)

(2) Example 2 We set the environment as 30 times 30 grids andthe length of the unit is 1 The starting point is (05 85) andthe end point is (255 285) (see Figure 6)

(3) Example 3 The experiment is made in model of 40 times 40grids and the length of each unit is 1 The starting point isset at (05 395) and the ending point is set at (395 05) (seeFigure 7)

52 Results Analysis As shown in Figure 5(a) it is a globaloptimal path in the case of Example 1 Similarly Figures6(a) and 7(a) are the global optimal paths in Examples 2and 3 respectively The obstacles in each environment arerandomly selected The three experiment results show thatthe ant colony algorithm and the improved ant colonyalgorithm both can find the global optimal paths in a varietyof environments

Figures 5(b) and 5(c) depict the path length iterationcurves in Example 1 by traditional ant colony algorithmand the improved ant colony algorithm And the searchprocess about the shortest path length of each round andthe current global optimal path length can be seen fromthe figures By comparison it can be seen that the shortestpath length is found in the 49th iteration by traditionalant colony algorithm while it is found in the fifth iterationby the improved algorithm And once the optimal solutionis obtained the search will converge to the shortest pathvalue by the improved algorithm Similarly Figures 6(b) and6(c) represent the search process of the two algorithms inExample 2 The traditional ant colony algorithm finds theshortest path in the 69th iteration with value of 3563 whilethe improved algorithm gets the shortest path in the ninthtimes In the same way Figures 7(b) and 7(c) show thatthe traditional ant colony algorithm and the improved antcolony algorithm find the shortest path in the 37th and 24thiterations respectively Therefore it can be seen that thesearch efficiency of the improved ant colony algorithm is

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Figure 1 The omnidirectional mobile vehicle

the scheduling problem effectively through a kind of specialants that remain alive till the fitter lazy ants are generatedin the successive generations Wang et al [19] proposed amodified ACS scheme that can identify the true attack patheven without the entire network routing information Jianget al [20] presented a coevolutionary improved multiantcolony optimization algorithm to design appropriate piperoutes in the layout space under various kinds of constraintsWhat is more Saidi-Mehrabad et al [21] also made someimprovements to the ant colony algorithm to solve theproblem Therefore solving the problem in accordance withthe feature of the control object environment and existingmethods is the best way

So accordingly ant colony algorithm is a key pointbecause it is the basis of the above methods Therefore basedon the motion characteristics of omnidirectional mobilevehicle this paper proposes a kind of improved ant colonyalgorithmThe initial pheromone distribution is nonuniformwhich helps reduce the blindness of the search In thestrategy of selection heuristic information with direction isintroduced to improve the probability of finding the opti-mal solution And piecewise adjusting pheromone behaviorhelps avoid falling into local optimal solution Finally thepheromone coverage and updating strategy can ensure thatthe algorithm can find the optimal path strictly in theory

2 Problem Statement

21 Control Object Aswe all know solving the path planningproblem in accordance with the feature of the control objectis the best way Different from mobile robot the omnidi-rectional mobile vehicle as shown in Figure 1 is equippedwith chassis and drive which can contribute to moving in anydirection In terms of its control system the omnidirectionalmobile vehicle has 8 basic motion units upper upper rightright lower right lower lower left left and upper left asshown in Figure 2Therefore its form of motion which is theoutput of the motion path can be composed of a number ofbasic motion units

1

2

3

4

5

6

7

8

Figure 2 Basic motion directions

22 Problem Description Without considering the omnidi-rectional mobile vehicle height the working environmentcan be considered as a two-dimensional plane So the pathplanning needs to solve three problems

(1) In a nonchanging work environment find a path toconnect the starting point and the ending point

(2) Find the shortest path with avoiding all obstacles(3) The algorithm should have certain simplicity low

complexity and good stability

3 Mathematical Model

31 Environment Model In practical applications the elec-tronic map is often needed to represent the working environ-ment This paper describes the work environment using thegrid method in which the grid coordinates are replaced bythe position of each center point So working environment isdivided into119872lowast119873 squares And the obstacles are representedby gray grids (less than one by one count) distinguishingthem from available parts The area is numbered from topto bottom and from left to right in the literature [22] whilethe squares can be represented by the set of numbered 119861 =

1 2 119887 119872 lowast 119873 which is shown in Figure 3 In orderto identify obstacles the white grid cell is represented by 0and the gray grid unit is represented by 1 Thus the workingenvironment is represented by119872 lowast 119873 binary matrix whichis denoted by 119866

32 Path Representation It can be known from the char-acteristics of the omnidirectional mobile vehicle and theenvironment representation that the resolution of the systemis related to the number of grids Finally the path is obtainedby adding the components of the solution step by step toobtain the solution The path of the vehicle can be expressedas 119871 = (119909

1 1199101) (119909

119894 119910119894) (119909

119905minus1 119910119905minus1) (119909119905 119910119905) the

starting point is (1199091 1199101) denoted as 119878 and the ending point

coordinate is (119909119905 119910119905) denoted as 119864 From the environmental

model we can know that the path can be expressed by the

Mathematical Problems in Engineering 3

[[[[[[[[[[

0 0 0 0 0

0 0 1

Start

End

6 7 8 9

1 2 3 4 5

1 0

0 0 1 1 0

0 0 0 0 0

0 0 0 0 0

]]]]]]]]]]

Figure 3 Mathematical model of working environment

grid number 119871 = 1198971 119897119894 119897119905minus1 119897119905 and (119909

119894 119910119894) is replaced

by 119897119894 119894 = 1 119905 minus 1 119905

4 Improved Ant Colony Algorithm

41 Improvement Aspects

(1)NonuniformDistribution of Initial Pheromone In the initialphase of traditional ant colony algorithm the pheromonedistribution is described by119872lowast119873 dimensionalmatrix wherethe element 120591

119887isin 119879 represents the initial value of pheromone

in 119887 grid As shown below

120591119887= 1198880

1198880is a constant (1)

To solve the problems of lack of initial pheromone andlow speed of search in literatures [10 11] particle swarmoptimization algorithm and genetic algorithm are proposedto generate an initial path which can be transformed into theinitial pheromone distribution so as to reduce the blindnessof the ant colony search In order to improve the efficiencyand simplicity of the algorithm the nonuniform distributionof initial pheromone is proposed in this paper So thepheromone distribution is described by matrix 119879 whereelement 120591

119887isin 119879 as shown below

120591119887=

1198880times 1205791

119887 isin 119860

1198880

otherwise(2)

where 119860 is the collection of all numbers between number1198971and 119897

119905 that is to say 119860 = min119897

1 119897119905min119897

1 119897119905 +

1 max1198971 119897119905 1205791is a constant greater than 1 Formula (2)

indicates that the number of pieces of information betweenthe starting point and the ending point is slightly higher thanothers Similar to the principle of the zero point theorem inview of the characteristics of the path the probability of anarbitrary path set between the starting point and the endingpoint is 1 Therefore the simple initial distribution strategyhas advantages to reduce the blindness of ant colony searchshorten the path search time And at the same time it doesnot increase the complexity of the algorithm

(2) Heuristic Strategy with Direction Information In the tradi-tional ant colony algorithm the probability of the next nodeis selected by rotating the roulette wheel method as follows

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1 119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(3)

where119901119897119894 119897119894+1

is the probability of the next node of 119897119894 120591119897119894119897119894+1

is thepheromone of the path from 119897

119894to 119897119894+1

and 120572 is the pheromonecoefficient Even 120578

119897119894 119897119894+1is the heuristic information of 119897

119894to 119897119894+1

and 120573 is the heuristic information parameter where 119889

119897119894+1 119897119905is

the distance between node 119897119894+1

and end node 119897119905 (119909119897119894+1 119910119897119894+1)

and (119909119897119905 119910119897119905) are the coordinates of 119897

119894+1and 119897119905

The direction information is proposed as the heuristicinformation of the node transfer strategy by Wang et al [19]to solve the path planning problem without obstacle

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120578119897119894 119897119894+1

=

1205792

if state is toward the goal

1 otherwise

(4)

where 120578119897119894 119897119894+1

is the direction of information and 1205792is a constant

little more than 1 In order to reduce the blindness of thesearch the paper puts forward the strategy of the directioninformation and the probability formula is

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

119897119894+1

isin the next node

0 otherwise

(5)

4 Mathematical Problems in Engineering

End

11

16

21 22

12 13

18

23

Figure 4 Directional information representation

where

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(6)

The direction information 120583119894119894+1

is expressed as follows

120583119897119894 119897119894+1

= 120579120574

3 (7)

where 120574 is the number of the same directions of node 119897119894to

next 119897119894+1

and node 119897119894to end 119897

119905 120574 = 0 1 2 And 120579

3is a constant

slightly greater than 1 As shown in Figure 4 the next node of17 may be 11 12 13 16 18 21 22 and 23 where node 13 hastwo same directions as the end so the direction informationis 12057923 And as such the direction information of nodes 11 12

