Get PDF - IOS Press
International Journal of Knowledge-based and Intelligent Engineering Systems 18 (2014) 11–21 DOI 10.3233/KES-130282 IOS Press
11
Pheromone trail initialization with local optimal solutions in ant colony optimization Hitoshi Kanoha,∗, Junichi Ochiaib and Yosuke Kamedab a
Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki, Japan b Department of Computer Science, Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki, Japan
Abstract. This paper presents a method to improve the search rate of Max-Min Ant System for the traveling salesman problem. The proposed method gives deviations from the initial pheromone trails by using a set of local optimal solutions calculated in advance. This method aims to build a near optimal solution at high speed by combining the candidate partial solutions contained in the set. Max-Min Ant System has demonstrated impressive performance, but the search rate is relatively low. Considering the generic purpose of stochastic search algorithms, which is to find near optimal solutions subject to time constraints, the search rate is important as well as the solution quality. The experimental results using benchmark problems with 51 to 1002 cities suggested that the proposed method has a faster search rate than Max-Min Ant System; the additional computation cost for calculating local optimal solutions is negligibly small. Keywords: Ant colony optimization, traveling salesman problem, search rate, local optimal solution
1. Introduction Ant colony optimization (ACO) is a stochastic search algorithm for problem solving that takes inspiration from the foraging behaviors of ants. ACO has been formalized into a metaheuristic for combinatorial optimization problems by Dorigo et al., and many successful applications are now available [1–4]. The main idea of ACO is the indirect communication among the individuals of a colony of ants based on an analogy with the pheromone trails that real ants use for communication. Many ants build solutions to a combinatorial optimization problem and exchange information on the quality of the solutions using pheromones. In ACO, the pheromone is the accumulation of information on a search space and the way ∗ Corresponding author: Hitoshi Kanoh, Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan. E-mail: [email protected].
of depositing the pheromone greatly influences search performance [1]. Many studies on pheromone update methods have been done to try to improve the performance of ACO. Max-Min Ant System (MMAS) has demonstrated especially impressive performance [5]. Its characterizing elements are that only the best ant in the colony updates the pheromone trails and that the value of the pheromone is bound. Since MMAS searches to maintain the diversity of solutions, it can find a solution near the optimal one. MMAS, however, cannot build a high-quality solution in the early search stages, and the search rate is relatively low. Earlier research focused on the quality of the final solution, and investigations on the search rate were seldom done. Considering the generic purpose of stochastic search algorithms, which is to find near optimal solutions subject to time constraints, the search rate is important as well as the solution quality (see Section 2.5). In this paper, we present a method to improve the search rate of MMAS for the traveling salesman problem (TSP) [6]. The proposed method gives deviations
c 2014 – IOS Press and the authors. All rights reserved ISSN 1327-2314/14/$27.50
12
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
from the initial pheromone trails by using a set of local optimal solutions calculated in advance. This method aims to build a near optimal solution at high speed by combining the candidate partial solutions contained in the set. In the field of evolutionary computation, improving the search rate by generating a proper initial population using domain-specific knowledge is generally known [7]. In contrast, the proposed method has the feature of not using such knowledge and has wide applicability. Some pheromone initialization methods have been proposed in order to improve the performance of Ant Colony Systems (ACS) for TSPs [9,10]. The improvement in search speed of these methods may not be enough. Furthermore, these methods are applied to only small problems. We have already reported the outline of a pheromone initialization method for MMAS [11]. In the present work, a systematic evaluation using large-scale problems was performed, and hybridizing with the local search and comparison with conventional techniques were also carried out. In the following section, we start by showing the problem description, an overview of MMAS, and related works. Then we describe the strategy and algorithm of the proposed method in detail. Finally, we give the results of experiments using benchmark problems with 51 to 1002 cities.
2. Overview 2.1. Traveling salesman problem (TSP) In this work, the proposed method was tested on the TSP, which is the most widely used benchmark for ACO. Here we express the TSP following the references for MMAS [5]. The TSP can be represented by a complete graph G = (N , A), where N is a set of nodes, i.e., cities, n = |N | is the number of cities, and A is the set of arcs fully connecting the nodes. Each arc (i, j) ∈ A is assigned a value di,j (= dj,i ), which represents the distance between cities i and j. The TSP then is the problem of finding a shortest closed tour visiting each of the cities of G exactly once. The TSP instances used in this paper are taken from the TSPLIB benchmark library [8]. 2.2. Basic algorithms for TSP In this section, we outline two basic algorithms for TSPs used in the proposed method.
