Research ArticleAnt Colony Optimization Using Common Social Informationand Self-Memory

Yoshiki Tamura1 Tomoko Sakiyama 2 and Ikuo Arizono 1

1Graduate School of Natural Science and Technology Okayama University Okayama 700-8530 Japan2Department of Information Systems Science Faculty of Science and Engineering Soka University Tokyo 192-8577 Japan

Correspondence should be addressed to Ikuo Arizono arizonookayama-uacjp

Received 3 November 2020 Revised 18 December 2020 Accepted 26 December 2020 Published 8 January 2021

Academic Editor Dimitri Volchenkov

Copyright copy 2021 Yoshiki Tamura et al -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Ant colony optimization (ACO) which is one of the metaheuristics imitating real ant foraging behavior is an effective method tofind a solution for the traveling salesman problem (TSP) -e rank-based ant system (ASrank) has been proposed as a developedversion of the fundamental model AS of ACO In the ASrank since only ant agents that have found one of some excellent solutionsare let to regulate the pheromone the pheromone concentrates on a specific route As a result although the ASrank can find arelatively good solution in a short time it has the disadvantage of being prone falling into a local solution because the pheromoneconcentrates on a specific route -is problem seems to come from the loss of diversity in route selection according to the rapidaccumulation of pheromones to the specific routes Some ACOmodels not just the ASrank also suffer from this problem of loss ofdiversity in route selection It can be considered that the diversity of solutions as well as the selection of solutions is an importantfactor in the solution system by swarm intelligence such as ACO In this paper to solve this problem we introduce the ant systemusing individual memories (ASIM) aiming to improve the ability to solve TSP while maintaining the diversity of the behavior ofeach ant We apply the existing ACO algorithms and ASIM to some TSP benchmarks and compare the ability to solve TSP

1 Introduction

Dorigo proposed ant colony optimization (ACO) as ametaheuristic for solving combination optimization prob-lems [1 2] ACO is one of the metaheuristics imitating thegroup intellectual behavior of social insects especially for-aging behavior of ants When real ants succeed in findingfood sources and so on they get back to the nest site whilelaying pheromones on the ground Other ants are inducedby pheromones and induced ants can reach a target placeFurthermore they will also back to the nest site while layingpheromones on the ground Since pheromones have vola-tility the pheromone of longer path will disappear comparedto the pheromone of shorter path In this way ant colony canfind a short path from the nest to the feeding site and theycan transport resources effectively ACO is a system de-veloped by imitating such a mechanism in ant foragingbehavior and is used as a method to find a solution for

combinatorial optimization problems represented by TSP-en ACO might be a useful solving tool for TSP and somemanufacturing problems in the real world [3ndash7]

Since the ant system (AS) which is the basic model ofACO has been proposed by Dorigo [1 2] various improvedmodels have been constructed -e rank-based ant system(ASrank) proposed by Bullnheimer is one of them [8] -ismodel has a feature at the procedure for pheromone reg-ulation In solving TSP by using Saran after completingmaking tours by all ant agents ant agents are sorted indescending order of tour lengths which have been found outby each ant agent -en only a certain number of superiorant agents called elite agents who have succeeded in findingshort tours are allowed to deposit pheromones Pheromoneson only the specific paths belonging to the shorter tours areadded by this feature As a result this feature that thepheromone converges on a specific route makes all antagents converge rapidly to a tour with short length

HindawiComplexityVolume 2021 Article ID 6610670 7 pageshttpsdoiorg10115520216610670

Although it contributes to finding the comparative rea-sonable solution within early time in solving TSP it has anissue that the system easily falls into a local solution becausepheromones on only some specific paths will be depositedFalling into a local optimal solution means finding no betternew solution compared to the current solution -at is itmeans that the ability to solve a problem in ASrank is im-paired because all ant agents converge rapidly to a tentativetour with comparatively shorter length

Such a trouble exists in many ACO algorithms not justASrank For this trouble Liu et al [9] have noted that particleswarm optimization (PSO) has a disadvantage of quicklylosing population diversity although it is a useful tool as ametaheuristic because of its simple concept and easyimplementation Also Mahapatra et al [10] have consideredthe diversity in the swarm intelligence- (SI-) based centralizedclustering solutions an important factor Hence it can beconsidered that the diversity in the selection of solutions is animportant factor in ACO as the solution system Furthermoreit might be considered for ASrank to have an incongruousmechanism as one of nature-inspired algorithms -at isalthough the ASrank has introduced the mechanism to selectelite ant agents from the outside viewpoint of the system thehabits of actual ants in the swarm do not have such amechanism based on the outside viewpoint

On the one hand Gruter et al [11] have reported thatindividual ants make decisions using both social informationas pheromones and own memories as individual informa-tion during foraging activities in the natural world -ere-fore it can be seen that the swarm of actual ants keeps thediversity in the decision making as a swarm In contrastwhile many existing ACO algorithms including AS andASrank imitate the behavior of ants all ant agents act usingonly social information shared by the system and not in-dividual information

-erefore in this paper we propose a novel ant systemmaintaining the diversity of the behavior of each ant agentby incorporating the mechanism of making decisions basedon onersquos own memory and social information -en thisnovel ant system would be named the ant system usingindividual memories (ASIM) Further ASIM does not re-quire the outside viewpoint of the system and agents inASIM build solutions using their ownmemory in addition tothe common social information such as heuristic andpheromone information -rough the unique behavior inASIM that all ant agents make decision using individual ownmemory in addition to the common social information itcan be expected for ASIM to keep diversity as solving systemand avoid falling into the local solution Based on numericalresults about four TSP benchmarks of different sizes [12] thesolving ability of ASIMwith the purpose of keeping diversityof solutions using nature-inspired memory information isevaluated by comparing with the existing ACO models thatis AS and ASrank

2 Materials and Methods

21 Ant System (AS) In this section we explain the antsystem (AS) that is the fundamental model of ACO proposed

by Dorigo [12] To solve TSP with AS the artificial antagents are placed one by one in all cities -en every antagent makes a tour under the condition of transiting everycity exactly once in one tour-e probability pk

ij(t) that city j

is selected to be visited immediately after city i can be writtenin a formula as follows

pkij(t)

τij(t)1113960 1113961αηij(t)1113960 1113961

β

1113936lisinNkτil(t)1113858 1113859

α ηil(t)1113858 1113859β if j isin Nk

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(1)

where t iteration counter τij intensity of pheromone trailbetween cities i and j α weight for τij ηij visibility of city i

from city j (ηij 1dij) β weight for ηij Nk set of cities thatthe ant agent k has not visited yet and dij distance cities i

and j-is city selection process is repeated until all ant agents

have completed a tour After creating one tour every antagent deposits pheromone on all paths where they havepassed-e amount of pheromone deposited depends on thelength of the tour -e pheromone regulation process isdefined as follows

τij(t + 1) (1 minus ρ)τij(t) + Δτij(t) (2)

where

Δτij(t) 1113944

m

k11Lk(t) if ant k travels on edge (i j)

0 otherwise

⎧⎪⎪⎨

⎪⎪⎩

ρ evaporation rate of the pheromone which is parameter toregulate τij Δτij total increase of trail level on edge (i j) mnumber of ant agents and Lk tour length found out by antagent k

-e amount of existing pheromone on edge τij is reducedby the evaporation rate of the pheromone ρ before the nexttour and the pheromones on the unpassed edge will onlyevaporate -erefore long-distance tours are eliminatedthrough the above pheromone regulation and this regulationpromotes early convergence for exploring tour -e bestsolution will be updated when the solution is newly found thathas the shorter tour length than the current best solution Intour exploration by using AS successive procedures oftraveling pheromone regulation and updating the best so-lution are repeated as mentioned above -e exploringsimulation finishes when the exit condition is completed