18 and 23 is 12057913 and for 16 21 and 22 it is 1205790

2

(3) Coverage and Updating Strategy In the traditional antcolony algorithm the next node position is decided by theroulette wheel method and repeated until the target point isobtained The pheromones of all nodes are updated by thefollowing rules after every ant 120585 searching

120591119897119894119897119894+1

(119896 120585 + 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 120585) + Δ120591119897119894119897119894+1

(119896 120585)

120591119897119894 119897119894+1

(119896 + 1 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 119877) + Δ120591119897119894119897119894+1

(119896 119877)

Δ120591119897119894119897119894+1

(119896 120585) =

1

PL119896120585

through 119897119894to 119897119894+1

0 otherwise

(8)

where 120588 is the evaporation rate of pheromone where thefunction is to avoid the pheromone accumulation 0 lt 120588 lt 1119877 is the number of the ants starting from the starting point inevery round Δ120591

119897119894119897119894+1(119896 120585) is the addition of the 119896th round 120585th

ant And PL119896120585

is the path length of the 120585th ant which can getto the end point in the 119896th round

In the improved ant colony algorithm the next nodeposition is decided by the roulette wheel method until the

target point is obtained And in a cycle the pheromone ofall the nodes is updated by following the rules in each roundof ants

1205911015840

119897119894119897119894+1(119896 + 1) = (1 minus 120588 (119896)) 120591

119897119894 119897119894+1(119896) + Δ120591

119897119894119897119894+1(119896)

12059110158401015840

119897119894119897119894+1(119896 + 1) =

1205794120591119897119894119897119894+1

(0)

ln (119896 + 1)+ Δ120591119897119894119897119894+1

(119896)

120591119897119894119897119894+1

(119896 + 1) = max 1205911015840119897119894 119897119894+1

(119896 + 1) 12059110158401015840

119897119894119897119894+1(119896 + 1)

(9)

Among them Δ120591119897119894119897119894+1

(119896) is the pheromone update part of119897119894to 119897119894+1

in the 119896th round 1205794is a constant and the maximum

value of the pheromone in the iterative process is the initialset value 120591max = 120591(0) 120588(119896) is the pheromone evaporationcoefficient 0 lt 120588(119896) lt 1 In the entire search space

Δ120591119897119894119897119894+1

(119896) = 119900 (Δ120591119897119894119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 120585)

Δ120591119897119894119897119894+1

(119896 119877 minus 1) Δ120591119897119894 119897119894+1

(119896 119877))

(10)

where 119900(Δ120591119897119894 119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 119877)) is a nonzero pherom-one space covering operation For example

119860 (1) =[[

[

1 2 4

0 0 1

3 0 0

]]

]

119860 (2) =[[

[

2 3 5

1 0 0

2 0 0

]]

]

so 119900 (119860 (1) 119860 (2)) =[[

[

2 3 5

1 0 1

2 0 0

]]

]

(11)

In formula (10)

Δ120591119897119894119897119894+1

(119896 120585) =

119876

PL119896120585

120591119897119894119897119894+1

(119896 120585) through 119897119894to 119897119894+1

0 otherwise(12)

Among them119876 is a constant It can be seen that if the antcan get to the end point the added part of the pheromone isin inverse proportion with the length and if the ant cannotreach the ending point it is recorded as 0 The rule canguide the search for the shortest path improve the speedof convergence and avoid the possibility of the pheromoneaccumulation caused by repeated search meanwhile theeffect on the performance of the algorithm caused by thequantity of the ants is reduced

(4) Evaporation Coefficient Segment When the problemscale is relatively large due to the presence of pheromoneevaporation the pheromone of some nodes will be reducedgreatly even close to zero which reduces the search ability ofthe algorithm When 120588 is large the search ability is affected

Mathematical Problems in Engineering 5

by repeated selection Meanwhile when 120588 is small the searchability of the algorithm is enhanced but the convergencespeed is decreased Therefore the heuristic informationcoefficient is adjusted as follows

120588 (119896 + 1) =

1205795120588 (119896) 120588 (119896) ge 120588min

120588min 120588 (119896) lt 120588min(13)

where 1205795is a constant less than 1 119896 is the number of

search rounds 119896 = 0 1 119870 120588(0) = 120588max 120588max and120588min are the maximum and minimum of the coefficientof evaporation The pheromone evaporation coefficient offragmentation can enhance the search ability in the initialstage of search increase the convergence speed later andimprove the performance of ant colony algorithm

42 Algorithm Steps The improved ant colony algorithm isto find the optimal path according to the following steps

Step 1 The nonuniform distribution of initial pheromone isproposed and pheromone matrix 119879 is constructed accordingto formula (2)

Step 2 Send 119870 rounds and each round of 119877 ants which areplaced at the starting point

Step 3 Send a round of ants and each ant to select the nodeaccording to the roulette wheel method with the probabilityof each point calculated by formulas (5) (6) and (7)

Step 4 After a round of ant search the pheromone wascalculated by formulas (10) and (12) the evaporation coeffi-cient was calculated by (13) and the pheromone was updatedaccording to formulas (9)

Step 5 Send the next round of ants and repeat above Steps 3and 4 until the end of the iteration

Step 6 Record the shortest path of each round then draw thecurve of the shortest path length of each round

Step 7 Compare the shortest paths of all the ants Finallyoutput the current global optimal path

43 Proof of Convergence The proof depends on a necessaryassumption there is a path between the starting point and theending at least According to formulas (10) and (12) the addedpart of pheromonemust be greater than or equal to 0That is

Δ120591119897119894119897119894+1

(119896) ge 0 (14)

Formulas (9) show that the pheromone can be expressedas follows

120591119897119894119897119894+1

(119896) ge max((1 minus 120588 (119896))119896 120591 (0) 1205794120591 (0)

ln (119896 + 1)) (15)

Event 119864119896means that the optimal solution is obtained for

the first time in the 119896th iteration Therefore event ⋀119870119896=1

119864119896

denotes that the algorithm can find the optimal solution forthe first time for 119870 iterations Then probability 119875(⋀119870

119896=1119864119896) is

satisfied

lim119896rarrinfin

119875(

119870

⋀

119896=1

119864119896) = 1 (16)

Proof 119897lowast119894+1

is the choice of the 119894th optimal solution and theprobability of finding the optimal solution for the 120585th ant inthe 119896th round 119901(119896 120585) can be obtained by formula (5) becausethe node selection is an independent event

1 ge 119901 (119896 120585) =

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

(120578119897119894 119897lowast

119894+1

)120573

120583119897119894 119897lowast

119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

=

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

120578119897119894 119897lowast

119894+1

)120573

(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

)

(17)

Define 120595(119897119894+1) = (120578

119897119894 119897119894+1120578119897119894 119897lowast

119894+1

)120573(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

) and 120595max =

max120595(119897119894+1) 119894 = 1 119905minus1 So formula (17) can be expressed

as

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

120595max (18)

Because of formula (12) it is known that the minimumvalue of pheromone in the 119896th iteration of the ant colonyalgorithm is

120591min (119896) ge1205794120591 (0)

ln (119896 + 1) forall119896 ge 1 (19)

Further the maximum value of the pheromone is theinitial value by setting some parameters So

120591max = 120591 (0) (20)

And the maximum number of 119873119888(119896 120585 (119897

119894 119897119894+1)) can be

expressed as

119873119888= max 119873

119888(119896 120577 (119897

119894 119897119894+1)) 119894 = 1 119905 minus 1 (21)

where119873119888(119896 120585 (119897

119894 119897119894+1)) is the number of options By (19) (20)

and (21) we can know that formula (18) meets the followingrelationship

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(120591119897119894 119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

120595max

gt ((1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

)

119905minus1

(22)

6 Mathematical Problems in Engineering

Record

119889 = ((1205794times 120591 (0))

120572

(120591 (0))120572times 119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

= ((1205794)120572

119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

(23)

So formula (22) is expressed as

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

=119889

(ln (119896 + 1))(119905minus1)120572

(24)

From the above we can know that

1 gt119889

(ln (119896 + 1))(119905minus1)120572gt 0 (25)

The probability that the optimal solution cannot be foundby any ants in the 119896th round is 119901(119896)

119901 (119896) = (1 minus 119901 (119896 120585))119877

(26)

Probability 119875(⋁119870119896=1

119864119896) that cannot find the optimal solu-

tion for the119870 rounds is given as follows

119875(

119870

⋁

119896=1

119864119896) =

119870

prod

119896=1

119901 (119896) =

119870

prod

119896=1

(1 minus 119901 (119896 120585))119877

lt

119870

prod

119896=1

(1 minus119889

(ln (119896 + 1))(119905minus1)120572)