(1) Greedy algorithm Construction algorithms build solutions to a problem under consideration incrementally, starting with an empty initial solution and iteratively adding appropriate solution components without backtracking until a complete solution is obtained. The greedy algorithm [1,6] is widely used in generating initial solutions for TSPs. A solution (tour) is built by adding city after city incrementally. A simple rule of thumb to build a tour is to start from an initial city and to always choose to go to the closest still unvisited city before returning to the start city. This algorithm is known as the nearest neighbor tour construction heuristic. An algorithmic outline is given below, where i, j, and k are city numbers, Sp is a partial solution (i.e., a list of cities visited), and N is a set of cities not yet visited. The procedure Greedy returns the solution with the best heuristic estimate as a function of the current Sp. This algorithm can generate n different solutions at maximum. Additionally, greedy decisions in early stages of the construction process constrain the available choices at later stages, often causing very poor moves in the final phases of the solution construction. Procedure Greedy(first-city) i = first-city; Sp = (i); N = N; While(N ’ is not empty){ N ← N − {i}; k = arg min di,j ; j ∈N /
Sp ← Sp ⊗ k; /*k is added to Sp*/ i = k; } return Sp as a solution; (2) 2-opt algorithm Local search algorithms start from a complete initial solution and try to find a better solution in an appropriately defined neighborhood of the current solution. An example neighborhood for the TSP is the 2-exchange neighborhood in which neighbor solutions differ by at most 2 edges. If such a solution is found, it replaces the current solution and the local search continues. These steps are repeated until no better neighbor solution can be found and the al-
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
gorithm ends in a local optimum. The 2-opt [1, 6] is widely used as a local search algorithm in TSPs since the calculation is quick. An outline is given below of an algorithm that performs a sequence of local modifications of the current tour to improve its length. The algorithm starts from an initial solution and repeatedly tries to improve the current solution through a local change. Given a candidate solution S, the output of the 2-opt is the shortest tour in the set of all candidate solutions S that can be obtained from S by exchanging two pairs of arcs in all the possible ways (for a detailed description, see [1] on page 32). Procedure 2-opt(S) S’ = Improve(S); while(S = S) { S = S ; S = Improve(S); } return S as a solution; 2.3. Ant colony optimization (ACO) The generic ACO metaheuristic for combinatorial optimization problems is shown below [2]. After initializing parameters and pheromone trails, the metaheuristic iterates over three phases. First, a number of solutions are constructed by ants, and then these solutions may be improved through an optional local search. Finally, before the start of the next iteration, the pheromone trails are adapted to reflect the search experience of the ants. Procedure ACO() Set parameters; Initialize pheromone trails; while(terminal condition not met){ Construct ant solutions; Apply local search; /* option */ Update pheromone; } Many versions of ACO algorithms have been proposed to try to improve their performance [1–4]. Dorigo et al.compare the development of the average solution quality of typical ACO algorithms as a function of the computation time [1]. They pointed out that Max-Min Ant System (MMAS) [5] initially produces rather poor solutions, and in the initial phases it is outperformed even by Ant System, which was the first
13
ACO algorithm to be proposed. Nevertheless, its final solution quality is among the compared ACO algorithms. These results are consistent with the findings of various published research papers on ACOs [1]. 2.4. Max-min ant system (MMAS) The MMAS constructs ant solutions and updates pheromone by following Eqs (1) and (3), respectively. The MMAS has two main characteristics [2]: only the best ant updates the pheromone trails, and the value of the pheromone is bound. Here, the best means that the length of the tour is shortest. The pheromone τi,j , associated with the edge joining cities i and j, is updated using Eq. (1): best τmax (1) τi,j ← (1 − ρ)τi,j + Δτi,j τmin ⎧ ⎪ ⎨a if x > a [x]ab = b if x < b ⎪ ⎩ x otherwise ⎧ ⎪ ⎨1/Lbest if (i, j) belongs to the best Δτi,j = best tour ⎪ ⎩ 0 otherwise where ρ is the evaporation rate, τmax and τmin are respectively the upper and lower bounds imposed on the pheromone, and Lbest is the length of the tour of the best ant. Some guidelines have been provided for defining τmax , τmin , and Lbest on the basis of analytical and empirical considerations. This paper defines these parameters by the following three formulas [5], where Lib is the length of the best tour found in the current iteration. τmax = τmin