22 Rank-Based Ant System (ASrank) In this section weexplain the rank-based ant system (ASrank) proposed byBullnheimer et al [8] as the improved algorithm from AS InASrank the way of determining the probability for nextdestination selection from the present city is same as that inAS shown in equation (1) However the way of pheromoneregulation is different from AS In pheromone regulationprocess of ASrank all ant agents are ranked by sorting inascending order of tour lengths At that time ant agentsranked between the top and the (σ minus 1)-th become elite antagents who can contribute to pheromone deposition In

2 Complexity

addition ant agent who has found the best so-far solutionuntil current exploring iteration has been identified as thevirtual ant agent and the virtual ant agent can also depositextra pheromones -ese superior ant agents and the virtualant agent are defined as the elite ant agents -e pheromoneregulation by the elite ant agent is defined as follows

τij(t + 1) (1 minus ρ)τij(t) + Δτij(t) + Δτlowastij(t) (3)

where

Δτij(t) 1113944σminus1

μ1Δτμij(t)

Δτμij(t)

(σ minus μ)Q

Lμ(t)1113888 1113889 if the μ minus th best ant travels on edge (i j)

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Δτlowastij(t)

σQ

Llowast1113874 1113875 if edge (i j) is part of best solution found so far

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(4)

σ number of elite ant agents μ ranking index Δτμij in-crease of trail level on the edge (i j) caused by the μ-th bestant agent Lμ tour length of the μ-th best ant agent Δτlowastijincrease of trail level on the edge (i j) caused by the elite antagents Llowast tour length of best solution found so far and Qquantity of pheromone laid by an ant agent per tour

ASrank intends to find a good solution efficiently bymaking a difference of pheromone strength between thepaths included in short traveling tours and nonpopularpaths

23 Ant System Using Individual Memories (ASIM) Asmentioned previously it can be considered that the diversityof solutions as well as the selection of solutions is an im-portant factor in the solution system by swarm intelligencesuch as ACO Hence we would like to propose the antsystem using individual memories (ASIM) as a new ACOalgorithm considering the diversity in the artificial ant be-havior-e heuristic information based on intercity distanceand the pheromone information applied to intercity routesmeans common social information which every ant agentuses commonly ASIM has a new aspect of using its ownmemory of each ant agent in addition to heuristic infor-mation based on intercity distance and pheromone

information applied to intercity routes In ASIM each antagent constructs a solution using both common social in-formation and personal information which is its ownmemory Specifically as personal information each antagent can remember the personal best solution it has foundso far In the case of regulating the pheromone the personalpheromone for each ant agent is applied additionally on thepath including the best solution that each ant agent hasfound out in the past tours apart from the pheromone associal information In the natural world it is known thatwhen real ants discover a shorter path from the nest to thefeeding site they increase the amount of pheromone theyapply along that route [13 14] On the one hand in ASIMwhen the best solution cannot be updated each ant agentcan regulate the personal pheromones on the path in the bestsolution that has been found so far Here note that each antagent cannot recognize the personal pheromones applied bythe other ant agents -e regulation of the personal pher-omone is defined as follows

ϕkij(t + 1) (1 minus θ)ϕk

ij(t) + Δϕkij(t) (5)

where

Δϕkij(t)

Z

Clowastk

1113888 1113889 if edge (i j) is part of ant krsquos best solution found so far

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(6)

ϕkij intensity of the ant agent kprime s personal pheromone trail

between cities i and j θ evaporation rate of the personalpheromone which is parameter to regulate ϕk

ij Δϕkij increase

of trail level by ant agent k on edge (i j) Z quantity of

Complexity 3

personal pheromone laid by an ant agent per tour and Clowastk tour length of the best solution of that ant agent k has everfound

In the pheromone regulation Z means the parameter toadjust the amount of the personal pheromone applied byeach ant agent In addition pheromones τij(t) are regulatedbased on equation (2) in the same way as AS In ASIM byincluding the personal pheromone ϕk

ij the probability pkij(t)

that city j is selected to be visited immediately after city i canbe rewritten in a formula as follows

pkij(t)

τij(t) + ϕkij(t)1113960 1113961

αηij(t)1113960 1113961

β

1113936lisinNkτil(t) + ϕk

il(t)1113960 1113961αηil(t)1113858 1113859

β if j isin Nk

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

Note that ASIM uses personal pheromones ϕkij(t) which

is the personal information in addition to the common socialinformation τij(t) and ηij(t) to select the next visited city Byusing the personal information which each ant agent hasindividually the diversity of tour in the ACO system can bemaintained and then it is expected that it is possible tosearch for a path while avoiding falling into a local solutionIn addition note that the computation complexity of theproposed algorithm ASIM is almost even to the existingalgorithms AS and ASrank except for memorizing the in-formation of the personal best solution in the past and thepersonal pheromone ϕk

ij(t)

3 Results and Discussion

To evaluate the solving ability of each ACO algorithm ASASrank and ASIM have been applied to four TSP bench-marks [12] -e four TSP benchmarks as shown in Table 1are taken from ldquoTSPLIBrdquo [12] In all three ACO algorithmsthe number of artificial ant agents has been set equal to thenumber of cities and one ant agent has been allocated in eachcity at the start of the tour-e parameters used in each ACOalgorithm are shown in Table 2 -e parameters used for ASand ASrank have been given the values that are generallyconsidered to give a good solution for the respective algo-rithms -e values of parameters in ASIM have been setbased on the results of preliminary experiments As men-tioned previously Z is the parameter to adjust the amount ofthe personal pheromone applied by each ant agent To adjustthe amount of personal pheromone according to the scale ofthe problem the value of Z has been set as the number ofcities m in the considered TSP Artificial ant agents insimulations have traveled 500 tours per one trial and 20trials have been conducted in each model and each TSPbenchmark

As mentioned previously the diversity of solutions is animportant factor in ACO In ASIM all ant agents use theirown memory information in addition to common socialinformation ASIM intends to maintain the diversity ofsolutions through this feature Figures 1ndash3 illustrate anexample of the transition of the best solution for each TSP

benchmark by respective ACO algorithms From Figure 1 itis found that the search of the best solution by using ASconverges rapidly to a solution -is feature seems to havearisen since all ant agents search the pathway using commonsocial information Figure 2 shows that the search of the bestsolution by using ASrank converges more rapidly to a so-lution than by using AS Since in ASrank algorithm only eliteant agents are allowed to update pheromone informationpheromones tend to concentrate to a specific path It can beseen that this trend has derived the feature that the search ofthe best solution by using ASrank converges more rapidly to asolution than by using AS On the other hand in Figure 3based on ASIM it can be seen that the best solution has beenupdated even as the tour time progresses and ASIM al-gorithm has increased the number of update times of thebest solution compared with AS and ASrank algorithms -isfeature has been caused from the ASIM algorithm with thediversity in the behavior of each ant -at is since all antagents use their ownmemories in addition to common socialinformation the ASIM algorithm maintains the diversity inthe behavior of each ant As a result the ASIM algorithmmaintains the diversity of solutions

In addition we compare the number of update times ofthe best solution in the three ACO algorithms -e averagenumber of update times of the best solution updating in 20trials of each ACO algorithm is shown in Table 3 -e factthat the number of update times of the best solution is largesuggests that ACO algorithm constructs the solution whileavoiding falling into a local solution Table 3 shows that theASIM algorithm has a larger number of update times of thebest solution than the conventional AS and ASrank algo-rithms and it is considered that ASIM has searched theoptimal solution while avoiding falling into a local solution