119877

(27)

After the logarithm to the above formula

ln119875(119870

⋁

119896=1

119864119896) lt

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

119877

= 119877

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

le minus119877 times 119889

119870

sum

119896=1

(1

ln (119896 + 1)(119905minus1)120572)

le minus119877 times 119889

119870

sum

119896=1

1

119896 + 1

lim119870rarrinfin

ln119875(119870

⋁

119896=1

119864119896) lt lim119870rarrinfin

ln(minus119877 times 119889119870

sum

119896=1

1

119896 + 1)

= minusinfin

(28)

That is 119875(⋁119870119896=1

119864119896) = 0 So lim

119896rarrinfin119875(⋀119870

119896=1119864119896) = 1

5 Numerical Simulations

The experiments are made to demonstrate the effectivenessof the proposed algorithm The algorithm is compiled inMATLAB software Experiments were conducted using acomparative method to be more persuasive along with thesame experimental conditions

51 Simulation Experiments The experiment was dividedinto three parts with two algorithms the traditional antcolony algorithm and the improved ant colony algorithm Inorder to compare the effects of the two algorithms they areused in the same environment

(1) Example 1 In this example set the simulation environ-ment as 20times20 grids and the length of each unit is 1The start-ing point is the upper left corner in the grid (05 195) and theend point is the lower right corner (195 05) (see Figure 5)

(2) Example 2 We set the environment as 30 times 30 grids andthe length of the unit is 1 The starting point is (05 85) andthe end point is (255 285) (see Figure 6)

(3) Example 3 The experiment is made in model of 40 times 40grids and the length of each unit is 1 The starting point isset at (05 395) and the ending point is set at (395 05) (seeFigure 7)

52 Results Analysis As shown in Figure 5(a) it is a globaloptimal path in the case of Example 1 Similarly Figures6(a) and 7(a) are the global optimal paths in Examples 2and 3 respectively The obstacles in each environment arerandomly selected The three experiment results show thatthe ant colony algorithm and the improved ant colonyalgorithm both can find the global optimal paths in a varietyof environments

Figures 5(b) and 5(c) depict the path length iterationcurves in Example 1 by traditional ant colony algorithmand the improved ant colony algorithm And the searchprocess about the shortest path length of each round andthe current global optimal path length can be seen fromthe figures By comparison it can be seen that the shortestpath length is found in the 49th iteration by traditionalant colony algorithm while it is found in the fifth iterationby the improved algorithm And once the optimal solutionis obtained the search will converge to the shortest pathvalue by the improved algorithm Similarly Figures 6(b) and6(c) represent the search process of the two algorithms inExample 2 The traditional ant colony algorithm finds theshortest path in the 69th iteration with value of 3563 whilethe improved algorithm gets the shortest path in the ninthtimes In the same way Figures 7(b) and 7(c) show thatthe traditional ant colony algorithm and the improved antcolony algorithm find the shortest path in the 37th and 24thiterations respectively Therefore it can be seen that thesearch efficiency of the improved ant colony algorithm is

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

[[[[[[[[[[

0 0 0 0 0

0 0 1

Start

End

6 7 8 9

1 2 3 4 5

1 0

0 0 1 1 0

0 0 0 0 0

0 0 0 0 0

]]]]]]]]]]

Figure 3 Mathematical model of working environment

grid number 119871 = 1198971 119897119894 119897119905minus1 119897119905 and (119909

119894 119910119894) is replaced

by 119897119894 119894 = 1 119905 minus 1 119905

4 Improved Ant Colony Algorithm

41 Improvement Aspects

(1)NonuniformDistribution of Initial Pheromone In the initialphase of traditional ant colony algorithm the pheromonedistribution is described by119872lowast119873 dimensionalmatrix wherethe element 120591

119887isin 119879 represents the initial value of pheromone

in 119887 grid As shown below

120591119887= 1198880

1198880is a constant (1)

To solve the problems of lack of initial pheromone andlow speed of search in literatures [10 11] particle swarmoptimization algorithm and genetic algorithm are proposedto generate an initial path which can be transformed into theinitial pheromone distribution so as to reduce the blindnessof the ant colony search In order to improve the efficiencyand simplicity of the algorithm the nonuniform distributionof initial pheromone is proposed in this paper So thepheromone distribution is described by matrix 119879 whereelement 120591

119887isin 119879 as shown below

120591119887=

1198880times 1205791

119887 isin 119860

1198880

otherwise(2)

where 119860 is the collection of all numbers between number1198971and 119897

119905 that is to say 119860 = min119897

1 119897119905min119897

1 119897119905 +

1 max1198971 119897119905 1205791is a constant greater than 1 Formula (2)

indicates that the number of pieces of information betweenthe starting point and the ending point is slightly higher thanothers Similar to the principle of the zero point theorem inview of the characteristics of the path the probability of anarbitrary path set between the starting point and the endingpoint is 1 Therefore the simple initial distribution strategyhas advantages to reduce the blindness of ant colony searchshorten the path search time And at the same time it doesnot increase the complexity of the algorithm

(2) Heuristic Strategy with Direction Information In the tradi-tional ant colony algorithm the probability of the next nodeis selected by rotating the roulette wheel method as follows

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1 119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(3)

where119901119897119894 119897119894+1

is the probability of the next node of 119897119894 120591119897119894119897119894+1

is thepheromone of the path from 119897

119894to 119897119894+1

and 120572 is the pheromonecoefficient Even 120578

119897119894 119897119894+1is the heuristic information of 119897

119894to 119897119894+1

and 120573 is the heuristic information parameter where 119889

119897119894+1 119897119905is

the distance between node 119897119894+1

and end node 119897119905 (119909119897119894+1 119910119897119894+1)

and (119909119897119905 119910119897119905) are the coordinates of 119897

119894+1and 119897119905

The direction information is proposed as the heuristicinformation of the node transfer strategy by Wang et al [19]to solve the path planning problem without obstacle

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120578119897119894 119897119894+1

=

1205792

if state is toward the goal

1 otherwise

(4)

where 120578119897119894 119897119894+1

is the direction of information and 1205792is a constant

little more than 1 In order to reduce the blindness of thesearch the paper puts forward the strategy of the directioninformation and the probability formula is

119901119897119894 119897119894+1

=

(120591119897119894119897119894+1

)120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

119897119894+1

isin the next node

0 otherwise

(5)

4 Mathematical Problems in Engineering

End

11

16

21 22

12 13

18

23

Figure 4 Directional information representation

where

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(6)

The direction information 120583119894119894+1

is expressed as follows

120583119897119894 119897119894+1

= 120579120574

3 (7)

where 120574 is the number of the same directions of node 119897119894to

next 119897119894+1

and node 119897119894to end 119897

119905 120574 = 0 1 2 And 120579

3is a constant

slightly greater than 1 As shown in Figure 4 the next node of17 may be 11 12 13 16 18 21 22 and 23 where node 13 hastwo same directions as the end so the direction informationis 12057923 And as such the direction information of nodes 11 12

18 and 23 is 12057913 and for 16 21 and 22 it is 1205790

2

(3) Coverage and Updating Strategy In the traditional antcolony algorithm the next node position is decided by theroulette wheel method and repeated until the target point isobtained The pheromones of all nodes are updated by thefollowing rules after every ant 120585 searching

120591119897119894119897119894+1

(119896 120585 + 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 120585) + Δ120591119897119894119897119894+1

(119896 120585)

120591119897119894 119897119894+1

(119896 + 1 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 119877) + Δ120591119897119894119897119894+1

(119896 119877)

Δ120591119897119894119897119894+1

(119896 120585) =

1

PL119896120585

through 119897119894to 119897119894+1

0 otherwise

(8)

where 120588 is the evaporation rate of pheromone where thefunction is to avoid the pheromone accumulation 0 lt 120588 lt 1119877 is the number of the ants starting from the starting point inevery round Δ120591

119897119894119897119894+1(119896 120585) is the addition of the 119896th round 120585th

ant And PL119896120585

is the path length of the 120585th ant which can getto the end point in the 119896th round

In the improved ant colony algorithm the next nodeposition is decided by the roulette wheel method until the

target point is obtained And in a cycle the pheromone ofall the nodes is updated by following the rules in each roundof ants

1205911015840

119897119894119897119894+1(119896 + 1) = (1 minus 120588 (119896)) 120591

119897119894 119897119894+1(119896) + Δ120591

119897119894119897119894+1(119896)

12059110158401015840

119897119894119897119894+1(119896 + 1) =

1205794120591119897119894119897119894+1

(0)

ln (119896 + 1)+ Δ120591119897119894119897119894+1

(119896)