1 ρLib
√ 1 − n 0.05 √ · τmax = (n/2 − 1) · n 0.05
(2)
Lbest = Lib In the construction of a solution, ants select the next city to be visited through a stochastic mechanism. When ant k is in city i and has so far constructed the partial solution, the probability of going to city j is given by Eq. (3). ⎧ α β ⎪ ⎨ [τi,j ] · [ηi,j ] if j ∈ N k k α β [τ ] · [η ] (3) pij = k i,l i,l l∈N ⎪ ⎩0 otherwise
14
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
Where N k is the set of cities not yet visited by the ant k. The parameters α and β control the relative importance of the pheromone versus the heuristic information ηi,j , which is given by ηi,j =
1 . di,j
2.5. Practical importance of search rate An effective solution of TSPs can be applied to tour planning problems in home-delivery services in the real world. This solution requires the early convergence for the reasons below. – It is necessary to calculate the optimal route between customers in practical problems, while the distance between cities is given in TSPs. Consequently, the order of visiting customers is calculated by ACO and the route between customers is calculated by Dijkstra’s algorithm, etc. – A practical problem has two or more objective functions including distance, travel time, safeness and comfortableness, and economy. Several nondominated solutions should be calculated. – When traffic congestion changes during traveling, the route should be reevaluated. Though the traffic congestion information on main roads is provided every five minutes in Japan, this calculation must be performed within five minutes. Thus, the tour planning problem in the real world can be formalized as a dynamic hybrid multi-objective problem and one iteration step of the solution takes much computational time, and moreover there is the time limit. The proposed method aims to improve the search rate of MMAS by giving the deviation from initial pheromone trails. 2.6. Related works In the field of evolutionary computation, generating an initial population using domain-specific knowledge is generally known. Bonissone et al. [7] pointed out that this knowledge may be incorporated either implicitly, in the design of data structures, encoding, and constraints representations, or explicitly, via the initialization of the first population and in the control of parameters. Kanoh et al. proposed a solution to timetabling problems using a genetic algorithm with prior knowledge [12]. The prior knowledge here is a set of candidate partial solutions of the final solution, and is built from timetables used in past years. In contrast, the pro-
posed method has the feature of not using domainspecific knowledge and has wide applicability. Tsai et al. proposed a solution to TSP using an Ant Colony System (ACS) with prior knowledge [9]. The prior knowledge here is the specific deviation of initial pheromone trails previously constructed by the greedy algorithm. The experimental results using TSP instances with 51 to 198 cities show that the proposed method improves the length of the solution. However, this paper does not present the formula to be used in the calculation of initial pheromone values and not discuss the search rate of the method. Furthermore, this paper is examining only small problems. Dai et al. proposed a new pheromone initialization strategy of ACS for TSP, which initializes the pheromone matrix in light of the minimal spanning tree (MST) information [10]. The MST describes the cheapest network to connect all of a given set of vertices. The purpose of this strategy is to give the edges belonging to the MST a higher probability of being chosen by ants at the start of the algorithm. The experimental results for instances with 14 to 136 cities show that the proposed strategy improves the convergence performance of ACS both in terms of solution quality and convergence speed. However, this strategy is applied to only small problems.
3. Proposed method 3.1. Strategy In this paper, we present a method to improve the search rate of MMAS according to the following strategies. 1) In addition to the usual initial pheromone trails, additional pheromones are deposited on the solution built before the first iteration by using local search. The proposed method helps to guide the search and so might reduce the number of points visited in the search space by giving deviations from the initial pheromone trails. 2) This reduction is conducted using local optimal solutions built by the greedy and the 2-opt algorithms. These solutions can include candidate partial solutions of the global optimal solution. Moreover, the search rate of 2-opt is very high; it can be considered to have no influence on the computational time of the proposed method.