Furthermore in order to confirm the solving ability of theproposed ASIM algorithm we compare the solutions by ASIMto the solutions by the conventional methods AS andASrank Foreach TSP benchmark the values of the best solution obtained in20 trials are shown in Table 4 In addition the average of the bestsolution obtained over 20 trials is shown in Table 5 From theseresults it is found that for all TSP benchmarks the proposedalgorithm ASIM obtained superior solutions compared to theconventional methods AS and ASrank -is is considered to bedue to the fact that ASIM maintains the diversity of behaviorwhile searching for a pathway thus producing a variety of

Table 1 TSP datasets for the evaluation experiment

TSP Number of citiesbayg29 29att48 48eil51 51berlin52 52

Table 2 Simulation condition and parameter settings

AS ASrank ASIM(α β) (1 5) (1 5) (1 5)ρ 05 05 05

Other parameters mdash σ 6 θ 01mdash Q 100 Z -e number of ants

4 Complexity

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

Figure 2 Continued

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

Tour

leng

th

Time (t)

32000

36000

40000

0 100 200 300 400 500

(b)

Tour

leng

th

Time (t)

400

450

500

0 100 200 300 400 500

(c)

Tour

leng

th

Time (t)

7000

8000

9000

0 100 200 300 400 500

(d)

Figure 1 Best solution update in AS (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Complexity 5

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 2 Best solution update in ASrank (a) bayg29 (b) att48 (c) eil51 (d) berlin52

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 3 Best solution update in ASIM (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Table 3 Average number of update times of the best solution in 20 trials

TSP ASIM AS ASrankbayg29 120 93 63att48 183 82 113eil51 149 86 117berlin52 207 91 113

6 Complexity

solutions As a result it can be said that the solving ability ofASIM has been superior to those of AS and ASrank

As a consequence from the comparisons about thenumber of update times of the best solution and thecomparisons about the best solutions it is seen that ASIMwith the superior escape ability from the local solution hasthe superior solving ability compared to the conventionalalgorithms AS and ASrank

4 Conclusions

In this paper we have proposed the novel ACO algorithmASIM using the individual personal information in additionto the common social information -en each ant agentconstructs a solution based on the common social infor-mation and its own memory In ASIM each ant agent re-members the best solution it has followed and applies apersonal pheromone regulation along the path -en thenext city to be selected is based on the common social in-formation and the personal pheromone information that isspecific to each ant agent -e experimental results haveshown that the proposed ACO algorithm ASIM is moreeffective than the conventional methods AS and ASrank insome TSP benchmarks We consider that this is because theproposed ACO algorithm ASIM has been able to search thepath while maintaining the diversity of the solutions byusing different personal information for each ant agent Itcan be concluded that ASIM is a more effective algorithmthan conventional algorithms such as AS and ASrank becauseit can search for paths while maintaining the diversity ofsolutions through autonomous process of decision making

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the Japan Society for thePromotion of Science (JSPS) (KAKENHI grant no

18K04611) ldquoEvaluation of system performance and re-liability under incomplete information environmentrdquo Wewould like to appreciate JSPS for the grant

References

[1] M Dorigo Optimization Learning and Natural AlgorithmsPhD -esis Politecnico di Milano Milan Italy 1992

[2] M Dorigo and L M Gambardella ldquoAnt colony system acooperative learning approach to the traveling salesmanproblemrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 53ndash66 1997

[3] A Prakash M K Tiwari and R Shankar ldquoOptimal job se-quence determination and operation machine allocation inflexible manufacturing systems an approach using adaptivehierarchical ant colony algorithmrdquo Journal of IntelligentManufacturing vol 19 no 2 pp 161ndash173 2008

[4] J-P Arnaout ldquoAnt colony optimization algorithm for theeuclidean location-allocation problem with unknown numberof facilitiesrdquo Journal of Intelligent Manufacturing vol 24no 1 pp 45ndash54 2013

[5] X-J Liu H Yi and Z-H Ni ldquoApplication of ant colonyoptimization algorithm in process planning optimizationrdquoJournal of Intelligent Manufacturing vol 24 no 1 pp 1ndash132013

[6] W Qin J Zhang and D Song ldquoAn improved ant colonyalgorithm for dynamic hybrid flow shop scheduling withuncertain processing timerdquo Journal of IntelligentManufacturing vol 29 no 4 pp 891ndash904 2018

[7] S Zhang and T N Wong ldquoIntegrated process planning andscheduling an enhanced ant colony optimization heuristicwith parameter tuningrdquo Journal of Intelligent Manufacturingvol 29 no 3 pp 585ndash601 2018

[8] B Bullnheimer R F Hartl and C Strauss ldquoA new rank basedversion of the ant system a computational studyrdquo CentralEuropean Journal of Operations Research vol 7 no 1pp 25ndash38 1999

[9] H Liu Y Wang L Tu G Ding and Y Hu ldquoA modifiedParticle swarm optimization for large-scale numerical opti-mizations and engineering design problemsrdquo Journal of In-telligent Manufacturing vol 30 no 6 pp 2407ndash2433 2019

[10] C Mahapatra A Payal and M Chopra ldquoSwarm intelligencebased centralized clustering a novel solutionrdquo Journal ofIntelligent Manufacturing vol 31 no 8 pp 1877ndash1888 2020

[11] C Gruter T J Czaczkes and F L W Ratnieks ldquoDecisionmaking in ant foragers (Lasius Niger) facing conflicting pri-vate and social informationrdquo Behavioral Ecology and Socio-biology vol 65 no 2 pp 141ndash148 2011

[12] Symmetric Traveling Salesman Problem TSPLIB httpelibzibdepubmp-testdatatsptsplibtsplibhtml

[13] R Beckers J L Deneubourg and S Goss ldquoModulation of traillaying in the ant Lasius Niger (Hymenoptera formicidae) andits role in the collective selection of a food sourcerdquo Journal ofInsect Behavior vol 6 no 6 pp 751ndash759 1993

[14] D E Jackson and N Chaline ldquoModulation of pheromone trailstrength with food quality in Pharaohrsquos ant Monomoriumpharaonisrdquo Animal Behaviour vol 74 no 3 pp 463ndash4702007

Table 4 -e best solution in 20 trials

TSP ASIM AS ASrankbayg29 9140 9195 9233att48 33549 35090 34106eil51 431 443 438berlin52 7544 7664 7549

Table 5 -e average of the best solutions in 20 trials

TSP ASIM AS ASrankbayg29 9146 9248 9349att48 33556 35234 34741eil51 434 451 447berlin52 7546 7681 7762

Complexity 7

addition ant agent who has found the best so-far solutionuntil current exploring iteration has been identified as thevirtual ant agent and the virtual ant agent can also depositextra pheromones -ese superior ant agents and the virtualant agent are defined as the elite ant agents -e pheromoneregulation by the elite ant agent is defined as follows

τij(t + 1) (1 minus ρ)τij(t) + Δτij(t) + Δτlowastij(t) (3)

where

Δτij(t) 1113944σminus1

μ1Δτμij(t)

Δτμij(t)

(σ minus μ)Q

Lμ(t)1113888 1113889 if the μ minus th best ant travels on edge (i j)

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Δτlowastij(t)

σQ

Llowast1113874 1113875 if edge (i j) is part of best solution found so far

0 otherwise

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(4)

σ number of elite ant agents μ ranking index Δτμij in-crease of trail level on the edge (i j) caused by the μ-th bestant agent Lμ tour length of the μ-th best ant agent Δτlowastijincrease of trail level on the edge (i j) caused by the elite antagents Llowast tour length of best solution found so far and Qquantity of pheromone laid by an ant agent per tour

ASrank intends to find a good solution efficiently bymaking a difference of pheromone strength between thepaths included in short traveling tours and nonpopularpaths