120591119897119894119897119894+1

(119896 + 1) = max 1205911015840119897119894 119897119894+1

(119896 + 1) 12059110158401015840

119897119894119897119894+1(119896 + 1)

(9)

Among them Δ120591119897119894119897119894+1

(119896) is the pheromone update part of119897119894to 119897119894+1

in the 119896th round 1205794is a constant and the maximum

value of the pheromone in the iterative process is the initialset value 120591max = 120591(0) 120588(119896) is the pheromone evaporationcoefficient 0 lt 120588(119896) lt 1 In the entire search space

Δ120591119897119894119897119894+1

(119896) = 119900 (Δ120591119897119894119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 120585)

Δ120591119897119894119897119894+1

(119896 119877 minus 1) Δ120591119897119894 119897119894+1

(119896 119877))

(10)

where 119900(Δ120591119897119894 119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 119877)) is a nonzero pherom-one space covering operation For example

119860 (1) =[[

[

1 2 4

0 0 1

3 0 0

]]

]

119860 (2) =[[

[

2 3 5

1 0 0

2 0 0

]]

]

so 119900 (119860 (1) 119860 (2)) =[[

[

2 3 5

1 0 1

2 0 0

]]

]

(11)

In formula (10)

Δ120591119897119894119897119894+1

(119896 120585) =

119876

PL119896120585

120591119897119894119897119894+1

(119896 120585) through 119897119894to 119897119894+1

0 otherwise(12)

Among them119876 is a constant It can be seen that if the antcan get to the end point the added part of the pheromone isin inverse proportion with the length and if the ant cannotreach the ending point it is recorded as 0 The rule canguide the search for the shortest path improve the speedof convergence and avoid the possibility of the pheromoneaccumulation caused by repeated search meanwhile theeffect on the performance of the algorithm caused by thequantity of the ants is reduced

(4) Evaporation Coefficient Segment When the problemscale is relatively large due to the presence of pheromoneevaporation the pheromone of some nodes will be reducedgreatly even close to zero which reduces the search ability ofthe algorithm When 120588 is large the search ability is affected

Mathematical Problems in Engineering 5

by repeated selection Meanwhile when 120588 is small the searchability of the algorithm is enhanced but the convergencespeed is decreased Therefore the heuristic informationcoefficient is adjusted as follows

120588 (119896 + 1) =

1205795120588 (119896) 120588 (119896) ge 120588min

120588min 120588 (119896) lt 120588min(13)

where 1205795is a constant less than 1 119896 is the number of

search rounds 119896 = 0 1 119870 120588(0) = 120588max 120588max and120588min are the maximum and minimum of the coefficientof evaporation The pheromone evaporation coefficient offragmentation can enhance the search ability in the initialstage of search increase the convergence speed later andimprove the performance of ant colony algorithm

42 Algorithm Steps The improved ant colony algorithm isto find the optimal path according to the following steps

Step 1 The nonuniform distribution of initial pheromone isproposed and pheromone matrix 119879 is constructed accordingto formula (2)

Step 2 Send 119870 rounds and each round of 119877 ants which areplaced at the starting point

Step 3 Send a round of ants and each ant to select the nodeaccording to the roulette wheel method with the probabilityof each point calculated by formulas (5) (6) and (7)

Step 4 After a round of ant search the pheromone wascalculated by formulas (10) and (12) the evaporation coeffi-cient was calculated by (13) and the pheromone was updatedaccording to formulas (9)

Step 5 Send the next round of ants and repeat above Steps 3and 4 until the end of the iteration

Step 6 Record the shortest path of each round then draw thecurve of the shortest path length of each round

Step 7 Compare the shortest paths of all the ants Finallyoutput the current global optimal path

43 Proof of Convergence The proof depends on a necessaryassumption there is a path between the starting point and theending at least According to formulas (10) and (12) the addedpart of pheromonemust be greater than or equal to 0That is

Δ120591119897119894119897119894+1

(119896) ge 0 (14)

Formulas (9) show that the pheromone can be expressedas follows

120591119897119894119897119894+1

(119896) ge max((1 minus 120588 (119896))119896 120591 (0) 1205794120591 (0)

ln (119896 + 1)) (15)

Event 119864119896means that the optimal solution is obtained for

the first time in the 119896th iteration Therefore event ⋀119870119896=1

119864119896

denotes that the algorithm can find the optimal solution forthe first time for 119870 iterations Then probability 119875(⋀119870

119896=1119864119896) is

satisfied

lim119896rarrinfin

119875(

119870

⋀

119896=1

119864119896) = 1 (16)

Proof 119897lowast119894+1

is the choice of the 119894th optimal solution and theprobability of finding the optimal solution for the 120585th ant inthe 119896th round 119901(119896 120585) can be obtained by formula (5) becausethe node selection is an independent event

1 ge 119901 (119896 120585) =

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

(120578119897119894 119897lowast

119894+1

)120573

120583119897119894 119897lowast

119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

=

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

120578119897119894 119897lowast

119894+1

)120573

(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

)

(17)

Define 120595(119897119894+1) = (120578

119897119894 119897119894+1120578119897119894 119897lowast

119894+1

)120573(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

) and 120595max =

max120595(119897119894+1) 119894 = 1 119905minus1 So formula (17) can be expressed

as

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

120595max (18)

Because of formula (12) it is known that the minimumvalue of pheromone in the 119896th iteration of the ant colonyalgorithm is

120591min (119896) ge1205794120591 (0)

ln (119896 + 1) forall119896 ge 1 (19)

Further the maximum value of the pheromone is theinitial value by setting some parameters So

120591max = 120591 (0) (20)

And the maximum number of 119873119888(119896 120585 (119897

119894 119897119894+1)) can be

expressed as

119873119888= max 119873

119888(119896 120577 (119897

119894 119897119894+1)) 119894 = 1 119905 minus 1 (21)

where119873119888(119896 120585 (119897

119894 119897119894+1)) is the number of options By (19) (20)

and (21) we can know that formula (18) meets the followingrelationship

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(120591119897119894 119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

120595max

gt ((1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

)

119905minus1

(22)

6 Mathematical Problems in Engineering

Record

119889 = ((1205794times 120591 (0))

120572

(120591 (0))120572times 119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

= ((1205794)120572

119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

(23)

So formula (22) is expressed as

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

=119889

(ln (119896 + 1))(119905minus1)120572

(24)

From the above we can know that

1 gt119889

(ln (119896 + 1))(119905minus1)120572gt 0 (25)

The probability that the optimal solution cannot be foundby any ants in the 119896th round is 119901(119896)

119901 (119896) = (1 minus 119901 (119896 120585))119877

(26)

Probability 119875(⋁119870119896=1

119864119896) that cannot find the optimal solu-

tion for the119870 rounds is given as follows

119875(

119870

⋁

119896=1

119864119896) =

119870

prod

119896=1

119901 (119896) =

119870

prod

119896=1

(1 minus 119901 (119896 120585))119877

lt

119870

prod

119896=1

(1 minus119889

(ln (119896 + 1))(119905minus1)120572)

119877

(27)

After the logarithm to the above formula

ln119875(119870

⋁

119896=1

119864119896) lt

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

119877

= 119877

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

le minus119877 times 119889

119870

sum

119896=1

(1

ln (119896 + 1)(119905minus1)120572)

le minus119877 times 119889

119870

sum

119896=1

1

119896 + 1

lim119870rarrinfin

ln119875(119870

⋁

119896=1

119864119896) lt lim119870rarrinfin

ln(minus119877 times 119889119870

sum

119896=1

1

119896 + 1)

= minusinfin

(28)

That is 119875(⋁119870119896=1

119864119896) = 0 So lim

119896rarrinfin119875(⋀119870

119896=1119864119896) = 1

5 Numerical Simulations

The experiments are made to demonstrate the effectivenessof the proposed algorithm The algorithm is compiled inMATLAB software Experiments were conducted using acomparative method to be more persuasive along with thesame experimental conditions

51 Simulation Experiments The experiment was dividedinto three parts with two algorithms the traditional antcolony algorithm and the improved ant colony algorithm Inorder to compare the effects of the two algorithms they areused in the same environment

(1) Example 1 In this example set the simulation environ-ment as 20times20 grids and the length of each unit is 1The start-ing point is the upper left corner in the grid (05 195) and theend point is the lower right corner (195 05) (see Figure 5)

(2) Example 2 We set the environment as 30 times 30 grids andthe length of the unit is 1 The starting point is (05 85) andthe end point is (255 285) (see Figure 6)

(3) Example 3 The experiment is made in model of 40 times 40grids and the length of each unit is 1 The starting point isset at (05 395) and the ending point is set at (395 05) (seeFigure 7)