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
3) First, many tours are generated by changing the starting city in the greedy algorithm. Then, the tours are improved by the 2-opt algorithm, and these are regarded as initial solutions. Finally, all the pheromone trails generated from the initial solutions are considered as the initial pheromone trails. The best-performing local search algorithm for TSPs is the Lin-Kernighan heuristic [13]. This heuristic, however, is much more complicated to implement than 2-opt and needs careful fine-tuning to obtain highquality solutions. Here, we use a 2-opt implementation considering computational time. 3.2. General procedure The general procedure of the proposed method is given below, where m is the number of ants, Sib is the best solution in the current iteration, Sgb is the best solution found since the start of the algorithm, and L(S) is the length of solution S. Procedure Initializepheromone will be shown in the next section. Procedure Main() Input TSP; Set parameters; Initialize-pheromone(); while(terminal condition not met){ for(k = 1; k
11
Pheromone trail initialization with local optimal solutions in ant colony optimization Hitoshi Kanoha,∗, Junichi Ochiaib and Yosuke Kamedab a
Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki, Japan b Department of Computer Science, Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki, Japan
Abstract. This paper presents a method to improve the search rate of Max-Min Ant System for the traveling salesman problem. The proposed method gives deviations from the initial pheromone trails by using a set of local optimal solutions calculated in advance. This method aims to build a near optimal solution at high speed by combining the candidate partial solutions contained in the set. Max-Min Ant System has demonstrated impressive performance, but the search rate is relatively low. Considering the generic purpose of stochastic search algorithms, which is to find near optimal solutions subject to time constraints, the search rate is important as well as the solution quality. The experimental results using benchmark problems with 51 to 1002 cities suggested that the proposed method has a faster search rate than Max-Min Ant System; the additional computation cost for calculating local optimal solutions is negligibly small. Keywords: Ant colony optimization, traveling salesman problem, search rate, local optimal solution
1. Introduction Ant colony optimization (ACO) is a stochastic search algorithm for problem solving that takes inspiration from the foraging behaviors of ants. ACO has been formalized into a metaheuristic for combinatorial optimization problems by Dorigo et al., and many successful applications are now available [1–4]. The main idea of ACO is the indirect communication among the individuals of a colony of ants based on an analogy with the pheromone trails that real ants use for communication. Many ants build solutions to a combinatorial optimization problem and exchange information on the quality of the solutions using pheromones. In ACO, the pheromone is the accumulation of information on a search space and the way ∗ Corresponding author: Hitoshi Kanoh, Division of Information Engineering, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8573, Japan. E-mail: [email protected].
of depositing the pheromone greatly influences search performance [1]. Many studies on pheromone update methods have been done to try to improve the performance of ACO. Max-Min Ant System (MMAS) has demonstrated especially impressive performance [5]. Its characterizing elements are that only the best ant in the colony updates the pheromone trails and that the value of the pheromone is bound. Since MMAS searches to maintain the diversity of solutions, it can find a solution near the optimal one. MMAS, however, cannot build a high-quality solution in the early search stages, and the search rate is relatively low. Earlier research focused on the quality of the final solution, and investigations on the search rate were seldom done. Considering the generic purpose of stochastic search algorithms, which is to find near optimal solutions subject to time constraints, the search rate is important as well as the solution quality (see Section 2.5). In this paper, we present a method to improve the search rate of MMAS for the traveling salesman problem (TSP) [6]. The proposed method gives deviations
c 2014 – IOS Press and the authors. All rights reserved ISSN 1327-2314/14/$27.50
12
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
from the initial pheromone trails by using a set of local optimal solutions calculated in advance. This method aims to build a near optimal solution at high speed by combining the candidate partial solutions contained in the set. In the field of evolutionary computation, improving the search rate by generating a proper initial population using domain-specific knowledge is generally known [7]. In contrast, the proposed method has the feature of not using such knowledge and has wide applicability. Some pheromone initialization methods have been proposed in order to improve the performance of Ant Colony Systems (ACS) for TSPs [9,10]. The improvement in search speed of these methods may not be enough. Furthermore, these methods are applied to only small problems. We have already reported the outline of a pheromone initialization method for MMAS [11]. In the present work, a systematic evaluation using large-scale problems was performed, and hybridizing with the local search and comparison with conventional techniques were also carried out. In the following section, we start by showing the problem description, an overview of MMAS, and related works. Then we describe the strategy and algorithm of the proposed method in detail. Finally, we give the results of experiments using benchmark problems with 51 to 1002 cities.
2. Overview 2.1. Traveling salesman problem (TSP) In this work, the proposed method was tested on the TSP, which is the most widely used benchmark for ACO. Here we express the TSP following the references for MMAS [5]. The TSP can be represented by a complete graph G = (N , A), where N is a set of nodes, i.e., cities, n = |N | is the number of cities, and A is the set of arcs fully connecting the nodes. Each arc (i, j) ∈ A is assigned a value di,j (= dj,i ), which represents the distance between cities i and j. The TSP then is the problem of finding a shortest closed tour visiting each of the cities of G exactly once. The TSP instances used in this paper are taken from the TSPLIB benchmark library [8]. 2.2. Basic algorithms for TSP In this section, we outline two basic algorithms for TSPs used in the proposed method.