23 Ant System Using Individual Memories (ASIM) Asmentioned previously it can be considered that the diversityof solutions as well as the selection of solutions is an im-portant factor in the solution system by swarm intelligencesuch as ACO Hence we would like to propose the antsystem using individual memories (ASIM) as a new ACOalgorithm considering the diversity in the artificial ant be-havior-e heuristic information based on intercity distanceand the pheromone information applied to intercity routesmeans common social information which every ant agentuses commonly ASIM has a new aspect of using its ownmemory of each ant agent in addition to heuristic infor-mation based on intercity distance and pheromone

information applied to intercity routes In ASIM each antagent constructs a solution using both common social in-formation and personal information which is its ownmemory Specifically as personal information each antagent can remember the personal best solution it has foundso far In the case of regulating the pheromone the personalpheromone for each ant agent is applied additionally on thepath including the best solution that each ant agent hasfound out in the past tours apart from the pheromone associal information In the natural world it is known thatwhen real ants discover a shorter path from the nest to thefeeding site they increase the amount of pheromone theyapply along that route [13 14] On the one hand in ASIMwhen the best solution cannot be updated each ant agentcan regulate the personal pheromones on the path in the bestsolution that has been found so far Here note that each antagent cannot recognize the personal pheromones applied bythe other ant agents -e regulation of the personal pher-omone is defined as follows

ϕkij(t + 1) (1 minus θ)ϕk

ij(t) + Δϕkij(t) (5)

where

Δϕkij(t)

Z

Clowastk

1113888 1113889 if edge (i j) is part of ant krsquos best solution found so far

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(6)

ϕkij intensity of the ant agent kprime s personal pheromone trail

between cities i and j θ evaporation rate of the personalpheromone which is parameter to regulate ϕk

ij Δϕkij increase

of trail level by ant agent k on edge (i j) Z quantity of

Complexity 3

personal pheromone laid by an ant agent per tour and Clowastk tour length of the best solution of that ant agent k has everfound

In the pheromone regulation Z means the parameter toadjust the amount of the personal pheromone applied byeach ant agent In addition pheromones τij(t) are regulatedbased on equation (2) in the same way as AS In ASIM byincluding the personal pheromone ϕk

ij the probability pkij(t)

that city j is selected to be visited immediately after city i canbe rewritten in a formula as follows

pkij(t)

τij(t) + ϕkij(t)1113960 1113961

αηij(t)1113960 1113961

β

1113936lisinNkτil(t) + ϕk

il(t)1113960 1113961αηil(t)1113858 1113859

β if j isin Nk

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

Note that ASIM uses personal pheromones ϕkij(t) which

is the personal information in addition to the common socialinformation τij(t) and ηij(t) to select the next visited city Byusing the personal information which each ant agent hasindividually the diversity of tour in the ACO system can bemaintained and then it is expected that it is possible tosearch for a path while avoiding falling into a local solutionIn addition note that the computation complexity of theproposed algorithm ASIM is almost even to the existingalgorithms AS and ASrank except for memorizing the in-formation of the personal best solution in the past and thepersonal pheromone ϕk

ij(t)

3 Results and Discussion

To evaluate the solving ability of each ACO algorithm ASASrank and ASIM have been applied to four TSP bench-marks [12] -e four TSP benchmarks as shown in Table 1are taken from ldquoTSPLIBrdquo [12] In all three ACO algorithmsthe number of artificial ant agents has been set equal to thenumber of cities and one ant agent has been allocated in eachcity at the start of the tour-e parameters used in each ACOalgorithm are shown in Table 2 -e parameters used for ASand ASrank have been given the values that are generallyconsidered to give a good solution for the respective algo-rithms -e values of parameters in ASIM have been setbased on the results of preliminary experiments As men-tioned previously Z is the parameter to adjust the amount ofthe personal pheromone applied by each ant agent To adjustthe amount of personal pheromone according to the scale ofthe problem the value of Z has been set as the number ofcities m in the considered TSP Artificial ant agents insimulations have traveled 500 tours per one trial and 20trials have been conducted in each model and each TSPbenchmark

As mentioned previously the diversity of solutions is animportant factor in ACO In ASIM all ant agents use theirown memory information in addition to common socialinformation ASIM intends to maintain the diversity ofsolutions through this feature Figures 1ndash3 illustrate anexample of the transition of the best solution for each TSP

benchmark by respective ACO algorithms From Figure 1 itis found that the search of the best solution by using ASconverges rapidly to a solution -is feature seems to havearisen since all ant agents search the pathway using commonsocial information Figure 2 shows that the search of the bestsolution by using ASrank converges more rapidly to a so-lution than by using AS Since in ASrank algorithm only eliteant agents are allowed to update pheromone informationpheromones tend to concentrate to a specific path It can beseen that this trend has derived the feature that the search ofthe best solution by using ASrank converges more rapidly to asolution than by using AS On the other hand in Figure 3based on ASIM it can be seen that the best solution has beenupdated even as the tour time progresses and ASIM al-gorithm has increased the number of update times of thebest solution compared with AS and ASrank algorithms -isfeature has been caused from the ASIM algorithm with thediversity in the behavior of each ant -at is since all antagents use their ownmemories in addition to common socialinformation the ASIM algorithm maintains the diversity inthe behavior of each ant As a result the ASIM algorithmmaintains the diversity of solutions

In addition we compare the number of update times ofthe best solution in the three ACO algorithms -e averagenumber of update times of the best solution updating in 20trials of each ACO algorithm is shown in Table 3 -e factthat the number of update times of the best solution is largesuggests that ACO algorithm constructs the solution whileavoiding falling into a local solution Table 3 shows that theASIM algorithm has a larger number of update times of thebest solution than the conventional AS and ASrank algo-rithms and it is considered that ASIM has searched theoptimal solution while avoiding falling into a local solution

Furthermore in order to confirm the solving ability of theproposed ASIM algorithm we compare the solutions by ASIMto the solutions by the conventional methods AS andASrank Foreach TSP benchmark the values of the best solution obtained in20 trials are shown in Table 4 In addition the average of the bestsolution obtained over 20 trials is shown in Table 5 From theseresults it is found that for all TSP benchmarks the proposedalgorithm ASIM obtained superior solutions compared to theconventional methods AS and ASrank -is is considered to bedue to the fact that ASIM maintains the diversity of behaviorwhile searching for a pathway thus producing a variety of

Table 1 TSP datasets for the evaluation experiment

TSP Number of citiesbayg29 29att48 48eil51 51berlin52 52

Table 2 Simulation condition and parameter settings

AS ASrank ASIM(α β) (1 5) (1 5) (1 5)ρ 05 05 05

Other parameters mdash σ 6 θ 01mdash Q 100 Z -e number of ants

4 Complexity

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

Figure 2 Continued

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

Tour

leng

th

Time (t)

32000

36000

40000

0 100 200 300 400 500

(b)

Tour

leng

th

Time (t)

400

450

500

0 100 200 300 400 500

(c)

Tour

leng

th

Time (t)

7000

8000

9000

0 100 200 300 400 500

(d)

Figure 1 Best solution update in AS (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Complexity 5

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 2 Best solution update in ASrank (a) bayg29 (b) att48 (c) eil51 (d) berlin52

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 3 Best solution update in ASIM (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Table 3 Average number of update times of the best solution in 20 trials

TSP ASIM AS ASrankbayg29 120 93 63att48 183 82 113eil51 149 86 117berlin52 207 91 113

6 Complexity

solutions As a result it can be said that the solving ability ofASIM has been superior to those of AS and ASrank

As a consequence from the comparisons about thenumber of update times of the best solution and thecomparisons about the best solutions it is seen that ASIMwith the superior escape ability from the local solution hasthe superior solving ability compared to the conventionalalgorithms AS and ASrank