52 Results Analysis As shown in Figure 5(a) it is a globaloptimal path in the case of Example 1 Similarly Figures6(a) and 7(a) are the global optimal paths in Examples 2and 3 respectively The obstacles in each environment arerandomly selected The three experiment results show thatthe ant colony algorithm and the improved ant colonyalgorithm both can find the global optimal paths in a varietyof environments

Figures 5(b) and 5(c) depict the path length iterationcurves in Example 1 by traditional ant colony algorithmand the improved ant colony algorithm And the searchprocess about the shortest path length of each round andthe current global optimal path length can be seen fromthe figures By comparison it can be seen that the shortestpath length is found in the 49th iteration by traditionalant colony algorithm while it is found in the fifth iterationby the improved algorithm And once the optimal solutionis obtained the search will converge to the shortest pathvalue by the improved algorithm Similarly Figures 6(b) and6(c) represent the search process of the two algorithms inExample 2 The traditional ant colony algorithm finds theshortest path in the 69th iteration with value of 3563 whilethe improved algorithm gets the shortest path in the ninthtimes In the same way Figures 7(b) and 7(c) show thatthe traditional ant colony algorithm and the improved antcolony algorithm find the shortest path in the 37th and 24thiterations respectively Therefore it can be seen that thesearch efficiency of the improved ant colony algorithm is

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

End

11

16

21 22

12 13

18

23

Figure 4 Directional information representation

where

120578119897119894 119897119894+1

=1

119889119897119894+1 119897119905

119889119897119894+1119897119905

= radic(119909119897119905minus 119909119897119894+1)2

+ (119910119897119905minus 119910119897119894+1)2

(6)

The direction information 120583119894119894+1

is expressed as follows

120583119897119894 119897119894+1

= 120579120574

3 (7)

where 120574 is the number of the same directions of node 119897119894to

next 119897119894+1

and node 119897119894to end 119897

119905 120574 = 0 1 2 And 120579

3is a constant

slightly greater than 1 As shown in Figure 4 the next node of17 may be 11 12 13 16 18 21 22 and 23 where node 13 hastwo same directions as the end so the direction informationis 12057923 And as such the direction information of nodes 11 12

18 and 23 is 12057913 and for 16 21 and 22 it is 1205790

2

(3) Coverage and Updating Strategy In the traditional antcolony algorithm the next node position is decided by theroulette wheel method and repeated until the target point isobtained The pheromones of all nodes are updated by thefollowing rules after every ant 120585 searching

120591119897119894119897119894+1

(119896 120585 + 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 120585) + Δ120591119897119894119897119894+1

(119896 120585)

120591119897119894 119897119894+1

(119896 + 1 1) = (1 minus 120588) 120591119897119894119897119894+1

(119896 119877) + Δ120591119897119894119897119894+1

(119896 119877)

Δ120591119897119894119897119894+1

(119896 120585) =

1

PL119896120585

through 119897119894to 119897119894+1

0 otherwise

(8)

where 120588 is the evaporation rate of pheromone where thefunction is to avoid the pheromone accumulation 0 lt 120588 lt 1119877 is the number of the ants starting from the starting point inevery round Δ120591

119897119894119897119894+1(119896 120585) is the addition of the 119896th round 120585th

ant And PL119896120585

is the path length of the 120585th ant which can getto the end point in the 119896th round

In the improved ant colony algorithm the next nodeposition is decided by the roulette wheel method until the

target point is obtained And in a cycle the pheromone ofall the nodes is updated by following the rules in each roundof ants

1205911015840

119897119894119897119894+1(119896 + 1) = (1 minus 120588 (119896)) 120591

119897119894 119897119894+1(119896) + Δ120591

119897119894119897119894+1(119896)

12059110158401015840

119897119894119897119894+1(119896 + 1) =

1205794120591119897119894119897119894+1

(0)

ln (119896 + 1)+ Δ120591119897119894119897119894+1

(119896)

120591119897119894119897119894+1

(119896 + 1) = max 1205911015840119897119894 119897119894+1

(119896 + 1) 12059110158401015840

119897119894119897119894+1(119896 + 1)

(9)

Among them Δ120591119897119894119897119894+1

(119896) is the pheromone update part of119897119894to 119897119894+1

in the 119896th round 1205794is a constant and the maximum

value of the pheromone in the iterative process is the initialset value 120591max = 120591(0) 120588(119896) is the pheromone evaporationcoefficient 0 lt 120588(119896) lt 1 In the entire search space

Δ120591119897119894119897119894+1

(119896) = 119900 (Δ120591119897119894119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 120585)

Δ120591119897119894119897119894+1

(119896 119877 minus 1) Δ120591119897119894 119897119894+1

(119896 119877))

(10)

where 119900(Δ120591119897119894 119897119894+1

(119896 1) Δ120591119897119894119897119894+1

(119896 119877)) is a nonzero pherom-one space covering operation For example

119860 (1) =[[

[

1 2 4

0 0 1

3 0 0

]]

]

119860 (2) =[[

[

2 3 5

1 0 0

2 0 0

]]

]

so 119900 (119860 (1) 119860 (2)) =[[

[

2 3 5

1 0 1

2 0 0

]]

]

(11)

In formula (10)

Δ120591119897119894119897119894+1

(119896 120585) =

119876

PL119896120585

120591119897119894119897119894+1

(119896 120585) through 119897119894to 119897119894+1

0 otherwise(12)

Among them119876 is a constant It can be seen that if the antcan get to the end point the added part of the pheromone isin inverse proportion with the length and if the ant cannotreach the ending point it is recorded as 0 The rule canguide the search for the shortest path improve the speedof convergence and avoid the possibility of the pheromoneaccumulation caused by repeated search meanwhile theeffect on the performance of the algorithm caused by thequantity of the ants is reduced

(4) Evaporation Coefficient Segment When the problemscale is relatively large due to the presence of pheromoneevaporation the pheromone of some nodes will be reducedgreatly even close to zero which reduces the search ability ofthe algorithm When 120588 is large the search ability is affected

Mathematical Problems in Engineering 5

by repeated selection Meanwhile when 120588 is small the searchability of the algorithm is enhanced but the convergencespeed is decreased Therefore the heuristic informationcoefficient is adjusted as follows

120588 (119896 + 1) =

1205795120588 (119896) 120588 (119896) ge 120588min

120588min 120588 (119896) lt 120588min(13)

where 1205795is a constant less than 1 119896 is the number of

search rounds 119896 = 0 1 119870 120588(0) = 120588max 120588max and120588min are the maximum and minimum of the coefficientof evaporation The pheromone evaporation coefficient offragmentation can enhance the search ability in the initialstage of search increase the convergence speed later andimprove the performance of ant colony algorithm

42 Algorithm Steps The improved ant colony algorithm isto find the optimal path according to the following steps

Step 1 The nonuniform distribution of initial pheromone isproposed and pheromone matrix 119879 is constructed accordingto formula (2)

Step 2 Send 119870 rounds and each round of 119877 ants which areplaced at the starting point

Step 3 Send a round of ants and each ant to select the nodeaccording to the roulette wheel method with the probabilityof each point calculated by formulas (5) (6) and (7)

Step 4 After a round of ant search the pheromone wascalculated by formulas (10) and (12) the evaporation coeffi-cient was calculated by (13) and the pheromone was updatedaccording to formulas (9)

Step 5 Send the next round of ants and repeat above Steps 3and 4 until the end of the iteration

Step 6 Record the shortest path of each round then draw thecurve of the shortest path length of each round

Step 7 Compare the shortest paths of all the ants Finallyoutput the current global optimal path

43 Proof of Convergence The proof depends on a necessaryassumption there is a path between the starting point and theending at least According to formulas (10) and (12) the addedpart of pheromonemust be greater than or equal to 0That is

Δ120591119897119894119897119894+1

(119896) ge 0 (14)

Formulas (9) show that the pheromone can be expressedas follows

120591119897119894119897119894+1

(119896) ge max((1 minus 120588 (119896))119896 120591 (0) 1205794120591 (0)

ln (119896 + 1)) (15)

Event 119864119896means that the optimal solution is obtained for

the first time in the 119896th iteration Therefore event ⋀119870119896=1

119864119896

denotes that the algorithm can find the optimal solution forthe first time for 119870 iterations Then probability 119875(⋀119870

119896=1119864119896) is

satisfied

lim119896rarrinfin

119875(

119870

⋀

119896=1

119864119896) = 1 (16)

Proof 119897lowast119894+1

is the choice of the 119894th optimal solution and theprobability of finding the optimal solution for the 120585th ant inthe 119896th round 119901(119896 120585) can be obtained by formula (5) becausethe node selection is an independent event

1 ge 119901 (119896 120585) =

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

(120578119897119894 119897lowast

119894+1

)120573

120583119897119894 119897lowast

119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

=

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

120578119897119894 119897lowast

119894+1

)120573

(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

)