(1) Greedy algorithm Construction algorithms build solutions to a problem under consideration incrementally, starting with an empty initial solution and iteratively adding appropriate solution components without backtracking until a complete solution is obtained. The greedy algorithm [1,6] is widely used in generating initial solutions for TSPs. A solution (tour) is built by adding city after city incrementally. A simple rule of thumb to build a tour is to start from an initial city and to always choose to go to the closest still unvisited city before returning to the start city. This algorithm is known as the nearest neighbor tour construction heuristic. An algorithmic outline is given below, where i, j, and k are city numbers, Sp is a partial solution (i.e., a list of cities visited), and N is a set of cities not yet visited. The procedure Greedy returns the solution with the best heuristic estimate as a function of the current Sp. This algorithm can generate n different solutions at maximum. Additionally, greedy decisions in early stages of the construction process constrain the available choices at later stages, often causing very poor moves in the final phases of the solution construction. Procedure Greedy(first-city) i = first-city; Sp = (i); N = N; While(N ’ is not empty){ N ← N − {i}; k = arg min di,j ; j ∈N /
Sp ← Sp ⊗ k; /*k is added to Sp*/ i = k; } return Sp as a solution; (2) 2-opt algorithm Local search algorithms start from a complete initial solution and try to find a better solution in an appropriately defined neighborhood of the current solution. An example neighborhood for the TSP is the 2-exchange neighborhood in which neighbor solutions differ by at most 2 edges. If such a solution is found, it replaces the current solution and the local search continues. These steps are repeated until no better neighbor solution can be found and the al-
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
gorithm ends in a local optimum. The 2-opt [1, 6] is widely used as a local search algorithm in TSPs since the calculation is quick. An outline is given below of an algorithm that performs a sequence of local modifications of the current tour to improve its length. The algorithm starts from an initial solution and repeatedly tries to improve the current solution through a local change. Given a candidate solution S, the output of the 2-opt is the shortest tour in the set of all candidate solutions S that can be obtained from S by exchanging two pairs of arcs in all the possible ways (for a detailed description, see [1] on page 32). Procedure 2-opt(S) S’ = Improve(S); while(S = S) { S = S ; S = Improve(S); } return S as a solution; 2.3. Ant colony optimization (ACO) The generic ACO metaheuristic for combinatorial optimization problems is shown below [2]. After initializing parameters and pheromone trails, the metaheuristic iterates over three phases. First, a number of solutions are constructed by ants, and then these solutions may be improved through an optional local search. Finally, before the start of the next iteration, the pheromone trails are adapted to reflect the search experience of the ants. Procedure ACO() Set parameters; Initialize pheromone trails; while(terminal condition not met){ Construct ant solutions; Apply local search; /* option */ Update pheromone; } Many versions of ACO algorithms have been proposed to try to improve their performance [1–4]. Dorigo et al.compare the development of the average solution quality of typical ACO algorithms as a function of the computation time [1]. They pointed out that Max-Min Ant System (MMAS) [5] initially produces rather poor solutions, and in the initial phases it is outperformed even by Ant System, which was the first
13
ACO algorithm to be proposed. Nevertheless, its final solution quality is among the compared ACO algorithms. These results are consistent with the findings of various published research papers on ACOs [1]. 2.4. Max-min ant system (MMAS) The MMAS constructs ant solutions and updates pheromone by following Eqs (1) and (3), respectively. The MMAS has two main characteristics [2]: only the best ant updates the pheromone trails, and the value of the pheromone is bound. Here, the best means that the length of the tour is shortest. The pheromone τi,j , associated with the edge joining cities i and j, is updated using Eq. (1): best τmax (1) τi,j ← (1 − ρ)τi,j + Δτi,j τmin ⎧ ⎪ ⎨a if x > a [x]ab = b if x < b ⎪ ⎩ x otherwise ⎧ ⎪ ⎨1/Lbest if (i, j) belongs to the best Δτi,j = best tour ⎪ ⎩ 0 otherwise where ρ is the evaporation rate, τmax and τmin are respectively the upper and lower bounds imposed on the pheromone, and Lbest is the length of the tour of the best ant. Some guidelines have been provided for defining τmax , τmin , and Lbest on the basis of analytical and empirical considerations. This paper defines these parameters by the following three formulas [5], where Lib is the length of the best tour found in the current iteration. τmax = τmin
1 ρLib
√ 1 − n 0.05 √ · τmax = (n/2 − 1) · n 0.05
(2)
Lbest = Lib In the construction of a solution, ants select the next city to be visited through a stochastic mechanism. When ant k is in city i and has so far constructed the partial solution, the probability of going to city j is given by Eq. (3). ⎧ α β ⎪ ⎨ [τi,j ] · [ηi,j ] if j ∈ N k k α β [τ ] · [η ] (3) pij = k i,l i,l l∈N ⎪ ⎩0 otherwise
14
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