4 Conclusions

In this paper we have proposed the novel ACO algorithmASIM using the individual personal information in additionto the common social information -en each ant agentconstructs a solution based on the common social infor-mation and its own memory In ASIM each ant agent re-members the best solution it has followed and applies apersonal pheromone regulation along the path -en thenext city to be selected is based on the common social in-formation and the personal pheromone information that isspecific to each ant agent -e experimental results haveshown that the proposed ACO algorithm ASIM is moreeffective than the conventional methods AS and ASrank insome TSP benchmarks We consider that this is because theproposed ACO algorithm ASIM has been able to search thepath while maintaining the diversity of the solutions byusing different personal information for each ant agent Itcan be concluded that ASIM is a more effective algorithmthan conventional algorithms such as AS and ASrank becauseit can search for paths while maintaining the diversity ofsolutions through autonomous process of decision making

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the Japan Society for thePromotion of Science (JSPS) (KAKENHI grant no

18K04611) ldquoEvaluation of system performance and re-liability under incomplete information environmentrdquo Wewould like to appreciate JSPS for the grant

References

[1] M Dorigo Optimization Learning and Natural AlgorithmsPhD -esis Politecnico di Milano Milan Italy 1992

[2] M Dorigo and L M Gambardella ldquoAnt colony system acooperative learning approach to the traveling salesmanproblemrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 53ndash66 1997

[3] A Prakash M K Tiwari and R Shankar ldquoOptimal job se-quence determination and operation machine allocation inflexible manufacturing systems an approach using adaptivehierarchical ant colony algorithmrdquo Journal of IntelligentManufacturing vol 19 no 2 pp 161ndash173 2008

[4] J-P Arnaout ldquoAnt colony optimization algorithm for theeuclidean location-allocation problem with unknown numberof facilitiesrdquo Journal of Intelligent Manufacturing vol 24no 1 pp 45ndash54 2013

[5] X-J Liu H Yi and Z-H Ni ldquoApplication of ant colonyoptimization algorithm in process planning optimizationrdquoJournal of Intelligent Manufacturing vol 24 no 1 pp 1ndash132013

[6] W Qin J Zhang and D Song ldquoAn improved ant colonyalgorithm for dynamic hybrid flow shop scheduling withuncertain processing timerdquo Journal of IntelligentManufacturing vol 29 no 4 pp 891ndash904 2018

[7] S Zhang and T N Wong ldquoIntegrated process planning andscheduling an enhanced ant colony optimization heuristicwith parameter tuningrdquo Journal of Intelligent Manufacturingvol 29 no 3 pp 585ndash601 2018

[8] B Bullnheimer R F Hartl and C Strauss ldquoA new rank basedversion of the ant system a computational studyrdquo CentralEuropean Journal of Operations Research vol 7 no 1pp 25ndash38 1999

[9] H Liu Y Wang L Tu G Ding and Y Hu ldquoA modifiedParticle swarm optimization for large-scale numerical opti-mizations and engineering design problemsrdquo Journal of In-telligent Manufacturing vol 30 no 6 pp 2407ndash2433 2019

[10] C Mahapatra A Payal and M Chopra ldquoSwarm intelligencebased centralized clustering a novel solutionrdquo Journal ofIntelligent Manufacturing vol 31 no 8 pp 1877ndash1888 2020

[11] C Gruter T J Czaczkes and F L W Ratnieks ldquoDecisionmaking in ant foragers (Lasius Niger) facing conflicting pri-vate and social informationrdquo Behavioral Ecology and Socio-biology vol 65 no 2 pp 141ndash148 2011

[12] Symmetric Traveling Salesman Problem TSPLIB httpelibzibdepubmp-testdatatsptsplibtsplibhtml

[13] R Beckers J L Deneubourg and S Goss ldquoModulation of traillaying in the ant Lasius Niger (Hymenoptera formicidae) andits role in the collective selection of a food sourcerdquo Journal ofInsect Behavior vol 6 no 6 pp 751ndash759 1993

[14] D E Jackson and N Chaline ldquoModulation of pheromone trailstrength with food quality in Pharaohrsquos ant Monomoriumpharaonisrdquo Animal Behaviour vol 74 no 3 pp 463ndash4702007

Table 4 -e best solution in 20 trials

TSP ASIM AS ASrankbayg29 9140 9195 9233att48 33549 35090 34106eil51 431 443 438berlin52 7544 7664 7549

Table 5 -e average of the best solutions in 20 trials

TSP ASIM AS ASrankbayg29 9146 9248 9349att48 33556 35234 34741eil51 434 451 447berlin52 7546 7681 7762

Complexity 7

personal pheromone laid by an ant agent per tour and Clowastk tour length of the best solution of that ant agent k has everfound

In the pheromone regulation Z means the parameter toadjust the amount of the personal pheromone applied byeach ant agent In addition pheromones τij(t) are regulatedbased on equation (2) in the same way as AS In ASIM byincluding the personal pheromone ϕk

ij the probability pkij(t)

that city j is selected to be visited immediately after city i canbe rewritten in a formula as follows

pkij(t)

τij(t) + ϕkij(t)1113960 1113961

αηij(t)1113960 1113961

β

1113936lisinNkτil(t) + ϕk

il(t)1113960 1113961αηil(t)1113858 1113859

β if j isin Nk

0 otherwise

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

Note that ASIM uses personal pheromones ϕkij(t) which

is the personal information in addition to the common socialinformation τij(t) and ηij(t) to select the next visited city Byusing the personal information which each ant agent hasindividually the diversity of tour in the ACO system can bemaintained and then it is expected that it is possible tosearch for a path while avoiding falling into a local solutionIn addition note that the computation complexity of theproposed algorithm ASIM is almost even to the existingalgorithms AS and ASrank except for memorizing the in-formation of the personal best solution in the past and thepersonal pheromone ϕk

ij(t)

3 Results and Discussion

To evaluate the solving ability of each ACO algorithm ASASrank and ASIM have been applied to four TSP bench-marks [12] -e four TSP benchmarks as shown in Table 1are taken from ldquoTSPLIBrdquo [12] In all three ACO algorithmsthe number of artificial ant agents has been set equal to thenumber of cities and one ant agent has been allocated in eachcity at the start of the tour-e parameters used in each ACOalgorithm are shown in Table 2 -e parameters used for ASand ASrank have been given the values that are generallyconsidered to give a good solution for the respective algo-rithms -e values of parameters in ASIM have been setbased on the results of preliminary experiments As men-tioned previously Z is the parameter to adjust the amount ofthe personal pheromone applied by each ant agent To adjustthe amount of personal pheromone according to the scale ofthe problem the value of Z has been set as the number ofcities m in the considered TSP Artificial ant agents insimulations have traveled 500 tours per one trial and 20trials have been conducted in each model and each TSPbenchmark

As mentioned previously the diversity of solutions is animportant factor in ACO In ASIM all ant agents use theirown memory information in addition to common socialinformation ASIM intends to maintain the diversity ofsolutions through this feature Figures 1ndash3 illustrate anexample of the transition of the best solution for each TSP

benchmark by respective ACO algorithms From Figure 1 itis found that the search of the best solution by using ASconverges rapidly to a solution -is feature seems to havearisen since all ant agents search the pathway using commonsocial information Figure 2 shows that the search of the bestsolution by using ASrank converges more rapidly to a so-lution than by using AS Since in ASrank algorithm only eliteant agents are allowed to update pheromone informationpheromones tend to concentrate to a specific path It can beseen that this trend has derived the feature that the search ofthe best solution by using ASrank converges more rapidly to asolution than by using AS On the other hand in Figure 3based on ASIM it can be seen that the best solution has beenupdated even as the tour time progresses and ASIM al-gorithm has increased the number of update times of thebest solution compared with AS and ASrank algorithms -isfeature has been caused from the ASIM algorithm with thediversity in the behavior of each ant -at is since all antagents use their ownmemories in addition to common socialinformation the ASIM algorithm maintains the diversity inthe behavior of each ant As a result the ASIM algorithmmaintains the diversity of solutions