(17)

Define 120595(119897119894+1) = (120578

119897119894 119897119894+1120578119897119894 119897lowast

119894+1

)120573(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

) and 120595max =

max120595(119897119894+1) 119894 = 1 119905minus1 So formula (17) can be expressed

as

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

120595max (18)

Because of formula (12) it is known that the minimumvalue of pheromone in the 119896th iteration of the ant colonyalgorithm is

120591min (119896) ge1205794120591 (0)

ln (119896 + 1) forall119896 ge 1 (19)

Further the maximum value of the pheromone is theinitial value by setting some parameters So

120591max = 120591 (0) (20)

And the maximum number of 119873119888(119896 120585 (119897

119894 119897119894+1)) can be

expressed as

119873119888= max 119873

119888(119896 120577 (119897

119894 119897119894+1)) 119894 = 1 119905 minus 1 (21)

where119873119888(119896 120585 (119897

119894 119897119894+1)) is the number of options By (19) (20)

and (21) we can know that formula (18) meets the followingrelationship

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(120591119897119894 119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

120595max

gt ((1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

)

119905minus1

(22)

6 Mathematical Problems in Engineering

Record

119889 = ((1205794times 120591 (0))

120572

(120591 (0))120572times 119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

= ((1205794)120572

119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

(23)

So formula (22) is expressed as

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

=119889

(ln (119896 + 1))(119905minus1)120572

(24)

From the above we can know that

1 gt119889

(ln (119896 + 1))(119905minus1)120572gt 0 (25)

The probability that the optimal solution cannot be foundby any ants in the 119896th round is 119901(119896)

119901 (119896) = (1 minus 119901 (119896 120585))119877

(26)

Probability 119875(⋁119870119896=1

119864119896) that cannot find the optimal solu-

tion for the119870 rounds is given as follows

119875(

119870

⋁

119896=1

119864119896) =

119870

prod

119896=1

119901 (119896) =

119870

prod

119896=1

(1 minus 119901 (119896 120585))119877

lt

119870

prod

119896=1

(1 minus119889

(ln (119896 + 1))(119905minus1)120572)

119877

(27)

After the logarithm to the above formula

ln119875(119870

⋁

119896=1

119864119896) lt

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

119877

= 119877

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

le minus119877 times 119889

119870

sum

119896=1

(1

ln (119896 + 1)(119905minus1)120572)

le minus119877 times 119889

119870

sum

119896=1

1

119896 + 1

lim119870rarrinfin

ln119875(119870

⋁

119896=1

119864119896) lt lim119870rarrinfin

ln(minus119877 times 119889119870

sum

119896=1

1

119896 + 1)

= minusinfin

(28)

That is 119875(⋁119870119896=1

119864119896) = 0 So lim

119896rarrinfin119875(⋀119870

119896=1119864119896) = 1

5 Numerical Simulations

The experiments are made to demonstrate the effectivenessof the proposed algorithm The algorithm is compiled inMATLAB software Experiments were conducted using acomparative method to be more persuasive along with thesame experimental conditions

51 Simulation Experiments The experiment was dividedinto three parts with two algorithms the traditional antcolony algorithm and the improved ant colony algorithm Inorder to compare the effects of the two algorithms they areused in the same environment

(1) Example 1 In this example set the simulation environ-ment as 20times20 grids and the length of each unit is 1The start-ing point is the upper left corner in the grid (05 195) and theend point is the lower right corner (195 05) (see Figure 5)

(2) Example 2 We set the environment as 30 times 30 grids andthe length of the unit is 1 The starting point is (05 85) andthe end point is (255 285) (see Figure 6)

(3) Example 3 The experiment is made in model of 40 times 40grids and the length of each unit is 1 The starting point isset at (05 395) and the ending point is set at (395 05) (seeFigure 7)

52 Results Analysis As shown in Figure 5(a) it is a globaloptimal path in the case of Example 1 Similarly Figures6(a) and 7(a) are the global optimal paths in Examples 2and 3 respectively The obstacles in each environment arerandomly selected The three experiment results show thatthe ant colony algorithm and the improved ant colonyalgorithm both can find the global optimal paths in a varietyof environments

Figures 5(b) and 5(c) depict the path length iterationcurves in Example 1 by traditional ant colony algorithmand the improved ant colony algorithm And the searchprocess about the shortest path length of each round andthe current global optimal path length can be seen fromthe figures By comparison it can be seen that the shortestpath length is found in the 49th iteration by traditionalant colony algorithm while it is found in the fifth iterationby the improved algorithm And once the optimal solutionis obtained the search will converge to the shortest pathvalue by the improved algorithm Similarly Figures 6(b) and6(c) represent the search process of the two algorithms inExample 2 The traditional ant colony algorithm finds theshortest path in the 69th iteration with value of 3563 whilethe improved algorithm gets the shortest path in the ninthtimes In the same way Figures 7(b) and 7(c) show thatthe traditional ant colony algorithm and the improved antcolony algorithm find the shortest path in the 37th and 24thiterations respectively Therefore it can be seen that thesearch efficiency of the improved ant colony algorithm is

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

by repeated selection Meanwhile when 120588 is small the searchability of the algorithm is enhanced but the convergencespeed is decreased Therefore the heuristic informationcoefficient is adjusted as follows

120588 (119896 + 1) =

1205795120588 (119896) 120588 (119896) ge 120588min

120588min 120588 (119896) lt 120588min(13)

where 1205795is a constant less than 1 119896 is the number of

search rounds 119896 = 0 1 119870 120588(0) = 120588max 120588max and120588min are the maximum and minimum of the coefficientof evaporation The pheromone evaporation coefficient offragmentation can enhance the search ability in the initialstage of search increase the convergence speed later andimprove the performance of ant colony algorithm

42 Algorithm Steps The improved ant colony algorithm isto find the optimal path according to the following steps

Step 1 The nonuniform distribution of initial pheromone isproposed and pheromone matrix 119879 is constructed accordingto formula (2)

Step 2 Send 119870 rounds and each round of 119877 ants which areplaced at the starting point

Step 3 Send a round of ants and each ant to select the nodeaccording to the roulette wheel method with the probabilityof each point calculated by formulas (5) (6) and (7)

Step 4 After a round of ant search the pheromone wascalculated by formulas (10) and (12) the evaporation coeffi-cient was calculated by (13) and the pheromone was updatedaccording to formulas (9)

Step 5 Send the next round of ants and repeat above Steps 3and 4 until the end of the iteration

Step 6 Record the shortest path of each round then draw thecurve of the shortest path length of each round

Step 7 Compare the shortest paths of all the ants Finallyoutput the current global optimal path

43 Proof of Convergence The proof depends on a necessaryassumption there is a path between the starting point and theending at least According to formulas (10) and (12) the addedpart of pheromonemust be greater than or equal to 0That is

Δ120591119897119894119897119894+1

(119896) ge 0 (14)

Formulas (9) show that the pheromone can be expressedas follows

120591119897119894119897119894+1

(119896) ge max((1 minus 120588 (119896))119896 120591 (0) 1205794120591 (0)

ln (119896 + 1)) (15)

Event 119864119896means that the optimal solution is obtained for

the first time in the 119896th iteration Therefore event ⋀119870119896=1

119864119896

denotes that the algorithm can find the optimal solution forthe first time for 119870 iterations Then probability 119875(⋀119870

119896=1119864119896) is

satisfied

lim119896rarrinfin

119875(

119870

⋀

119896=1

119864119896) = 1 (16)

Proof 119897lowast119894+1

is the choice of the 119894th optimal solution and theprobability of finding the optimal solution for the 120585th ant inthe 119896th round 119901(119896 120585) can be obtained by formula (5) becausethe node selection is an independent event

1 ge 119901 (119896 120585) =

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

(120578119897119894 119897lowast

119894+1

)120573

120583119897119894 119897lowast

119894+1

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

(120578119897119894 119897119894+1

)120573

120583119897119894 119897119894+1

=

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

(120578119897119894 119897119894+1

120578119897119894 119897lowast

119894+1

)120573

(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

)

(17)

Define 120595(119897119894+1) = (120578

119897119894 119897119894+1120578119897119894 119897lowast

119894+1

)120573(120583119897119894 119897119894+1

120583119897119894 119897lowast

119894+1

) and 120595max =

max120595(119897119894+1) 119894 = 1 119905minus1 So formula (17) can be expressed

as

119905minus1

prod

119894=1

(120591119897119894119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894 119897119894+1)

120572

120595max (18)

Because of formula (12) it is known that the minimumvalue of pheromone in the 119896th iteration of the ant colonyalgorithm is

120591min (119896) ge1205794120591 (0)

ln (119896 + 1) forall119896 ge 1 (19)

Further the maximum value of the pheromone is theinitial value by setting some parameters So