Where N k is the set of cities not yet visited by the ant k. The parameters α and β control the relative importance of the pheromone versus the heuristic information ηi,j , which is given by ηi,j =
1 . di,j
2.5. Practical importance of search rate An effective solution of TSPs can be applied to tour planning problems in home-delivery services in the real world. This solution requires the early convergence for the reasons below. – It is necessary to calculate the optimal route between customers in practical problems, while the distance between cities is given in TSPs. Consequently, the order of visiting customers is calculated by ACO and the route between customers is calculated by Dijkstra’s algorithm, etc. – A practical problem has two or more objective functions including distance, travel time, safeness and comfortableness, and economy. Several nondominated solutions should be calculated. – When traffic congestion changes during traveling, the route should be reevaluated. Though the traffic congestion information on main roads is provided every five minutes in Japan, this calculation must be performed within five minutes. Thus, the tour planning problem in the real world can be formalized as a dynamic hybrid multi-objective problem and one iteration step of the solution takes much computational time, and moreover there is the time limit. The proposed method aims to improve the search rate of MMAS by giving the deviation from initial pheromone trails. 2.6. Related works In the field of evolutionary computation, generating an initial population using domain-specific knowledge is generally known. Bonissone et al. [7] pointed out that this knowledge may be incorporated either implicitly, in the design of data structures, encoding, and constraints representations, or explicitly, via the initialization of the first population and in the control of parameters. Kanoh et al. proposed a solution to timetabling problems using a genetic algorithm with prior knowledge [12]. The prior knowledge here is a set of candidate partial solutions of the final solution, and is built from timetables used in past years. In contrast, the pro-
posed method has the feature of not using domainspecific knowledge and has wide applicability. Tsai et al. proposed a solution to TSP using an Ant Colony System (ACS) with prior knowledge [9]. The prior knowledge here is the specific deviation of initial pheromone trails previously constructed by the greedy algorithm. The experimental results using TSP instances with 51 to 198 cities show that the proposed method improves the length of the solution. However, this paper does not present the formula to be used in the calculation of initial pheromone values and not discuss the search rate of the method. Furthermore, this paper is examining only small problems. Dai et al. proposed a new pheromone initialization strategy of ACS for TSP, which initializes the pheromone matrix in light of the minimal spanning tree (MST) information [10]. The MST describes the cheapest network to connect all of a given set of vertices. The purpose of this strategy is to give the edges belonging to the MST a higher probability of being chosen by ants at the start of the algorithm. The experimental results for instances with 14 to 136 cities show that the proposed strategy improves the convergence performance of ACS both in terms of solution quality and convergence speed. However, this strategy is applied to only small problems.
3. Proposed method 3.1. Strategy In this paper, we present a method to improve the search rate of MMAS according to the following strategies. 1) In addition to the usual initial pheromone trails, additional pheromones are deposited on the solution built before the first iteration by using local search. The proposed method helps to guide the search and so might reduce the number of points visited in the search space by giving deviations from the initial pheromone trails. 2) This reduction is conducted using local optimal solutions built by the greedy and the 2-opt algorithms. These solutions can include candidate partial solutions of the global optimal solution. Moreover, the search rate of 2-opt is very high; it can be considered to have no influence on the computational time of the proposed method.
H. Kanoh et al. / Pheromone trail initialization with local optimal solutions in ant colony optimization
3) First, many tours are generated by changing the starting city in the greedy algorithm. Then, the tours are improved by the 2-opt algorithm, and these are regarded as initial solutions. Finally, all the pheromone trails generated from the initial solutions are considered as the initial pheromone trails. The best-performing local search algorithm for TSPs is the Lin-Kernighan heuristic [13]. This heuristic, however, is much more complicated to implement than 2-opt and needs careful fine-tuning to obtain highquality solutions. Here, we use a 2-opt implementation considering computational time. 3.2. General procedure The general procedure of the proposed method is given below, where m is the number of ants, Sib is the best solution in the current iteration, Sgb is the best solution found since the start of the algorithm, and L(S) is the length of solution S. Procedure Initializepheromone will be shown in the next section. Procedure Main() Input TSP; Set parameters; Initialize-pheromone(); while(terminal condition not met){ for(k = 1; k