In addition we compare the number of update times ofthe best solution in the three ACO algorithms -e averagenumber of update times of the best solution updating in 20trials of each ACO algorithm is shown in Table 3 -e factthat the number of update times of the best solution is largesuggests that ACO algorithm constructs the solution whileavoiding falling into a local solution Table 3 shows that theASIM algorithm has a larger number of update times of thebest solution than the conventional AS and ASrank algo-rithms and it is considered that ASIM has searched theoptimal solution while avoiding falling into a local solution

Furthermore in order to confirm the solving ability of theproposed ASIM algorithm we compare the solutions by ASIMto the solutions by the conventional methods AS andASrank Foreach TSP benchmark the values of the best solution obtained in20 trials are shown in Table 4 In addition the average of the bestsolution obtained over 20 trials is shown in Table 5 From theseresults it is found that for all TSP benchmarks the proposedalgorithm ASIM obtained superior solutions compared to theconventional methods AS and ASrank -is is considered to bedue to the fact that ASIM maintains the diversity of behaviorwhile searching for a pathway thus producing a variety of

Table 1 TSP datasets for the evaluation experiment

TSP Number of citiesbayg29 29att48 48eil51 51berlin52 52

Table 2 Simulation condition and parameter settings

AS ASrank ASIM(α β) (1 5) (1 5) (1 5)ρ 05 05 05

Other parameters mdash σ 6 θ 01mdash Q 100 Z -e number of ants

4 Complexity

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

Figure 2 Continued

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

Tour

leng

th

Time (t)

32000

36000

40000

0 100 200 300 400 500

(b)

Tour

leng

th

Time (t)

400

450

500

0 100 200 300 400 500

(c)

Tour

leng

th

Time (t)

7000

8000

9000

0 100 200 300 400 500

(d)

Figure 1 Best solution update in AS (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Complexity 5

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 2 Best solution update in ASrank (a) bayg29 (b) att48 (c) eil51 (d) berlin52

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 3 Best solution update in ASIM (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Table 3 Average number of update times of the best solution in 20 trials

TSP ASIM AS ASrankbayg29 120 93 63att48 183 82 113eil51 149 86 117berlin52 207 91 113

6 Complexity

solutions As a result it can be said that the solving ability ofASIM has been superior to those of AS and ASrank

As a consequence from the comparisons about thenumber of update times of the best solution and thecomparisons about the best solutions it is seen that ASIMwith the superior escape ability from the local solution hasthe superior solving ability compared to the conventionalalgorithms AS and ASrank

4 Conclusions

In this paper we have proposed the novel ACO algorithmASIM using the individual personal information in additionto the common social information -en each ant agentconstructs a solution based on the common social infor-mation and its own memory In ASIM each ant agent re-members the best solution it has followed and applies apersonal pheromone regulation along the path -en thenext city to be selected is based on the common social in-formation and the personal pheromone information that isspecific to each ant agent -e experimental results haveshown that the proposed ACO algorithm ASIM is moreeffective than the conventional methods AS and ASrank insome TSP benchmarks We consider that this is because theproposed ACO algorithm ASIM has been able to search thepath while maintaining the diversity of the solutions byusing different personal information for each ant agent Itcan be concluded that ASIM is a more effective algorithmthan conventional algorithms such as AS and ASrank becauseit can search for paths while maintaining the diversity ofsolutions through autonomous process of decision making

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the Japan Society for thePromotion of Science (JSPS) (KAKENHI grant no

18K04611) ldquoEvaluation of system performance and re-liability under incomplete information environmentrdquo Wewould like to appreciate JSPS for the grant

References

[1] M Dorigo Optimization Learning and Natural AlgorithmsPhD -esis Politecnico di Milano Milan Italy 1992

[2] M Dorigo and L M Gambardella ldquoAnt colony system acooperative learning approach to the traveling salesmanproblemrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 53ndash66 1997

[3] A Prakash M K Tiwari and R Shankar ldquoOptimal job se-quence determination and operation machine allocation inflexible manufacturing systems an approach using adaptivehierarchical ant colony algorithmrdquo Journal of IntelligentManufacturing vol 19 no 2 pp 161ndash173 2008

[4] J-P Arnaout ldquoAnt colony optimization algorithm for theeuclidean location-allocation problem with unknown numberof facilitiesrdquo Journal of Intelligent Manufacturing vol 24no 1 pp 45ndash54 2013

[5] X-J Liu H Yi and Z-H Ni ldquoApplication of ant colonyoptimization algorithm in process planning optimizationrdquoJournal of Intelligent Manufacturing vol 24 no 1 pp 1ndash132013

[6] W Qin J Zhang and D Song ldquoAn improved ant colonyalgorithm for dynamic hybrid flow shop scheduling withuncertain processing timerdquo Journal of IntelligentManufacturing vol 29 no 4 pp 891ndash904 2018

[7] S Zhang and T N Wong ldquoIntegrated process planning andscheduling an enhanced ant colony optimization heuristicwith parameter tuningrdquo Journal of Intelligent Manufacturingvol 29 no 3 pp 585ndash601 2018

[8] B Bullnheimer R F Hartl and C Strauss ldquoA new rank basedversion of the ant system a computational studyrdquo CentralEuropean Journal of Operations Research vol 7 no 1pp 25ndash38 1999

[9] H Liu Y Wang L Tu G Ding and Y Hu ldquoA modifiedParticle swarm optimization for large-scale numerical opti-mizations and engineering design problemsrdquo Journal of In-telligent Manufacturing vol 30 no 6 pp 2407ndash2433 2019

[10] C Mahapatra A Payal and M Chopra ldquoSwarm intelligencebased centralized clustering a novel solutionrdquo Journal ofIntelligent Manufacturing vol 31 no 8 pp 1877ndash1888 2020

[11] C Gruter T J Czaczkes and F L W Ratnieks ldquoDecisionmaking in ant foragers (Lasius Niger) facing conflicting pri-vate and social informationrdquo Behavioral Ecology and Socio-biology vol 65 no 2 pp 141ndash148 2011

[12] Symmetric Traveling Salesman Problem TSPLIB httpelibzibdepubmp-testdatatsptsplibtsplibhtml

[13] R Beckers J L Deneubourg and S Goss ldquoModulation of traillaying in the ant Lasius Niger (Hymenoptera formicidae) andits role in the collective selection of a food sourcerdquo Journal ofInsect Behavior vol 6 no 6 pp 751ndash759 1993

[14] D E Jackson and N Chaline ldquoModulation of pheromone trailstrength with food quality in Pharaohrsquos ant Monomoriumpharaonisrdquo Animal Behaviour vol 74 no 3 pp 463ndash4702007

Table 4 -e best solution in 20 trials

TSP ASIM AS ASrankbayg29 9140 9195 9233att48 33549 35090 34106eil51 431 443 438berlin52 7544 7664 7549

Table 5 -e average of the best solutions in 20 trials

TSP ASIM AS ASrankbayg29 9146 9248 9349att48 33556 35234 34741eil51 434 451 447berlin52 7546 7681 7762

Complexity 7

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

Figure 2 Continued

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

Tour

leng

th

Time (t)

32000

36000

40000

0 100 200 300 400 500

(b)

Tour

leng

th

Time (t)

400

450

500

0 100 200 300 400 500

(c)

Tour

leng

th

Time (t)

7000

8000

9000

0 100 200 300 400 500

(d)

Figure 1 Best solution update in AS (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Complexity 5

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 2 Best solution update in ASrank (a) bayg29 (b) att48 (c) eil51 (d) berlin52

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 3 Best solution update in ASIM (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Table 3 Average number of update times of the best solution in 20 trials