120591max = 120591 (0) (20)

And the maximum number of 119873119888(119896 120585 (119897

119894 119897119894+1)) can be

expressed as

119873119888= max 119873

119888(119896 120577 (119897

119894 119897119894+1)) 119894 = 1 119905 minus 1 (21)

where119873119888(119896 120585 (119897

119894 119897119894+1)) is the number of options By (19) (20)

and (21) we can know that formula (18) meets the followingrelationship

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(120591119897119894 119897lowast

119894+1

)120572

sum119897119894+1isinallowed(119887) (120591119897119894119897119894+1)

120572

120595max

gt ((1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

)

119905minus1

(22)

6 Mathematical Problems in Engineering

Record

119889 = ((1205794times 120591 (0))

120572

(120591 (0))120572times 119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

= ((1205794)120572

119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

(23)

So formula (22) is expressed as

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

=119889

(ln (119896 + 1))(119905minus1)120572

(24)

From the above we can know that

1 gt119889

(ln (119896 + 1))(119905minus1)120572gt 0 (25)

The probability that the optimal solution cannot be foundby any ants in the 119896th round is 119901(119896)

119901 (119896) = (1 minus 119901 (119896 120585))119877

(26)

Probability 119875(⋁119870119896=1

119864119896) that cannot find the optimal solu-

tion for the119870 rounds is given as follows

119875(

119870

⋁

119896=1

119864119896) =

119870

prod

119896=1

119901 (119896) =

119870

prod

119896=1

(1 minus 119901 (119896 120585))119877

lt

119870

prod

119896=1

(1 minus119889

(ln (119896 + 1))(119905minus1)120572)

119877

(27)

After the logarithm to the above formula

ln119875(119870

⋁

119896=1

119864119896) lt

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

119877

= 119877

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

le minus119877 times 119889

119870

sum

119896=1

(1

ln (119896 + 1)(119905minus1)120572)

le minus119877 times 119889

119870

sum

119896=1

1

119896 + 1

lim119870rarrinfin

ln119875(119870

⋁

119896=1

119864119896) lt lim119870rarrinfin

ln(minus119877 times 119889119870

sum

119896=1

1

119896 + 1)

= minusinfin

(28)

That is 119875(⋁119870119896=1

119864119896) = 0 So lim

119896rarrinfin119875(⋀119870

119896=1119864119896) = 1

5 Numerical Simulations

The experiments are made to demonstrate the effectivenessof the proposed algorithm The algorithm is compiled inMATLAB software Experiments were conducted using acomparative method to be more persuasive along with thesame experimental conditions

51 Simulation Experiments The experiment was dividedinto three parts with two algorithms the traditional antcolony algorithm and the improved ant colony algorithm Inorder to compare the effects of the two algorithms they areused in the same environment

(1) Example 1 In this example set the simulation environ-ment as 20times20 grids and the length of each unit is 1The start-ing point is the upper left corner in the grid (05 195) and theend point is the lower right corner (195 05) (see Figure 5)

(2) Example 2 We set the environment as 30 times 30 grids andthe length of the unit is 1 The starting point is (05 85) andthe end point is (255 285) (see Figure 6)

(3) Example 3 The experiment is made in model of 40 times 40grids and the length of each unit is 1 The starting point isset at (05 395) and the ending point is set at (395 05) (seeFigure 7)

52 Results Analysis As shown in Figure 5(a) it is a globaloptimal path in the case of Example 1 Similarly Figures6(a) and 7(a) are the global optimal paths in Examples 2and 3 respectively The obstacles in each environment arerandomly selected The three experiment results show thatthe ant colony algorithm and the improved ant colonyalgorithm both can find the global optimal paths in a varietyof environments

Figures 5(b) and 5(c) depict the path length iterationcurves in Example 1 by traditional ant colony algorithmand the improved ant colony algorithm And the searchprocess about the shortest path length of each round andthe current global optimal path length can be seen fromthe figures By comparison it can be seen that the shortestpath length is found in the 49th iteration by traditionalant colony algorithm while it is found in the fifth iterationby the improved algorithm And once the optimal solutionis obtained the search will converge to the shortest pathvalue by the improved algorithm Similarly Figures 6(b) and6(c) represent the search process of the two algorithms inExample 2 The traditional ant colony algorithm finds theshortest path in the 69th iteration with value of 3563 whilethe improved algorithm gets the shortest path in the ninthtimes In the same way Figures 7(b) and 7(c) show thatthe traditional ant colony algorithm and the improved antcolony algorithm find the shortest path in the 37th and 24thiterations respectively Therefore it can be seen that thesearch efficiency of the improved ant colony algorithm is

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Record

119889 = ((1205794times 120591 (0))

120572

(120591 (0))120572times 119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

= ((1205794)120572

119873119888(119896 120585 (119897

119894 119897119894+1)) times 120595max

)

119905minus1

(23)

So formula (22) is expressed as

1 ge 119875 (119896 120577) gt

119905minus1

prod

119894=1

(1205794times 120591 (0) ln (119896 + 1))120572

(120591 (0))120572times 119873119888times 120595max

=119889

(ln (119896 + 1))(119905minus1)120572

(24)

From the above we can know that

1 gt119889

(ln (119896 + 1))(119905minus1)120572gt 0 (25)

The probability that the optimal solution cannot be foundby any ants in the 119896th round is 119901(119896)

119901 (119896) = (1 minus 119901 (119896 120585))119877

(26)

Probability 119875(⋁119870119896=1

119864119896) that cannot find the optimal solu-

tion for the119870 rounds is given as follows

119875(

119870

⋁

119896=1

119864119896) =

119870

prod

119896=1

119901 (119896) =

119870

prod

119896=1

(1 minus 119901 (119896 120585))119877

lt

119870

prod

119896=1

(1 minus119889

(ln (119896 + 1))(119905minus1)120572)

119877

(27)

After the logarithm to the above formula

ln119875(119870

⋁

119896=1

119864119896) lt

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

119877

= 119877

119870

sum

119896=1

ln[1 minus ( 119889

ln (119896 + 1)(119905minus1)120572)]

le minus119877 times 119889

119870

sum

119896=1

(1

ln (119896 + 1)(119905minus1)120572)

le minus119877 times 119889

119870

sum

119896=1

1

119896 + 1

lim119870rarrinfin

ln119875(119870

⋁

119896=1

119864119896) lt lim119870rarrinfin

ln(minus119877 times 119889119870

sum

119896=1

1

119896 + 1)

= minusinfin

(28)

That is 119875(⋁119870119896=1

119864119896) = 0 So lim

119896rarrinfin119875(⋀119870

119896=1119864119896) = 1

5 Numerical Simulations

The experiments are made to demonstrate the effectivenessof the proposed algorithm The algorithm is compiled inMATLAB software Experiments were conducted using acomparative method to be more persuasive along with thesame experimental conditions

51 Simulation Experiments The experiment was dividedinto three parts with two algorithms the traditional antcolony algorithm and the improved ant colony algorithm Inorder to compare the effects of the two algorithms they areused in the same environment

(1) Example 1 In this example set the simulation environ-ment as 20times20 grids and the length of each unit is 1The start-ing point is the upper left corner in the grid (05 195) and theend point is the lower right corner (195 05) (see Figure 5)

(2) Example 2 We set the environment as 30 times 30 grids andthe length of the unit is 1 The starting point is (05 85) andthe end point is (255 285) (see Figure 6)

(3) Example 3 The experiment is made in model of 40 times 40grids and the length of each unit is 1 The starting point isset at (05 395) and the ending point is set at (395 05) (seeFigure 7)

52 Results Analysis As shown in Figure 5(a) it is a globaloptimal path in the case of Example 1 Similarly Figures6(a) and 7(a) are the global optimal paths in Examples 2and 3 respectively The obstacles in each environment arerandomly selected The three experiment results show thatthe ant colony algorithm and the improved ant colonyalgorithm both can find the global optimal paths in a varietyof environments

Figures 5(b) and 5(c) depict the path length iterationcurves in Example 1 by traditional ant colony algorithmand the improved ant colony algorithm And the searchprocess about the shortest path length of each round andthe current global optimal path length can be seen fromthe figures By comparison it can be seen that the shortestpath length is found in the 49th iteration by traditionalant colony algorithm while it is found in the fifth iterationby the improved algorithm And once the optimal solutionis obtained the search will converge to the shortest pathvalue by the improved algorithm Similarly Figures 6(b) and6(c) represent the search process of the two algorithms inExample 2 The traditional ant colony algorithm finds theshortest path in the 69th iteration with value of 3563 whilethe improved algorithm gets the shortest path in the ninthtimes In the same way Figures 7(b) and 7(c) show thatthe traditional ant colony algorithm and the improved antcolony algorithm find the shortest path in the 37th and 24thiterations respectively Therefore it can be seen that thesearch efficiency of the improved ant colony algorithm is