TSP ASIM AS ASrankbayg29 120 93 63att48 183 82 113eil51 149 86 117berlin52 207 91 113

6 Complexity

solutions As a result it can be said that the solving ability ofASIM has been superior to those of AS and ASrank

As a consequence from the comparisons about thenumber of update times of the best solution and thecomparisons about the best solutions it is seen that ASIMwith the superior escape ability from the local solution hasthe superior solving ability compared to the conventionalalgorithms AS and ASrank

4 Conclusions

In this paper we have proposed the novel ACO algorithmASIM using the individual personal information in additionto the common social information -en each ant agentconstructs a solution based on the common social infor-mation and its own memory In ASIM each ant agent re-members the best solution it has followed and applies apersonal pheromone regulation along the path -en thenext city to be selected is based on the common social in-formation and the personal pheromone information that isspecific to each ant agent -e experimental results haveshown that the proposed ACO algorithm ASIM is moreeffective than the conventional methods AS and ASrank insome TSP benchmarks We consider that this is because theproposed ACO algorithm ASIM has been able to search thepath while maintaining the diversity of the solutions byusing different personal information for each ant agent Itcan be concluded that ASIM is a more effective algorithmthan conventional algorithms such as AS and ASrank becauseit can search for paths while maintaining the diversity ofsolutions through autonomous process of decision making

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the Japan Society for thePromotion of Science (JSPS) (KAKENHI grant no

18K04611) ldquoEvaluation of system performance and re-liability under incomplete information environmentrdquo Wewould like to appreciate JSPS for the grant

References

[1] M Dorigo Optimization Learning and Natural AlgorithmsPhD -esis Politecnico di Milano Milan Italy 1992

[2] M Dorigo and L M Gambardella ldquoAnt colony system acooperative learning approach to the traveling salesmanproblemrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 53ndash66 1997

[3] A Prakash M K Tiwari and R Shankar ldquoOptimal job se-quence determination and operation machine allocation inflexible manufacturing systems an approach using adaptivehierarchical ant colony algorithmrdquo Journal of IntelligentManufacturing vol 19 no 2 pp 161ndash173 2008

[4] J-P Arnaout ldquoAnt colony optimization algorithm for theeuclidean location-allocation problem with unknown numberof facilitiesrdquo Journal of Intelligent Manufacturing vol 24no 1 pp 45ndash54 2013

[5] X-J Liu H Yi and Z-H Ni ldquoApplication of ant colonyoptimization algorithm in process planning optimizationrdquoJournal of Intelligent Manufacturing vol 24 no 1 pp 1ndash132013

[6] W Qin J Zhang and D Song ldquoAn improved ant colonyalgorithm for dynamic hybrid flow shop scheduling withuncertain processing timerdquo Journal of IntelligentManufacturing vol 29 no 4 pp 891ndash904 2018

[7] S Zhang and T N Wong ldquoIntegrated process planning andscheduling an enhanced ant colony optimization heuristicwith parameter tuningrdquo Journal of Intelligent Manufacturingvol 29 no 3 pp 585ndash601 2018

[8] B Bullnheimer R F Hartl and C Strauss ldquoA new rank basedversion of the ant system a computational studyrdquo CentralEuropean Journal of Operations Research vol 7 no 1pp 25ndash38 1999

[9] H Liu Y Wang L Tu G Ding and Y Hu ldquoA modifiedParticle swarm optimization for large-scale numerical opti-mizations and engineering design problemsrdquo Journal of In-telligent Manufacturing vol 30 no 6 pp 2407ndash2433 2019

[10] C Mahapatra A Payal and M Chopra ldquoSwarm intelligencebased centralized clustering a novel solutionrdquo Journal ofIntelligent Manufacturing vol 31 no 8 pp 1877ndash1888 2020

[11] C Gruter T J Czaczkes and F L W Ratnieks ldquoDecisionmaking in ant foragers (Lasius Niger) facing conflicting pri-vate and social informationrdquo Behavioral Ecology and Socio-biology vol 65 no 2 pp 141ndash148 2011

[12] Symmetric Traveling Salesman Problem TSPLIB httpelibzibdepubmp-testdatatsptsplibtsplibhtml

[13] R Beckers J L Deneubourg and S Goss ldquoModulation of traillaying in the ant Lasius Niger (Hymenoptera formicidae) andits role in the collective selection of a food sourcerdquo Journal ofInsect Behavior vol 6 no 6 pp 751ndash759 1993

[14] D E Jackson and N Chaline ldquoModulation of pheromone trailstrength with food quality in Pharaohrsquos ant Monomoriumpharaonisrdquo Animal Behaviour vol 74 no 3 pp 463ndash4702007

Table 4 -e best solution in 20 trials

TSP ASIM AS ASrankbayg29 9140 9195 9233att48 33549 35090 34106eil51 431 443 438berlin52 7544 7664 7549

Table 5 -e average of the best solutions in 20 trials

TSP ASIM AS ASrankbayg29 9146 9248 9349att48 33556 35234 34741eil51 434 451 447berlin52 7546 7681 7762

Complexity 7

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 2 Best solution update in ASrank (a) bayg29 (b) att48 (c) eil51 (d) berlin52

9000

10000

11000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(a)

32000

36000

40000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(b)

400

450

500

0 100 200 300 400 500

Tour

leng

th

Time (t)

(c)

7000

8000

9000

0 100 200 300 400 500

Tour

leng

th

Time (t)

(d)

Figure 3 Best solution update in ASIM (a) bayg29 (b) att48 (c) eil51 (d) berlin52

Table 3 Average number of update times of the best solution in 20 trials

TSP ASIM AS ASrankbayg29 120 93 63att48 183 82 113eil51 149 86 117berlin52 207 91 113

6 Complexity

solutions As a result it can be said that the solving ability ofASIM has been superior to those of AS and ASrank

As a consequence from the comparisons about thenumber of update times of the best solution and thecomparisons about the best solutions it is seen that ASIMwith the superior escape ability from the local solution hasthe superior solving ability compared to the conventionalalgorithms AS and ASrank

4 Conclusions

In this paper we have proposed the novel ACO algorithmASIM using the individual personal information in additionto the common social information -en each ant agentconstructs a solution based on the common social infor-mation and its own memory In ASIM each ant agent re-members the best solution it has followed and applies apersonal pheromone regulation along the path -en thenext city to be selected is based on the common social in-formation and the personal pheromone information that isspecific to each ant agent -e experimental results haveshown that the proposed ACO algorithm ASIM is moreeffective than the conventional methods AS and ASrank insome TSP benchmarks We consider that this is because theproposed ACO algorithm ASIM has been able to search thepath while maintaining the diversity of the solutions byusing different personal information for each ant agent Itcan be concluded that ASIM is a more effective algorithmthan conventional algorithms such as AS and ASrank becauseit can search for paths while maintaining the diversity ofsolutions through autonomous process of decision making

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the Japan Society for thePromotion of Science (JSPS) (KAKENHI grant no

18K04611) ldquoEvaluation of system performance and re-liability under incomplete information environmentrdquo Wewould like to appreciate JSPS for the grant

References

[1] M Dorigo Optimization Learning and Natural AlgorithmsPhD -esis Politecnico di Milano Milan Italy 1992

[2] M Dorigo and L M Gambardella ldquoAnt colony system acooperative learning approach to the traveling salesmanproblemrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 53ndash66 1997

[3] A Prakash M K Tiwari and R Shankar ldquoOptimal job se-quence determination and operation machine allocation inflexible manufacturing systems an approach using adaptivehierarchical ant colony algorithmrdquo Journal of IntelligentManufacturing vol 19 no 2 pp 161ndash173 2008

[4] J-P Arnaout ldquoAnt colony optimization algorithm for theeuclidean location-allocation problem with unknown numberof facilitiesrdquo Journal of Intelligent Manufacturing vol 24no 1 pp 45ndash54 2013

[5] X-J Liu H Yi and Z-H Ni ldquoApplication of ant colonyoptimization algorithm in process planning optimizationrdquoJournal of Intelligent Manufacturing vol 24 no 1 pp 1ndash132013

[6] W Qin J Zhang and D Song ldquoAn improved ant colonyalgorithm for dynamic hybrid flow shop scheduling withuncertain processing timerdquo Journal of IntelligentManufacturing vol 29 no 4 pp 891ndash904 2018

[7] S Zhang and T N Wong ldquoIntegrated process planning andscheduling an enhanced ant colony optimization heuristicwith parameter tuningrdquo Journal of Intelligent Manufacturingvol 29 no 3 pp 585ndash601 2018

[8] B Bullnheimer R F Hartl and C Strauss ldquoA new rank basedversion of the ant system a computational studyrdquo CentralEuropean Journal of Operations Research vol 7 no 1pp 25ndash38 1999

[9] H Liu Y Wang L Tu G Ding and Y Hu ldquoA modifiedParticle swarm optimization for large-scale numerical opti-mizations and engineering design problemsrdquo Journal of In-telligent Manufacturing vol 30 no 6 pp 2407ndash2433 2019

[10] C Mahapatra A Payal and M Chopra ldquoSwarm intelligencebased centralized clustering a novel solutionrdquo Journal ofIntelligent Manufacturing vol 31 no 8 pp 1877ndash1888 2020

[11] C Gruter T J Czaczkes and F L W Ratnieks ldquoDecisionmaking in ant foragers (Lasius Niger) facing conflicting pri-vate and social informationrdquo Behavioral Ecology and Socio-biology vol 65 no 2 pp 141ndash148 2011

[12] Symmetric Traveling Salesman Problem TSPLIB httpelibzibdepubmp-testdatatsptsplibtsplibhtml

[13] R Beckers J L Deneubourg and S Goss ldquoModulation of traillaying in the ant Lasius Niger (Hymenoptera formicidae) andits role in the collective selection of a food sourcerdquo Journal ofInsect Behavior vol 6 no 6 pp 751ndash759 1993

[14] D E Jackson and N Chaline ldquoModulation of pheromone trailstrength with food quality in Pharaohrsquos ant Monomoriumpharaonisrdquo Animal Behaviour vol 74 no 3 pp 463ndash4702007

Table 4 -e best solution in 20 trials

TSP ASIM AS ASrankbayg29 9140 9195 9233att48 33549 35090 34106eil51 431 443 438berlin52 7544 7664 7549

Table 5 -e average of the best solutions in 20 trials

TSP ASIM AS ASrankbayg29 9146 9248 9349att48 33556 35234 34741eil51 434 451 447berlin52 7546 7681 7762

Complexity 7

solutions As a result it can be said that the solving ability ofASIM has been superior to those of AS and ASrank

As a consequence from the comparisons about thenumber of update times of the best solution and thecomparisons about the best solutions it is seen that ASIMwith the superior escape ability from the local solution hasthe superior solving ability compared to the conventionalalgorithms AS and ASrank

4 Conclusions

In this paper we have proposed the novel ACO algorithmASIM using the individual personal information in additionto the common social information -en each ant agentconstructs a solution based on the common social infor-mation and its own memory In ASIM each ant agent re-members the best solution it has followed and applies apersonal pheromone regulation along the path -en thenext city to be selected is based on the common social in-formation and the personal pheromone information that isspecific to each ant agent -e experimental results haveshown that the proposed ACO algorithm ASIM is moreeffective than the conventional methods AS and ASrank insome TSP benchmarks We consider that this is because theproposed ACO algorithm ASIM has been able to search thepath while maintaining the diversity of the solutions byusing different personal information for each ant agent Itcan be concluded that ASIM is a more effective algorithmthan conventional algorithms such as AS and ASrank becauseit can search for paths while maintaining the diversity ofsolutions through autonomous process of decision making

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is study was supported by the Japan Society for thePromotion of Science (JSPS) (KAKENHI grant no

18K04611) ldquoEvaluation of system performance and re-liability under incomplete information environmentrdquo Wewould like to appreciate JSPS for the grant

References

[1] M Dorigo Optimization Learning and Natural AlgorithmsPhD -esis Politecnico di Milano Milan Italy 1992

[2] M Dorigo and L M Gambardella ldquoAnt colony system acooperative learning approach to the traveling salesmanproblemrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 53ndash66 1997

[3] A Prakash M K Tiwari and R Shankar ldquoOptimal job se-quence determination and operation machine allocation inflexible manufacturing systems an approach using adaptivehierarchical ant colony algorithmrdquo Journal of IntelligentManufacturing vol 19 no 2 pp 161ndash173 2008

[4] J-P Arnaout ldquoAnt colony optimization algorithm for theeuclidean location-allocation problem with unknown numberof facilitiesrdquo Journal of Intelligent Manufacturing vol 24no 1 pp 45ndash54 2013

[5] X-J Liu H Yi and Z-H Ni ldquoApplication of ant colonyoptimization algorithm in process planning optimizationrdquoJournal of Intelligent Manufacturing vol 24 no 1 pp 1ndash132013

[6] W Qin J Zhang and D Song ldquoAn improved ant colonyalgorithm for dynamic hybrid flow shop scheduling withuncertain processing timerdquo Journal of IntelligentManufacturing vol 29 no 4 pp 891ndash904 2018

[7] S Zhang and T N Wong ldquoIntegrated process planning andscheduling an enhanced ant colony optimization heuristicwith parameter tuningrdquo Journal of Intelligent Manufacturingvol 29 no 3 pp 585ndash601 2018

[8] B Bullnheimer R F Hartl and C Strauss ldquoA new rank basedversion of the ant system a computational studyrdquo CentralEuropean Journal of Operations Research vol 7 no 1pp 25ndash38 1999

[9] H Liu Y Wang L Tu G Ding and Y Hu ldquoA modifiedParticle swarm optimization for large-scale numerical opti-mizations and engineering design problemsrdquo Journal of In-telligent Manufacturing vol 30 no 6 pp 2407ndash2433 2019

[10] C Mahapatra A Payal and M Chopra ldquoSwarm intelligencebased centralized clustering a novel solutionrdquo Journal ofIntelligent Manufacturing vol 31 no 8 pp 1877ndash1888 2020

[11] C Gruter T J Czaczkes and F L W Ratnieks ldquoDecisionmaking in ant foragers (Lasius Niger) facing conflicting pri-vate and social informationrdquo Behavioral Ecology and Socio-biology vol 65 no 2 pp 141ndash148 2011

[12] Symmetric Traveling Salesman Problem TSPLIB httpelibzibdepubmp-testdatatsptsplibtsplibhtml

[13] R Beckers J L Deneubourg and S Goss ldquoModulation of traillaying in the ant Lasius Niger (Hymenoptera formicidae) andits role in the collective selection of a food sourcerdquo Journal ofInsect Behavior vol 6 no 6 pp 751ndash759 1993

[14] D E Jackson and N Chaline ldquoModulation of pheromone trailstrength with food quality in Pharaohrsquos ant Monomoriumpharaonisrdquo Animal Behaviour vol 74 no 3 pp 463ndash4702007

Table 4 -e best solution in 20 trials

TSP ASIM AS ASrankbayg29 9140 9195 9233att48 33549 35090 34106eil51 431 443 438berlin52 7544 7664 7549

Table 5 -e average of the best solutions in 20 trials

TSP ASIM AS ASrankbayg29 9146 9248 9349att48 33556 35234 34741eil51 434 451 447berlin52 7546 7681 7762

Complexity 7