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

2

4

6

8

10

12

14

16

18

20

5 10 150 20

(a)

X 49Y 2921

The current global optimal pathThe shortest path of each round

10 20 30 40 50 60 70 800Iteration number

28

30

32

34

36

38

40

42

44

Path

leng

th(b)

X 5Y 2921

The current global optimal pathThe shortest path of each round

28

30

32

34

36

38

40

42

44

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

25

30

35

40

45

50

55

The p

ath

aver

age

10 20 30 40 50 60 70 800Iterative number

The traditional algorithmThe improved algorithm

(d)

Figure 5 (a) Simulation results in 20 times 20 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

significantly higher compared to the traditional ant colonyalgorithm in a variety of environment

The mean value of path iterative length often representsthe convergence ability of the algorithm Figures 5(d) 6(d)and 7(d) present the contrast of the two algorithms on thesearch path average valueThe improved algorithmnot only isfaster than the ant colony algorithm but also can avoid fallinginto local minimum point and obtain the global optimalsolution effectively either in simple environment or in acomplex one

6 Conclusion

In this paper an improved ant colony algorithm is proposedfor the 8 control operating units of the omnidirectionalmobile vehicle The grid method is used to establish theenvironment model and the tabu list is introduced By usingthe tabu list to show the obstacles and the units that havepassed through it is flexible to deal with obstacles and avoidduplicating the path In this paper the initial distributionof nonuniformity pheromone is presented which improves

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

5

10

15

20

25

30

5 10 15 20 250 30

(a)

X 69Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

X 9Y 3563

The current global optimal pathThe shortest path of each round

35

40

45

50

55

60

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(c)

The traditional algorithmThe improved algorithm

35

40

45

50

55

60

65

70

75

The p

ath

aver

age

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 6 (a) Simulation results in 30 times 30 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

the time efficiency and the simplicity of the algorithm andreduces the search space of ant colony algorithm Addingthe direction of the selection strategy can get more effectiveinformation as the heuristic information more actively guidethe search behavior of ants and reduce the blindness Byusing the rule of coverage the search probability is reducedthe stability of the algorithm is guaranteed and the effectof the quantity of the ants on the performance of the algo-rithm is guaranteed The pheromone evaporation coefficientis segmented and adjusted which can effectively balancethe convergence speed and search ability of the algorithm

Finally it is strictly proven that the probability of finding anoptimal solution is limited to 1 by the improved algorithmThe improved ant colony algorithm in solving the shortestpath planning problem of the omnidirectional mobile vehiclehas very good performance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

5 10 15 20 25 30 35 4000

5

10

15

20

25

30

35

40

(a)

The current global optimal pathThe shortest path of each round

X 37Y 5984

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

(b)

The current global optimal pathThe shortest path of each round

55

60

65

70

75

80

85

90

95

100

105

Path

leng

th

10 20 30 40 50 60 70 800Iteration number

X 24Y 5984

(c)

The traditional algorithmThe improved algorithm

60

70

80

90

100

110

120

130

140Th

e pat

h av

erag

e

10 20 30 40 50 60 70 800Iteration number

(d)

Figure 7 (a) Simulation results in 40 times 40 grids (b) The iterative curve by the traditional algorithm (c) The iterative curve by the improvedalgorithm (d) The path average curve length by two algorithms

Acknowledgments

This research is supported by Science and TechnologyResearch Project of Colleges and Universities in HebeiProvince (Grant no ZD2016142) Natural Science Foundationof Hebei Province (Grant no F2014208013) and DoctoralScientific Research Startup Foundation of Hebei Universityof Science and Technology (Grant no QD201302)

References

[1] Y Y Gao X G Ruan H J Song and J J Yu ldquoPath planningmethod for mobile robot based on a hybrid learning approachrdquoControl and Decision vol 27 no 12 pp 1822ndash1827 2012

[2] T Lozano-Perez and M A Wesley ldquoAn algorithm for planningcollision-free paths among polyedral obstaclesrdquo Communica-tions of ACM vol 2 pp 959ndash962 1979

[3] E Masehian and D Sedighizadeh ldquoClassic and heuristicapproaches in robotmotion planningmdasha chronological reviewrdquoInternational Journal of Mechanical Industrial Science andEngineering vol 1 no 5 pp 101ndash106 2007

[4] W Lu G Zhang and S Ferrari ldquoAn information potentialapproach to integrated sensor path planning and controlrdquo IEEETransactions on Robotics vol 30 no 4 pp 919ndash934 2014

[5] J Lee B-Y Kang and D-W Kim ldquoFast genetic algorithm forrobot path planningrdquo Electronics Letters vol 49 no 23 pp1449ndash1451 2013

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[6] Y Fu M Ding C Zhou and H Hu ldquoRoute planning forunmanned aerial vehicle (UAV) on the sea using hybrid dif-ferential evolution and quantum-behaved particle swarm opti-mizationrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 43 no 6 pp 1451ndash1465 2013

[7] M A P Garcia O Montiel O Castillo R Sepulveda and PMelin ldquoPath planning for autonomous mobile robot navigationwith ant colony optimization and fuzzy cost function evalua-tionrdquo Applied Soft Computing vol 9 no 3 pp 1102ndash1110 2009

[8] H Mo and L Xu ldquoResearch of biogeography particle swarmoptimization for robot path planningrdquo Neurocomputing vol148 pp 91ndash99 2015

[9] R-E Precup E M Petriu M-B Radac E-I Voisan and FDragan ldquoAdaptive charged system search approach to pathplanning for multiple mobile robotsrdquo in Proceedings of the 16thInternational Federation of Automatic Control pp 294ndash299Sozopol Bulgaria September 2015

[10] F Ducho A Babineca M Kajana et al ldquoPath planningwith modified a star algorithm for a mobile robotrdquo ProcediaEngineering vol 96 pp 59ndash69 2014

[11] A K Guruji H Agarwal and D K Parsediya ldquoTime-efficientAlowast algorithm for robot path planningrdquoProcedia Technology vol23 pp 144ndash149 2016

[12] A Hidalgo-Paniagua M A Vega-Rodrıguez and J FerruzldquoApplying theMOVNS (multi-objective variable neighborhoodsearch) algorithm to solve the path planning problem inmobileroboticsrdquo Expert Systems with Applications vol 58 pp 20ndash352016

[13] A T Rashid A A Ali M Frasca and L Fortuna ldquoPathplanning with obstacle avoidance based on visibility binary treealgorithmrdquoRobotics andAutonomous Systems vol 61 no 12 pp1440ndash1449 2013

[14] M Davoodi F Panahi A Mohades and S N HashemildquoMulti-objective path planning in discrete spacerdquo Applied SoftComputing vol 13 no 1 pp 709ndash720 2013

[15] A Hidalgo-Paniagua M A Vega-Rodrıguez J Ferruz and NPavon ldquoMOSFLA-MRPP multi-objective shuffled frog-leapingalgorithm applied to mobile robot path planningrdquo EngineeringApplications of Artificial Intelligence vol 44 no 2342 pp 123ndash136 2015

[16] G-F Deng X-P Zhang andY-P Liu ldquoAnt colony optimizationand particle swarm optimization for robot-path planning inobstacle environmentrdquo Control Theory amp Applications vol 26no 8 pp 879ndash883 2009

[17] J He Z Tu and Y Niu ldquoA method of mobile robotic pathplanning based on integrating of GA and ACOrdquo ComputerSimulation vol 27 no 3 pp 170ndash174 2010

[18] P K Tiwari and D P Vidyarthi ldquoImproved auto control antcolony optimization using lazy ant approach for grid schedulingproblemrdquo Future Generation Computer Systems vol 60 pp 78ndash89 2016

[19] P Wang H-T Lin and T-S Wang ldquoAn improved ant colonysystem algorithm for solving the IP traceback problemrdquo Infor-mation Sciences vol 326 pp 172ndash187 2016

[20] W-Y Jiang Y Lin M Chen and Y-Y Yu ldquoA co-evolutionaryimproved multi-ant colony optimization for ship multiple andbranch pipe route designrdquo Ocean Engineering vol 102 pp 63ndash70 2015

[21] M Saidi-Mehrabad S Dehnavi-Arani F Evazabadian and VMahmoodian ldquoAn Ant Colony Algorithm (ACA) for solving

the new integrated model of job shop scheduling and conflict-free routing of AGVsrdquo Computers and Industrial Engineeringvol 86 pp 2ndash13 2015

[22] Z Wang X Zhu and Q Han ldquoMobile robot path planningbased on parameter optimization ant colony algorithmrdquo Proce-dia Engineering vol 15 pp 2738ndash2741 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of